Zero-Shot Weight-Space Programming Across a Difficulty-Graded Ladder of Two-Player Games

Large language models (LLMs) have recently been reported to read and write the raw weights of neural networks directly, a capability termed weight-space programming. We ask whether the reading side of this capability reflects reasoning about weight flows or a combination of local arithmetic at the network's edges and memorized priors over public datasets. We test Claude Opus 5.5 on small convolutional networks of about 125,000 parameters trained on five image datasets, providing every weight as plain text with no tools, no forward passes, and no dataset or class names. The model performs accurate local calculations: batch-normalization statistics lead it to infer about three optimizer steps on networks stopped after three updates, and output-layer sign patterns recover the vehicle/animal split of CIFAR-10. The sessions report reading the first and last layers closely and sampling the middle, and several claims built on partial tracing fail verification, although we did not run a direct forward-pass test. Semantic judgments lean on familiar datasets: the model names CIFAR-10 for CIFAR-10 networks but also for all three fast-food networks, relabels KMNIST as MNIST, reads a 99.57%-accurate SAT-6 network as CIFAR-10 animal photos, and calls "frog" the output favored by a feature that fires most on trees. Permuting CIFAR-10's labels preserves the output split at the new indices, but every class pair the model interprets is then wrong, indicating reliance on remembered label order. These are exploratory results from mostly single runs. We conclude that zero-shot weight reading by this model is better described as local arithmetic plus dataset priors than as weight-flow tracing, and we provide a protocol for auditing future weight-space claims.

Why can neural networks struggle to learn optimal play in a game with a simple, efficient solution algorithm? Motivated by empirical difficulties with parity in impartial games, we use NIM to study a representational bottleneck and three ways around it. We show that a restricted model of shallow, fixed-scale quantised networks lies in $\mathrm{AC}^0$, a circuit class unable to compute parity on arbitrarily large inputs. This prevents optimal single-position play across all board sizes within the model. The obstruction also guides architectural design: two-frame inputs allow restoration from a known zero-nim-sum position; complementary rollout policies reveal the missing invariant; and persistent recurrence accumulates it row by row. Each route retains shallow network components while supplying additional information or processing time. Supervised experiments illustrate the benefits of two-frame restoration and multiple rollout heads; recurrence is a theoretical construction. These results establish representational possibilities, not guarantees of successful training: $\mathrm{AC}^0$ does not characterise practical learnability. Core mathematical results and constructions have been checked in Lean, with classical complexity results as explicit external inputs. The interactive edition and complete Lean sources are available online at https://sr123.github.io/impartial-games-learning-lean/.

Recent work has fine-tuned language models on chess data and reported high benchmark scores as evidence that the resulting models can understand the rules of chess, play full chess games at a professional level, or generate human-readable explanations grounded in expert knowledge. We train KinGPT, a 25M-parameter character-level language model trained only on (position, best-move) pairs, who exceeds 3B-parameter ChessGPT on a 600-puzzle mate-in-N suite and 4B-parameter C1-4B over a 20-theme puzzle benchmark. We examine several claims made in existing literature regarding chess-trained language models and assert that their impressive benchmark performance is largely explained by pattern-matching. We also demonstrate how LLM-Modulo, a verifier-in-the-loop framework, raises RedPajama 3B's best move accuracy from 1.2% to 21.2% and move generation validity from 19.3% to 95.3% on mate-in-N chess puzzles, comparable to gains achieved from ChessGPT's fine-tuning on chess-specific web corpora at a fraction of the cost. Our results illustrate how pairing a general LLM with an external verifier offers a more flexible alternative to directly training on synthetic data for well-defined domains. We open source all training/evaluation code, datasets, puzzle samples, and KinGPT model checkpoints for reproducibility.