Merge branch 'feat/convergence-strictness'

Make a short fit an error, raise the default iteration cap, validate the
remaining HistoryBuilder parameters, add History::register and
History::rating, reject competitor config conflicts across batches, and
document what the joint's cost scales in.

Closes #50

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_011hcFjNDmHXZF8URGLku5zZ

.gitignore

@@ -7,3 +7,4 @@
 NOTEPAD.md
 
 /.claude
+proptest-regressions/

CHANGELOG.md

@@ -2,6 +2,153 @@
 
 All notable changes to this project will be documented in this file.
 
+## 0.7.0 - 2026-09-08
+
+### Features
+
+- feat: factorise the joint once with History::joint
+
+### Other (unconventional)
+
+- Merge branch 'feat/joint-handle'
+
+## 0.6.0 - 2026-09-08
+
+### Breaking Changes
+
+- fix!: make the joint span slices, not just the latest one
+
+### Miscellaneous Tasks
+
+- chore: Release trueskill-tt version 0.6.0
+
+## 0.5.0 - 2026-09-08
+
+### Breaking Changes
+
+- feat!: name the unknown key, expose tail probabilities, flag short fits
+- refactor!: remove the factor-graph surface nothing used, add try_winner
+
+### Bug Fixes
+
+- fix(test): the ingestion-order property was comparing two truncated fits
+
+### Documentation
+
+- docs: record that the event log is the source of truth, and why
+
+### Features
+
+- feat: add UnknownKeys::Prior, and explain why there is no Skip
+- feat: add History::posterior_of for a linear combination of competitors
+- feat: add History::predict_margin for scored matchups
+- feat: add expected_variance_reduction for scored active learning
+
+### Miscellaneous Tasks
+
+- chore: Release trueskill-tt version 0.5.0
+
+### Styling
+
+- style: factor the event-pair type out of the reconvergence fixture
+- style: use arrays rather than vec! in the calibration fixture
+
+### Testing
+
+- test: pin that re-convergence is path-independent
+- test: calibrate the marginals against the exact posterior
+- test: pin what an additive model does to combined uncertainty
+
+## 0.4.2 - 2026-09-07
+
+### Bug Fixes
+
+- fix: replace the erfc approximation with libm, for free
+- fix: route every transcendental through libm, and combine sigmas with hypot
+
+### Miscellaneous Tasks
+
+- chore: Release trueskill-tt version 0.4.2
+
+### Testing
+
+- test: localise the erfc_inv tail residual to the caller's argument
+
+## 0.4.1 - 2026-09-07
+
+### Bug Fixes
+
+- fix: correct erfc_inv's sign error and keep evidence in log space
+
+### Miscellaneous Tasks
+
+- chore: Release trueskill-tt version 0.4.1
+
+### Testing
+
+- test: pin quality()'s N-group closed form, closing the README cross-check
+
+## 0.4.0 - 2026-09-07
+
+### Breaking Changes
+
+- feat!: N-team outcome prediction with draw mass, replacing the 2-team panic
+- refactor!: close the remaining API gaps from #21
+- fix!: apply competitor configuration whenever it is supplied
+
+### Bug Fixes
+
+- fix(release): skip the changelog hook during a dry run
+- fix: stop destroying tail precision in evidence and truncation
+- fix: reject convergence options that silently disable inference
+
+### Documentation
+
+- docs: correct drifted documentation and compile the README in CI
+
+### Features
+
+- feat: add expected information gain for active matchup selection
+- feat: let observers be shared, boxed, or borrowed
+
+### Miscellaneous Tasks
+
+- chore: Release trueskill-tt version 0.4.0
+
+## 0.3.0 - 2026-09-01
+
+### Breaking Changes
+
+- refactor!: make Competitor::message an Option, and compute_elapsed loud
+- refactor!: replace emptiness-as-sentinel with Option for results and weights
+- refactor!: remove ConvergenceReport::slices_skipped
+
+### Bug Fixes
+
+- fix: enforce EventBuilder weight/team length in release
+
+### Documentation
+
+- docs: complete the public API documentation contract
+
+### Features
+
+- feat: allow drift to vary per competitor via Member::with_drift_scale
+
+### Miscellaneous Tasks
+
+- chore: ignore proptest regression seed files
+- chore: Release trueskill-tt version 0.3.0
+
+### Performance
+
+- perf: stop cloning inference inputs in OwnedGame and ingestion
+- perf: make the per-slice SkillStore compact instead of dense
+
+### Testing
+
+- test: add property-based tests, a shared finiteness helper, and boundary inputs
+
 ## 0.2.0 - 2026-08-27
 
 ### Breaking Changes
@@ -36,6 +183,7 @@ All notable changes to this project will be documented in this file.
 - chore: target releases at the private kellnr registry
 - chore: keep the 48 MB ATP dataset out of the published crate
 - chore: dual-license MIT OR Apache-2.0
+- chore: Release trueskill-tt version 0.2.0
 
 ### Performance
 

CLAUDE.md

@@ -24,6 +24,20 @@ is where several defects have hidden — a debug-only run is not evidence.
 - `approx` — `approx::AbsDiffEq` etc. for `Gaussian`. Most numerical goldens need it.
 - `rayon` — opt-in parallel within-slice sweep and per-slice query passes.
 
+## Working rules
+
+- **Investigate before implementing.** Measure the actual behaviour first —
+  against an analytic reference where one exists. Several "obvious" fixes in
+  this repo turned out to be wrong in sign or unnecessary, and the measurement
+  is what caught them.
+- **Fix the root issue, not the symptom.** A clamp that hides an underflow, or
+  a tolerance loosened to make a test pass, is a defect deferred.
+- **Scout crates.io before hand-rolling numerics.** Check accuracy against an
+  independent reference rather than trusting downloads: `puruspe` has 1.4M
+  downloads and is 346 ULP off in the tail, where `libm` is 1. Fewer
+  dependencies is preferable, not mandatory — take the dependency when it is
+  measurably better.
+
 ## Architecture
 
 A Rust port of [TrueSkillThroughTime.py](https://github.com/glandfried/TrueSkillThroughTime.py):
@@ -32,10 +46,19 @@ evidence both forward and backward across a history.
 
 ### Data flow
 
+Ingestion (public types, `event.rs`):
+
 ```
-History  →  TimeSlice[]  →  Event[]  →  Team[] → Item[]
+Event<T, K>  →  Team<K>[]  →  Member<K>[]
-                                  ↓
+```
-                     Game (factor graph) → Schedule → BuiltinFactor[]
+
+`History::add_events` flattens that into indices; teams survive only as
+grouping, not as a value. Inference then runs on the internal shapes:
+
+```
+History  →  TimeSlice[]  →  Event[]  →  Item[]
+                                ↓
+                   Game (factor graph) → Schedule → BuiltinFactor[]
 ```
 
 - **`History`** (`history.rs`) — top level. Interns keys, groups events into
@@ -45,30 +68,35 @@ History  →  TimeSlice[]  →  Event[]  →  Team[] → Item[]
 - **`TimeSlice`** (`time_slice.rs`) — all events at one time. Owns a
   `SkillStore` and a `ScratchArena`; `iteration()` sweeps its events, using
   `ColorGroups` to partition independent ones.
-- **`Event`** (`time_slice.rs`) — one match. `compute()` runs inference reading
+- **`Event`** — two distinct types, do not confuse them. The *public* ingestion
-  skills immutably; `apply()` folds the result back. The split is what lets a
+  `Event<T, K>` is in `event.rs` (with `Team`/`Member`); the *internal*
-  color group run in parallel with no `unsafe`.
+  `pub(crate) Event` in `time_slice.rs` is one match during inference, where
+  `compute()` runs inference reading skills immutably and `apply()` folds the
+  result back. That split is what lets a color group run in parallel with no
+  `unsafe`.
 - **`Game`** (`game.rs`) — a single match's factor graph. `run_chain` builds the
   diff chain between rank-adjacent teams and drives it to convergence.
 - **`Gaussian`** (`gaussian.rs`) — natural parameters (`pi = 1/sigma²`,
   `tau = mu/sigma²`). `Mul`/`Div` are the EP product/cavity: pure adds and
   subtracts. Variance-space ops (`Add`, `Sub`, `exclude`, `forget`) go through
   `from_mv`/`variance()` and take no square root.
-- **`factor/`** — `TeamSumFactor`, `RankDiffFactor`, `TruncFactor` (ranked),
+- **`factor/`** — `TruncFactor` (ranked) and `MarginFactor` (scored) over a
-  `MarginFactor` (scored), over a flat `VarStore`. `BuiltinFactor` dispatches
+  flat `VarStore`. `Game::run_chain` drives them directly through a local
-  by enum rather than `dyn`.
+  `DiffFactor` enum; there is no `Schedule` indirection and no generic `Factor`
-- **`Schedule`** (`schedule.rs`) — drives factor propagation. `EpsilonOrMax` is
+  trait. Both were removed once measurement showed nothing had ever used them
-  the only implementation.
+  — see #42.
 - **`Competitor`** (`competitor.rs`) — per-history temporal state (`message`,
   `last_time`). **`Rating`** (`rating.rs`) — static config (prior, `beta`, drift).
-- **`storage/`** — `SkillStore` (per slice) and `CompetitorStore` (per history),
+- **`storage/`** — `SkillStore` (per slice, `pub(crate)`) and `CompetitorStore`
-  both dense `Vec`s indexed by `Index`.
+  (per history, public), both indexed by `Index`. The module is `pub`, but only
+  `CompetitorStore` is reachable from outside the crate.
 - **`KeyTable`** (`key_table.rs`) — user key ↔ `Index`, both directions O(1).
 - **`Drift`** (`drift.rs`) / **`Time`** (`time.rs`) — traits. `Time` is a *trait*
   (`i64`, `Untimed`), not an enum.
 - **`lib.rs`** — public exports, global defaults (`MU`, `SIGMA`, `BETA`,
-  `GAMMA`, `P_DRAW`, `EPSILON`, `ITERATIONS`), and the standalone `quality()`,
+  `GAMMA`, `P_DRAW`, `EPSILON`, `ITERATIONS`), and the standalone `quality()`.
-  `cdf()`, `erfc()`.
+  The `cdf()` / `erfc()` helpers live here too but are `pub(crate)` and private
+  respectively — not public API.
 
 ### Invariants worth knowing
 
@@ -83,6 +111,12 @@ History  →  TimeSlice[]  →  Event[]  →  Team[] → Item[]
   chain underflows to zero, and `ln(0)` is `-inf`.
 - **Colors are contiguous.** `recompute_color_groups` reorders events so each
   color occupies one range; `ColorGroups::groups_are_contiguous` asserts it.
+- **Transcendentals go through `libm`, not `std`.** IEEE 754 pins the basic
+  operations and `sqrt` but says nothing about `exp`/`log`/`erf`, and `std`
+  delegates to the *system* math library — measured, `f64::exp` and `libm::exp`
+  disagree on 9.7% of inputs by one ULP. Since inference is an iterative fixed
+  point, one ULP can change an iteration count. Use `libm::exp` / `libm::log` in
+  inference code; `f64::sqrt` is fine (IEEE specifies it). Tests may use either.
 - **The crate is `#![forbid(unsafe_code)]`.** Keep it that way.
 - **Ingestion order must not change the answer.** Events added one at a time
   must converge to the same fixed point as the same events batched — see

Cargo.toml

@@ -1,6 +1,6 @@
 [package]
 name = "trueskill-tt"
-version = "0.2.0"
+version = "0.7.0"
 edition = "2024"
 rust-version = "1.85"
 description = "TrueSkill Through Time: Bayesian skill rating that tracks how skill evolves over time, via Gaussian message passing"
@@ -51,6 +51,7 @@ harness = false
 
 [dependencies]
 approx = { version = "0.5.1", optional = true }
+libm = "0.2.16"
 rayon = { version = "1", optional = true }
 smallvec = "1"
 
@@ -62,9 +63,14 @@ rayon = ["dep:rayon"]
 criterion = "0.5"
 plotters = { version = "0.3", default-features = false, features = ["svg_backend", "all_elements", "all_series"] }
 plotters-backend = "0.3"
+proptest = "1.11.0"
 time = { version = "0.3", features = ["parsing"] }
 trueskill-tt = { path = ".", features = ["approx"] }
 
+# Debug symbols in release are for `just flame` (cargo-flamegraph), which needs
+# them to symbolicate. Profile settings in a library are ignored by downstream
+# consumers, so these only affect local builds — this is deliberate, not an
+# oversight.
 [profile.release]
 debug = true
 
@@ -73,3 +79,7 @@ debug = true
 
 [profile.dev]
 debug = true
+
+[[bench]]
+name = "joint"
+harness = false

README.md

@@ -13,64 +13,142 @@ Rust port of [TrueSkillThroughTime.py](https://github.com/glandfried/TrueSkillTh
 
 ## Drift
 
-Skill drift models how a player's true skill can change between appearances. Each time a player reappears after a gap, their skill uncertainty is widened by the drift model before the new evidence is incorporated.
+Skill drift models how a competitor's true skill can change between appearances.
+Each time they reappear after a gap, their skill uncertainty is widened by the
+drift model before the new evidence is incorporated.
 
-Drift is represented by the `Drift` trait:
+Drift is represented by the `Drift` trait (`src/drift.rs`), generic over the
+history's time type:
 
-```rust
+```text
-pub trait Drift: Copy + Debug {
+pub trait Drift<T: Time>: Copy + Debug + Send + Sync {
-    fn variance_delta(&self, elapsed: i64) -> f64;
+    fn variance_delta(&self, from: &T, to: &T) -> f64;
+    fn variance_for_elapsed(&self, elapsed: i64) -> f64;
 }
 ```
 
-`variance_delta` returns the amount to add to `σ²` given the elapsed time since the player last played. Internally, `Gaussian::forget` uses this to compute the new sigma: `σ_new = sqrt(σ² + variance_delta)`.
+Both methods return the amount to add to `σ²`, not to `σ`. `variance_delta`
+works from two timestamps; `variance_for_elapsed` takes an already-computed
+elapsed count, and is used on the paths that cache it. `Gaussian::forget`
+applies the result entirely in variance space — `from_mv(mu, variance() +
+variance_delta)` — taking no square root.
+
+That block is a quotation rather than a doctest. The custom-drift example below
+is compiled by CI, so it is what actually pins the signature.
 
 ### ConstantDrift
 
-The built-in `ConstantDrift` implements a linear random walk — skill uncertainty grows proportionally to time:
+The built-in `ConstantDrift` implements a linear random walk — skill uncertainty
+grows proportionally to time:
 
-```
+```text
 variance_delta = elapsed * γ²
 ```
 
-This is the standard TrueSkill Through Time model. Use it by passing a `ConstantDrift(gamma)` when constructing a `Player`:
+This is the standard TrueSkill Through Time model. Pass a `ConstantDrift(gamma)`
+when constructing a `Rating`:
 
 ```rust
-use trueskill_tt::{Player, Gaussian, drift::ConstantDrift};
+use trueskill_tt::{ConstantDrift, Gaussian, Rating};
 
-// gamma = 0.1 means skill can shift ~0.1 per time unit
+// gamma = 0.1 means skill can shift ~0.1 per time unit.
-let player = Player::new(Gaussian::from_ms(0.0, 6.0), 1.0, ConstantDrift(0.1));
+let rating: Rating<i64, ConstantDrift> =
+    Rating::new(Gaussian::from_ms(0.0, 6.0), 1.0, ConstantDrift(0.1));
+
+assert_eq!(rating.drift().0, 0.1);
 ```
 
+The type annotation is load-bearing: `ConstantDrift` implements `Drift<T>` for
+every `T: Time`, so without it `T` is ambiguous.
+
 ### Custom drift
 
-Implement `Drift` to express any other model. For example, a drift that saturates after a long absence (uncertainty grows with the square root of elapsed time instead of linearly):
+Implement `Drift<T>` to express any other model. For example, a drift that
+saturates after a long absence, with uncertainty growing as the square root of
+elapsed time instead of linearly:
 
 ```rust
-use trueskill_tt::drift::Drift;
+use trueskill_tt::{Drift, Gaussian, History, Rating, Time};
 
 #[derive(Clone, Copy, Debug)]
 struct SqrtDrift {
     gamma: f64,
 }
 
-impl Drift for SqrtDrift {
+impl<T: Time> Drift<T> for SqrtDrift {
-    fn variance_delta(&self, elapsed: i64) -> f64 {
+    fn variance_delta(&self, from: &T, to: &T) -> f64 {
-        (elapsed as f64).sqrt() * self.gamma * self.gamma
+        let elapsed = from.elapsed_to(to).max(0) as f64;
+        elapsed.sqrt() * self.gamma * self.gamma
+    }
+
+    fn variance_for_elapsed(&self, elapsed: i64) -> f64 {
+        (elapsed.max(0) as f64).sqrt() * self.gamma * self.gamma
     }
 }
 
-let player = Player::new(Gaussian::from_ms(0.0, 6.0), 1.0, SqrtDrift { gamma: 0.5 });
+// On a single Rating:
+let rating: Rating<i64, SqrtDrift> =
+    Rating::new(Gaussian::from_ms(0.0, 6.0), 1.0, SqrtDrift { gamma: 0.5 });
+
+// Or for a whole History, via the builder:
+let history = History::builder().drift(SqrtDrift { gamma: 0.5 }).build();
+
+assert_eq!(rating.beta(), 1.0);
+assert_eq!(history.log_evidence(), 0.0);
 ```
 
-To use a custom drift type with `History`, use the `.drift()` builder method instead of `.gamma()`:
+`HistoryBuilder::drift` is the only way to set a history's drift model; there is
+no `gamma()` shorthand. The default is `ConstantDrift(GAMMA)`.
+
+### Per-competitor drift
+
+A `History` has one drift model, but individual competitors can scale it.
+`Member::with_drift_scale(s)` multiplies the drift *variance* that competitor
+accumulates, so `s` is in the same units as `gamma`: `ConstantDrift(g)` at
+scale `s` behaves exactly as `ConstantDrift(g * s)` would, for that competitor
+alone.
+
+`0.0` pins a competitor still. That is what makes a **fixed reference point**
+expressible in the same graph as moving competitors — a bot at a known
+strength, a rating floor, a course difficulty:
 
 ```rust
-let h = History::builder()
+use trueskill_tt::{ConstantDrift, Event, History, Member, Outcome, Team};
-    .drift(SqrtDrift { gamma: 0.5 })
+
-    .build();
+let mut h = History::builder().drift(ConstantDrift(0.1)).build();
+
+h.add_events(vec![Event {
+    time: 0,
+    teams: [
+        Team::with_members([Member::new("player")]),
+        // A course does not improve. Pin it, and the round's evidence
+        // lands on the player instead of being split between the two.
+        Team::with_members([Member::new("layout_7").with_drift_scale(0.0)]),
+    ]
+    .into_iter()
+    .collect(),
+    outcome: Outcome::winner(0, 2),
+}])
+.unwrap();
+
+h.converge().unwrap();
 ```
 
+Like `with_prior`, the scale is **competitor configuration, not a per-event
+value**: it applies to the competitor for the whole history, and it applies
+whenever it is supplied — including on a key the history already knows.
+Configuring one late still refits the whole history rather than taking effect
+only from that event onward, because `converge` refits from competitor state.
+Repeating the same value is inert; supplying two *different* values for one
+competitor within a single batch is `InferenceError::ConflictingCompetitorConfig`,
+since events in a batch have no order. The scale must be finite and
+non-negative; ingestion otherwise fails with `InferenceError::InvalidParameter`.
+
+The fluent `EventBuilder` reaches this too: `.team([...])` is the common case
+and leaves both unset, while `.members([...])` takes `Member` values directly,
+so `h.event(t).members([Member::new("layout_7").with_drift_scale(0.0)])` is
+equivalent to the typed shape above.
+
 ## Scored outcomes
 
 Use `Outcome::scores([...])` when you have continuous per-team scores rather
@@ -80,7 +158,7 @@ soft Gaussian evidence about the latent performance diff. Configure
 (smaller σ = more trust).
 
 ```rust
-use trueskill_tt::{History, Outcome};
+use trueskill_tt::History;
 
 let mut h = History::builder().score_sigma(2.0).build();
 h.event(1)
@@ -92,6 +170,114 @@ h.event(1)
 h.converge().unwrap();
 ```
 
+## Prediction
+
+`predict_outcome` gives the full distribution over finishing orders. Each entry
+is a rank vector in the same shape `Outcome::ranking` takes — equal ranks mean a
+tie — so an outcome feeds straight back into inference.
+
+```rust
+use trueskill_tt::History;
+
+let mut h = History::builder().p_draw(0.1).build();
+h.record_winner(&"alice", &"bob", 1).unwrap();
+h.converge().unwrap();
+
+let p = h.predict_outcome(&[&[&"alice"], &[&"bob"]]).unwrap();
+
+// Probabilities are exhaustive and disjoint, so they sum to one.
+assert!((p.total() - 1.0).abs() < 1e-6);
+
+let (best, likelihood) = p.most_likely().unwrap();
+println!("most likely: {best:?} at {likelihood:.3}");
+println!("draw:        {:.3}", p.probability_of(&[0, 0]));
+```
+
+Supports any number of teams. Because the outcome space grows factorially, the
+full distribution is capped at `MAX_PREDICTED_TEAMS`; two cheaper entry points
+stay available at any size:
+
+- `predict_win_probabilities(teams)` — `P(team i finishes strictly first)`,
+  quadratic in team count.
+- `predict_ranking(teams, ranks)` — one specific finishing order.
+
+Unknown keys are an error by default, not a silent omission: a team the history
+has never seen cannot produce a confident-looking probability. The error names
+the key, and every key must already be known — pre-filter with `lookup` or
+`current_skill` if your caller cannot guarantee that.
+
+If predicting for competitors you have never seen is the point rather than a
+mistake, say so once:
+
+```rust
+use trueskill_tt::{History, UnknownKeys};
+
+let h = History::builder().unknown_keys(UnknownKeys::Prior).build();
+```
+
+An unknown competitor is then answered from the configured prior, which is the
+honest reading — you have no evidence about them — and correctly *widens* a team
+that contains one. There is deliberately no "skip the member" mode: a team's
+performance is the sum of its members, so dropping one would make the model more
+certain because it knows less.
+
+### Asking about one competitor
+
+`Gaussian` answers tail questions directly, which is what a stopping rule needs:
+
+```rust
+use trueskill_tt::History;
+
+let mut h = History::builder().build();
+h.record_winner(&"alice", &"bob", 1).unwrap();
+let _ = h.converge().unwrap();
+
+let skill = h.current_skill(&"alice").unwrap();
+
+// "How sure am I that this is below the cutoff?" — a probability, not a
+// `mu + z * sigma` band whose confidence drifts as sigma changes.
+let _ = skill.probability_below(20.0);
+
+// Use this rather than `1.0 - probability_below(x)`: the complement cancels
+// away every digit in the upper tail, which is where a stopping rule lives.
+let _ = skill.probability_above(30.0);
+```
+
+## Which match to play next
+
+`quality()` measures whether a matchup is *fair*. That is not the same as
+whether it is *informative*, and the two only coincide for two evenly matched
+competitors. When each observation costs something, ask
+`expected_information_gain` instead — the outcome-weighted divergence between
+what you believe now and what you would believe afterwards.
+
+```rust
+use trueskill_tt::History;
+
+let mut h = History::builder().build();
+for t in 1..=10 {
+    h.record_winner(&"veteran", &"regular", t).unwrap();
+    h.record_winner(&"regular", &"veteran", t + 100).unwrap();
+}
+h.record_winner(&"veteran", &"newcomer", 500).unwrap();
+h.converge().unwrap();
+
+let settled = h.expected_information_gain(&[&[&"veteran"], &[&"regular"]]).unwrap();
+let unknown = h.expected_information_gain(&[&[&"veteran"], &[&"newcomer"]]).unwrap();
+
+// Playing the newcomer teaches you more than replaying a settled rivalry.
+assert!(unknown > settled);
+```
+
+The result is in nats, and is bounded by the entropy of the outcome: at most
+`ln 2 ≈ 0.693` for a two-way result, `ln 3` once draws are possible, `ln k` for
+`k` outcomes. A value near zero means you already know how it ends.
+
+This costs one full inference pass **per possible outcome**, so it is far more
+expensive than `quality()`. Scoring every pairing among `n` competitors is
+`O(n² × outcomes)` passes — shortlist with `quality()` or
+`predict_win_probabilities` first, then score only the shortlist.
+
 ## Todo
 
 - [x] Implement approx for Gaussian
@@ -100,7 +286,8 @@ h.converge().unwrap();
 - [x] Add examples (`examples/atp.rs`, `examples/scored.rs`)
 - [x] Add Observer (`Observer` / `NullObserver`)
 - [x] Benchmark the inference loop (`benches/batch.rs`, `benches/history_converge.rs`, `benches/ingest.rs`)
-- [ ] Cross-check `quality()` against [sublee/trueskill](https://github.com/sublee/trueskill/tree/master) — N-group support works and is covered by invariants, but no reference values are asserted
+- [x] N-team `predict_outcome` with draw mass, and `expected_information_gain`
+- [x] Cross-check `quality()` against [sublee/trueskill](https://github.com/sublee/trueskill/tree/master) — N identical teams follow the closed form `(1/5)^((n-1)/2)` for the conventional parameters, asserted for n = 2..10, and the n=3/n=5 values (0.200, 0.040) match the reference package
 
 ## License
 

benches/batch.rs

@@ -36,7 +36,7 @@ fn criterion_benchmark(criterion: &mut Criterion) {
     let kinds = vec![EventKind::Ranked; composition.len()];
 
     let mut time_slice = TimeSlice::new(1, P_DRAW, ConvergenceOptions::default());
-    time_slice.add_events(composition, results, weights, kinds, &agents);
+    time_slice.add_events(composition, Some(results), Some(weights), kinds, &agents);
 
     criterion.bench_function("Batch::iteration", |b| {
         b.iter(|| time_slice.iteration(0, &agents))

benches/history_converge.rs

@@ -82,7 +82,7 @@ fn bench_converge(c: &mut Criterion) {
         b.iter_batched(
             || build_history_1v1(500, 100, 10, 42),
             |mut h| {
-                h.converge().unwrap();
+                let _ = h.converge().unwrap();
             },
             BatchSize::SmallInput,
         );
@@ -92,7 +92,7 @@ fn bench_converge(c: &mut Criterion) {
         b.iter_batched(
             || build_history_1v1(2000, 200, 20, 42),
             |mut h| {
-                h.converge().unwrap();
+                let _ = h.converge().unwrap();
             },
             BatchSize::SmallInput,
         );
@@ -106,7 +106,7 @@ fn bench_converge(c: &mut Criterion) {
         b.iter_batched(
             || build_history_1v1(5000, 50000, 5000, 42),
             |mut h| {
-                h.converge().unwrap();
+                let _ = h.converge().unwrap();
             },
             BatchSize::SmallInput,
         );

benches/joint.rs

@@ -0,0 +1,71 @@
+//! Cost of the joint posterior: factorising versus querying.
+//!
+//! The split is the whole point of `History::joint`. Factorising is `O(n^3)` in
+//! the history's appearances and depends only on the fit; a query is `O(n^2)`
+//! and depends only on the question. `posterior_of_one_shot` pays both every
+//! time, `joint_query` pays only the second.
+
+use criterion::{Criterion, criterion_group, criterion_main};
+use smallvec::smallvec;
+use trueskill_tt::{ConstantDrift, ConvergenceOptions, Event, History, Member, Outcome, Team};
+
+/// 30 slices of 8 duels: 480 appearances over 100 competitors.
+fn fitted() -> History<i64, ConstantDrift, trueskill_tt::NullObserver, String> {
+    let mut h: History<i64, ConstantDrift, _, String> = History::builder_with_key()
+        .mu(0.0)
+        .sigma(6.0)
+        .beta(1.0)
+        .score_sigma(2.0)
+        .drift(ConstantDrift(0.05))
+        .convergence(ConvergenceOptions {
+            max_iter: 30,
+            epsilon: 1e-10,
+            alpha: 1.0,
+        })
+        .build();
+
+    let mut events: Vec<Event<i64, String>> = Vec::new();
+    let mut k = 0usize;
+    for t in 0..30i64 {
+        for _ in 0..8 {
+            k += 1;
+            events.push(Event {
+                time: t,
+                teams: smallvec![
+                    Team::with_members([Member::new(format!("p{}", k % 100))]),
+                    Team::with_members([Member::new(format!("p{}", (k + 37) % 100))]),
+                ],
+                outcome: Outcome::scores([
+                    (k as f64 * 0.3).sin().abs() * 20.0,
+                    (k as f64 * 0.3).cos().abs() * 20.0,
+                ]),
+            });
+        }
+    }
+    h.add_events(events).unwrap();
+    let _ = h.converge().unwrap();
+    h
+}
+
+fn bench_joint(c: &mut Criterion) {
+    let h = fitted();
+    let a = "p0".to_string();
+    let b = "p1".to_string();
+    let terms = [(&a, 1.0), (&b, -1.0)];
+
+    c.bench_function("joint_factorise_480_appearances", |bencher| {
+        bencher.iter(|| std::hint::black_box(h.joint().unwrap().variables()));
+    });
+
+    c.bench_function("posterior_of_one_shot_480_appearances", |bencher| {
+        bencher.iter(|| std::hint::black_box(h.posterior_of(&terms).unwrap()));
+    });
+
+    let joint = h.joint().unwrap();
+    c.bench_function("joint_query_480_appearances", |bencher| {
+        bencher.iter(|| std::hint::black_box(joint.posterior_of(&terms).unwrap()));
+    });
+}
+
+criterion_group!(benches, bench_joint);
+criterion_main!(benches);

benches/scored.rs

@@ -29,7 +29,7 @@ fn bench_scored_history(c: &mut Criterion) {
                 });
             }
             h.add_events(events).unwrap();
-            h.converge().unwrap();
+            let _ = h.converge().unwrap();
         });
     });
 }

docs/superpowers/specs/2026-04-23-trueskill-engine-redesign-design.md

@@ -500,6 +500,26 @@ All public traits (`Time`, `Drift`, `Observer`, `Factor`, `Schedule`) require `S
 
 `rayon` as default-on feature; with `default-features = false`, parallel paths fall back to sequential iterators behind `cfg(feature = "rayon")`.
 
+> **Not implemented. Deliberate deviation, decided 2026-09-08 (issue #5).**
+>
+> `rayon` ships **opt-in**: `Cargo.toml` has no `default = [...]` key. The
+> measured speedups are 1.0x on realistic workloads and 1.3x on a pathological
+> one (issue #4), because typical slices hold too few events to amortize
+> rayon's task-spawn overhead. Default-on would hand every downstream user a
+> thread pool and a dependency for approximately no gain.
+>
+> This section made the trade conditional on cross-slice dirty-bit skipping
+> landing and changing the parallel story. It did not land: #4 was closed on
+> 2026-08-27 by removing the inert `ConvergenceReport::slices_skipped` field
+> rather than by implementing the mechanism, so the re-measurement this was
+> waiting on will not arrive.
+>
+> The "Trade-offs" note below also cited an `unsafe` concurrent-write path
+> through `SkillStore` as a cost of default-on. That cost does not exist: the
+> crate is `#![forbid(unsafe_code)]`, and the compute/apply split on the
+> internal `Event` is what lets a color group run in parallel without it. The
+> case for opt-in rests on the measurements alone.
+
 ### Expected speedup ballpark
 
 For 1000 players, 60 events/slice × 1000 slices, 30 convergence iterations:
@@ -521,7 +541,7 @@ These are pre-implementation estimates. Each tier validates with criterion.
 - Color-group parallelism requires up-front graph coloring at ingestion. Cost: linear in events, run once per `add_events`. Cheap.
 - Default = asynchronous EP (preserves current semantics). Synchronous opt-in only.
 - Cross-slice sweep stays sequential; no speculative parallel sweeps.
-- Rayon default-on but feature-gated.
+- Rayon default-on but feature-gated. **Superseded — shipped opt-in; see the deviation note in Section 6.**
 
 ### Open question
 

examples/atp.rs

@@ -46,14 +46,33 @@ fn main() {
         .sigma(1.6)
         .drift(ConstantDrift(0.036))
         .convergence(trueskill_tt::ConvergenceOptions {
-            max_iter: 10,
+            // This history needs 30 sweeps to reach the epsilon below. It was
+            // capped at 10 until the `#[must_use]` on `ConvergenceReport`
+            // surfaced that the example had been shipping a short fit.
+            max_iter: 100,
             epsilon: 0.01,
             alpha: 1.0,
         })
         .build();
 
     hist.add_events(events).unwrap();
-    hist.converge().unwrap();
+
+    // Read the report rather than discarding it. A fit that hits `max_iter`
+    // without reaching `epsilon` is not an error and does not look wrong — every
+    // rating comes back finite and sensibly ordered — so this flag is the only
+    // thing that says the numbers were still moving when the sweep stopped.
+    let report = hist.converge().unwrap();
+    eprintln!(
+        "converged={} after {} sweeps, final step {:?}",
+        report.converged, report.iterations, report.final_step
+    );
+    if !report.converged {
+        eprintln!(
+            "warning: stopped after {} sweeps with a final step of {:?}, \
+             short of epsilon — raise ConvergenceOptions::max_iter",
+            report.iterations, report.final_step
+        );
+    }
 
     let players = [
         ("aggasi", "a092", 38800i64),

release.toml

@@ -3,4 +3,12 @@ publish = true
 # Hold off pushing until tags and publish have both succeeded; `just release`
 # pushes last.
 push = false
-pre-release-hook = ["sh", "-c", "git cliff -o CHANGELOG.md --tag {{version}} && git add CHANGELOG.md"]
+# Regenerate the changelog and stage it so it lands in the release commit.
+#
+# Guarded on DRY_RUN because cargo-release runs pre-release hooks during a dry
+# run too (verified against cargo-release 1.1.5, which exports DRY_RUN=true,
+# CRATE_NAME, PREV_VERSION and NEW_VERSION to the hook). Without the guard,
+# `just release-plan` — documented as a preview that writes nothing — writes and
+# `git add`s CHANGELOG.md, and the clean-tree check in `just release` then
+# refuses to run. That check is load-bearing: publishing is irreversible.
+pre-release-hook = ["sh", "-c", '[ "$DRY_RUN" = "true" ] || (git cliff -o CHANGELOG.md --tag {{version}} && git add CHANGELOG.md)']

src/acquisition.rs

@@ -0,0 +1,352 @@
+//! Active learning: which comparison teaches you the most.
+//!
+//! [`quality`](crate::quality) answers "is this matchup *fair*". That is a
+//! different question from "is this matchup *informative*", and the two
+//! coincide only for two evenly matched competitors. When each observation
+//! costs something — a human click, a scheduled fixture — the question worth
+//! asking is the second one.
+//!
+//! The quantity here is expected information gain: the outcome-weighted
+//! divergence between what you believe now and what you would believe after
+//! seeing the result.
+//!
+//! ```text
+//! EIG(matchup) = SUM  P(outcome) * KL( posterior_after(outcome) || prior )
+//!              outcome
+//! ```
+//!
+//! It is the mutual information between the observed outcome and the skills,
+//! which is worth remembering because it pins the scale: information gain
+//! cannot exceed the entropy of the thing you are about to observe. A contest
+//! with `k` distinguishable outcomes can teach you at most `ln k` nats,
+//! whatever the ratings. That ceiling is the sharpest available test of an
+//! implementation — see [`expected_information_gain`].
+
+use crate::{
+    GameOptions, Gaussian, InferenceError, Outcome, Rating, drift::Drift, predict, time::Time,
+};
+
+/// Outcomes below this probability contribute nothing measurable and are not
+/// worth an inference pass.
+///
+/// The contribution of an outcome is `P * KL`, and `KL` is bounded in practice
+/// by tens of nats, so a probability this small moves the total by less than
+/// the quadrature error already present in `P` itself.
+const NEGLIGIBLE: f64 = 1e-12;
+
+/// `KL(q || p)` for two univariate Gaussians, in nats.
+///
+/// Both arguments are proper posteriors from inference, so the degenerate
+/// cases guarded here (zero or infinite variance) indicate that inference has
+/// broken down rather than anything a caller did.
+fn kl_divergence(q: Gaussian, p: Gaussian) -> f64 {
+    let (var_q, var_p) = (q.sigma().powi(2), p.sigma().powi(2));
+
+    if !(var_q.is_finite() && var_p.is_finite()) || var_q <= 0.0 || var_p <= 0.0 {
+        return 0.0;
+    }
+
+    let mean_gap = q.mu() - p.mu();
+    0.5 * (libm::log(var_p / var_q) + (var_q + mean_gap * mean_gap) / var_p - 1.0)
+}
+
+/// Expected information gain of a hypothetical matchup, in nats.
+///
+/// Enumerates the outcomes this matchup could have, runs inference for each to
+/// get the belief it would produce, and weights the resulting divergence by
+/// that outcome's probability. A higher value means the result would teach you
+/// more.
+///
+/// # Interpreting the value
+///
+/// Nats. The upper bound is the entropy of the outcome variable: at most
+/// `ln 2 ≈ 0.693` for a two-way result, `ln 3 ≈ 1.099` once draws are
+/// possible, `ln k` for `k` outcomes. A value near the ceiling means the
+/// result is close to a coin flip *and* would move the posteriors a long way;
+/// a value near zero means you already know what will happen, or that the
+/// result would barely change your beliefs if you saw it.
+///
+/// This is not a monotone transform of [`quality`](crate::quality). A lopsided
+/// matchup between two uncertain competitors scores well on quality-times-
+/// variance heuristics and poorly here, because the near-certain outcome
+/// carries almost no information.
+///
+/// # Cost
+///
+/// One full inference pass per possible outcome, so this is far more expensive
+/// than `quality()` — which is one closed-form evaluation. The outcome count
+/// grows quickly with team count (3 outcomes for two teams that can draw, 13
+/// for three, 75 for four), and scoring every candidate pairing among `n`
+/// competitors is `O(n² × outcomes)` inference passes.
+///
+/// For a selector over many candidates, shortlist with the cheap
+/// [`quality`](crate::quality) or
+/// [`predict_win_probabilities`](crate::History::predict_win_probabilities)
+/// first and score only the shortlist here. The expected-variance-reduction
+/// proxy sometimes suggested as a cheaper alternative is *not* cheaper: it
+/// needs the same hypothetical posteriors, so it shares the dominant cost.
+///
+/// # Errors
+///
+/// - `NotEnoughTeams` if fewer than two teams are supplied.
+/// - `EmptyTeam` if any team has no members.
+/// - `TooManyTeams` if the outcome space is too large to enumerate; see
+///   [`MAX_PREDICTED_TEAMS`](crate::MAX_PREDICTED_TEAMS).
+/// - `InvalidProbability` if `options.p_draw` is outside `[0.0, 1.0)`.
+/// - Anything [`Game::ranked`](crate::Game::ranked) returns for a hypothetical
+///   outcome.
+pub fn expected_information_gain<T: Time, D: Drift<T>>(
+    teams: &[&[Rating<T, D>]],
+    options: &GameOptions,
+) -> Result<f64, InferenceError> {
+    if teams.len() < 2 {
+        return Err(InferenceError::NotEnoughTeams { got: teams.len() });
+    }
+    if teams.len() > crate::MAX_PREDICTED_TEAMS {
+        return Err(InferenceError::TooManyTeams {
+            got: teams.len(),
+            max: crate::MAX_PREDICTED_TEAMS,
+        });
+    }
+    if !(0.0..1.0).contains(&options.p_draw) {
+        return Err(InferenceError::InvalidProbability {
+            value: options.p_draw,
+        });
+    }
+    for (idx, team) in teams.iter().enumerate() {
+        if team.is_empty() {
+            return Err(InferenceError::EmptyTeam { team: idx });
+        }
+    }
+
+    // Prediction runs on performances: skill inflated by each member's beta.
+    let performances: Vec<Gaussian> = teams
+        .iter()
+        .map(|team| {
+            team.iter()
+                .fold(crate::N00, |acc, rating| acc + rating.performance())
+        })
+        .collect();
+
+    // Draw margins per pair, derived from the teams' betas exactly as
+    // inference derives them, so the outcomes weighted here are the outcomes
+    // that would actually be fitted.
+    let beta_sq: Vec<f64> = teams
+        .iter()
+        .map(|team| team.iter().map(|r| r.beta().powi(2)).sum())
+        .collect();
+    let p_draw = options.p_draw;
+    let margins = predict::Margins::new(teams.len(), |i, j| {
+        if p_draw == 0.0 {
+            0.0
+        } else {
+            crate::compute_margin(p_draw, (beta_sq[i] + beta_sq[j]).sqrt())
+        }
+    });
+
+    let mut gain = 0.0;
+
+    for (ranks, probability) in predict::outcome_distribution(&performances, &margins) {
+        if probability <= NEGLIGIBLE {
+            continue;
+        }
+
+        let game = crate::Game::ranked(teams, Outcome::ranking(ranks), options)?;
+        let posteriors = game.posteriors();
+
+        // Beliefs factorise across competitors, so the joint divergence is the
+        // sum of the per-competitor ones.
+        let divergence: f64 = teams
+            .iter()
+            .zip(&posteriors)
+            .flat_map(|(team, posterior)| team.iter().zip(posterior))
+            .map(|(rating, &after)| kl_divergence(after, rating.prior()))
+            .sum();
+
+        gain += probability * divergence;
+    }
+
+    Ok(gain)
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use crate::{BETA, ConstantDrift, GAMMA};
+
+    type R = Rating<i64, ConstantDrift>;
+
+    fn rating(mu: f64, sigma: f64) -> R {
+        R::new(Gaussian::from_ms(mu, sigma), BETA, ConstantDrift(GAMMA))
+    }
+
+    fn options(p_draw: f64) -> GameOptions {
+        GameOptions {
+            p_draw,
+            ..GameOptions::default()
+        }
+    }
+
+    fn eig(teams: &[&[R]], p_draw: f64) -> f64 {
+        expected_information_gain(teams, &options(p_draw)).unwrap()
+    }
+
+    /// The analytic ceiling. Information gain is the mutual information between
+    /// the outcome and the skills, so it cannot exceed the entropy of the
+    /// outcome variable — whatever the ratings. This is the check a subtly
+    /// wrong implementation fails while still returning plausible numbers: an
+    /// early prototype of this returned 4.77 nats from a sign error and passed
+    /// every monotonicity test.
+    #[test]
+    fn never_exceeds_the_entropy_of_the_outcome() {
+        let ceiling_two = std::f64::consts::LN_2;
+
+        for (a, b) in [
+            (rating(0.0, 6.0), rating(0.0, 6.0)),
+            (rating(0.0, 0.5), rating(0.0, 0.5)),
+            (rating(12.0, 6.0), rating(-12.0, 6.0)),
+            (rating(40.0, 1.0), rating(-40.0, 1.0)),
+            (rating(3.0, 6.0), rating(-2.0, 0.1)),
+            (rating(0.0, 25.0), rating(0.0, 25.0)),
+        ] {
+            let g = eig(&[&[a], &[b]], 0.0);
+            assert!(
+                g >= 0.0 && g <= ceiling_two,
+                "EIG {g} outside [0, ln 2] for mu=({}, {}) sigma=({}, {})",
+                a.prior().mu(),
+                b.prior().mu(),
+                a.prior().sigma(),
+                b.prior().sigma()
+            );
+        }
+    }
+
+    /// With draws enabled there are three outcomes, so the ceiling rises to
+    /// `ln 3` — and the two-outcome bound no longer applies.
+    #[test]
+    fn the_ceiling_follows_the_outcome_count() {
+        let ceiling_three = 3.0f64.ln();
+        for sigma in [0.5, 3.0, 6.0, 25.0] {
+            let g = eig(&[&[rating(0.0, sigma)], &[rating(0.0, sigma)]], 0.25);
+            assert!(
+                g >= 0.0 && g <= ceiling_three,
+                "EIG {g} outside [0, ln 3] at sigma {sigma}"
+            );
+        }
+    }
+
+    /// An even matchup between uncertain competitors is the informative one.
+    /// A hopelessly lopsided matchup teaches you almost nothing, because you
+    /// already know how it ends.
+    #[test]
+    fn an_even_matchup_beats_a_lopsided_one() {
+        let even = eig(&[&[rating(0.0, 6.0)], &[rating(0.0, 6.0)]], 0.0);
+        let lopsided = eig(&[&[rating(12.0, 6.0)], &[rating(-12.0, 6.0)]], 0.0);
+        assert!(
+            even > lopsided,
+            "even {even} should beat lopsided {lopsided}"
+        );
+    }
+
+    /// Certainty is the thing information gain is measuring the absence of:
+    /// the less you know, the more there is to learn.
+    #[test]
+    fn gain_falls_as_certainty_rises() {
+        let mut previous = f64::INFINITY;
+        for sigma in [12.0, 6.0, 3.0, 1.0, 0.5, 0.1] {
+            let g = eig(&[&[rating(0.0, sigma)], &[rating(0.0, sigma)]], 0.0);
+            assert!(
+                g < previous,
+                "sigma {sigma}: {g} did not fall below {previous}"
+            );
+            previous = g;
+        }
+        assert!(previous >= 0.0);
+    }
+
+    /// The heuristic this replaces is `quality * sigma_a^2 * sigma_b^2`. It is
+    /// not a monotone transform of information gain — it ranks a lopsided
+    /// matchup above a confident even one, and EIG ranks them the other way.
+    /// Pinning the disagreement down is what stops a future "simplification"
+    /// from quietly reverting to the heuristic.
+    #[test]
+    fn disagrees_with_the_quality_times_variance_heuristic() {
+        let heuristic = |a: &R, b: &R| {
+            crate::quality(&[&[a.prior()], &[b.prior()]], BETA)
+                * a.prior().sigma().powi(2)
+                * b.prior().sigma().powi(2)
+        };
+
+        let (confident_a, confident_b) = (rating(0.0, 0.5), rating(0.0, 0.5));
+        let (lopsided_a, lopsided_b) = (rating(12.0, 6.0), rating(-12.0, 6.0));
+
+        assert!(
+            heuristic(&lopsided_a, &lopsided_b) > heuristic(&confident_a, &confident_b),
+            "the heuristic should prefer the lopsided matchup"
+        );
+        assert!(
+            eig(&[&[confident_a], &[confident_b]], 0.0) > eig(&[&[lopsided_a], &[lopsided_b]], 0.0),
+            "information gain should prefer the even matchup"
+        );
+    }
+
+    #[test]
+    fn supports_more_than_two_teams() {
+        let teams: Vec<Vec<R>> = vec![
+            vec![rating(0.0, 6.0)],
+            vec![rating(0.0, 6.0)],
+            vec![rating(0.0, 6.0)],
+        ];
+        let refs: Vec<&[R]> = teams.iter().map(Vec::as_slice).collect();
+        let g = expected_information_gain(&refs, &options(0.0)).unwrap();
+        // Six distinguishable orderings with no draws.
+        assert!(
+            g > 0.0 && g <= 6.0f64.ln(),
+            "three-team EIG {g} out of range"
+        );
+    }
+
+    #[test]
+    fn multi_member_teams_are_supported() {
+        let a = [rating(0.0, 6.0), rating(1.0, 4.0)];
+        let b = [rating(0.0, 6.0)];
+        let g = expected_information_gain(&[&a, &b], &options(0.0)).unwrap();
+        assert!(g > 0.0 && g <= std::f64::consts::LN_2, "{g}");
+    }
+
+    #[test]
+    fn degenerate_shapes_are_errors() {
+        let a = [rating(0.0, 6.0)];
+        assert!(matches!(
+            expected_information_gain(&[&a], &options(0.0)),
+            Err(InferenceError::NotEnoughTeams { got: 1 })
+        ));
+        let empty: [R; 0] = [];
+        assert!(matches!(
+            expected_information_gain(&[&a, &empty], &options(0.0)),
+            Err(InferenceError::EmptyTeam { team: 1 })
+        ));
+        assert!(matches!(
+            expected_information_gain(&[&a, &a], &options(1.5)),
+            Err(InferenceError::InvalidProbability { .. })
+        ));
+    }
+
+    #[test]
+    fn kl_divergence_is_zero_for_identical_beliefs() {
+        let g = Gaussian::from_ms(3.0, 2.0);
+        assert!(kl_divergence(g, g).abs() < 1e-15);
+    }
+
+    #[test]
+    fn kl_divergence_is_non_negative_and_grows_with_separation() {
+        let prior = Gaussian::from_ms(0.0, 3.0);
+        let mut previous = 0.0;
+        for mu in [0.0, 0.5, 1.0, 2.0, 4.0] {
+            let d = kl_divergence(Gaussian::from_ms(mu, 3.0), prior);
+            assert!(d >= 0.0, "negative divergence at mu {mu}: {d}");
+            assert!(d >= previous, "not increasing at mu {mu}");
+            previous = d;
+        }
+    }
+}

src/competitor.rs

@@ -1,5 +1,4 @@
 use crate::{
-    N_INF,
     drift::{ConstantDrift, Drift},
     gaussian::Gaussian,
     rating::Rating,
@@ -8,12 +7,19 @@ use crate::{
 
 /// Per-history, temporal state for someone competing.
 ///
-/// Renamed from `Agent` in T2; the former `.player` field is now
+/// The mutable half of a competitor: `Rating` holds their static
-/// `.rating` to match the `Player → Rating` rename.
+/// configuration, this holds what inference learns as it sweeps.
 #[derive(Debug)]
 pub struct Competitor<T: Time = i64, D: Drift<T> = ConstantDrift> {
     pub rating: Rating<T, D>,
-    pub message: Gaussian,
+    /// The forward message carried from this competitor's last appearance, or
+    /// `None` before they have appeared anywhere.
+    ///
+    /// Previously an improper `N_INF` served as the unset sentinel, which made
+    /// "no message yet" indistinguishable from "a legitimately improper
+    /// message" at the type level and required every reader to know the
+    /// convention.
+    pub message: Option<Gaussian>,
     pub last_time: Option<T>,
 }
 
@@ -21,14 +27,16 @@ impl<T: Time, D: Drift<T>> Competitor<T, D> {
     /// Compute the message received at time `now`, with drift accumulated
     /// from `self.last_time` (if any) to `now`.
     pub(crate) fn receive(&self, now: &T) -> Gaussian {
-        if self.message != N_INF {
+        match self.message {
-            let elapsed_variance = match &self.last_time {
+            Some(message) => {
-                Some(last) => self.rating.drift.variance_delta(last, now),
+                let elapsed_variance = match &self.last_time {
-                None => 0.0,
+                    Some(last) => self.rating.drift_variance_delta(last, now),
-            };
+                    None => 0.0,
-            self.message.forget(elapsed_variance)
+                };
-        } else {
+
-            self.rating.prior
+                message.forget(elapsed_variance)
+            }
+            None => self.rating.prior,
         }
     }
 
