Canonical Separable Reciprocal Cost

• Canonical separable reciprocal cost is a unique penalty function that measures deviation from a zero-defect configuration in (ℝ₊)ⁿ via a normalized reciprocal law.
• It extends a one-dimensional function to multivariate settings using separable sums and weighted geometric means, ensuring permutation symmetry.
• The induced Hessian geometry and its links to divergence measures and the Fisher–Rao metric underpin optimal finite-data decision procedures in statistical contexts.

The canonical separable reciprocal cost is a uniquely specified, mathematically rigid penalty function for quantifying the deviation of positive vectors from a neutral (zero-defect) configuration in the positive orthant $(\mathbb{R}_{>0})^n$. Its fundamental form is characterized by a combination of normalization, a nonlinear composition law, and local quadratic calibration. This structure grounds optimal finite-data decision procedures and induces a singular Hessian geometry in both statistical and information-theoretic contexts.

1. Definition and Uniqueness

The canonical separable reciprocal cost arises from the one-dimensional function

$$J: (0, \infty) \longrightarrow [0, \infty), \quad J(x) = \frac{1}{2}(x + x^{-1}) - 1.$$

This function is then extended to $(\mathbb{R}_{>0})^n$ in a separable manner: $J_n(x_1, ..., x_n) = \sum_{i=1}^n J(x_i).$ Given scale maps $\iota_S$ and $\iota_O$, the cost between a signal $s$ and an observation $o$ is

$$c(s, o) = J\left( \frac{\iota_S(s)}{\iota_O(o)} \right).$$

The uniqueness of $J$ is proved by requiring:

• Normalization: $$J: (0, \infty) \longrightarrow [0, \infty), \quad J(x) = \frac{1}{2}(x + x^{-1}) - 1.$$0
• Recognition Composition Law (RCL): For all $$J: (0, \infty) \longrightarrow [0, \infty), \quad J(x) = \frac{1}{2}(x + x^{-1}) - 1.$$1,

$$J: (0, \infty) \longrightarrow [0, \infty), \quad J(x) = \frac{1}{2}(x + x^{-1}) - 1.$$2

• Local quadratic calibration: $$J: (0, \infty) \longrightarrow [0, \infty), \quad J(x) = \frac{1}{2}(x + x^{-1}) - 1.$$3 with $$J: (0, \infty) \longrightarrow [0, \infty), \quad J(x) = \frac{1}{2}(x + x^{-1}) - 1.$$4

The only continuous, non-constant solution to these conditions is $$J: (0, \infty) \longrightarrow [0, \infty), \quad J(x) = \frac{1}{2}(x + x^{-1}) - 1.$$5 above (Washburn et al., 5 Feb 2026, Washburn et al., 27 Feb 2026).

2. Axiomatic Foundation: Recognition Composition Law and Rigidity

The Recognition Composition Law (RCL) is the central axiom that rigorously determines the admissible cost function. RCL, together with normalization, enforces reciprocity: $$J: (0, \infty) \longrightarrow [0, \infty), \quad J(x) = \frac{1}{2}(x + x^{-1}) - 1.$$6.

Expressing $$J: (0, \infty) \longrightarrow [0, \infty), \quad J(x) = \frac{1}{2}(x + x^{-1}) - 1.$$7 in logarithmic coordinates and applying the RCL, one finds the associated function $$J: (0, \infty) \longrightarrow [0, \infty), \quad J(x) = \frac{1}{2}(x + x^{-1}) - 1.$$8 must solve the functional equation

$$J: (0, \infty) \longrightarrow [0, \infty), \quad J(x) = \frac{1}{2}(x + x^{-1}) - 1.$$9

Quadratic calibration selects the specific hyperbolic-cosine solution $(\mathbb{R}_{>0})^n$0, so $(\mathbb{R}_{>0})^n$1. This is the sole admissible function with the prescribed normalization and calibration—demonstrating that the cost is rigidly determined by the axioms, and any deviation (e.g., dropping calibration or continuity) results in a broader family or pathological solutions (Washburn et al., 5 Feb 2026).

