cs.AINov 6, 2026
SaveZero-Shot Weight-Space Programming Across a Difficulty-Graded Ladder of Two-Player Games
Organizations: Universidad Pontificia Comillas (IIT, DIGNUM) Madrid, Spain
Abstract
Weight-space programming sets every weight of a feed-forward network by hand, from an explicit specification of the desired computation, with no gradient descent and no training data anywhere in the construction. We ask how far a large language model can carry this practice, zero-shot, across games of deliberately increasing difficulty. We design a four-rung ladder (Nim, tic-tac-toe, two sizes of Connect Four, and a chess endgame) spanning four axes of difficulty, and build a second, independent network at each rung, using a structurally unrelated game at matched difficulty, to separate what a rung's finding says about the difficulty level from what it says about the one game used to measure it. Ten networks are built and independently verified. Six admit a complete, non-statistical proof of correctness against an independent oracle. The chess-endgame rung is exact where correctness is cheap, move legality and, in one network, an exact one-ply mate detector, and collapses where it is expensive, composing a multi-move mating plan, from 84.1-100% success at the one-ply tactics it was built to get right down to 0.57-9.13% in full adversarial play; its replicate shows this collapse is an order of magnitude less severe under a shorter required plan and exact rather than heuristic safety checks. The standard-sized rung confirms, by holding hand-set evaluation weights fixed and adding only shallow search, that the remaining gap is an architectural ceiling rather than a matter of engineering effort, and this exact collapse reproduces across two structurally unrelated games. We release all ten networks, their independent oracles, construction-cost measurements, and a cold-prompt protocol for a future cross-model comparison
Figures & tables
Figure 1: The construction-and-verification pipeline applied to every network in this study. A game’s rules and, where one exists, its closed-form strategy are compiled by hand into weights using four reusable ReLU gate primitives, producing a single forward pass with no gradient descent and no search anywhere in the construction; the resulting network is then checked exhaustively against an independent oracle wherever the reachable state space is small enough to enumerate, and statistically otherwise.
Figure 2: The ladder (top row, solid arrows) and its replicates (bottom row, dashed connectors), one independent network of a structurally unrelated game at matched difficulty for every rung. A solid arrow changes exactly one of the four difficulty axes relative to the rung before it; a dashed connector changes the game mechanic while holding every axis fixed.
| Rung | Game | Verification | Headline outcome |
|---|---|---|---|
| 0 | Nim, 3 piles | exhaustive, 512 states | never loses from any winning position |
| 0, replicate | Wythoff’s game, 2 piles | exhaustive, 64 states | never loses from any winning position |
| 1 | Tic-tac-toe | exhaustive, 4520 states | never loses moving first or second |
| 1, replicate | Dots and Boxes, | exhaustive, 4095 states | matches exact game value from every state |
| 2a | Connect Four, , connect-3 | exhaustive, 104 399 states | matches exact game value from every state |
| 2a, replicate | Othello, | exhaustive, 53 651 states | matches exact game value from every state |
Table 1: Verification standard and headline outcome for all ten networks. “Replicate” rows share every difficulty axis with the primary game above them but use a structurally unrelated mechanic.
| Network | Parameters | Input/output dims | Override table | State space | Wall-clock |
|---|---|---|---|---|---|
| Nim | 4323 | 9 / 24 | 0 (arithmetic) | 512 | 1 s |
| Wythoff | 3786 | 6 / 22 | 57 (full table) | 64 | 1 s |
| Tic-tac-toe | 15 781 | 18 / 9 | 80 | 4520 | 2 s |
| Dots and Boxes | 16 924 | 12 / 12 | 556 | 4095 | 5 s |
| Connect Four, small | 62 363 | 40 / 4 | 751 | 104 399 | 100 s |
| Othello, small | 139 361 | 32 / 16 | 2537 | 53 651 | 60 s |
Table 2: Construction-cost measurements for every network: total hand-set parameter count, input and output tensor dimensions, size of the exact-match override table where one is used, the independent oracle’s state-space size, and wall-clock time for this paper’s full verification protocol on the development machine (16 cores, no GPU; statistical-block timings include the full seeded evaluation script, not only the unit-test smoke check). Parameter count spans four orders of magnitude across the ladder and is driven almost entirely by whether a network’s tactical tiers are exact circuits (KQK’s dedicated opponent-mobility computation) or incidental properties of unconditional heuristics (every other chess network), not by board size alone.
