# Interpolation Is Not Invariance: Pair Count Is Not Coverage in Transformation Audits

arXiv: https://arxiv.org/abs/2609.14870 · HTML: https://mlsanity.com/paper/2609.14870/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 19 references that are in the map.
- First posted on arXiv 14 Sep 2026.

## Builds on (most cited first, 7 of 7)

- [Null It Out: Guarding Protected Attributes by Iterative Nullspace Projection](https://mlsanity.com/paper/2004.07667/) (2020)
- [Low-Resource Languages Jailbreak GPT-4](https://mlsanity.com/paper/2310.02446/) (2023)
- [LEACE: Perfect linear concept erasure in closed form](https://mlsanity.com/paper/2306.03819/) (2023)
- [Multilingual Jailbreak Challenges in Large Language Models](https://mlsanity.com/paper/2310.06474/) (2023)
- [Preserving Task-Relevant Information Under Linear Concept Removal](https://mlsanity.com/paper/2506.10703/) (2025)
- [Data Augmentation: A Fourier Analysis Perspective](https://mlsanity.com/paper/2606.24418/) (2026)
- [Efficient Learning and Symmetry Discovery under Exact Invariances](https://mlsanity.com/paper/2609.07031/) (2026)

## Abstract (arXiv)

Counting equivalent pairs is a common way to report transformation-audit coverage, but it can substantially overstate the constraints imposed by an audit: pairs generated from the same semantic object are correlated, and complete orbit graphs contain algebraically redundant edges. We therefore distinguish four complementary quantities---edge count m, effective contrast rank s, population support rank r, and graph spectral gap η---and characterize their roles in audit coverage and deployment reliability. Under a rank-r Gaussian contrast model, a population-invariant calibrated reader exists exactly when the anchor has a component in ker T. When an audit has rank s < r, its unobserved risk is R^star/U, with U ~ operatornameBeta((r-s)/2,s/2); when s ≥ r, exact calibrated interpolation is infeasible. The same distinction appears in orbit topology: a spanning tree imposes the same exact-null constraints as a complete graph, while a sharp graph Poincare inequality propagates edge-level drift to an entire orbit at a cost proportional to 1/η. Cyclic audits can additionally yield zero pair-level leave-one-out error without holding out any semantic object. To address these failures, we derive exact block-Woodbury leave-one-orbit-out updates and introduce a source-disjoint deployment gate over finitely many candidate readers. The gate retains the original reader unless uncertainty bounds certify lower drift within a prescribed clean-utility budget. etc..