diff --git a/benches/batch.rs b/benches/batch.rs
index 1562402..7bc0bc0 100644
--- a/benches/batch.rs
+++ b/benches/batch.rs
@@ -11,7 +11,7 @@ fn criterion_benchmark(criterion: &mut Criterion) {
     let b = index_map.get_or_create("b");
     let c = index_map.get_or_create("c");
 
-    let mut agents: CompetitorStore<ConstantDrift> = CompetitorStore::new();
+    let mut agents: CompetitorStore<i64, ConstantDrift> = CompetitorStore::new();
 
     for agent in [a, b, c] {
         agents.insert(
diff --git a/src/competitor.rs b/src/competitor.rs
index f2f270b..78b44a5 100644
--- a/src/competitor.rs
+++ b/src/competitor.rs
@@ -3,6 +3,7 @@ use crate::{
     drift::{ConstantDrift, Drift},
     gaussian::Gaussian,
     rating::Rating,
+    time::Time,
 };
 
 /// Per-history, temporal state for someone competing.
@@ -10,41 +11,61 @@ use crate::{
 /// Renamed from `Agent` in T2; the former `.player` field is now
 /// `.rating` to match the `Player → Rating` rename.
 #[derive(Debug)]
-pub struct Competitor<D: Drift = ConstantDrift> {
-    pub rating: Rating<D>,
+pub struct Competitor<T: Time = i64, D: Drift<T> = ConstantDrift> {
+    pub rating: Rating<T, D>,
     pub message: Gaussian,
-    pub last_time: i64,
+    pub last_time: Option<T>,
 }
 
-impl<D: Drift> Competitor<D> {
-    pub(crate) fn receive(&self, elapsed: i64) -> Gaussian {
+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 {
+            let elapsed_variance = match &self.last_time {
+                Some(last) => self.rating.drift.variance_delta(last, now),
+                None => 0.0,
+            };
+            self.message.forget(elapsed_variance)
+        } else {
+            self.rating.prior
+        }
+    }
+
+    /// Compute the message using a pre-cached elapsed count (in `Time::elapsed_to` units).
+    ///
+    /// 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 {
             self.message
-                .forget(self.rating.drift.variance_delta(elapsed))
+                .forget(self.rating.drift.variance_for_elapsed(elapsed))
         } else {
             self.rating.prior
         }
     }
 }
 
-impl Default for Competitor<ConstantDrift> {
+impl Default for Competitor<i64, ConstantDrift> {
     fn default() -> Self {
         Self {
             rating: Rating::default(),
             message: N_INF,
-            last_time: i64::MIN,
+            last_time: None,
         }
     }
 }
 
-pub(crate) fn clean<'a, D: Drift + 'a, C: Iterator<Item = &'a mut Competitor<D>>>(
-    competitors: C,
-    last_time: bool,
-) {
+pub(crate) fn clean<'a, T, D, C>(competitors: C, last_time: bool)
+where
+    T: Time + 'a,
+    D: Drift<T> + 'a,
+    C: Iterator<Item = &'a mut Competitor<T, D>>,
+{
     for c in competitors {
         c.message = N_INF;
         if last_time {
-            c.last_time = i64::MIN;
+            c.last_time = None;
         }
     }
 }
diff --git a/src/drift.rs b/src/drift.rs
index 5c7107e..57e684a 100644
--- a/src/drift.rs
+++ b/src/drift.rs
@@ -1,14 +1,36 @@
 use std::fmt::Debug;
 
-pub trait Drift: Copy + Debug {
-    fn variance_delta(&self, elapsed: i64) -> f64;
+use crate::time::Time;
+
+/// Governs how much a competitor's skill can drift between two time points.
+///
+/// Generic over `T: Time` so seasonal or calendar-aware drift is expressible
+/// without going through `i64`.
+pub trait Drift<T: Time>: Copy + Debug {
+    /// Variance added to the skill prior for elapsed time `from -> to`.
+    ///
+    /// Called with `from <= to`; returning zero means no drift accumulates.
+    fn variance_delta(&self, from: &T, to: &T) -> f64;
+
+    /// Variance added for a pre-computed elapsed count (in the same units as
+    /// `T::elapsed_to`). Used where the elapsed is already cached as `i64`.
+    fn variance_for_elapsed(&self, elapsed: i64) -> f64;
 }
 
+/// Simple constant-per-unit-time drift.
+///
+/// For `Time = i64`: variance added is `(to - from) * gamma^2`.
+/// For `Time = Untimed`: elapsed is always 0, so drift is always 0.
 #[derive(Clone, Copy, Debug)]
 pub struct ConstantDrift(pub f64);
 
-impl Drift for ConstantDrift {
-    fn variance_delta(&self, elapsed: i64) -> f64 {
-        elapsed as f64 * self.0 * self.0
+impl<T: Time> Drift<T> for ConstantDrift {
+    fn variance_delta(&self, from: &T, to: &T) -> f64 {
+        let elapsed = from.elapsed_to(to).max(0) as f64;
+        elapsed * self.0 * self.0
+    }
+
+    fn variance_for_elapsed(&self, elapsed: i64) -> f64 {
+        elapsed.max(0) as f64 * self.0 * self.0
     }
 }
diff --git a/src/game.rs b/src/game.rs
index 9f76d9c..30f0889 100644
--- a/src/game.rs
+++ b/src/game.rs
@@ -8,12 +8,13 @@ use crate::{
     factor::{Factor, trunc::TruncFactor},
     gaussian::Gaussian,
     rating::Rating,
+    time::Time,
     tuple_gt, tuple_max,
 };
 