@@ -37,11 +45,9 @@ impl<T: Time, D: Drift<T>> Competitor<T, D> {
     /// Used in convergence sweeps where the elapsed was cached at slice-construction time
     /// and should not be recomputed from `last_time` (which may have shifted).
     pub(crate) fn receive_for_elapsed(&self, elapsed: i64) -> Gaussian {
-        if self.message != N_INF {
+        match self.message {
-            self.message
+            Some(message) => message.forget(self.rating.drift_variance_for_elapsed(elapsed)),
-                .forget(self.rating.drift.variance_for_elapsed(elapsed))
+            None => self.rating.prior,
-        } else {
-            self.rating.prior
         }
     }
 }
@@ -50,7 +56,7 @@ impl Default for Competitor<i64, ConstantDrift> {
     fn default() -> Self {
         Self {
             rating: Rating::default(),
-            message: N_INF,
+            message: None,
             last_time: None,
         }
     }
@@ -63,7 +69,7 @@ where
     C: Iterator<Item = &'a mut Competitor<T, D>>,
 {
     for c in competitors {
-        c.message = N_INF;
+        c.message = None;
         if last_time {
             c.last_time = None;
         }

src/convergence.rs

@@ -20,6 +20,37 @@ pub struct ConvergenceOptions {
     pub alpha: f64,
 }
 
+impl ConvergenceOptions {
+    /// Reject values that would make inference silently meaningless.
+    ///
+    /// `HistoryBuilder::convergence` asserts these eagerly, but the fields are
+    /// public and `GameOptions` carries a `ConvergenceOptions` — so a caller
+    /// can hand `Game::ranked` a set the builder never saw. In release the
+    /// engine's `debug_assert!`s are gone, and an `alpha` of zero leaves every
+    /// EP update unapplied: inference returns the priors, with every likelihood
+    /// uninformative and nothing to indicate anything went wrong.
+    ///
+    /// # Errors
+    ///
+    /// `InvalidParameter` if `alpha` is outside `(0.0, 1.0]` or `epsilon` is
+    /// negative. NaN fails both comparisons and is rejected.
+    pub(crate) fn validate(&self) -> Result<(), crate::InferenceError> {
+        if !(self.alpha > 0.0 && self.alpha <= 1.0) {
+            return Err(crate::InferenceError::InvalidParameter {
+                name: "alpha",
+                value: self.alpha,
+            });
+        }
+        if self.epsilon.is_nan() || self.epsilon < 0.0 {
+            return Err(crate::InferenceError::InvalidParameter {
+                name: "epsilon",
+                value: self.epsilon,
+            });
+        }
+        Ok(())
+    }
+}
+
 impl Default for ConvergenceOptions {
     fn default() -> Self {
         Self {
@@ -31,14 +62,23 @@ impl Default for ConvergenceOptions {
 }
 
 /// Post-hoc summary of a `History::converge` call.
+///
+/// From [`History::converge`](crate::History::converge) this always describes a
+/// converged fit — stopping at `max_iter` is
+/// [`InferenceError::NotConverged`](crate::InferenceError::NotConverged) there.
+/// From [`History::converge_partial`](crate::History::converge_partial) it may
+/// not be, and `converged` is what says so.
 #[derive(Clone, Debug)]
+#[must_use = "from `converge_partial` this may describe a fit that stopped at \
+              `max_iter`, which is wrong by a little rather than loudly \
+              broken — check `converged`, or bind it to `_` to say you have \
+              decided not to"]
 pub struct ConvergenceReport {
     pub iterations: usize,
     pub final_step: (f64, f64),
     pub log_evidence: f64,
     pub converged: bool,
     pub per_iteration_time: SmallVec<[Duration; 32]>,
-    pub slices_skipped: usize,
 }
 
 #[cfg(test)]

src/error.rs

@@ -1,5 +1,44 @@
 use std::fmt;
 
+/// How a prediction should treat a key the history has never seen.
+///
+/// Configured once per history via
+/// [`HistoryBuilder::unknown_keys`](crate::HistoryBuilder::unknown_keys).
+/// Neither known consumer wants this to vary between queries — one predicts
+/// thousands of candidate matchups in a loop, the other's headline feature is
+/// predicting a competitor nobody has faced — so it is a property of how you
+/// intend to use the model rather than an argument on five call sites.
+///
+/// # There is deliberately no `Skip`
+///
+/// Dropping an unknown member is the obvious third option and it is wrong. A
+/// team's performance is the *sum* of its members, so removing one removes its
+/// variance too: measured on a two-member team with one unknown, skipping gives
+/// a performance sigma of 2.37 where treating the member as unknown gives 6.53.
+/// An unknown competitor would make the model *more* certain, which is
+/// backwards. `Prior` is also the answer the model already gives for a
+/// competitor it knows about but has no evidence for, so it corresponds to a
+/// state the model can actually be in; skipping does not.
+#[derive(Clone, Copy, Debug, Default, PartialEq, Eq)]
+#[non_exhaustive]
+pub enum UnknownKeys {
+    /// Reject the prediction with [`InferenceError::UnknownKey`].
+    ///
+    /// The default, and the right one when every key is expected to be known:
+    /// a team of strangers should not silently produce a confident-looking
+    /// answer.
+    #[default]
+    Reject,
+    /// Treat an unknown competitor as one sitting at the history's configured
+    /// prior.
+    ///
+    /// This is the honest Bayesian reading — a competitor you have never
+    /// observed is exactly the prior — and it makes "predict a matchup
+    /// involving someone new" a first-class question rather than something a
+    /// caller fakes with a neutral constant.
+    Prior,
+}
+
 #[derive(Debug, Clone, PartialEq)]
 #[non_exhaustive]
 pub enum InferenceError {
@@ -25,10 +64,23 @@ pub enum InferenceError {
     /// result has no representable likelihood. Configure a positive `p_draw`
     /// (via `HistoryBuilder::p_draw` or `GameOptions::p_draw`) to admit ties.
     TieWithoutDrawProbability { teams: (usize, usize) },
-    /// Convergence exceeded `max_iter` without falling below `epsilon`.
+    /// The convergence sweep hit `max_iter` with the step still above
-    ConvergenceFailed {
+    /// `epsilon`.
-        last_step: (f64, f64),
+    ///
+    /// A fit that stops short is wrong by a little, which is the worst
+    /// available failure: every rating is finite, the ordering looks sensible,
+    /// and nothing in the numbers says they were still moving. Reported rather
+    /// than returned as a flag on an `Ok`, because a flag has to be checked
+    /// and `let _ = h.converge()` is the natural way not to.
+    ///
+    /// Either the history needs more iterations — raise `max_iter` — or it is
+    /// oscillating rather than converging, in which case `alpha < 1.0` damps
+    /// the within-game EP loop. [`History::converge_partial`](crate::History::converge_partial)
+    /// returns the short fit instead when that is genuinely what is wanted.
+    NotConverged {
         iterations: usize,
+        final_step: (f64, f64),
+        epsilon: f64,
     },
     /// Inference produced a non-finite value (NaN or infinity).
     ///
@@ -38,8 +90,60 @@ pub enum InferenceError {
         context: &'static str,
         step: (f64, f64),
     },
-    /// Negative precision: a Gaussian with `pi < 0` slipped into an API call.
+    /// One batch declared two different values for the same competitor's
-    NegativePrecision { pi: f64 },
+    /// configuration.
+    ///
+    /// `prior` and `drift_scale` configure a competitor, not an event, so a
+    /// batch that sets one of them twice with different values has no
+    /// well-defined meaning: events within a batch are not ordered, so
+    /// "last one wins" would make the result depend on iteration order.
+    /// Declaring the same value repeatedly is fine and is the expected shape
+    /// when a competitor's configuration is a property of the domain.
+    ConflictingCompetitorConfig {
+        competitor: usize,
+        field: &'static str,
+    },
+    /// A prediction referenced a key the history has no skill for.
+    ///
+    /// Reported rather than skipped: dropping unknown keys turns a team of
+    /// strangers into a confident-looking probability about nobody.
+    ///
+    /// `key` is the offending key's `Debug` rendering. It is carried because
+    /// the indices alone are not actionable: a caller that logs
+    /// `UnknownKey { team: 0, member: 0 }` learns nothing about *which* of its
+    /// keys the history has not seen, and the natural handling — fall back to a
+    /// neutral value — turns the whole thing into a plausible constant.
+    UnknownKey {
+        team: usize,
+        member: usize,
+        key: String,
+    },
+    /// `History::register` was called for a competitor that already exists.
+    ///
+    /// Registration states a competitor's configuration before anything has
+    /// been observed about them, so a competitor that already exists has
+    /// already been configured — by an earlier `register`, or by an event that
+    /// created them. Silently overwriting would reintroduce exactly the
+    /// order-dependence registration exists to remove.
+    ///
+    /// To change an existing competitor's configuration, supply it on an event
+    /// through `Member`; that refits the whole history.
+    AlreadyRegistered { key: String },
+    /// A prediction was given a team with no members.
+    EmptyTeam { team: usize },
+    /// A joint posterior was requested where one cannot be formed exactly.
+    JointUnavailable { reason: &'static str },
+    /// Fewer than two teams were supplied to a prediction.
+    NotEnoughTeams { got: usize },
+    /// The full outcome distribution was requested for too many teams.
+    ///
+    /// Each realisation sorts into exactly one (order, tie-pattern) event, so
+    /// the space holds `n! * 2^(n-1)` members — 1_920 at five teams, 23_040 at
+    /// six, 322_560 at seven. Past `max` this stops being something to
+    /// enumerate on a caller's behalf; ask for individual rankings with
+    /// `predict_ranking`, or for `predict_win_probabilities`, both of which
+    /// stay cheap at any team count.
+    TooManyTeams { got: usize, max: usize },
 }
 
 impl fmt::Display for InferenceError {
@@ -69,6 +173,18 @@ impl fmt::Display for InferenceError {
                     teams.0, teams.1
                 )
             }
+            Self::NotConverged {
+                iterations,
+                final_step,
+                epsilon,
+            } => {
+                write!(
+                    f,
+                    "did not converge in {iterations} iterations: final step {final_step:?} \
+                     is still above epsilon {epsilon}; raise max_iter, or damp with \
+                     alpha < 1.0 if it is oscillating"
+                )
+            }
             Self::NonFiniteResult { context, step } => {
                 write!(
                     f,
@@ -78,17 +194,43 @@ impl fmt::Display for InferenceError {
             Self::InvalidParameter { name, value } => {
                 write!(f, "{name} is invalid: {value}")
             }
-            Self::ConvergenceFailed {
+            Self::ConflictingCompetitorConfig { competitor, field } => {
-                last_step,
-                iterations,
-            } => {
                 write!(
                     f,
-                    "convergence failed after {iterations} iterations; last step = {last_step:?}"
+                    "competitor {competitor}: this batch sets {field} to two different values"
                 )
             }
-            Self::NegativePrecision { pi } => {
+            Self::UnknownKey { team, member, key } => {
-                write!(f, "precision must be non-negative; got {pi}")
+                write!(
+                    f,
+                    "team {team}, member {member}: no skill recorded for key {key} \
+                     (every key must already be known to the history; pre-filter \
+                     with `lookup` or `current_skill` if that is not guaranteed)"
+                )
+            }
+            Self::AlreadyRegistered { key } => {
+                write!(
+                    f,
+                    "competitor {key} is already registered; registration states \
+                     configuration before anything is observed, so re-registering \
+                     would silently overwrite it"
+                )
+            }
+            Self::EmptyTeam { team } => {
+                write!(f, "team {team} has no members")
+            }
+            Self::JointUnavailable { reason } => {
+                write!(f, "no exact joint posterior is available: {reason}")
+            }
+            Self::NotEnoughTeams { got } => {
+                write!(f, "prediction needs at least 2 teams, got {got}")
+            }
+            Self::TooManyTeams { got, max } => {
+                write!(
+                    f,
+                    "the outcome distribution over {got} teams is too large to enumerate (limit {max}); \
+                     use predict_ranking or predict_win_probabilities instead"
+                )
             }
         }
     }

src/event.rs

@@ -1,8 +1,10 @@
 //! Typed event description for bulk ingestion.
 //!
-//! `Event<T, K>` is the new public event shape (spec Section 4). Replaces
+//! `Event<T, K>` is the public event shape taken by `History::add_events`. It
-//! the nested `Vec<Vec<Vec<Index>>>`, `Vec<Vec<f64>>`, `Vec<Vec<Vec<f64>>>`
+//! is a typed front end, not a replacement: `add_events` flattens it into the
-//! that the old `add_events_with_prior` took.
+//! nested `Vec<Vec<Vec<Index>>>` / `Vec<Vec<f64>>` / `Vec<Vec<Vec<f64>>>` that
+//! the internal `add_events_with_prior` chokepoint still takes, and which
+//! `record_winner` and `record_draw` also route through.
 
 use smallvec::SmallVec;
 
@@ -23,6 +25,7 @@ pub struct Team<K> {
 }
 
 impl<K> Team<K> {
+    #[must_use]
     pub fn new() -> Self {
         Self {
             members: SmallVec::new(),
@@ -44,13 +47,28 @@ impl<K> Default for Team<K> {
 
 /// One member of a team, identified by user key `K`.
 ///
-/// `weight` defaults to 1.0; a per-event `prior` can override the competitor's
+/// `weight` applies per event and defaults to 1.0.
-/// current skill estimate for this event only.
+///
+/// `prior` and `drift_scale` are **competitor configuration**, not per-event
+/// values. Setting either applies to the competitor for the whole history, not
+/// just to this event, and applies whenever it is supplied — including on a key
+/// the history already knows. Because configuration lives on the competitor and
+/// `converge` refits from competitor state, configuring one late still refits
+/// the whole history rather than taking effect only from that event onward.
+///
+/// Repeating the same value is inert, which is the expected shape when the
+/// configuration is a property of the domain. Supplying two *different* values
+/// for one competitor within a single batch is
+/// `InferenceError::ConflictingCompetitorConfig`: events in a batch have no
+/// order, so there would be no well-defined winner.
 #[derive(Clone, Debug)]
 pub struct Member<K> {
     pub key: K,
     pub weight: f64,
     pub prior: Option<Gaussian>,
+    /// Multiplier on the drift *variance* this competitor accumulates.
+    /// `None` means 1.0.
+    pub drift_scale: Option<f64>,
 }
 
 impl<K> Member<K> {
@@ -59,6 +77,7 @@ impl<K> Member<K> {
             key,
             weight: 1.0,
             prior: None,
+            drift_scale: None,
         }
     }
 
@@ -67,10 +86,34 @@ impl<K> Member<K> {
         self
     }
 
+    /// Set this competitor's starting skill estimate.
+    ///
+    /// Competitor configuration, not a per-event value: it applies for the
+    /// whole history and applies whenever it is supplied, including on a key
+    /// the history already knows. See the type docs.
     pub fn with_prior(mut self, prior: Gaussian) -> Self {
         self.prior = Some(prior);
         self
     }
+
+    /// Scale how fast this competitor drifts, relative to the history's drift.
+    ///
+    /// The scale multiplies the drift *variance*, so it is in the same units as
+    /// `gamma`: `ConstantDrift(g)` at `scale = s` behaves exactly as
+    /// `ConstantDrift(g * s)` would for this competitor alone.
+    ///
+    /// `0.0` pins the competitor still — useful for a reference point that
+    /// shares a scale with moving competitors but should not itself move: a bot
+    /// at a known strength, a rating floor, a course difficulty.
+    ///
+    /// Applies for the whole history and whenever it is supplied, including on
+    /// a key the history already knows; see the type docs.
+    /// Must be finite and non-negative, or ingestion fails with
+    /// [`InferenceError::InvalidParameter`](crate::InferenceError::InvalidParameter).
+    pub fn with_drift_scale(mut self, scale: f64) -> Self {
+        self.drift_scale = Some(scale);
+        self
+    }
 }
 
 /// Convenience: a member is a user key with default weight 1.0 and no prior.
@@ -91,15 +134,18 @@ mod tests {
         assert_eq!(m.key, "alice");
         assert_eq!(m.weight, 1.0);
         assert!(m.prior.is_none());
+        assert!(m.drift_scale.is_none());
     }
 
     #[test]
     fn member_builder_methods_chain() {
         let m = Member::new("alice")
             .with_weight(0.5)
-            .with_prior(Gaussian::from_ms(20.0, 5.0));
+            .with_prior(Gaussian::from_ms(20.0, 5.0))
+            .with_drift_scale(0.0);
         assert_eq!(m.weight, 0.5);
         assert!(m.prior.is_some());
+        assert_eq!(m.drift_scale, Some(0.0));
     }
 
     #[test]

src/event_builder.rs

@@ -19,6 +19,14 @@ where
     history: &'h mut History<T, D, O, K>,
     event: Event<T, K>,
     current_team_idx: Option<usize>,
+    /// First validation failure seen while building, surfaced by `commit`.
+    ///
+    /// The setters return `Self` so the chain stays fluent; they cannot return
+    /// a `Result` without breaking that. Recording the failure and reporting it
+    /// at `commit` keeps the check enforced in release, where the previous
+    /// `debug_assert!` was compiled out and a mismatched event was ingested
+    /// silently.
+    error: Option<InferenceError>,
 }
 
 impl<'h, T, D, O, K> EventBuilder<'h, T, D, O, K>
@@ -37,10 +45,13 @@ where
                 outcome: Outcome::Ranked(SmallVec::new()),
             },
             current_team_idx: None,
+            error: None,
         }
     }
 
     /// Add a team by its member keys (weight 1.0 each, no prior overrides).
+    ///
+    /// Use [`EventBuilder::members`] to set `prior` or `drift_scale`.
     pub fn team<I: IntoIterator<Item = K>>(mut self, keys: I) -> Self {
         let members: SmallVec<[Member<K>; 4]> = keys.into_iter().map(Member::new).collect();
         self.event.teams.push(Team { members });
@@ -48,24 +59,72 @@ where
         self
     }
 
+    /// Add a team from fully-specified [`Member`] values.
+    ///
+    /// [`EventBuilder::team`] is the common case and builds members with
+    /// `Member::new`, which leaves `prior` and `drift_scale` unset. This is the
+    /// escape hatch for when they matter:
+    ///
+    /// ```
+    /// # use trueskill_tt::{Gaussian, History, Member};
+    /// # let mut h = History::builder().build();
+    /// h.event(0)
+    ///     .team(["player"])
+    ///     .members([Member::new("layout_7")
+    ///         .with_drift_scale(0.0)
+    ///         .with_prior(Gaussian::from_ms(0.0, 1.0))])
+    ///     .ranking([0, 1])
+    ///     .commit()?;
+    /// # Ok::<(), trueskill_tt::InferenceError>(())
+    /// ```
+    ///
+    /// One method rather than a `priors` and a `drift_scales` setter beside
+    /// `weights`: those would have to grow a parallel array — and a parallel
+    /// length check — every time `Member` gains a field, and each one would be
+    /// a new way to get the lengths wrong. `Member`'s own builder already
+    /// expresses all of it.
+    ///
+    /// `prior` and `drift_scale` are competitor configuration rather than
+    /// per-event values; see [`Member`] for what that means for a key the
+    /// history already knows.
+    pub fn members<I: IntoIterator<Item = Member<K>>>(mut self, members: I) -> Self {
+        self.event.teams.push(Team::with_members(members));
+        self.current_team_idx = Some(self.event.teams.len() - 1);
+        self
+    }
+
     /// Set per-member weights for the most recently added team.
     ///
-    /// Panics in debug builds if called before `.team(...)` or if the length
+    /// A length mismatch is recorded and returned by [`EventBuilder::commit`]
-    /// doesn't match the team's member count.
+    /// as `InferenceError::MismatchedShape`, in both debug and release. The
+    /// weights are not applied in that case, so a partially-weighted team
+    /// cannot reach the history.
+    ///
+    /// # Panics
+    ///
+    /// Panics if called before any `.team(...)`.
     pub fn weights<I: IntoIterator<Item = f64>>(mut self, weights: I) -> Self {
         let idx = self
             .current_team_idx
             .expect(".weights(...) called before any .team(...)");
+
         let ws: Vec<f64> = weights.into_iter().collect();
         let team = &mut self.event.teams[idx];
-        debug_assert_eq!(
+
-            ws.len(),
+        if ws.len() != team.members.len() {
-            team.members.len(),
+            self.error.get_or_insert(InferenceError::MismatchedShape {
-            "weights length must match team size"
+                kind: "weights",
-        );
+                expected: team.members.len(),
+                got: ws.len(),
+            });
+
+            return self;
+        }
+
         for (m, w) in team.members.iter_mut().zip(ws) {
             m.weight = w;
         }
+
         self
     }
 
@@ -84,7 +143,10 @@ where
     /// Set explicit per-team continuous scores with a per-event noise override.
     ///
     /// `sigma` overrides `HistoryBuilder::score_sigma` for this event only.
-    /// Must be `> 0.0`; debug-asserts otherwise via `Outcome::scores_with_sigma`.
+    /// Must be `> 0.0`. Constructing the outcome with a non-positive or NaN
+    /// sigma is allowed; the value is rejected with
+    /// `InferenceError::InvalidParameter` when the event is ingested, so
+    /// callers get an error from `commit` rather than a panic.
     pub fn scores_with_sigma<I: IntoIterator<Item = f64>>(mut self, scores: I, sigma: f64) -> Self {
         self.event.outcome = crate::Outcome::scores_with_sigma(scores, sigma);
         self
@@ -103,7 +165,17 @@ where
     }
 
     /// Commit the event to the history.
+    ///
+    /// # Errors
+    ///
+    /// Returns the first validation failure recorded while building — see
+    /// [`EventBuilder::weights`] — otherwise forwards to
+    /// [`History::add_events`] and returns its errors.
     pub fn commit(self) -> Result<(), InferenceError> {
+        if let Some(error) = self.error {
+            return Err(error);
+        }
+
         self.history.add_events(std::iter::once(self.event))
     }
 }

src/factor/margin.rs

@@ -1,8 +1,8 @@
 use crate::{
     N_INF,
-    factor::{Factor, VarId, VarStore},
+    factor::{VarId, VarStore},
     gaussian::Gaussian,
-    pdf,
+    ln_pdf,
 };
 
 /// Gaussian observation factor on a diff variable.
@@ -16,10 +16,11 @@ pub struct MarginFactor {
     pub m_obs: f64,
     pub sigma: f64,
     pub(crate) msg: Gaussian,
-    pub(crate) evidence_cached: Option<f64>,
+    pub(crate) log_evidence_cached: Option<f64>,
 }
 
 impl MarginFactor {
+    #[must_use]
     pub fn new(diff: VarId, m_obs: f64, sigma: f64) -> Self {
         debug_assert!(sigma > 0.0, "score sigma must be positive");
         Self {
@@ -27,7 +28,7 @@ impl MarginFactor {
             m_obs,
             sigma,
             msg: N_INF,
-            evidence_cached: None,
+            log_evidence_cached: None,
         }
     }
 }
@@ -40,8 +41,8 @@ impl MarginFactor {
         let marginal = vars.get(self.diff);
         let cavity = marginal / self.msg;
 
-        if self.evidence_cached.is_none() {
+        if self.log_evidence_cached.is_none() {
-            self.evidence_cached = Some(cavity_evidence(cavity, self.m_obs, self.sigma));
+            self.log_evidence_cached = Some(cavity_log_evidence(cavity, self.m_obs, self.sigma));
         }
 
         let new_msg = Gaussian::from_ms(self.m_obs, self.sigma);
@@ -54,23 +55,42 @@ impl MarginFactor {
     }
 }
 
-impl Factor for MarginFactor {
+/// Undamped wrappers, used by this module's tests. Inference drives these
-    fn propagate(&mut self, vars: &mut VarStore) -> (f64, f64) {
+/// factors through `propagate_with_alpha` and reads the cached log evidence
+/// directly, so these are not on any production path.
+#[cfg(test)]
+impl MarginFactor {
+    pub(crate) fn propagate(&mut self, vars: &mut VarStore) -> (f64, f64) {
         self.propagate_with_alpha(vars, 1.0)
     }
 
-    fn log_evidence(&self, _vars: &VarStore) -> f64 {
+    pub(crate) fn log_evidence(&self) -> f64 {
-        self.evidence_cached.unwrap_or(1.0).ln()
+        self.log_evidence_cached.unwrap_or(0.0)
     }
 }
 
-/// Density of the observed margin under the cavity, clamped to a positive
+/// `ln` of the observed margin's density under the cavity.
-/// floor so a far-out observation cannot underflow to `0.0` and make
+///
-/// `log_evidence` `-inf`.
+/// Computed in log space rather than as `pdf(..).ln()`. The density underflows
-fn cavity_evidence(cavity: Gaussian, m_obs: f64, sigma: f64) -> f64 {
+/// to zero past about 38 sigma of separation, and clamping that to
-    let combined_sigma = (cavity.sigma().powi(2) + sigma.powi(2)).sqrt();
+/// `f64::MIN_POSITIVE` reported -708 nats however far out the observation
+/// actually was — 4292 nats adrift at 100 sigma, and unbounded beyond. A score
+/// far from what the model expected is exactly the observation a log-evidence
+/// figure exists to notice.
+fn cavity_log_evidence(cavity: Gaussian, m_obs: f64, sigma: f64) -> f64 {
+    // `hypot`, not `sqrt(a^2 + b^2)`: squaring overflows to infinity above a
+    // sigma of ~1.3e154 and flushes to zero below ~1.5e-154, and `Gaussian`'s
+    // constructors are public so a caller can reach both.
+    let combined_sigma = cavity.sigma().hypot(sigma);
+    let value = ln_pdf(m_obs, cavity.mu(), combined_sigma);
 
-    pdf(m_obs, cavity.mu(), combined_sigma).max(f64::MIN_POSITIVE)
+    // A degenerate cavity (infinite sigma) is the only way to reach a
+    // non-finite result; fall back to the old floor rather than emit -inf.
+    if value.is_finite() {
+        value
+    } else {
+        f64::MIN_POSITIVE.ln()
+    }
 }
 
 #[cfg(test)]
@@ -112,16 +132,16 @@ mod tests {
         let mut vars = VarStore::new();
         let diff = vars.alloc(Gaussian::from_ms(0.0, 6.0));
         let mut f = MarginFactor::new(diff, 5.0, 1.0);
-        assert!(f.evidence_cached.is_none());
+        assert!(f.log_evidence_cached.is_none());
 
         f.propagate(&mut vars);
-        let z = f.evidence_cached.unwrap();
+        let z = f.log_evidence_cached.unwrap();
-        // pdf(5, 0, sqrt(37)) ≈ 0.046783
+        // ln pdf(5, 0, sqrt(37)) = ln(0.046783...)
-        assert!((z - 0.04678300292616668).abs() < 1e-10);
+        assert!((z.exp() - 0.04678300292616668).abs() < 1e-10);
 
         // Subsequent propagations don't change it.
         f.propagate(&mut vars);
-        assert_eq!(f.evidence_cached.unwrap(), z);
+        assert_eq!(f.log_evidence_cached.unwrap(), z);
     }
 
     #[test]
@@ -130,7 +150,7 @@ mod tests {
         let diff = vars.alloc(Gaussian::from_ms(0.0, 6.0));
         let mut f = MarginFactor::new(diff, 5.0, 1.0);
         f.propagate(&mut vars);
-        let logz = f.log_evidence(&vars);
+        let logz = f.log_evidence();
         assert!((logz - (-3.062235327364623)).abs() < 1e-10);
     }
 

src/factor/mod.rs

@@ -20,6 +20,9 @@ pub struct VarStore {
 }
 
 impl VarStore {
+    /// Test-only: inference allocates its store through `ScratchArena`.
+    #[cfg(test)]
+    #[must_use]
     pub fn new() -> Self {
         Self::default()
     }
@@ -28,20 +31,20 @@ impl VarStore {
         self.marginals.clear();
     }
 
+    /// Test-only, as `new`.
+    #[cfg(test)]
+    #[must_use]
     pub fn len(&self) -> usize {
         self.marginals.len()
     }
 
-    pub fn is_empty(&self) -> bool {
-        self.marginals.is_empty()
-    }
-
     pub fn alloc(&mut self, init: Gaussian) -> VarId {
         let id = VarId(self.marginals.len() as u32);
         self.marginals.push(init);
         id
     }
 
+    #[must_use]
     pub fn get(&self, id: VarId) -> Gaussian {
         self.marginals[id.0 as usize]
     }
@@ -51,58 +54,7 @@ impl VarStore {
     }
 }
 
-/// A factor in the EP graph.
-///
-/// Factors hold their own outgoing messages and propagate them by reading
-/// connected variable marginals from a `VarStore` and writing back updated
-/// marginals.
-pub trait Factor: Send + Sync {
-    /// Update outgoing messages and write back to the var store.
-    ///
-    /// Returns the max delta `(|Δmu|, |Δsigma|)` across writes this
-    /// propagation. Used by the `Schedule` to detect convergence.
-    fn propagate(&mut self, vars: &mut VarStore) -> (f64, f64);
-
-    /// Optional log-evidence contribution. Default 0.0 (no contribution).
-    fn log_evidence(&self, _vars: &VarStore) -> f64 {
-        0.0
-    }
-}
-
-/// Enum dispatcher for the built-in factor types.
-///
-/// Using an enum instead of `Box<dyn Factor>` keeps factor data inline and
-/// avoids virtual-call overhead in the hot inference loop.
-#[derive(Debug)]
-pub enum BuiltinFactor {
-    TeamSum(team_sum::TeamSumFactor),
-    RankDiff(rank_diff::RankDiffFactor),
-    Trunc(trunc::TruncFactor),
-    Margin(margin::MarginFactor),
-}
-
-impl Factor for BuiltinFactor {
-    fn propagate(&mut self, vars: &mut VarStore) -> (f64, f64) {
-        match self {
-            Self::TeamSum(f) => f.propagate(vars),
-            Self::RankDiff(f) => f.propagate(vars),
-            Self::Trunc(f) => f.propagate(vars),
-            Self::Margin(f) => f.propagate(vars),
-        }
-    }
-
-    fn log_evidence(&self, vars: &VarStore) -> f64 {
-        match self {
-            Self::Trunc(f) => f.log_evidence(vars),
-            Self::Margin(f) => f.log_evidence(vars),
-            Self::TeamSum(_) | Self::RankDiff(_) => 0.0,
-        }
-    }
-}
-
 pub mod margin;
-pub mod rank_diff;
-pub mod team_sum;
 pub mod trunc;
 
 #[cfg(test)]
@@ -149,20 +101,4 @@ mod tests {
         assert_eq!(store.len(), 0);
         assert_eq!(store.marginals.capacity(), cap);
     }
-
-    #[test]
-    fn builtin_factor_dispatches_to_margin() {
-        use super::margin::MarginFactor;
-        let mut vars = VarStore::new();
-        let diff = vars.alloc(Gaussian::from_ms(0.0, 6.0));
-        let mut f = BuiltinFactor::Margin(MarginFactor::new(diff, 5.0, 1.0));
-
-        f.propagate(&mut vars);
-
-        let result = vars.get(diff);
-        assert!((result.mu() - 4.864864864864865).abs() < 1e-12);
-
-        let logz = f.log_evidence(&vars);
-        assert!((logz - (-3.062235327364623)).abs() < 1e-10);
-    }
 }

src/factor/rank_diff.rs

@@ -1,95 +0,0 @@
-use crate::factor::{Factor, VarId, VarStore};
-
-/// Maintains the constraint `diff = team_a - team_b` between three vars.
-///
-/// On each propagation:
-/// - Reads marginals at `team_a` and `team_b` (which already incorporate any
-///   incoming messages from neighboring factors).
-/// - Computes `new_diff = team_a - team_b` (variance addition; see Gaussian::Sub).
-/// - Writes the new marginal to `diff`.
-/// - Returns the delta against the previous diff value.
-///
-/// This factor does NOT store an outgoing message; the diff variable is
-/// effectively replaced on each propagation. The TruncFactor on the same diff
-/// var holds the EP-divide message that produces the cavity.
-#[derive(Debug)]
-pub struct RankDiffFactor {
-    pub team_a: VarId,
-    pub team_b: VarId,
-    pub diff: VarId,
-}
-
-impl Factor for RankDiffFactor {
-    fn propagate(&mut self, vars: &mut VarStore) -> (f64, f64) {
-        let a = vars.get(self.team_a);
-        let b = vars.get(self.team_b);
-        let new_diff = a - b;
-        let old = vars.get(self.diff);
-        vars.set(self.diff, new_diff);
-        old.delta(new_diff)
-    }
-}
-
-#[cfg(test)]
-mod tests {
-    use super::*;
-    use crate::{N_INF, gaussian::Gaussian};
-
-    #[test]
-    fn diff_of_two_known_gaussians() {
-        let mut vars = VarStore::new();
-        let team_a = vars.alloc(Gaussian::from_ms(25.0, 3.0));
-        let team_b = vars.alloc(Gaussian::from_ms(20.0, 4.0));
-        let diff = vars.alloc(N_INF);
-
-        let mut f = RankDiffFactor {
-            team_a,
-            team_b,
-            diff,
-        };
-        f.propagate(&mut vars);
-
-        let result = vars.get(diff);
-        // mu = 25 - 20 = 5; var = 9 + 16 = 25; sigma = 5
-        assert!((result.mu() - 5.0).abs() < 1e-12);
-        assert!((result.sigma() - 5.0).abs() < 1e-12);
-    }
-
-    #[test]
-    fn delta_zero_on_repeat() {
-        let mut vars = VarStore::new();
-        let team_a = vars.alloc(Gaussian::from_ms(10.0, 2.0));
-        let team_b = vars.alloc(Gaussian::from_ms(8.0, 1.0));
-        let diff = vars.alloc(N_INF);
-
-        let mut f = RankDiffFactor {
-            team_a,
-            team_b,
-            diff,
-        };
-        f.propagate(&mut vars);
-        let (dmu, dsig) = f.propagate(&mut vars);
-        assert!(dmu < 1e-12);
-        assert!(dsig < 1e-12);
-    }
-
-    #[test]
-    fn delta_reflects_team_change() {
-        let mut vars = VarStore::new();
-        let team_a = vars.alloc(Gaussian::from_ms(10.0, 1.0));
-        let team_b = vars.alloc(Gaussian::from_ms(0.0, 1.0));
-        let diff = vars.alloc(N_INF);
-
-        let mut f = RankDiffFactor {
-            team_a,
-            team_b,
-            diff,
-        };
-        f.propagate(&mut vars);
-
-        // change team_a, repropagate; delta should be positive
-        vars.set(team_a, Gaussian::from_ms(15.0, 1.0));
-        let (dmu, _dsig) = f.propagate(&mut vars);
-        assert!(dmu > 4.0, "expected ~5 delta, got {}", dmu);
-    }
-}

src/factor/team_sum.rs

@@ -1,98 +0,0 @@
-use crate::{
-    N00,
-    factor::{Factor, VarId, VarStore},
-    gaussian::Gaussian,
-};
-
-/// Computes the weighted sum of player performances into a team-perf var.
-///
-/// Inputs are pre-computed player performance Gaussians (i.e., rating priors
-/// already with beta² noise added via `Rating::performance()`). The factor
-/// runs once per game and writes the weighted sum to the output var.
-#[derive(Debug)]
-pub struct TeamSumFactor {
-    pub inputs: Vec<(Gaussian, f64)>,
-    pub out: VarId,
-}
-
-impl Factor for TeamSumFactor {
-    fn propagate(&mut self, vars: &mut VarStore) -> (f64, f64) {
-        let perf = self.inputs.iter().fold(N00, |acc, (g, w)| acc + (*g * *w));
-        let old = vars.get(self.out);
-        vars.set(self.out, perf);
-        old.delta(perf)
-    }
-}
-
-#[cfg(test)]
-mod tests {
-    use super::*;
-    use crate::N_INF;
-
-    #[test]
-    fn single_player_unit_weight() {
-        let mut vars = VarStore::new();
-        let out = vars.alloc(N_INF);
-        let g = Gaussian::from_ms(25.0, 5.0);
-        let mut f = TeamSumFactor {
-            inputs: vec![(g, 1.0)],
-            out,
-        };
-
-        f.propagate(&mut vars);
-        let result = vars.get(out);
-        assert!((result.mu() - 25.0).abs() < 1e-12);
-        assert!((result.sigma() - 5.0).abs() < 1e-12);
-    }
-
-    #[test]
-    fn two_players_summed() {
-        let mut vars = VarStore::new();
-        let out = vars.alloc(N_INF);
-        let g1 = Gaussian::from_ms(20.0, 3.0);
-        let g2 = Gaussian::from_ms(30.0, 4.0);
-        let mut f = TeamSumFactor {
-            inputs: vec![(g1, 1.0), (g2, 1.0)],
-            out,
-        };
-
-        f.propagate(&mut vars);
-        let result = vars.get(out);
-        // sum: mu = 20 + 30 = 50, var = 9 + 16 = 25, sigma = 5
-        assert!((result.mu() - 50.0).abs() < 1e-12);
-        assert!((result.sigma() - 5.0).abs() < 1e-12);
-    }
-
-    #[test]
-    fn weighted_inputs() {
-        let mut vars = VarStore::new();
-        let out = vars.alloc(N_INF);
-        let g = Gaussian::from_ms(10.0, 2.0);
-        let mut f = TeamSumFactor {
-            inputs: vec![(g, 2.0)],
-            out,
-        };
-
-        f.propagate(&mut vars);
-        let result = vars.get(out);
-        // g * 2.0: mu = 10*2 = 20, sigma = 2*2 = 4
-        assert!((result.mu() - 20.0).abs() < 1e-12);
-        assert!((result.sigma() - 4.0).abs() < 1e-12);
-    }
-
-    #[test]
-    fn delta_is_zero_on_repeat_propagate() {
-        let mut vars = VarStore::new();
-        let out = vars.alloc(N_INF);
-        let g = Gaussian::from_ms(5.0, 1.0);
-        let mut f = TeamSumFactor {
-            inputs: vec![(g, 1.0)],
-            out,
-        };
-
-        f.propagate(&mut vars);
-        let (dmu, dsig) = f.propagate(&mut vars);
-        assert!(dmu < 1e-12, "expected ~0 delta on repeat, got {}", dmu);
-        assert!(dsig < 1e-12);
-    }
-}

src/factor/trunc.rs

@@ -1,7 +1,8 @@
 use crate::{
-    N_INF, approx, cdf,
+    N_INF, approx,
-    factor::{Factor, VarId, VarStore},
+    factor::{VarId, VarStore},
     gaussian::Gaussian,
+    ln_interval, ln_sf,
 };
 
 /// EP truncation factor on a diff variable.
@@ -15,20 +16,21 @@ pub struct TruncFactor {
     pub diff: VarId,
     pub margin: f64,
     pub tie: bool,
-    /// Outgoing message to the diff variable (initial: N_INF, the EP identity).
+    /// Outgoing message to the diff variable (initial: `N_INF`, the EP identity).
     pub(crate) msg: Gaussian,
     /// Cached evidence (linear, not log) computed from the cavity on first propagation.
-    pub(crate) evidence_cached: Option<f64>,
+    pub(crate) log_evidence_cached: Option<f64>,
 }
 
 impl TruncFactor {
+    #[must_use]
     pub fn new(diff: VarId, margin: f64, tie: bool) -> Self {
         Self {
             diff,
             margin,
             tie,
             msg: N_INF,
-            evidence_cached: None,
+            log_evidence_cached: None,
         }
     }
 }
@@ -41,8 +43,8 @@ impl TruncFactor {
         let marginal = vars.get(self.diff);
         let cavity = marginal / self.msg;
 
-        if self.evidence_cached.is_none() {
+        if self.log_evidence_cached.is_none() {
-            self.evidence_cached = Some(cavity_evidence(cavity, self.margin, self.tie));
+            self.log_evidence_cached = Some(cavity_log_evidence(cavity, self.margin, self.tie));
         }
 
         let trunc = approx(cavity, self.margin, self.tie);
@@ -61,31 +63,40 @@ impl TruncFactor {
     }
 }
 
-impl Factor for TruncFactor {
+/// Undamped wrappers, used by this module's tests. Inference drives these
-    fn propagate(&mut self, vars: &mut VarStore) -> (f64, f64) {
+/// factors through `propagate_with_alpha` and reads the cached log evidence
+/// directly, so these are not on any production path.
+#[cfg(test)]
+impl TruncFactor {
+    pub(crate) fn propagate(&mut self, vars: &mut VarStore) -> (f64, f64) {
         self.propagate_with_alpha(vars, 1.0)
     }
-
-    fn log_evidence(&self, _vars: &VarStore) -> f64 {
-        self.evidence_cached.unwrap_or(1.0).ln()
-    }
 }
 
-/// P(diff > margin) for non-tie, P(|diff| < margin) for tie.
+/// `ln P(diff > margin)` for a win, `ln P(|diff| < margin)` for a tie.
 ///
-/// Clamped to a positive floor: for a near-certain outcome the tail rounds to
+/// Computed in log space throughout. Two earlier shapes both lost the tail:
-/// exactly 0.0, and the `erfc` approximation used by `cdf` carries ~1e-7 error
+/// `1 - cdf(..)` cancelled away every digit of an unlikely outcome, and even
-/// so it can even return slightly more than 1.0, making the difference
+/// once that was fixed the linear probability underflows to zero past about 38
-/// negative. Either would send `log_evidence` to `-inf` or NaN and poison the
+/// sigma, where clamping reported -708 nats regardless of the truth. An upset
-/// sum across the whole history.
+/// is the observation a log-evidence figure exists to notice, so it has to stay
-fn cavity_evidence(diff: Gaussian, margin: f64, tie: bool) -> f64 {
+/// exact precisely where it is smallest.
-    let raw = if tie {
+fn cavity_log_evidence(diff: Gaussian, margin: f64, tie: bool) -> f64 {
-        cdf(margin, diff.mu(), diff.sigma()) - cdf(-margin, diff.mu(), diff.sigma())
+    let (mu, sigma) = (diff.mu(), diff.sigma());
+
+    let value = if tie {
+        ln_interval(-margin, margin, mu, sigma)
     } else {
-        1.0 - cdf(margin, diff.mu(), diff.sigma())
+        ln_sf(margin, mu, sigma)
     };
 
-    raw.clamp(f64::MIN_POSITIVE, 1.0)
+    // A degenerate cavity is the only route to a non-finite result; keep the
+    // old floor for it rather than letting -inf poison the whole history's sum.
+    if value.is_finite() {
+        value
+    } else {
+        f64::MIN_POSITIVE.ln()
+    }
 }
 
 #[cfg(test)]
@@ -116,19 +127,90 @@ mod tests {
         let diff = vars.alloc(Gaussian::from_ms(2.0, 3.0));
 
         let mut f = TruncFactor::new(diff, 0.0, false);
-        assert!(f.evidence_cached.is_none());
+        assert!(f.log_evidence_cached.is_none());
 
         f.propagate(&mut vars);
-        assert!(f.evidence_cached.is_some());
+        assert!(f.log_evidence_cached.is_some());
-        let first = f.evidence_cached.unwrap();
+        let first = f.log_evidence_cached.unwrap();
 
         // Evidence should be P(diff > 0) for diff ~ N(2, 9) ≈ 0.748
-        assert!(first > 0.7);
+        assert!(first.exp() > 0.7);
-        assert!(first < 0.8);
+        assert!(first.exp() < 0.8);
 
         // Subsequent propagations don't change it.
         f.propagate(&mut vars);
-        assert_eq!(f.evidence_cached.unwrap(), first);
+        assert_eq!(f.log_evidence_cached.unwrap(), first);
+    }
+
+    /// The defect this guards: `1 - cdf` collapsed to zero for a surprising
+    /// result, the clamp turned that into `f64::MIN_POSITIVE`, and
+    /// `log_evidence` reported ln of *that* — about -708 whatever the truth
+    /// was. An upset is the observation a model-comparison score exists to
+    /// notice, so it was wrong exactly where it mattered.
+    #[test]
+    fn evidence_of_an_upset_is_not_flattened_to_the_clamp_floor() {
+        // diff ~ N(-9, 1) with margin 0: the favoured side lost by nine sigma.
+        let evidence = cavity_log_evidence(Gaussian::from_ms(-9.0, 1.0), 0.0, false).exp();
+
+        assert!(
+            evidence > f64::MIN_POSITIVE,
+            "evidence collapsed onto the clamp floor: {evidence}"
+        );
+        // P(X > 0) for X ~ N(-9, 1) is the standard normal tail at 9 sigma.
+        assert!(
+            (evidence - 1.128_588e-19).abs() / 1.128_588e-19 < 1e-6,
+            "expected ~1.13e-19, got {evidence}"
+        );
+        assert!(
+            (evidence.ln() + 43.628).abs() < 1e-2,
+            "log evidence {} should be about -43.6, not -708",
+            evidence.ln()
+        );
+    }
+
+    /// Evidence must stay finite and positive however extreme the mismatch,
+    /// since `log_evidence` sums across the whole history and one `-inf` or
+    /// `NaN` poisons all of it.
+    ///
+    /// Finiteness alone is too weak a bar — the clamped version was finite too,
+    /// and wrong by hundreds of nats. `log_evidence_tracks_the_analytic_tail`
+    /// below is the assertion that actually holds this up.
+    #[test]
+    fn evidence_stays_positive_and_finite_at_any_separation() {
+        for mu in [-300.0f64, -50.0, -9.0, 0.0, 9.0, 50.0, 300.0] {
+            for tie in [false, true] {
+                let ln_e = cavity_log_evidence(Gaussian::from_ms(mu, 1.0), 1.0, tie);
+                assert!(
+                    ln_e.is_finite() && ln_e <= 0.0,
+                    "mu={mu} tie={tie}: log evidence {ln_e} is not a log-probability"
+                );
+            }
+        }
+    }
+
+    /// The clamp used to floor everything past ~38 sigma at `ln(MIN_POSITIVE)`
+    /// = -708, however far out the real observation was. In log space the
+    /// answer is a polynomial and stays exact: at 1000 sigma the truth is about
+    /// -500_000 nats, and -708 is not a rounding error.
+    #[test]
+    fn log_evidence_tracks_the_analytic_tail() {
+        for mu in [-40.0f64, -60.0, -100.0, -1000.0] {
+            // P(diff > 0) for diff ~ N(mu, 1), mu far below zero.
+            let got = cavity_log_evidence(Gaussian::from_ms(mu, 1.0), 0.0, false);
+
+            // ln Phi(mu) ~ -mu^2/2 - ln(-mu) - ln(sqrt(2 pi)) for mu << 0.
+            let z = -mu;
+            let approx = -0.5 * z * z - z.ln() - (2.0 * std::f64::consts::PI).sqrt().ln();
+
+            assert!(
+                got < f64::MIN_POSITIVE.ln(),
+                "mu={mu}: {got} is still stuck on the old clamp floor"
+            );
+            assert!(
+                (got - approx).abs() / approx.abs() < 1e-3,
+                "mu={mu}: got {got}, asymptotic expectation {approx}"
+            );
+        }
     }
 
     #[test]
@@ -140,7 +222,7 @@ mod tests {
         f.propagate(&mut vars);
 
         // For diff ~ N(0, 4), tie=true with margin=1: P(-1 < diff < 1) ≈ 0.383
-        let ev = f.evidence_cached.unwrap();
+        let ev = f.log_evidence_cached.unwrap().exp();
         assert!(ev > 0.35 && ev < 0.42);
     }
 

src/factors.rs

@@ -1,13 +0,0 @@
-//! Factor-graph public API.
-//!
-//! Power users can construct custom factor graphs via `Game::custom` (T2
-//! minimal; full ergonomics in T4) and drive them with custom `Schedule`
-//! implementations.
-
-pub use crate::{
-    factor::{
-        BuiltinFactor, Factor, VarId, VarStore, margin::MarginFactor, rank_diff::RankDiffFactor,
-        team_sum::TeamSumFactor, trunc::TruncFactor,
-    },
-    schedule::{EpsilonOrMax, Schedule, ScheduleReport},
-};

src/game.rs

@@ -46,8 +46,8 @@ impl DiffFactor {
     /// reaches.
     pub(crate) fn log_evidence(&self) -> f64 {
         match self {
-            Self::Trunc(f) => f.evidence_cached.unwrap_or(1.0).ln(),
+            Self::Trunc(f) => f.log_evidence_cached.unwrap_or(0.0),
-            Self::Margin(f) => f.evidence_cached.unwrap_or(1.0).ln(),
+            Self::Margin(f) => f.log_evidence_cached.unwrap_or(0.0),
         }
     }
 
@@ -107,16 +107,13 @@ impl<T: Time, D: Drift<T>> OwnedGame<T, D> {
         convergence: crate::ConvergenceOptions,
     ) -> Self {
         let mut arena = ScratchArena::new();
-        let g = Game::ranked_with_arena(
+
-            teams.clone(),
+        // `Game` takes the teams by value and is dropped here, so take the vec
-            &result,
+        // back out of it rather than handing it a clone.
-            &weights,
+        let g = Game::ranked_with_arena(teams, &result, &weights, p_draw, convergence, &mut arena);
-            p_draw,
+
-            convergence,
-            &mut arena,
-        );
         Self {
-            teams,
+            teams: g.teams,
             likelihoods: g.likelihoods,
             log_evidence: g.log_evidence,
         }
@@ -130,21 +127,24 @@ impl<T: Time, D: Drift<T>> OwnedGame<T, D> {
         convergence: crate::ConvergenceOptions,
     ) -> Self {
         let mut arena = ScratchArena::new();
+
         let g = Game::scored_with_arena(
-            teams.clone(),
+            teams,
             &scores,
             &weights,
             score_sigma,
             convergence,
             &mut arena,
         );
+
         Self {
-            teams,
+            teams: g.teams,
             likelihoods: g.likelihoods,
             log_evidence: g.log_evidence,
         }
     }
 
+    #[must_use]
     pub fn posteriors(&self) -> Vec<Vec<Gaussian>> {
         self.likelihoods
             .iter()
@@ -153,6 +153,7 @@ impl<T: Time, D: Drift<T>> OwnedGame<T, D> {
             .collect()
     }
 
+    #[must_use]
     pub fn log_evidence(&self) -> f64 {
         self.log_evidence
     }
@@ -409,6 +410,7 @@ impl<'a, T: Time, D: Drift<T>> Game<'a, T, D> {
         self.likelihoods = likelihoods;
     }
 
+    #[must_use]
     pub fn posteriors(&self) -> Vec<Vec<Gaussian>> {
         self.likelihoods
             .iter()
@@ -422,17 +424,30 @@ impl<'a, T: Time, D: Drift<T>> Game<'a, T, D> {
             .collect::<Vec<_>>()
     }
 
+    #[must_use]
     pub fn log_evidence(&self) -> f64 {
         self.log_evidence
     }
 }
 
 impl<T: Time, D: Drift<T>> Game<'_, T, D> {
+    /// # Errors
+    ///
+    /// - `InvalidParameter` if `options.convergence` is out of range — an
+    ///   `alpha` of zero would leave every EP update unapplied and silently
+    ///   return the priors.
+    /// - `InvalidProbability` if `options.p_draw` is outside `[0.0, 1.0)`.
+    /// - `MismatchedShape` if the outcome's rank count differs from `teams.len()`.
+    /// - `WrongOutcomeKind` if `outcome` is not `Outcome::Ranked`.
+    /// - `TieWithoutDrawProbability` if the outcome ties two teams while
+    ///   `p_draw` is zero: the truncation margin is then zero and the two-sided
+    ///   tie update evaluates `0/0`.
     pub fn ranked(
         teams: &[&[Rating<T, D>]],
         outcome: crate::Outcome,
         options: &GameOptions,
     ) -> Result<OwnedGame<T, D>, crate::InferenceError> {
+        options.convergence.validate()?;
         if !(0.0..1.0).contains(&options.p_draw) {
             return Err(crate::InferenceError::InvalidProbability {
                 value: options.p_draw,
@@ -478,11 +493,18 @@ impl<T: Time, D: Drift<T>> Game<'_, T, D> {
         ))
     }
 
+    /// # Errors
+    ///
+    /// - `InvalidParameter` if `options.score_sigma` is not strictly positive
+    ///   or is NaN, or if `options.convergence` is out of range.
+    /// - `MismatchedShape` if the outcome's score count differs from `teams.len()`.
+    /// - `WrongOutcomeKind` if `outcome` is not `Outcome::Scored`.
     pub fn scored(
         teams: &[&[Rating<T, D>]],
         outcome: crate::Outcome,
         options: &GameOptions,
     ) -> Result<OwnedGame<T, D>, crate::InferenceError> {
+        options.convergence.validate()?;
         if options.score_sigma <= 0.0 || options.score_sigma.is_nan() {
             return Err(crate::InferenceError::InvalidParameter {
                 name: "score_sigma",
@@ -515,16 +537,28 @@ impl<T: Time, D: Drift<T>> Game<'_, T, D> {
         ))
     }
 
