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PHASE4_HC_ATTRIBUTION.md

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+# Phase 4 — H-C refinement & attribution: the gain is *learnability*, not Calabi–Yau
+
+After the autoresearch sweep found that a **learnable per-head γ** "Holonomic-ALiBi"
+bias beats baselines, we asked the question that decides what we actually learned:
+**is the gain from the Calabi–Yau curve shape, or just from making the slope
+learnable?** Two runs answer it (2026-06-22).
+
+## Result 1 — v2 GPU full run confirms the gain is REAL and holds at scale (PASS)
+
+Regularized (γ-L2 + val early-stop + 8M-char corpus), 8000 steps, real WikiText-2:
+
+| scheme | val ppl@512 |
+|---|---|
+| **holo_ladder_pos** | **3.324** |
+| holo_ladder | 3.336 |
+| holo_tiny | 3.576 |
+| learned (baseline) | 3.617 |
+| sliding (baseline) | 3.702 |
+
+holo_ladder_pos vs best baseline = **+8.1%**. The v1 overfitting inversion is gone;
+the learnable Holonomic bias genuinely beats the baselines at scale. **The original
+§5 KILL is overturned — this configuration PASSES.**
+
+## Result 2 — attribution (local, controlled): it's LEARNABILITY, not the CY shape
+
+A nested bias $\text{bias}_h(d) = -a_h\,d + b_h\log d$ ($a,b$ learnable per head)
+lets us separate the linear slope from the log-curvature, with controls:
+
+| scheme | ppl@512 | vs fixed ALiBi | isolates |
+|---|---|---|---|
+| **nested_curv0** | **8.548** | **+6.9%** | learnable slope + free log-curvature (linear init) |
+| nested_free | 8.680 | +5.5% | learnable slope + free curvature (CY init) |
+| **alibi_learn** | 8.835 | **+3.8%** | learnable slope, **no curvature** (control) |
+| holo_fixed | 8.998 | +2.0% | fixed CY shape (the autoresearch "winner") |
+| log_only | 8.998 | +2.0% | fixed CY shape (exact tiered) |
+| alibi (fixed) | 9.181 | — | baseline |
+
+**Three conclusions, stated plainly:**
+
+1. **Learnability is the driver, not the geometry.** A plain *learnable* ALiBi slope
+   with zero Calabi–Yau content (`alibi_learn`, +3.8%) beats the fixed holonomic
+   bias (`holo_fixed`, +2.0%). The headline "CY-Sieve beats ALiBi" was really
+   "learnable beats fixed."
+
+2. **Given freedom, the model moves AWAY from the CY curvature.** `nested_free`
+   starts at the Calabi–Yau prior (log-curvature $b=\beta=2$ per unit slope) and
+   gradient descent drives $b$ **negative** ($b\approx-0.2$). The $\beta=2$
+   curvature is not what the data wants — the geometry was a prior, and not a
+   well-pointed one.
+
+3. **The actual best bias drops the holonomic sequence entirely.** `nested_curv0`
+   (+6.9%) — a learnable linear slope plus a small *learned* log-correction — wins,
+   and needs no $S_{20}$, no recurrence, no Calabi–Yau structure. It is essentially
+   **learnable-ALiBi + a learnable log term**.
+
+## Honest consolidation — what H-C actually is
+
+- **A real, modest quality win exists** (+6.9% local / +8.1% at GPU scale vs
+  baselines) and it is **shippable** — but its source is *parameterizing the
+  positional slope (and a log term) as learnable per-head*, a known-good idea in the
+  ALiBi/learnable-bias family. The Calabi–Yau sequence is **not** the source and is
+  best dropped.
+- **No memory win** (see `CYSIEVE_FUTURE_DIRECTIONS.md`): a learnable per-head
+  scalar bias is O(1) state and O(L) to apply — same as ALiBi — so there is nothing
+  to save over FlashAttention+ALiBi.
+- **The most defensible contribution** from this whole applied arc is therefore a
+  small, honest one: *a learnable per-head linear+log positional bias that edges
+  ALiBi by a few %, with stable length-extrapolation.* We will not attach the
+  Calabi–Yau story to it, because the controlled experiment says the geometry does
+  not earn its place.
+
+## What to ship / next
+
+- **Ship `nested_curv0` (learnable slope + learnable log term)** as the positional
+  scheme, named for what it is (not "CY-Sieve"). Re-run it at GPU scale alongside
+  learnable-ALiBi to confirm the local ranking.
+- The Calabi–Yau mathematics remains the project's real contribution; the attention
+  kernel is, honestly, *a learnable-bias result that the geometry inspired but does
+  not explain*.