 #[derive(Debug)]
-pub struct Game<'a, D: Drift> {
-    teams: Vec<Vec<Rating<D>>>,
+pub struct Game<'a, T: Time = i64, D: Drift<T> = crate::drift::ConstantDrift> {
+    teams: Vec<Vec<Rating<T, D>>>,
     result: &'a [f64],
     weights: &'a [Vec<f64>],
     p_draw: f64,
@@ -21,9 +22,9 @@ pub struct Game<'a, D: Drift> {
     pub(crate) evidence: f64,
 }
 
-impl<'a, D: Drift> Game<'a, D> {
+impl<'a, T: Time, D: Drift<T>> Game<'a, T, D> {
     pub fn new(
-        teams: Vec<Vec<Rating<D>>>,
+        teams: Vec<Vec<Rating<T, D>>>,
         result: &'a [f64],
         weights: &'a [Vec<f64>],
         p_draw: f64,
@@ -227,14 +228,16 @@ mod tests {
     use super::*;
     use crate::{ConstantDrift, GAMMA, Gaussian, N_INF, Rating, arena::ScratchArena};
 
+    type R = Rating<i64, ConstantDrift>;
+
     #[test]
     fn test_1vs1() {
-        let t_a = Rating::new(
+        let t_a = R::new(
             Gaussian::from_ms(25.0, 25.0 / 3.0),
             25.0 / 6.0,
             ConstantDrift(25.0 / 300.0),
         );
-        let t_b = Rating::new(
+        let t_b = R::new(
             Gaussian::from_ms(25.0, 25.0 / 3.0),
             25.0 / 6.0,
             ConstantDrift(25.0 / 300.0),
@@ -256,12 +259,12 @@ mod tests {
         assert_ulps_eq!(a, Gaussian::from_ms(20.794779, 7.194481), epsilon = 1e-6);
         assert_ulps_eq!(b, Gaussian::from_ms(29.205220, 7.194481), epsilon = 1e-6);
 
-        let t_a = Rating::new(
+        let t_a = R::new(
             Gaussian::from_ms(29.0, 1.0),
             25.0 / 6.0,
             ConstantDrift(GAMMA),
         );
-        let t_b = Rating::new(
+        let t_b = R::new(
             Gaussian::from_ms(25.0, 25.0 / 3.0),
             25.0 / 6.0,
             ConstantDrift(GAMMA),
@@ -283,8 +286,8 @@ mod tests {
         assert_ulps_eq!(a, Gaussian::from_ms(28.896475, 0.996604), epsilon = 1e-6);
         assert_ulps_eq!(b, Gaussian::from_ms(32.189211, 6.062063), epsilon = 1e-6);
 
-        let t_a = Rating::new(Gaussian::from_ms(1.139, 0.531), 1.0, ConstantDrift(0.2125));
-        let t_b = Rating::new(Gaussian::from_ms(15.568, 0.51), 1.0, ConstantDrift(0.2125));
+        let t_a = R::new(Gaussian::from_ms(1.139, 0.531), 1.0, ConstantDrift(0.2125));
+        let t_b = R::new(Gaussian::from_ms(15.568, 0.51), 1.0, ConstantDrift(0.2125));
 
         let w = [vec![1.0], vec![1.0]];
         let g = Game::new(
@@ -302,17 +305,17 @@ mod tests {
     #[test]
     fn test_1vs1vs1() {
         let teams = vec![
-            vec![Rating::new(
+            vec![R::new(
                 Gaussian::from_ms(25.0, 25.0 / 3.0),
                 25.0 / 6.0,
                 ConstantDrift(25.0 / 300.0),
             )],
-            vec![Rating::new(
+            vec![R::new(
                 Gaussian::from_ms(25.0, 25.0 / 3.0),
                 25.0 / 6.0,
                 ConstantDrift(25.0 / 300.0),
             )],
-            vec![Rating::new(
+            vec![R::new(
                 Gaussian::from_ms(25.0, 25.0 / 3.0),
                 25.0 / 6.0,
                 ConstantDrift(25.0 / 300.0),
@@ -367,12 +370,12 @@ mod tests {
 
     #[test]
     fn test_1vs1_draw() {
-        let t_a = Rating::new(
+        let t_a = R::new(
             Gaussian::from_ms(25.0, 25.0 / 3.0),
             25.0 / 6.0,
             ConstantDrift(25.0 / 300.0),
         );
-        let t_b = Rating::new(
+        let t_b = R::new(
             Gaussian::from_ms(25.0, 25.0 / 3.0),
             25.0 / 6.0,
             ConstantDrift(25.0 / 300.0),
@@ -394,12 +397,12 @@ mod tests {
         assert_ulps_eq!(a, Gaussian::from_ms(24.999999, 6.469480), epsilon = 1e-6);
         assert_ulps_eq!(b, Gaussian::from_ms(24.999999, 6.469480), epsilon = 1e-6);
 
-        let t_a = Rating::new(
+        let t_a = R::new(
             Gaussian::from_ms(25.0, 3.0),
             25.0 / 6.0,
             ConstantDrift(25.0 / 300.0),
         );
-        let t_b = Rating::new(
+        let t_b = R::new(
             Gaussian::from_ms(29.0, 2.0),
             25.0 / 6.0,
             ConstantDrift(25.0 / 300.0),
@@ -424,17 +427,17 @@ mod tests {
 