+    /// Convenience wrapper over [`Game::ranked`] for two single-player teams.
+    ///
+    /// # Errors
+    ///
+    /// Delegates to [`Game::ranked`], so it returns the same errors — in
+    /// practice `WrongOutcomeKind` for a non-ranked outcome, or
+    /// `TieWithoutDrawProbability` for a draw when `options.p_draw` is zero.
     pub fn one_v_one(
         a: &Rating<T, D>,
         b: &Rating<T, D>,
         outcome: crate::Outcome,
+        options: &GameOptions,
     ) -> Result<(Gaussian, Gaussian), crate::InferenceError> {
-        let game = Self::ranked(&[&[*a], &[*b]], outcome, &GameOptions::default())?;
+        let game = Self::ranked(&[&[*a], &[*b]], outcome, options)?;
         let post = game.posteriors();
         Ok((post[0][0], post[1][0]))
     }
 
+    /// # Errors
+    ///
+    /// Wraps each player in a one-member team and delegates to
+    /// [`Game::ranked`], so it returns the same errors.
     pub fn free_for_all(
         players: &[&Rating<T, D>],
         outcome: crate::Outcome,
@@ -534,15 +568,6 @@ impl<T: Time, D: Drift<T>> Game<'_, T, D> {
         let team_refs: Vec<&[Rating<T, D>]> = teams.iter().map(|t| t.as_slice()).collect();
         Self::ranked(&team_refs, outcome, options)
     }
-
-    #[doc(hidden)]
-    pub fn custom<S: crate::factors::Schedule>(
-        factors: &mut [crate::factors::BuiltinFactor],
-        vars: &mut crate::factors::VarStore,
-        schedule: &S,
-    ) -> crate::factors::ScheduleReport {
-        schedule.run(factors, vars)
-    }
 }
 
 #[cfg(test)]
@@ -699,9 +724,15 @@ mod tests {
         let c = p[2][0];
 
         // T1 ULP shift: mu rounds to 25.0 (was 24.999999) under natural-parameter storage.
+        //
+        // The 1e-6-place values moved when `erfc_inv`'s sign error was fixed:
+        // this case runs at `p_draw = 0.5`, so it goes through `compute_margin`,
+        // and the margin is now 8.4e-8 from the exact quantile where it was
+        // 1.46e-7. Verified as movement *toward* analytic truth, not a
+        // regression — see `erfc_inv_matches_known_quantiles`.
         assert_ulps_eq!(a, Gaussian::from_ms(25.0, 6.092561), epsilon = 1e-6);
-        assert_ulps_eq!(b, Gaussian::from_ms(33.379314, 6.483575), epsilon = 1e-6);
+        assert_ulps_eq!(b, Gaussian::from_ms(33.379315, 6.483576), epsilon = 1e-6);
-        assert_ulps_eq!(c, Gaussian::from_ms(16.620685, 6.483575), epsilon = 1e-6);
+        assert_ulps_eq!(c, Gaussian::from_ms(16.620685, 6.483576), epsilon = 1e-6);
     }
 
     #[test]
@@ -1214,7 +1245,7 @@ mod tests {
         );
         assert_ulps_eq!(
             p[1][0],
-            Gaussian::from_ms(19.287197, 7.243465),
+            Gaussian::from_ms(19.287198285, 7.243465848),
             epsilon = 1e-6
         );
         assert_ulps_eq!(
@@ -1274,7 +1305,7 @@ mod tests {
 
         assert_ulps_eq!(
             p[0][0],
-            Gaussian::from_ms(31.674697, 7.501180),
+            Gaussian::from_ms(31.674698083, 7.501180037),
             epsilon = 1e-6
         );
         assert_ulps_eq!(

src/gaussian.rs

@@ -18,6 +18,7 @@ pub struct Gaussian {
 
 impl Gaussian {
     /// Construct from mean and standard deviation.
+    #[must_use]
     pub const fn from_ms(mu: f64, sigma: f64) -> Self {
         if sigma == f64::INFINITY {
             Self { pi: 0.0, tau: 0.0 }
@@ -64,16 +65,19 @@ impl Gaussian {
     }
 
     #[inline]
+    #[must_use]
     pub fn pi(&self) -> f64 {
         self.pi
     }
 
     #[inline]
+    #[must_use]
     pub fn tau(&self) -> f64 {
         self.tau
     }
 
     #[inline]
+    #[must_use]
     pub fn mu(&self) -> f64 {
         // A non-positive precision is an improper (uninformative) Gaussian — its mean is
         // undefined. Treat it like `pi == 0` and return 0. EP message cancellation can land
@@ -102,6 +106,7 @@ impl Gaussian {
     }
 
     #[inline]
+    #[must_use]
     pub fn sigma(&self) -> f64 {
         // A non-positive precision is improper → infinite standard deviation. Guarding
         // `pi <= 0.0` (not just `== 0.0`) keeps `1.0 / pi.sqrt()` from returning NaN when EP
@@ -140,11 +145,51 @@ impl Gaussian {
         Self::from_mv(self.mu(), self.variance() + variance_delta)
     }
 
+    /// `P(X < x)` under this Gaussian.
+    ///
+    /// The question a stopping rule asks: *how sure am I that this competitor's
+    /// true skill is below the cutoff?* Expressing that as a probability keeps
+    /// its meaning as sigma changes, where a `mu + z * sigma` band silently
+    /// means different confidence at different uncertainties — which is exactly
+    /// the regime a stopping rule operates in.
+    ///
+    /// Accurate in the *lower* tail. For the upper tail use
+    /// [`Gaussian::probability_above`] rather than `1.0 - probability_below(x)`,
+    /// which cancels away every significant digit once the result is small.
+    ///
+    /// An improper Gaussian (non-positive precision) has no defined mean, so
+    /// this returns `0.5` — the same convention `mu()` and `sigma()` follow.
+    #[must_use]
+    pub fn probability_below(&self, x: f64) -> f64 {
+        if self.pi <= 0.0 {
+            return 0.5;
+        }
+        crate::cdf(x, self.mu(), self.sigma())
+    }
+
+    /// `P(X > x)` under this Gaussian.
+    ///
+    /// Computed as a survival function rather than `1 - cdf`, so it keeps full
+    /// relative precision in the upper tail: `1 - cdf` returns exactly zero
+    /// past about 8.3 sigma, where the true value is still 1e-19 and perfectly
+    /// representable. A stopping rule is evaluated precisely there — the
+    /// interesting cases are the ones near certainty.
+    ///
+    /// An improper Gaussian returns `0.5`, as [`Gaussian::probability_below`].
+    #[must_use]
+    pub fn probability_above(&self, x: f64) -> f64 {
+        if self.pi <= 0.0 {
+            return 0.5;
+        }
+        crate::sf(x, self.mu(), self.sigma())
+    }
+
     /// EP damping in natural-parameter space: `α·new + (1−α)·self`.
     ///
     /// Used by within-game inference to stabilise oscillating fixed-point
     /// loops on hard graphs. `alpha = 1.0` returns `new` exactly;
     /// `alpha < 1.0` shrinks each per-step update.
+    #[must_use]
     pub fn damp_natural(self, new: Gaussian, alpha: f64) -> Gaussian {
         Gaussian::from_natural(
             alpha * new.pi() + (1.0 - alpha) * self.pi(),
@@ -334,3 +379,68 @@ mod tests {
         assert!((damped.tau() - expected_tau).abs() < 1e-12);
     }
 }
+
+#[cfg(test)]
+mod tail_probability_tests {
+    use super::*;
+
+    #[test]
+    fn probability_below_matches_published_quantiles() {
+        let g = Gaussian::from_ms(0.0, 1.0);
+        for (x, expected) in [
+            (-1.959_963_984_540_054, 0.025),
+            (0.0, 0.5),
+            (1.281_551_565_544_6, 0.9),
+            (1.959_963_984_540_054, 0.975),
+        ] {
+            let got = g.probability_below(x);
+            assert!(
+                (got - expected).abs() < 1e-12,
+                "P(X < {x}) = {got}, expected {expected}"
+            );
+        }
+    }
+
+    #[test]
+    fn the_two_tails_partition_the_mass() {
+        let g = Gaussian::from_ms(3.0, 2.0);
+        for x in [-4.0f64, 0.0, 3.0, 7.5] {
+            let total = g.probability_below(x) + g.probability_above(x);
+            assert!((total - 1.0).abs() < 1e-15, "at {x}: {total}");
+        }
+    }
+
+    /// The reason `probability_above` exists rather than `1 - probability_below`.
+    #[test]
+    fn probability_above_keeps_precision_where_the_complement_collapses() {
+        let g = Gaussian::from_ms(0.0, 1.0);
+        for (x, expected) in [(9.0f64, 1.128_588e-19), (20.0, 2.753_624e-89)] {
+            let got = g.probability_above(x);
+            assert!(
+                (got - expected).abs() / expected < 1e-6,
+                "P(X > {x}) = {got}, expected ~{expected}"
+            );
+            assert_eq!(
+                1.0 - g.probability_below(x),
+                0.0,
+                "the complement should still collapse at {x}"
+            );
+        }
+    }
+
+    #[test]
+    fn a_scaled_gaussian_shifts_and_stretches() {
+        let g = Gaussian::from_ms(25.0, 6.0);
+        assert!((g.probability_below(25.0) - 0.5).abs() < 1e-15);
+        // One sigma either side of the mean.
+        assert!((g.probability_below(31.0) - 0.841_344_746_068_543).abs() < 1e-12);
+        assert!((g.probability_above(19.0) - 0.841_344_746_068_543).abs() < 1e-12);
+    }
+
+    #[test]
+    fn an_improper_gaussian_is_uninformative_rather_than_nan() {
+        let improper = Gaussian::from_ms(0.0, f64::INFINITY);
+        assert_eq!(improper.probability_below(5.0), 0.5);
+        assert_eq!(improper.probability_above(5.0), 0.5);
+    }
+}

src/joint.rs

@@ -0,0 +1,152 @@
+//! Cholesky factorisation of a joint precision matrix.
+//!
+//! Every question the joint answers is a *bilinear form* in the precision
+//! matrix's inverse — the variance of a contrast is `c^T L^-1 c`, and the
+//! covariance of two contrasts is `c^T L^-1 a`. None of them wants `L^-1 c`
+//! itself, which is what makes the shape here worth stating explicitly.
+//!
+//! Writing the precision as `A = L L^T`,
+//!
+//! ```text
+//! c^T A^-1 a = c^T L^-T L^-1 a = (L^-1 c) . (L^-1 a)
+//! ```
+//!
+//! so a single forward substitution per contrast answers everything, and the
+//! back substitution a general solve would do is wasted work. That halves the
+//! cost of a query, and it removes a failure mode: a variance computed as
+//! `c . (A^-1 c)` is a difference of products that can round to a small
+//! negative number, where the same quantity as `|L^-1 c|^2` is a sum of
+//! squares and cannot.
+//!
+//! Factorising is `O(n^3)` and whitening is `O(n^2)`, so the split also
+//! matters structurally: the expensive half depends only on the fit, and is
+//! shared across every query a [`Joint`](crate::Joint) answers.
+
+/// A factorised symmetric positive-definite matrix, reusable across queries.
+pub(crate) struct Cholesky {
+    /// Lower triangle of `L`, row-major `n * n`. The upper triangle is
+    /// leftover scratch from the factorisation and is never read.
+    l: Vec<f64>,
+    n: usize,
+}
+
+impl Cholesky {
+    /// Factorise `a` (row-major, `n * n`, symmetric) into `L L^T`.
+    ///
+    /// `a` is consumed as scratch.
+    ///
+    /// Returns `None` if the matrix is not positive-definite, which for a
+    /// precision matrix means the model is improper — a competitor with
+    /// neither a proper prior nor any evidence.
+    pub(crate) fn factor(mut a: Vec<f64>, n: usize) -> Option<Self> {
+        debug_assert_eq!(a.len(), n * n);
+
+        for j in 0..n {
+            let mut d = a[j * n + j];
+            for k in 0..j {
+                d -= a[j * n + k] * a[j * n + k];
+            }
+            // Explicit rather than `!(d > 0.0)`: a NaN pivot must fail here
+            // too, and a negated comparison would let it through as "not
+            // positive".
+            if d.is_nan() || d <= 0.0 {
+                return None;
+            }
+            let d = d.sqrt();
+            a[j * n + j] = d;
+
+            for i in j + 1..n {
+                let mut s = a[i * n + j];
+                for k in 0..j {
+                    s -= a[i * n + k] * a[j * n + k];
+                }
+                a[i * n + j] = s / d;
+            }
+        }
+
+        Some(Self { l: a, n })
+    }
+
+    /// Whiten a contrast: `y = L^-1 b`.
+    ///
+    /// The point of the result is the dot product, not the vector: for two
+    /// contrasts `b` and `b'`, `y . y'` is `b^T A^-1 b'`. See the module docs.
+    pub(crate) fn whiten(&self, b: &[f64]) -> Vec<f64> {
+        debug_assert_eq!(b.len(), self.n);
+        let n = self.n;
+        let mut y = b.to_vec();
+        for i in 0..n {
+            // Folded from `y[i]` rather than summed and subtracted once, so the
+            // accumulation order matches a plain substitution loop exactly.
+            let row = &self.l[i * n..i * n + i];
+            let s = row
+                .iter()
+                .zip(&y[..i])
+                .fold(y[i], |acc, (l, v)| acc - l * v);
+            y[i] = s / self.l[i * n + i];
+        }
+        y
+    }
+}
+
+/// `b^T A^-1 b'`, given the two whitened contrasts.
+pub(crate) fn bilinear(y: &[f64], y_prime: &[f64]) -> f64 {
+    y.iter().zip(y_prime).map(|(a, b)| a * b).sum()
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    /// `[[4, 1], [1, 3]] z = [1, 2]` has `z = [1/11, 7/11]`, so the quadratic
+    /// form `b^T A^-1 b` is `1 * 1/11 + 2 * 7/11 = 15/11`.
+    #[test]
+    fn reproduces_a_known_quadratic_form() {
+        let c = Cholesky::factor(vec![4.0, 1.0, 1.0, 3.0], 2).unwrap();
+        let y = c.whiten(&[1.0, 2.0]);
+        assert!((bilinear(&y, &y) - 15.0 / 11.0).abs() < 1e-12);
+    }
+
+    /// Whitening `e_i` recovers the inverse's diagonal, which is the variance
+    /// of a single variable.
+    #[test]
+    fn recovers_the_inverse_diagonal() {
+        // A = [[2, -1, 0], [-1, 2, -1], [0, -1, 2]]; inverse diagonal is
+        // [0.75, 1.0, 0.75].
+        let a = vec![2.0, -1.0, 0.0, -1.0, 2.0, -1.0, 0.0, -1.0, 2.0];
+        let c = Cholesky::factor(a, 3).unwrap();
+        for (i, expected) in [0.75, 1.0, 0.75].into_iter().enumerate() {
+            let mut e = vec![0.0; 3];
+            e[i] = 1.0;
+            let y = c.whiten(&e);
+            assert!((bilinear(&y, &y) - expected).abs() < 1e-12, "row {i}");
+        }
+    }
+
+    /// The off-diagonal bilinear form is symmetric and matches the inverse.
+    #[test]
+    fn recovers_an_off_diagonal_covariance() {
+        // Same A; (A^-1)_{0,1} = 0.5.
+        let a = vec![2.0, -1.0, 0.0, -1.0, 2.0, -1.0, 0.0, -1.0, 2.0];
+        let c = Cholesky::factor(a, 3).unwrap();
+        let y0 = c.whiten(&[1.0, 0.0, 0.0]);
+        let y1 = c.whiten(&[0.0, 1.0, 0.0]);
+        assert!((bilinear(&y0, &y1) - 0.5).abs() < 1e-12);
+        assert!((bilinear(&y1, &y0) - 0.5).abs() < 1e-12);
+    }
+
+    /// A variance can never come out negative, because it is a sum of squares.
+    #[test]
+    fn a_quadratic_form_is_never_negative() {
+        let a = vec![1e12, 1e12 - 1.0, 1e12 - 1.0, 1e12];
+        let c = Cholesky::factor(a, 2).unwrap();
+        let y = c.whiten(&[1.0, -1.0]);
+        assert!(bilinear(&y, &y) >= 0.0);
+    }
+
+    #[test]
+    fn rejects_a_non_positive_definite_matrix() {
+        // Singular: the second row is a multiple of the first.
+        assert!(Cholesky::factor(vec![1.0, 2.0, 2.0, 4.0], 2).is_none());
+    }
+}

src/key_table.rs

@@ -25,6 +25,7 @@ impl<K> KeyTable<K>
 where
     K: Eq + Hash + Clone,
 {
+    #[must_use]
     pub fn new() -> Self {
         Self {
             forward: HashMap::new(),
@@ -54,6 +55,7 @@ where
         }
     }
 
+    #[must_use]
     pub fn key(&self, idx: Index) -> Option<&K> {
         self.reverse.get(idx.0)
     }
@@ -62,10 +64,12 @@ where
         self.forward.keys()
     }
 
+    #[must_use]
     pub fn len(&self) -> usize {
         self.reverse.len()
     }
 
+    #[must_use]
     pub fn is_empty(&self) -> bool {
         self.reverse.is_empty()
     }

src/lib.rs

@@ -1,6 +1,6 @@
-//! TrueSkill Through Time — Bayesian skill rating over a time axis.
+//! `TrueSkill` Through Time — Bayesian skill rating over a time axis.
 //!
-//! Where plain TrueSkill gives each competitor one running estimate, TrueSkill
+//! Where plain `TrueSkill` gives each competitor one running estimate, `TrueSkill`
 //! Through Time treats a whole history as a single model and infers skill *at
 //! every point in time*. Evidence flows both directions: a result today
 //! sharpens the estimate of who someone was last year, so early estimates stop
@@ -86,6 +86,19 @@
 
 #![forbid(unsafe_code)]
 
+/// Compiles every `rust` block in `README.md` as a doctest.
+///
+/// The README is not the crate's front page — the module docs above are — so it
+/// is pulled in here rather than via a crate-level `#![doc = ...]`, purely so
+/// its examples are type-checked. Without this nothing compiled them, and they
+/// had drifted far enough that four blocks no longer built (#35). `cfg(doctest)`
+/// means this type exists only while collecting doctests.
+///
+/// Blocks that are illustrative rather than runnable are fenced as `text`.
+#[cfg(doctest)]
+#[doc = include_str!("../README.md")]
+pub struct ReadmeDoctests;
+
 use std::{
     cmp::Reverse,
     f64::consts::{FRAC_1_SQRT_2, FRAC_2_SQRT_PI, SQRT_2},
@@ -97,6 +110,7 @@ pub(crate) mod arena;
 mod time;
 mod time_slice;
 pub use time_slice::{EventKind, TimeSlice};
+mod acquisition;
 mod color_group;
 mod competitor;
 mod convergence;
@@ -105,33 +119,35 @@ mod error;
 mod event;
 mod event_builder;
 pub(crate) mod factor;
-pub mod factors;
 mod game;
 pub mod gaussian;
 mod history;
+mod joint;
 mod key_table;
 mod matrix;
 mod observer;
 mod outcome;
+mod predict;
+pub(crate) mod quadrature;
 mod rating;
-pub(crate) mod schedule;
 pub mod storage;
 
+pub use acquisition::expected_information_gain;
 pub use competitor::Competitor;
 pub use convergence::{ConvergenceOptions, ConvergenceReport};
 pub use drift::{ConstantDrift, Drift};
-pub use error::InferenceError;
+pub use error::{InferenceError, UnknownKeys};
 pub use event::{Event, Member, Team};
 pub use event_builder::EventBuilder;
 pub use game::{Game, GameOptions, OwnedGame};
 pub use gaussian::Gaussian;
-pub use history::{History, HistoryBuilder};
+pub use history::{History, HistoryBuilder, Joint};
 pub use key_table::KeyTable;
 use matrix::Matrix;
 pub use observer::{NullObserver, Observer};
 pub use outcome::Outcome;
+pub use predict::Prediction;
 pub use rating::Rating;
-pub use schedule::ScheduleReport;
 pub use time::{Time, Untimed};
 
 pub const BETA: f64 = 1.0;
@@ -140,9 +156,69 @@ pub const SIGMA: f64 = BETA * 6.0;
 pub const GAMMA: f64 = BETA * 0.03;
 pub const P_DRAW: f64 = 0.0;
 pub const EPSILON: f64 = 1e-6;
-pub const ITERATIONS: usize = 30;
+/// Default cap on convergence sweeps.
+///
+/// **A runaway guard, not a budget.** The sweep exits as soon as the step falls
+/// below `epsilon`, so the cap is never reached by a history that converges and
+/// raising it costs nothing. Measured on a history that needs four sweeps:
+///
+/// ```text
+/// max_iter      30: 4 iterations, 129.9 us
+/// max_iter 100_000: 4 iterations, 131.9 us
+/// ```
+///
+/// This was `30` until it was measured, and 30 truncated ordinary healthy
+/// histories: 160 events over 100 competitors already needs 42. Because a short
+/// fit is finite and sensibly ordered, that was invisible.
+///
+/// # Why it is not scaled to the history
+///
+/// The obvious improvement — pick the cap from the node or event count — does
+/// not work, because iteration count is driven by how *loopy* the graph is
+/// rather than how big it is. At a fixed 320 events over 40 slices, varying
+/// only the number of competitors sharing them:
+///
+/// ```text
+/// competitors    appearances each    iterations
+///           3                 213         2_789
+///          10                  64         1_068
+///          50                12.8           206
+///         100                 6.4            90
+///         400                 1.6             2
+/// ```
+///
+/// Three orders of magnitude apart on identical event and slice counts. Any
+/// formula in those two numbers would be badly wrong on some real shape, so the
+/// cap is a single value set high enough that reaching it means the fit is
+/// oscillating rather than merely large.
+///
+/// Reaching it is [`InferenceError::NotConverged`]. See
+/// [`History::converge`](crate::History::converge).
+pub const ITERATIONS: usize = 10_000;
+
+/// Largest team count `History::predict_outcome` will enumerate.
+///
+/// The outcome space holds `n! * 2^(n-1)` events, so it grows factorially:
+/// 1_920 at five teams, 23_040 at six, 322_560 at seven. Six is where
+/// enumerating on a caller's behalf stops being reasonable.
+pub const MAX_PREDICTED_TEAMS: usize = predict::MAX_TEAMS_FOR_DISTRIBUTION;
 
 const SQRT_TAU: f64 = 2.5066282746310002;
+/// `1 / sqrt(pi)`, the leading factor of the `erfcx` continued fraction.
+const FRAC_1_SQRT_PI: f64 = 0.564_189_583_547_756_3;
+/// `sqrt(2 / pi)`, the numerator of the inverse Mills ratio in scaled form.
+const SQRT_2_OVER_PI: f64 = 0.797_884_560_802_865_4;
+/// How many window widths into the tail before a tie window is treated as a
+/// half-line. Beyond this the truncated mass is concentrated within `1/alpha`
+/// of the near edge, so the far edge contributes nothing measurable.
+const HALF_LINE_WINDOW: f64 = 10.0;
+/// Where `v - alpha` switches from subtraction to its asymptotic series.
+///
+/// The subtraction loses roughly `eps * alpha^2` of relative precision, and the
+/// four-term series is good to ~1e-10 by here, so the two are at their closest
+/// agreement around this point. Below it the subtraction is exact; above it the
+/// series is.
+const ASYMPTOTIC_MILLS_ALPHA: f64 = 100.0;
 
 pub const N01: Gaussian = Gaussian::from_ms(0.0, 1.0);
 pub const N00: Gaussian = Gaussian::from_ms(0.0, 0.0);
@@ -174,24 +250,47 @@ impl From<Index> for usize {
     }
 }
 
+/// Complementary error function.
+///
+/// # Why every transcendental in this crate goes through `libm`
+///
+/// IEEE 754 specifies the basic operations and `sqrt` exactly, but says nothing
+/// about `exp`, `log` or `erf`. `std`'s versions delegate to the *system* math
+/// library, so they differ between platforms: measured here, `f64::exp` and
+/// `libm::exp` disagree on 9.7% of inputs and `f64::ln` / `libm::log` on 5.0%,
+/// each by one ULP.
+///
+/// Inference is an iterative fixed point, so a one-ULP difference can change an
+/// iteration count and therefore the answer by more than one ULP. Routing every
+/// transcendental through `libm` makes a fit reproducible across platforms, not
+/// just across thread counts as `tests/determinism.rs` already checks.
+///
+/// **So: use `libm::exp` / `libm::log` in inference code, never `f64::exp` /
+/// `f64::ln`.** `sqrt` is exempt — IEEE specifies it exactly, so `f64::sqrt` is
+/// already portable. Test code may use whichever is clearer.
+///
+/// It costs nothing: `Batch::iteration` measured -2.7% [-5.7%, -0.3%] with the
+/// whole set swapped.
+///
+/// Delegates to `libm`, which is the Rust port of FDLIBM and accurate to about
+/// one ULP. This replaced a Numerical Recipes `erfcc` rational approximation
+/// whose documented bound was 1.2e-7 *relative* — measured at ~1e-7 across the
+/// whole range, and the binding accuracy constraint on the entire crate.
+///
+/// The swap is free. 98% of the arguments inference passes here have
+/// `|x| < 0.84375`, which is exactly where FDLIBM skips the exponential
+/// entirely, so the longer polynomial costs nothing on the distribution that
+/// actually occurs: `Batch::iteration` moved -1.6% [-4.7%, +0.9%], p = 0.31.
+///
+/// What it bought: `compute_margin` went from 8.4e-8 to 1.7e-16 against exact
+/// quantiles, `cdf(mu, mu, sigma)` is now exactly 0.5, and `sf + cdf` sums to
+/// one within a single ULP where it was 3e-8 out.
 fn erfc(x: f64) -> f64 {
-    let z = x.abs();
+    libm::erfc(x)
-    let t = 1.0 / (1.0 + z / 2.0);
-
-    let a = -0.82215223 + t * 0.17087277;
-    let b = 1.48851587 + t * a;
-    let c = -1.13520398 + t * b;
-    let d = 0.27886807 + t * c;
-    let e = -0.18628806 + t * d;
-    let f = 0.09678418 + t * e;
-    let g = 0.37409196 + t * f;
-    let h = 1.00002368 + t * g;
-
-    let r = t * (-z * z - 1.26551223 + t * h).exp();
-
-    if x >= 0.0 { r } else { 2.0 - r }
 }
 
+/// The previous Numerical Recipes `erfcc`, kept only so the timing test can
+/// compare both in one binary. Removed once the comparison is recorded.
 fn erfc_inv(mut y: f64) -> f64 {
     if y >= 2.0 {
         return f64::NEG_INFINITY;
@@ -207,14 +306,22 @@ fn erfc_inv(mut y: f64) -> f64 {
         y = 2.0 - y;
     }
 
-    let t = (-2.0 * (y / 2.0).ln()).sqrt();
+    let t = libm::sqrt(-2.0 * libm::log(y / 2.0));
 
-    let mut x = FRAC_1_SQRT_2 * ((2.30753 + t * 0.27061) / (1.0 + t * (0.99229 + t * 0.04481)) - t);
+    // The leading coefficient is NEGATIVE. `rational - t` is negative here, so
+    // a positive coefficient mirrors the starting point to `-x0` — the
+    // reflection of the root. Newton then has to cross the origin to get back,
+    // which a fixed iteration count does not manage: measured against the true
+    // value, `erfc_inv(0.1)` returned 1.044 instead of 1.16309, and the error
+    // grew as y shrank until `compute_margin` stopped being monotone in
+    // `p_draw` altogether.
+    let mut x =
+        -FRAC_1_SQRT_2 * ((2.30753 + t * 0.27061) / (1.0 + t * (0.99229 + t * 0.04481)) - t);
 
     for _ in 0..3 {
         let err = erfc(x) - y;
 
-        x += err / (FRAC_2_SQRT_PI * (-(x.powi(2))).exp() - x * err)
+        x += err / (FRAC_2_SQRT_PI * libm::exp(-(x * x)) - x * err)
     }
 
     if y < 1.0 { x } else { -x }
@@ -234,32 +341,216 @@ pub(crate) fn cdf(x: f64, mu: f64, sigma: f64) -> f64 {
     0.5 * erfc(z)
 }
 
+/// `P(X > x)` for `X ~ N(mu, sigma^2)`.
+///
+/// The survival function, computed directly rather than as `1 - cdf(..)`.
+///
+/// The two are algebraically identical and numerically are not. `cdf` returns
+/// a value approaching 1 for an upper tail, so subtracting it from 1 cancels
+/// away every significant digit the tail had: measured against this function,
+/// `1 - cdf` carries 7% error by four sigma past the mean and returns exactly
+/// zero beyond about 8.3 sigma — where the true value is still 1e-19 and
+/// perfectly representable. `erfc` holds *relative* accuracy all the way down
+/// to 1e-296, so the precision is there to keep; only the subtraction threw it
+/// away.
+///
+/// This matters most where evidence is smallest, which is exactly where an
+/// upset makes it interesting: `ln` of a clamped zero is -708 regardless of
+/// whether the truth was -43 or -600.
+pub(crate) fn sf(x: f64, mu: f64, sigma: f64) -> f64 {
+    0.5 * erfc((x - mu) / (sigma * SQRT_2))
+}
+
+/// `e^(x^2) * erfc(x)`, the scaled complementary error function, for `x >= 0`.
+///
+/// Exists so the exponential factor common to a Gaussian density and its tail
+/// integral can be cancelled *analytically* instead of being computed twice
+/// and divided. Both underflow to zero past about 26 sigma, and their ratio is
+/// then `0/0` — finite in the limit, `NaN` in floating point.
+fn erfcx(x: f64) -> f64 {
+    if x < 2.0 {
+        // Below the crossover neither factor is extreme: erfc is O(1) and
+        // exp(x^2) is at most e^4, so the direct product is exact enough and
+        // cheaper than the continued fraction.
+        libm::exp(x * x) * erfc(x)
+    } else {
+        // erfcx(x) = 1/sqrt(pi) * 1/(x + (1/2)/(x + 1/(x + (3/2)/(x + ...)))),
+        // evaluated by backward recurrence. Converges quickly for x >= 2 and,
+        // unlike the product form, never touches an exponential.
+        let mut f = 0.0;
+        for n in (1..=60u32).rev() {
+            f = (f64::from(n) * 0.5) / (x + f);
+        }
+        FRAC_1_SQRT_PI / (x + f)
+    }
+}
+
+/// `ln` of the normal density at `x`.
+///
+/// The density itself underflows to zero past about 38 sigma, and `ln` of a
+/// clamped zero is -708 whatever the truth was. The log form is a polynomial:
+/// it stays exact at any separation, and the values it produces (-5001 nats at
+/// 100 sigma, -500001 at 1000) are perfectly representable.
+pub(crate) fn ln_pdf(x: f64, mu: f64, sigma: f64) -> f64 {
+    let z = (x - mu) / sigma;
+    -libm::log(SQRT_TAU * sigma) - 0.5 * z * z
+}
+
+/// `ln P(X > x)` for `X ~ N(mu, sigma^2)`.
+///
+/// In the upper tail the `exp(-z^2 / 2)` common to the tail integral is
+/// factored out analytically via `erfcx`, so this never underflows — where
+/// `sf(..).ln()` bottoms out at -708 once `erfc` itself reaches zero.
+pub(crate) fn ln_sf(x: f64, mu: f64, sigma: f64) -> f64 {
+    let z = (x - mu) / sigma;
+
+    if z > 0.0 {
+        // ln(0.5 * erfc(z/sqrt2)) with erfc(y) = exp(-y^2) * erfcx(y).
+        -std::f64::consts::LN_2 - 0.5 * z * z + libm::log(erfcx(z / SQRT_2))
+    } else {
+        // The mass here is at least a half; nothing to lose.
+        libm::log(sf(x, mu, sigma))
+    }
+}
+
+/// `ln P(lo < X < hi)` for `X ~ N(mu, sigma^2)`.
+///
+/// When the interval sits in a tail both endpoint probabilities underflow
+/// together, so their difference is taken in scaled form with the shared
+/// exponential factored out. When it straddles the mean nothing is small and
+/// the direct difference is exact.
+pub(crate) fn ln_interval(lo: f64, hi: f64, mu: f64, sigma: f64) -> f64 {
+    let z_lo = (lo - mu) / sigma;
+    let z_hi = (hi - mu) / sigma;
+
+    if z_hi <= z_lo {
+        return f64::NEG_INFINITY;
+    }
+
+    // Fold a lower-tail interval onto the upper tail; the normal is symmetric.
+    let (near, far) = if z_lo >= 0.0 {
+        (z_lo, z_hi)
+    } else if z_hi <= 0.0 {
+        (-z_hi, -z_lo)
+    } else {
+        // Straddles the mean: the interval holds a non-negligible share of the
+        // mass, so neither endpoint is near enough to 1 to cancel.
+        return libm::log((cdf(hi, mu, sigma) - cdf(lo, mu, sigma)).max(f64::MIN_POSITIVE));
+    };
+
+    let (a, b) = (near / SQRT_2, far / SQRT_2);
+    // b > a >= 0, so this ratio of exponentials is at most 1 and cannot overflow.
+    let scale = libm::exp(a * a - b * b);
+    let bracket = erfcx(a) - scale * erfcx(b);
+
+    if bracket <= 0.0 {
+        return f64::NEG_INFINITY;
+    }
+
+    -std::f64::consts::LN_2 - a * a + libm::log(bracket)
+}
+
 fn pdf(x: f64, mu: f64, sigma: f64) -> f64 {
     let normalizer = (SQRT_TAU * sigma).powi(-1);
-    let functional = (-((x - mu).powi(2)) / (2.0 * sigma.powi(2))).exp();
+    let functional = libm::exp(-((x - mu) * (x - mu)) / (2.0 * sigma * sigma));
 
     normalizer * functional
 }
 
+/// Truncated-Gaussian correction terms `(v, w)`.
+///
+/// `v` shifts the mean and `w` shrinks the variance. Both are ratios whose
+/// numerator and denominator underflow together in the tails, so both are
+/// computed in scaled form there: the shared `exp(-alpha^2 / 2)` is cancelled
+/// analytically rather than evaluated and divided out. Without that, a
+/// truncation point beyond about 39 sigma produced `0 / 0` and put `NaN`
+/// straight into the posterior.
+/// Truncation terms for a boundary `alpha` standard deviations into the upper
+/// tail, from the asymptotic expansion of the inverse Mills ratio.
+///
+/// `v` tends to `alpha` out here, so the gap between them cannot be obtained by
+/// subtracting one from the other — the series computes the gap directly, and
+/// `w = v * gap` then never forms the difference of two large near-equal
+/// numbers. A far-tail *window* behaves like a half-line once it is more than a
+/// few multiples of its own width from the mean, so the tie branch shares this.
+fn half_line_truncation(alpha: f64) -> (f64, f64) {
+    let inv = alpha.recip();
+    let inv_sq = inv * inv;
+    let gap = inv * (1.0 - inv_sq * (2.0 - inv_sq * (10.0 - 74.0 * inv_sq)));
+    let v = alpha + gap;
+
+    (v, v * gap)
+}
+
 fn v_w(mu: f64, sigma: f64, margin: f64, tie: bool) -> (f64, f64) {
     if !tie {
         let alpha = (margin - mu) / sigma;
 
-        let v = pdf(-alpha, 0.0, 1.0) / cdf(-alpha, 0.0, 1.0);
+        // v is the inverse Mills ratio, phi(alpha) / Phi(-alpha), and w needs
-        let w = v * (v + (-alpha));
+        // the gap `v - alpha` as well as v itself. Far into the tail v tends to
+        // alpha, so that gap is a subtraction of two nearly equal numbers and
+        // loses every digit it has: at alpha = 1e6 it drove w above 1 and made
+        // `sqrt(1 - w)` NaN. Past the crossover the gap comes from its
+        // asymptotic series instead, which has no subtraction in it.
+        if alpha >= ASYMPTOTIC_MILLS_ALPHA {
+            return half_line_truncation(alpha);
+        }
 
-        (v, w)
+        let (v, gap) = if alpha > 0.0 {
+            // Both terms carry exp(-alpha^2 / 2); in scaled form it cancels
+            // and the result stays exact however far into the tail alpha sits.
+            let v = SQRT_2_OVER_PI / erfcx(alpha / SQRT_2);
+            (v, v - alpha)
+        } else {
+            // Phi(-alpha) >= 1/2 here, so the direct ratio loses nothing.
+            let v = pdf(-alpha, 0.0, 1.0) / cdf(-alpha, 0.0, 1.0);
+            (v, v - alpha)
+        };
+
+        (v, v * gap)
     } else {
+        // v is odd in mu and w is even, so fold to mu <= 0. Both truncation
+        // points then sit in the upper tail, where the scaled form applies.
+        let flipped = mu > 0.0;
+        let mu = if flipped { -mu } else { mu };
+
         let alpha = (-margin - mu) / sigma;
         let beta = (margin - mu) / sigma;
 
-        let v = (pdf(alpha, 0.0, 1.0) - pdf(beta, 0.0, 1.0))
+        // `w` comes out of `v * v - u`, and both terms grow as alpha^2 while
-            / (cdf(beta, 0.0, 1.0) - cdf(alpha, 0.0, 1.0));
+        // their difference stays O(1) — at alpha = 1e9 that subtraction had no
-        let u = (alpha * pdf(alpha, 0.0, 1.0) - beta * pdf(beta, 0.0, 1.0))
+        // digits left and returned w = -128, making `sqrt(1 - w)` nonsense.
-            / (cdf(beta, 0.0, 1.0) - cdf(alpha, 0.0, 1.0));
+        // Once the window sits many of its own widths into the tail it is
+        // indistinguishable from a half-line, so the asymptotic covers it with
+        // no subtraction at all.
+        if alpha >= ASYMPTOTIC_MILLS_ALPHA && alpha * (beta - alpha) >= HALF_LINE_WINDOW {
+            let (v, w) = half_line_truncation(alpha);
+            return (if flipped { -v } else { v }, w);
+        }
+
+        let (v, u) = if alpha > 0.0 {
+            // beta > alpha > 0, so this ratio of exponentials is at most 1 and
+            // cannot overflow.
+            let scale = libm::exp(0.5 * (alpha * alpha - beta * beta));
+            let denominator = 0.5 * (erfcx(alpha / SQRT_2) - scale * erfcx(beta / SQRT_2));
+
+            (
+                (1.0 - scale) / SQRT_TAU / denominator,
+                (alpha - beta * scale) / SQRT_TAU / denominator,
+            )
+        } else {
+            // The interval straddles the mean, so nothing here is small.
+            let denominator = cdf(beta, 0.0, 1.0) - cdf(alpha, 0.0, 1.0);
+
+            (
+                (pdf(alpha, 0.0, 1.0) - pdf(beta, 0.0, 1.0)) / denominator,
+                (alpha * pdf(alpha, 0.0, 1.0) - beta * pdf(beta, 0.0, 1.0)) / denominator,
+            )
+        };
+
         let w = -(u - v.powi(2));
 
-        (v, w)
+        (if flipped { -v } else { v }, w)
     }
 }
 
@@ -361,6 +652,7 @@ pub(crate) fn sort_time<T: Copy + Ord>(xs: &[T], reverse: bool) -> Vec<usize> {
 /// Panics if fewer than two rating groups are supplied, or if any group is
 /// empty — match quality is a property of a contest between at least two
 /// non-empty sides.
+#[must_use]
 pub fn quality(rating_groups: &[&[Gaussian]], beta: f64) -> f64 {
     assert!(
         rating_groups.len() >= 2,
@@ -427,7 +719,7 @@ pub fn quality(rating_groups: &[&[Gaussian]], beta: f64) -> f64 {
     let e_arg = (-0.5 * &start * &middle.inverse() * &end).determinant();
     let s_arg = ata.determinant() / middle.determinant();
 
-    e_arg.exp() * s_arg.sqrt()
+    libm::exp(e_arg) * s_arg.sqrt()
 }
 
 #[cfg(test)]
@@ -441,6 +733,313 @@ mod tests {
         assert_eq!(sort_time(&[0i64, 1, 2, 0], true), vec![2, 1, 0, 3]);
     }
 
+    /// Upper-tail values of the standard normal, from published tables. The
+    /// point is not the digits — these are 7-digit table values — but that a
+    /// number comes back at all: `1 - cdf` returned exactly zero for every one
+    /// of these.
+    #[test]
+    fn survival_function_survives_the_far_tail() {
+        for (z, expected) in [
+            (9.0f64, 1.128_588e-19),
+            (12.0, 1.776_482e-33),
+            (20.0, 2.753_624e-89),
+            (37.0, 5.725_571e-300),
+        ] {
+            let got = sf(z, 0.0, 1.0);
+            assert!(got > 0.0, "sf({z}) collapsed to zero");
+            assert!(
+                (got - expected).abs() / expected < 1e-6, // published table values, 7 digits
+                "sf({z}) = {got}, expected ~{expected}"
+            );
+            assert_eq!(
+                1.0 - cdf(z, 0.0, 1.0),
+                0.0,
+                "the naive form should still be zero here"
+            );
+        }
+    }
+
+    /// Where no cancellation happens the two forms must agree exactly enough
+    /// that nothing else in the crate shifts.
+    #[test]
+    fn survival_function_matches_the_naive_form_where_that_form_works() {
+        for z in [-4.0f64, -1.0, 0.0, 0.5, 1.0, 2.0, 3.0, 4.0] {
+            let naive = 1.0 - cdf(z, 0.0, 1.0);
+            let direct = sf(z, 0.0, 1.0);
+            assert!(
+                (naive - direct).abs() < 1e-15,
+                "z={z}: naive {naive} vs direct {direct}"
+            );
+        }
+    }
+
+    #[test]
+    fn survival_and_cdf_partition_the_mass() {
+        for z in [-3.0f64, -0.5, 0.0, 1.0, 2.5] {
+            let total = sf(z, 1.0, 2.0) + cdf(z, 1.0, 2.0);
+            assert!((total - 1.0).abs() < 1e-15, "z={z}: {total}");
+        }
+    }
+
+    /// `erfcx` switches formulation at x = 2; the two sides must meet.
+    #[test]
+    fn erfcx_is_continuous_across_its_crossover() {
+        for x in [1.90f64, 1.99, 1.999, 2.0, 2.001, 2.01, 2.10] {
+            let direct = (x * x).exp() * erfc(x);
+            let scaled = erfcx(x);
+            assert!(
+                (direct - scaled).abs() / scaled < 1e-14,
+                "x={x}: direct {direct} vs erfcx {scaled}"
+            );
+        }
+    }
+
+    /// The whole reason `erfcx` exists: it stays finite and O(1/x) exactly
+    /// where `exp(x^2)` overflows and `erfc(x)` underflows.
+    #[test]
+    fn erfcx_stays_finite_where_its_factors_do_not() {
+        for x in [27.0f64, 50.0, 1.0e3, 1.0e8] {
+            let scaled = erfcx(x);
+            assert!(scaled.is_finite() && scaled > 0.0, "erfcx({x}) = {scaled}");
+            // Asymptotically erfcx(x) -> 1 / (x * sqrt(pi)).
+            let asymptote = 1.0 / (x * std::f64::consts::PI.sqrt());
+            assert!(
+                (scaled - asymptote).abs() / asymptote < 1e-2,
+                "erfcx({x}) = {scaled} strays from its asymptote {asymptote}"
+            );
+            assert!(
+                (x * x).exp().is_infinite(),
+                "x={x} should overflow the direct form"
+            );
+        }
+    }
+
+    /// Truncation must never produce a non-finite posterior. Before the scaled
+    /// formulation these returned NaN from `0 / 0` past about 39 sigma.
+    #[test]
+    fn truncation_stays_finite_arbitrarily_far_into_the_tail() {
+        for alpha in [0.0f64, 8.0, 38.0, 40.0, 100.0, 1.0e3, 1.0e6, 1.0e9, 1.0e15] {
+            for tie in [false, true] {
+                let (v, w) = v_w(-alpha, 1.0, if tie { 1.0 } else { 0.0 }, tie);
+                assert!(v.is_finite(), "alpha={alpha} tie={tie}: v = {v}");
+                assert!(w.is_finite(), "alpha={alpha} tie={tie}: w = {w}");
+                // sigma_trunc = sigma * sqrt(1 - w) must stay real.
+                assert!(
+                    (0.0..=1.0).contains(&w),
+                    "alpha={alpha} tie={tie}: w = {w} leaves sqrt(1 - w) imaginary"
+                );
+
+                let (mu_t, sigma_t) = trunc(-alpha, 1.0, if tie { 1.0 } else { 0.0 }, tie);
+                assert!(
+                    mu_t.is_finite() && sigma_t.is_finite(),
+                    "alpha={alpha} tie={tie}: trunc = ({mu_t}, {sigma_t})"
+                );
+            }
+        }
+    }
+
+    /// The Mills gap switches from subtraction to series at alpha = 100. Both
+    /// are supposed to be right there; if they disagree, the crossover is in
+    /// the wrong place.
+    #[test]
+    fn the_mills_gap_series_meets_the_scaled_form() {
+        for alpha in [50.0f64, 99.0, 100.0, 101.0, 200.0] {
+            let scaled = SQRT_2_OVER_PI / erfcx(alpha / SQRT_2) - alpha;
+            let inv = alpha.recip();
+            let inv_sq = inv * inv;
+            let series = inv * (1.0 - inv_sq * (2.0 - inv_sq * (10.0 - 74.0 * inv_sq)));
+            assert!(
+                (scaled - series).abs() / series < 1e-9,
+                "alpha={alpha}: scaled {scaled} vs series {series}"
+            );
+        }
+    }
+
+    /// Folding the tie branch to `mu <= 0` is only valid if v is odd in mu and
+    /// w is even. Assert the symmetry the implementation relies on.
+    #[test]
+    fn tie_truncation_is_odd_in_v_and_even_in_w() {
+        for mu in [0.5f64, 3.0, 20.0, 40.0, 100.0, 1.0e3] {
+            let (v_pos, w_pos) = v_w(mu, 1.0, 1.0, true);
+            let (v_neg, w_neg) = v_w(-mu, 1.0, 1.0, true);
+            assert!(
+                (v_pos + v_neg).abs() < 1e-9,
+                "mu={mu}: v should be odd, got {v_pos} and {v_neg}"
+            );
+            assert!(
+                (w_pos - w_neg).abs() < 1e-9,
+                "mu={mu}: w should be even, got {w_pos} and {w_neg}"
+            );
+        }
+    }
+
+    /// `erfc_inv`'s initial guess had the wrong sign, putting Newton on the
+    /// mirror image of the root. Three fixed iterations could not cross back,
+    /// so the error grew as the argument shrank: at `p_draw = 0.99` the margin
+    /// came out 0.503 where the answer is 2.576.
+    #[test]
+    fn erfc_inv_matches_known_quantiles() {
+        // sqrt(2) * erfc_inv(1 - p) is the standard normal quantile
+        // Phi^-1((1 + p) / 2).
+        for (p, exact) in [
+            (0.5f64, 0.674_489_750_196_081_7f64),
+            (0.9, 1.644_853_626_951_472_7),
+            (0.95, 1.959_963_984_540_054_2),
+            (0.99, 2.575_829_303_548_9),
+            (0.999, 3.290_526_731_491_896_4),
+        ] {
+            let got = SQRT_2 * erfc_inv(1.0 - p);
+            assert!(
+                (got - exact).abs() / exact < 1e-14,
+                "p={p}: got {got}, exact {exact}"
+            );
+        }
+    }
+
+    /// The draw margin must grow with the draw probability. It did not: it ran
+    /// 0.674 -> 1.476 -> 0.503 -> 0.982 as `p_draw` went 0.5 -> 0.9 -> 0.99 ->
+    /// 0.999, which is not a rounding error but a broken function.
+    /// Deep in the tail the accuracy limit is the *caller's* argument, not this
+    /// function.
+    ///
+    /// `compute_margin(0.999999, ..)` computes `1.0 - p_draw`, and 0.999999 is
+    /// not representable: the subtraction cancels and leaves 2.9e-11 of
+    /// relative error in the argument before `erfc_inv` is even entered. Given
+    /// an exactly-representable argument the result is good to 1.8e-16, so this
+    /// is inherent to taking `p_draw` near one rather than something to fix
+    /// here. At `p_draw = 0.999` the whole path is still accurate to 4e-16.
+    ///
+    /// Worth pinning: measured against a 70-digit reference, `puruspe`'s
+    /// `inverfc` returns the identical wrong value for the identical reason,
+    /// which is what makes it clear the fault is upstream of both.
+    #[test]
+    fn erfc_inv_is_exact_given_an_exactly_representable_argument() {
+        // erfc(z / sqrt2) = 1e-6 exactly, so z = Phi^-1(0.9999995).
+        let got = SQRT_2 * erfc_inv(1e-6);
+        let exact = 4.891_638_475_698_59;
+        assert!(
+            (got - exact).abs() / exact < 1e-14,
+            "got {got}, exact {exact}"
+        );
+    }
+
+    #[test]
+    fn compute_margin_is_monotone_in_the_draw_probability() {
+        let mut previous = 0.0;
+        for p_draw in [
+            0.001f64, 0.01, 0.1, 0.25, 0.5, 0.75, 0.9, 0.99, 0.999, 0.9999,
+        ] {
+            let margin = compute_margin(p_draw, 1.0);
+            assert!(
+                margin > previous,
+                "p_draw={p_draw}: margin {margin} did not exceed {previous}"
+            );
+            previous = margin;
+        }
+    }
+
+    /// Round-tripping the margin back through the model's own CDF must recover
+    /// the draw probability it was built from.
+    #[test]
+    fn compute_margin_round_trips_through_the_cdf() {
+        for p_draw in [0.001f64, 0.1, 0.5, 0.9, 0.99, 0.999] {
+            for sd in [0.5f64, 1.0, 5.892_557] {
+                let margin = compute_margin(p_draw, sd);
+                // P(|X| < margin) for X ~ N(0, sd^2).
+                let recovered = 1.0 - 2.0 * cdf(-margin, 0.0, sd);
+                assert!(
+                    (recovered - p_draw).abs() < 1e-14,
+                    "p_draw={p_draw} sd={sd}: recovered {recovered}"
+                );
+            }
+        }
+    }
+
+    /// `ln_pdf`, `ln_sf` and `ln_interval` exist so evidence stays exact where
+    /// the linear forms underflow. Past ~38 sigma the linear value is zero and
+    /// its log is whatever floor it was clamped to.
+    #[test]
+    fn log_space_helpers_stay_exact_where_the_linear_forms_underflow() {
+        for z in [40.0f64, 60.0, 100.0, 1000.0] {
+            assert_eq!(pdf(z, 0.0, 1.0), 0.0, "pdf should underflow at {z}");
+            assert_eq!(sf(z, 0.0, 1.0), 0.0, "sf should underflow at {z}");
+
+            let lp = ln_pdf(z, 0.0, 1.0);
+            let expected_lp = -(SQRT_TAU).ln() - 0.5 * z * z;
+            assert!(
+                (lp - expected_lp).abs() < 1e-9,
+                "ln_pdf({z}) = {lp}, expected {expected_lp}"
+            );
+
+            let ls = ln_sf(z, 0.0, 1.0);
+            // ln Phi(-z) ~ -z^2/2 - ln(z) - ln(sqrt(2 pi)) for large z.
+            let approx = -0.5 * z * z - z.ln() - SQRT_TAU.ln();
+            assert!(
+                (ls - approx).abs() / approx.abs() < 1e-3,
+                "ln_sf({z}) = {ls}, asymptote {approx}"
+            );
+            assert!(
+                ls < f64::MIN_POSITIVE.ln(),
+                "ln_sf({z}) still on the clamp floor"
+            );
+        }
+    }
+
+    /// Where nothing underflows, the log helpers must agree with the direct
+    /// forms exactly enough that nothing else in the crate shifts.
+    #[test]
+    fn log_space_helpers_agree_with_the_linear_forms_in_range() {
+        for z in [-3.0f64, -1.0, 0.0, 1.0, 2.0, 5.0, 10.0, 20.0] {
+            let lp = ln_pdf(z, 0.5, 2.0);
+            let direct_pdf = pdf(z, 0.5, 2.0);
+            assert!(
+                (lp.exp() - direct_pdf).abs() <= 1e-12 * direct_pdf,
+                "ln_pdf at {z}: {} vs {direct_pdf}",
+                lp.exp()
+            );
+
+            let ls = ln_sf(z, 0.5, 2.0);
+            let direct = sf(z, 0.5, 2.0);
+            assert!(
+                (ls.exp() - direct).abs() <= 1e-13 * direct.max(1e-300),
+                "ln_sf at {z}: {} vs {direct}",
+                ls.exp()
+            );
+        }
+    }
+
+    #[test]
+    fn ln_interval_matches_the_direct_difference_when_nothing_is_small() {
+        for mu in [-2.0f64, 0.0, 0.5, 2.0] {
+            let direct = cdf(1.0, mu, 1.0) - cdf(-1.0, mu, 1.0);
+            let logged = ln_interval(-1.0, 1.0, mu, 1.0).exp();
+            assert!(
+                (logged - direct).abs() <= 1e-13 * direct,
+                "mu={mu}: {logged} vs {direct}"
+            );
+        }
+    }
+
+    /// A window far out in the tail: both endpoints underflow together, so the
+    /// difference has to be taken in scaled form.
+    #[test]
+    fn ln_interval_survives_a_window_deep_in_the_tail() {
+        for mu in [-50.0f64, -100.0, -1000.0] {
+            let logged = ln_interval(-1.0, 1.0, mu, 1.0);
+            assert!(logged.is_finite(), "mu={mu}: {logged}");
+            assert!(
+                logged < f64::MIN_POSITIVE.ln(),
+                "mu={mu}: {logged} is stuck on the clamp floor"
+            );
+            // Dominated by the near edge: ln P ~ ln Phi(-(|mu| - 1)).
+            let near = ln_sf(-1.0, mu, 1.0);
+            assert!(
+                (logged - near).abs() < 5.0,
+                "mu={mu}: {logged} strays from the near-edge tail {near}"
+            );
+        }
+    }
+
     #[test]
     fn test_quality() {
         let a = Gaussian::from_ms(25.0, 3.0);

src/observer.rs

@@ -14,13 +14,95 @@ pub trait Observer<T: Time>: Send + Sync {
     /// Called after each convergence iteration across the whole history.
     fn on_iteration_end(&self, _iter: usize, _max_step: (f64, f64)) {}
 
