The Deformation Laws of Neural Identity

Neural network identity is not monolithic. Different observables — hidden-state geometry, pre-softmax logit statistics, and behavioral output templates — sit at different depths in the forward computation and respond to perturbation on different timescales. This paper shows that three identity layers — structural, thermodynamic, and functional — each obey a distinct validated deformation law. The structural layer is model-specific, stable under non-destructive training interventions, and inert under same-family direct targeting in the observed regime. The thermodynamic layer is approximately universal across a validated 22-model Transformer cross-section. The functional layer is volatile, transferring through distillation and eroding under continued fine-tuning. We resolve the carrier of the structural layer as a two-channel geometric observable requiring both token-level magnitude and token-level direction, and we falsify two natural simplifications: that the structural fingerprint reduces to a gauge projection, and that it is predictable from coarse architecture features. Together these results define an admissibility condition for neural identity claims: such claims must specify which layer they address, because the layers do not share a deformation law. 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) 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).