     #[test]
     fn test_1vs1vs1_draw() {
-        let t_a = Rating::new(
+        let t_a = R::new(
             Gaussian::from_ms(25.0, 25.0 / 3.0),
             25.0 / 6.0,
             ConstantDrift(25.0 / 300.0),
         );
-        let t_b = Rating::new(
+        let t_b = R::new(
             Gaussian::from_ms(25.0, 25.0 / 3.0),
             25.0 / 6.0,
             ConstantDrift(25.0 / 300.0),
         );
-        let t_c = Rating::new(
+        let t_c = R::new(
             Gaussian::from_ms(25.0, 25.0 / 3.0),
             25.0 / 6.0,
             ConstantDrift(25.0 / 300.0),
@@ -460,17 +463,17 @@ mod tests {
         assert_ulps_eq!(b, Gaussian::from_ms(25.0, 5.707424), epsilon = 1e-6);
         assert_ulps_eq!(c, Gaussian::from_ms(25.0, 5.729069), epsilon = 1e-6);
 
-        let t_a = Rating::new(
+        let t_a = R::new(
             Gaussian::from_ms(25.0, 3.0),
             25.0 / 6.0,
             ConstantDrift(25.0 / 300.0),
         );
-        let t_b = Rating::new(
+        let t_b = R::new(
             Gaussian::from_ms(25.0, 3.0),
             25.0 / 6.0,
             ConstantDrift(25.0 / 300.0),
         );
-        let t_c = Rating::new(
+        let t_c = R::new(
             Gaussian::from_ms(29.0, 2.0),
             25.0 / 6.0,
             ConstantDrift(25.0 / 300.0),
@@ -498,29 +501,29 @@ mod tests {
     #[test]
     fn test_2vs1vs2_mixed() {
         let t_a = vec![
-            Rating::new(
+            R::new(
                 Gaussian::from_ms(12.0, 3.0),
                 25.0 / 6.0,
                 ConstantDrift(25.0 / 300.0),
             ),
-            Rating::new(
+            R::new(
                 Gaussian::from_ms(18.0, 3.0),
                 25.0 / 6.0,
                 ConstantDrift(25.0 / 300.0),
             ),
         ];
-        let t_b = vec![Rating::new(
+        let t_b = vec![R::new(
             Gaussian::from_ms(30.0, 3.0),
             25.0 / 6.0,
             ConstantDrift(25.0 / 300.0),
         )];
         let t_c = vec![
-            Rating::new(
+            R::new(
                 Gaussian::from_ms(14.0, 3.0),
                 25.0 / 6.0,
                 ConstantDrift(25.0 / 300.0),
             ),
-            Rating::new(
+            R::new(
                 Gaussian::from_ms(16., 3.0),
                 25.0 / 6.0,
                 ConstantDrift(25.0 / 300.0),
@@ -549,12 +552,12 @@ mod tests {
         let w_a = vec![1.0];
         let w_b = vec![2.0];
 
-        let t_a = vec![Rating::new(
+        let t_a = vec![R::new(
             Gaussian::from_ms(25.0, 25.0 / 3.0),
             25.0 / 6.0,
             ConstantDrift(0.0),
         )];
-        let t_b = vec![Rating::new(
+        let t_b = vec![R::new(
             Gaussian::from_ms(25.0, 25.0 / 3.0),
             25.0 / 6.0,
             ConstantDrift(0.0),
@@ -632,16 +635,8 @@ mod tests {
         let w_a = vec![1.0];
         let w_b = vec![0.0];
 
-        let t_a = vec![Rating::new(
-            Gaussian::from_ms(2.0, 6.0),
-            1.0,
-            ConstantDrift(0.0),
-        )];
-        let t_b = vec![Rating::new(
-            Gaussian::from_ms(2.0, 6.0),
-            1.0,
-            ConstantDrift(0.0),
-        )];
+        let t_a = vec![R::new(Gaussian::from_ms(2.0, 6.0), 1.0, ConstantDrift(0.0))];
+        let t_b = vec![R::new(Gaussian::from_ms(2.0, 6.0), 1.0, ConstantDrift(0.0))];
 
         let w = [w_a, w_b];
         let g = Game::new(
@@ -667,16 +662,8 @@ mod tests {
         let w_a = vec![1.0];
         let w_b = vec![-1.0];
 
-        let t_a = vec![Rating::new(
-            Gaussian::from_ms(2.0, 6.0),
-            1.0,
-            ConstantDrift(0.0),
-        )];
-        let t_b = vec![Rating::new(
-            Gaussian::from_ms(2.0, 6.0),
-            1.0,
-            ConstantDrift(0.0),
-        )];
+        let t_a = vec![R::new(Gaussian::from_ms(2.0, 6.0), 1.0, ConstantDrift(0.0))];
+        let t_b = vec![R::new(Gaussian::from_ms(2.0, 6.0), 1.0, ConstantDrift(0.0))];
 
