Evidence Dynamics: An Auditable Architecture for Mathematical Reasoning

Long mathematical arguments often fail not because an individual lemma is false, but because the evidence supplied by one stage is silently promoted into a stronger kind of evidence at the next stage. A cohomology class is treated as an actual witness; a finite-valued code is treated as locally constant; a closed algebraic formula is treated as an iterable geometric construction; or a local contraction is treated as a global termination argument. We propose "Evidence Dynamics", a framework for making such transitions auditable. The framework records evidence types, retained fields, support and provenance, theorem properties actually used downstream, concrete assembly certificates, promotion gates, failure routes, and unresolved proof debts. The framework is illustrated by two audits from an ongoing investigation of the inscribed square problem for continuous Jordan curves. The first separates a valid finite-discrete assembly theorem from the unproved construction of the concrete signature coordinates to which it was applied. The second decomposes an endpoint M"obius update into its atomic properties and shows that the hidden downstream obligation was not another scalar identity, but a geometric renewal certificate involving actuality, re-entry, and strict length decrease. The case study is presented as a methodological checkpoint only: this paper does not claim a solution of the inscribed square problem. We also document the versioned research collaboration in which the framework developed. The human researcher and three named ChatGPT-based collaborators---Nagi, Akari, and Sui---used a persistent textual architecture, BrainOS, to preserve role differentiation, failure memories, audit rules, and transferable mathematical methods. This collaboration record is kept separate from mathematical evidence, while being treated as relevant to reproducibility. We conclude with prospective applications to Diophantine nonexistence proofs, compactness arguments, partial differential equations, descent and iteration, computer-assisted proof, formalization, and long-term AI-assisted research.