BUILD LOG · SUDOKU · few-shot recursion

Energy as an OBSERVATION lifts held-out exact 0 → 0.09 — the first liftoff in the learned recursion.

Feed the ruler's violation-map to GRAM as a detached input channel — never as a gradient. Loss stays CE-only. Result: energy's first net-positive contribution to a weight-shared K=10 recursion, generalizing past pure memorization to held-out + weak-OOD puzzles.

held-out exact

0 → 0.090 ✓

first sustained liftoff in the recursion (was 0 = memorized)

weak-OOD exact

0 → 0.095 ✓

still climbing at 8000 steps

target-OOD exact

0 ✗

17.9-clue hard CSP — needs guessing + backtracking

the unified law

observation lives.
gradient dies (8×).

energy read at discrete grids — never descended

§0

UPDATE — VERIFIED SEARCH

심야 갱신 · ①×② 탐색이 신기록

After this page was first built, three verdicts landed the same day. (1) P/Q split: champion = Q (init-div + per-step re-damage; clue-p was the memorization culprit). (2) Observation ablation: the 4-ch violation-map observation is a modest but real contributor (held-out exact 0.040→0.070 paired); the main driver is operator learning. A richer 13-ch candidate-mask observation LOWERED the ceiling (crutch effect) — yet the same information consumed as search is decisive: the value of a signal is set by where it is consumed, not what it contains. (3) C2 verified search (inference-only, no training, gram frozen): the model orders MRV branches, E_c-derived dead-ends prune, sound verification accepts.

decode	held-out exact	weak-OOD exact	target-OOD exact	note
model alone (greedy)	0.070	0.100	0.005	BoN-16 ≡ greedy (confident model)
MRV search, no model @64	0.21	0.55	0.02	blind backtracking
model-guided @16	0.19	0.49	0.03	≈ blind@64 with 4× fewer nodes
model-guided @64	0.31	0.75	0.09	project record — beats the old hybrid BoN (0.7)

① supplies sound pruning/acceptance, ② supplies branch ordering worth ~4× the search budget; wrong-accepts 0 everywhere. Next: distill multiple verified paths back into the operator (CA-2, running).

§1

RESULT

결론 — 무엇이 됐나 (당일 오전 기준)

system	held-out exact	weak-OOD exact	target-OOD exact	note
2026-06-26 pathwise −∇E DEAD	0	0	0	DEAD method (gradient) — died 8×
C-gold (obs, no init-div)	0	0	0	first energy net-positive, but memorizes clue→gold
CA-1a (obs + init-div + re-damage)	0.090	0.095	0	first held-out + OOD liftoff
CA-1b (+ pool)	~0.060	~0	0	pool net-negative (more memorization)
E0 propagation ceiling	0.150	0.427	0.326	logical-only baseline (the wall to beat)

Held-out exact-match over training steps: C-gold flat at 0, CA-1a lifting to 0.090. — **Held-out exact over 8000 steps.** C-gold stays flat at 0 (pure memorization); CA-1a lifts off near step ~1000 and is still climbing at 0.090 when training ends.

Learned operator vs the E0 propagation ceiling across held-out, weak-OOD and target-OOD. — **Against the propagation ceiling.** The learned operator (0.090) is real but still *below* logic-only propagation (0.150) — a soft corrector, not yet a sound propagator.

§2

HOW IT WORKS

관측 루프 — 에너지를 읽되 미분하지 않는다

① ruler E_chand-written row/col/box conflict count — exactly 0 only at a valid solution

② GRAMweight-shared recursion, builds the answer over K=10 steps

→

follow ① from few datano augmentation; generalize to held-out + OOD

Each GRAM step reads the E_c violation-map as a detached no-grad input channel; the candidate is refined by CE only.

The observation loop (centerpiece). Every one of K=10 steps re-reads the E_c violation-map — row/col/box conflicts + candidate count — as a detached, no_grad input channel. The candidate moves by CE only; no energy gradient ever flows.

