README: v2 family rows + dense-supervision finding in headline

README.md

@@ -54,8 +54,12 @@ cosine decay, batch 256, 25k steps (100k for fixed4-long).
 | rec-1×dynamic | 1 × R~U{2..8} | 234,880 | 2.73e15 |
 | rec-4×4 | 4 × 4 | 829,696 | 8.68e15 |
 | fixed-4-long | 4 × 1, 100k steps | 829,696 | 8.74e15 |
+| rec-1×aligned | 1 × k (R = k) | 240,256 | 2.46e15 |
+| rec-1×m4 | 1 × (k+4) | 242,048 | 4.62e15 |
+| rec-1×m4+dense | 1 × (k+4), stem, dense loss | 440,320 | 4.62e15* |
 
 Seeds: 3 per family (2 for fixed-4-long). All checkpoints in `ckpts/`.
+(*The m4+dense row's stored FLOPs under-count its stem by ~12%.)
 
 ## Headline results (mean over seeds, exact-match accuracy)
 
@@ -67,6 +71,9 @@ Seeds: 3 per family (2 for fixed-4-long). All checkpoints in `ckpts/`.
 | rec-1×dynamic | 0.899±0.008 | 0.103±0.036 | 0.195 |
 | rec-4×4 | **1.000** | 0.251±0.099 | 0.269 |
 | fixed-4-long (iso-FLOP) | 0.716±0.009 | 0.171±0.018 | 0.172 |
+| rec-1×aligned (R=k) | 0.106±0.003 | 0.017±0.003 | 0.139 |
+| rec-1×m4 (R=k+4) | 0.262±0.003 | 0.029±0.017 | 0.196 |
+| rec-1×m4+dense | **1.000±0.000** | 0.075±0.032 | **0.437** |
 
 Key observations (empirical):
 
@@ -80,8 +87,18 @@ Key observations (empirical):
 3. **More inference-time unrolls than trained sometimes help a lot** —
    rec-4×4 seed2: 0.394→0.401 at R=5; rec-1×dynamic seed1: 0.088→0.319 at
    R=5 — but the effect is seed-dependent, not systematic at this scale.
-4. **Large seed variance in OOD** (rec-4×4 k=64: 0.18/0.18/0.39) means
-   single-seed comparisons of length generalisation are unreliable here.
+4. **Large seed variance in OOD** (rec-4×4 k=64: 0.18/0.18/0.39;
+   m4+dense k=64: 0.116/0.038/0.070) means single-seed comparisons of
+   length generalisation are unreliable here.
+5. **The v2 ablation (§4.7-4.8) decomposes what makes aligned training
+   work**: R=k with end-of-chain loss fails (0.106 — credit assignment
+   across depths); a +4 unroll margin alone only partially helps (0.262);
+   adding an untied stem block and dense per-iteration supervision with
+   hold-after-k targets f^min(t,k)(x) fully rescues it (1.000), and its
+   fixed-R sweep shows a staircase (+0.634) — the model genuinely does
+   one hop per unroll. **But even this model does not extend past k=8**
+   (best k=64 over R: 0.437) — the frontier tracks the training-data
+   limit, making curriculum/data coverage the most promising next lever.
 
 See REPORT.md for the full analysis, figures, and limitations.