What Counts as Proof? — Admissible Evidence for Neural Network Identity Claims

We know how to document an AI system. We know how to test it, log what it did, and report when something goes wrong. What current governance practice does not clearly tell us is how to verify which model is actually computing. This is not a hypothetical gap. When an organization says "this is the model we evaluated," that claim is typically supported by a model card, a registry entry, or a hash of a weight file — evidence about a *file*, not about the system that is running. A neural network is not a static document. A weight file stores the network; the model is what appears when that file is loaded and begins transforming inputs into outputs. The file and the running model are related, but they are not the same thing — and current governance practice rarely distinguishes between them. This paper proposes a framework for doing so. It identifies three kinds of evidence that can support model identity claims, each answering a different question. Structural evidence — drawn from the model's internal computations during live operation — can verify which specific model is running, and is the most resistant to tampering. Thermodynamic evidence — drawn from the model's output statistics — can verify that the system is a genuine neural network rather than a substitute, but cannot distinguish one model from another. Functional evidence — drawn from patterns in the model's outputs over an API — can detect whether a model was copied from another, but this signal fades quickly: routine model updates can erase it within days to weeks of continued training. The paper shows that inspecting the model's files alone is insufficient for verifying which specific model is running. The identity-bearing signal cannot be recovered from the tested static properties of those files; it is most reliably established by observing the model while it operates. The paper formally proves that these three kinds of evidence cannot substitute for one another. Verifying that a system is genuine does not tell you which specific model it is. Detecting that a model was copied does not tell you the identity of the copy. The practical consequence is a standard for identity claims: any claim should declare which kind of evidence supports it, because borrowing evidence from the wrong category produces unreliable conclusions. The framework maps directly to compliance questions raised by current AI governance obligations, including those under the EU AI Act. It provides the missing evidentiary specification for model identity claims: which kind of evidence is admissible for which identity question. Supplementary Material This paper is accompanied by EvidenceSufficiency.v, a Coq proof file that formally verifies the cross-layer inadmissibility results described in §4. The proof mechanically checks each logical step of the observation-limited verification impossibility theorem and its three directional corollaries. The file contains no unresolved obligations (Admitted) and compiles cleanly under the Rocq Prover 9.1.1 (the current release of the Coq proof assistant, compiled with OCaml 5.4.0). It is available for download as a supplementary file attached to this record. The Neural Network Identity Series — Mathematical foundations, empirical validation, and governance frameworks for verifying which model is running Paper 1: The δ-Gene: Inference-Time Physical Unclonable Functions from Architecture-Invariant Output Geometry (DOI: 10.5281/zenodo.18704275) Paper 2: Template-Based Endpoint Verification via Logprob Order-Statistic Geometry (DOI: 10.5281/zenodo.18776711) Paper 3: The Geometry of Model Theft: Distillation Forensics, Adversarial Erasure, and the Illusion of Spoofing (DOI: 10.5281/zenodo.18818608) Paper 4: Provenance Generalization and Verification Scaling for Neural Network Forensics (DOI: 10.5281/zenodo.18872071) Paper 5: Beneath the Character: The Structural Identity of Neural Networks — Mathematical Evidence for a Non-Narrative Layer of AI Identity (DOI: 10.5281/zenodo.18907292) Paper 6: Which Model Is Running?: Structural Identity as a Prerequisite for Trustworthy Zero-Knowledge Machine Learning (DOI: 10.5281/zenodo.19008116) Paper 7: The Deformation Laws of Neural Identity (DOI: 10.5281/zenodo.19055966) Paper 8: What Counts as Proof? — Admissible Evidence for Neural Network Identity Claims (DOI: 10.5281/zenodo.19058540) Paper 9: Composable Model Identity — Formal Hardening of Structural Attestations in the Enterprise Identity Stack (DOI: 10.5281/zenodo.19099911) Formal Verification Stack for Neural Network Structural Identity (IT-PUF Coq Proofs) (DOI: 10.5281/zenodo.18930621) Copyright (c) 2026 Anthony Ray Coslett / Fall Risk AI, LLC. All Rights Reserved. Confidential and Proprietary. Patent Pending (Applications 63/982,893, 63/990,487, 63/996,680, 64/003,244).