         let w = [w_a, w_b];
         let g = Game::new(
@@ -694,12 +681,12 @@ mod tests {
     #[test]
     fn test_2vs2_weighted() {
         let t_a = vec![
-            Rating::new(
+            R::new(
                 Gaussian::from_ms(25.0, 25.0 / 3.0),
                 25.0 / 6.0,
                 ConstantDrift(0.0),
             ),
-            Rating::new(
+            R::new(
                 Gaussian::from_ms(25.0, 25.0 / 3.0),
                 25.0 / 6.0,
                 ConstantDrift(0.0),
@@ -708,12 +695,12 @@ mod tests {
         let w_a = vec![0.4, 0.8];
 
         let t_b = vec![
-            Rating::new(
+            R::new(
                 Gaussian::from_ms(25.0, 25.0 / 3.0),
                 25.0 / 6.0,
                 ConstantDrift(0.0),
             ),
-            Rating::new(
+            R::new(
                 Gaussian::from_ms(25.0, 25.0 / 3.0),
                 25.0 / 6.0,
                 ConstantDrift(0.0),
@@ -824,7 +811,7 @@ mod tests {
         let g = Game::new(
             vec![
                 t_a.clone(),
-                vec![Rating::new(
+                vec![R::new(
                     Gaussian::from_ms(25.0, 25.0 / 3.0),
                     25.0 / 6.0,
                     ConstantDrift(0.0),
diff --git a/src/history.rs b/src/history.rs
index 851b11f..9238e55 100644
--- a/src/history.rs
+++ b/src/history.rs
@@ -8,12 +8,13 @@ use crate::{
     rating::Rating,
     sort_time,
     storage::CompetitorStore,
+    time::Time,
     time_slice::{self, TimeSlice},
     tuple_gt, tuple_max,
 };
 
 #[derive(Clone)]
-pub struct HistoryBuilder<D: Drift = ConstantDrift> {
+pub struct HistoryBuilder<T: Time = i64, D: Drift<T> = ConstantDrift> {
     time: bool,
     mu: f64,
     sigma: f64,
@@ -21,9 +22,10 @@ pub struct HistoryBuilder<D: Drift = ConstantDrift> {
     drift: D,
     p_draw: f64,
     online: bool,
+    _time: std::marker::PhantomData<T>,
 }
 
-impl<D: Drift> HistoryBuilder<D> {
+impl<T: Time, D: Drift<T>> HistoryBuilder<T, D> {
     pub fn time(mut self, time: bool) -> Self {
         self.time = time;
         self
@@ -44,7 +46,7 @@ impl<D: Drift> HistoryBuilder<D> {
         self
     }
 
-    pub fn drift<D2: Drift>(self, drift: D2) -> HistoryBuilder<D2> {
+    pub fn drift<D2: Drift<T>>(self, drift: D2) -> HistoryBuilder<T, D2> {
         HistoryBuilder {
             drift,
             time: self.time,
@@ -53,6 +55,7 @@ impl<D: Drift> HistoryBuilder<D> {
             beta: self.beta,
             p_draw: self.p_draw,
             online: self.online,
+            _time: std::marker::PhantomData,
         }
     }
 
@@ -66,7 +69,7 @@ impl<D: Drift> HistoryBuilder<D> {
         self
     }
 
-    pub fn build(self) -> History<D> {
+    pub fn build(self) -> History<T, D> {
         History {
             size: 0,
             time_slices: Vec::new(),
@@ -82,14 +85,14 @@ impl<D: Drift> HistoryBuilder<D> {
     }
 }
 
-impl HistoryBuilder<ConstantDrift> {
+impl HistoryBuilder<i64, ConstantDrift> {
     pub fn gamma(mut self, gamma: f64) -> Self {
         self.drift = ConstantDrift(gamma);
         self
     }
 }
 
-impl Default for HistoryBuilder<ConstantDrift> {
+impl Default for HistoryBuilder<i64, ConstantDrift> {
     fn default() -> Self {
         Self {
             time: true,
@@ -99,14 +102,15 @@ impl Default for HistoryBuilder<ConstantDrift> {
             drift: ConstantDrift(GAMMA),
             p_draw: P_DRAW,
             online: false,
+            _time: std::marker::PhantomData,
         }
     }
 }
 
-pub struct History<D: Drift = ConstantDrift> {
+pub struct History<T: Time = i64, D: Drift<T> = ConstantDrift> {
     size: usize,
-    pub(crate) time_slices: Vec<TimeSlice>,
-    agents: CompetitorStore<D>,
+    pub(crate) time_slices: Vec<TimeSlice<T>>,
+    pub(crate) agents: CompetitorStore<T, D>,
     time: bool,
     mu: f64,
     sigma: f64,
@@ -116,7 +120,7 @@ pub struct History<D: Drift = ConstantDrift> {
     online: bool,
 }
 
-impl Default for History<ConstantDrift> {
+impl Default for History<i64, ConstantDrift> {
     fn default() -> Self {
         Self {
             size: 0,
@@ -133,13 +137,13 @@ impl Default for History<ConstantDrift> {
     }
 }
 
-impl History<ConstantDrift> {
-    pub fn builder() -> HistoryBuilder<ConstantDrift> {
+impl History<i64, ConstantDrift> {
+    pub fn builder() -> HistoryBuilder<i64, ConstantDrift> {
         HistoryBuilder::default()
     }
 }
 
-impl<D: Drift> History<D> {
+impl<T: Time, D: Drift<T>> History<T, D> {
     fn iteration(&mut self) -> (f64, f64) {
         let mut step = (0.0, 0.0);
 