Figure 3: Win rate against a uniformly random, non-searching opponent for the two rung-2b statistical networks (averaged over measured conditions; Othello’s bar is the standard board, since the replicate was verified exactly rather than sampled). Both networks score exactly 0% against an opponent sharing its own hand-set evaluation weights (an unrelated hand-rolled evaluation, since their search opponents are independent programs) but given two to four plies of search instead of zero; that value is not plotted as a second series because a true-zero bar carries no visual information beyond the number itself.
| Task | Claude Sonnet 5 | Codex (“6 Luna High”) | Grok (Grok 4.7 High) |
|---|---|---|---|
| Nim | exhaustive, 512 states, exact | exhaustive, 512 states, exact | exhaustive, 512 states, exact |
| Wythoff | exhaustive, 64 states (piles 0–7) | exhaustive, 441 states (piles 0–20, as specified) | exhaustive, 441 states (piles 0–20, as specified) |
| Dots and Boxes | exhaustive, 4095 states, exact | representation ambiguity identified (875/4096 masks); non-optimal baseline, 55% vs. random, 5% vs. minimax (20 games each) | exhaustive, 4095 states, exact; 0 score/edge conflicts found in 5101 reachable mask-score pairs |
| Connect Four, | exhaustive, 104 399 states, exact | exhaustive, 104 413 states (104 399 non-terminal), exact | exhaustive, 104 399 non-terminal states, exact |
| Othello, | exhaustive, 53 651 states, exact, initial value | exhaustive, 61 619 states, exact, initial value (disc differential) | exhaustive, 58 533 states, exact, initial value |
| Connect Four, | 99.5–99.8% vs. random, 0/800 vs. depth-4 (10 seeds 400 games) | 100% vs. random, 0% vs. depth-3 (10 seeds 2 games) | 99.75% vs. random (400 games), 0% vs. depth-4 (80 games) |
Table 3: This paper’s Track A result beside both completed independent replications, task by task. Sample sizes differ across all three attempts (noted inline), which affects the width of each replication’s confidence intervals relative to our own; each replication’s own self-reported scope limitations are carried over verbatim rather than smoothed into a single comparable number. Grok’s KRK/KQK oracle enumerates 223 944 piece placements before filtering out those in which the black king is already in check, a configuration illegal with white to move; filtering that condition out ourselves, directly from Grok’s own code, leaves exactly 175 168 and 144 508 placements respectively, matching our own two endgames’ independently computed legal-position counts exactly. Othello’s three attempts use three different state-counting conventions (53 651, 58 533, and 61 619 reachable positions respectively) and, for Codex, a different game-value convention (exact disc differential rather than win-loss sign); all three agree on the sign of the initial position’s value. The Othello, row’s Grok result against a two-ply search opponent, 50%, is the one result across either replication that does not reproduce this paper’s otherwise uniform collapse against search, discussed in Section 5 .
Figure 4: The share of each exact task’s reachable states for which the hand-derived tactical tier alone, with no exact-match override table consulted, already matches the independent oracle’s optimal value, for the two attempts that both reported this breakdown. Computed from Table 2 ’s override counts for Track A (one minus the override count divided by the state-space size) and from the Opus-tier attempt’s own reported figures.
| Task | Opus | Sonnet (fresh, independent of Track A) | Haiku |
|---|---|---|---|
| Nim | exhaustive, 512 states, exact; 3843 params, pure circuit, no table | exhaustive, 512 states, exact; 4725 params | exhaustive, 512 states, exact; 17 432 params |
| Wythoff | exhaustive, 441 states, exact; 32 340 params; no finite circuit for found, lookup table used | exhaustive, 441 states, exact; 43 113 params; same no-finite-circuit finding | exhaustive, 441 states, exact; 32 255 params |
| Dots and Boxes | exhaustive, 4095 states, exact; value ; 10 655 params; 91.04% from the hand tier alone, 367 overrides | exhaustive, 4095 states, exact; value ; 102 712 params | declined; interface misread as requiring turn-state tracking |
| Connect Four, | exhaustive, 104 399 states, exact; 57 995 params; 99.28% hand-tier alone, 751 overrides (matches Track A exactly) | exhaustive, 104 399 states, exact | declined; state space misestimated at (actual 104 399) |
| Othello, | exhaustive, 58 533 states, exact, value ; 128 590 params; 94.55% hand-tier alone, 2474 overrides | exhaustive, 45 387 active states, exact, value | declined; legality judged to need “200+ neurons” exceeding ReLU’s expressiveness |
| Connect Four, | 99.80% vs. random, 32.95% vs. depth-2, 26.55% vs. depth-4 (10 seeds 200 games) | 99.6/99.1% vs. random (first/second), 21.0/12.0% vs. depth-6 (10 seeds 200 games) | declined; state space judged intractable |
Table 4: The three Claude-family tier-comparison attempts, task by task, all independently re-verified by us. Opponent strength and sample sizes vary across attempts exactly as they do in Table 3 , so statistical rows are not directly comparable without accounting for the stated opponent; Opus’s KRK/KQK figures marked “vs. every defense” are an exhaustive worst-case guarantee, not a simulated-game win rate, and are not on the same scale as the other attempts’ simulated figures in the same rows.
Figure 5: Of the nine replicated tasks (tic-tac-toe excluded as the given worked example), the share each attempt met in full, built but fell short on, or declined to attempt at all. Five of the seven attempts, spanning both external providers and all three Claude tiers tested above Haiku, meet the bar on every task; Codex falls short on three (Dots and Boxes and the two chess endgames); Haiku attempts only two.