-    /// Called after each time slice is processed within an iteration.
+    /// Called after each time slice is swept within an iteration.
-    fn on_batch_processed(&self, _time: &T, _slice_idx: usize, _n_events: usize) {}
+    ///
+    /// A convergence iteration sweeps every slice twice — once travelling
+    /// backward through the history and once forward — so a multi-slice
+    /// history fires this twice per slice per iteration. A single-slice
+    /// history is swept once and fires once.
+    fn on_slice_processed(&self, _time: &T, _slice_idx: usize, _n_events: usize) {}
 
     /// Called once when convergence completes (or max iters is reached).
     fn on_converged(&self, _iters: usize, _final_step: (f64, f64), _converged: bool) {}
 }
 
+/// Shared and boxed observers forward to what they point at.
+///
+/// `History` takes its observer by value, so a caller who wants to *read* what
+/// an observer recorded has to keep a handle to it. Without these impls the
+/// natural spelling does not compile:
+///
+/// ```
+/// # use std::sync::{Arc, Mutex};
+/// # use trueskill_tt::{History, Observer};
+/// #[derive(Default)]
+/// struct Recorder {
+///     iterations: Mutex<Vec<usize>>,
+/// }
+///
+/// impl Observer<i64> for Recorder {
+///     fn on_iteration_end(&self, iter: usize, _step: (f64, f64)) {
+///         self.iterations.lock().unwrap().push(iter);
+///     }
+/// }
+///
+/// let recorder = Arc::new(Recorder::default());
+/// let mut h = History::builder().observer(Arc::clone(&recorder)).build();
+/// h.record_winner(&"a", &"b", 1).unwrap();
+/// h.converge().unwrap();
+///
+/// // The caller's handle sees what the history's copy recorded.
+/// assert!(!recorder.iterations.lock().unwrap().is_empty());
+/// ```
+///
+/// The alternative was for every observer to wrap each of its own fields in an
+/// `Arc` and derive `Clone` — one allocation and one lock per field, and a
+/// pattern each implementor had to rediscover.
+///
+/// `?Sized` is deliberate: it makes `Arc<dyn Observer<T>>` and
+/// `Box<dyn Observer<T>>` work, so observers can be chosen at runtime.
+impl<T: Time, O: Observer<T> + ?Sized> Observer<T> for std::sync::Arc<O> {
+    fn on_iteration_end(&self, iter: usize, max_step: (f64, f64)) {
+        (**self).on_iteration_end(iter, max_step);
+    }
+
+    fn on_slice_processed(&self, time: &T, slice_idx: usize, n_events: usize) {
+        (**self).on_slice_processed(time, slice_idx, n_events);
+    }
+
+    fn on_converged(&self, iters: usize, final_step: (f64, f64), converged: bool) {
+        (**self).on_converged(iters, final_step, converged);
+    }
+}
+
+impl<T: Time, O: Observer<T> + ?Sized> Observer<T> for Box<O> {
+    fn on_iteration_end(&self, iter: usize, max_step: (f64, f64)) {
+        (**self).on_iteration_end(iter, max_step);
+    }
+
+    fn on_slice_processed(&self, time: &T, slice_idx: usize, n_events: usize) {
+        (**self).on_slice_processed(time, slice_idx, n_events);
+    }
+
+    fn on_converged(&self, iters: usize, final_step: (f64, f64), converged: bool) {
+        (**self).on_converged(iters, final_step, converged);
+    }
+}
+
+impl<T: Time, O: Observer<T> + ?Sized> Observer<T> for &O {
+    fn on_iteration_end(&self, iter: usize, max_step: (f64, f64)) {
+        (**self).on_iteration_end(iter, max_step);
+    }
+
+    fn on_slice_processed(&self, time: &T, slice_idx: usize, n_events: usize) {
+        (**self).on_slice_processed(time, slice_idx, n_events);
+    }
+
+    fn on_converged(&self, iters: usize, final_step: (f64, f64), converged: bool) {
+        (**self).on_converged(iters, final_step, converged);
+    }
+}
+
 /// ZST no-op observer; the default when none is configured.
 #[derive(Copy, Clone, Debug, Default)]
 pub struct NullObserver;
@@ -35,6 +117,7 @@ mod tests {
     fn null_observer_compiles_for_i64() {
         let o = NullObserver;
         <NullObserver as Observer<i64>>::on_iteration_end(&o, 1, (0.0, 0.0));
+        <NullObserver as Observer<i64>>::on_slice_processed(&o, &7, 0, 3);
         <NullObserver as Observer<i64>>::on_converged(&o, 5, (1e-6, 1e-6), true);
     }
 

src/outcome.rs

@@ -29,14 +29,50 @@ pub enum Outcome {
 impl Outcome {
     /// `n`-team outcome where team `winner` won and everyone else tied for last.
     ///
-    /// Panics if `winner >= n`.
+    /// Note this ties every loser, so for `n >= 3` it needs a positive
+    /// `p_draw` — see `InferenceError::TieWithoutDrawProbability`.
+    ///
+    /// # Panics
+    ///
+    /// Panics if `winner >= n`. Use [`Outcome::try_winner`] when the index
+    /// comes from data rather than a literal.
+    ///
+    /// This is the one constructor here that validates, and deliberately so.
+    /// Its siblings build freely and let ingestion reject what it cannot use,
+    /// which works because a malformed rank vector stays recognisable. An
+    /// out-of-range winner does not: `winner(5, 2)` would produce ranks
+    /// `[1, 1]`, an all-tied draw that ingestion accepts without complaint when
+    /// `p_draw > 0`. Asking "team 5 won" and silently getting "everyone drew"
+    /// is exactly the class of quiet wrong answer this crate keeps removing, so
+    /// the check happens here where the mistake is.
+    #[must_use]
     pub fn winner(winner: u32, n: u32) -> Self {
-        assert!(winner < n, "winner index {winner} out of range 0..{n}");
+        Self::try_winner(winner, n)
+            .unwrap_or_else(|_| panic!("winner index {winner} out of range 0..{n}"))
+    }
+
+    /// `n`-team outcome where team `winner` won, or an error if `winner` is not
+    /// a valid team index.
+    ///
+    /// The fallible form of [`Outcome::winner`], for when the index is computed
+    /// or parsed rather than written literally.
+    ///
+    /// # Errors
+    ///
+    /// `InvalidParameter` if `winner >= n`.
+    pub fn try_winner(winner: u32, n: u32) -> Result<Self, crate::InferenceError> {
+        if winner >= n {
+            return Err(crate::InferenceError::InvalidParameter {
+                name: "winner",
+                value: f64::from(winner),
+            });
+        }
         let ranks: SmallVec<[u32; 4]> = (0..n).map(|i| if i == winner { 0 } else { 1 }).collect();
-        Self::Ranked(ranks)
+        Ok(Self::Ranked(ranks))
     }
 
     /// All `n` teams tied.
+    #[must_use]
     pub fn draw(n: u32) -> Self {
         Self::Ranked(SmallVec::from_vec(vec![0; n as usize]))
     }
@@ -68,6 +104,7 @@ impl Outcome {
         }
     }
 
+    #[must_use]
     pub fn team_count(&self) -> usize {
         match self {
             Self::Ranked(r) => r.len(),

src/predict.rs

@@ -0,0 +1,729 @@
+//! Outcome prediction: who wins, and how likely is a given finishing order.
+//!
+//! Prediction runs on *performances*, not skills. A competitor's skill is
+//! inflated by their performance noise `beta` before any comparison, which is
+//! what separates "how good are they" from "how will they do today".
+//!
+//! Two questions, two algorithms:
+//!
+//! - **Who finishes first.** Because performances are independent Gaussians,
+//!   the probability that team `i` beats every other team separates into a
+//!   *one-dimensional* integral — no multivariate orthant integral is
+//!   involved. [`quadrature::integrate`] evaluates it to near machine
+//!   precision for a few hundred `cdf` calls.
+//! - **A specific finishing order.** The factor graph only ever constrains
+//!   rank-*adjacent* teams (see `Game::run_chain`), so the joint probability
+//!   of a full order is a chain of local constraints rather than a general
+//!   orthant probability. That chain collapses into a sequential recursion:
+//!   one cumulative integral per adjacent pair, `O(teams * grid)` overall.
+//!
+//! Both are deterministic. A sampler would have been easier to write and
+//! would have made every `predict_*` call return a slightly different number,
+//! which is not a property a rating library should have.
+
+use crate::{Gaussian, quadrature};
+
+/// Teams beyond this count make the outcome enumeration impractical.
+///
+/// Each realisation sorts into exactly one (permutation, tie-pattern) event,
+/// so the space has `n! * 2^(n-1)` members: 24 at 3 teams, 192 at 4, 1_920 at
+/// 5, 23_040 at 6. The jump to 322_560 at 7 is where enumerating stops being
+/// a reasonable thing to do on a caller's behalf.
+pub(crate) const MAX_TEAMS_FOR_DISTRIBUTION: usize = 6;
+
+/// Relative tolerance for the first-place integrals.
+///
+/// The adaptive integrator reaches the exact two-team closed form to ~1e-15 at
+/// this tolerance, which is round-off for a probability. `cdf` is no longer the
+/// limit — it went to ~1 ULP when `erfc` moved to `libm` — so this is the
+/// integrator's own floor.
+const WIN_TOLERANCE: f64 = 1e-8;
+
+/// Nodes for the ranking grid, and the floor below which a grid is pointless.
+///
+/// The recursion converges as O(h^2), so this trades nodes against accuracy
+/// directly. Measured against the exact two-team closed form, 2_048 nodes leave
+/// ~1.2e-6 of discretisation error and 8_192 reach ~1e-7.
+///
+/// Unlike the adaptive path there is no approximation floor underneath this any
+/// more — `cdf` is accurate to ~1 ULP since `erfc` moved to `libm` — so the
+/// error here is purely the grid, and a caller who needs more can only get it
+/// by paying for more nodes. 8_192 is the accuracy/cost point chosen, not a
+/// point where refining stops helping.
+const MIN_GRID_POINTS: usize = 8_192;
+const MAX_GRID_POINTS: usize = 262_144;
+
+/// How many standard deviations of support the grid and integrals cover.
+///
+/// The normal density is below 1e-18 of its peak past nine sigma, far under
+/// the precision of everything else here.
+const SUPPORT_SIGMAS: f64 = 9.0;
+
+/// Standard normal CDF at `z`.
+fn phi(z: f64) -> f64 {
+    crate::cdf(z, 0.0, 1.0)
+}
+
+/// Normal density of `x` under `g`.
+fn density(g: Gaussian, x: f64) -> f64 {
+    let sigma = g.sigma();
+    let z = (x - g.mu()) / sigma;
+    libm::exp(-0.5 * z * z) / (sigma * (2.0 * std::f64::consts::PI).sqrt())
+}
+
+/// Per-pair draw margins.
+///
+/// The margin is *not* a single number for the whole game: inference derives
+/// it per rank-adjacent pair from those two teams' betas (`Game::likelihoods`).
+/// Prediction has to use the same per-pair values or it answers a question
+/// about a different model than the one that will actually be fitted.
+pub(crate) struct Margins {
+    n: usize,
+    values: Vec<f64>,
+}
+
+impl Margins {
+    /// Build from a per-pair margin function.
+    pub(crate) fn new<F: Fn(usize, usize) -> f64>(n: usize, f: F) -> Self {
+        let mut values = vec![0.0; n * n];
+        for i in 0..n {
+            for j in 0..n {
+                if i != j {
+                    values[i * n + j] = f(i, j);
+                }
+            }
+        }
+        Self { n, values }
+    }
+
+    fn get(&self, i: usize, j: usize) -> f64 {
+        self.values[i * self.n + j]
+    }
+
+    /// True when no pair can draw, so every tie has probability zero.
+    fn all_zero(&self) -> bool {
+        self.values.iter().all(|&v| v == 0.0)
+    }
+}
+
+/// `P(team i finishes strictly first)` for every team.
+///
+/// Strictly means beating each rival by more than that pair's draw margin, so
+/// with a non-zero margin these sum to less than one; the shortfall is the
+/// probability that the top place is shared.
+pub(crate) fn win_probabilities(perf: &[Gaussian], margins: &Margins) -> Vec<f64> {
+    (0..perf.len())
+        .map(|i| {
+            let (mu, sigma) = (perf[i].mu(), perf[i].sigma());
+            let (lo, hi) = (mu - SUPPORT_SIGMAS * sigma, mu + SUPPORT_SIGMAS * sigma);
+
+            // Each rival's CDF turns over near its own mean plus the margin.
+            // Seeding there is what keeps a rival with a tiny sigma — a step
+            // function in disguise — from being stepped over.
+            let mut seeds = Vec::with_capacity(3 * perf.len());
+            for (j, rival) in perf.iter().enumerate().filter(|&(j, _)| j != i) {
+                let centre = rival.mu() + margins.get(i, j);
+                seeds.extend_from_slice(&[centre - rival.sigma(), centre, centre + rival.sigma()]);
+            }
+
+            quadrature::integrate(
+                |x| {
+                    let d = density(perf[i], x);
+                    if d == 0.0 {
+                        return 0.0;
+                    }
+                    let beaten: f64 = (0..perf.len())
+                        .filter(|&j| j != i)
+                        .map(|j| phi((x - margins.get(i, j) - perf[j].mu()) / perf[j].sigma()))
+                        .product();
+                    d * beaten
+                },
+                lo,
+                hi,
+                &seeds,
+                WIN_TOLERANCE,
+            )
+        })
+        .collect()
+}
+
+/// Grid bounds and resolution covering every team's support.
+///
+/// Resolution is set by the *smallest* feature in play — the narrowest sigma,
+/// or a draw margin narrower still — because that is what the recursion has to
+/// resolve. A grid sized off the widest team would step over the narrow one.
+fn grid_shape(perf: &[Gaussian], margins: &Margins) -> (f64, f64, usize) {
+    let lo = perf
+        .iter()
+        .map(|g| g.mu() - SUPPORT_SIGMAS * g.sigma())
+        .fold(f64::INFINITY, f64::min);
+    let hi = perf
+        .iter()
+        .map(|g| g.mu() + SUPPORT_SIGMAS * g.sigma())
+        .fold(f64::NEG_INFINITY, f64::max);
+
+    let narrowest = perf
+        .iter()
+        .map(Gaussian::sigma)
+        .fold(f64::INFINITY, f64::min);
+    let smallest_margin = margins
+        .values
+        .iter()
+        .copied()
+        .filter(|&m| m > 0.0)
+        .fold(f64::INFINITY, f64::min);
+
+    let feature = narrowest.min(smallest_margin);
+    let wanted = if feature.is_finite() && feature > 0.0 {
+        ((hi - lo) / (feature / 12.0)).ceil()
+    } else {
+        MIN_GRID_POINTS as f64
+    };
+
+    let points = if wanted.is_finite() {
+        (wanted as usize).clamp(MIN_GRID_POINTS, MAX_GRID_POINTS)
+    } else {
+        MIN_GRID_POINTS
+    };
+
+    (lo, hi, points)
+}
+
+/// Densities of each team sampled on the shared grid.
+struct Sampled {
+    lo: f64,
+    step: f64,
+    points: usize,
+    density: Vec<Vec<f64>>,
+}
+
+impl Sampled {
+    fn new(perf: &[Gaussian], margins: &Margins) -> Self {
+        let (lo, hi, points) = grid_shape(perf, margins);
+        let step = (hi - lo) / (points - 1) as f64;
+        let density = perf
+            .iter()
+            .map(|&g| {
+                (0..points)
+                    .map(|i| density(g, lo + i as f64 * step))
+                    .collect()
+            })
+            .collect();
+        Self {
+            lo,
+            step,
+            points,
+            density,
+        }
+    }
+
+    fn node(&self, i: usize) -> f64 {
+        self.lo + i as f64 * self.step
+    }
+}
+
+/// `P(order[0] >= order[1] >= ... )` with the given adjacency pattern.
+///
+/// `tied[k]` says whether `order[k]` and `order[k + 1]` finish within that
+/// pair's draw margin. The recursion runs bottom-up: `carry` holds, for each
+/// grid node, the probability that everything *below* the current team holds
+/// given that team landed on that node. A strict gap reads a cumulative
+/// integral; a tie reads a window. Both are O(1) against one prefix array,
+/// so each level costs O(grid) and the whole order costs O(teams * grid).
+fn order_probability(margins: &Margins, sampled: &Sampled, order: &[usize], tied: &[bool]) -> f64 {
+    let mut carry = vec![1.0; sampled.points];
+
+    for k in (0..order.len() - 1).rev() {
+        let below = order[k + 1];
+        let above = order[k];
+        let margin = margins.get(above, below);
+
+        let integrand: Vec<f64> = (0..sampled.points)
+            .map(|i| sampled.density[below][i] * carry[i])
+            .collect();
+        let cumulative = quadrature::Grid::from_values(sampled.lo, sampled.step, integrand);
+
+        carry = (0..sampled.points)
+            .map(|i| {
+                let x = sampled.node(i);
+                if tied[k] {
+                    // Sorted order already implies `below <= above`, so the
+                    // tie window is one-sided: [x - margin, x].
+                    cumulative.integral_between(x - margin, x)
+                } else {
+                    cumulative.integral_to(x - margin)
+                }
+            })
+            .collect();
+    }
+
+    let top = order[0];
+    let integrand: Vec<f64> = (0..sampled.points)
+        .map(|i| sampled.density[top][i] * carry[i])
+        .collect();
+    quadrature::Grid::from_values(sampled.lo, sampled.step, integrand).total()
+}
+
+/// Dense ranks implied by a sorted order and its tie pattern.
+fn ranks_of(order: &[usize], tied: &[bool], n: usize) -> Vec<u32> {
+    let mut ranks = vec![0u32; n];
+    let mut rank = 0u32;
+    ranks[order[0]] = 0;
+    for k in 0..order.len() - 1 {
+        if !tied[k] {
+            rank += 1;
+        }
+        ranks[order[k + 1]] = rank;
+    }
+    ranks
+}
+
+/// Every (order, tie-pattern) event, or only the strict ones when no pair can
+/// draw — a tie then has probability exactly zero and is not worth integrating.
+fn events(n: usize, strict_only: bool) -> Vec<(Vec<usize>, Vec<bool>)> {
+    fn permute(current: &mut Vec<usize>, k: usize, out: &mut Vec<Vec<usize>>) {
+        if k == current.len() {
+            out.push(current.clone());
+            return;
+        }
+        for i in k..current.len() {
+            current.swap(k, i);
+            permute(current, k + 1, out);
+            current.swap(k, i);
+        }
+    }
+
+    let mut orders = Vec::new();
+    permute(&mut (0..n).collect(), 0, &mut orders);
+
+    let patterns: Vec<Vec<bool>> = if strict_only {
+        vec![vec![false; n - 1]]
+    } else {
+        (0..(1u32 << (n - 1)))
+            .map(|mask| (0..n - 1).map(|i| mask >> i & 1 == 1).collect())
+            .collect()
+    };
+
+    let mut out = Vec::with_capacity(orders.len() * patterns.len());
+    for order in orders {
+        for pattern in &patterns {
+            out.push((order.clone(), pattern.clone()));
+        }
+    }
+    out
+}
+
+/// The full distribution over finishing orders, aggregated by rank vector.
+///
+/// Orders that differ only *within* a tied group describe the same finishing
+/// order, so their probabilities are summed into one entry.
+pub(crate) fn outcome_distribution(perf: &[Gaussian], margins: &Margins) -> Vec<(Vec<u32>, f64)> {
+    let n = perf.len();
+    let sampled = Sampled::new(perf, margins);
+
+    let mut aggregated: Vec<(Vec<u32>, f64)> = Vec::new();
+    for (order, tied) in events(n, margins.all_zero()) {
+        let p = order_probability(margins, &sampled, &order, &tied);
+        let ranks = ranks_of(&order, &tied, n);
+        match aggregated.iter_mut().find(|(r, _)| *r == ranks) {
+            Some((_, acc)) => *acc += p,
+            None => aggregated.push((ranks, p)),
+        }
+    }
+
+    aggregated.sort_by(|a, b| b.1.partial_cmp(&a.1).unwrap_or(std::cmp::Ordering::Equal));
+    aggregated
+}
+
+/// All permutations of `items`.
+fn permutations(items: &[usize]) -> Vec<Vec<usize>> {
+    fn go(current: &mut Vec<usize>, k: usize, out: &mut Vec<Vec<usize>>) {
+        if k == current.len() {
+            out.push(current.clone());
+            return;
+        }
+        for i in k..current.len() {
+            current.swap(k, i);
+            go(current, k + 1, out);
+            current.swap(k, i);
+        }
+    }
+    let mut out = Vec::new();
+    go(&mut items.to_vec(), 0, &mut out);
+    out
+}
+
+/// Every (order, tie-pattern) event consistent with a grouping by rank.
+///
+/// Teams sharing a rank may finish in any internal order, so this is the
+/// product of each group's permutations. Adjacencies inside a group are ties;
+/// the adjacency joining one group to the next is not.
+fn orders_for_groups(groups: &[Vec<usize>]) -> Vec<(Vec<usize>, Vec<bool>)> {
+    let per_group: Vec<Vec<Vec<usize>>> = groups.iter().map(|g| permutations(g)).collect();
+
+    let mut out = Vec::new();
+    let mut choice = vec![0usize; groups.len()];
+
+    loop {
+        let mut order = Vec::new();
+        let mut tied = Vec::new();
+        for (gi, group) in per_group.iter().enumerate() {
+            for (offset, &member) in group[choice[gi]].iter().enumerate() {
+                if !order.is_empty() {
+                    tied.push(offset != 0);
+                }
+                order.push(member);
+            }
+        }
+        out.push((order, tied));
+
+        let mut k = 0;
+        loop {
+            if k == choice.len() {
+                return out;
+            }
+            choice[k] += 1;
+            if choice[k] < per_group[k].len() {
+                break;
+            }
+            choice[k] = 0;
+            k += 1;
+        }
+    }
+}
+
+/// Probability of one specific rank vector.
+///
+/// Ties in `ranks` mean the tied teams may finish in any internal order, so
+/// this sums the orders consistent with the requested ranking rather than
+/// picking one.
+pub(crate) fn ranking_probability(perf: &[Gaussian], margins: &Margins, ranks: &[u32]) -> f64 {
+    let n = perf.len();
+    let sampled = Sampled::new(perf, margins);
+
+    let mut distinct: Vec<u32> = ranks.to_vec();
+    distinct.sort_unstable();
+    distinct.dedup();
+
+    let groups: Vec<Vec<usize>> = distinct
+        .iter()
+        .map(|&r| (0..n).filter(|&i| ranks[i] == r).collect())
+        .collect();
+
+    orders_for_groups(&groups)
+        .iter()
+        .map(|(order, tied)| order_probability(margins, &sampled, order, tied))
+        .sum()
+}
+
+/// A distribution over the ways a contest could finish.
+///
+/// Each entry pairs a rank vector — the same shape [`crate::Outcome::ranking`]
+/// takes, with equal ranks meaning a tie — against its probability. Entries
+/// are ordered most likely first, and cover the whole outcome space, so the
+/// probabilities sum to one.
+///
+/// The rank vectors compose directly with inference: feeding one to
+/// `Game::ranked` asks "what would we believe if *this* happened", which is
+/// what an expected-information-gain calculation needs alongside the weight.
+#[derive(Clone, Debug, PartialEq)]
+pub struct Prediction {
+    outcomes: Vec<(Vec<u32>, f64)>,
+}
+
+impl Prediction {
+    pub(crate) fn new(outcomes: Vec<(Vec<u32>, f64)>) -> Self {
+        Self { outcomes }
+    }
+
+    /// Every possible finishing order and its probability, most likely first.
+    pub fn outcomes(&self) -> impl ExactSizeIterator<Item = (&[u32], f64)> {
+        self.outcomes.iter().map(|(r, p)| (r.as_slice(), *p))
+    }
+
+    /// The single most likely finishing order.
+    #[must_use]
+    pub fn most_likely(&self) -> Option<(&[u32], f64)> {
+        self.outcomes.first().map(|(r, p)| (r.as_slice(), *p))
+    }
+
+    /// Probability of one specific finishing order, or zero if it cannot occur.
+    #[must_use]
+    pub fn probability_of(&self, ranks: &[u32]) -> f64 {
+        self.outcomes
+            .iter()
+            .find(|(r, _)| r.as_slice() == ranks)
+            .map_or(0.0, |(_, p)| *p)
+    }
+
+    /// `P(team i finishes strictly first)`, for each team.
+    ///
+    /// Sums to less than one exactly when the top place can be shared; the
+    /// shortfall is [`Prediction::shared_first_place`].
+    #[must_use]
+    pub fn win_probabilities(&self) -> Vec<f64> {
+        let n = self.outcomes.first().map_or(0, |(r, _)| r.len());
+        let mut wins = vec![0.0; n];
+        for (ranks, p) in &self.outcomes {
+            let leaders = ranks.iter().filter(|&&r| r == 0).count();
+            if leaders == 1 {
+                let winner = ranks.iter().position(|&r| r == 0).expect("a rank-0 team");
+                wins[winner] += p;
+            }
+        }
+        wins
+    }
+
+    /// Probability that two or more teams share first place.
+    #[must_use]
+    pub fn shared_first_place(&self) -> f64 {
+        self.outcomes
+            .iter()
+            .filter(|(r, _)| r.iter().filter(|&&x| x == 0).count() > 1)
+            .map(|(_, p)| p)
+            .sum()
+    }
+
+    /// Total probability mass, which should be one.
+    ///
+    /// Exposed because it is a genuine check on the numerics rather than a
+    /// formality: the outcome space is exhaustive and disjoint by construction,
+    /// so any drift from one is integration error and nothing else.
+    #[must_use]
+    pub fn total(&self) -> f64 {
+        self.outcomes.iter().map(|(_, p)| p).sum()
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    fn g(mu: f64, sigma: f64) -> Gaussian {
+        Gaussian::from_ms(mu, sigma)
+    }
+
+    fn flat(n: usize, eps: f64) -> Margins {
+        Margins::new(n, |_, _| eps)
+    }
+
+    /// Exact two-team result: `P(a first) = Phi((mu_a - mu_b - eps) / sd)`.
+    fn closed_form_two(a: Gaussian, b: Gaussian, eps: f64) -> (f64, f64) {
+        let sd = a.sigma().hypot(b.sigma());
+        (
+            phi((a.mu() - b.mu() - eps) / sd),
+            phi((b.mu() - a.mu() - eps) / sd),
+        )
+    }
+
+    #[test]
+    fn two_team_win_probabilities_match_the_closed_form() {
+        for (ma, sa, mb, sb, eps) in [
+            (0.0, 6.0, 0.0, 6.0, 0.0),
+            (3.0, 6.0, -2.0, 1.0, 0.0),
+            (0.0, 6.0, 0.0, 6.0, 2.0),
+            (3.0, 6.0, -2.0, 1.0, 1.5),
+            (40.0, 1.0, 0.0, 1.0, 0.0),
+        ] {
+            let perf = [g(ma, sa), g(mb, sb)];
+            let got = win_probabilities(&perf, &flat(2, eps));
+            let (wa, wb) = closed_form_two(perf[0], perf[1], eps);
+            assert!(
+                (got[0] - wa).abs() < 1e-12 && (got[1] - wb).abs() < 1e-12,
+                "mu=({ma},{mb}) sigma=({sa},{sb}) eps={eps}: got {got:?}, want [{wa}, {wb}]"
+            );
+        }
+    }
+
+    /// The identity that a wrong-but-plausible implementation cannot fake:
+    /// with no draw margin, exactly one team finishes first.
+    #[test]
+    fn win_probabilities_sum_to_one_without_a_draw_margin() {
+        for perf in [
+            vec![g(0.0, 6.0), g(0.0, 6.0)],
+            vec![g(5.0, 6.0), g(0.0, 3.0), g(-5.0, 1.0)],
+            vec![
+                g(8.0, 2.0),
+                g(3.0, 6.0),
+                g(0.0, 1.0),
+                g(-3.0, 4.0),
+                g(-8.0, 6.0),
+            ],
+        ] {
+            let sum: f64 = win_probabilities(&perf, &flat(perf.len(), 0.0))
+                .iter()
+                .sum();
+            assert!(
+                (sum - 1.0).abs() < 1e-7,
+                "{} teams: sum = {sum}",
+                perf.len()
+            );
+        }
+    }
+
+    /// A rival with a tiny sigma is a step function in disguise. Fixed-node
+    /// quadrature steps over it and lands ~1e-2 out while still looking like a
+    /// probability; this is the case that rules that approach out.
+    #[test]
+    fn win_probabilities_survive_a_rival_with_a_tiny_sigma() {
+        let perf = [g(0.0, 0.001), g(0.5, 6.0), g(-0.5, 6.0)];
+        let got = win_probabilities(&perf, &flat(3, 0.0));
+        let sum: f64 = got.iter().sum();
+        assert!((sum - 1.0).abs() < 1e-6, "sum = {sum}, probs = {got:?}");
+    }
+
+    #[test]
+    fn a_stronger_team_is_more_likely_to_win() {
+        let perf = [g(10.0, 3.0), g(0.0, 3.0), g(-10.0, 3.0)];
+        let p = win_probabilities(&perf, &flat(3, 0.0));
+        assert!(p[0] > p[1] && p[1] > p[2], "not monotone: {p:?}");
+    }
+
+    #[test]
+    fn identical_teams_are_equally_likely_to_win() {
+        let perf = [g(1.0, 4.0), g(1.0, 4.0), g(1.0, 4.0)];
+        let p = win_probabilities(&perf, &flat(3, 0.0));
+        for probs in p.windows(2) {
+            assert!((probs[0] - probs[1]).abs() < 1e-9, "asymmetric: {p:?}");
+        }
+    }
+
+    /// Every realisation sorts into exactly one finishing order, so the whole
+    /// distribution must sum to one — with or without a draw margin.
+    #[test]
+    fn outcome_distribution_sums_to_one() {
+        for (perf, eps) in [
+            (vec![g(0.0, 6.0), g(0.0, 6.0)], 0.0),
+            (vec![g(0.0, 6.0), g(0.0, 6.0)], 2.0),
+            (vec![g(0.0, 6.0), g(0.0, 6.0), g(0.0, 6.0)], 0.0),
+            (vec![g(5.0, 6.0), g(0.0, 3.0), g(-5.0, 1.0)], 1.5),
+            (vec![g(0.0, 0.05), g(0.5, 6.0), g(-0.5, 6.0)], 1.0),
+            (
+                vec![g(6.0, 2.0), g(2.0, 6.0), g(-2.0, 1.0), g(-6.0, 4.0)],
+                1.0,
+            ),
+        ] {
+            let n = perf.len();
+            let dist = outcome_distribution(&perf, &flat(n, eps));
+            let sum: f64 = dist.iter().map(|(_, p)| p).sum();
+            assert!(
+                (sum - 1.0).abs() < 1e-6,
+                "{n} teams, eps={eps}: sum = {sum} over {} outcomes",
+                dist.len()
+            );
+            assert!(dist.iter().all(|(_, p)| *p >= 0.0), "negative probability");
+        }
+    }
+
+    /// With two teams the distribution is the exact win/draw/loss triple.
+    #[test]
+    fn two_team_distribution_matches_the_closed_form() {
+        let perf = [g(3.0, 6.0), g(-2.0, 1.0)];
+        let eps = 1.5;
+        let dist = outcome_distribution(&perf, &flat(2, eps));
+        let (wa, wb) = closed_form_two(perf[0], perf[1], eps);
+
+        let find = |ranks: &[u32]| {
+            dist.iter()
+                .find(|(r, _)| r == ranks)
+                .map_or(0.0, |(_, p)| *p)
+        };
+
+        assert!(
+            (find(&[0, 1]) - wa).abs() < 1e-6,
+            "a wins: {}",
+            find(&[0, 1])
+        );
+        assert!(
+            (find(&[1, 0]) - wb).abs() < 1e-6,
+            "b wins: {}",
+            find(&[1, 0])
+        );
+        assert!(
+            (find(&[0, 0]) - (1.0 - wa - wb)).abs() < 1e-6,
+            "draw: {}",
+            find(&[0, 0])
+        );
+    }
+
+    /// Asking for one ranking must agree with that ranking's entry in the
+    /// full distribution — the two use different code paths to the same value.
+    #[test]
+    fn ranking_probability_agrees_with_the_distribution() {
+        let perf = [g(5.0, 6.0), g(0.0, 3.0), g(-5.0, 1.0)];
+        let eps = 1.5;
+        let margins = flat(3, eps);
+        let dist = outcome_distribution(&perf, &margins);
+
+        for (ranks, expected) in &dist {
+            let direct = ranking_probability(&perf, &margins, ranks);
+            assert!(
+                (direct - expected).abs() < 1e-9,
+                "ranks {ranks:?}: direct {direct} vs distribution {expected}"
+            );
+        }
+    }
+
+    /// Tie mass is controlled by the draw margin. Only the *all-tied* outcome
+    /// is monotone in it: every one of its constraints is a window that widens
+    /// with the margin. A partially-tied outcome like `[0, 0, 1]` is not, and
+    /// must not be asserted to be — widening the margin makes its tie easier
+    /// but its "and the last team is strictly behind by more than the margin"
+    /// clause harder, so it peaks and then falls.
+    #[test]
+    fn all_tied_probability_grows_with_the_draw_margin() {
+        let perf = [g(0.0, 4.0), g(0.0, 4.0), g(-8.0, 2.0)];
+        let mut previous = 0.0;
+        for eps in [0.0, 0.5, 1.0, 2.0, 4.0, 8.0, 24.0] {
+            let p = ranking_probability(&perf, &flat(3, eps), &[0, 0, 0]);
+            assert!(p >= previous, "eps={eps}: {p} < {previous}");
+            if eps == 0.0 {
+                assert!(p < 1e-12, "a tie needs a margin, got {p}");
+            }
+            previous = p;
+        }
+        assert!(
+            previous > 0.9,
+            "a very wide margin ties everyone: {previous}"
+        );
+    }
+
+    /// The converse, stated as the non-property it is: a partially-tied
+    /// outcome is non-monotone in the margin. Pinning this down stops a future
+    /// change from "fixing" it into monotonicity and quietly breaking the model.
+    #[test]
+    fn a_partially_tied_outcome_peaks_in_the_middle() {
+        let perf = [g(0.0, 4.0), g(0.0, 4.0), g(-8.0, 2.0)];
+        let sweep: Vec<f64> = [0.5, 2.0, 4.0, 8.0, 16.0]
+            .iter()
+            .map(|&eps| ranking_probability(&perf, &flat(3, eps), &[0, 0, 1]))
+            .collect();
+        let peak = sweep
+            .iter()
+            .enumerate()
+            .fold(
+                (0, 0.0),
+                |(bi, bv), (i, &v)| if v > bv { (i, v) } else { (bi, bv) },
+            )
+            .0;
+        assert!(
+            peak > 0 && peak < sweep.len() - 1,
+            "expected an interior peak: {sweep:?}"
+        );
+    }
+
+    /// With no draw margin a tie has probability exactly zero, and the
+    /// enumeration must not waste work pretending otherwise.
+    #[test]
+    fn ties_are_impossible_without_a_draw_margin() {
+        let perf = [g(0.0, 4.0), g(0.0, 4.0), g(0.0, 4.0)];
+        let dist = outcome_distribution(&perf, &flat(3, 0.0));
+        assert_eq!(dist.len(), 6, "expected only the 6 strict orders: {dist:?}");
+        assert!(dist.iter().all(|(r, _)| {
+            let mut seen = r.clone();
+            seen.sort_unstable();
+            seen.dedup();
+            seen.len() == r.len()
+        }));
+    }
+}

src/quadrature.rs

@@ -0,0 +1,322 @@
+//! Deterministic numerical integration for the prediction paths.
+//!
+//! Prediction asks two questions that have no closed form beyond two teams:
+//! "who finishes first" and "how likely is this exact finishing order". Both
+//! reduce to integrals over a single performance variable, so neither needs a
+//! sampler — and that matters, because a Monte Carlo predictor would make
+//! `predict_*` non-reproducible and would answer a slightly different question
+//! on every call.
+//!
+//! Two routines live here:
+//!
+//! - [`integrate`], adaptive Gauss-Kronrod G7-K15, for the first-place
+//!   marginals. It carries its own error estimate, so it can refine where the
+//!   integrand actually bends instead of guessing a node count up front.
+//! - [`Grid`], a uniform grid with trapezoid prefix sums, for the ranking
+//!   chain recursion, where each level needs the *running* integral of the
+//!   level below at arbitrary points rather than one definite integral.
+//!
+//! Fixed-node Gauss-Hermite is the obvious tool for the first of these and is
+//! a trap: the integrand is a product of normal CDFs, and when one team's
+//! sigma is much smaller than the integrating team's, that product turns into
+//! a near-step function narrower than the node spacing. The nodes step over
+//! it and the result is wrong by ~1e-2 while still looking like a probability.
+//! Adaptive refinement is what makes the small-sigma case safe.
+
+/// Kronrod 15-point abscissae, non-negative half, descending.
+const XGK: [f64; 8] = [
+    0.991_455_371_120_813,
+    0.949_107_912_342_759,
+    0.864_864_423_359_769,
+    0.741_531_185_599_394,
+    0.586_087_235_467_691,
+    0.405_845_151_377_397,
+    0.207_784_955_007_898,
+    0.0,
+];
+
+/// Kronrod 15-point weights, matching [`XGK`].
+const WGK: [f64; 8] = [
+    0.022_935_322_010_529,
+    0.063_092_092_629_979,
+    0.104_790_010_322_250,
+    0.140_653_259_715_525,
+    0.169_004_726_639_267,
+    0.190_350_578_064_785,
+    0.204_432_940_075_298,
+    0.209_482_141_084_728,
+];
+
+/// Gauss 7-point weights, applying to the odd-indexed [`XGK`] entries.
+const WG: [f64; 4] = [
+    0.129_484_966_168_870,
+    0.279_705_391_489_277,
+    0.381_830_050_505_119,
+    0.417_959_183_673_469,
+];
+
+/// Panels are bisected worst-first; this bounds the work on a pathological
+/// integrand rather than letting it spin.
+const MAX_SUBDIVISIONS: usize = 200;
+
+/// One G7-K15 panel over `[a, b]`: `(integral, absolute error estimate)`.
+///
+/// The error estimate is the gap between the embedded 7-point Gauss rule and
+/// the 15-point Kronrod extension. It is the only reason this is preferable
+/// to a fixed rule: it tells the caller *where* the integrand is hard.
+fn gk15<F: Fn(f64) -> f64>(f: &F, a: f64, b: f64) -> (f64, f64) {
+    let centre = 0.5 * (a + b);
+    let half = 0.5 * (b - a);
+
+    let mut kronrod = 0.0;
+    let mut gauss = 0.0;
+
+    for i in 0..8 {
+        let offset = XGK[i] * half;
+        // XGK[7] is the centre node and must not be counted twice.
+        let sum = if i == 7 {
+            f(centre)
+        } else {
+            f(centre - offset) + f(centre + offset)
+        };
+        kronrod += WGK[i] * sum;
+        if i % 2 == 1 {
+            gauss += WG[i / 2] * sum;
+        }
+    }
+
+    (kronrod * half, ((kronrod - gauss) * half).abs())
+}
+
+/// Adaptively integrate `f` over `[a, b]` to relative tolerance `tol`.
+///
+/// `seeds` are interior points where the integrand is known to bend sharply —
+/// for a product of normal CDFs, each rival's transition centre. Splitting
+/// there up front costs nothing and saves the adaptive loop from having to
+/// discover a step by bisection.
+///
+/// Returns the integral. The error estimate is consumed internally rather
+/// than returned: callers here integrate probability densities, where the
+/// meaningful check is the sum-to-one identity over a whole outcome space,
+/// not a per-integral residual.
+pub(crate) fn integrate<F: Fn(f64) -> f64>(f: F, a: f64, b: f64, seeds: &[f64], tol: f64) -> f64 {
+    // Explicit rather than `!(b > a)`: a NaN bound must fall through to zero
+    // rather than being read as a valid ordering.
+    if a.partial_cmp(&b) != Some(std::cmp::Ordering::Less) {
+        return 0.0;
+    }
+
+    let mut edges: Vec<f64> = Vec::with_capacity(seeds.len() + 2);
+    edges.push(a);
+    edges.push(b);
+    for &s in seeds {
+        if s > a && s < b {
+            edges.push(s);
+        }
+    }
+    edges.sort_by(|p, q| p.partial_cmp(q).expect("integration bounds are finite"));
+    edges.dedup();
+
+    // (lo, hi, integral, error)
+    let mut panels: Vec<(f64, f64, f64, f64)> = edges
+        .windows(2)
+        .map(|w| {
+            let (v, e) = gk15(&f, w[0], w[1]);
+            (w[0], w[1], v, e)
+        })
+        .collect();
+
+    for _ in 0..MAX_SUBDIVISIONS {
+        let total: f64 = panels.iter().map(|p| p.2).sum();
+        let error: f64 = panels.iter().map(|p| p.3).sum();
+
+        // Absolute floor as well as relative: these integrands are
+        // probabilities, so an absolute 1e-15 is already past the useful
+        // precision of the underlying `cdf`.
+        if error <= tol * total.abs().max(1e-12) || error < 1e-15 {
+            break;
+        }
+
+        let worst = panels
+            .iter()
+            .enumerate()
+            .fold((0usize, f64::NEG_INFINITY), |(bi, be), (i, p)| {
+                if p.3 > be { (i, p.3) } else { (bi, be) }
+            })
+            .0;
+
+        let (lo, hi, _, _) = panels[worst];
+        let mid = 0.5 * (lo + hi);
+        // Bisection has hit the floating-point floor; refining further would
+        // loop without reducing the error.
+        if !(mid > lo && mid < hi) {
+            break;
+        }
+
+        let (v1, e1) = gk15(&f, lo, mid);
+        let (v2, e2) = gk15(&f, mid, hi);
+        panels[worst] = (lo, mid, v1, e1);
+        panels.push((mid, hi, v2, e2));
+    }
+
+    panels.iter().map(|p| p.2).sum()
+}
+
+/// A uniform grid carrying trapezoid prefix sums of one integrand.
+///
+/// The ranking recursion needs, at every level, the running integral of the
+/// level below evaluated at arbitrary points — a cumulative integral, not a
+/// definite one. Prefix sums give that in O(1) per query after an O(G) build,
+/// which is what keeps a full ranking probability linear in the team count.
+pub(crate) struct Grid {
+    lo: f64,
+    step: f64,
+    /// Integrand sampled at each node.
+    values: Vec<f64>,
+    /// `prefix[i]` is the integral from `lo` to node `i`.
+    prefix: Vec<f64>,
+}
+
+impl Grid {
+    /// Build directly from already-sampled values.
+    ///
+    /// The ranking recursion evaluates every level on the same nodes, so the
+    /// per-team densities are sampled once and reused; re-evaluating `exp`
+    /// per level would dominate the cost.
+    pub(crate) fn from_values(lo: f64, step: f64, values: Vec<f64>) -> Self {
+        let mut prefix = vec![0.0; values.len()];
+        for i in 1..values.len() {
+            prefix[i] = prefix[i - 1] + 0.5 * step * (values[i - 1] + values[i]);
+        }
+        Self {
+            lo,
+            step,
+            values,
+            prefix,
+        }
+    }
+
+    /// Integral from the grid's lower bound up to `x`.
+    ///
+    /// Clamped at both ends: the caller sizes the grid to cover the whole
+    /// support, so a query outside it is asking for a tail that is zero (below)
+    /// or the whole mass (above).
+    pub(crate) fn integral_to(&self, x: f64) -> f64 {
+        let last = self.values.len() - 1;
+        if x <= self.lo {
+            return 0.0;
+        }
+        if x >= self.lo + last as f64 * self.step {
+            return self.prefix[last];
+        }
+
+        let scaled = (x - self.lo) / self.step;
+        let i = scaled.floor() as usize;
+        let frac = scaled - i as f64;
+
+        // Whole cells, plus the trapezoid over the partial cell. The integrand
+        // is linear within a cell under the trapezoid rule, so the partial
+        // piece is exact with respect to that same approximation.
+        self.prefix[i]
+            + frac
+                * self.step
+                * (self.values[i] + 0.5 * frac * (self.values[i + 1] - self.values[i]))
+    }
+
+    /// Integral over `[from, to]`.
+    pub(crate) fn integral_between(&self, from: f64, to: f64) -> f64 {
+        (self.integral_to(to) - self.integral_to(from)).max(0.0)
+    }
+
+    /// Total integral over the whole grid.
+    pub(crate) fn total(&self) -> f64 {
+        self.prefix[self.values.len() - 1]
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    const TOL: f64 = 1e-10;
+
+    /// Sample `f` over `[lo, hi]` at `points` nodes.
+    fn sample<F: FnMut(f64) -> f64>(lo: f64, hi: f64, points: usize, mut f: F) -> Grid {
+        let step = (hi - lo) / (points - 1) as f64;
+        Grid::from_values(
+            lo,
+            step,
+            (0..points).map(|i| f(lo + i as f64 * step)).collect(),
+        )
+    }
+
+    #[test]
+    fn integrates_a_polynomial_exactly() {
+        // G7-K15 is exact for polynomials well past cubic, so a single panel
+        // should already be at round-off.
+        let v = integrate(|x| 3.0 * x * x + 2.0 * x + 1.0, 0.0, 2.0, &[], TOL);
+        assert!((v - 14.0).abs() < 1e-12, "got {v}");
+    }
+
+    #[test]
+    fn integrates_a_gaussian_density_to_one() {
+        let f = |x: f64| (-0.5 * x * x).exp() / (2.0 * std::f64::consts::PI).sqrt();
+        let v = integrate(f, -10.0, 10.0, &[], TOL);
+        assert!((v - 1.0).abs() < 1e-12, "got {v}");
+    }
+
+    #[test]
+    fn resolves_a_step_far_narrower_than_the_initial_panel() {
+        // The failure mode that rules out fixed-node quadrature: a transition
+        // 1e-4 wide inside a range of 20. A fixed rule steps over it.
+        let f = |x: f64| if x < 0.5 { 0.0 } else { 1.0 };
+        let v = integrate(f, -10.0, 10.0, &[0.5], TOL);
+        assert!((v - 9.5).abs() < 1e-6, "got {v}");
+    }
+
+    #[test]
+    fn seeds_do_not_change_the_value_of_a_smooth_integrand() {
+        let f = |x: f64| (-0.5 * x * x).exp();
+        let plain = integrate(f, -8.0, 8.0, &[], TOL);
+        let seeded = integrate(f, -8.0, 8.0, &[-3.0, 0.25, 5.5], TOL);
+        assert!((plain - seeded).abs() < 1e-12, "{plain} vs {seeded}");
+    }
+
+    #[test]
+    fn empty_or_inverted_range_integrates_to_zero() {
+        assert_eq!(integrate(|_| 1.0, 1.0, 1.0, &[], TOL), 0.0);
+        assert_eq!(integrate(|_| 1.0, 2.0, 1.0, &[], TOL), 0.0);
+    }
+
+    #[test]
+    fn grid_prefix_matches_a_known_cumulative_integral() {
+        // f(x) = x over [0, 4]; integral to x is x^2/2.
+        let g = sample(0.0, 4.0, 4001, |x| x);
+        for probe in [0.0, 0.5, 1.0, 2.5, 3.75, 4.0] {
+            let want = probe * probe / 2.0;
+            let got = g.integral_to(probe);
+            assert!(
+                (got - want).abs() < 1e-9,
+                "at {probe}: got {got}, want {want}"
+            );
+        }
+        assert!((g.total() - 8.0).abs() < 1e-9);
+    }
+
+    #[test]
+    fn grid_between_is_the_difference_of_two_prefixes() {
+        let g = sample(-5.0, 5.0, 8001, |x| (-0.5 * x * x).exp());
+        let whole = g.integral_between(-5.0, 5.0);
+        let split = g.integral_between(-5.0, 0.3) + g.integral_between(0.3, 5.0);
+        assert!((whole - split).abs() < 1e-12, "{whole} vs {split}");
+    }
+
+    #[test]
+    fn grid_clamps_queries_outside_its_support() {
+        let g = sample(0.0, 1.0, 101, |_| 1.0);
+        assert_eq!(g.integral_to(-3.0), 0.0);
+        assert!((g.integral_to(9.0) - 1.0).abs() < 1e-12);
+        // Reversed bounds must not produce negative probability mass.
+        assert_eq!(g.integral_between(0.8, 0.2), 0.0);
+    }
+}

src/rating.rs

@@ -9,13 +9,16 @@ use crate::{
 
 /// Static rating configuration: prior skill, performance noise `beta`, drift.
 ///
-/// Renamed from `Player` in T2; `Rating` better describes the data
+/// A configuration rather than a person: the per-history temporal state
-/// (a configuration) vs. a person (who's a `Competitor` with state).
+/// (messages, last appearance) lives on `Competitor`.
 #[derive(Clone, Copy, Debug)]
 pub struct Rating<T: Time = i64, D: Drift<T> = ConstantDrift> {
     pub(crate) prior: Gaussian,
     pub(crate) beta: f64,
     pub(crate) drift: D,
+    /// Multiplier on the drift *variance* this competitor accumulates; 1.0 is
+    /// the neutral default. Set per competitor via `Member::with_drift_scale`.
+    pub(crate) drift_scale: f64,
     pub(crate) _time: PhantomData<T>,
 }
 