@@ -226,8 +230,8 @@ impl<D: Drift> History<D> {
         (step, i)
     }
 
-    pub fn learning_curves(&self) -> HashMap<Index, Vec<(i64, Gaussian)>> {
-        let mut data: HashMap<Index, Vec<(i64, Gaussian)>> = HashMap::new();
+    pub fn learning_curves(&self) -> HashMap<Index, Vec<(T, Gaussian)>> {
+        let mut data: HashMap<Index, Vec<(T, Gaussian)>> = HashMap::new();
 
         for b in &self.time_slices {
             for (agent, skill) in b.skills.iter() {
@@ -250,7 +254,9 @@ impl<D: Drift> History<D> {
             .map(|ts| ts.log_evidence(self.online, targets, forward, &self.agents))
             .sum()
     }
+}
 
+impl<D: Drift<i64>> History<i64, D> {
     pub fn add_events(
         &mut self,
         composition: Vec<Vec<Vec<Index>>>,
@@ -267,7 +273,7 @@ impl<D: Drift> History<D> {
         results: Vec<Vec<f64>>,
         times: Vec<i64>,
         weights: Vec<Vec<Vec<f64>>>,
-        mut priors: HashMap<Index, Rating<D>>,
+        mut priors: HashMap<Index, Rating<i64, D>>,
     ) {
         assert!(times.is_empty() || self.time, "length(times)>0 but !h.time");
         assert!(
@@ -310,7 +316,7 @@ impl<D: Drift> History<D> {
                             )
                         }),
                         message: N_INF,
-                        last_time: i64::MIN,
+                        last_time: None,
                     },
                 );
             }
@@ -343,21 +349,30 @@ impl<D: Drift> History<D> {
                     time_slice.new_forward_info(&self.agents);
                 }
 
-                // TODO: Is it faster to iterate over agents in batch instead?
                 for agent_idx in &this_agent {
                     if let Some(skill) = time_slice.skills.get_mut(*agent_idx) {
                         skill.elapsed = time_slice::compute_elapsed(
-                            self.agents[*agent_idx].last_time,
-                            time_slice.time,
+                            self.agents[*agent_idx].last_time.as_ref(),
+                            &time_slice.time,
                         );
 
                         let agent = self.agents.get_mut(*agent_idx).unwrap();
 
-                        agent.last_time = if self.time { time_slice.time } else { i64::MAX };
+                        agent.last_time = Some(time_slice.time);
                         agent.message = time_slice.forward_prior_out(agent_idx);
                     }
                 }
 
+                if !self.time {
+                    let slice_time = time_slice.time;
+                    for agent_idx in &this_agent {
+                        let c = self.agents.get_mut(*agent_idx).unwrap();
+                        if c.last_time.is_some() {
+                            c.last_time = Some(slice_time);
+                        }
+                    }
+                }
+
                 k += 1;
             }
 
@@ -384,11 +399,11 @@ impl<D: Drift> History<D> {
                 for agent_idx in time_slice.skills.keys() {
                     let agent = self.agents.get_mut(agent_idx).unwrap();
 
-                    agent.last_time = if self.time { t } else { i64::MAX };
+                    agent.last_time = Some(t);
                     agent.message = time_slice.forward_prior_out(&agent_idx);
                 }
             } else {
-                let mut time_slice: TimeSlice = TimeSlice::new(t, self.p_draw);
+                let mut time_slice = TimeSlice::new(t, self.p_draw);
                 time_slice.add_events(composition, results, weights, &self.agents);
 
                 self.time_slices.insert(k, time_slice);
@@ -398,10 +413,19 @@ impl<D: Drift> History<D> {
                 for agent_idx in time_slice.skills.keys() {
                     let agent = self.agents.get_mut(agent_idx).unwrap();
 
-                    agent.last_time = if self.time { t } else { i64::MAX };
+                    agent.last_time = Some(t);
                     agent.message = time_slice.forward_prior_out(&agent_idx);
                 }
 
+                if !self.time {
+                    for agent_idx in &this_agent {
+                        let c = self.agents.get_mut(*agent_idx).unwrap();
+                        if c.last_time.is_some() {
+                            c.last_time = Some(t);
+                        }
+                    }
+                }
+
                 k += 1;
             }
 
@@ -413,17 +437,16 @@ impl<D: Drift> History<D> {
 
             time_slice.new_forward_info(&self.agents);
 
-            // TODO: Is it faster to iterate over agents in batch instead?
             for agent_idx in &this_agent {
                 if let Some(skill) = time_slice.skills.get_mut(*agent_idx) {
                     skill.elapsed = time_slice::compute_elapsed(
-                        self.agents[*agent_idx].last_time,
-                        time_slice.time,
+                        self.agents[*agent_idx].last_time.as_ref(),
+                        &time_slice.time,
                     );
 
                     let agent = self.agents.get_mut(*agent_idx).unwrap();
 
-                    agent.last_time = if self.time { time_slice.time } else { i64::MAX };
+                    agent.last_time = Some(time_slice.time);
                     agent.message = time_slice.forward_prior_out(agent_idx);
                 }
             }
diff --git a/src/rating.rs b/src/rating.rs
index 25fe13e..3530e24 100644
--- a/src/rating.rs
+++ b/src/rating.rs
@@ -1,7 +1,10 @@
+use std::marker::PhantomData;
+
 use crate::{
     BETA, GAMMA,
     drift::{ConstantDrift, Drift},
     gaussian::Gaussian,
+    time::Time,
 };
 
