Template-Based Endpoint Verification via Logprob Order-Statistic Geometry

Abstract

We study what model-identifying information leaks through commercial language-model APIs that expose top-k token log probabilities. Building on extreme-value theory predictions for logit order-statistic gaps, we confirm that the normalized third logit gap (δ norm) remains near the Gumbel-class constant ≈0.318 across 6 models from 3 providers (OpenAI, Google Vertex AI, xAI) and 3 independent measurement sessions, demonstrating that output-layer universality persists through API truncation and quantization. We introduce a PPP-residualization transform that removes the dominant tail scale factor and reveals a low-dimensional but stable endpoint-specific geometry in the remaining gap spectrum. Contrary to common assumption, "provider" is not a geometrically coherent label: models do not cluster by corporate origin under these observables, but they do separate by model identity across independent sessions. Using a challenge-response protocol with centroid averaging and per-model thresholds, we demonstrate cross-session endpoint verification with a 0.83% breach rate (119/120 correct identifications across three temporal sessions); per-model thresholds eliminate all breaches on this dataset. We observe a robustness phase transition governed by enrollment depth. Under single-session enrollment, prompt selection is load-bearing: the majority of bootstrapped banks fail to separate the six endpoints. Under two-session enrollment, bank sensitivity collapses on this dataset, and a bank compiler produces small compiled banks that exceed the margin of larger uncompiled banks. A dimensionless robustness parameter SNR(K,S) unifies both axes: prompt count K and enrollment depth S jointly govern the transition from bank-sensitive to bank-robust verification. We discuss operational implications for re-enrollment cadence and template management in production deployments. Addendum (02/26/2026): Post-publication results extend this framework in two directions. A distillation experiment across six training protocols demonstrates that a model's structural fingerprint (weight-geometry regime) is completely invariant to knowledge distillation, while its functional fingerprint (PPP-residual template) converges 31--52% toward the teacher's — enabling forensic detection of distillation provenance through API measurements alone. A conditional impossibility theorem, machine-checked in Coq (41 theorems, 0 Admitted), proves that no standalone model can spoof another's PPP-residual template across independent challenge prompts without exhausting its KL divergence budget, under four explicit trust assumptions. Technical Note: Gap Invariance: Why PPP Measurements Are Domain-Independent by Construction (DOI: 10.5281/zenodo.19275524) The Neural Network Identity Series — Mathematical foundations, empirical validation, and governance frameworks for verifying which model is running Newest addition: Technical Note: The Disappearing Window — AI Logprob Access Withdrawal and the Structural Verifiability of Frontier Model Contracts (DOI: 10.5281/zenodo.20362098) 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) Paper 10:Where Identity Comes From: Path Sensitivity and Endpoint Underdetermination in Neural Network Training (DOI: 10.5281/zenodo.19118807) Paper 11: Post-Hoc Disclosure Is Not Runtime Proof: Model Identity at Frontier Scale (DOI: 10.5281/zenodo.19216634) Paper 12: Family-Dependent Response to Reasoning Distillation Across Structural and Functional Identity Layers (DOI: 10.5281/zenodo.19298857) Paper 13: Safety-Alignment Removal as a Model-Identity Failure — Structural Evidence from Published Weight-Level Mutation Checkpoints (DOI: 10.5281/zenodo.19383019) Technical Note: Agent Identity Is Not Model Identity (DOI: 10.5281/zenodo.19240883) Technical Note: Measured Model Substitution Under Valid Agent Credentials (DOI: 10.5281/zenodo.19342848) Technical Note: Artifact Identity Is Not Runtime Identity — Trustfall Lite and the Boundary of File-Level Model Verification (DOI: 10.5281/zenodo.20019127) 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).