Interpolation Is Not Invariance: Pair Count Is Not Coverage in Transformation Audits
TL;DR
Four complementary quantities are distinguished---edge count $m$ effective contrast rank $s$, effective contrast rank $s$, population support rank $r$, and graph spectral gap $\eta$---and characterize their roles in audit coverage and deployment reliability.
Abstract
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 $\eta$---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 $U \sim \operatorname{Beta}((r-s)/2,s/2)$; when $s \ge r$, 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 $1/\eta$. 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..