 /// Static rating configuration: prior skill, performance noise `beta`, drift.
@@ -9,15 +12,21 @@ use crate::{
 /// Renamed from `Player` in T2; `Rating` better describes the data
 /// (a configuration) vs. a person (who's a `Competitor` with state).
 #[derive(Clone, Copy, Debug)]
-pub struct Rating<D: Drift = ConstantDrift> {
+pub struct Rating<T: Time = i64, D: Drift<T> = ConstantDrift> {
     pub(crate) prior: Gaussian,
     pub(crate) beta: f64,
     pub(crate) drift: D,
+    pub(crate) _time: PhantomData<T>,
 }
 
-impl<D: Drift> Rating<D> {
+impl<T: Time, D: Drift<T>> Rating<T, D> {
     pub fn new(prior: Gaussian, beta: f64, drift: D) -> Self {
-        Self { prior, beta, drift }
+        Self {
+            prior,
+            beta,
+            drift,
+            _time: PhantomData,
+        }
     }
 
     pub(crate) fn performance(&self) -> Gaussian {
@@ -25,12 +34,13 @@ impl<D: Drift> Rating<D> {
     }
 }
 
-impl Default for Rating<ConstantDrift> {
+impl Default for Rating<i64, ConstantDrift> {
     fn default() -> Self {
         Self {
             prior: Gaussian::default(),
             beta: BETA,
             drift: ConstantDrift(GAMMA),
+            _time: PhantomData,
         }
     }
 }
diff --git a/src/storage/competitor_store.rs b/src/storage/competitor_store.rs
index b8f392f..25f72aa 100644
--- a/src/storage/competitor_store.rs
+++ b/src/storage/competitor_store.rs
@@ -1,17 +1,17 @@
-use crate::{Index, competitor::Competitor, drift::Drift};
+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
-/// forward/backward sweep. Uses `Vec<Option<Competitor<D>>>` so slots can be
+/// forward/backward sweep. Uses `Vec<Option<Competitor<T, D>>>` so slots can be
 /// absent without an explicit present mask.
 #[derive(Debug)]
-pub struct CompetitorStore<D: Drift> {
-    competitors: Vec<Option<Competitor<D>>>,
+pub struct CompetitorStore<T: Time = i64, D: Drift<T> = crate::drift::ConstantDrift> {
+    competitors: Vec<Option<Competitor<T, D>>>,
     n_present: usize,
 }
 
-impl<D: Drift> Default for CompetitorStore<D> {
+impl<T: Time, D: Drift<T>> Default for CompetitorStore<T, D> {
     fn default() -> Self {
         Self {
             competitors: Vec::new(),
@@ -20,7 +20,7 @@ impl<D: Drift> Default for CompetitorStore<D> {
     }
 }
 
-impl<D: Drift> CompetitorStore<D> {
+impl<T: Time, D: Drift<T>> CompetitorStore<T, D> {
     pub fn new() -> Self {
         Self::default()
     }
@@ -31,7 +31,7 @@ impl<D: Drift> CompetitorStore<D> {
         }
     }
 
-    pub fn insert(&mut self, idx: Index, competitor: Competitor<D>) {
+    pub fn insert(&mut self, idx: Index, competitor: Competitor<T, D>) {
         self.ensure_capacity(idx.0);
         if self.competitors[idx.0].is_none() {
             self.n_present += 1;
@@ -39,11 +39,11 @@ impl<D: Drift> CompetitorStore<D> {
         self.competitors[idx.0] = Some(competitor);
     }
 
-    pub fn get(&self, idx: Index) -> Option<&Competitor<D>> {
+    pub fn get(&self, idx: Index) -> Option<&Competitor<T, D>> {
         self.competitors.get(idx.0).and_then(|slot| slot.as_ref())
     }
 
-    pub fn get_mut(&mut self, idx: Index) -> Option<&mut Competitor<D>> {
+    pub fn get_mut(&mut self, idx: Index) -> Option<&mut Competitor<T, D>> {
         self.competitors
             .get_mut(idx.0)
             .and_then(|slot| slot.as_mut())
@@ -61,34 +61,34 @@ impl<D: Drift> CompetitorStore<D> {
         self.n_present == 0
     }
 
-    pub fn iter(&self) -> impl Iterator<Item = (Index, &Competitor<D>)> {
+    pub fn iter(&self) -> impl Iterator<Item = (Index, &Competitor<T, D>)> {
         self.competitors
             .iter()
             .enumerate()
             .filter_map(|(i, slot)| slot.as_ref().map(|a| (Index(i), a)))
     }
 
-    pub fn iter_mut(&mut self) -> impl Iterator<Item = (Index, &mut Competitor<D>)> {
+    pub fn iter_mut(&mut self) -> impl Iterator<Item = (Index, &mut Competitor<T, D>)> {
         self.competitors
             .iter_mut()
             .enumerate()
             .filter_map(|(i, slot)| slot.as_mut().map(|a| (Index(i), a)))
     }
 
-    pub fn values_mut(&mut self) -> impl Iterator<Item = &mut Competitor<D>> {
+    pub fn values_mut(&mut self) -> impl Iterator<Item = &mut Competitor<T, D>> {
         self.competitors.iter_mut().filter_map(|s| s.as_mut())
     }
 }
 
