⚠ ARCHIVED SNAPSHOT (2026-06-26) — this describes the now-SUPERSEDED "descend the energy" thesis, later found to be a dead gradient path. See the latest: /2026-07-06/

Solve a Sudoku by following the energy downhill

Sudoku-Extreme, ~56 blanks per puzzle, valid-or-not is unambiguous.

1.0 ✓
Verifier AUC (held-out)
556 < 891
gold E < random (healthy)
≈1.0
exact on training
open ✗
brand-new puzzles
✓ works energy is a near-perfect judge; pathwise solves the trained puzzles (generated E ~571 ~ gold 556) ✗ open does the generator reach brand-new puzzles? · lever = data augmentation
Intent

Fill a partly-blank grid so every row, column, and box holds 1–9 exactly once.

Sudoku-Extreme, ~56 blanks per puzzle; valid-or-not is unambiguous.

Hypothesis

If the energy is a near-perfect verifier, a generator can be trained to descend it to valid completions.

And ideally generalize the RULE to unseen puzzles.

Method

Energy is the judge, the GRAM generator is the player, pathwise sends −∇E into the player; the judge never changes.

Energy = a conv net reading each cell + clue + digit into one number, DSM-trained then frozen. GRAM is recursive: keeps an answer + a scratchpad, nudges toward lower energy over cycles, decodes a digit per cell.

Energy ❄ frozen −∇E downhill Generator GRAM produces candidate solution judged again — energy never changes
Sudoku module pipeline: a frozen energy feeds the GRAM generator.
The concrete pipeline. A frozen DSM-trained energy reads each cell + clue + digit; the recursive GRAM generator descends it via pathwise −∇E.
Result

The generator solves its TRAINING puzzles per-cell near-exact; generated grids score E ~571 ~ gold 556.

Verifier is correct on HELD-OUT too (AUC 1.0). Open: does the generator reach brand-new puzzles?

WhatResultNote
Energy sanity556 < 891gold below random — healthy judge
Score-matching loss0.694 → 0.017converges
Verifier AUC (held-out)1.0perfect valid / invalid separation
Generator on training≈1.0solves; generated E ~571 ~ gold 556
Generator on brand-new puzzlesopenthe data-efficiency question
Training accuracy rises while held-out accuracy stays low.
The train-vs-held-out gap. Training accuracy rises while held-out stays low — exactly the data-efficiency question the open result names.
Why

A perfect verifier does NOT imply the generator can REACH good solutions; the train-vs-held-out gap is the data-efficiency question.

Annealing sub-note. A small barrier sits before the answer; straight descent stalls in the trap, while annealing smooths-then-sharpens to cross it.
Annealing

Greedy stalls in a nearby trap; high noise flattens the barrier so the annealed run crosses to gold, then the surface sharpens back.

energy trap barrier gold greedy annealed sharp surface — greedy is trapped high noise smooths it flat — annealed crosses the barrier
A 1D slice between a trap and the answer with a small barrier.
The barrier is small. A 1D slice between a trap and the answer shows a low ridge that straight descent cannot cross.
Greedy descent stalls in a trap while the annealed run reaches the global minimum.
Greedy vs annealed. On a two-valley landscape greedy stalls; annealing smooths-then-sharpens to reach the global minimum.