diag(policy-quality): wire per-branch action entropy (ent_mag, ent_dir)

Adds module-level shannon_entropy_normalized helper and computes per-branch
action entropy from the existing monitor.action_counts data:
  * ent_mag = entropy over Quarter/Half/Full distribution
  * ent_dir = entropy over direction-bin distribution (indices 0-2 / 3-5 / 6-8)

Normalized to [0, 1] so 0 = full collapse, 1 = uniform.

Does not complete Task 0.10: sigma_mean (NoisyNets σ overall) remains 0.0
until Task 0.6 wires the per-branch NoisyNets σ readback (separate commit).

crates/ml/src/trainers/dqn/trainer/training_loop.rs

@@ -30,6 +30,27 @@ use super::super::financials::compute_epoch_financials;
 use super::super::monitoring::TrainingMonitor;
 use super::DQNTrainer;
 
+/// Normalized Shannon entropy of a discrete probability distribution.
+///
+/// Returns the entropy divided by log(N) where N is the number of bins, so
+/// the result is in [0.0, 1.0]: 0.0 = policy fully collapsed to one bin,
+/// 1.0 = uniform. Inputs need not sum to exactly 1 — the function
+/// re-normalizes defensively.
+fn shannon_entropy_normalized(dist: &[f32]) -> f32 {
+    let sum: f32 = dist.iter().sum();
+    if sum <= 0.0 || dist.len() < 2 {
+        return 0.0;
+    }
+    let mut h = 0.0_f32;
+    for &p in dist {
+        let q = p / sum;
+        if q > 1e-9 {
+            h -= q * q.ln();
+        }
+    }
+    h / (dist.len() as f32).ln()
+}
+
 /// Metrics returned from `process_epoch_boundary` for downstream logging.
 pub(crate) struct EpochBoundaryMetrics {
     pub avg_loss: f32,
@@ -2174,6 +2195,22 @@ impl DQNTrainer {
             // Cache for smoke-test accessors (magnitude_distribution test).
             self.last_magnitude_dist = [dist_q, dist_h, dist_f];
 
+            // Track 4 exploration entropy — per-branch action entropy normalized
+            // to [0, 1]. Magnitude branch uses the [dist_q, dist_h, dist_f] above;
+            // direction is summed across magnitudes per direction bin.
+            // Layout (dir*3 + mag): dir 0 = indices 0..3, dir 1 (Flat) = 3..6, dir 2 = 6..9.
+            let ent_mag = shannon_entropy_normalized(&[dist_q, dist_h, dist_f]);
+            let (dir0, dir1, dir2) = if action_total_epoch > 0 {
+                let t = action_total_epoch as f32;
+                let d0 = (monitor.action_counts[0] + monitor.action_counts[1] + monitor.action_counts[2]) as f32 / t;
+                let d1 = (monitor.action_counts[3] + monitor.action_counts[4] + monitor.action_counts[5]) as f32 / t;
+                let d2 = (monitor.action_counts[6] + monitor.action_counts[7] + monitor.action_counts[8]) as f32 / t;
+                (d0, d1, d2)
+            } else {
+                (0.0_f32, 0.0_f32, 0.0_f32)
+            };
+            let ent_dir = shannon_entropy_normalized(&[dir0, dir1, dir2]);
+
             tracing::info!(
                 "HEALTH_DIAG[{}]: health={:.2} components [q_gap={:.2} q_var={:.2} atoms={:.2} grad_stable={:.2} ens_agree={:.2} grad_cos={:.2} spectral={:.2}] effective [cql_alpha={:.4} iqn_budget={:.2} cql_budget={:.2} c51_budget={:.2} tau={:.5} sarsa_tau={:.2} gamma={:.3} cf_ratio={:.2}] novels [distill={} barrier={:.3} plasticity={} ib={:.3} ensemble_collapse={:.2} contrarian={} meta_q_pred={:.2}] diag [sharpe_ema={:.3} action_entropy={:.2}] gems [g12_predictive={:.4}] mag [q_full={:.3} q_half={:.3} q_quarter={:.3} var_scale={:.3} kelly_f={:.3} avg_win_ratio={:.3} grad_ratio_mag_dir={:.4} dist_q={:.3} dist_h={:.3} dist_f={:.3}] trail [fire_q={:.3} fire_h={:.3} fire_f={:.3} hold_q={:.2} hold_h={:.2} hold_f={:.2}] noisy [vsn_mag={:.3} vsn_dir={:.3} sigma_mag={:.4} sigma_dir={:.4} drift_mag={:.3} drift_dir={:.3}] eval_dist [eq={:.3} eh={:.3} ef={:.3}] reward_contrib [popart={:.3} cf={:.3} trail_r={:.3} micro={:.3} la={:.3} seg={:.3}] controller [anti_lr={} tau={} gamma={} clip={} cql={} cost={} fire_frac={:.2}] explore [ent_mag={:.2} ent_dir={:.2} sigma_mean={:.4}]",
                 epoch,
@@ -2219,8 +2256,9 @@ impl DQNTrainer {
                 0.0_f32, 0.0_f32, 0.0_f32, 0.0_f32, 0.0_f32, 0.0_f32,
                 // Track 3 — controllers (6 bool, 1 f32)
                 false, false, false, false, false, false, 0.0_f32,
-                // Track 4 — explore (3 f32)
-                0.0_f32, 0.0_f32, 0.0_f32,
+                // Track 4 — explore (3 f32): ent_mag, ent_dir, sigma_mean.
+                // sigma_mean still stubbed (needs NoisyNets σ readback, Task 0.6).
+                ent_mag, ent_dir, 0.0_f32,
             );
 
             // C1/P1: propagate health to GPU replay buffer for diversity-weighted priorities.