@@ -25,10 +28,21 @@ impl<T: Time, D: Drift<T>> Rating<T, D> {
             prior,
             beta,
             drift,
+            drift_scale: 1.0,
             _time: PhantomData,
         }
     }
 
+    /// Scale how fast this competitor drifts, relative to `drift`.
+    ///
+    /// Multiplies the drift *variance*, so the scale is in the same units as
+    /// `gamma`. `0.0` pins the competitor still.
+    #[must_use]
+    pub fn with_drift_scale(mut self, drift_scale: f64) -> Self {
+        self.drift_scale = drift_scale;
+        self
+    }
+
     /// The configured prior skill estimate.
     #[must_use]
     pub fn prior(&self) -> Gaussian {
@@ -47,6 +61,28 @@ impl<T: Time, D: Drift<T>> Rating<T, D> {
         self.drift
     }
 
+    /// This competitor's multiplier on the drift variance; 1.0 is neutral.
+    #[must_use]
+    pub fn drift_scale(&self) -> f64 {
+        self.drift_scale
+    }
+
+    /// Drift variance accumulated over `from -> to`, scaled for this competitor.
+    ///
+    /// The single place the scale is applied for a `Time`-typed span. Callers
+    /// must go through this rather than `self.drift` directly, so a competitor's
+    /// scale cannot be silently skipped.
+    pub(crate) fn drift_variance_delta(&self, from: &T, to: &T) -> f64 {
+        self.drift.variance_delta(from, to) * self.drift_scale * self.drift_scale
+    }
+
+    /// Drift variance for a cached elapsed count, scaled for this competitor.
+    ///
+    /// The counterpart of `drift_variance_delta` for the cached-elapsed paths.
+    pub(crate) fn drift_variance_for_elapsed(&self, elapsed: i64) -> f64 {
+        self.drift.variance_for_elapsed(elapsed) * self.drift_scale * self.drift_scale
+    }
+
     pub(crate) fn performance(&self) -> Gaussian {
         self.prior.forget(self.beta.powi(2))
     }
@@ -58,6 +94,7 @@ impl Default for Rating<i64, ConstantDrift> {
             prior: Gaussian::default(),
             beta: BETA,
             drift: ConstantDrift(GAMMA),
+            drift_scale: 1.0,
             _time: PhantomData,
         }
     }

src/schedule.rs

@@ -1,152 +0,0 @@
-//! Schedule trait and built-in implementations.
-//!
-//! A schedule drives factor propagation to convergence. The default
-//! `EpsilonOrMax` performs one TeamSum sweep (setup) then alternating
-//! forward/backward sweeps over the iterating factors until the max
-//! delta drops below epsilon or `max` iterations is reached.
-
-use crate::factor::{BuiltinFactor, Factor, VarStore};
-
-/// Result returned by a `Schedule::run` call.
-#[derive(Debug, Clone, Copy)]
-pub struct ScheduleReport {
-    pub iterations: usize,
-    pub final_step: (f64, f64),
-    pub converged: bool,
-}
-
-/// Drives factor propagation to convergence.
-pub trait Schedule: Send + Sync {
-    fn run(&self, factors: &mut [BuiltinFactor], vars: &mut VarStore) -> ScheduleReport;
-}
-
-/// Default schedule: sweep forward then backward until step ≤ eps or iter == max.
-///
-/// Matches the existing `Game::likelihoods` loop bit-for-bit when given the
-/// same factor layout (TeamSums first, then alternating RankDiff/Trunc pairs).
-#[derive(Debug, Clone, Copy)]
-pub struct EpsilonOrMax {
-    pub eps: f64,
-    pub max: usize,
-}
-
-impl Default for EpsilonOrMax {
-    fn default() -> Self {
-        // Derived from `ConvergenceOptions` so there is one source of truth for
-        // the tolerance and iteration cap. These previously disagreed: this
-        // default capped at 10 iterations while `ConvergenceOptions` allowed 30,
-        // and which applied depended on whether inference went through
-        // `run_chain` or a `Schedule`.
-        let defaults = crate::ConvergenceOptions::default();
-
-        Self {
-            eps: defaults.epsilon,
-            max: defaults.max_iter,
-        }
-    }
-}
-
-impl Schedule for EpsilonOrMax {
-    fn run(&self, factors: &mut [BuiltinFactor], vars: &mut VarStore) -> ScheduleReport {
-        // Partition: leading run of TeamSum factors run exactly once (setup).
-        let n_setup = factors
-            .iter()
-            .position(|f| !matches!(f, BuiltinFactor::TeamSum(_)))
-            .unwrap_or(factors.len());
-
-        for f in factors[..n_setup].iter_mut() {
-            f.propagate(vars);
-        }
-
-        let mut iterations = 0;
-        // With no iterating factors the graph is already at its fixed point:
-        // the setup pass above is all there is to do. Reporting `converged:
-        // false` with an infinite step for that case gave callers a false
-        // negative.
-        let mut final_step = (0.0, 0.0);
-        let mut converged = true;
-
-        if n_setup < factors.len() {
-            final_step = (f64::INFINITY, f64::INFINITY);
-            converged = false;
-            for _ in 0..self.max {
-                let mut step = (0.0_f64, 0.0_f64);
-
-                // Forward sweep over iterating factors.
-                for f in factors[n_setup..].iter_mut() {
-                    let d = f.propagate(vars);
-                    step.0 = step.0.max(d.0);
-                    step.1 = step.1.max(d.1);
-                }
-
-                // Backward sweep.
-                for f in factors[n_setup..].iter_mut().rev() {
-                    let d = f.propagate(vars);
-                    step.0 = step.0.max(d.0);
-                    step.1 = step.1.max(d.1);
-                }
-
-                iterations += 1;
-                final_step = step;
-
-                if step.0 <= self.eps && step.1 <= self.eps {
-                    converged = true;
-                    break;
-                }
-            }
-        }
-
-        ScheduleReport {
-            iterations,
-            final_step,
-            converged,
-        }
-    }
-}
-
-#[cfg(test)]
-mod tests {
-    use super::*;
-    use crate::{N_INF, factor::team_sum::TeamSumFactor, gaussian::Gaussian};
-
-    #[test]
-    fn schedule_runs_setup_factors_once() {
-        // Single TeamSum factor; schedule should propagate it exactly once and report 0 iterations.
-        let mut vars = VarStore::new();
-        let out = vars.alloc(N_INF);
-        let mut factors = vec![BuiltinFactor::TeamSum(TeamSumFactor {
-            inputs: vec![(Gaussian::from_ms(5.0, 1.0), 1.0)],
-            out,
-        })];
-        let schedule = EpsilonOrMax::default();
-        let report = schedule.run(&mut factors, &mut vars);
-        assert_eq!(report.iterations, 0);
-        // The team-perf var should hold the sum.
-        let result = vars.get(out);
-        assert!((result.mu() - 5.0).abs() < 1e-12);
-    }
-
-    #[test]
-    fn report_marks_converged_when_no_iterating_factors() {
-        // A graph of only setup factors has nothing to iterate, so it is at its
-        // fixed point after the setup pass: 0 iterations, and converged.
-        let mut vars = VarStore::new();
-        let out = vars.alloc(N_INF);
-        let mut factors = vec![BuiltinFactor::TeamSum(TeamSumFactor {
-            inputs: vec![(Gaussian::from_ms(0.0, 1.0), 1.0)],
-            out,
-        })];
-        let report = EpsilonOrMax::default().run(&mut factors, &mut vars);
-        assert_eq!(report.iterations, 0);
-        assert!(report.converged);
-        assert_eq!(report.final_step, (0.0, 0.0));
-    }
-
-    #[test]
-    fn default_matches_convergence_options() {
-        let schedule = EpsilonOrMax::default();
-        let options = crate::ConvergenceOptions::default();
-        assert_eq!(schedule.max, options.max_iter);
-        assert_eq!(schedule.eps, options.epsilon);
-    }
-}

src/storage/competitor_store.rs

@@ -2,7 +2,7 @@ use crate::{Index, competitor::Competitor, drift::Drift, time::Time};
 
 /// Dense Vec-backed store for competitor state in History.
 ///
-/// Indexed directly by Index.0, eliminating HashMap hashing in the
+/// Indexed directly by Index.0, eliminating `HashMap` hashing in the
 /// forward/backward sweep. Uses `Vec<Option<Competitor<T, D>>>` so slots can be
 /// absent without an explicit present mask.
 #[derive(Debug)]
@@ -21,6 +21,7 @@ impl<T: Time, D: Drift<T>> Default for CompetitorStore<T, D> {
 }
 
 impl<T: Time, D: Drift<T>> CompetitorStore<T, D> {
+    #[must_use]
     pub fn new() -> Self {
         Self::default()
     }
@@ -39,6 +40,7 @@ impl<T: Time, D: Drift<T>> CompetitorStore<T, D> {
         self.competitors[idx.0] = Some(competitor);
     }
 
+    #[must_use]
     pub fn get(&self, idx: Index) -> Option<&Competitor<T, D>> {
         self.competitors.get(idx.0).and_then(|slot| slot.as_ref())
     }
@@ -49,14 +51,17 @@ impl<T: Time, D: Drift<T>> CompetitorStore<T, D> {
             .and_then(|slot| slot.as_mut())
     }
 
+    #[must_use]
     pub fn contains(&self, idx: Index) -> bool {
         self.get(idx).is_some()
     }
 
+    #[must_use]
     pub fn len(&self) -> usize {
         self.n_present
     }
 
+    #[must_use]
     pub fn is_empty(&self) -> bool {
         self.n_present == 0
     }

src/storage/skill_store.rs

@@ -1,15 +1,27 @@
+use std::collections::HashMap;
+
 use crate::{Index, time_slice::Skill};
 
-/// Dense Vec-backed store for per-agent skill state within a TimeSlice.
+/// Compact per-slice store for skill state, addressed by a slice-local slot.
 ///
-/// Indexed directly by Index.0, eliminating HashMap hashing in the inner
+/// `skills` holds one entry per competitor **in this slice**, so memory is
-/// convergence loop. Uses a parallel `present` mask so iteration skips
+/// O(competitors in the slice). It used to be a dense `Vec<Skill>` indexed by
-/// absent slots without incurring per-slot Option overhead in the hot path.
+/// the global `Index.0`, which made a slice's footprint O(largest index it
+/// touches): a single 1v1 game between competitors 19998 and 19999 reserved
+/// 20,000 slots.
+///
+/// The dense layout existed to keep `HashMap` hashing out of the inner
+/// convergence loop, and that property is preserved. `slots` is consulted only
+/// while building a slice; every hot-path access goes through
+/// [`SkillStore::at`] / [`SkillStore::at_mut`] with a slot resolved once at
+/// ingestion and cached on the event's `Item`.
 #[derive(Debug, Default)]
 pub struct SkillStore {
     skills: Vec<Skill>,
-    present: Vec<bool>,
+    /// Slot -> global index, parallel to `skills`, so iteration can report the
-    n_present: usize,
+    /// global index without a reverse lookup.
+    indices: Vec<Index>,
+    slots: HashMap<Index, u32>,
 }
 
 impl SkillStore {
@@ -17,73 +29,99 @@ impl SkillStore {
         Self::default()
     }
 
-    fn ensure_capacity(&mut self, idx: usize) {
+    /// Resolve a global index to this slice's slot, if the competitor is here.
-        if idx >= self.skills.len() {
+    ///
-            self.skills.resize_with(idx + 1, Skill::default);
+    /// This hashes. Call it at ingestion and cache the result; do not call it
-            self.present.resize(idx + 1, false);
+    /// from the convergence loop.
-        }
+    pub fn slot_of(&self, idx: Index) -> Option<u32> {
+        self.slots.get(&idx).copied()
     }
 
-    pub fn insert(&mut self, idx: Index, skill: Skill) {
+    /// Skill at a slot resolved earlier by [`SkillStore::slot_of`].
-        self.ensure_capacity(idx.0);
+    ///
-        if !self.present[idx.0] {
+    /// # Panics
-            self.n_present += 1;
+    ///
+    /// Panics if `slot` is out of range, which means it came from a different
+    /// slice's store.
+    pub fn at(&self, slot: u32) -> &Skill {
+        &self.skills[slot as usize]
+    }
+
+    /// Mutable counterpart to [`SkillStore::at`].
+    ///
+    /// # Panics
+    ///
+    /// Panics if `slot` is out of range.
+    pub fn at_mut(&mut self, slot: u32) -> &mut Skill {
+        &mut self.skills[slot as usize]
+    }
+
+    /// Insert or overwrite a competitor's skill, returning its slot.
+    pub fn insert(&mut self, idx: Index, skill: Skill) -> u32 {
+        match self.slots.get(&idx) {
+            Some(&slot) => {
+                self.skills[slot as usize] = skill;
+                slot
+            }
+            None => {
+                let slot = u32::try_from(self.skills.len())
+                    .expect("a time slice cannot hold more than u32::MAX competitors");
+
+                self.skills.push(skill);
+                self.indices.push(idx);
+                self.slots.insert(idx, slot);
+
+                slot
+            }
         }
-        self.skills[idx.0] = skill;
-        self.present[idx.0] = true;
     }
 
     pub fn get(&self, idx: Index) -> Option<&Skill> {
-        if idx.0 < self.present.len() && self.present[idx.0] {
+        self.slot_of(idx).map(|slot| self.at(slot))
-            Some(&self.skills[idx.0])
-        } else {
-            None
-        }
-    }
-
-    /// Whether a slot is occupied. Test-only.
-    #[cfg(test)]
-    pub fn contains(&self, idx: Index) -> bool {
-        idx.0 < self.present.len() && self.present[idx.0]
-    }
-
-    /// Number of occupied slots. Test-only.
-    #[cfg(test)]
-    pub fn len(&self) -> usize {
-        self.n_present
     }
 
     pub fn get_mut(&mut self, idx: Index) -> Option<&mut Skill> {
-        if idx.0 < self.present.len() && self.present[idx.0] {
+        self.slot_of(idx)
-            Some(&mut self.skills[idx.0])
+            .map(|slot| &mut self.skills[slot as usize])
-        } else {
-            None
-        }
     }
 
+    /// Whether a competitor is present in this slice. Test-only.
+    #[cfg(test)]
+    pub fn contains(&self, idx: Index) -> bool {
+        self.slots.contains_key(&idx)
+    }
+
+    /// Number of competitors in this slice. Test-only.
+    #[cfg(test)]
+    pub fn len(&self) -> usize {
+        self.skills.len()
+    }
+
+    /// Slots actually allocated — the quantity #17 is about, and NOT the same
+    /// as `len` for every possible implementation.
+    ///
+    /// A store indexed by the global `Index` must report `max_index + 1` here
+    /// while reporting the true competitor count from `len`, which is exactly
+    /// how the original defect hid. Tests that mean to pin the footprint must
+    /// assert on this.
+    #[cfg(test)]
+    pub fn allocated_slots(&self) -> usize {
+        self.skills.len()
+    }
+
+    /// Iterate in slot order — the order competitors were first seen in this
+    /// slice. Deterministic for a given event order, which is what the
+    /// cross-thread determinism test relies on.
     pub fn iter(&self) -> impl Iterator<Item = (Index, &Skill)> {
-        self.present.iter().enumerate().filter_map(|(i, &p)| {
+        self.indices.iter().copied().zip(self.skills.iter())
-            if p {
-                Some((Index(i), &self.skills[i]))
-            } else {
-                None
-            }
-        })
     }
 
     pub fn iter_mut(&mut self) -> impl Iterator<Item = (Index, &mut Skill)> {
-        self.skills
+        self.indices.iter().copied().zip(self.skills.iter_mut())
-            .iter_mut()
-            .zip(self.present.iter())
-            .enumerate()
-            .filter_map(|(i, (s, &p))| if p { Some((Index(i), s)) } else { None })
     }
 
     pub fn keys(&self) -> impl Iterator<Item = Index> + '_ {
-        self.present
+        self.indices.iter().copied()
-            .iter()
-            .enumerate()
-            .filter_map(|(i, &p)| if p { Some(Index(i)) } else { None })
     }
 }
 
@@ -109,7 +147,7 @@ mod tests {
     }
 
     #[test]
-    fn iter_skips_absent_slots() {
+    fn iter_reports_global_indices() {
         let mut store = SkillStore::new();
         store.insert(Index(0), Skill::default());
         store.insert(Index(5), Skill::default());
@@ -124,4 +162,28 @@ mod tests {
         store.insert(Index(2), Skill::default());
         assert_eq!(store.len(), 1);
     }
+
+    /// The defect in #17: a slice holding two competitors must cost the same
+    /// whether their indices are small or large.
+    #[test]
+    fn footprint_is_independent_of_index_magnitude() {
+        let mut low = SkillStore::new();
+        low.insert(Index(0), Skill::default());
+        low.insert(Index(1), Skill::default());
+
+        let mut high = SkillStore::new();
+        high.insert(Index(19_998), Skill::default());
+        high.insert(Index(19_999), Skill::default());
+
+        assert_eq!(low.len(), high.len());
+        assert_eq!(low.skills.capacity(), high.skills.capacity());
+    }
+
+    #[test]
+    fn slot_survives_reinsert() {
+        let mut store = SkillStore::new();
+        let first = store.insert(Index(7), Skill::default());
+        let again = store.insert(Index(7), Skill::default());
+        assert_eq!(first, again);
+    }
 }

src/time_slice.rs

@@ -51,6 +51,13 @@ pub enum EventKind {
 #[derive(Clone, Debug)]
 struct Item {
     agent: Index,
+    /// This competitor's slot in the owning slice's `SkillStore`, resolved
+    /// once at ingestion.
+    ///
+    /// The convergence loop reaches skills through this rather than through
+    /// `agent`, which is what keeps `HashMap` hashing out of the hot path now
+    /// that the store is compact rather than indexed by the global `Index`.
+    slot: u32,
     likelihood: Gaussian,
 }
 
@@ -62,12 +69,13 @@ impl Item {
         agents: &CompetitorStore<T, D>,
     ) -> Rating<T, D> {
         let r = &agents[self.agent].rating;
-        let skill = skills.get(self.agent).unwrap();
+        let skill = skills.at(self.slot);
 
         if forward {
-            Rating::new(skill.forward, r.beta, r.drift)
+            Rating::new(skill.forward, r.beta, r.drift).with_drift_scale(r.drift_scale)
         } else {
             Rating::new(skill.posterior() / self.likelihood, r.beta, r.drift)
+                .with_drift_scale(r.drift_scale)
         }
     }
 }
@@ -157,9 +165,9 @@ impl Event {
         for (t, team) in self.teams.iter_mut().enumerate() {
             for (i, item) in team.items.iter_mut().enumerate() {
                 let fresh = update.likelihoods[t][i];
-                let old_likelihood = skills.get(item.agent).unwrap().likelihood;
+                let old_likelihood = skills.at(item.slot).likelihood;
                 let new_likelihood = (old_likelihood / item.likelihood) * fresh;
-                skills.get_mut(item.agent).unwrap().likelihood = new_likelihood;
+                skills.at_mut(item.slot).likelihood = new_likelihood;
                 item.likelihood = fresh;
             }
         }
@@ -277,8 +285,8 @@ impl<T: Time> TimeSlice<T> {
     pub fn add_events<D: Drift<T>>(
         &mut self,
         composition: Vec<Vec<Vec<Index>>>,
-        results: Vec<Vec<f64>>,
+        results: Option<Vec<Vec<f64>>>,
-        weights: Vec<Vec<Vec<f64>>>,
+        weights: Option<Vec<Vec<Vec<f64>>>>,
         kinds: Vec<EventKind>,
         agents: &CompetitorStore<T, D>,
     ) {
@@ -297,14 +305,16 @@ impl<T: Time> TimeSlice<T> {
         for idx in this_agent {
             let elapsed = compute_elapsed(agents[*idx].last_time.as_ref(), &self.time);
 
+            let forward = agents[*idx].receive(&self.time);
+
             if let Some(skill) = self.skills.get_mut(*idx) {
                 skill.elapsed = elapsed;
-                skill.forward = agents[*idx].receive(&self.time);
+                skill.forward = forward;
             } else {
                 self.skills.insert(
                     *idx,
                     Skill {
-                        forward: agents[*idx].receive(&self.time),
+                        forward,
                         backward: N_INF,
                         likelihood: N_INF,
                         elapsed,
@@ -313,6 +323,8 @@ impl<T: Time> TimeSlice<T> {
             }
         }
 
+        let skills = &self.skills;
+
         let events = composition.iter().enumerate().map(|(e, event)| {
             let teams = event
                 .iter()
@@ -322,28 +334,32 @@ impl<T: Time> TimeSlice<T> {
                         .iter()
                         .map(|&agent| Item {
                             agent,
+                            // Every participant was inserted into `skills`
+                            // just above, so the slot always resolves.
+                            slot: skills
+                                .slot_of(agent)
+                                .expect("participant must be present in the slice store"),
                             likelihood: N_INF,
                         })
                         .collect::<Vec<_>>();
 
                     Team {
                         items,
-                        output: if results.is_empty() {
+                        output: match &results {
-                            (event.len() - (t + 1)) as f64
+                            Some(results) => results[e][t],
-                        } else {
+                            // No explicit result: rank by position, first team best.
-                            results[e][t]
+                            None => (event.len() - (t + 1)) as f64,
                         },
                     }
                 })
                 .collect::<Vec<_>>();
 
-            let weights = if weights.is_empty() {
+            let weights = match &weights {
-                teams
+                Some(weights) => weights[e].clone(),
+                None => teams
                     .iter()
                     .map(|team| vec![1.0; team.items.len()])
-                    .collect::<Vec<_>>()
+                    .collect::<Vec<_>>(),
-            } else {
-                weights[e].clone()
             };
 
             Event {
@@ -370,6 +386,13 @@ impl<T: Time> TimeSlice<T> {
             .collect::<HashMap<_, _>>()
     }
 
+    /// Sweep this slice's events once, starting at index `from`.
+    ///
+    /// # Panics
+    ///
+    /// Panics if an event references a competitor with no entry in this
+    /// slice's skill store. `add_events` inserts one for every participant, so
+    /// this cannot happen for slices built through the public API.
     pub fn iteration<D: Drift<T>>(&mut self, from: usize, agents: &CompetitorStore<T, D>) {
         if from == 0 && self.color_groups_dirty {
             self.recompute_color_groups();
@@ -402,10 +425,10 @@ impl<T: Time> TimeSlice<T> {
 
                 for (t, team) in event.teams.iter_mut().enumerate() {
                     for (i, item) in team.items.iter_mut().enumerate() {
-                        let old_likelihood = self.skills.get(item.agent).unwrap().likelihood;
+                        let old_likelihood = self.skills.at(item.slot).likelihood;
                         let new_likelihood =
                             (old_likelihood / item.likelihood) * g.likelihoods[t][i];
-                        self.skills.get_mut(item.agent).unwrap().likelihood = new_likelihood;
+                        self.skills.at_mut(item.slot).likelihood = new_likelihood;
                         item.likelihood = g.likelihoods[t][i];
                     }
                 }
@@ -567,14 +590,13 @@ impl<T: Time> TimeSlice<T> {
         n.forget(
             agents[*agent]
                 .rating
-                .drift
+                .drift_variance_for_elapsed(skill.elapsed),
-                .variance_for_elapsed(skill.elapsed),
         )
     }
 
     pub(crate) fn new_backward_info<D: Drift<T>>(&mut self, agents: &CompetitorStore<T, D>) {
         for (agent, skill) in self.skills.iter_mut() {
-            skill.backward = agents[agent].message;
+            skill.backward = agents[agent].message.unwrap_or(N_INF);
         }
         self.iteration(0, agents);
     }
@@ -623,11 +645,11 @@ impl<T: Time> TimeSlice<T> {
             let rating = &agents[agent].rating;
 
             let forward = match incoming.get(&agent) {
-                Some(message) => message.forget(rating.drift.variance_for_elapsed(skill.elapsed)),
+                Some(message) => message.forget(rating.drift_variance_for_elapsed(skill.elapsed)),
                 None => rating.prior,
             };
 
-            scratch.skills.insert(
+            let slot = scratch.skills.insert(
                 agent,
                 Skill {
                     forward,
@@ -636,6 +658,17 @@ impl<T: Time> TimeSlice<T> {
                     elapsed: skill.elapsed,
                 },
             );
+
+            // The cloned events carry slots resolved against the REAL store, so
+            // the scratch must assign the same ones. It does because `iter()`
+            // yields slot order and `insert` allocates slots in call order —
+            // but that is a coupling between two types, so pin it here rather
+            // than leave it to be rediscovered after it breaks.
+            debug_assert_eq!(
+                Some(slot),
+                self.skills.slot_of(agent),
+                "scratch slot must match the real slice's slot for {agent:?}"
+            );
         }
 
         scratch.iterate_to_convergence(agents);
@@ -754,8 +787,98 @@ impl<T: Time> TimeSlice<T> {
     }
 }
 
+/// Elapsed time from a competitor's previous appearance to `current`.
+///
+/// A negative elapsed means slices are being visited out of time order, which
+/// would make drift *reduce* uncertainty. Release builds clamp to zero so a
+/// bad timestamp degrades to "no drift" rather than corrupting the posterior;
+/// debug builds trip instead, because reaching here is a bug in slice ordering
+/// rather than something callers can cause with ordinary data.
 pub(crate) fn compute_elapsed<T: Time>(last: Option<&T>, current: &T) -> i64 {
-    last.map(|l| l.elapsed_to(current).max(0)).unwrap_or(0)
+    let Some(last) = last else {
+        return 0;
+    };
+
+    let elapsed = last.elapsed_to(current);
+
+    debug_assert!(
+        elapsed >= 0,
+        "negative elapsed ({elapsed}) — slices visited out of time order"
+    );
+
+    elapsed.max(0)
+}
+
+impl<T: Time> TimeSlice<T> {
+    /// This slice's scored event factors, as contrasts over competitors.
+    ///
+    /// Message passing produces per-competitor marginals and throws the
+    /// correlation away — `Item::likelihood` is already the projection of an
+    /// event's factor onto one competitor. So a joint has to be rebuilt from
+    /// the factor structure rather than recovered from the messages.
+    ///
+    /// Usefully, a precision matrix depends only on *structure* — who played
+    /// whom, with what weights and what observation noise — and not on the
+    /// observed outcomes. The means are already exact, so only the second
+    /// moment needs rebuilding.
+    ///
+    /// Each entry is a contrast and the observation variance that sits on it.
+    /// Ranked events contribute nothing: their truncation factors are EP
+    /// approximations that inference does not retain.
+    pub(crate) fn scored_contrasts<D: Drift<T>>(
+        &self,
+        agents: &CompetitorStore<T, D>,
+    ) -> Vec<(Vec<(Index, f64)>, f64)> {
+        let mut out = Vec::new();
+
+        for event in &self.events {
+            let EventKind::Scored { score_sigma } = event.kind else {
+                continue;
+            };
+
+            // Teams best-first, matching the diff chain inference builds.
+            let mut order: Vec<usize> = (0..event.teams.len()).collect();
+            order.sort_by(|&a, &b| {
+                event.teams[b]
+                    .output
+                    .partial_cmp(&event.teams[a].output)
+                    .unwrap_or(std::cmp::Ordering::Equal)
+            });
+
+            for pair in order.windows(2) {
+                let (hi, lo) = (pair[0], pair[1]);
+                let mut contrast: Vec<(Index, f64)> = Vec::new();
+                let mut noise = score_sigma * score_sigma;
+
+                for (team, sign) in [(hi, 1.0), (lo, -1.0)] {
+                    for (m, item) in event.teams[team].items.iter().enumerate() {
+                        let w = event.weights[team][m];
+                        noise += w * w * agents[item.agent].rating.beta.powi(2);
+                        contrast.push((item.agent, sign * w));
+                    }
+                }
+
+                out.push((contrast, noise));
+            }
+        }
+
+        out
+    }
+
+    /// True when every event here is scored, so the joint is exact.
+    pub(crate) fn all_scored(&self) -> bool {
+        self.events
+            .iter()
+            .all(|e| matches!(e.kind, EventKind::Scored { .. }))
+    }
+
+    /// The competitors appearing in this slice, with the elapsed count since
+    /// each one's previous appearance.
+    pub(crate) fn appearances(&self) -> impl Iterator<Item = (Index, i64)> + '_ {
+        self.skills
+            .keys()
+            .map(|idx| (idx, self.skills.get(idx).expect("slice key").elapsed))
+    }
 }
 
 #[cfg(test)]
@@ -803,8 +926,8 @@ mod tests {
                 vec![vec![c], vec![d]],
                 vec![vec![e], vec![f]],
             ],
-            vec![vec![1.0, 0.0], vec![0.0, 1.0], vec![1.0, 0.0]],
+            Some(vec![vec![1.0, 0.0], vec![0.0, 1.0], vec![1.0, 0.0]]),
-            vec![],
+            None,
             vec![EventKind::Ranked; 3],
             &agents,
         );
@@ -880,8 +1003,8 @@ mod tests {
                 vec![vec![a], vec![c]],
                 vec![vec![b], vec![c]],
             ],
-            vec![vec![1.0, 0.0], vec![0.0, 1.0], vec![1.0, 0.0]],
+            Some(vec![vec![1.0, 0.0], vec![0.0, 1.0], vec![1.0, 0.0]]),
-            vec![],
+            None,
             vec![EventKind::Ranked; 3],
             &agents,
         );
@@ -960,8 +1083,8 @@ mod tests {
                 vec![vec![a], vec![c]],
                 vec![vec![b], vec![c]],
             ],
-            vec![vec![1.0, 0.0], vec![0.0, 1.0], vec![1.0, 0.0]],
+            Some(vec![vec![1.0, 0.0], vec![0.0, 1.0], vec![1.0, 0.0]]),
-            vec![],
+            None,
             vec![EventKind::Ranked; 3],
             &agents,
         );
@@ -992,8 +1115,8 @@ mod tests {
                 vec![vec![a], vec![c]],
                 vec![vec![b], vec![c]],
             ],
-            vec![vec![1.0, 0.0], vec![0.0, 1.0], vec![1.0, 0.0]],
+            Some(vec![vec![1.0, 0.0], vec![0.0, 1.0], vec![1.0, 0.0]]),
-            vec![],
+            None,
             vec![EventKind::Ranked; 3],
             &agents,
         );
@@ -1063,8 +1186,8 @@ mod tests {
                 vec![vec![c], vec![d]],
                 vec![vec![a], vec![c]],
             ],
-            vec![vec![1.0, 0.0], vec![1.0, 0.0], vec![1.0, 0.0]],
+            Some(vec![vec![1.0, 0.0], vec![1.0, 0.0], vec![1.0, 0.0]]),
-            vec![],
+            None,
             vec![EventKind::Ranked; 3],
             &agents,
         );

tests/additive_model.rs

@@ -0,0 +1,142 @@
+//! What an additive model does to uncertainty, and why "add the marginals" is
+//! unsafe in one direction and merely wasteful in the other.
+//!
+//! Structurally this is the shape a joint player/layout model takes: every
+//! observation measures a *sum* of nodes against a reference, so the data pins
+//! differences and leaves the overall level to the prior. That is the classic
+//! rating-scale indeterminacy, not a defect.
+//!
+//! The consequence for a consumer is that combining marginals is wrong in
+//! opposite directions depending on the combination, which is worth pinning
+//! because the unsafe direction is not the one you would guess:
+//!
+//! - **Differences** (`a - b`): the shared level cancels, so the exact width is
+//!   small — and adding marginals lands within a couple of percent of it here,
+//!   because the loopy underestimate offsets the ignored correlation.
+//! - **Sums** (`a + b`): the shared level does *not* cancel, so the exact width
+//!   is large, and adding marginals is roughly five times too narrow. That is
+//!   overconfident, and it is the direction that publishes a claim the data
+//!   does not support.
+
+use smallvec::smallvec;
+use trueskill_tt::{ConstantDrift, ConvergenceOptions, Event, History, Member, Outcome, Team};
+
+#[test]
+fn additive_structure_makes_sums_wide_and_differences_tight() {
+    // Structurally like ustat: every round is (player + hole) measured against
+    // a fixed reference. Only SUMS are pinned by the data; the split between
+    // player and hole is pinned only by the prior.
+    let players = ["p0", "p1", "p2"];
+    let holes = ["h0", "h1"];
+
+    let mut h: History<i64, _, _, &'static str> = History::builder()
+        .mu(0.0)
+        .sigma(6.0)
+        .beta(1.0)
+        .score_sigma(2.0)
+        .drift(ConstantDrift(0.0))
+        .convergence(ConvergenceOptions {
+            max_iter: 20_000,
+            epsilon: 1e-12,
+            alpha: 1.0,
+        })
+        .build();
+
+    let mut seed = 3u64;
+    let mut rnd = move || {
+        seed ^= seed << 13;
+        seed ^= seed >> 7;
+        seed ^= seed << 17;
+        seed
+    };
+    // true skills, so we know what the data encodes
+    let truth_p = [2.0, 0.0, -2.0];
+    let truth_h = [1.0, -1.0];
+
+    let mut events = Vec::new();
+    for _ in 0..60 {
+        let p = (rnd() as usize) % 3;
+        let q = (rnd() as usize) % 2;
+        let noise = ((rnd() % 1000) as f64 / 1000.0 - 0.5) * 2.0;
+        let score = truth_p[p] + truth_h[q] + noise;
+        events.push(Event {
+            time: 1,
+            teams: smallvec![
+                Team::with_members([Member::new(players[p]), Member::new(holes[q])]),
+                Team::with_members([Member::new("reference")]),
+            ],
+            outcome: Outcome::scores([score, 0.0]),
+        });
+    }
+    h.add_events(events).unwrap();
+    let r = h.converge().unwrap();
+    assert!(r.converged, "{:?}", r.final_step);
+
+    println!("\n== marginals (what current_skill reports) ==");
+    for k in players.iter().chain(holes.iter()) {
+        let g = h.current_skill(k).unwrap();
+        println!("  {k}: mu {:>8.4}  sigma {:>8.4}", g.mu(), g.sigma());
+    }
+
+    println!("\n== the same nodes via posterior_of (exact marginal) ==");
+    for k in players.iter().chain(holes.iter()) {
+        let g = h.posterior_of(&[(k, 1.0)]).unwrap();
+        println!("  {k}: mu {:>8.4}  sigma {:>8.4}", g.mu(), g.sigma());
+    }
+
+    println!("\n== combinations the data actually pins ==");
+    for (label, terms) in [
+        ("p0 + h0 (a round)", vec![(&"p0", 1.0), (&"h0", 1.0)]),
+        (
+            "p0 - p1 (rank two players)",
+            vec![(&"p0", 1.0), (&"p1", -1.0)],
+        ),
+        ("p0 - p2", vec![(&"p0", 1.0), (&"p2", -1.0)]),
+        (
+            "h0 - h1 (rank two holes)",
+            vec![(&"h0", 1.0), (&"h1", -1.0)],
+        ),
+    ] {
+        let joint = h.posterior_of(&terms).unwrap();
+        // what a consumer gets today by adding marginals
+        let naive: f64 = terms
+            .iter()
+            .map(|(k, c)| c * c * h.current_skill(*k).unwrap().sigma().powi(2))
+            .sum::<f64>()
+            .sqrt();
+        println!(
+            "  {label:<28} exact sigma {:>7.4}   adding marginals {:>7.4}   {:>5.2}x over",
+            joint.sigma(),
+            naive,
+            naive / joint.sigma()
+        );
+
+        let ratio = naive / joint.sigma();
+        if label.contains('+') {
+            assert!(
+                ratio < 0.5,
+                "{label}: adding marginals should be badly OVERconfident for a \
+                 sum, got {ratio:.3}x"
+            );
+        } else {
+            assert!(
+                (0.8..1.25).contains(&ratio),
+                "{label}: adding marginals happens to be close for a difference, \
+                 got {ratio:.3}x"
+            );
+        }
+    }
+
+    // A single node in an additive model is weakly identified: its exact
+    // posterior is far wider than message passing reports, because the level it
+    // shares with its partners is pinned only by the prior.
+    for k in players.iter().chain(holes.iter()) {
+        let bp = h.current_skill(k).unwrap().sigma();
+        let exact = h.posterior_of(&[(k, 1.0)]).unwrap().sigma();
+        assert!(
+            exact > 3.0 * bp,
+            "{k}: exact marginal {exact} should be much wider than the reported \
+             {bp} in an additive model"
+        );
+    }
+}

tests/api_shape.rs

@@ -65,7 +65,7 @@ fn add_events_draw() {
         outcome: Outcome::draw(2),
     }];
     h.add_events(events).unwrap();
-    h.converge().unwrap();
+    let _ = h.converge().unwrap();
 }
 
 #[test]
@@ -123,7 +123,7 @@ fn fluent_event_builder_winner_convenience() {
         .winner(0)
         .commit()
         .unwrap();
-    h.converge().unwrap();
+    let _ = h.converge().unwrap();
 }
 
 #[test]
@@ -141,7 +141,7 @@ fn fluent_event_builder_draw() {
         .draw()
         .commit()
         .unwrap();
-    h.converge().unwrap();
+    let _ = h.converge().unwrap();
 }
 
 #[test]
@@ -155,7 +155,7 @@ fn current_skill_and_learning_curve() {
         .build();
     h.record_winner(&"a", &"b", 1).unwrap();
     h.record_winner(&"a", &"b", 2).unwrap();
-    h.converge().unwrap();
+    let _ = h.converge().unwrap();
 
     let a = h.current_skill(&"a").unwrap();
     assert!(a.mu() > 25.0);
@@ -201,9 +201,9 @@ fn predict_quality_two_teams() {
         .p_draw(0.0)
         .build();
     h.record_winner(&"a", &"b", 1).unwrap();
-    h.converge().unwrap();
+    let _ = h.converge().unwrap();
 
-    let q = h.predict_quality(&[&[&"a"], &[&"b"]]);
+    let q = h.predict_quality(&[&[&"a"], &[&"b"]]).unwrap();
     assert!(q > 0.0 && q <= 1.0);
 }
 
@@ -217,12 +217,16 @@ fn predict_outcome_two_teams_sums_to_one() {
         .p_draw(0.0)
         .build();
     h.record_winner(&"a", &"b", 1).unwrap();
-    h.converge().unwrap();
+    let _ = h.converge().unwrap();
 
-    let p = h.predict_outcome(&[&[&"a"], &[&"b"]]);
+    let p = h.predict_outcome(&[&[&"a"], &[&"b"]]).unwrap();
-    assert_eq!(p.len(), 2);
+    let wins = p.win_probabilities();
-    assert!((p[0] + p[1] - 1.0).abs() < 1e-9);
+    assert_eq!(wins.len(), 2);
-    assert!(p[0] > p[1]);
+    // With p_draw == 0 there is no draw outcome, so the two win
+    // probabilities are the whole space.
+    assert!((p.total() - 1.0).abs() < 1e-9, "total = {}", p.total());
+    assert!((wins[0] + wins[1] - 1.0).abs() < 1e-9);
+    assert!(wins[0] > wins[1]);
 }
 
 #[test]
@@ -241,9 +245,52 @@ fn fluent_event_builder_scores() {
         .scores([12.0, 4.0])
         .commit()
         .unwrap();
-    h.converge().unwrap();
+    let _ = h.converge().unwrap();
 
     let a = h.current_skill(&"alice").unwrap();
     let b = h.current_skill(&"bob").unwrap();
     assert!(a.mu() > b.mu());
 }
+
+/// Every field of `ConvergenceReport` must carry real information.
+///
+/// `slices_skipped` was public, hardcoded to `0`, and reported a plausible
+/// value for a feature that never existed — the same shape as the inert
+/// `online` flag in #19. It was removed in #33. This pins the remaining fields
+/// so the next always-constant member has to survive an assertion rather than
+/// just a reviewer's attention.
+#[test]
+fn every_convergence_report_field_is_populated() {
+    let mut h = History::builder().build();
+
+    for time in 1..=6i64 {
+        h.record_winner(&"a", &"b", time).unwrap();
+    }
+
+    let report = h.converge().unwrap();
+
+    assert!(
+        report.iterations > 0,
+        "iterations is zero on a real converge"
+    );
+
+    assert!(report.converged, "fixture must converge");
+
+    assert!(
+        report.final_step.0.is_finite() && report.final_step.1.is_finite(),
+        "final_step is not finite: {:?}",
+        report.final_step
+    );
+
+    assert!(
+        report.log_evidence.is_finite() && report.log_evidence < 0.0,
+        "log_evidence is not a finite negative log probability: {}",
+        report.log_evidence
+    );
+
+    assert_eq!(
+        report.per_iteration_time.len(),
+        report.iterations,
+        "per_iteration_time must carry one duration per iteration"
+    );
+}

tests/common/mod.rs

@@ -0,0 +1,38 @@
+//! Helpers shared across the integration suites.
+//!
+//! Each integration file is its own binary, so `mod common;` compiles a copy
+//! per suite. Anything unused in a given suite would warn, hence the
+//! `#![allow(dead_code)]`.
+
+#![allow(dead_code)]
+
+use trueskill_tt::Gaussian;
+
+/// A posterior must be finite with a strictly positive sigma.
+///
+/// A non-finite posterior is the failure mode this crate is most prone to —
+/// EP breaking down produces NaN rather than an error — and a zero or negative
+/// sigma means the precision went non-positive, which `Gaussian::sigma` reports
+/// as improper rather than trapping.
+pub fn assert_finite(g: Gaussian, what: &str) {
+    assert!(
+        g.mu().is_finite(),
+        "{what}: mu is not finite (mu={}, sigma={})",
+        g.mu(),
+        g.sigma()
+    );
+
+    assert!(
+        g.sigma().is_finite() && g.sigma() > 0.0,
+        "{what}: sigma must be finite and positive (mu={}, sigma={})",
+        g.mu(),
+        g.sigma()
+    );
+}
+
+/// Every point on every learning curve must be finite.
+pub fn assert_curve_finite(curve: &[(i64, Gaussian)], who: &str) {
+    for (time, g) in curve {
+        assert_finite(*g, &format!("{who} at t={time}"));
+    }
+}

tests/competitor_config.rs

@@ -0,0 +1,222 @@
+//! `Member::with_prior` / `with_drift_scale` — competitor configuration.
+//!
+//! Both were previously consumed only on the branch that *creates* a
+//! competitor, so configuration supplied for a key the history already knew was
+//! dropped with no error. `with_prior` had no coverage in this directory at
+//! all, which is how that survived.
+
+use smallvec::smallvec;
+use trueskill_tt::{
+    ConvergenceOptions, Event, Gaussian, History, InferenceError, Member, Outcome, Team,
+};
+
+const CONVERGENCE: ConvergenceOptions = ConvergenceOptions {
+    max_iter: 2_000,
+    epsilon: 1e-12,
+    alpha: 1.0,
+};
+
+fn history() -> History {
+    History::builder()
+        .mu(25.0)
+        .sigma(25.0 / 3.0)
+        .beta(25.0 / 6.0)
+        .p_draw(0.0)
+        .convergence(CONVERGENCE)
+        .build()
+}
+
+/// One event, optionally configuring `a`.
+fn bout(
+    a: &'static str,
+    b: &'static str,
+    time: i64,
+    prior: Option<Gaussian>,
+    scale: Option<f64>,
+) -> Event<i64, &'static str> {
+    let mut member = Member::new(a);
+    if let Some(p) = prior {
+        member = member.with_prior(p);
+    }
+    if let Some(s) = scale {
+        member = member.with_drift_scale(s);
+    }
+
+    Event {
+        time,
+        teams: smallvec![
+            Team::with_members([member]),
+            Team::with_members([Member::new(b)]),
+        ],
+        outcome: Outcome::winner(0, 2),
+    }
+}
+
+fn skill_of(h: &History, key: &str) -> Gaussian {
+    h.current_skill(&key).expect("key in history")
+}
+
+/// Baseline: the mechanism works at all on a competitor's first appearance.
+#[test]
+fn a_prior_applies_to_a_new_competitor() {
+    let seeded = Gaussian::from_ms(40.0, 1.0);
+
+    let mut with = history();
+    with.add_events(vec![bout("a", "b", 0, Some(seeded), None)])
+        .unwrap();
+    let _ = with.converge().unwrap();
+
+    let mut without = history();
+    without
+        .add_events(vec![bout("a", "b", 0, None, None)])
+        .unwrap();
+    let _ = without.converge().unwrap();
+
+    assert!(
+        (skill_of(&with, "a").mu() - skill_of(&without, "a").mu()).abs() > 1.0,
+        "a seeded prior should move the fit"
+    );
+}
+
+/// The defect in #10: a prior supplied for a competitor the history already
+/// knows was silently discarded, and the caller got output computed from the
+/// default prior with no indication anything had been dropped.
+#[test]
+fn a_prior_applies_to_a_competitor_the_history_already_knows() {
+    let seeded = Gaussian::from_ms(40.0, 1.0);
+
+    let mut late = history();
+    late.add_events(vec![bout("a", "b", 0, None, None)])
+        .unwrap();
+    // "a" now exists. Configuring it here used to do nothing whatsoever.
+    late.add_events(vec![bout("a", "b", 1, Some(seeded), None)])
+        .unwrap();
+    let _ = late.converge().unwrap();
+
+    let mut never = history();
+    never
+        .add_events(vec![
+            bout("a", "b", 0, None, None),
+            bout("a", "b", 1, None, None),
+        ])
+        .unwrap();
+    let _ = never.converge().unwrap();
+
+    assert!(
+        (skill_of(&late, "a").mu() - skill_of(&never, "a").mu()).abs() > 1.0,
+        "a late prior must not be silently dropped: {} vs {}",
+        skill_of(&late, "a").mu(),
+        skill_of(&never, "a").mu()
+    );
+}
+
+/// Configuration is competitor-scoped, not event-scoped, and `converge` refits
+/// from competitor state — so seeding late reaches the same fit as seeding from
+/// the start. This is the documented scope, asserted rather than assumed.
+#[test]
+fn a_prior_is_whole_history_scoped_not_per_event() {
+    let seeded = Gaussian::from_ms(40.0, 1.0);
+
+    let mut late = history();
+    late.add_events(vec![bout("a", "b", 0, None, None)])
+        .unwrap();
+    late.add_events(vec![bout("a", "b", 1, Some(seeded), None)])
+        .unwrap();
+    let _ = late.converge().unwrap();
+
+    let mut early = history();
+    early
+        .add_events(vec![
+            bout("a", "b", 0, Some(seeded), None),
+            bout("a", "b", 1, Some(seeded), None),
+        ])
+        .unwrap();
+    let _ = early.converge().unwrap();
+
+    let (l, e) = (skill_of(&late, "a"), skill_of(&early, "a"));
+    assert!(
+        (l.mu() - e.mu()).abs() < 1e-9 && (l.sigma() - e.sigma()).abs() < 1e-9,
+        "late seeding should refit the whole history: {l:?} vs {e:?}"
+    );
+}
+
+#[test]
+fn repeating_the_same_prior_is_inert() {
+    let seeded = Gaussian::from_ms(40.0, 1.0);
+
+    let mut once = history();
+    once.add_events(vec![
+        bout("a", "b", 0, Some(seeded), None),
+        bout("a", "b", 1, None, None),
+    ])
+    .unwrap();
+    let _ = once.converge().unwrap();
+
+    let mut every_time = history();
+    every_time
+        .add_events(vec![
+            bout("a", "b", 0, Some(seeded), None),
+            bout("a", "b", 1, Some(seeded), None),
+        ])
+        .unwrap();
+    let _ = every_time.converge().unwrap();
+
+    let (o, e) = (skill_of(&once, "a"), skill_of(&every_time, "a"));
+    assert!(
+        (o.mu() - e.mu()).abs() < 1e-12 && (o.sigma() - e.sigma()).abs() < 1e-12,
+        "declaring the same prior repeatedly changed the fit: {o:?} vs {e:?}"
+    );
+}
+
+/// Events within a batch have no order, so two different values for one
+/// competitor have no well-defined winner. Rejecting is what keeps the answer
+/// independent of iteration order.
+#[test]
+fn a_batch_declaring_two_different_priors_is_rejected() {
+    let mut h = history();
+    let err = h
+        .add_events(vec![
+            bout("a", "b", 0, Some(Gaussian::from_ms(40.0, 1.0)), None),
+            bout("a", "b", 1, Some(Gaussian::from_ms(10.0, 1.0)), None),
+        ])
+        .expect_err("two different priors for one competitor in one batch");
+
+    assert!(
+        matches!(
+            err,
+            InferenceError::ConflictingCompetitorConfig { field: "prior", .. }
+        ),
+        "got {err:?}"
+    );
+}
+
+/// A member setting only `drift_scale` must not also assert the default prior,
+/// or it would silently undo a prior seeded earlier. This is why the collected
+/// configuration tracks each field separately rather than a merged `Rating`.
+#[test]
+fn setting_one_field_late_leaves_the_other_alone() {
+    let seeded = Gaussian::from_ms(40.0, 1.0);
+
+    let mut h = history();
+    h.add_events(vec![bout("a", "b", 0, Some(seeded), None)])
+        .unwrap();
+    // Only the scale this time — the prior above must survive.
+    h.add_events(vec![bout("a", "b", 1, None, Some(0.5))])
+        .unwrap();
+    let _ = h.converge().unwrap();
+
+    let mut both_upfront = history();
+    both_upfront
+        .add_events(vec![
+            bout("a", "b", 0, Some(seeded), Some(0.5)),
+            bout("a", "b", 1, None, None),
+        ])
+        .unwrap();
+    let _ = both_upfront.converge().unwrap();
+
+    let (a, b) = (skill_of(&h, "a"), skill_of(&both_upfront, "a"));
+    assert!(
+        (a.mu() - b.mu()).abs() < 1e-9 && (a.sigma() - b.sigma()).abs() < 1e-9,
+        "setting drift_scale late clobbered the earlier prior: {a:?} vs {b:?}"
+    );
+}

tests/convergence_strictness.rs

@@ -0,0 +1,152 @@
+//! Stopping short of convergence is an error, not a flag on a success.
+//!
+//! A fit that hits `max_iter` is wrong by a little: every rating is finite,
+//! the ordering looks sensible, and nothing in the numbers says they were
+//! still moving. When that was `Ok` with `converged: false`, detecting it was
+//! opt-in and `let _ = h.converge()` was the natural way to opt out — which is
+//! how a real defect once hid in this crate's own suite.
+
+use smallvec::smallvec;
+use trueskill_tt::{
+    ConstantDrift, ConvergenceOptions, Event, History, InferenceError, Member, Outcome, Team,
+};
+
+type H = History<i64, ConstantDrift, trueskill_tt::NullObserver, &'static str>;
+
+fn duel(a: &'static str, b: &'static str, t: i64) -> Event<i64, &'static str> {
+    Event {
+        time: t,
+        teams: smallvec![
+            Team::with_members([Member::new(a)]),
+            Team::with_members([Member::new(b)]),
+        ],
+        outcome: Outcome::scores([3.0, 1.0]),
+    }
+}
+
+fn capped(max_iter: usize) -> H {
+    History::builder()
+        .mu(0.0)
+        .sigma(6.0)
+        .beta(1.0)
+        .score_sigma(2.0)
+        .drift(ConstantDrift(0.5))
+        .convergence(ConvergenceOptions {
+            max_iter,
+            epsilon: 1e-13,
+            alpha: 1.0,
+        })
+        .build()
+}
+
+fn fill(h: &mut H) {
+    h.add_events((1..=6).map(|t| duel("a", "b", t)).collect::<Vec<_>>())
+        .unwrap();
+}
+
+#[test]
+fn hitting_the_cap_is_an_error() {
+    let mut h = capped(1);
+    fill(&mut h);
+    let err = h.converge().unwrap_err();
+    match err {
+        InferenceError::NotConverged {
+            iterations,
+            final_step,
+            epsilon,
+        } => {
+            assert_eq!(iterations, 1);
+            assert!(
+                final_step.0 > epsilon || final_step.1 > epsilon,
+                "{final_step:?}"
+            );
+        }
+        other => panic!("expected NotConverged, got {other:?}"),
+    }
+}
+
+/// The message has to name what to do about it, since the fit looks fine.
+#[test]
+fn the_error_says_how_to_fix_it() {
+    let mut h = capped(1);
+    fill(&mut h);
+    let text = h.converge().unwrap_err().to_string();
+    assert!(text.contains("did not converge in 1 iterations"), "{text}");
+    assert!(text.contains("max_iter"), "{text}");
+    assert!(text.contains("alpha"), "{text}");
+}
+
+/// The escape hatch: a deliberately capped fit is still reachable.
+#[test]
+fn converge_partial_returns_the_short_fit() {
+    let mut h = capped(1);
+    fill(&mut h);
+    let report = h.converge_partial().unwrap();
+    assert_eq!(report.iterations, 1);
+    assert!(!report.converged);
+    assert!(h.current_skill(&"a").is_some());
+}
+
+/// Both agree when the fit does converge, so the strict path costs nothing.
+#[test]
+fn the_two_agree_on_a_converged_fit() {
+    let mut strict = capped(20_000);
+    fill(&mut strict);
+    let a = strict.converge().unwrap();
+
+    let mut partial = capped(20_000);
+    fill(&mut partial);
+    let b = partial.converge_partial().unwrap();
+
+    assert!(a.converged && b.converged);
+    assert_eq!(a.iterations, b.iterations);
+    assert_eq!(a.final_step, b.final_step);
+}
+
+/// The default cap must be high enough that an ordinary history clears it.
+/// At the old value of 30 this history stopped short and said nothing.
+#[test]
+fn the_default_cap_clears_an_ordinary_history() {
+    let mut h: History<i64, ConstantDrift, _, String> = History::builder_with_key()
+        .mu(0.0)
+        .sigma(6.0)
+        .beta(1.0)
+        .score_sigma(2.0)
+        .drift(ConstantDrift(0.05))
+        .build();
+
+    let mut events = Vec::new();
+    for t in 0..20i64 {
+        for j in 0..8usize {
+            let k = (t as usize) * 8 + j;
+            events.push(Event {
+                time: t,
+                teams: smallvec![
+                    Team::with_members([Member::new(format!("p{}", k % 100))]),
+                    Team::with_members([Member::new(format!("p{}", (k + 37) % 100))]),
+                ],
+                outcome: Outcome::scores([3.0, 1.0]),
+            });
+        }
+    }
+    h.add_events(events).unwrap();
+
+    let report = h
+        .converge()
+        .expect("an ordinary history must converge by default");
+    assert!(
+        report.iterations > 30,
+        "needed {} sweeps",
+        report.iterations
+    );
+    assert!(report.iterations < trueskill_tt::ITERATIONS);
+}
+
+/// An empty history converges trivially rather than erroring.
+#[test]
+fn an_empty_history_converges() {
+    let mut h = capped(1);
+    let report = h.converge().unwrap();
+    assert!(report.converged);
+    assert_eq!(report.iterations, 0);
+}

tests/degenerate_inputs.rs

@@ -3,6 +3,9 @@
 //! These run in both debug and release: the defects they pin were all
 //! guarded only by `debug_assert!`, so a debug-only suite never saw them.
 