NCA-style damage-regenerate: a partially filled grid is corrupted and the operator regrows the solution.

NCA-style damage → regenerate. Training init is diversified (partial gold fill + corrupt); the operator learns to repair arbitrary states, not just replay clean clue trajectories. This kills the exposure bias.

The unified law · 프로젝트의 하드코어

Energy must be consumed as a discrete-state OBSERVATION / verifier / filter — never descended as a continuous gradient field.

✓ energy as OBSERVATION — read at one-hot grids, where it is perfect ✗ energy as GRADIENT — the interior has a Jensen-variance trap (died 8×)

§3

PRIOR WORKS

흡수한 통찰 — 다섯 개의 막다른 길과 씨앗

Recursive weight-shared refinement building an answer over steps.

Recursive refine

Weight-shared K-step recursion is the right backbone — this is GRAM.

Neural cellular automata regrowing a pattern from damage.

NCA regenerate

Damage→regrow diversifies init state and kills exposure bias.

Min-conflicts

Conflict count is a faithful discrete signal — reuse it as the ruler.

Energy descent

−∇E is non-navigable: interior blur = low-energy trap. Dead.

CSP backtrack

Hard puzzles need guess+backtrack — a fixed-depth operator cannot.

§4

METHOD — WHY IT LIFTS

두 독립 레버: 기억(gap) vs 커버리지(OOD)

The CA-1 battery moved three things at once. Diagnosed by mechanism into two independent levers — one drives the memorization gap, the other drives OOD liftoff.

clue-p 0.25 — the gap lever

Re-opens the clue→gold lookup. Drives the train/held-out cell gap (memorization). Net-negative for generalization — drop it.

per-step re-damage — the OOD lever

Wide state-space coverage. Drives the sustained held-out + weak-OOD exact liftoff. This is the real engine — keep it.

§5

P/Q SPLIT — SETTLED

완주 · 챔피언 = Q

FINAL · champion = Q: held-out 0.070 / weak-OOD 0.100 > P 0.045 / 0.090 — re-damage carries OOD, clue-p was the memorization culprit

Isolating the two levers. P (champion) = pure init-diversification, clue-p 0 + no re-damage (the zero-gap candidate). Q = CA-1a minus clue-p (keeps re-damage, drops the memorization path). Q–vs–CA-1a isolates clue-p's gap; P–vs–Q isolates re-damage's OOD. Predicted winner if the mechanism holds: Q.

Train-vs-held-out cell gap over steps for P, Q and CA-1a arms. — **Cell gap (train − held-out).** The memorization signature clue-p leaves behind — P is designed to flatten it, Q to shrink it while keeping OOD coverage.

arm @ step ~750	loss	reads as
P — champion	0.64	easier objective
Q — minus clue-p	0.81	predicted winner
CA-1a — reference	baseline	3-lever mix

Both exact 0 is normal this early — the pilot only lifted near step ~1000. P loss 0.64 < Q loss 0.81 confirms P is the easier objective. Climbing.

§6

LIMITS

정직한 벽 — 무엇이 아직 안 되나

ABelow the propagation ceiling

Best held-out exact 0.090 is under logic-only 0.150 — the operator is a soft statistical corrector, not a sound propagator. Tested fix (candidate-SET as observation) FAILED — it lowered the ceiling (crutch). What worked: consume the same information as SEARCH (§0), which passes the ceiling at 0.31.

Btarget-OOD exact = 0

Ceiling is 0.326 but a fixed-depth monotone operator cannot guess + backtrack. Fix: CA-2 verified-rollout distillation — operator proposes, sound solver filters, distill the successes.

CMemorization gap persists

CA-1a still carries a large train/held-out cell gap (0.343); clue-p and pool both widen it. The P/Q split is the live attempt to close it.

DSingle benchmark

Everything here is Sudoku with a hand-written ruler. Transfer to other verifiable structures (e.g. crystal band-gap) is untested at this stage.