PHASE4_HC_ATTRIBUTION.md

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.