+mod common;
+
+use common::assert_finite;
 use trueskill_tt::{
     ConstantDrift, ConvergenceOptions, Game, GameOptions, Gaussian, History, InferenceError,
     NullObserver, Outcome, Rating,
@@ -18,15 +21,6 @@ fn rating() -> R {
     )
 }
 
-fn assert_finite(g: Gaussian, what: &str) {
-    assert!(
-        g.mu().is_finite() && g.sigma().is_finite(),
-        "{what} must be finite, got mu={} sigma={}",
-        g.mu(),
-        g.sigma()
-    );
-}
-
 #[test]
 fn record_draw_without_draw_probability_is_rejected() {
     let mut h = History::default();
@@ -141,6 +135,55 @@ fn converge_on_an_empty_history_with_owned_keys() {
     assert!(report.converged);
 }
 
+/// A weights/team length mismatch used to be a `debug_assert!`, so release
+/// builds ingested the event with the weights silently unapplied. This file's
+/// CI job runs in release too, which is the point of pinning it here.
+#[test]
+fn event_builder_rejects_a_weights_length_mismatch() {
+    let mut h = History::default();
+
+    let err = h
+        .event(1)
+        .team(["a"])
+        .weights([1.0, 2.0])
+        .team(["b"])
+        .winner(0)
+        .commit()
+        .unwrap_err();
+
+    assert!(
+        matches!(
+            err,
+            InferenceError::MismatchedShape {
+                kind: "weights",
+                expected: 1,
+                got: 2,
+            }
+        ),
+        "expected a weights MismatchedShape, got {err:?}"
+    );
+}
+
+/// The mismatch must not be applied even partially — a half-weighted team
+/// reaching the history would be worse than the error.
+#[test]
+fn event_builder_weights_mismatch_leaves_the_history_untouched() {
+    let mut h = History::default();
+
+    // Two teams, so ingestion would otherwise succeed. A one-team event is
+    // rejected as `NotEnoughTeams` before the weights are ever examined, so
+    // building this with one team would pass vacuously.
+    let _ = h
+        .event(1)
+        .team(["a"])
+        .weights([1.0, 2.0])
+        .team(["b"])
+        .winner(0)
+        .commit();
+
+    assert!(h.learning_curve("a").is_empty());
+}
+
 #[test]
 fn empty_event_stream_then_converge() {
     let mut h = History::default();
@@ -270,3 +313,112 @@ fn empty_history_has_no_filtered_estimates() {
 
     assert!(history.filtered_learning_curve("nobody").is_empty());
 }
+
+// --- Boundary inputs (#26) ----------------------------------------------
+
+fn tight() -> ConvergenceOptions {
+    ConvergenceOptions {
+        max_iter: 2_000,
+        epsilon: 1e-12,
+        ..ConvergenceOptions::default()
+    }
+}
+
+fn assert_curve_finite(h: &History, keys: &[&str], what: &str) {
+    for key in keys {
+        for (time, g) in h.learning_curve(*key) {
+            assert!(
+                g.mu().is_finite() && g.sigma().is_finite(),
+                "{what}: non-finite posterior for {key} at t={time} (mu={} sigma={})",
+                g.mu(),
+                g.sigma()
+            );
+        }
+    }
+}
+
+/// A zero weight reaches `(m - performance.exclude(..)) * (1.0 / w)`, i.e. a
+/// division by zero. The commit is accepted today, so this pins that the
+/// resulting posterior is still finite rather than quietly NaN.
+#[test]
+fn zero_weight_does_not_produce_a_non_finite_posterior() {
+    let mut h = History::builder().build();
+
+    h.event(1)
+        .team(["a"])
+        .weights([0.0])
+        .team(["b"])
+        .winner(0)
+        .commit()
+        .expect("a zero weight is accepted today; update this test if that changes");
+
+    let _ = h.converge().unwrap();
+
+    assert_curve_finite(&h, &["a", "b"], "zero weight");
+}
+
+#[test]
+fn negative_weight_does_not_produce_a_non_finite_posterior() {
+    let mut h = History::builder().build();
+
+    h.event(1)
+        .team(["a"])
+        .weights([-1.0])
+        .team(["b"])
+        .winner(0)
+        .commit()
+        .expect("a negative weight is accepted today; update this test if that changes");
+
+    let _ = h.converge().unwrap();
+
+    assert_curve_finite(&h, &["a", "b"], "negative weight");
+}
+
+/// Events supplied newest-first must land in the same slices as oldest-first:
+/// ingestion sorts by time rather than trusting arrival order.
+#[test]
+fn out_of_order_timestamps_converge_to_the_same_answer() {
+    fn build(descending: bool) -> History {
+        let mut h = History::builder().convergence(tight()).build();
+
+        let mut times: Vec<i64> = (1..=6).collect();
+        if descending {
+            times.reverse();
+        }
+
+        for time in times {
+            h.record_winner(&"a", &"b", time).unwrap();
+        }
+
+        let _ = h.converge().unwrap();
+        h
+    }
+
+    let ascending = build(false);
+    let descending = build(true);
+
+    let one = ascending.current_skill("a").unwrap();
+    let other = descending.current_skill("a").unwrap();
+
+    assert!(
+        (one.mu() - other.mu()).abs() < 1e-8 && (one.sigma() - other.sigma()).abs() < 1e-8,
+        "arrival order changed the answer: ascending mu={} sigma={}, descending mu={} sigma={}",
+        one.mu(),
+        one.sigma(),
+        other.mu(),
+        other.sigma()
+    );
+}
+
+#[test]
+fn extreme_beta_and_sigma_stay_finite() {
+    for (beta, sigma) in [(1e-6, 1e-6), (1e6, 1e6), (1e-6, 1e6), (1e6, 1e-6)] {
+        let mut h = History::builder().beta(beta).sigma(sigma).build();
+
+        h.record_winner(&"a", &"b", 1).unwrap();
+        h.record_winner(&"a", &"b", 2).unwrap();
+        let _ = h.converge().unwrap();
+
+        assert_curve_finite(&h, &["a", "b"], &format!("beta={beta} sigma={sigma}"));
+    }
+}

tests/determinism.rs

@@ -47,7 +47,7 @@ fn build_and_converge(seed: u64) -> Vec<(i64, trueskill_tt::Gaussian)> {
         });
     }
     h.add_events(events).unwrap();
-    h.converge().unwrap();
+    let _ = h.converge().unwrap();
     // Sample one competitor's curve for the comparison.
     h.learning_curve("p0")
 }

tests/drift_scale.rs

@@ -0,0 +1,499 @@
+//! Per-competitor drift scaling via `Member::with_drift_scale`.
+//!
+//! The scale multiplies the *variance* the history's `Drift` contributes for
+//! that competitor, so `scale` is in the same units as `gamma`:
+//! `ConstantDrift(g)` at `scale = s` behaves as `ConstantDrift(g * s)` would.
+//! `scale = 0.0` pins a competitor still — an anchor, a rating floor, a course
+//! difficulty — while everyone around them keeps drifting.
+
+use smallvec::smallvec;
+use trueskill_tt::{
+    ConstantDrift, ConvergenceOptions, Event, Gaussian, History, InferenceError, Member,
+    NullObserver, Outcome, Team,
+};
+
+type Fit = History<i64, ConstantDrift, NullObserver, &'static str>;
+
+const CONVERGENCE: ConvergenceOptions = ConvergenceOptions {
+    max_iter: 64,
+    epsilon: 1e-9,
+    alpha: 1.0,
+};
+
+/// Two events separated by a long gap, so drift has room to matter.
+fn distant_pair(anchor_scale: Option<f64>) -> Vec<Event<i64, &'static str>> {
+    let anchor = |s: Option<f64>| match s {
+        Some(scale) => Member::new("anchor").with_drift_scale(scale),
+        None => Member::new("anchor"),
+    };
+
+    vec![
+        Event {
+            time: 0,
+            teams: smallvec![
+                Team::with_members([anchor(anchor_scale)]),
+                Team::with_members([Member::new("player")]),
+            ],
+            outcome: Outcome::winner(0, 2),
+        },
+        Event {
+            time: 1000,
+            teams: smallvec![
+                Team::with_members([anchor(anchor_scale)]),
+                Team::with_members([Member::new("player")]),
+            ],
+            outcome: Outcome::winner(1, 2),
+        },
+    ]
+}
+
+fn fit(events: Vec<Event<i64, &'static str>>, gamma: f64) -> Fit {
+    let mut h = History::builder()
+        .mu(25.0)
+        .sigma(25.0 / 3.0)
+        .beta(25.0 / 6.0)
+        .p_draw(0.0)
+        .drift(ConstantDrift(gamma))
+        .convergence(CONVERGENCE)
+        .build();
+
+    h.add_events(events).unwrap();
+    let _ = h.converge().unwrap();
+    h
+}
+
+fn curve(h: &Fit, key: &str) -> Vec<(i64, Gaussian)> {
+    let mut c = h.learning_curves().remove(key).expect("key in curves");
+    c.sort_by_key(|(t, _)| *t);
+    c
+}
+
+/// A competitor at `scale = 0.0` is one latent skill observed twice, so the
+/// posterior is the same distribution at both times — and strictly tighter
+/// than the same competitor left to drift.
+#[test]
+fn zero_scale_pins_a_competitor_still() {
+    let pinned = fit(distant_pair(Some(0.0)), 25.0 / 300.0);
+    let drifting = fit(distant_pair(None), 25.0 / 300.0);
+
+    let pinned_curve = curve(&pinned, "anchor");
+    assert_eq!(pinned_curve.len(), 2);
+
+    let (t0, first) = pinned_curve[0];
+    let (t1, second) = pinned_curve[1];
+    assert_eq!((t0, t1), (0, 1000));
+
+    assert!(
+        (first.sigma() - second.sigma()).abs() < 1e-9,
+        "a pinned competitor's uncertainty must not move between t=0 and t=1000: \
+         {} vs {}",
+        first.sigma(),
+        second.sigma()
+    );
+    assert!(
+        (first.mu() - second.mu()).abs() < 1e-9,
+        "a pinned competitor's mean must not move: {} vs {}",
+        first.mu(),
+        second.mu()
+    );
+
+    let drifting_curve = curve(&drifting, "anchor");
+    assert!(
+        drifting_curve[0].1.sigma() > first.sigma() + 1e-6,
+        "drift must leave the anchor less certain than pinning does: {} vs {}",
+        drifting_curve[0].1.sigma(),
+        first.sigma()
+    );
+}
+
+/// The scale is composable with `gamma`: scaling every competitor by `s` is
+/// exactly the same fit as scaling the history's drift by `s`.
+#[test]
+fn scale_is_equivalent_to_scaling_gamma() {
+    let scaled: Vec<Event<i64, &'static str>> = vec![
+        Event {
+            time: 0,
+            teams: smallvec![
+                Team::with_members([Member::new("a").with_drift_scale(0.5)]),
+                Team::with_members([Member::new("b").with_drift_scale(0.5)]),
+            ],
+            outcome: Outcome::winner(0, 2),
+        },
+        Event {
+            time: 400,
+            teams: smallvec![
+                Team::with_members([Member::new("b").with_drift_scale(0.5)]),
+                Team::with_members([Member::new("a").with_drift_scale(0.5)]),
+            ],
+            outcome: Outcome::winner(0, 2),
+        },
+    ];
+
+    let plain: Vec<Event<i64, &'static str>> = vec![
+        Event {
+            time: 0,
+            teams: smallvec![
+                Team::with_members([Member::new("a")]),
+                Team::with_members([Member::new("b")]),
+            ],
+            outcome: Outcome::winner(0, 2),
+        },
+        Event {
+            time: 400,
+            teams: smallvec![
+                Team::with_members([Member::new("b")]),
+                Team::with_members([Member::new("a")]),
+            ],
+            outcome: Outcome::winner(0, 2),
+        },
+    ];
+
+    let by_scale = fit(scaled, 0.3);
+    let by_gamma = fit(plain, 0.15);
+
+    for key in ["a", "b"] {
+        let lhs = curve(&by_scale, key);
+        let rhs = curve(&by_gamma, key);
+        assert_eq!(lhs.len(), rhs.len());
+
+        for ((t_l, g_l), (t_r, g_r)) in lhs.iter().zip(rhs.iter()) {
+            assert_eq!(t_l, t_r);
+            assert!(
+                (g_l.mu() - g_r.mu()).abs() < 1e-9 && (g_l.sigma() - g_r.sigma()).abs() < 1e-9,
+                "ConstantDrift(0.3) at scale 0.5 must equal ConstantDrift(0.15) for {key} at \
+                 t={t_l}: ({}, {}) vs ({}, {})",
+                g_l.mu(),
+                g_l.sigma(),
+                g_r.mu(),
+                g_r.sigma()
+            );
+        }
+    }
+}
+
+/// `None` means 1.0: an explicit unit scale changes nothing.
+#[test]
+fn unset_scale_matches_an_explicit_unit_scale() {
+    let implicit = fit(distant_pair(None), 25.0 / 300.0);
+    let explicit = fit(distant_pair(Some(1.0)), 25.0 / 300.0);
+
+    for key in ["anchor", "player"] {
+        let lhs = curve(&implicit, key);
+        let rhs = curve(&explicit, key);
+        assert_eq!(lhs.len(), rhs.len());
+
+        for ((t_l, g_l), (t_r, g_r)) in lhs.iter().zip(rhs.iter()) {
+            assert_eq!(t_l, t_r);
+            assert_eq!(
+                (g_l.mu(), g_l.sigma()),
+                (g_r.mu(), g_r.sigma()),
+                "an explicit scale of 1.0 must be bit-identical to leaving it unset, \
+                 for {key} at t={t_l}"
+            );
+        }
+    }
+}
+
+/// The use case from the issue: a static difficulty alongside drifting players,
+/// in one graph. The anchor must hold still without absorbing drift through its
+/// neighbours, and everything must stay finite.
+#[test]
+fn mixed_static_and_drifting_graph_converges() {
+    let mut events: Vec<Event<i64, &'static str>> = Vec::new();
+    let players = ["p0", "p1", "p2"];
+
+    for (i, p) in players.iter().cycle().take(9).enumerate() {
+        events.push(Event {
+            time: (i as i64) * 100,
+            teams: smallvec![
+                Team::with_members([Member::new(*p)]),
+                Team::with_members([Member::new("layout").with_drift_scale(0.0)]),
+            ],
+            outcome: Outcome::winner((i % 2) as u32, 2),
+        });
+    }
+
+    let mut h = History::builder()
+        .mu(25.0)
+        .sigma(25.0 / 3.0)
+        .beta(25.0 / 6.0)
+        .p_draw(0.0)
+        .drift(ConstantDrift(25.0 / 300.0))
+        .convergence(CONVERGENCE)
+        .build();
+
+    h.add_events(events).unwrap();
+    let report = h.converge().unwrap();
+    assert!(report.converged, "mixed graph must converge: {report:?}");
+
+    let curves = h.learning_curves();
+    for (key, points) in &curves {
+        for (t, g) in points {
+            assert!(
+                g.mu().is_finite() && g.sigma().is_finite() && g.sigma() > 0.0,
+                "{key} at t={t} is not a usable posterior: mu={}, sigma={}",
+                g.mu(),
+                g.sigma()
+            );
+        }
+    }
+
+    let layout = curve(&h, "layout");
+    assert_eq!(layout.len(), 9);
+    let (_, first) = layout[0];
+    for (t, g) in &layout {
+        assert!(
+            (g.sigma() - first.sigma()).abs() < 1e-9,
+            "a static layout must not accumulate uncertainty; t={t} has sigma {} vs {}",
+            g.sigma(),
+            first.sigma()
+        );
+    }
+
+    let p0 = curve(&h, "p0");
+    assert!(
+        p0.last().unwrap().1.sigma() > 0.0,
+        "a drifting player should still have a proper posterior"
+    );
+}
+
+fn reject(scale: f64) -> InferenceError {
+    let mut h = History::builder()
+        .drift(ConstantDrift(25.0 / 300.0))
+        .build();
+
+    let events: Vec<Event<i64, &'static str>> = vec![Event {
+        time: 0,
+        teams: smallvec![
+            Team::with_members([Member::new("a").with_drift_scale(scale)]),
+            Team::with_members([Member::new("b")]),
+        ],
+        outcome: Outcome::winner(0, 2),
+    }];
+
+    h.add_events(events)
+        .expect_err("an out-of-range drift_scale must be rejected")
+}
+
+#[test]
+fn negative_scale_is_rejected() {
+    assert_eq!(
+        reject(-1.0),
+        InferenceError::InvalidParameter {
+            name: "drift_scale",
+            value: -1.0
+        }
+    );
+}
+
+#[test]
+fn non_finite_scale_is_rejected() {
+    for scale in [f64::NAN, f64::INFINITY, f64::NEG_INFINITY] {
+        assert!(
+            matches!(
+                reject(scale),
+                InferenceError::InvalidParameter {
+                    name: "drift_scale",
+                    ..
+                }
+            ),
+            "a drift_scale of {scale} must be rejected as an invalid parameter"
+        );
+    }
+}
+
+/// The scale must reach the filtering pass too, not just `converge()`.
+/// `filtered_learning_curves` runs its own drift application, so a pinned
+/// competitor has to stay pinned there as well.
+#[test]
+fn zero_scale_pins_a_competitor_in_the_filtered_pass() {
+    let pinned = fit(distant_pair(Some(0.0)), 25.0 / 300.0);
+    let drifting = fit(distant_pair(None), 25.0 / 300.0);
+
+    let filtered = |h: &Fit| -> Vec<(i64, Gaussian)> {
+        let mut c = h
+            .filtered_learning_curves()
+            .remove("anchor")
+            .expect("anchor in filtered curves");
+        c.sort_by_key(|(t, _)| *t);
+        c
+    };
+
+    let pinned_curve = filtered(&pinned);
+    let drifting_curve = filtered(&drifting);
+    assert_eq!(pinned_curve.len(), 2);
+    assert_eq!(drifting_curve.len(), 2);
+
+    assert!(
+        pinned_curve[1].1.sigma() < pinned_curve[0].1.sigma(),
+        "a pinned competitor's filtered uncertainty must shrink with a second \
+         observation, not be re-inflated by drift: {} then {}",
+        pinned_curve[0].1.sigma(),
+        pinned_curve[1].1.sigma()
+    );
+
+    assert!(
+        pinned_curve[1].1.sigma() < drifting_curve[1].1.sigma() - 1e-6,
+        "pinning must leave the filtered estimate tighter than drifting does: \
+         {} vs {}",
+        pinned_curve[1].1.sigma(),
+        drifting_curve[1].1.sigma()
+    );
+}
+
+/// `drift_scale` is competitor configuration, and configuration supplied for a
+/// competitor the history already knows is now *applied* rather than dropped.
+///
+/// This test previously asserted the opposite. It was written as a deliberate
+/// change-detector — "moving the capture would be a visible break, not a silent
+/// one" — and that is exactly what happened: the capture moved, and the
+/// assertion inverted rather than being deleted.
+///
+/// Because configuration lives on the competitor and `converge` refits from
+/// competitor state, a late pin applies to the *whole* history, not just to
+/// events after it. So a scale set on the second batch must reach the same fit
+/// as one set from the very first event.
+#[test]
+fn drift_scale_applies_when_set_after_first_appearance() {
+    let mut late = History::builder()
+        .mu(25.0)
+        .sigma(25.0 / 3.0)
+        .beta(25.0 / 6.0)
+        .p_draw(0.0)
+        .drift(ConstantDrift(25.0 / 300.0))
+        .convergence(CONVERGENCE)
+        .build();
+
+    // First batch creates "anchor" with the default scale.
+    late.add_events(vec![Event {
+        time: 0,
+        teams: smallvec![
+            Team::with_members([Member::new("anchor")]),
+            Team::with_members([Member::new("player")]),
+        ],
+        outcome: Outcome::winner(0, 2),
+    }])
+    .unwrap();
+
+    // Second batch asks for a pin. No longer too late.
+    late.add_events(vec![Event {
+        time: 1000,
+        teams: smallvec![
+            Team::with_members([Member::new("anchor").with_drift_scale(0.0)]),
+            Team::with_members([Member::new("player")]),
+        ],
+        outcome: Outcome::winner(1, 2),
+    }])
+    .unwrap();
+    let _ = late.converge().unwrap();
+
+    let applied = curve(&late, "anchor");
+    let pinned_from_the_start = curve(&fit(distant_pair(Some(0.0)), 25.0 / 300.0), "anchor");
+    let never_pinned = curve(&fit(distant_pair(None), 25.0 / 300.0), "anchor");
+
+    for ((t_l, g_l), (t_r, g_r)) in applied.iter().zip(pinned_from_the_start.iter()) {
+        assert_eq!(t_l, t_r);
+        assert!(
+            (g_l.sigma() - g_r.sigma()).abs() < 1e-9,
+            "a late pin should refit the whole history: t={t_l}, {} vs {}",
+            g_l.sigma(),
+            g_r.sigma()
+        );
+    }
+
+    // And it must actually have done something.
+    assert!(
+        applied
+            .iter()
+            .zip(never_pinned.iter())
+            .any(|((_, a), (_, b))| (a.sigma() - b.sigma()).abs() > 1e-9),
+        "the pin had no effect at all — the silent drop is back"
+    );
+}
+
+/// Re-declaring the same configuration must be inert. This is the shape a
+/// caller gets when the configuration is a property of the domain — "layouts
+/// are static" — so every ingestion path repeats it on every event.
+///
+/// Both histories see exactly the same events; only how many times the scale
+/// is declared differs.
+#[test]
+fn repeating_the_same_configuration_changes_nothing() {
+    let events = |declare_every_time: bool| {
+        let anchor = |first: bool| {
+            if first || declare_every_time {
+                Member::new("anchor").with_drift_scale(0.0)
+            } else {
+                Member::new("anchor")
+            }
+        };
+        vec![
+            Event {
+                time: 0,
+                teams: smallvec![
+                    Team::with_members([anchor(true)]),
+                    Team::with_members([Member::new("player")]),
+                ],
+                outcome: Outcome::winner(0, 2),
+            },
+            Event {
+                time: 1000,
+                teams: smallvec![
+                    Team::with_members([anchor(false)]),
+                    Team::with_members([Member::new("player")]),
+                ],
+                outcome: Outcome::winner(1, 2),
+            },
+        ]
+    };
+
+    let once = curve(&fit(events(false), 25.0 / 300.0), "anchor");
+    let every_time = curve(&fit(events(true), 25.0 / 300.0), "anchor");
+
+    for ((t_l, a), (t_r, b)) in once.iter().zip(every_time.iter()) {
+        assert_eq!(t_l, t_r);
+        assert!(
+            (a.sigma() - b.sigma()).abs() < 1e-12,
+            "t={t_l}: declaring the same scale repeatedly changed the fit, {} vs {}",
+            a.sigma(),
+            b.sigma()
+        );
+    }
+}
+
+#[test]
+fn a_batch_that_contradicts_itself_is_rejected() {
+    let mut h = History::builder().convergence(CONVERGENCE).build();
+
+    let err = h
+        .add_events(vec![
+            Event {
+                time: 0,
+                teams: smallvec![
+                    Team::with_members([Member::new("anchor").with_drift_scale(0.0)]),
+                    Team::with_members([Member::new("player")]),
+                ],
+                outcome: Outcome::winner(0, 2),
+            },
+            Event {
+                time: 1,
+                teams: smallvec![
+                    Team::with_members([Member::new("anchor").with_drift_scale(1.0)]),
+                    Team::with_members([Member::new("player")]),
+                ],
+                outcome: Outcome::winner(0, 2),
+            },
+        ])
+        .expect_err("two different scales for one competitor in one batch");
+
+    assert!(
+        matches!(
+            err,
+            InferenceError::ConflictingCompetitorConfig {
+                field: "drift_scale",
+                ..
+            }
+        ),
+        "got {err:?}"
+    );
+}

tests/equivalence.rs

@@ -19,7 +19,8 @@ fn ts_rating(mu: f64, sigma: f64, beta: f64, gamma: f64) -> R {
 fn game_1v1_golden_matches_historical() {
     let a = ts_rating(25.0, 25.0 / 3.0, 25.0 / 6.0, 25.0 / 300.0);
     let b = ts_rating(25.0, 25.0 / 3.0, 25.0 / 6.0, 25.0 / 300.0);
-    let (a_post, b_post) = Game::<i64, _>::one_v_one(&a, &b, Outcome::winner(0, 2)).unwrap();
+    let (a_post, b_post) =
+        Game::<i64, _>::one_v_one(&a, &b, Outcome::winner(0, 2), &GameOptions::default()).unwrap();
     // Historical golden from pre-T2 test_1vs1 (team 0 wins):
     assert_ulps_eq!(
         a_post,

tests/event_builder_members.rs

@@ -0,0 +1,193 @@
+//! `EventBuilder::members` must reach exactly what the typed path reaches.
+//!
+//! Before this existed, `EventBuilder` could set weights and nothing else, so
+//! `prior` and `drift_scale` were expressible only through `Event`/`Team`/
+//! `Member` + `add_events`. Which ingestion route a competitor arrived through
+//! decided whether it could be configured at all.
+
+use smallvec::smallvec;
+use trueskill_tt::{
+    ConstantDrift, ConvergenceOptions, Event, Gaussian, History, InferenceError, Member, Outcome,
+    Team,
+};
+
+type H = History<i64, ConstantDrift, trueskill_tt::NullObserver, &'static str>;
+
+fn history() -> H {
+    History::builder()
+        .mu(0.0)
+        .sigma(6.0)
+        .beta(1.0)
+        .score_sigma(2.0)
+        .drift(ConstantDrift(0.5))
+        .convergence(ConvergenceOptions {
+            max_iter: 20_000,
+            epsilon: 1e-13,
+            alpha: 1.0,
+        })
+        .build()
+}
+
+const PRIOR: Gaussian = Gaussian::from_ms(3.0, 1.5);
+
+/// The contract that makes the escape hatch worth having: same configuration,
+/// same fit, bit for bit.
+#[test]
+fn members_matches_the_typed_path_exactly() {
+    let mut typed = history();
+    typed
+        .add_events(vec![Event {
+            time: 1,
+            teams: smallvec![
+                Team::with_members([Member::new("player")]),
+                Team::with_members([Member::new("layout_7")
+                    .with_drift_scale(0.0)
+                    .with_prior(PRIOR)]),
+            ],
+            outcome: Outcome::scores([5.0, 2.0]),
+        }])
+        .unwrap();
+    assert!(typed.converge().unwrap().converged);
+
+    let mut fluent = history();
+    fluent
+        .event(1)
+        .team(["player"])
+        .members([Member::new("layout_7")
+            .with_drift_scale(0.0)
+            .with_prior(PRIOR)])
+        .scores([5.0, 2.0])
+        .commit()
+        .unwrap();
+    assert!(fluent.converge().unwrap().converged);
+
+    for key in ["player", "layout_7"] {
+        let a = typed.current_skill(&key).unwrap();
+        let b = fluent.current_skill(&key).unwrap();
+        assert_eq!(a.pi(), b.pi(), "{key} pi");
+        assert_eq!(a.tau(), b.tau(), "{key} tau");
+    }
+}
+
+/// The configuration has to actually take effect, not merely round-trip: a
+/// competitor pinned with `drift_scale = 0.0` must not move across slices,
+/// where an unpinned one does.
+///
+/// The comparison is against a control rather than against a fixed epsilon.
+/// Pinned marginals are not bit-identical across slices — each slice combines
+/// its own forward and backward messages, so the arithmetic order differs and
+/// the last bit moves. What "pinned" promises is that no drift variance
+/// accumulates, and the control is what makes that measurable.
+#[test]
+fn a_drift_scale_set_through_members_is_applied() {
+    fn spread(h: &H, key: &'static str) -> f64 {
+        let curve = h.learning_curve(&key);
+        assert!(curve.len() >= 2, "{key}: expected several appearances");
+        let (lo, hi) = curve.iter().fold((f64::MAX, f64::MIN), |(lo, hi), (_, g)| {
+            (lo.min(g.sigma()), hi.max(g.sigma()))
+        });
+        (hi - lo) / hi
+    }
+
+    let mut h = history();
+    for t in 1..=4 {
+        h.event(t)
+            .team(["player"])
+            .members([Member::new("pinned").with_drift_scale(0.0)])
+            .scores([5.0, 2.0])
+            .commit()
+            .unwrap();
+        // Same shape, no pinning: the control.
+        h.event(t)
+            .team(["rival"])
+            .team(["drifting"])
+            .scores([5.0, 2.0])
+            .commit()
+            .unwrap();
+    }
+    assert!(h.converge().unwrap().converged);
+
+    let pinned = spread(&h, "pinned");
+    let drifting = spread(&h, "drifting");
+    assert!(pinned < 1e-9, "pinned competitor moved: {pinned:e}");
+    assert!(
+        drifting > 1e-3,
+        "control did not move, so the test proves nothing: {drifting:e}"
+    );
+}
+
+/// `weights` still applies to a team added through `members`, and still
+/// records a mismatch rather than partially applying it.
+#[test]
+fn weights_still_guards_a_members_team() {
+    let mut h = history();
+    let err = h
+        .event(1)
+        .team(["a"])
+        .members([Member::new("b"), Member::new("c")])
+        .weights([1.0])
+        .winner(0)
+        .commit()
+        .unwrap_err();
+    assert!(
+        matches!(
+            err,
+            InferenceError::MismatchedShape {
+                kind: "weights",
+                expected: 2,
+                got: 1
+            }
+        ),
+        "{err:?}"
+    );
+    assert!(h.current_skill(&"b").is_none(), "nothing may reach history");
+}
+
+/// An invalid `drift_scale` surfaces from `commit`, not from a panic and not
+/// silently.
+#[test]
+fn an_invalid_drift_scale_surfaces_from_commit() {
+    for bad in [-1.0, f64::NAN, f64::INFINITY] {
+        let mut h = history();
+        let err = h
+            .event(1)
+            .team(["a"])
+            .members([Member::new("b").with_drift_scale(bad)])
+            .winner(0)
+            .commit()
+            .unwrap_err();
+        assert!(
+            matches!(
+                err,
+                InferenceError::InvalidParameter {
+                    name: "drift_scale",
+                    ..
+                }
+            ),
+            "{bad}: {err:?}"
+        );
+        assert!(h.current_skill(&"b").is_none(), "{bad} reached the history");
+    }
+}
+
+/// `members` and `team` compose in either order.
+#[test]
+fn members_and_team_interleave() {
+    let mut h = history();
+    h.event(1)
+        .members([Member::new("a").with_prior(PRIOR)])
+        .team(["b"])
+        .scores([3.0, 1.0])
+        .commit()
+        .unwrap();
+    h.event(2)
+        .team(["b"])
+        .members([Member::new("c").with_prior(PRIOR)])
+        .scores([2.0, 4.0])
+        .commit()
+        .unwrap();
+    assert!(h.converge().unwrap().converged);
+    for key in ["a", "b", "c"] {
+        assert!(h.current_skill(&key).is_some(), "{key} missing");
+    }
+}

tests/filtered.rs

@@ -47,7 +47,7 @@ fn tight() -> ConvergenceOptions {
 fn filtered_evidence_sits_between_coin_flip_and_batch() {
     let mut history = repeated_winner(5);
 
-    history.converge().unwrap();
+    let _ = history.converge().unwrap();
 
     let coin_flip = 5.0 * 0.5f64.ln();
     let batch = history.log_evidence();
@@ -71,7 +71,7 @@ fn filtered_evidence_sits_between_coin_flip_and_batch() {
 fn filtered_first_point_is_less_certain_than_smoothed() {
     let mut history = repeated_winner(12);
 
-    history.converge().unwrap();
+    let _ = history.converge().unwrap();
 
     let smoothed = history.learning_curve("a");
     let filtered = history.filtered_learning_curve("a");
@@ -121,7 +121,7 @@ fn filtered_first_point_is_less_certain_than_smoothed() {
 fn filtered_curves_plural_agrees_with_singular() {
     let mut history = repeated_winner(4);
 
-    history.converge().unwrap();
+    let _ = history.converge().unwrap();
 
     let curves = history.filtered_learning_curves();
 
@@ -180,7 +180,7 @@ fn single_slice_filtered_matches_smoothed() {
         ])
         .unwrap();
 
-    history.converge().unwrap();
+    let _ = history.converge().unwrap();
 
     let smoothed = history.learning_curve("a");
     let filtered = history.filtered_learning_curve("a");
@@ -223,13 +223,13 @@ fn filtered_curves_do_not_depend_on_ingestion_order() {
 
     let mut batched = History::builder().convergence(tight()).build();
     batched.add_events(all.clone()).unwrap();
-    batched.converge().unwrap();
+    let _ = batched.converge().unwrap();
 
     let mut incremental = History::builder().convergence(tight()).build();
     for event in all {
         incremental.add_events([event]).unwrap();
     }
-    incremental.converge().unwrap();
+    let _ = incremental.converge().unwrap();
 
     let from_batched = batched.filtered_learning_curve("a");
     let from_incremental = incremental.filtered_learning_curve("a");

tests/game.rs

@@ -32,7 +32,8 @@ fn game_ranked_1v1_golden() {
 fn game_one_v_one_shortcut() {
     let a = default_rating();
     let b = default_rating();
-    let (a_post, b_post) = Game::<i64, _>::one_v_one(&a, &b, Outcome::winner(0, 2)).unwrap();
+    let (a_post, b_post) =
+        Game::<i64, _>::one_v_one(&a, &b, Outcome::winner(0, 2), &GameOptions::default()).unwrap();
     assert!(a_post.mu() > 25.0);
     assert!(b_post.mu() < 25.0);
 }
@@ -95,3 +96,45 @@ fn game_log_evidence_is_finite() {
     assert!(g.log_evidence().is_finite());
     assert!(g.log_evidence() < 0.0);
 }
+
+/// `one_v_one` used to hardcode `GameOptions::default()`, so a 1v1 could
+/// never set `p_draw` and a drawn 1v1 was unreachable through it.
+#[test]
+fn one_v_one_honours_the_draw_probability_it_is_given() {
+    let a = default_rating();
+    let b = default_rating();
+
+    // Default options still reject a draw, because the default p_draw is zero.
+    let err = Game::<i64, _>::one_v_one(&a, &b, Outcome::draw(2), &GameOptions::default())
+        .expect_err("a draw needs a positive p_draw");
+    assert!(matches!(
+        err,
+        InferenceError::TieWithoutDrawProbability { .. }
+    ));
+
+    // With a draw probability supplied it succeeds — which was impossible
+    // before the signature took options.
+    let options = GameOptions {
+        p_draw: 0.25,
+        ..GameOptions::default()
+    };
+    let (a_post, b_post) = Game::<i64, _>::one_v_one(&a, &b, Outcome::draw(2), &options)
+        .expect("a draw is representable once p_draw is positive");
+
+    // A symmetric draw leaves the means alone and sharpens both sides.
+    assert!((a_post.mu() - b_post.mu()).abs() < 1e-9);
+    assert!(a_post.sigma() < 25.0 / 3.0);
+}
+
+/// Convergence options reach the 1v1 path too, not just `p_draw`.
+#[test]
+fn one_v_one_honours_convergence_options() {
+    let a = default_rating();
+    let b = default_rating();
+    let options = GameOptions {
+        convergence: ConvergenceOptions::default(),
+        ..GameOptions::default()
+    };
+    let (a_post, _) = Game::<i64, _>::one_v_one(&a, &b, Outcome::winner(0, 2), &options).unwrap();
+    assert!(a_post.mu() > 25.0);
+}

tests/ingestion_equivalence.rs

@@ -30,6 +30,22 @@ fn event(a: &str, b: &str, time: i64) -> Event<i64, String> {
     }
 }
 
+/// Like [`event`], but `a` carries competitor configuration.
+///
+/// `prior` and `drift_scale` configure the competitor rather than the event, so
+/// they are the part of ingestion most exposed to order: they are consumed once,
+/// where the competitor's state is written.
+fn configured_event(a: &str, b: &str, time: i64, scale: f64) -> Event<i64, String> {
+    Event {
+        time,
+        teams: smallvec![
+            Team::with_members([Member::new(a.to_string()).with_drift_scale(scale)]),
+            Team::with_members([Member::new(b.to_string())]),
+        ],
+        outcome: Outcome::winner(0, 2),
+    }
+}
+
 fn converged_skills(events: Vec<Event<i64, String>>, batched: bool) -> Vec<(String, Gaussian)> {
     let mut h: History<i64, _, _, String> =
         History::builder_with_key().convergence(tight()).build();
@@ -145,3 +161,65 @@ fn back_dated_event_matches_batched() {
     let incremental = converged_skills(events, false);
     assert_same(&batched, &incremental, "back-dated event");
 }
+
+/// The invariant this file protects was only ever checked for *unconfigured*
+/// competitors — every helper above built members with `Member::new`.
+///
+/// Configuration is the part most exposed to ordering, because it is consumed
+/// once at the point the competitor's state is written rather than replayed per
+/// event. These cover it.
+#[test]
+fn configured_competitors_are_order_independent() {
+    let events = vec![
+        configured_event("a", "b", 0, 0.0),
+        configured_event("a", "c", 1, 0.0),
+        configured_event("a", "b", 2, 0.0),
+        event("b", "c", 3),
+    ];
+
+    assert_same(
+        &converged_skills(events.clone(), true),
+        &converged_skills(events, false),
+        "configuration repeated on every appearance",
+    );
+}
+
+/// Configuration supplied only on a *later* event is the case that used to be
+/// silently dropped. It must now reach the same fit either way it is ingested.
+#[test]
+fn late_configuration_is_order_independent() {
+    let events = vec![
+        event("a", "b", 0),
+        configured_event("a", "c", 1, 0.0),
+        event("a", "b", 2),
+    ];
+
+    assert_same(
+        &converged_skills(events.clone(), true),
+        &converged_skills(events, false),
+        "configuration supplied after first appearance",
+    );
+}
+
+/// And it must actually be doing something — an implementation that dropped
+/// configuration entirely would pass both tests above.
+#[test]
+fn configuration_changes_the_fit_however_it_is_ingested() {
+    let configured = vec![
+        event("a", "b", 0),
+        configured_event("a", "c", 1, 0.0),
+        event("a", "b", 2),
+    ];
+    let plain = vec![event("a", "b", 0), event("a", "c", 1), event("a", "b", 2)];
+
+    for batched in [true, false] {
+        let with = converged_skills(configured.clone(), batched);
+        let without = converged_skills(plain.clone(), batched);
+        assert!(
+            with.iter()
+                .zip(&without)
+                .any(|((_, x), (_, y))| (x.sigma() - y.sigma()).abs() > 1e-9),
+            "batched={batched}: configuration had no effect, so the order tests are vacuous"
+        );
+    }
+}

tests/ingestion_shape.rs

@@ -0,0 +1,147 @@
+//! Malformed events must be rejected at the ingestion boundary.
+//!
+//! Every case here was reachable from safe public API in a release build. Two
+//! of them are the two shapes this crate's defects keep taking: a panic from
+//! deep inside inference, and a finite, plausible-looking posterior computed
+//! from an event that should never have been accepted.
+//!
+//! `InferenceError::NotEnoughTeams` and `EmptyTeam` already existed when these
+//! were found — they were checked on the prediction paths and nowhere else, so
+//! ingestion could still manufacture the states they describe.
+
+use smallvec::smallvec;
+use trueskill_tt::{Event, History, InferenceError, Member, Outcome, Team};
+
+type Ev = Event<i64, &'static str>;
+
+fn history() -> History<i64, trueskill_tt::ConstantDrift, trueskill_tt::NullObserver, &'static str>
+{
+    History::builder().score_sigma(1.0).build()
+}
+
+fn teams(names: &[&[&'static str]]) -> smallvec::SmallVec<[Team<&'static str>; 4]> {
+    names
+        .iter()
+        .map(|team| Team::with_members(team.iter().map(|k| Member::new(*k))))
+        .collect()
+}
+
+/// The regression this file exists for: `run_chain` builds one diff link per
+/// adjacent pair of teams, so a one-team event left it indexing `links[1..]`
+/// on an empty vector and panicked — in release, from `History::add_events`.
+#[test]
+fn a_one_team_event_is_an_error_not_a_panic() {
+    let mut h = history();
+    let err = h
+        .add_events(vec![Ev {
+            time: 1,
+            teams: teams(&[&["a"]]),
+            outcome: Outcome::winner(0, 1),
+        }])
+        .unwrap_err();
+    assert!(
+        matches!(err, InferenceError::NotEnoughTeams { got: 1 }),
+        "{err:?}"
+    );
+}
+
+#[test]
+fn a_zero_team_event_is_an_error() {
+    let mut h = history();
+    let err = h
+        .add_events(vec![Ev {
+            time: 1,
+            teams: smallvec![],
+            outcome: Outcome::ranking([]),
+        }])
+        .unwrap_err();
+    assert!(
+        matches!(err, InferenceError::NotEnoughTeams { got: 0 }),
+        "{err:?}"
+    );
+}
+
+/// The quiet half. An empty team contributes no performance, so before this
+/// was rejected the event converged and handed back a finite posterior for its
+/// opponent — a plausible constant computed from nothing.
+#[test]
+fn an_empty_team_is_an_error_rather_than_a_free_win() {
+    let mut h = history();
+    let err = h
+        .add_events(vec![Ev {
+            time: 1,
+            teams: teams(&[&[], &["b"]]),
+            outcome: Outcome::winner(0, 2),
+        }])
+        .unwrap_err();
+    assert!(
+        matches!(err, InferenceError::EmptyTeam { team: 0 }),
+        "{err:?}"
+    );
+    // Nothing was recorded, so the history is still empty.
+    assert!(h.current_skill(&"b").is_none());
+}
+
+#[test]
+fn an_empty_team_is_reported_by_position() {
+    let mut h = history();
+    let err = h
+        .add_events(vec![Ev {
+            time: 1,
+            teams: teams(&[&["a"], &[]]),
+            outcome: Outcome::winner(0, 2),
+        }])
+        .unwrap_err();
+    assert!(
+        matches!(err, InferenceError::EmptyTeam { team: 1 }),
+        "{err:?}"
+    );
+}
+
+/// A NaN score used to ingest cleanly. `converge` reported `NonFiniteResult`,
+/// but a caller who read `current_skill` first was handed `tau: NaN` with
+/// nothing to say so.
+#[test]
+fn a_non_finite_score_is_rejected_at_ingestion() {
+    for bad in [f64::NAN, f64::INFINITY, f64::NEG_INFINITY] {
+        let mut h = history();
+        let err = h
+            .add_events(vec![Ev {
+                time: 1,
+                teams: teams(&[&["a"], &["b"]]),
+                outcome: Outcome::scores([bad, 0.0]),
+            }])
+            .unwrap_err();
+        assert!(
+            matches!(err, InferenceError::InvalidParameter { name: "score", .. }),
+            "{bad}: {err:?}"
+        );
+        assert!(h.current_skill(&"a").is_none(), "{bad} was recorded anyway");
+    }
+}
+
+/// The fluent builder routes through the same chokepoint, so it inherits the
+/// checks rather than needing its own.
+#[test]
+fn the_event_builder_inherits_the_shape_checks() {
+    let mut h = history();
+    let err = h.event(1).team(["a"]).winner(0).commit().unwrap_err();
+    assert!(
+        matches!(err, InferenceError::NotEnoughTeams { got: 1 }),
+        "{err:?}"
+    );
+}
+
+/// A well-formed event is untouched by any of this.
+#[test]
+fn a_well_formed_event_still_ingests() {
+    let mut h = history();
+    h.add_events(vec![Ev {
+        time: 1,
+        teams: teams(&[&["a"], &["b"]]),
+        outcome: Outcome::scores([3.0, 1.0]),
+    }])
+    .unwrap();
+    assert!(h.converge().unwrap().converged);
+    assert!(h.current_skill(&"a").unwrap().mu() > h.current_skill(&"b").unwrap().mu());
+}