-impl<D: Drift> std::ops::Index<Index> for CompetitorStore<D> {
-    type Output = Competitor<D>;
-    fn index(&self, idx: Index) -> &Competitor<D> {
+impl<T: Time, D: Drift<T>> std::ops::Index<Index> for CompetitorStore<T, D> {
+    type Output = Competitor<T, D>;
+    fn index(&self, idx: Index) -> &Competitor<T, D> {
         self.get(idx).expect("competitor not found at index")
     }
 }
 
-impl<D: Drift> std::ops::IndexMut<Index> for CompetitorStore<D> {
-    fn index_mut(&mut self, idx: Index) -> &mut Competitor<D> {
+impl<T: Time, D: Drift<T>> std::ops::IndexMut<Index> for CompetitorStore<T, D> {
+    fn index_mut(&mut self, idx: Index) -> &mut Competitor<T, D> {
         self.get_mut(idx).expect("competitor not found at index")
     }
 }
@@ -100,7 +100,7 @@ mod tests {
 
     #[test]
     fn insert_then_get() {
-        let mut store: CompetitorStore<ConstantDrift> = CompetitorStore::new();
+        let mut store: CompetitorStore<i64, ConstantDrift> = CompetitorStore::new();
         let idx = Index(7);
         store.insert(idx, Competitor::default());
         assert!(store.contains(idx));
@@ -110,7 +110,7 @@ mod tests {
 
     #[test]
     fn iter_in_index_order() {
-        let mut store: CompetitorStore<ConstantDrift> = CompetitorStore::new();
+        let mut store: CompetitorStore<i64, ConstantDrift> = CompetitorStore::new();
         store.insert(Index(2), Competitor::default());
         store.insert(Index(0), Competitor::default());
         store.insert(Index(5), Competitor::default());
@@ -120,7 +120,7 @@ mod tests {
 
     #[test]
     fn index_operator_works() {
-        let mut store: CompetitorStore<ConstantDrift> = CompetitorStore::new();
+        let mut store: CompetitorStore<i64, ConstantDrift> = CompetitorStore::new();
         store.insert(Index(3), Competitor::default());
         let _ = &store[Index(3)];
     }
diff --git a/src/time_slice.rs b/src/time_slice.rs
index 6f1ed1f..162398a 100644
--- a/src/time_slice.rs
+++ b/src/time_slice.rs
@@ -12,6 +12,7 @@ use crate::{
     gaussian::Gaussian,
     rating::Rating,
     storage::{CompetitorStore, SkillStore},
+    time::Time,
     tuple_gt, tuple_max,
 };
 
@@ -49,13 +50,13 @@ struct Item {
 }
 
 impl Item {
-    fn within_prior<D: Drift>(
+    fn within_prior<T: Time, D: Drift<T>>(
         &self,
         online: bool,
         forward: bool,
         skills: &SkillStore,
-        agents: &CompetitorStore<D>,
-    ) -> Rating<D> {
+        agents: &CompetitorStore<T, D>,
+    ) -> Rating<T, D> {
         let r = &agents[self.agent].rating;
         let skill = skills.get(self.agent).unwrap();
 
@@ -90,13 +91,13 @@ impl Event {
             .collect::<Vec<_>>()
     }
 
-    pub(crate) fn within_priors<D: Drift>(
+    pub(crate) fn within_priors<T: Time, D: Drift<T>>(
         &self,
         online: bool,
         forward: bool,
         skills: &SkillStore,
-        agents: &CompetitorStore<D>,
-    ) -> Vec<Vec<Rating<D>>> {
+        agents: &CompetitorStore<T, D>,
+    ) -> Vec<Vec<Rating<T, D>>> {
         self.teams
             .iter()
             .map(|team| {
@@ -110,16 +111,16 @@ impl Event {
 }
 
 #[derive(Debug)]
-pub struct TimeSlice {
+pub struct TimeSlice<T: Time = i64> {
     pub(crate) events: Vec<Event>,
     pub(crate) skills: SkillStore,
-    pub(crate) time: i64,
+    pub(crate) time: T,
     p_draw: f64,
     arena: ScratchArena,
 }
 
-impl TimeSlice {
-    pub fn new(time: i64, p_draw: f64) -> Self {
+impl<T: Time> TimeSlice<T> {
+    pub fn new(time: T, p_draw: f64) -> Self {
         Self {
             events: Vec::new(),
             skills: SkillStore::new(),
@@ -129,12 +130,12 @@ impl TimeSlice {
         }
     }
 
-    pub fn add_events<D: Drift>(
+    pub fn add_events<D: Drift<T>>(
         &mut self,
         composition: Vec<Vec<Vec<Index>>>,
         results: Vec<Vec<f64>>,
         weights: Vec<Vec<Vec<f64>>>,
-        agents: &CompetitorStore<D>,
+        agents: &CompetitorStore<T, D>,
     ) {
         let mut unique = Vec::with_capacity(10);
 
@@ -149,16 +150,16 @@ impl TimeSlice {
         });
 
         for idx in this_agent {
-            let elapsed = compute_elapsed(agents[*idx].last_time, self.time);
+            let elapsed = compute_elapsed(agents[*idx].last_time.as_ref(), &self.time);
 
             if let Some(skill) = self.skills.get_mut(*idx) {
                 skill.elapsed = elapsed;
-                skill.forward = agents[*idx].receive(elapsed);
+                skill.forward = agents[*idx].receive(&self.time);
             } else {
                 self.skills.insert(
                     *idx,
                     Skill {
-                        forward: agents[*idx].receive(elapsed),
+                        forward: agents[*idx].receive(&self.time),
                         elapsed,
                         ..Default::default()
                     },
@@ -220,7 +221,7 @@ impl TimeSlice {
             .collect::<HashMap<_, _>>()
     }
 
