fix!: propagate NaN through the convergence reduction

`tuple_max` compared with a plain `>`, which is false against NaN, so a
NaN accumulator was replaced by the next finite delta. The fold runs over
`TimeSlice::posteriors()`, a HashMap, so whether a NaN survived to `step`
depended on per-process hash order.

Measured before, four competitors in one slice with one pathological
pair, same binary and input, 30 separate processes:

  16  Ok  converged=true, iterations=1, a = Gaussian { pi: NaN, tau: NaN }
  14  Err NonFiniteResult

After: 30/30 Err. A coin flip on whether a NaN fit was reported as an
error or as a successful, converged fit — inside the guard whose entire
purpose is "NaN is never convergence".

`f64::max` would not have fixed it. It also ignores NaN by design, which
is the same defect wearing a standard-library name, and a test pins that
we do not use it.

`Gaussian::delta` had to be fixed FIRST, and that ordering is the whole
subtlety. Two identical improper messages produced `(0.0, NaN)` — not
from `mu()`, which is guarded and returns 0.0, but from `inf - inf` in
the sigma component. That NaN is reachable in ordinary healthy inference:
once a pairing is more than about nine cavity-sigma apart the truncation
is a no-op and the chain compares one identity message against another.
Propagating NaN without fixing `delta` would therefore have turned
correct fits into NonFiniteResult errors. `delta` now answers the
identical-message case in natural space before touching the accessors.

My first version of the `delta` test asserted `mu()` was NaN. It is not;
the accessor guards `pi <= 0.0`. The test caught my own wrong premise,
and the doc comment is corrected to match.

BREAKING CHANGE: a fit that produced NaN in a non-final reduction
position previously returned `Ok` with `converged: true` and a NaN
posterior; it now returns `Err(NonFiniteResult)`. That was always the
documented intent.

Closes #58

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

src/gaussian.rs

@@ -120,7 +120,25 @@ impl Gaussian {
         }
     }
 
+    /// How far this Gaussian moved from `other`, as `(|d mu|, |d sigma|)`.
+    ///
+    /// Identical messages have not moved, whatever their parameters, and that
+    /// case is answered in natural space before touching `mu()`/`sigma()`. An
+    /// improper message has `pi == 0`, so `sigma()` is infinite — and
+    /// `inf - inf` is NaN, a NaN *change* for a message that did not change at
+    /// all. (`mu()` is guarded and returns 0.0 here, so the mean component was
+    /// never the problem; the sigma component alone produced `(0.0, NaN)`.)
+    ///
+    /// That is reachable in ordinary inference: once a pairing is more than
+    /// about nine cavity-sigma apart the truncation is a no-op, `trunc / cavity`
+    /// is exactly the identity message, and the chain compares one identity
+    /// against another. Before this guard that produced `(0.0, NaN)`, which
+    /// silently disabled the sigma half of the convergence test.
     pub(crate) fn delta(&self, other: Gaussian) -> (f64, f64) {
+        if self.pi == other.pi && self.tau == other.tau {
+            return (0.0, 0.0);
+        }
+
         (
             (self.mu() - other.mu()).abs(),
             (self.sigma() - other.sigma()).abs(),
@@ -256,6 +274,42 @@ impl ops::Div<Gaussian> for Gaussian {
 
 #[cfg(test)]
 mod tests {
+    /// A message that did not change must report no change, even when it is
+    /// improper. `mu()` of an improper Gaussian is `0/0 = NaN` and `sigma()` is
+    /// infinite, so the mean/sigma form reported `(NaN, NaN)` for two identical
+    /// identity messages — which silently disabled the sigma half of the
+    /// convergence test in `run_chain`.
+    #[test]
+    fn delta_of_two_identical_improper_messages_is_zero() {
+        let improper = crate::N_INF;
+        // `mu()` is guarded and returns 0.0 for an improper Gaussian, so the
+        // mean component was always fine. The NaN came from the sigma
+        // component alone: `inf - inf`. The pre-fix value was `(0.0, NaN)`.
+        assert!(improper.sigma().is_infinite(), "premise: sigma is infinite");
+        assert_eq!(improper.mu(), 0.0, "premise: mu is guarded, not NaN");
+        assert!(
+            (improper.sigma() - improper.sigma()).is_nan(),
+            "premise: the unguarded sigma difference is NaN"
+        );
+        assert_eq!(improper.delta(improper), (0.0, 0.0));
+    }
+
+    #[test]
+    fn delta_of_identical_proper_messages_is_zero() {
+        let g = Gaussian::from_ms(25.0, 8.0);
+        assert_eq!(g.delta(g), (0.0, 0.0));
+    }
+
+    /// The shortcut must not swallow a real difference.
+    #[test]
+    fn delta_still_measures_a_real_move() {
+        let a = Gaussian::from_ms(25.0, 8.0);
+        let b = Gaussian::from_ms(26.0, 9.0);
+        let (dmu, dsigma) = a.delta(b);
+        assert!((dmu - 1.0).abs() < 1e-12, "{dmu}");
+        assert!((dsigma - 1.0).abs() < 1e-12, "{dsigma}");
+    }
+
     use super::*;
 
     #[test]