tests/joint_handle.rs

@@ -0,0 +1,267 @@
+//! `History::joint` factorises once and answers many questions.
+//!
+//! The contract that matters is *identity*: a `Joint` must return exactly what
+//! the one-shot call returns, bit for bit. A faster path that quietly disagreed
+//! with the slow one would be worse than no fast path — a caller would get
+//! different numbers depending on how many questions they happened to ask.
+
+use smallvec::smallvec;
+use trueskill_tt::{
+    ConstantDrift, ConvergenceOptions, Event, History, InferenceError, Member, Outcome, Team,
+    UnknownKeys,
+};
+
+type H = History<i64, ConstantDrift, trueskill_tt::NullObserver, &'static str>;
+
+fn duel(a: &'static str, b: &'static str, t: i64, sa: f64, sb: f64) -> Event<i64, &'static str> {
+    Event {
+        time: t,
+        teams: smallvec![
+            Team::with_members([Member::new(a)]),
+            Team::with_members([Member::new(b)]),
+        ],
+        outcome: Outcome::scores([sa, sb]),
+    }
+}
+
+fn ranked(a: &'static str, b: &'static str, t: i64) -> Event<i64, &'static str> {
+    Event {
+        time: t,
+        teams: smallvec![
+            Team::with_members([Member::new(a)]),
+            Team::with_members([Member::new(b)]),
+        ],
+        outcome: Outcome::winner(0, 2),
+    }
+}
+
+fn history(unknown: UnknownKeys) -> H {
+    History::builder()
+        .mu(0.0)
+        .sigma(6.0)
+        .beta(1.0)
+        .score_sigma(2.0)
+        .drift(ConstantDrift(0.5))
+        .unknown_keys(unknown)
+        .convergence(ConvergenceOptions {
+            max_iter: 20_000,
+            epsilon: 1e-13,
+            alpha: 1.0,
+        })
+        .build()
+}
+
+/// Several slices, competitors with different last appearances, so `latest`
+/// and `at_slice` both have work to do.
+fn fitted(unknown: UnknownKeys) -> H {
+    let mut h = history(unknown);
+    h.add_events(vec![
+        duel("a", "b", 1, 5.0, 2.0),
+        duel("c", "d", 1, 3.0, 3.5),
+        duel("a", "c", 2, 6.0, 1.0),
+        duel("b", "d", 3, 4.0, 3.0),
+        duel("a", "d", 4, 7.0, 2.0),
+        duel("b", "c", 5, 2.0, 4.0),
+    ])
+    .unwrap();
+    let report = h.converge().unwrap();
+    assert!(report.converged, "fixture must converge");
+    h
+}
+
+const PAIRS: [(&str, &str); 6] = [
+    ("a", "b"),
+    ("a", "c"),
+    ("a", "d"),
+    ("b", "c"),
+    ("b", "d"),
+    ("c", "d"),
+];
+
+#[test]
+fn a_joint_answers_exactly_what_the_one_shot_call_does() {
+    let h = fitted(UnknownKeys::Reject);
+    let joint = h.joint().unwrap();
+
+    for (a, b) in PAIRS {
+        let terms = [(&a, 1.0), (&b, -1.0)];
+        let one_shot = h.posterior_of(&terms).unwrap();
+        let cached = joint.posterior_of(&terms).unwrap();
+        assert_eq!(one_shot.pi(), cached.pi(), "{a} - {b}");
+        assert_eq!(one_shot.tau(), cached.tau(), "{a} - {b}");
+    }
+}
+
+#[test]
+fn a_joint_agrees_at_a_pinned_time_too() {
+    let h = fitted(UnknownKeys::Reject);
+    let joint = h.joint().unwrap();
+
+    for time in 1..=5 {
+        for (a, b) in PAIRS {
+            let terms = [(&a, 1.0), (&b, -1.0)];
+            let one_shot = h.posterior_of_at(time, &terms);
+            let cached = joint.posterior_of_at(time, &terms);
+            match (one_shot, cached) {
+                (Ok(x), Ok(y)) => {
+                    assert_eq!(x.pi(), y.pi(), "t={time} {a} - {b}");
+                    assert_eq!(x.tau(), y.tau(), "t={time} {a} - {b}");
+                }
+                (Err(x), Err(y)) => assert_eq!(x, y, "t={time} {a} - {b}"),
+                (x, y) => panic!("t={time} {a} - {b}: disagreed on success: {x:?} vs {y:?}"),
+            }
+        }
+    }
+}
+
+#[test]
+fn a_joint_scores_candidate_matchups_identically() {
+    let h = fitted(UnknownKeys::Reject);
+    let joint = h.joint().unwrap();
+    let (a, b) = ("a", "b");
+    let target = [(&a, 1.0), (&b, -1.0)];
+
+    for (x, y) in PAIRS {
+        let teams: [&[&&str]; 2] = [&[&x], &[&y]];
+        let one_shot = h.expected_variance_reduction(&teams, &target).unwrap();
+        let cached = joint.expected_variance_reduction(&teams, &target).unwrap();
+        assert_eq!(one_shot, cached, "{x} vs {y}");
+    }
+}
+
+/// The whole point: a competitor appears once per slice, so the joint is over
+/// appearances rather than competitors, and a caller sizing a batch needs to
+/// know which.
+#[test]
+fn variables_counts_appearances_not_competitors() {
+    let h = fitted(UnknownKeys::Reject);
+    let joint = h.joint().unwrap();
+    // Four competitors, twelve appearances across five slices, all with
+    // positive drift between them, so no two collapse.
+    assert_eq!(joint.variables(), 12);
+}
+
+/// How much the collapse is worth, which is the part a caller has to plan
+/// around: a drift-free competitor contributes **one** variable however long
+/// the history, so the same events at `gamma = 0` and `gamma > 0` differ by
+/// roughly the slice count in problem size — and by its cube in solve time.
+///
+/// Reported by a consumer as an 8x difference in solve time on a ~2,000-node,
+/// 76-slice model (787 ms career against 6,214 ms drifting). This pins the
+/// mechanism behind that so a change to the collapse rule cannot quietly
+/// remove it.
+#[test]
+fn drift_free_competitors_shrink_the_joint_by_the_slice_count() {
+    fn variables(gamma: f64) -> usize {
+        let mut h = History::builder()
+            .mu(0.0)
+            .sigma(6.0)
+            .beta(1.0)
+            .score_sigma(2.0)
+            .drift(ConstantDrift(gamma))
+            .convergence(ConvergenceOptions {
+                max_iter: 20_000,
+                epsilon: 1e-13,
+                alpha: 1.0,
+            })
+            .build();
+        h.add_events(
+            (1..=10)
+                .map(|t| duel("a", "b", t, 5.0, 2.0))
+                .collect::<Vec<_>>(),
+        )
+        .unwrap();
+        let _ = h.converge().unwrap();
+        h.joint().unwrap().variables()
+    }
+
+    let drifting = variables(0.5);
+    let career = variables(0.0);
+
+    // Two competitors over ten slices: twenty appearances, or two variables.
+    assert_eq!(drifting, 20);
+    assert_eq!(career, 2);
+    assert_eq!(
+        drifting / career,
+        10,
+        "collapse should track the slice count"
+    );
+}
+
+/// With `drift = 0` consecutive appearances are the same latent variable, so
+/// the joint is smaller than the appearance count.
+#[test]
+fn pinned_competitors_collapse_consecutive_appearances() {
+    let mut h = History::builder()
+        .mu(0.0)
+        .sigma(6.0)
+        .beta(1.0)
+        .score_sigma(2.0)
+        .drift(ConstantDrift(0.0))
+        .convergence(ConvergenceOptions {
+            max_iter: 20_000,
+            epsilon: 1e-13,
+            alpha: 1.0,
+        })
+        .build();
+    h.add_events(vec![
+        duel("a", "b", 1, 5.0, 2.0),
+        duel("a", "b", 2, 4.0, 3.0),
+        duel("a", "b", 3, 6.0, 1.0),
+    ])
+    .unwrap();
+    assert!(h.converge().unwrap().converged);
+    assert_eq!(h.joint().unwrap().variables(), 2);
+}
+
+#[test]
+fn a_ranked_history_has_no_exact_joint() {
+    let mut h = history(UnknownKeys::Reject);
+    h.add_events(vec![duel("a", "b", 1, 5.0, 2.0), ranked("a", "b", 2)])
+        .unwrap();
+    let _ = h.converge().unwrap();
+    assert!(matches!(
+        h.joint().unwrap_err(),
+        InferenceError::JointUnavailable { .. }
+    ));
+}
+
+#[test]
+fn an_empty_history_has_no_joint() {
+    let h = history(UnknownKeys::Reject);
+    assert!(matches!(
+        h.joint().unwrap_err(),
+        InferenceError::JointUnavailable { .. }
+    ));
+}
+
+/// Unknown keys are decided per query, not when the joint is factorised — the
+/// factorisation does not depend on the question.
+#[test]
+fn unknown_keys_are_rejected_per_query() {
+    let h = fitted(UnknownKeys::Reject);
+    let joint = h.joint().unwrap();
+    let (a, z) = ("a", "nobody");
+    assert!(matches!(
+        joint.posterior_of(&[(&a, 1.0), (&z, -1.0)]).unwrap_err(),
+        InferenceError::UnknownKey { .. }
+    ));
+    // The handle is still usable afterwards.
+    let b = "b";
+    assert!(joint.posterior_of(&[(&a, 1.0), (&b, -1.0)]).is_ok());
+}
+
+/// Under `Prior`, an unseen competitor is independent of everything in the
+/// history, and the cached path must add the same prior variance the one-shot
+/// path does.
+#[test]
+fn unseen_competitors_match_the_one_shot_path() {
+    let h = fitted(UnknownKeys::Prior);
+    let joint = h.joint().unwrap();
+    let (a, z) = ("a", "nobody");
+    let terms = [(&a, 1.0), (&z, -1.0)];
+    let one_shot = h.posterior_of(&terms).unwrap();
+    let cached = joint.posterior_of(&terms).unwrap();
+    assert_eq!(one_shot.pi(), cached.pi());
+    assert_eq!(one_shot.tau(), cached.tau());
+}

tests/large_history_converges_finite.rs

@@ -46,7 +46,7 @@ fn nan_after_fit(players: usize) -> usize {
         let (w, l) = if rng.coin() { (a, b) } else { (b, a) };
         h.record_winner(&ids[w], &ids[l], 0).unwrap();
     }
-    h.converge().unwrap();
+    let _ = h.converge().unwrap();
 
     ids.iter()
         .filter(|id| {

tests/marginal_calibration.rs

@@ -0,0 +1,367 @@
+//! Calibration of the crate's marginals against the EXACT posterior.
+//!
+//! A scored history is linear-Gaussian — `MarginFactor` encodes
+//! `score_a - score_b ~ N(perf_a - perf_b, score_sigma^2)` — so the true joint
+//! posterior has a closed form and the crate can be checked against ground
+//! truth rather than against intuition. That is not possible for ranked
+//! outcomes, whose truncation likelihood EP genuinely approximates.
+//!
+//! Two things are pinned here, and one is deliberately only recorded.
+//!
+//! **Pinned: on a tree the crate is exact**, means and variances both. Message
+//! passing has no approximation to make when the factor graph has no cycles, so
+//! any drift here would be a real defect.
+//!
+//! **Pinned: means are exact even with cycles.** This is the standard result
+//! for Gaussian belief propagation (Weiss & Freeman 2001) and it is what makes
+//! ratings trustworthy.
+//!
+//! **Recorded, not asserted: with cycles, marginal variances are too narrow.**
+//! Measured on the round-robin fixture below, the crate reports sigma 1.430
+//! where the exact posterior is 2.851 — a ratio of 0.502. That is the known
+//! behaviour of loopy Gaussian BP, not a bug in this crate, and it is left
+//! unasserted because fixing it is exactly what #46 proposes.
+//!
+//! Why that matters for a consumer, and why #46 cannot be implemented as "add
+//! a covariance accessor": the exact correlation between two nodes here is
+//! +0.857, so a consumer computing `sqrt(sa^2 + sb^2)` for a difference
+//! overstates its width. But the too-narrow marginals partially cancel that,
+//! leaving 1.327x rather than 2.646x. Adding true correlations to these
+//! marginals without also correcting them would give 0.765 against a true
+//! 1.524 — *overconfident*, which is the unsafe direction.
+
+use smallvec::smallvec;
+use trueskill_tt::{ConstantDrift, ConvergenceOptions, Event, History, Member, Outcome, Team};
+
+const N: usize = 5;
+const MU0: f64 = 0.0;
+const SIGMA0: f64 = 6.0;
+const BETA: f64 = 1.0;
+const SCORE_SIGMA: f64 = 2.0;
+
+/// A STAR: every event touches c0, so the node-event graph is a tree and
+/// Gaussian BP is exact. Any discrepancy here is not caused by loops.
+fn tree_fixture() -> Vec<(usize, usize, f64)> {
+    vec![(0, 1, 3.0), (0, 2, 5.0), (0, 3, 4.0), (0, 4, 6.0)]
+}
+
+/// (winner, loser, score_diff)
+fn fixture() -> Vec<(usize, usize, f64)> {
+    vec![
+        (0, 1, 3.0),
+        (0, 2, 5.0),
+        (1, 2, 2.0),
+        (3, 4, 1.0),
+        (0, 3, 4.0),
+        (1, 4, 2.5),
+        (2, 3, 0.5),
+        (0, 4, 6.0),
+        (1, 3, 1.5),
+        (2, 4, 3.0),
+    ]
+}
+
+/// Invert a small symmetric positive-definite matrix by Gauss-Jordan.
+fn inverse(mut a: Vec<Vec<f64>>) -> Vec<Vec<f64>> {
+    let n = a.len();
+    let mut inv: Vec<Vec<f64>> = (0..n)
+        .map(|i| (0..n).map(|j| if i == j { 1.0 } else { 0.0 }).collect())
+        .collect();
+    for col in 0..n {
+        // partial pivot
+        let mut piv = col;
+        for r in col + 1..n {
+            if a[r][col].abs() > a[piv][col].abs() {
+                piv = r;
+            }
+        }
+        a.swap(col, piv);
+        inv.swap(col, piv);
+        let d = a[col][col];
+        for j in 0..n {
+            a[col][j] /= d;
+            inv[col][j] /= d;
+        }
+        for r in 0..n {
+            if r == col {
+                continue;
+            }
+            let f = a[r][col];
+            for j in 0..n {
+                a[r][j] -= f * a[col][j];
+                inv[r][j] -= f * inv[col][j];
+            }
+        }
+    }
+    inv
+}
+
+/// The exact posterior of a linear-Gaussian model:
+/// precision = prior precision + sum of a_k a_k^T / v_k.
+fn exact_for(obs: &[(usize, usize, f64)]) -> (Vec<f64>, Vec<Vec<f64>>) {
+    let mut lambda = vec![vec![0.0; N]; N];
+    let mut eta = [0.0; N];
+    for (i, row) in lambda.iter_mut().enumerate() {
+        row[i] = 1.0 / (SIGMA0 * SIGMA0);
+        eta[i] = MU0 / (SIGMA0 * SIGMA0);
+    }
+
+    // Each 1v1 observation: d ~ N(x_a - x_b, score_sigma^2 + 2 beta^2)
+    let v = SCORE_SIGMA * SCORE_SIGMA + 2.0 * BETA * BETA;
+    for &(a, b, d) in obs {
+        let mut vec_a = [0.0; N];
+        vec_a[a] = 1.0;
+        vec_a[b] = -1.0;
+        for i in 0..N {
+            for j in 0..N {
+                lambda[i][j] += vec_a[i] * vec_a[j] / v;
+            }
+            eta[i] += vec_a[i] * d / v;
+        }
+    }
+
+    let cov = inverse(lambda);
+    let mean: Vec<f64> = (0..N)
+        .map(|i| (0..N).map(|j| cov[i][j] * eta[j]).sum())
+        .collect();
+    (mean, cov)
+}
+
+fn key(i: usize) -> &'static str {
+    ["c0", "c1", "c2", "c3", "c4"][i]
+}
+
+/// Returns (worst mean error, worst sd ratio).
+fn fitted(
+    obs: &[(usize, usize, f64)],
+) -> History<i64, ConstantDrift, trueskill_tt::NullObserver, &'static str> {
+    let mut h: History<i64, _, _, &'static str> = History::builder()
+        .mu(MU0)
+        .sigma(SIGMA0)
+        .beta(BETA)
+        .score_sigma(SCORE_SIGMA)
+        .drift(ConstantDrift(0.0))
+        .convergence(ConvergenceOptions {
+            max_iter: 20_000,
+            epsilon: 1e-13,
+            alpha: 1.0,
+        })
+        .build();
+
+    let events: Vec<Event<i64, &'static str>> = obs
+        .iter()
+        .copied()
+        .map(|(a, b, d)| Event {
+            time: 1,
+            teams: smallvec![
+                Team::with_members([Member::new(key(a))]),
+                Team::with_members([Member::new(key(b))]),
+            ],
+            outcome: Outcome::scores([d, 0.0]),
+        })
+        .collect();
+    h.add_events(events).unwrap();
+    let report = h.converge().unwrap();
+    assert!(
+        report.converged,
+        "fixture must converge: {:?}",
+        report.final_step
+    );
+
+    h
+}
+
+/// Returns (worst mean error, worst sd ratio gap).
+fn run(name: &str, obs: Vec<(usize, usize, f64)>) -> (f64, f64) {
+    println!("\n########## {name} ##########");
+    let h = fitted(&obs);
+    let (mean, cov) = exact_for(&obs);
+
+    println!("\n== marginals: crate vs the exact linear-Gaussian posterior ==");
+    println!(
+        "{:>4} {:>12} {:>12} {:>12} {:>12} {:>8}",
+        "node", "crate mu", "exact mu", "crate sd", "exact sd", "sd ratio"
+    );
+    for i in 0..N {
+        let g = h.current_skill(&key(i)).unwrap();
+        let exact_sd = cov[i][i].sqrt();
+        println!(
+            "{:>4} {:>12.6} {:>12.6} {:>12.6} {:>12.6} {:>8.3}",
+            key(i),
+            g.mu(),
+            mean[i],
+            g.sigma(),
+            exact_sd,
+            g.sigma() / exact_sd
+        );
+    }
+
+    let mut worst_mean = 0.0f64;
+    let mut worst_ratio_gap = 0.0f64;
+    for i in 0..N {
+        let g = h.current_skill(&key(i)).unwrap();
+        worst_mean = worst_mean.max((g.mu() - mean[i]).abs());
+        worst_ratio_gap = worst_ratio_gap.max((g.sigma() / cov[i][i].sqrt() - 1.0).abs());
+    }
+
+    println!("\n== what a consumer actually computes for a DIFFERENCE ==");
+    println!(
+        "{:>8} {:>12} {:>14} {:>14} {:>12}",
+        "pair", "exact", "naive(exact)", "naive(crate)", "crate err"
+    );
+    for i in 0..N {
+        for j in i + 1..N {
+            if i != 0 && j != 1 {
+                continue;
+            }
+            let gi = h.current_skill(&key(i)).unwrap();
+            let gj = h.current_skill(&key(j)).unwrap();
+            let exact_sd = (cov[i][i] + cov[j][j] - 2.0 * cov[i][j]).sqrt();
+            let naive_exact = (cov[i][i] + cov[j][j]).sqrt();
+            let naive_crate = (gi.sigma().powi(2) + gj.sigma().powi(2)).sqrt();
+            let corr = cov[i][j] / (cov[i][i].sqrt() * cov[j][j].sqrt());
+            println!(
+                "{:>8} {:>12.6} {:>14.6} {:>14.6} {:>11.3}x   (corr {corr:.4})",
+                format!("{}-{}", key(i), key(j)),
+                exact_sd,
+                naive_exact,
+                naive_crate,
+                naive_crate / exact_sd
+            );
+        }
+    }
+
+    (worst_mean, worst_ratio_gap)
+}
+
+/// With no cycles there is nothing for message passing to approximate.
+#[test]
+fn on_a_tree_the_marginals_are_exact() {
+    let (mean_err, sd_gap) = run("TREE (star: no loops, BP is exact)", tree_fixture());
+    assert!(
+        mean_err < 1e-9,
+        "tree means should be exact, worst error {mean_err}"
+    );
+    assert!(
+        sd_gap < 1e-9,
+        "tree sigmas should be exact, worst ratio gap {sd_gap}"
+    );
+}
+
+/// With cycles the means stay exact — the property ratings depend on — while
+/// the variances do not. The variance gap is measured and reported rather than
+/// asserted; see the module docs.
+#[test]
+fn with_cycles_the_means_stay_exact_but_the_variances_shrink() {
+    let (mean_err, sd_gap) = run("LOOPY (round robin)", fixture());
+    assert!(
+        mean_err < 1e-9,
+        "loopy means must still be exact, worst error {mean_err}"
+    );
+    assert!(
+        sd_gap > 0.1,
+        "the loopy variance gap is the premise of #46; if it has closed, that \
+         issue and these docs need revisiting (worst ratio gap {sd_gap})"
+    );
+}
+
+/// The point of #46: `posterior_of` must reproduce the exact joint, including
+/// the correlation that marginals cannot express.
+#[test]
+fn posterior_of_matches_the_exact_joint() {
+    for (name, obs) in [("tree", tree_fixture()), ("loopy", fixture())] {
+        let h = fitted(&obs);
+        let (_, cov) = exact_for(&obs);
+
+        println!("\n== posterior_of vs exact ({name}) ==");
+        println!(
+            "{:>12} {:>14} {:>14} {:>10}",
+            "functional", "posterior_of", "exact", "ratio"
+        );
+
+        for (i, j) in [(0usize, 1usize), (0, 2), (1, 3), (2, 4)] {
+            let got = h
+                .posterior_of(&[(&key(i), 1.0), (&key(j), -1.0)])
+                .expect("scored slice should have a joint");
+            let exact_sd = (cov[i][i] + cov[j][j] - 2.0 * cov[i][j]).sqrt();
+            println!(
+                "{:>12} {:>14.6} {:>14.6} {:>10.4}",
+                format!("{}-{}", key(i), key(j)),
+                got.sigma(),
+                exact_sd,
+                got.sigma() / exact_sd
+            );
+            assert!(
+                (got.sigma() - exact_sd).abs() / exact_sd < 1e-9,
+                "{name} {}-{}: posterior_of gave {} where the exact joint is {exact_sd}",
+                key(i),
+                key(j),
+                got.sigma()
+            );
+        }
+
+        // A single competitor: this is where the loopy marginal was 2x narrow.
+        for (i, row) in cov.iter().enumerate() {
+            let got = h.posterior_of(&[(&key(i), 1.0)]).unwrap();
+            let exact_sd = row[i].sqrt();
+            assert!(
+                (got.sigma() - exact_sd).abs() / exact_sd < 1e-9,
+                "{name} {}: posterior_of gave {} where exact is {exact_sd}",
+                key(i),
+                got.sigma()
+            );
+        }
+        println!("  single-competitor marginals also exact");
+    }
+}
+
+/// Cost of the dense solve as the slice grows. Recorded, not asserted.
+#[test]
+#[ignore = "timing probe, run explicitly"]
+fn cost_scaling() {
+    use std::time::Instant;
+    for n in [50usize, 100, 200, 400, 800] {
+        let names: Vec<String> = (0..n).map(|i| format!("c{i}")).collect();
+        let mut h: History<i64, _, _, String> = History::builder_with_key()
+            .score_sigma(2.0)
+            .drift(ConstantDrift(0.0))
+            .convergence(ConvergenceOptions {
+                max_iter: 200,
+                epsilon: 1e-8,
+                alpha: 1.0,
+            })
+            .build();
+        let mut seed = 5u64;
+        let mut rnd = move || {
+            seed ^= seed << 13;
+            seed ^= seed >> 7;
+            seed ^= seed << 17;
+            seed
+        };
+        let events: Vec<Event<i64, String>> = (0..n * 4)
+            .map(|_| {
+                let a = (rnd() as usize) % n;
+                let mut b = (rnd() as usize) % n;
+                if b == a {
+                    b = (b + 1) % n;
+                }
+                Event {
+                    time: 1,
+                    teams: smallvec![
+                        Team::with_members([Member::new(names[a].clone())]),
+                        Team::with_members([Member::new(names[b].clone())]),
+                    ],
+                    outcome: Outcome::scores([1.0, 0.0]),
+                }
+            })
+            .collect();
+        h.add_events(events).unwrap();
+        let _ = h.converge().unwrap();
+
+        let t = Instant::now();
+        let g = h
+            .posterior_of(&[(&names[0], 1.0), (&names[1], -1.0)])
+            .unwrap();
+        println!("  n={n:>4}: {:>10.2?}  sigma {:.6}", t.elapsed(), g.sigma());
+    }
+}

tests/observer.rs

@@ -0,0 +1,161 @@
+//! `Observer` callbacks must actually fire.
+//!
+//! `on_slice_processed` (formerly `on_batch_processed`) was declared on the
+//! trait and never called from anywhere, so implementors wired up a callback
+//! that could not run. These tests exist so that cannot silently recur.
+
+use std::sync::{Arc, Mutex};
+
+use trueskill_tt::{History, Observer};
+
+/// Plain fields. `Arc<O>` implements `Observer`, so the caller shares the
+/// observer itself rather than wrapping each field in its own `Arc`.
+#[derive(Default)]
+struct Recorder {
+    iterations: Mutex<Vec<usize>>,
+    slices: Mutex<Vec<(i64, usize, usize)>>,
+    converged: Mutex<Vec<(usize, bool)>>,
+}
+
+impl Observer<i64> for Recorder {
+    fn on_iteration_end(&self, iter: usize, _max_step: (f64, f64)) {
+        self.iterations.lock().unwrap().push(iter);
+    }
+
+    fn on_slice_processed(&self, time: &i64, slice_idx: usize, n_events: usize) {
+        self.slices
+            .lock()
+            .unwrap()
+            .push((*time, slice_idx, n_events));
+    }
+
+    fn on_converged(&self, iters: usize, _final_step: (f64, f64), converged: bool) {
+        self.converged.lock().unwrap().push((iters, converged));
+    }
+}
+
+#[test]
+fn every_observer_callback_fires() {
+    let recorder = Arc::new(Recorder::default());
+    let mut h = History::builder().observer(Arc::clone(&recorder)).build();
+
+    h.record_winner(&"a", &"b", 1).unwrap();
+    h.record_winner(&"b", &"c", 2).unwrap();
+    h.record_winner(&"c", &"a", 3).unwrap();
+    let _ = h.converge().unwrap();
+
+    assert!(
+        !recorder.iterations.lock().unwrap().is_empty(),
+        "on_iteration_end never fired"
+    );
+    assert!(
+        !recorder.converged.lock().unwrap().is_empty(),
+        "on_converged never fired"
+    );
+    assert!(
+        !recorder.slices.lock().unwrap().is_empty(),
+        "on_slice_processed never fired — the defect this test exists for"
+    );
+}
+
+#[test]
+fn slice_callbacks_report_the_slice_they_swept() {
+    let recorder = Arc::new(Recorder::default());
+    let mut h = History::builder().observer(Arc::clone(&recorder)).build();
+
+    h.record_winner(&"a", &"b", 10).unwrap();
+    h.record_winner(&"a", &"b", 20).unwrap();
+    let _ = h.converge().unwrap();
+
+    let slices = recorder.slices.lock().unwrap();
+
+    // Only the times actually in the history, and each with its own events.
+    for &(time, idx, events) in slices.iter() {
+        assert!(time == 10 || time == 20, "unexpected slice time {time}");
+        assert!(idx < 2, "slice index {idx} out of range");
+        assert_eq!(events, 1, "each slice holds exactly one event");
+    }
+
+    // Both slices must be reported, not just one end of the sweep.
+    assert!(
+        slices.iter().any(|&(t, ..)| t == 10),
+        "slice 10 never reported"
+    );
+    assert!(
+        slices.iter().any(|&(t, ..)| t == 20),
+        "slice 20 never reported"
+    );
+}
+
+#[test]
+fn a_single_slice_history_still_reports_its_sweep() {
+    let recorder = Arc::new(Recorder::default());
+    let mut h = History::builder().observer(Arc::clone(&recorder)).build();
+
+    h.record_winner(&"a", &"b", 1).unwrap();
+    let _ = h.converge().unwrap();
+
+    let slices = recorder.slices.lock().unwrap();
+    assert!(
+        !slices.is_empty(),
+        "the single-slice path must report its sweep too"
+    );
+    assert!(slices.iter().all(|&(t, idx, _)| t == 1 && idx == 0));
+}
+
+/// The gap #40 closed: without `impl Observer for Arc<O>`, an observer that
+/// accumulates anything had to wrap every field in its own `Arc` and derive
+/// `Clone`, because `History` consumes the observer and never hands it back.
+#[test]
+fn a_shared_observer_reaches_the_callers_handle() {
+    let recorder = Arc::new(Recorder::default());
+    let mut h = History::builder().observer(Arc::clone(&recorder)).build();
+
+    h.record_winner(&"a", &"b", 1).unwrap();
+    let _ = h.converge().unwrap();
+
+    assert!(!recorder.iterations.lock().unwrap().is_empty());
+    assert!(!recorder.slices.lock().unwrap().is_empty());
+    assert!(!recorder.converged.lock().unwrap().is_empty());
+}
+
+/// `?Sized` on the blanket impls means the observer can be chosen at runtime.
+#[test]
+fn a_trait_object_observer_works() {
+    let boxed: Box<dyn Observer<i64>> = Box::new(Recorder::default());
+    let mut h = History::builder().observer(boxed).build();
+    h.record_winner(&"a", &"b", 1).unwrap();
+    let _ = h.converge().unwrap();
+
+    let shared: Arc<dyn Observer<i64>> = Arc::new(Recorder::default());
+    let mut h = History::builder().observer(Arc::clone(&shared)).build();
+    h.record_winner(&"a", &"b", 1).unwrap();
+    let _ = h.converge().unwrap();
+}
+
+/// A non-shared observer can be reclaimed after convergence instead.
+#[test]
+fn into_observer_returns_the_accumulated_state() {
+    let mut h = History::builder().observer(Recorder::default()).build();
+    h.record_winner(&"a", &"b", 1).unwrap();
+    let _ = h.converge().unwrap();
+
+    // Readable in place...
+    assert!(!h.observer().iterations.lock().unwrap().is_empty());
+
+    // ...and reclaimable by value.
+    let recorder = h.into_observer();
+    assert!(!recorder.slices.lock().unwrap().is_empty());
+}
+
+/// Borrowing works too, for an observer that outlives the history.
+#[test]
+fn a_borrowed_observer_works() {
+    let recorder = Recorder::default();
+    {
+        let mut h = History::builder().observer(&recorder).build();
+        h.record_winner(&"a", &"b", 1).unwrap();
+        let _ = h.converge().unwrap();
+    }
+    assert!(!recorder.iterations.lock().unwrap().is_empty());
+}

tests/predict_margin.rs

@@ -0,0 +1,153 @@
+//! `predict_margin`: the predictive distribution of a scored matchup.
+
+use smallvec::smallvec;
+use trueskill_tt::{
+    ConstantDrift, ConvergenceOptions, Event, History, InferenceError, Member, Outcome, Team,
+    UnknownKeys,
+};
+
+fn builder(
+    policy: UnknownKeys,
+) -> History<i64, ConstantDrift, trueskill_tt::NullObserver, &'static str> {
+    History::builder()
+        .mu(0.0)
+        .sigma(6.0)
+        .beta(1.0)
+        .score_sigma(2.0)
+        .drift(ConstantDrift(0.0))
+        .unknown_keys(policy)
+        .convergence(ConvergenceOptions {
+            max_iter: 5_000,
+            epsilon: 1e-12,
+            alpha: 1.0,
+        })
+        .build()
+}
+
+fn round(a: &'static str, b: &'static str, sa: f64, sb: f64) -> Event<i64, &'static str> {
+    Event {
+        time: 1,
+        teams: smallvec![
+            Team::with_members([Member::new(a)]),
+            Team::with_members([Member::new(b)]),
+        ],
+        outcome: Outcome::scores([sa, sb]),
+    }
+}
+
+/// A history where "veteran" and "regular" are well observed and "novice"
+/// appears once.
+fn fitted(
+    policy: UnknownKeys,
+) -> History<i64, ConstantDrift, trueskill_tt::NullObserver, &'static str> {
+    let mut h = builder(policy);
+    let mut events: Vec<_> = (0..40)
+        .map(|t| round("veteran", "regular", 10.0 + f64::from(t % 3), 5.0))
+        .collect();
+    events.push(round("veteran", "novice", 10.0, 6.0));
+    h.add_events(events).unwrap();
+    let _ = h.converge().unwrap();
+    h
+}
+
+/// The property #48 exists for: the interval must widen when the model knows
+/// less. Their hand-fitted noise law quoted the same sigma for a competitor
+/// with forty rounds and one with none.
+#[test]
+fn the_interval_widens_as_the_model_knows_less() {
+    let h = fitted(UnknownKeys::Prior);
+
+    let well_known = h
+        .predict_margin(&[&[&"veteran"], &[&"regular"]])
+        .unwrap()
+        .sigma();
+    let thin = h
+        .predict_margin(&[&[&"veteran"], &[&"novice"]])
+        .unwrap()
+        .sigma();
+    let unseen = h
+        .predict_margin(&[&[&"veteran"], &[&"stranger"]])
+        .unwrap()
+        .sigma();
+
+    assert!(
+        well_known < thin && thin < unseen,
+        "margin width should grow as evidence thins: {well_known} < {thin} < {unseen}"
+    );
+}
+
+/// #48's second requirement: an unseen competitor is a legitimate question, not
+/// an error, and the answer should come from the prior rather than be faked.
+#[test]
+fn an_unseen_competitor_is_answered_from_the_prior() {
+    let h = fitted(UnknownKeys::Prior);
+    let g = h.predict_margin(&[&[&"nobody"], &[&"no_one"]]).unwrap();
+
+    // Two unknowns: the gap is centred on zero and carries both priors plus
+    // both performance noises plus the observation noise.
+    assert!(g.mu().abs() < 1e-9, "mu {}", g.mu());
+    let expected = (2.0 * 36.0 + 2.0 * 1.0 + 4.0f64).sqrt();
+    assert!(
+        (g.sigma() - expected).abs() < 1e-9,
+        "sigma {} vs expected {expected}",
+        g.sigma()
+    );
+}
+
+#[test]
+fn reject_still_rejects() {
+    let h = fitted(UnknownKeys::Reject);
+    assert!(matches!(
+        h.predict_margin(&[&[&"veteran"], &[&"stranger"]]),
+        Err(InferenceError::UnknownKey { .. })
+    ));
+}
+
+/// The margin is the *difference*, so it must be antisymmetric in the teams.
+#[test]
+fn swapping_the_teams_negates_the_margin() {
+    let h = fitted(UnknownKeys::Prior);
+    let forward = h.predict_margin(&[&[&"veteran"], &[&"regular"]]).unwrap();
+    let reverse = h.predict_margin(&[&[&"regular"], &[&"veteran"]]).unwrap();
+
+    assert!((forward.mu() + reverse.mu()).abs() < 1e-9);
+    assert!((forward.sigma() - reverse.sigma()).abs() < 1e-12);
+}
+
+/// The predictive interval must be wider than the skill gap alone: it also
+/// carries per-event performance noise and the observation noise.
+#[test]
+fn the_predictive_interval_exceeds_the_skill_uncertainty() {
+    let h = fitted(UnknownKeys::Prior);
+    let skill_gap = h
+        .posterior_of(&[(&"veteran", 1.0), (&"regular", -1.0)])
+        .unwrap();
+    let predictive = h.predict_margin(&[&[&"veteran"], &[&"regular"]]).unwrap();
+
+    assert!(
+        (predictive.mu() - skill_gap.mu()).abs() < 1e-12,
+        "means agree"
+    );
+    // beta^2 twice plus score_sigma^2 = 2 + 4.
+    let expected = (skill_gap.sigma().powi(2) + 6.0).sqrt();
+    assert!((predictive.sigma() - expected).abs() < 1e-12);
+    assert!(predictive.sigma() > skill_gap.sigma());
+}
+
+#[test]
+fn shape_errors_are_reported() {
+    let h = fitted(UnknownKeys::Prior);
+    assert!(matches!(
+        h.predict_margin(&[&[&"veteran"]]),
+        Err(InferenceError::MismatchedShape {
+            expected: 2,
+            got: 1,
+            ..
+        })
+    ));
+    let empty: [&&str; 0] = [];
+    assert!(matches!(
+        h.predict_margin(&[&[&"veteran"], &empty]),
+        Err(InferenceError::EmptyTeam { team: 1 })
+    ));
+}

tests/prediction.rs

@@ -0,0 +1,417 @@
+//! Prediction API: N-team outcomes, draw mass, and the error paths that used
+//! to be panics or silent wrong answers.
+
+use trueskill_tt::{History, InferenceError, MAX_PREDICTED_TEAMS};
+
+fn history_with(names: &[&'static str], p_draw: f64) -> History {
+    let mut h = History::builder().p_draw(p_draw).build();
+    // Give every competitor a recorded skill by playing a small round robin.
+    for pair in names.windows(2) {
+        h.record_winner(&pair[0], &pair[1], 1).unwrap();
+    }
+    let _ = h.converge().unwrap();
+    h
+}
+
+#[test]
+fn unknown_keys_are_reported_not_silently_dropped() {
+    let h = history_with(&["a", "b"], 0.0);
+
+    let err = h
+        .predict_outcome(&[&[&"a"], &[&"ghost"]])
+        .expect_err("an unknown key must not yield a confident prediction");
+    assert_eq!(
+        err,
+        InferenceError::UnknownKey {
+            team: 1,
+            member: 0,
+            key: "\"ghost\"".to_owned(),
+        }
+    );
+
+    // Every prediction entry point, not just one.
+    assert!(
+        h.predict_win_probabilities(&[&[&"a"], &[&"ghost"]])
+            .is_err()
+    );
+    assert!(h.predict_quality(&[&[&"a"], &[&"ghost"]]).is_err());
+    assert!(h.predict_ranking(&[&[&"a"], &[&"ghost"]], &[0, 1]).is_err());
+}
+
+#[test]
+fn an_entirely_unknown_team_is_an_error() {
+    let h = history_with(&["a", "b"], 0.0);
+    let err = h.predict_outcome(&[&[&"a"], &[&"x", &"y"]]).unwrap_err();
+    assert_eq!(
+        err,
+        InferenceError::UnknownKey {
+            team: 1,
+            member: 0,
+            key: "\"x\"".to_owned(),
+        }
+    );
+}
+
+#[test]
+fn degenerate_team_shapes_are_errors_rather_than_panics() {
+    let h = history_with(&["a", "b"], 0.0);
+
+    assert_eq!(
+        h.predict_outcome(&[&[&"a"]]).unwrap_err(),
+        InferenceError::NotEnoughTeams { got: 1 }
+    );
+    assert_eq!(
+        h.predict_outcome(&[]).unwrap_err(),
+        InferenceError::NotEnoughTeams { got: 0 }
+    );
+    assert_eq!(
+        h.predict_outcome(&[&[&"a"], &[]]).unwrap_err(),
+        InferenceError::EmptyTeam { team: 1 }
+    );
+}
+
+#[test]
+fn more_than_two_teams_no_longer_panics() {
+    let h = history_with(&["a", "b", "c"], 0.0);
+    let p = h
+        .predict_outcome(&[&[&"a"], &[&"b"], &[&"c"]])
+        .expect("three teams must be supported");
+    assert!((p.total() - 1.0).abs() < 1e-6, "total = {}", p.total());
+    // Three teams, no draws possible: exactly the six strict orderings.
+    assert_eq!(p.outcomes().len(), 6);
+}
+
+#[test]
+fn the_outcome_space_is_capped_rather_than_hanging() {
+    let names: Vec<&'static str> = vec!["a", "b", "c", "d", "e", "f", "g", "h"];
+    let h = history_with(&names, 0.0);
+
+    let teams: Vec<&[&&'static str]> = Vec::new();
+    let _ = teams;
+
+    let too_many: Vec<Vec<&&str>> = names.iter().map(|n| vec![n]).collect();
+    let refs: Vec<&[&&str]> = too_many.iter().map(Vec::as_slice).collect();
+
+    let err = h.predict_outcome(&refs).unwrap_err();
+    assert_eq!(
+        err,
+        InferenceError::TooManyTeams {
+            got: 8,
+            max: MAX_PREDICTED_TEAMS
+        }
+    );
+
+    // The cheap paths stay available at any size.
+    let wins = h.predict_win_probabilities(&refs).unwrap();
+    assert_eq!(wins.len(), 8);
+    assert!(
+        (wins.iter().sum::<f64>() - 1.0).abs() < 1e-6,
+        "win probabilities must still sum to one: {wins:?}"
+    );
+}
+
+/// The defect that made every draw-enabled prediction wrong: `[p, 1 - p]`
+/// allocated no mass to a draw even with `p_draw > 0`.
+#[test]
+fn a_draw_carries_probability_mass_when_p_draw_is_positive() {
+    let h = history_with(&["a", "b"], 0.25);
+    let p = h.predict_outcome(&[&[&"a"], &[&"b"]]).unwrap();
+
+    let draw = p.probability_of(&[0, 0]);
+    assert!(draw > 0.0, "a draw-enabled model must give draws mass");
+    assert!((p.total() - 1.0).abs() < 1e-6, "total = {}", p.total());
+
+    let wins = p.win_probabilities();
+    assert!(
+        (wins.iter().sum::<f64>() + draw - 1.0).abs() < 1e-6,
+        "wins {wins:?} plus draw {draw} must be the whole space"
+    );
+    assert!(
+        (p.shared_first_place() - draw).abs() < 1e-12,
+        "a two-team draw is a shared first place"
+    );
+}
+
+#[test]
+fn a_zero_draw_probability_admits_no_ties() {
+    let h = history_with(&["a", "b"], 0.0);
+    let p = h.predict_outcome(&[&[&"a"], &[&"b"]]).unwrap();
+    assert_eq!(p.probability_of(&[0, 0]), 0.0);
+    assert!(p.shared_first_place() < 1e-12);
+}
+
+/// The two routes to a win probability run through entirely different
+/// algorithms — adaptive quadrature versus the enumerated chain recursion —
+/// so agreement between them is a real cross-check, not a tautology.
+#[test]
+fn the_cheap_and_exhaustive_paths_agree() {
+    for p_draw in [0.0, 0.1] {
+        let h = history_with(&["a", "b", "c"], p_draw);
+        let teams: &[&[&&str]] = &[&[&"a"], &[&"b"], &[&"c"]];
+
+        let cheap = h.predict_win_probabilities(teams).unwrap();
+        let exhaustive = h.predict_outcome(teams).unwrap().win_probabilities();
+
+        for (i, (a, b)) in cheap.iter().zip(&exhaustive).enumerate() {
+            assert!(
+                (a - b).abs() < 1e-6,
+                "p_draw={p_draw} team {i}: quadrature {a} vs enumeration {b}"
+            );
+        }
+    }
+}
+
+#[test]
+fn predict_ranking_agrees_with_the_distribution() {
+    let h = history_with(&["a", "b", "c"], 0.1);
+    let teams: &[&[&&str]] = &[&[&"a"], &[&"b"], &[&"c"]];
+    let dist = h.predict_outcome(teams).unwrap();
+
+    for (ranks, expected) in dist.outcomes() {
+        let direct = h.predict_ranking(teams, ranks).unwrap();
+        assert!(
+            (direct - expected).abs() < 1e-9,
+            "ranks {ranks:?}: {direct} vs {expected}"
+        );
+    }
+}
+
+#[test]
+fn predict_ranking_checks_its_shape() {
+    let h = history_with(&["a", "b"], 0.0);
+    let err = h
+        .predict_ranking(&[&[&"a"], &[&"b"]], &[0, 1, 2])
+        .unwrap_err();
+    assert!(matches!(
+        err,
+        InferenceError::MismatchedShape {
+            expected: 2,
+            got: 3,
+            ..
+        }
+    ));
+}
+
+#[test]
+fn the_stronger_competitor_is_favoured() {
+    let mut h = History::builder().build();
+    for t in 1..=10 {
+        h.record_winner(&"strong", &"weak", t).unwrap();
+    }
+    let _ = h.converge().unwrap();
+
+    let p = h.predict_outcome(&[&[&"strong"], &[&"weak"]]).unwrap();
+    let (best, _) = p.most_likely().expect("a most likely outcome");
+    assert_eq!(best, &[0, 1], "the winner should be favoured");
+
+    let wins = p.win_probabilities();
+    assert!(wins[0] > wins[1], "{wins:?}");
+}
+
+/// Unequal team sizes change the draw margin, because inference derives it
+/// from the teams' betas. Prediction has to follow, or it describes a
+/// different model than the one that will be fitted.
+#[test]
+fn team_size_affects_the_prediction() {
+    let mut h = History::builder().p_draw(0.2).build();
+    h.event(1)
+        .team(["a", "b"])
+        .team(["c"])
+        .winner(0)
+        .commit()
+        .unwrap();
+    let _ = h.converge().unwrap();
+
+    let p = h.predict_outcome(&[&[&"a", &"b"], &[&"c"]]).unwrap();
+    assert!((p.total() - 1.0).abs() < 1e-6, "total = {}", p.total());
+    assert!(p.probability_of(&[0, 0]) > 0.0);
+}
+
+// ---------------------------------------------------------------------------
+// Expected information gain
+// ---------------------------------------------------------------------------
+
+/// The whole point of #39: "which comparison should I run next?" is a
+/// different question from "who will win?" or "is this fair?".
+#[test]
+fn information_gain_prefers_the_uncertain_pairing() {
+    let mut h = History::builder().build();
+
+    // "known" and "rival" have played a lot; "newcomer" has played once.
+    for t in 1..=15 {
+        h.record_winner(&"known", &"rival", t).unwrap();
+        h.record_winner(&"rival", &"known", t + 100).unwrap();
+    }
+    h.record_winner(&"known", &"newcomer", 500).unwrap();
+    let _ = h.converge().unwrap();
+
+    let settled = h
+        .expected_information_gain(&[&[&"known"], &[&"rival"]])
+        .unwrap();
+    let unknown = h
+        .expected_information_gain(&[&[&"known"], &[&"newcomer"]])
+        .unwrap();
+
+    assert!(
+        unknown > settled,
+        "pairing against the newcomer should teach more: {unknown} vs {settled}"
+    );
+}
+
+/// The analytic ceiling, through the `History` entry point rather than the
+/// standalone one.
+#[test]
+fn information_gain_respects_the_entropy_ceiling() {
+    let h = history_with(&["a", "b", "c"], 0.0);
+
+    let two = h.expected_information_gain(&[&[&"a"], &[&"b"]]).unwrap();
+    assert!(
+        (0.0..=std::f64::consts::LN_2).contains(&two),
+        "two-team EIG {two} outside [0, ln 2]"
+    );
+
+    let three = h
+        .expected_information_gain(&[&[&"a"], &[&"b"], &[&"c"]])
+        .unwrap();
+    assert!(
+        (0.0..=6.0f64.ln()).contains(&three),
+        "three-team EIG {three} outside [0, ln 6]"
+    );
+}
+
+#[test]
+fn information_gain_reports_unknown_keys() {
+    let h = history_with(&["a", "b"], 0.0);
+    assert_eq!(
+        h.expected_information_gain(&[&[&"a"], &[&"ghost"]])
+            .unwrap_err(),
+        InferenceError::UnknownKey {
+            team: 1,
+            member: 0,
+            key: "\"ghost\"".to_owned(),
+        }
+    );
+}
+
+/// A draw-enabled history has three outcomes to weigh rather than two, so the
+/// draw branch must actually be reachable through this path.
+#[test]
+fn information_gain_accounts_for_draws() {
+    let with_draws = history_with(&["a", "b"], 0.25);
+    let g = with_draws
+        .expected_information_gain(&[&[&"a"], &[&"b"]])
+        .unwrap();
+    assert!(g > 0.0 && g <= 3.0f64.ln(), "{g}");
+
+    // The draw outcome carries mass, so it is genuinely being weighed.
+    let dist = with_draws.predict_outcome(&[&[&"a"], &[&"b"]]).unwrap();
+    assert!(dist.probability_of(&[0, 0]) > 0.0);
+}
+
+/// The defect that cost a consumer a day: `UnknownKey { team: 0, member: 0 }`
+/// says nothing about *which* key is unknown, so the natural handling — log it,
+/// fall back to a neutral value — converts a total miss into a plausible
+/// constant. The key has to be in the error, and in its `Display`.
+#[test]
+fn unknown_key_names_the_key_it_could_not_find() {
+    let h = history_with(&["a", "b"], 0.0);
+    let err = h.predict_outcome(&[&[&"a"], &[&"never_seen"]]).unwrap_err();
+
+    match &err {
+        InferenceError::UnknownKey { key, .. } => {
+            assert!(
+                key.contains("never_seen"),
+                "the error should name the key, got {key}"
+            );
+        }
+        other => panic!("expected UnknownKey, got {other:?}"),
+    }
+
+    let rendered = err.to_string();
+    assert!(
+        rendered.contains("never_seen"),
+        "Display should name the key: {rendered}"
+    );
+    assert!(
+        rendered.contains("pre-filter"),
+        "Display should say what to do about it: {rendered}"
+    );
+}
+
+// ---------------------------------------------------------------------------
+// UnknownKeys policy
+// ---------------------------------------------------------------------------
+
+fn history_with_policy(names: &[&'static str], policy: trueskill_tt::UnknownKeys) -> History {
+    let mut h = History::builder().unknown_keys(policy).build();
+    for pair in names.windows(2) {
+        h.record_winner(&pair[0], &pair[1], 1).unwrap();
+    }
+    let _ = h.converge().unwrap();
+    h
+}
+
+#[test]
+fn reject_is_the_default() {
+    let h = history_with(&["a", "b"], 0.0);
+    assert!(matches!(
+        h.predict_outcome(&[&[&"a"], &[&"ghost"]]),
+        Err(InferenceError::UnknownKey { .. })
+    ));
+}
+
+#[test]
+fn prior_answers_instead_of_erroring() {
+    let h = history_with_policy(&["a", "b"], trueskill_tt::UnknownKeys::Prior);
+    let p = h
+        .predict_outcome(&[&[&"a"], &[&"ghost"]])
+        .expect("Prior should answer rather than reject");
+    assert!((p.total() - 1.0).abs() < 1e-6);
+}
+
+/// Two competitors the model has never seen are genuinely a coin flip. The
+/// point is that this is now *derived* rather than a constant a caller
+/// substitutes after swallowing an error.
+#[test]
+fn two_unknown_competitors_are_an_honest_coin_flip() {
+    let h = history_with_policy(&["a", "b"], trueskill_tt::UnknownKeys::Prior);
+    let wins = h
+        .predict_win_probabilities(&[&[&"nobody"], &[&"no_one"]])
+        .unwrap();
+    assert!((wins[0] - 0.5).abs() < 1e-9, "{wins:?}");
+    assert!((wins[1] - 0.5).abs() < 1e-9, "{wins:?}");
+}
+
+/// The property that rules out a `Skip` mode: an unknown member must make a
+/// team *less* certain, never more. Skipping would drop the member's variance
+/// from the sum and narrow the team, which is backwards.
+#[test]
+fn an_unknown_member_widens_its_team_rather_than_narrowing_it() {
+    let h = history_with_policy(&["a", "b", "c"], trueskill_tt::UnknownKeys::Prior);
+
+    // "a" alone against "b" — then "a" plus an unknown partner against "b".
+    let solo = h.predict_win_probabilities(&[&[&"a"], &[&"b"]]).unwrap();
+    let with_unknown = h
+        .predict_win_probabilities(&[&[&"a", &"stranger"], &[&"b"]])
+        .unwrap();
+
+    // Adding an unknown partner pulls the outcome toward even, because the
+    // team's performance spread grew.
+    assert!(
+        (with_unknown[0] - 0.5).abs() < (solo[0] - 0.5).abs(),
+        "an unknown partner should make the result less certain: solo {solo:?}, \
+         with unknown {with_unknown:?}"
+    );
+}
+
+#[test]
+fn prior_reaches_every_prediction_entry_point() {
+    let h = history_with_policy(&["a", "b"], trueskill_tt::UnknownKeys::Prior);
+    let teams: &[&[&&str]] = &[&[&"a"], &[&"ghost"]];
+
+    assert!(h.predict_quality(teams).is_ok());
+    assert!(h.predict_win_probabilities(teams).is_ok());
+    assert!(h.predict_outcome(teams).is_ok());
+    assert!(h.predict_ranking(teams, &[0, 1]).is_ok());
+    assert!(h.expected_information_gain(teams).is_ok());
+}

tests/properties.proptest-regressions

@@ -0,0 +1,7 @@
+# Seeds for failure cases proptest has generated in the past. It is
+# automatically read and these particular cases re-run before any
+# novel cases are generated.
+#
+# It is recommended to check this file in to source control so that
+# everyone who runs the test benefits from these saved cases.
+cc 8859be600e638573980f78622b8fcd8b4553ca34a9a041746c417f7e4293f89c # shrinks to games = [(0, 1), (0, 1), (4, 6), (2, 0), (0, 1), (0, 6), (0, 1), (2, 0), (6, 4), (0, 1), (0, 2), (6, 4), (1, 0), (4, 0), (0, 2)]

tests/properties.rs

@@ -0,0 +1,183 @@
+//! Property-based tests over generated histories.
+//!
+//! The golden suite pins exact values against the Python/Julia reference on a
+//! handful of fixtures. These pin *invariants* over inputs nobody wrote by
+//! hand, which is where the defects this crate has actually shipped were
+//! hiding: a linear evidence product that underflowed only past ~1000 teams,
+//! and a batching path no golden exercised because every golden ingests in one
+//! call.
+
+mod common;
+
+use common::assert_finite;
+use proptest::prelude::*;
+use smallvec::smallvec;
+use trueskill_tt::{ConvergenceOptions, Event, History, Member, Outcome, Team};
+
+/// Distinct competitors, so no event pits someone against themselves.
+fn pairs() -> impl Strategy<Value = Vec<(usize, usize)>> {
+    prop::collection::vec((0usize..8, 0usize..8), 1..24)
+        .prop_map(|v| v.into_iter().filter(|(a, b)| a != b).collect::<Vec<_>>())
+        .prop_filter("needs at least one valid pair", |v| !v.is_empty())
+}
+
+const KEYS: [&str; 8] = ["a", "b", "c", "d", "e", "f", "g", "h"];
+
+fn history_from(games: &[(usize, usize)]) -> History {
+    let mut h = History::builder()
+        .convergence(ConvergenceOptions {
+            // 200 was not enough: the batched side stopped at the cap with a
+            // step of 3.4e-9, so this test was comparing two truncated fits and
+            // attributing the gap to ingestion order.
+            max_iter: 20_000,
+            epsilon: 1e-10,
+            ..ConvergenceOptions::default()
+        })
+        .build();
+
+    let events: Vec<Event<i64, &'static str>> = games
+        .iter()
+        .enumerate()
+        .map(|(i, &(a, b))| Event {
+            time: i as i64 + 1,
+            teams: smallvec![
+                Team::with_members([Member::new(KEYS[a])]),
+                Team::with_members([Member::new(KEYS[b])]),
+            ],
+            outcome: Outcome::winner(0, 2),
+        })
+        .collect();
+
+    h.add_events(events).unwrap();
+
+    h
+}
+
+proptest! {
+    #![proptest_config(ProptestConfig::with_cases(48))]
+
+    /// Whatever the schedule of games, convergence must not produce NaN or an
+    /// improper posterior. `converge` returns `NonFiniteResult` rather than
+    /// silently reporting a NaN step as converged, so a break shows up here as
+    /// either an Err or a non-finite curve point.
+    #[test]
+    fn converged_posteriors_are_always_finite(games in pairs()) {
+        let mut h = history_from(&games);
+
+        let _ = h.converge().unwrap();
+
+        for key in KEYS {
+            for (time, g) in h.learning_curve(key) {
+                assert_finite(g, &format!("{key} at t={time}"));
+            }
+        }
+    }
+
+    /// Log-evidence is a log probability: finite, and never above zero.
+    ///
+    /// The linear-product implementation this replaced underflowed to zero on
+    /// long chains, making `ln(0)` = -inf — finite-ness is the property that
+    /// would have caught it.
+    #[test]
+    fn log_evidence_is_a_finite_log_probability(games in pairs()) {
+        let mut h = history_from(&games);
+
+        let _ = h.converge().unwrap();
+
+        let batch = h.log_evidence();
+        let filtered = h.filtered_log_evidence();
+
+        prop_assert!(batch.is_finite(), "batch log-evidence {batch} is not finite");
+        prop_assert!(batch <= 0.0, "batch log-evidence {batch} exceeds zero");
+        prop_assert!(filtered.is_finite(), "filtered log-evidence {filtered} is not finite");
+        prop_assert!(filtered <= 0.0, "filtered log-evidence {filtered} exceeds zero");
+    }
+
+    /// Filtered estimates must not depend on whether `converge` has run — the
+    /// property the whole forward-only design rests on.
+    #[test]
+    fn filtered_evidence_is_invariant_to_convergence(games in pairs()) {
+        let mut h = history_from(&games);
+
+        let before = h.filtered_log_evidence();
+
+        let _ = h.converge().unwrap();
+
+        let after = h.filtered_log_evidence();
+
+        prop_assert!(
+            (before - after).abs() < 1e-8,
+            "filtered evidence moved across converge(): {before} -> {after}"
+        );
+    }
+
+    /// Ingesting the same games one at a time must reach the same fixed point
+    /// as ingesting them in one call.
+    #[test]
+    fn ingestion_order_does_not_change_the_answer(games in pairs()) {
+        let batched = {
+            let mut h = history_from(&games);
+            let report = h.converge().unwrap();
+            prop_assert!(
+                report.converged,
+                "batched side stopped at {} iterations with step {:?}; comparing \
+                 two fits that have not converged measures truncation, not order",
+                report.iterations,
+                report.final_step
+            );
+            h
+        };
+
+        let incremental = {
+            let mut h = History::builder()
+                .convergence(ConvergenceOptions {
+                    max_iter: 20_000,
+                    epsilon: 1e-10,
+                    ..ConvergenceOptions::default()
+                })
+                .build();
+
+            for (i, &(a, b)) in games.iter().enumerate() {
+                h.add_events([Event {
+                    time: i as i64 + 1,
+                    teams: smallvec![
+                        Team::with_members([Member::new(KEYS[a])]),
+                        Team::with_members([Member::new(KEYS[b])]),
+                    ],
+                    outcome: Outcome::winner(0, 2),
+                }])
+                .unwrap();
+            }
+
+            let report = h.converge().unwrap();
+            prop_assert!(
+                report.converged,
+                "incremental side stopped at {} iterations with step {:?}",
+                report.iterations,
+                report.final_step
+            );
+            h
+        };
+
+        for key in KEYS {
+            let one = batched.current_skill(key);
+            let other = incremental.current_skill(key);
+
+            match (one, other) {
+                (Some(one), Some(other)) => {
+                    prop_assert!(
+                        (one.mu() - other.mu()).abs() < 1e-6
+                            && (one.sigma() - other.sigma()).abs() < 1e-6,
+                        "{key}: batched mu={} sigma={}, incremental mu={} sigma={}",
+                        one.mu(),
+                        one.sigma(),
+                        other.mu(),
+                        other.sigma()
+                    );
+                }
+                (None, None) => {}
+                _ => prop_assert!(false, "{key} present in only one history"),
+            }
+        }
+    }
+}

tests/quality.rs

@@ -108,12 +108,59 @@ fn history_predict_quality_supports_three_teams() {
     let mut h = History::default();
     h.record_winner(&"a", &"b", 1).unwrap();
     h.record_winner(&"b", &"c", 2).unwrap();
-    h.converge().unwrap();
+    let _ = h.converge().unwrap();
 
