Symmetry-Protected Geometric Separation for Verified Algebraic Reasoning

TMLR Paper8126 Authors

Abstract: Many reasoning problems are defined by algebraic invariance: multiple surface forms represent the same computation, and correctness must remain stable under semantics-preserving rewrites and increasing compositional depth---a regime where standard sequence models can exhibit severe drift. We study a mechanism that targets this setting through explicit operator composition. Holonomic Networks associate tokens with learned geometric actions in $SO(d)$ and implement computation by their ordered composition (holonomy). We formalize the Holonomic Network variable-binding swap-program instantiation in this operator language and show that the same compositional engine supports multiple decoders. Replacing variable readout by an identity-energy decoder yields a unified verifier for algebraic word identity (whether a sequence of generators evaluates to the identity element), scoring a word by the normalized Frobenius distance of its composed operator to the identity. We instantiate the template for finite Coxeter words in $S_{32}$ and infinite braid words in $B_8$, organizing specialization through a compact set of enforcement modes: hypothesis-class restriction to $SO(d)$, architectural tying where available (inverse-by-transpose), symmetry-alignment losses with anti-collapse guardrails, and oracle-audited evaluation. Trained only on short words ($L\le 50$), the resulting verifiers achieve perfect sampled classification under a $100\times$ length extrapolation to $L=5000$ in both settings. In the Coxeter case, identity and non-identity energies remain separated by a persistent positive gap, yielding $\mathrm{TPR}=1$ and $\mathrm{FPR}=0$ on the reported evaluations. For braid words, an independent SageMath audit confirms agreement with the oracle on a 14,000-example evaluation dump with zero detected label mismatches.

Submission Type: Long submission (more than 12 pages of main content)

Previous TMLR Submission Url: https://openreview.net/forum?id=PcZLsikHfN

Changes Since Last Submission: Thank you for your message, Editors in Chief. I have uploaded a revised version of the manuscript where the references are now placed before the appendix, as requested. I also acknowledge and agree with the re-classification of our paper as a long submission.

Assigned Action Editor: ~Junyuan_Hong1

Submission Number: 8126