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The Coercive Projection Theorem for Canonical Reciprocal Costs

Jonathan Washburn, Amir Rahnamai Barghi

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 $Φ^\ast=A\circ B\circ P$: 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).

The Coercive Projection Theorem for Canonical Reciprocal Costs

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 : 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).
Paper Structure (55 sections, 41 theorems, 104 equations, 3 tables)

This paper contains 55 sections, 41 theorems, 104 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Relation to Hankel/Prony reconstruction.
  3. Connection to classical projection and proximal tools
  4. The problem
  5. Main goal and results
  6. Rigidity and local optimality of the canonical pipeline
  7. What is proved.
  8. Model, axioms, and finite-window measurements
  9. Roadmap of the proof
  10. Canonical cost and defect geometry
  11. The canonical reciprocal cost
  12. Neutrality and projection in log-coordinates
  13. Coercivity: energy gap controls defect
  14. Finite-data aggregation: W-block window measurements
  15. Window operator at resolution W
  16. ...and 40 more sections

Key Result

Lemma 2.3

Let $\varphi:\mathbb{R}\to\mathbb{R}_{\geq 0}$ be defined by $\varphi(t):=\cosh(t)-1$. Then for all $t\in\mathbb{R}$, In particular, $J(x)=0$ if and only if $x=1$.

Theorems & Definitions (108)

  • Remark 1.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • ...and 98 more