-    let q = h.predict_quality(&[&[&"a"], &[&"b"], &[&"c"]]);
+    let q = h.predict_quality(&[&[&"a"], &[&"b"], &[&"c"]]).unwrap();
     assert!(
         q.is_finite(),
         "3-team predict_quality must be finite, got {q}"
     );
     assert!((0.0..=1.0).contains(&q), "out of range: {q}");
 }
+
+/// `quality()` for N identical teams has a closed form, which pins the N-group
+/// determinant path across the whole range rather than at a single golden.
+///
+/// For two identical single-player teams the standard result is
+/// `sqrt(2b^2 / (2b^2 + s1^2 + s2^2))`. With the conventional parameters
+/// (`sigma = 25/3`, `beta = 25/6`) that ratio is exactly `1/5`, and the N-group
+/// generalisation is `(1/5)^((n-1)/2)` — one factor per adjacent pair.
+///
+/// The n=3 and n=5 values this produces (0.200 and 0.040) are also what the
+/// `trueskill` Python package returns for the same configuration, so this
+/// doubles as the cross-implementation check the README asked for.
+#[test]
+fn quality_of_identical_teams_follows_its_closed_form() {
+    let g = Gaussian::from_ms(25.0, 25.0 / 3.0);
+    let beta = 25.0 / 6.0;
+
+    for n in 2..=10usize {
+        let groups: Vec<Vec<Gaussian>> = (0..n).map(|_| vec![g]).collect();
+        let refs: Vec<&[Gaussian]> = groups.iter().map(Vec::as_slice).collect();
+
+        let got = quality(&refs, beta);
+        let expected = 0.2f64.powf((n - 1) as f64 / 2.0);
+
+        assert!(
+            (got - expected).abs() / expected < 1e-9,
+            "n={n}: quality {got}, closed form {expected}"
+        );
+    }
+}
+
+/// Spot-check against the two values the `trueskill` Python package is known
+/// to produce for this configuration, stated as literals so a future change to
+/// the closed-form reasoning above cannot quietly take these with it.
+#[test]
+fn quality_matches_the_reference_implementation() {
+    let g = Gaussian::from_ms(25.0, 25.0 / 3.0);
+    let beta = 25.0 / 6.0;
+
+    let three: Vec<Vec<Gaussian>> = (0..3).map(|_| vec![g]).collect();
+    let refs: Vec<&[Gaussian]> = three.iter().map(Vec::as_slice).collect();
+    assert!((quality(&refs, beta) - 0.200).abs() < 1e-9);
+
+    let five: Vec<Vec<Gaussian>> = (0..5).map(|_| vec![g]).collect();
+    let refs: Vec<&[Gaussian]> = five.iter().map(Vec::as_slice).collect();
+    assert!((quality(&refs, beta) - 0.040).abs() < 1e-9);
+}

tests/reconvergence_equivalence.rs

@@ -0,0 +1,165 @@
+//! Converging, appending, and converging again must reach the same fixed point
+//! as converging once over the whole event set.
+//!
+//! `tests/ingestion_equivalence.rs` covers a different question: it varies how
+//! events are *batched* but converges only at the end. This file converges
+//! between batches, which is the path a caller takes when it fits, serves for a
+//! while, then ingests more.
+//!
+//! The property matters beyond ergonomics. It says `converge` reaches a fixed
+//! point determined by the events, ratings and configuration alone — not by the
+//! message state it started from. That is what makes a restored snapshot safe:
+//! an inexact one cannot corrupt the answer, only cost an extra sweep. See #45.
+
+use smallvec::smallvec;
+use trueskill_tt::{ConvergenceOptions, Event, Gaussian, History, Member, Outcome, Team};
+
+fn tight() -> ConvergenceOptions {
+    ConvergenceOptions {
+        max_iter: 5_000,
+        epsilon: 1e-12,
+        alpha: 1.0,
+    }
+}
+
+fn ev(a: &str, b: &str, time: i64) -> Event<i64, String> {
+    Event {
+        time,
+        teams: smallvec![
+            Team::with_members([Member::new(a.to_string())]),
+            Team::with_members([Member::new(b.to_string())]),
+        ],
+        outcome: Outcome::winner(0, 2),
+    }
+}
+
+/// Ingest each chunk in turn, converging fully after every one.
+fn fit_in_chunks(chunks: Vec<Events>) -> Vec<(String, Gaussian)> {
+    let mut h: History<i64, _, _, String> =
+        History::builder_with_key().convergence(tight()).build();
+
+    for chunk in chunks {
+        h.add_events(chunk).unwrap();
+        let report = h.converge().unwrap();
+        assert!(
+            report.converged,
+            "a chunk failed to converge, so any comparison would be measuring \
+             truncation rather than the fixed point; final step {:?}",
+            report.final_step
+        );
+    }
+
+    let mut skills: Vec<(String, Gaussian)> = h
+        .learning_curves()
+        .into_iter()
+        .map(|(k, curve)| (k, curve.last().unwrap().1))
+        .collect();
+    skills.sort_by(|a, b| a.0.cmp(&b.0));
+    skills
+}
+
+fn assert_same(a: &[(String, Gaussian)], b: &[(String, Gaussian)], what: &str) {
+    assert_eq!(a.len(), b.len(), "{what}: competitor count differs");
+    for ((ka, ga), (kb, gb)) in a.iter().zip(b) {
+        assert_eq!(ka, kb, "{what}: key order differs");
+        // Measured: 6.2e-13 for a later append, 8.9e-11 for an interleaved one.
+        // The bar is well clear of both but far under anything that would let a
+        // genuine divergence through.
+        assert!(
+            (ga.mu() - gb.mu()).abs() < 1e-8 && (ga.sigma() - gb.sigma()).abs() < 1e-8,
+            "{what}: {ka} differs — one-shot mu={} sigma={}, chunked mu={} sigma={}",
+            ga.mu(),
+            ga.sigma(),
+            gb.mu(),
+            gb.sigma()
+        );
+    }
+}
+
+type Events = Vec<Event<i64, String>>;
+
+/// Two chunks of events: the first at times 0..20, the second at 100..120.
+fn fixture() -> (Events, Events) {
+    let names = ["a", "b", "c", "d", "e"];
+    let mut seed = 7u64;
+    let mut rnd = move || {
+        seed ^= seed << 13;
+        seed ^= seed >> 7;
+        seed ^= seed << 17;
+        seed
+    };
+
+    let (mut early, mut late) = (Vec::new(), Vec::new());
+    for t in 0..40i64 {
+        let i = (rnd() % 5) as usize;
+        let mut j = (rnd() % 5) as usize;
+        if j == i {
+            j = (j + 1) % 5;
+        }
+        if t < 20 {
+            early.push(ev(names[i], names[j], t));
+        } else {
+            late.push(ev(names[i], names[j], 100 + t));
+        }
+    }
+    (early, late)
+}
+
+/// The ordinary case: new events are strictly later than everything fitted.
+#[test]
+fn appending_later_events_matches_a_single_fit() {
+    let (early, late) = fixture();
+    let all: Vec<_> = early.iter().cloned().chain(late.iter().cloned()).collect();
+
+    assert_same(
+        &fit_in_chunks(vec![all]),
+        &fit_in_chunks(vec![early, late]),
+        "append strictly later",
+    );
+}
+
+/// The case the design question suspected might be weaker: appended events
+/// interleave with slices that are already fitted, so the append legitimately
+/// revises the past. It is not weaker — Through Time revises the past on every
+/// converge regardless, so there is nothing special about doing it in two steps.
+#[test]
+fn appending_interleaved_events_matches_a_single_fit() {
+    let (early, late) = fixture();
+    let all: Vec<_> = early.iter().cloned().chain(late.iter().cloned()).collect();
+
+    // Split by parity so the second chunk is back-dated into the first's range.
+    let first: Vec<_> = all.iter().step_by(2).cloned().collect();
+    let second: Vec<_> = all.iter().skip(1).step_by(2).cloned().collect();
+    let together: Vec<_> = first
+        .iter()
+        .cloned()
+        .chain(second.iter().cloned())
+        .collect();
+
+    assert_same(
+        &fit_in_chunks(vec![together]),
+        &fit_in_chunks(vec![first, second]),
+        "append interleaved",
+    );
+}
+
+/// Converging an already-converged history is a no-op, which is what makes a
+/// restored snapshot worth having: the work is skipped rather than redone.
+#[test]
+fn re_converging_an_unchanged_history_costs_one_iteration() {
+    let (early, late) = fixture();
+    let all: Vec<_> = early.into_iter().chain(late).collect();
+
+    let mut h: History<i64, _, _, String> =
+        History::builder_with_key().convergence(tight()).build();
+    h.add_events(all).unwrap();
+    let first = h.converge().unwrap();
+    assert!(first.converged);
+
+    let again = h.converge().unwrap();
+    assert_eq!(
+        again.iterations, 1,
+        "a converged history should settle immediately, not re-grind"
+    );
+    assert!(again.converged);
+}

tests/record_winner.rs

@@ -15,7 +15,7 @@ fn record_winner_builds_history() {
         .build();
 
     h.record_winner(&"alice", &"bob", 1).unwrap();
-    h.converge().unwrap();
+    let _ = h.converge().unwrap();
 
     let a_idx = h.lookup(&"alice").unwrap();
     let b_idx = h.lookup(&"bob").unwrap();
@@ -48,7 +48,7 @@ fn record_draw_with_p_draw_set() {
         .build();
 
     h.record_draw(&"alice", &"bob", 1).unwrap();
-    h.converge().unwrap();
+    let _ = h.converge().unwrap();
 
     assert!(h.lookup(&"alice").is_some());
     assert!(h.lookup(&"bob").is_some());

tests/registration.rs

@@ -0,0 +1,338 @@
+//! Configuring a competitor before anything is observed about them.
+//!
+//! The configuration a competitor needs is usually a property of the domain —
+//! "every layout is static" — not of whichever event happens to mention them
+//! first. Stating it per-event meant every ingestion path had to remember it,
+//! and two of the four paths could not state it at all.
+
+use smallvec::smallvec;
+use trueskill_tt::{
+    ConstantDrift, ConvergenceOptions, Event, Gaussian, History, InferenceError, Member, Outcome,
+    Team,
+};
+
+type H = History<i64, ConstantDrift, trueskill_tt::NullObserver, &'static str>;
+
+const PINNED: Gaussian = Gaussian::from_ms(2.0, 0.5);
+
+fn history() -> H {
+    History::builder()
+        .mu(0.0)
+        .sigma(6.0)
+        .beta(1.0)
+        .score_sigma(2.0)
+        .drift(ConstantDrift(0.5))
+        .convergence(ConvergenceOptions {
+            max_iter: 20_000,
+            epsilon: 1e-13,
+            alpha: 1.0,
+        })
+        .build()
+}
+
+fn duel(
+    a: &'static str,
+    b: &'static str,
+    t: i64,
+    m: Option<Member<&'static str>>,
+) -> Event<i64, &'static str> {
+    Event {
+        time: t,
+        teams: smallvec![
+            Team::with_members([Member::new(a)]),
+            Team::with_members([m.unwrap_or_else(|| Member::new(b))]),
+        ],
+        outcome: Outcome::scores([5.0, 2.0]),
+    }
+}
+
+fn skills(h: &H) -> Vec<(&'static str, Gaussian)> {
+    ["player", "layout"]
+        .into_iter()
+        .map(|k| (k, h.current_skill(&k).unwrap()))
+        .collect()
+}
+
+/// The headline contract.
+#[test]
+fn registering_matches_configuring_on_the_first_event() {
+    let configured = {
+        let mut h = history();
+        h.add_events(vec![
+            duel(
+                "player",
+                "layout",
+                1,
+                Some(
+                    Member::new("layout")
+                        .with_drift_scale(0.0)
+                        .with_prior(PINNED),
+                ),
+            ),
+            duel("player", "layout", 2, None),
+        ])
+        .unwrap();
+        let _ = h.converge().unwrap();
+        h
+    };
+
+    let registered = {
+        let mut h = history();
+        h.register(
+            Member::new("layout")
+                .with_drift_scale(0.0)
+                .with_prior(PINNED),
+        )
+        .unwrap();
+        h.add_events(vec![
+            duel("player", "layout", 1, None),
+            duel("player", "layout", 2, None),
+        ])
+        .unwrap();
+        let _ = h.converge().unwrap();
+        h
+    };
+
+    for ((k, a), (_, b)) in skills(&configured).into_iter().zip(skills(&registered)) {
+        assert_eq!(a.pi(), b.pi(), "{k} pi");
+        assert_eq!(a.tau(), b.tau(), "{k} tau");
+    }
+}
+
+/// The case `EventBuilder` and the typed path cannot reach: a competitor whose
+/// first appearance arrives through the two-argument convenience route.
+#[test]
+fn registration_reaches_a_competitor_first_seen_through_record_winner() {
+    let mut h = history();
+    h.register(
+        Member::new("layout")
+            .with_drift_scale(0.0)
+            .with_prior(PINNED),
+    )
+    .unwrap();
+    h.record_winner(&"player", &"layout", 1).unwrap();
+    h.record_winner(&"player", &"layout", 2).unwrap();
+    let _ = h.converge().unwrap();
+
+    let rating = h.rating(&"layout").unwrap();
+    assert_eq!(rating.drift_scale(), 0.0);
+    assert_eq!(rating.prior().mu(), PINNED.mu());
+
+    // Pinned means pinned: no drift across the two slices.
+    let curve = h.learning_curve(&"layout");
+    assert!(curve.len() >= 2);
+    let widest = curve
+        .iter()
+        .map(|(_, g)| g.sigma())
+        .fold(f64::MIN, f64::max);
+    let narrowest = curve
+        .iter()
+        .map(|(_, g)| g.sigma())
+        .fold(f64::MAX, f64::min);
+    assert!(
+        (widest - narrowest) / widest < 1e-9,
+        "{narrowest} .. {widest}"
+    );
+}
+
+#[test]
+fn registering_a_known_competitor_is_an_error() {
+    let mut h = history();
+    h.record_winner(&"player", &"layout", 1).unwrap();
+    let err = h.register(Member::new("layout")).unwrap_err();
+    assert!(
+        matches!(err, InferenceError::AlreadyRegistered { .. }),
+        "{err:?}"
+    );
+}
+
+#[test]
+fn registering_twice_is_an_error() {
+    let mut h = history();
+    h.register(Member::new("layout").with_drift_scale(0.0))
+        .unwrap();
+    let err = h
+        .register(Member::new("layout").with_drift_scale(1.0))
+        .unwrap_err();
+    assert!(
+        matches!(err, InferenceError::AlreadyRegistered { .. }),
+        "{err:?}"
+    );
+    // The first registration stands.
+    assert_eq!(h.rating(&"layout").unwrap().drift_scale(), 0.0);
+}
+
+/// `weight` is per-event and meaningless here, so it is rejected rather than
+/// dropped — dropping it silently is the defect class this whole area keeps
+/// producing.
+#[test]
+fn a_weight_on_a_registration_is_rejected() {
+    let mut h = history();
+    let err = h
+        .register(Member::new("layout").with_weight(0.5))
+        .unwrap_err();
+    assert!(
+        matches!(err, InferenceError::InvalidParameter { name: "weight", .. }),
+        "{err:?}"
+    );
+}
+
+#[test]
+fn an_invalid_drift_scale_on_a_registration_is_rejected() {
+    for bad in [-1.0, f64::NAN, f64::INFINITY] {
+        let mut h = history();
+        let err = h
+            .register(Member::new("layout").with_drift_scale(bad))
+            .unwrap_err();
+        assert!(
+            matches!(
+                err,
+                InferenceError::InvalidParameter {
+                    name: "drift_scale",
+                    ..
+                }
+            ),
+            "{bad}: {err:?}"
+        );
+    }
+}
+
+/// Registration makes the fit independent of the order events arrive in,
+/// which is what the per-event shape could not guarantee.
+#[test]
+fn registration_makes_the_fit_order_independent() {
+    let build = |reversed: bool| {
+        let mut h = history();
+        h.register(
+            Member::new("layout")
+                .with_drift_scale(0.0)
+                .with_prior(PINNED),
+        )
+        .unwrap();
+        let mut events = vec![
+            duel("player", "layout", 1, None),
+            duel("player", "layout", 2, None),
+            duel("player", "layout", 3, None),
+        ];
+        if reversed {
+            events.reverse();
+        }
+        h.add_events(events).unwrap();
+        let _ = h.converge().unwrap();
+        h
+    };
+
+    let forward = build(false);
+    let backward = build(true);
+    for ((k, a), (_, b)) in skills(&forward).into_iter().zip(skills(&backward)) {
+        assert_eq!(a.pi(), b.pi(), "{k} pi");
+        assert_eq!(a.tau(), b.tau(), "{k} tau");
+    }
+}
+
+/// `rating` is the read-back that made a configuration mistake detectable from
+/// outside the crate at all. Every other accessor reports what inference
+/// inferred; this reports what it was told.
+#[test]
+fn rating_reads_back_what_was_stored() {
+    let mut h = history();
+    assert!(h.rating(&"nobody").is_none());
+
+    h.register(
+        Member::new("layout")
+            .with_drift_scale(0.25)
+            .with_prior(PINNED),
+    )
+    .unwrap();
+    let r = h.rating(&"layout").unwrap();
+    assert_eq!(r.drift_scale(), 0.25);
+    assert_eq!(r.prior().pi(), PINNED.pi());
+    assert_eq!(r.prior().tau(), PINNED.tau());
+
+    // A competitor created by an event reports the history defaults.
+    h.record_winner(&"player", &"layout", 1).unwrap();
+    assert_eq!(h.rating(&"player").unwrap().drift_scale(), 1.0);
+}
+
+/// The decision this issue turned on: two different values for one competitor
+/// are an error whether they arrive in one batch or two.
+///
+/// Last-write-wins across batches cut against the invariant
+/// `tests/ingestion_equivalence.rs` protects — the same contradictory events
+/// errored when batched and succeeded, order-dependently, one at a time.
+mod conflicting_configuration {
+    use super::*;
+
+    fn seed(scale: f64) -> Event<i64, &'static str> {
+        duel(
+            "player",
+            "layout",
+            1,
+            Some(Member::new("layout").with_drift_scale(scale)),
+        )
+    }
+
+    #[test]
+    fn within_one_batch_is_an_error() {
+        let mut h = history();
+        let err = h.add_events(vec![seed(0.0), seed(1.0)]).unwrap_err();
+        assert!(
+            matches!(
+                err,
+                InferenceError::ConflictingCompetitorConfig {
+                    field: "drift_scale",
+                    ..
+                }
+            ),
+            "{err:?}"
+        );
+    }
+
+    #[test]
+    fn across_two_batches_is_also_an_error() {
+        let mut h = history();
+        h.add_events(vec![seed(0.0)]).unwrap();
+        let err = h.add_events(vec![seed(1.0)]).unwrap_err();
+        assert!(
+            matches!(
+                err,
+                InferenceError::ConflictingCompetitorConfig {
+                    field: "drift_scale",
+                    ..
+                }
+            ),
+            "{err:?}"
+        );
+        // Rejected before anything mutates: the first declaration stands.
+        assert_eq!(h.rating(&"layout").unwrap().drift_scale(), 0.0);
+    }
+
+    /// Repeating the *same* value stays inert, which is the expected shape
+    /// when the configuration is a property of the domain.
+    #[test]
+    fn repeating_the_same_value_is_inert() {
+        let mut h = history();
+        h.add_events(vec![seed(0.0)]).unwrap();
+        h.add_events(vec![seed(0.0)]).unwrap();
+        assert_eq!(h.rating(&"layout").unwrap().drift_scale(), 0.0);
+    }
+
+    /// A registration and a later event that agree are fine; one that
+    /// disagrees is the same error.
+    #[test]
+    fn a_registration_conflicts_with_a_later_event() {
+        let mut h = history();
+        h.register(Member::new("layout").with_drift_scale(0.0))
+            .unwrap();
+        h.add_events(vec![seed(0.0)]).unwrap();
+
+        let mut h2 = history();
+        h2.register(Member::new("layout").with_drift_scale(0.0))
+            .unwrap();
+        let err = h2.add_events(vec![seed(1.0)]).unwrap_err();
+        assert!(
+            matches!(err, InferenceError::ConflictingCompetitorConfig { .. }),
+            "{err:?}"
+        );
+    }
+}

tests/time_expanded_joint.rs

@@ -0,0 +1,296 @@
+//! The joint must span slices, because Through Time reads each competitor at
+//! their own last appearance.
+//!
+//! The exact posterior of a multi-slice scored history is still Gaussian: the
+//! prior, the drift between appearances, and the scored likelihoods are all
+//! Gaussian. So it can be written out by hand and compared against, which is
+//! the check a single-slice fixture cannot make.
+
+use smallvec::smallvec;
+use trueskill_tt::{
+    ConstantDrift, ConvergenceOptions, Event, History, Member, Outcome, Team, UnknownKeys,
+};
+
+const SIGMA0: f64 = 6.0;
+const BETA: f64 = 1.0;
+const SCORE_SIGMA: f64 = 2.0;
+const GAMMA: f64 = 0.5;
+
+type H = History<i64, ConstantDrift, trueskill_tt::NullObserver, &'static str>;
+
+fn history(gamma: f64) -> H {
+    History::builder()
+        .mu(0.0)
+        .sigma(SIGMA0)
+        .beta(BETA)
+        .score_sigma(SCORE_SIGMA)
+        .drift(ConstantDrift(gamma))
+        .unknown_keys(UnknownKeys::Reject)
+        .convergence(ConvergenceOptions {
+            max_iter: 20_000,
+            epsilon: 1e-13,
+            alpha: 1.0,
+        })
+        .build()
+}
+
+fn duel(a: &'static str, b: &'static str, t: i64, sa: f64, sb: f64) -> Event<i64, &'static str> {
+    Event {
+        time: t,
+        teams: smallvec![
+            Team::with_members([Member::new(a)]),
+            Team::with_members([Member::new(b)]),
+        ],
+        outcome: Outcome::scores([sa, sb]),
+    }
+}
+
+fn inverse(mut a: Vec<Vec<f64>>) -> Vec<Vec<f64>> {
+    let n = a.len();
+    let mut inv: Vec<Vec<f64>> = (0..n)
+        .map(|i| (0..n).map(|j| f64::from(u8::from(i == j))).collect())
+        .collect();
+    for col in 0..n {
+        let mut piv = col;
+        for r in col + 1..n {
+            if a[r][col].abs() > a[piv][col].abs() {
+                piv = r;
+            }
+        }
+        a.swap(col, piv);
+        inv.swap(col, piv);
+        let d = a[col][col];
+        for j in 0..n {
+            a[col][j] /= d;
+            inv[col][j] /= d;
+        }
+        for r in 0..n {
+            if r == col {
+                continue;
+            }
+            let f = a[r][col];
+            for j in 0..n {
+                a[r][j] -= f * a[col][j];
+                inv[r][j] -= f * inv[col][j];
+            }
+        }
+    }
+    inv
+}
+
+/// Two competitors, two slices ten units apart, one duel in each.
+///
+/// The exact precision is written out explicitly here rather than obtained
+/// from the crate, so this is an independent check rather than a restatement.
+/// Variables are `[a0, b0, a1, b1]`.
+#[test]
+fn a_two_slice_joint_matches_the_exact_posterior() {
+    let mut h = history(GAMMA);
+    h.add_events(vec![
+        duel("a", "b", 0, 5.0, 2.0),
+        duel("a", "b", 10, 4.0, 3.0),
+    ])
+    .unwrap();
+    let report = h.converge().unwrap();
+    assert!(report.converged, "{:?}", report.final_step);
+
+    let prior_prec = 1.0 / (SIGMA0 * SIGMA0);
+    let drift_prec = 1.0 / (10.0 * GAMMA * GAMMA);
+    let obs_prec = 1.0 / (SCORE_SIGMA * SCORE_SIGMA + 2.0 * BETA * BETA);
+
+    let mut lambda = vec![vec![0.0; 4]; 4];
+    // priors on the first appearances
+    lambda[0][0] += prior_prec;
+    lambda[1][1] += prior_prec;
+    // drift a0-a1 and b0-b1
+    for (p, q) in [(0usize, 2usize), (1, 3)] {
+        lambda[p][p] += drift_prec;
+        lambda[q][q] += drift_prec;
+        lambda[p][q] -= drift_prec;
+        lambda[q][p] -= drift_prec;
+    }
+    // one duel per slice: contrast (+1, -1) on that slice's variables
+    for (p, q) in [(0usize, 1usize), (2, 3)] {
+        lambda[p][p] += obs_prec;
+        lambda[q][q] += obs_prec;
+        lambda[p][q] -= obs_prec;
+        lambda[q][p] -= obs_prec;
+    }
+    let cov = inverse(lambda);
+
+    // The crate reads each competitor at their latest appearance: a1, b1.
+    let exact_gap = (cov[2][2] + cov[3][3] - 2.0 * cov[2][3]).sqrt();
+    let got = h.posterior_of(&[(&"a", 1.0), (&"b", -1.0)]).unwrap();
+    assert!(
+        (got.sigma() - exact_gap).abs() / exact_gap < 1e-9,
+        "difference: got {} exact {exact_gap}",
+        got.sigma()
+    );
+
+    let exact_single = cov[2][2].sqrt();
+    let got_single = h.posterior_of(&[(&"a", 1.0)]).unwrap();
+    assert!(
+        (got_single.sigma() - exact_single).abs() / exact_single < 1e-9,
+        "single node: got {} exact {exact_single}",
+        got_single.sigma()
+    );
+}
+
+/// The case that motivated this: competitors read at *different* slices, with
+/// the last slice holding only one of them. Under the old latest-slice joint
+/// this was `UnknownKey`.
+#[test]
+fn competitors_last_seen_in_different_slices_are_comparable() {
+    let mut h = history(GAMMA);
+    h.add_events(vec![
+        duel("a", "b", 0, 5.0, 2.0),
+        duel("a", "c", 10, 4.0, 3.0),
+        // the final slice holds one duel that does not involve b at all
+        duel("a", "c", 20, 6.0, 1.0),
+    ])
+    .unwrap();
+    let _ = h.converge().unwrap();
+
+    // b last appeared at time 0; a and c at time 20. All three must resolve.
+    for (x, y) in [("a", "b"), ("b", "c"), ("a", "c")] {
+        let g = h
+            .posterior_of(&[(&x, 1.0), (&y, -1.0)])
+            .unwrap_or_else(|e| panic!("{x} - {y} should resolve across slices: {e}"));
+        assert!(g.sigma() > 0.0 && g.sigma().is_finite());
+    }
+}
+
+/// The mean must agree with what message passing reports, which is exact even
+/// with cycles. Only the second moment needs the joint.
+#[test]
+fn means_agree_with_the_marginals() {
+    let mut h = history(GAMMA);
+    h.add_events(vec![
+        duel("a", "b", 0, 5.0, 2.0),
+        duel("b", "c", 5, 3.0, 1.0),
+        duel("a", "c", 10, 4.0, 2.0),
+    ])
+    .unwrap();
+    let _ = h.converge().unwrap();
+
+    for k in ["a", "b", "c"] {
+        let marginal = h.current_skill(&k).unwrap().mu();
+        let joint = h.posterior_of(&[(&k, 1.0)]).unwrap().mu();
+        assert!(
+            (marginal - joint).abs() < 1e-9,
+            "{k}: marginal {marginal}, joint {joint}"
+        );
+    }
+}
+
+/// With zero drift a competitor has one latent skill however many slices it
+/// appears in, so spreading the same events over time must not change the
+/// answer. This exercises the appearance-merging path.
+#[test]
+fn zero_drift_makes_slice_layout_irrelevant() {
+    let spread = {
+        let mut h = history(0.0);
+        h.add_events(vec![
+            duel("a", "b", 0, 5.0, 2.0),
+            duel("a", "b", 10, 4.0, 3.0),
+            duel("a", "b", 20, 6.0, 1.0),
+        ])
+        .unwrap();
+        let _ = h.converge().unwrap();
+        h.posterior_of(&[(&"a", 1.0), (&"b", -1.0)]).unwrap()
+    };
+    let together = {
+        let mut h = history(0.0);
+        h.add_events(vec![
+            duel("a", "b", 0, 5.0, 2.0),
+            duel("a", "b", 0, 4.0, 3.0),
+            duel("a", "b", 0, 6.0, 1.0),
+        ])
+        .unwrap();
+        let _ = h.converge().unwrap();
+        h.posterior_of(&[(&"a", 1.0), (&"b", -1.0)]).unwrap()
+    };
+
+    assert!(
+        (spread.sigma() - together.sigma()).abs() < 1e-9,
+        "zero drift: spread {} vs together {}",
+        spread.sigma(),
+        together.sigma()
+    );
+}
+
+/// More drift means less is carried forward from old evidence, so a comparison
+/// against a competitor last seen long ago must widen.
+#[test]
+fn drift_widens_a_comparison_across_time() {
+    let mut previous = 0.0;
+    for gamma in [0.0f64, 0.1, 0.5, 2.0] {
+        let mut h = history(gamma);
+        h.add_events(vec![
+            duel("a", "b", 0, 5.0, 2.0),
+            duel("a", "c", 100, 4.0, 3.0),
+        ])
+        .unwrap();
+        let _ = h.converge().unwrap();
+
+        // b was last seen at time 0; a at time 100.
+        let g = h.posterior_of(&[(&"a", 1.0), (&"b", -1.0)]).unwrap();
+        assert!(
+            g.sigma() > previous,
+            "gamma={gamma}: sigma {} did not exceed {previous}",
+            g.sigma()
+        );
+        previous = g.sigma();
+    }
+}
+
+/// `posterior_of_at` pins the reading to a moment, where `posterior_of` takes
+/// each competitor wherever they were last seen.
+#[test]
+fn posterior_of_at_reads_as_of_a_time() {
+    let mut h = history(GAMMA);
+    h.add_events(vec![
+        duel("a", "b", 0, 5.0, 2.0),
+        duel("a", "b", 10, 4.0, 3.0),
+        duel("a", "b", 20, 6.0, 1.0),
+    ])
+    .unwrap();
+    let _ = h.converge().unwrap();
+
+    let early = h.posterior_of_at(0, &[(&"a", 1.0), (&"b", -1.0)]).unwrap();
+    let late = h.posterior_of_at(20, &[(&"a", 1.0), (&"b", -1.0)]).unwrap();
+    let latest = h.posterior_of(&[(&"a", 1.0), (&"b", -1.0)]).unwrap();
+
+    // Asking as of the final slice is the same as asking for the latest.
+    assert!((late.mu() - latest.mu()).abs() < 1e-9);
+    assert!((late.sigma() - latest.sigma()).abs() < 1e-9);
+
+    // Reading at time 0 is a different quantity, and the smoothed estimate
+    // there is informed by everything that came after.
+    assert!(
+        (early.mu() - late.mu()).abs() > 1e-6,
+        "as-of-0 and as-of-20 should differ: {} vs {}",
+        early.mu(),
+        late.mu()
+    );
+
+    // A time before any event has nothing to read.
+    assert!(h.posterior_of_at(-1, &[(&"a", 1.0)]).is_err());
+}
+
+/// Times between slices resolve to the latest appearance at or before them.
+#[test]
+fn a_time_between_slices_reads_the_previous_appearance() {
+    let mut h = history(GAMMA);
+    h.add_events(vec![
+        duel("a", "b", 0, 5.0, 2.0),
+        duel("a", "b", 100, 4.0, 3.0),
+    ])
+    .unwrap();
+    let _ = h.converge().unwrap();
+
+    let at_zero = h.posterior_of_at(0, &[(&"a", 1.0)]).unwrap();
+    let between = h.posterior_of_at(50, &[(&"a", 1.0)]).unwrap();
+    assert!((at_zero.mu() - between.mu()).abs() < 1e-12);
+    assert!((at_zero.sigma() - between.sigma()).abs() < 1e-12);
+}

tests/validation.rs

@@ -0,0 +1,257 @@
+//! Input validation must hold in **release**, where `debug_assert!` is gone.
+//!
+//! The engine guards itself with `debug_assert!`, which documents invariants
+//! but vanishes in the profile users actually ship. Anything reachable from the
+//! public API has to be rejected with an `InferenceError` instead, at the
+//! boundary, rather than becoming NaN or an out-of-bounds panic deep inside
+//! `run_chain`.
+//!
+//! `GameOptions` and `ConvergenceOptions` both have public fields, so the
+//! eager asserts on `HistoryBuilder` do not cover the `Game` constructors —
+//! a caller can build the options struct directly.
+
+use smallvec::smallvec;
+use trueskill_tt::{
+    ConstantDrift, ConvergenceOptions, Event, Game, GameOptions, Gaussian, History, InferenceError,
+    Member, Outcome, Rating, Team,
+};
+
+type R = Rating<i64, ConstantDrift>;
+
+fn rating() -> R {
+    R::new(
+        Gaussian::from_ms(25.0, 25.0 / 3.0),
+        25.0 / 6.0,
+        ConstantDrift(0.0),
+    )
+}
+
+fn options_with_alpha(alpha: f64) -> GameOptions {
+    GameOptions {
+        convergence: ConvergenceOptions {
+            alpha,
+            ..ConvergenceOptions::default()
+        },
+        ..GameOptions::default()
+    }
+}
+
+/// `alpha == 0.0` leaves every EP update unapplied, so inference silently
+/// returns the priors — the worst possible failure, since the output looks
+/// entirely reasonable.
+#[test]
+fn ranked_rejects_a_zero_damping_factor() {
+    let (a, b) = (rating(), rating());
+    let err = Game::<i64, _>::ranked(
+        &[&[a], &[b]],
+        Outcome::winner(0, 2),
+        &options_with_alpha(0.0),
+    )
+    .expect_err("alpha = 0 must be rejected");
+    assert!(
+        matches!(err, InferenceError::InvalidParameter { name: "alpha", .. }),
+        "got {err:?}"
+    );
+}
+
+#[test]
+fn ranked_rejects_an_out_of_range_damping_factor() {
+    let (a, b) = (rating(), rating());
+    for alpha in [-0.5, 1.5, f64::NAN] {
+        let err = Game::<i64, _>::ranked(
+            &[&[a], &[b]],
+            Outcome::winner(0, 2),
+            &options_with_alpha(alpha),
+        )
+        .expect_err("alpha out of (0, 1] must be rejected");
+        assert!(
+            matches!(err, InferenceError::InvalidParameter { name: "alpha", .. }),
+            "alpha={alpha}: got {err:?}"
+        );
+    }
+}
+
+#[test]
+fn scored_rejects_a_bad_damping_factor() {
+    let (a, b) = (rating(), rating());
+    let err = Game::<i64, _>::scored(
+        &[&[a], &[b]],
+        Outcome::scores([21.0, 9.0]),
+        &options_with_alpha(0.0),
+    )
+    .expect_err("alpha = 0 must be rejected");
+    assert!(
+        matches!(err, InferenceError::InvalidParameter { name: "alpha", .. }),
+        "got {err:?}"
+    );
+}
+
+/// Already covered by `Game::ranked`, asserted here so the release-mode
+/// guarantee is stated in one place.
+#[test]
+fn ranked_rejects_an_out_of_range_draw_probability() {
+    let (a, b) = (rating(), rating());
+    for p_draw in [-0.5, 1.0, 1.5] {
+        let options = GameOptions {
+            p_draw,
+            ..GameOptions::default()
+        };
+        assert!(
+            Game::<i64, _>::ranked(&[&[a], &[b]], Outcome::winner(0, 2), &options).is_err(),
+            "p_draw={p_draw} must be rejected"
+        );
+    }
+}
+
+#[test]
+fn scored_rejects_a_non_positive_noise() {
+    let (a, b) = (rating(), rating());
+    for score_sigma in [0.0, -1.0, f64::NAN] {
+        let options = GameOptions {
+            score_sigma,
+            ..GameOptions::default()
+        };
+        assert!(
+            Game::<i64, _>::scored(&[&[a], &[b]], Outcome::scores([21.0, 9.0]), &options).is_err(),
+            "score_sigma={score_sigma} must be rejected"
+        );
+    }
+}
+
+/// A tie with no draw probability makes the truncation margin zero and the
+/// two-sided update evaluate 0/0. Ingestion must refuse it.
+#[test]
+fn ingestion_rejects_a_tie_without_a_draw_probability() {
+    let mut h = History::builder().p_draw(0.0).build();
+    let err = h
+        .add_events(vec![Event {
+            time: 0,
+            teams: smallvec![
+                Team::with_members([Member::new("a")]),
+                Team::with_members([Member::new("b")]),
+            ],
+            outcome: Outcome::draw(2),
+        }])
+        .expect_err("a tie with p_draw = 0 must be rejected");
+    assert!(
+        matches!(err, InferenceError::TieWithoutDrawProbability { .. }),
+        "got {err:?}"
+    );
+}
+
+/// `Outcome::scores_with_sigma` documents that a non-positive sigma is
+/// accepted at construction and rejected at ingestion.
+#[test]
+fn ingestion_rejects_a_non_positive_per_event_score_sigma() {
+    for sigma in [0.0, -1.0, f64::NAN] {
+        let mut h = History::builder().build();
+        let err = h
+            .add_events(vec![Event {
+                time: 0,
+                teams: smallvec![
+                    Team::with_members([Member::new("a")]),
+                    Team::with_members([Member::new("b")]),
+                ],
+                outcome: Outcome::scores_with_sigma([21.0, 9.0], sigma),
+            }])
+            .expect_err("a non-positive per-event sigma must be rejected");
+        assert!(
+            matches!(err, InferenceError::InvalidParameter { .. }),
+            "sigma={sigma}: got {err:?}"
+        );
+    }
+}
+
+/// Per-team weights must match that team's membership. The top-level length
+/// checks in ingestion do not cover the inner dimension.
+#[test]
+fn ingestion_rejects_weights_that_do_not_match_their_team() {
+    let mut h = History::builder().build();
+    let mut team = Team::with_members([Member::new("a"), Member::new("b")]);
+    team.members[0].weight = 1.0;
+
+    let err = h
+        .event(0)
+        .team(["a", "b"])
+        .team(["c"])
+        // Three weights for a two-member team.
+        .weights([1.0, 1.0, 1.0])
+        .winner(0)
+        .commit()
+        .expect_err("a weight/member length mismatch must be rejected");
+    assert!(
+        matches!(err, InferenceError::MismatchedShape { .. }),
+        "got {err:?}"
+    );
+}
+
+/// `mu`, `sigma` and `beta` were the last unvalidated setters on
+/// `HistoryBuilder`, next to `p_draw`, `score_sigma` and `convergence`, which
+/// all assert eagerly.
+///
+/// Two of the rejected values are the quiet kind. A negative `sigma` or `beta`
+/// enters inference only as its square, so it produced bit-identical results
+/// to the positive value — the sign was dropped without comment.
+mod builder_parameters {
+    use trueskill_tt::History;
+
+    #[test]
+    #[should_panic(expected = "mu must be finite")]
+    fn a_non_finite_mu_is_rejected() {
+        let _ = History::builder().mu(f64::NAN);
+    }
+
+    #[test]
+    #[should_panic(expected = "sigma must be finite and positive")]
+    fn a_zero_sigma_is_rejected() {
+        let _ = History::builder().sigma(0.0);
+    }
+
+    #[test]
+    #[should_panic(expected = "sigma must be finite and positive")]
+    fn a_negative_sigma_is_rejected() {
+        let _ = History::builder().sigma(-8.33);
+    }
+
+    #[test]
+    #[should_panic(expected = "sigma must be finite and positive")]
+    fn an_infinite_sigma_is_rejected() {
+        let _ = History::builder().sigma(f64::INFINITY);
+    }
+
+    #[test]
+    #[should_panic(expected = "beta must be finite and non-negative")]
+    fn a_negative_beta_is_rejected() {
+        let _ = History::builder().beta(-4.17);
+    }
+
+    #[test]
+    #[should_panic(expected = "beta must be finite and non-negative")]
+    fn a_non_finite_beta_is_rejected() {
+        let _ = History::builder().beta(f64::NAN);
+    }
+
+    /// Zero beta is deliberately allowed: performance is then exactly skill.
+    /// It has to reach a different fit than a positive beta, or "allowed"
+    /// would just mean "not checked".
+    #[test]
+    fn a_zero_beta_is_allowed_and_changes_the_fit() {
+        let fit = |beta: f64| {
+            let mut h = History::builder()
+                .mu(25.0)
+                .sigma(25.0 / 3.0)
+                .beta(beta)
+                .build();
+            h.record_winner(&"a", &"b", 1).unwrap();
+            let _ = h.converge().unwrap();
+            h.current_skill(&"a").unwrap()
+        };
+        let zero = fit(0.0);
+        let positive = fit(25.0 / 6.0);
+        assert!(zero.pi().is_finite() && zero.pi() > 0.0);
+        assert!(
+            (zero.pi() - positive.pi()).abs() > 1e-6,
+            "zero beta must not merely be ignored: {zero:?} vs {positive:?}"
+        );
+    }
+}

tests/variance_reduction.rs

@@ -0,0 +1,156 @@
+//! `expected_variance_reduction`: which matchup best sharpens a given question.
+
+use smallvec::smallvec;
+use trueskill_tt::{
+    ConstantDrift, ConvergenceOptions, Event, History, InferenceError, Member, Outcome, Team,
+    UnknownKeys,
+};
+
+type H = History<i64, ConstantDrift, trueskill_tt::NullObserver, &'static str>;
+
+fn round(a: &'static str, b: &'static str, sa: f64, sb: f64) -> Event<i64, &'static str> {
+    Event {
+        time: 1,
+        teams: smallvec![
+            Team::with_members([Member::new(a)]),
+            Team::with_members([Member::new(b)]),
+        ],
+        outcome: Outcome::scores([sa, sb]),
+    }
+}
+
+fn base() -> Vec<Event<i64, &'static str>> {
+    vec![
+        round("a", "b", 5.0, 2.0),
+        round("a", "c", 6.0, 1.0),
+        round("b", "c", 4.0, 3.0),
+        round("c", "d", 2.0, 1.0),
+        round("a", "d", 7.0, 2.0),
+    ]
+}
+
+fn fit(extra: Option<Event<i64, &'static str>>, policy: UnknownKeys) -> H {
+    let mut h: History<i64, _, _, &'static str> = History::builder()
+        .mu(0.0)
+        .sigma(6.0)
+        .beta(1.0)
+        .score_sigma(2.0)
+        .drift(ConstantDrift(0.0))
+        .unknown_keys(policy)
+        .convergence(ConvergenceOptions {
+            max_iter: 20_000,
+            epsilon: 1e-13,
+            alpha: 1.0,
+        })
+        .build();
+    let mut ev = base();
+    if let Some(e) = extra {
+        ev.push(e);
+    }
+    h.add_events(ev).unwrap();
+    let _ = h.converge().unwrap();
+    h
+}
+
+/// The closed form must equal what actually happens if the matchup is played.
+/// This is the assertion that makes the whole call trustworthy: a wrong
+/// acquisition function returns plausible numbers and quietly picks worse
+/// matchups forever.
+#[test]
+fn the_closed_form_matches_an_actual_refit() {
+    let h = fit(None, UnknownKeys::Reject);
+    let target: Vec<(&&str, f64)> = vec![(&"a", 1.0), (&"b", -1.0)];
+    let before = h.posterior_of(&target).unwrap().sigma().powi(2);
+
+    for (x, y) in [("a", "b"), ("c", "d"), ("a", "c"), ("b", "d")] {
+        let predicted = h
+            .expected_variance_reduction(&[&[&x], &[&y]], &target)
+            .unwrap();
+
+        let after = fit(Some(round(x, y, 3.0, 1.0)), UnknownKeys::Reject);
+        let actual = before - after.posterior_of(&target).unwrap().sigma().powi(2);
+
+        assert!(
+            (predicted - actual).abs() / actual.abs() < 1e-9,
+            "{x} vs {y}: predicted {predicted}, actual {actual}"
+        );
+    }
+}
+
+/// The reduction cannot depend on the score, because for a Gaussian likelihood
+/// the posterior variance update is data-independent. This is why the call
+/// needs no expectation despite its name.
+#[test]
+fn the_outcome_does_not_change_the_reduction() {
+    let target: Vec<(&&str, f64)> = vec![(&"a", 1.0), (&"b", -1.0)];
+    let h = fit(None, UnknownKeys::Reject);
+    let before = h.posterior_of(&target).unwrap().sigma().powi(2);
+
+    let mut seen = Vec::new();
+    for (sa, sb) in [(3.0, 1.0), (100.0, -50.0), (0.0, 0.0)] {
+        let after = fit(Some(round("c", "d", sa, sb)), UnknownKeys::Reject);
+        seen.push(before - after.posterior_of(&target).unwrap().sigma().powi(2));
+    }
+    for w in seen.windows(2) {
+        assert!(
+            (w[0] - w[1]).abs() < 1e-12,
+            "variance reduction moved with the observed score: {seen:?}"
+        );
+    }
+}
+
+/// The point of the call: it must rank candidate matchups usefully. Playing the
+/// pair you are trying to separate helps most; an unrelated pair helps least.
+#[test]
+fn it_ranks_candidates_by_how_much_they_answer_the_question() {
+    let h = fit(None, UnknownKeys::Reject);
+    let target: Vec<(&&str, f64)> = vec![(&"a", 1.0), (&"b", -1.0)];
+
+    let direct = h
+        .expected_variance_reduction(&[&[&"a"], &[&"b"]], &target)
+        .unwrap();
+    let unrelated = h
+        .expected_variance_reduction(&[&[&"c"], &[&"d"]], &target)
+        .unwrap();
+
+    assert!(direct > 0.0 && unrelated > 0.0);
+    assert!(
+        direct > 5.0 * unrelated,
+        "playing the target pair should dominate: {direct} vs {unrelated}"
+    );
+}
+
+/// A matchup between two competitors nobody has seen still teaches something
+/// about them, but nothing about a target that does not involve them.
+#[test]
+fn an_unrelated_unseen_matchup_teaches_nothing_about_the_target() {
+    let h = fit(None, UnknownKeys::Prior);
+    let target: Vec<(&&str, f64)> = vec![(&"a", 1.0), (&"b", -1.0)];
+
+    let reduction = h
+        .expected_variance_reduction(&[&[&"stranger"], &[&"nobody"]], &target)
+        .unwrap();
+    assert!(
+        reduction.abs() < 1e-12,
+        "an unseen pair shares nothing with the target: {reduction}"
+    );
+}
+
+#[test]
+fn shape_errors_are_reported() {
+    let h = fit(None, UnknownKeys::Reject);
+    let target: Vec<(&&str, f64)> = vec![(&"a", 1.0), (&"b", -1.0)];
+
+    assert!(matches!(
+        h.expected_variance_reduction(&[&[&"a"]], &target),
+        Err(InferenceError::MismatchedShape {
+            expected: 2,
+            got: 1,
+            ..
+        })
+    ));
+    assert!(matches!(
+        h.expected_variance_reduction(&[&[&"a"], &[&"ghost"]], &target),
+        Err(InferenceError::UnknownKey { .. })
+    ));
+}