Equilibrium Residuals Expose Three Regimes of Matrix-Game Strategic Reasoning in Language Models

Large language models can score well on named game-theory benchmarks while failing on the same strategic computation once semantic cues are removed. We show this gap with procedurally generated zero-sum matrix games: a model that recognizes familiar games drops to 34%, 18%, and 2% success on anonymous $2{\times}2$, $3{\times}3$, and $5{\times}5$ payoff matrices. The benchmark separates semantic recall, learned approximate Nash computation, and an output-interface bottleneck that limits scale. Training only on $2{\times}2$ and $3{\times}3$ games, supervised fine-tuning raises unseen $5{\times}5$--$7{\times}7$ success from 2% to 61%, while exploitability-reward training averages 37% with high seed variance. We prove that the exploitability residual is $2$-Lipschitz in payoff perturbations, unlike discontinuous vertex-returning LP equilibrium selectors, explaining why residual training can transfer under payoff shifts even when formatting instability limits mean performance. A dominated-action padding experiment provides causal evidence: trained models solve $3{\times}3$ games embedded in much larger matrices, while random-padded controls fail and dense $12{\times}12$ games remain near failure. Procedural evaluation is therefore necessary for measuring strategic reasoning, and residual rewards expose a real but format-limited route to approximate equilibrium computation.

Identified-Set Geometry of Distributional Model Extraction under Top-$K$ Censored API Access

Modern LLM APIs often reveal only top-$K$ logit scores and censor the remaining vocabulary. We study the per-position distribution-recovery limits of this access model. For censoring threshold $τ$, the compatible teacher distributions form an identified set whose total-variation diameter is exactly $U_K=(V-K)\exp(τ)/(Z_A+(V-K)\exp(τ))$, where $Z_A$ is the observed partition function. For KL recovery, we give a computable binary-endpoint lower bound and an asymptotically matching small-ambiguity upper bound, with an extension to reference-aware attackers. Experiments on a Qwen3 math-reasoning teacher reveal a layered extraction hierarchy: on-task top-$K$ distillation recovers 12% of private capability, full-logit distillation recovers 56% despite 99% KL closure, and generation-based extraction recovers 96%. Top-$K$ censoring therefore limits per-position distribution recovery but does not by itself prevent capability extraction, separating fidelity from transfer in prompt-only logit distillation.

CodaRAG: Connecting the Dots with Associativity Inspired by Complementary Learning

Large Language Models (LLMs) struggle with knowledge-intensive tasks due to hallucinations and fragmented reasoning over dispersed information. While Retrieval-Augmented Generation (RAG) grounds generation in external sources, existing methods often treat evidence as isolated units, failing to reconstruct the logical chains that connect these dots. Inspired by Complementary Learning Systems (CLS), we propose CodaRAG, a framework that evolves retrieval from passive lookup into active associative discovery. CodaRAG operates via a three-stage pipeline: (1) Knowledge Consolidation to unify fragmented extractions into a stable memory substrate; (2) Associative Navigation to traverse the graph via multi-dimensional pathways-semantic, contextualized, and functional-explicitly recovering dispersed evidence chains; and (3) Interference Elimination to prune hyper-associative noise, ensuring a coherent, high-precision reasoning context. On GraphRAG-Bench, CodaRAG achieves absolute gains of 7-10% in retrieval recall and 3-11% in generation accuracy. These results demonstrate CodaRAG's superior ability to systematically robustify associative evidence retrieval for factual, reasoning, and creative tasks.