Published March 28, 2024 | Version v2
Technical note Open

Gap Invariance: Why PPP Measurements Are Domain-Independent by Construction

Authors/Creators

  • 1. Fall Risk Research

Description

The order-statistic gaps that underlie PPP-residualized functional identity measurement are exactly invariant to log-softmax transformation, exactly equivariant under positive scaling (including temperature), and exactly invariant to any position-independent constant shift applied to the logit vector. These are not empirical approximations — they are mathematical identities that hold for any logit vector over any vocabulary size. The result has been formally verified in Coq (GapInvariance.v: 5 theorems, 2 axioms, 0 Admitted). It retroactively strengthens the empirical API-wall finding reported in earlier work: the order-statistic gap geometry measured through API logprobs does not merely "survive" the log-softmax transformation — it is mathematically immune to it. Any deviation attributable to the API boundary must come from truncation, quantization, or coverage limitations, not from the probability-domain transformation itself.
 
Why this matters. Earlier work showed empirically that PPP-based measurements remained stable when models were accessed through APIs that expose log-probabilities instead of raw logits. This note upgrades that result from empirical robustness to mathematical invariance. It removes the probability-domain transformation itself from the list of plausible failure modes. If an API-based PPP measurement deviates from a weights-based measurement, the cause must lie in truncation, quantization, coverage limitations, or the model — not in log-softmax. The API wall is narrower than previously understood, and the space of plausible objections to API-domain model identity measurement has shrunk by one major category.
 
Supplementary Material. This note is accompanied by GapInvariance.v, a Coq proof file that formally verifies the five gap-invariance theorems described in §2: constant-shift invariance, positive-scale equivariance, affine scaling, log-softmax invariance, and general position-independent shift invariance. The file proves 5 theorems from 2 named axioms (OS1 and OS2), with 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

  1. Paper 1: The δ-Gene: Inference-Time Physical Unclonable Functions from Architecture-Invariant Output Geometry (DOI: 10.5281/zenodo.18704275)

  2. Paper 2: Template-Based Endpoint Verification via Logprob Order-Statistic Geometry (DOI: 10.5281/zenodo.18776711)

  3. Paper 3: The Geometry of Model Theft: Distillation Forensics, Adversarial Erasure, and the Illusion of Spoofing (DOI: 10.5281/zenodo.18818608)

  4. Paper 4: Provenance Generalization and Verification Scaling for Neural Network Forensics (DOI: 10.5281/zenodo.18872071)

  5. 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)

  6. Paper 6: Which Model Is Running?: Structural Identity as a Prerequisite for Trustworthy Zero-Knowledge Machine Learning (DOI: 10.5281/zenodo.19008116)

  7. Paper 7: The Deformation Laws of Neural Identity (DOI: 10.5281/zenodo.19055966) 

  8. Paper 8: What Counts as Proof? — Admissible Evidence for Neural Network Identity Claims (DOI: 10.5281/zenodo.19058540)

  9. Paper 9: Composable Model Identity — Formal Hardening of Structural Attestations in the Enterprise Identity Stack (DOI: 10.5281/zenodo.19099911) 

  10. Paper 10:Where Identity Comes From: Path Sensitivity and Endpoint Underdetermination in Neural Network Training (DOI: 10.5281/zenodo.19118807)

  11. Paper 11: Post-Hoc Disclosure Is Not Runtime Proof: Model Identity at Frontier Scale (DOI: 10.5281/zenodo.19216634)

  12. Paper 12: Family-Dependent Response to Reasoning Distillation Across Structural and Functional Identity Layers (DOI: 10.5281/zenodo.19298857)

  13. Paper 13: Safety-Alignment Removal as a Model-Identity Failure — Structural Evidence from Published Weight-Level Mutation Checkpoints (DOI: 10.5281/zenodo.19383019)

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).

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Additional details

Identifiers

URL
https://fallrisk.ai

Is supplement to
Preprint: 10.5281/zenodo.18704275 (DOI)
Preprint: 10.5281/zenodo.18818608 (DOI)
Preprint: 10.5281/zenodo.18872071 (DOI)
Preprint: 10.5281/zenodo.18907293 (DOI)
Preprint: 10.5281/zenodo.19008116 (DOI)
Preprint: 10.5281/zenodo.19055966 (DOI)
Preprint: 10.5281/zenodo.19058540 (DOI)
Preprint: 10.5281/zenodo.19099911 (DOI)
Preprint: 10.5281/zenodo.19118807 (DOI)
Preprint: 10.5281/zenodo.19216634 (DOI)
Technical note: 10.5281/zenodo.19240883 (DOI)
Preprint: 10.5281/zenodo.18776711 (DOI)
Preprint: 10.5281/zenodo.19298857 (DOI)
Technical note: 10.5281/zenodo.19342848 (DOI)
Preprint: 10.5281/zenodo.19383019 (DOI)
Is supplemented by
Software: 10.5281/zenodo.18930621 (DOI)