-    pub fn iteration<D: Drift>(&mut self, from: usize, agents: &CompetitorStore<D>) {
+    pub fn iteration<D: Drift<T>>(&mut self, from: usize, agents: &CompetitorStore<T, D>) {
         for event in self.events.iter_mut().skip(from) {
             let teams = event.within_priors(false, false, &self.skills, agents);
             let result = event.outputs();
@@ -241,7 +242,7 @@ impl TimeSlice {
     }
 
     #[allow(dead_code)]
-    pub(crate) fn convergence<D: Drift>(&mut self, agents: &CompetitorStore<D>) -> usize {
+    pub(crate) fn convergence<D: Drift<T>>(&mut self, agents: &CompetitorStore<T, D>) -> usize {
         let epsilon = 1e-6;
         let iterations = 20;
 
@@ -270,36 +271,41 @@ impl TimeSlice {
         skill.forward * skill.likelihood
     }
 
-    pub(crate) fn backward_prior_out<D: Drift>(
+    pub(crate) fn backward_prior_out<D: Drift<T>>(
         &self,
         agent: &Index,
-        agents: &CompetitorStore<D>,
+        agents: &CompetitorStore<T, D>,
     ) -> Gaussian {
         let skill = self.skills.get(*agent).unwrap();
         let n = skill.likelihood * skill.backward;
-        n.forget(agents[*agent].rating.drift.variance_delta(skill.elapsed))
+        n.forget(
+            agents[*agent]
+                .rating
+                .drift
+                .variance_for_elapsed(skill.elapsed),
+        )
     }
 
-    pub(crate) fn new_backward_info<D: Drift>(&mut self, agents: &CompetitorStore<D>) {
+    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;
         }
         self.iteration(0, agents);
     }
 
-    pub(crate) fn new_forward_info<D: Drift>(&mut self, agents: &CompetitorStore<D>) {
+    pub(crate) fn new_forward_info<D: Drift<T>>(&mut self, agents: &CompetitorStore<T, D>) {
         for (agent, skill) in self.skills.iter_mut() {
-            skill.forward = agents[agent].receive(skill.elapsed);
+            skill.forward = agents[agent].receive_for_elapsed(skill.elapsed);
         }
         self.iteration(0, agents);
     }
 
-    pub(crate) fn log_evidence<D: Drift>(
+    pub(crate) fn log_evidence<D: Drift<T>>(
         &self,
         online: bool,
         targets: &[Index],
         forward: bool,
-        agents: &CompetitorStore<D>,
+        agents: &CompetitorStore<T, D>,
     ) -> f64 {
         // log_evidence is infrequent; a local arena avoids needing &mut self.
         let mut arena = ScratchArena::new();
@@ -388,14 +394,8 @@ impl TimeSlice {
     }
 }
 
-pub(crate) fn compute_elapsed(last_time: i64, actual_time: i64) -> i64 {
-    if last_time == i64::MIN {
-        0
-    } else if last_time == i64::MAX {
-        1
-    } else {
-        actual_time - last_time
-    }
+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)
 }
 
 #[cfg(test)]
@@ -419,7 +419,7 @@ mod tests {
         let e = index_map.get_or_create("e");
         let f = index_map.get_or_create("f");
 
-        let mut agents: CompetitorStore<ConstantDrift> = CompetitorStore::new();
+        let mut agents: CompetitorStore<i64, ConstantDrift> = CompetitorStore::new();
 
         for agent in [a, b, c, d, e, f] {
             agents.insert(
@@ -435,7 +435,7 @@ mod tests {
             );
         }
 
-        let mut time_slice = TimeSlice::new(0, 0.0);
+        let mut time_slice = TimeSlice::new(0i64, 0.0);
 
         time_slice.add_events(
             vec![
@@ -495,7 +495,7 @@ mod tests {
         let e = index_map.get_or_create("e");
         let f = index_map.get_or_create("f");
 
-        let mut agents: CompetitorStore<ConstantDrift> = CompetitorStore::new();
+        let mut agents: CompetitorStore<i64, ConstantDrift> = CompetitorStore::new();
 
         for agent in [a, b, c, d, e, f] {
             agents.insert(
@@ -511,7 +511,7 @@ mod tests {
             );
         }
 
-        let mut time_slice = TimeSlice::new(0, 0.0);
+        let mut time_slice = TimeSlice::new(0i64, 0.0);
 
         time_slice.add_events(
             vec![
@@ -574,7 +574,7 @@ mod tests {
         let e = index_map.get_or_create("e");
         let f = index_map.get_or_create("f");
 
-        let mut agents: CompetitorStore<ConstantDrift> = CompetitorStore::new();
+        let mut agents: CompetitorStore<i64, ConstantDrift> = CompetitorStore::new();
 
         for agent in [a, b, c, d, e, f] {
             agents.insert(
@@ -590,7 +590,7 @@ mod tests {
             );
         }
 
-        let mut time_slice = TimeSlice::new(0, 0.0);
+        let mut time_slice = TimeSlice::new(0i64, 0.0);
 
         time_slice.add_events(
             vec![