The Coercive Projection Theorem for Canonical Reciprocal Costs

TL;DR

This paper introduces a finite-data framework for certifying zero-defect configurations in positive vectors using a canonical reciprocal cost, establishing a decision procedure that is locally maximal among sound methods.

Contribution

It develops a novel finite-data certification method based on the canonical reciprocal cost, with a unique decision procedure that is locally maximal on the identifiability locus.

Findings

Characterizes the canonical reciprocal cost among continuous costs.

Constructs a decision procedure that is locally maximal on the identifiability locus.

Shows reciprocity symmetry and normalization follow from the composition law.

Abstract

We develop a finite-data framework for certifying \emph{zero-defect} (neutral) configurations of positive vectors under the canonical separable reciprocal cost. We show that this scalar cost is characterized among non-constant continuous costs by the Recognition Composition Law together with a local quadratic calibration at balance; in particular, reciprocity symmetry and the normalization at the neutral point follow from the composition law. Under a conservation constraint and short-window observations of a rational (finite--state) signal class, we construct a canonical decision procedure that is \emph{locally maximal on the identifiability locus} among all sound procedures: any sound rule that resolves a datum must agree with the canonical output, and cannot resolve strictly more cases. The method is organized as $\Phi^\ast=A\circ B\circ P$: the projection/coercivity core is forced by the canonical-cost axioms, while the aggregation/reconstruction step is specified on a non-degenerate identifiability locus (e.g.\ a Hankel invertibility condition).