Minimal Topologies of Forward-Local Failure in AI Systems: The Hamiltonian Microscope

Cross-provider empirical pilot extending the Sudoku Microscope methodology to a structurally distinct problem class — the 4×4 Hamiltonian-path traversal task — providing a second independent lens on forward-local commitment failure in large language models. Identifies the Layer 2 silent-commit signature: across all 195 catastrophic commitments observed in the cross-provider corpus (GPT-5.5 and Claude Opus 4.7), the model committed the failing move without having acquired positive admissibility evidence for the committed move tuple. Abstract We introduce the Hamiltonian Microscope: a custom-domain empirical instrument for probing forward-local commitment failure in large language models, instantiated as a 4×4 Hamiltonian-path traversal task. The microscope methodology extends the Global Admissibility Filtering (GAF) framework's Sudoku Microscope to a structurally distinct problem class, providing a second independent lens on the same architectural failure mode. We characterize the empirical behavior of two frontier models — GPT-5.5 and Claude Opus 4.7 — across five condition variants (no-tool, dummy-tools, named-dummy, optional admissibility verification, forced admissibility verification). The two models exhibit empirically distinct regimes: GPT-5.5 expresses the failure distributionally across stochastic trajectories, with 125 catastrophic commitments observed across 220 runs; Claude Opus 4.7 expresses it through stable deterministic trajectories in some conditions and perturbation-sensitive diversification in others, characterized through a 100-run primary corpus across default-mode and nonce-perturbation experimental phases. The central architectural finding unifies both providers: across all 195 catastrophic commitments observed in the cross-provider corpus, the model committed the failing move without having acquired positive admissibility evidence for the committed move tuple — the Layer 2 silent-commit signature, satisfied 195/195. Under forced admissibility verification (the forced-gate positive control), both models reliably solve the canonical instance: 50/50 for GPT-5.5 and 15/15 across all Claude batches. We additionally document two instance-bound Claude observations: Path D convergence under the tested canonical instance (20/20 solving runs in the nonce and expansion batches produce the same 16-cell Hamiltonian trajectory) and a verification-pattern observation in the voluntary-verification condition where all four solving runs in the diagnostic batch engage admissibility queries at the same three commit positions. The paper is positioned as a custom-domain pilot and mechanism-discovery study, not a comprehensive cross-provider benchmark. We document limitations explicitly: single-instance generalization is stronger on the GPT side than on the Claude side; sample sizes were chosen for regime classification rather than distributional inference; and the relationship to the prior Sudoku Microscope is restricted to architectural-signature replication, not behavioral transfer. The work presents the Hamiltonian Microscope as a working empirical instrument providing mechanism-discovery evidence consistent with the GAF prediction outside the Sudoku exact-constraint domain. A Lean 4 companion formalization verifies the abstract Hamiltonian constraint-system instantiation, Layer 2 signature definition, positive-evidence criterion, signature locality lemma, and evidence-status riders used by the paper; it does not certify empirical transcripts or provider behavior. Companion Lean 4 formalization: https://doi.org/10.5281/zenodo.20073715 GitHub repository: https://github.com/shawnjason/Hamiltonian-Microscope Related papers in the program: PIT (foundational projection-theoretic result): https://doi.org/10.5281/zenodo.19633241NEO (forward-case impossibility theorem): https://doi.org/10.5281/zenodo.19688367IA (admissibility-dynamics framework): https://doi.org/10.5281/zenodo.19688628HAL (language-model specialization): https://doi.org/10.5281/zenodo.19715059RLM (recursive language models via admissibility dynamics): https://doi.org/10.5281/zenodo.19753549OOL (OOLONG-Pairs empirical companion to RLM): https://doi.org/10.5281/zenodo.20277804SUD (Sudoku Microscope — direct predecessor extends to Hamiltonian microscope): https://doi.org/10.5281/zenodo.20277939