3. Multidimensional Generalization and Hessian Geometry

The cost is extended to the multidimensional setting using a weighted geometric mean: $(\mathbb{R}_{>0})^n$2 Setting $(\mathbb{R}_{>0})^n$3 yields the permutation-symmetric form,

$(\mathbb{R}_{>0})^n$4

In logarithmic coordinates $(\mathbb{R}_{>0})^n$5, defining $(\mathbb{R}_{>0})^n$6, one finds $(\mathbb{R}_{>0})^n$7, so the potential depends only on the scalar $(\mathbb{R}_{>0})^n$8 (Washburn et al., 8 Apr 2026).

The induced Hessian metric in log-coordinates, $(\mathbb{R}_{>0})^n$9, has rank one everywhere, with the null distribution $J_n(x_1, ..., x_n) = \sum_{i=1}^n J(x_i).$0 forming an $J_n(x_1, ..., x_n) = \sum_{i=1}^n J(x_i).$1-dimensional integrable subspace. In the original coordinates, the corresponding Hessian metric is generically pseudo-Riemannian, degenerating along hypersurfaces defined by $J_n(x_1, ..., x_n) = \sum_{i=1}^n J(x_i).$2 and $J_n(x_1, ..., x_n) = \sum_{i=1}^n J(x_i).$3 (Washburn et al., 8 Apr 2026).

4. Decision Procedures and the Coercive Projection Theorem

A canonical maximal procedure for certifying neutral configurations from aggregated window-sum data is constructed and proven optimal within its class. For a configuration $J_n(x_1, ..., x_n) = \sum_{i=1}^n J(x_i).$4 and observed "window-sums" $J_n(x_1, ..., x_n) = \sum_{i=1}^n J(x_i).$5, the procedure $J_n(x_1, ..., x_n) = \sum_{i=1}^n J(x_i).$6 decomposes as

$J_n(x_1, ..., x_n) = \sum_{i=1}^n J(x_i).$7

where:

• $J_n(x_1, ..., x_n) = \sum_{i=1}^n J(x_i).$8: Aggregation/reconstruction—locally inverts the window map to obtain $J_n(x_1, ..., x_n) = \sum_{i=1}^n J(x_i).$9 if the data lie on the identifiability locus (full-rank Jacobian or invertible Hankel matrix).
• $\iota_S$0: Projects $\iota_S$1 to mean-zero under the conservation constraint: $\iota_S$2.
• $\iota_S$3: Applies the separable cost and decides:

$\iota_S$4

with $\iota_S$5 (Washburn et al., 27 Feb 2026).

On the identifiability locus, $\iota_S$6 resolves all and only those cases that can be decided from finite data, and agrees with any other sound certification rule wherever defined. This maximality is a direct consequence of the structure forced by the canonical cost.

5. Relations to Divergences and Information Geometry

The canonical separable reciprocal cost connects to several general divergence concepts:

• Symmetrized Itakura–Saito divergence: For $\iota_S$7 as above,

$\iota_S$8

where $\iota_S$9.

• Bregman divergence: In logarithmic coordinates, $\iota_O$0 is convex, and its Bregman divergence locally agrees with the Hessian metric.
• Fisher–Rao metric realization: The Hessian metric $\iota_O$1 arises as the Fisher information metric for Gaussian models whose mean depends on $\iota_O$2 via a suitable transformation (Washburn et al., 8 Apr 2026).

6. Illustrative Special Cases and Stability

Special instances include:

• Exponential sum signals: For $\iota_O$3, Prony methods demonstrate invertibility of the window map.
• $\iota_O$4-tolerant noise model: If observed windows deviate by at most $\iota_O$5, the inverse function theorem and Lipschitz properties of $\iota_O$6 yield explicit stability bounds: “zero-defect” is certified only if the true defect falls within $\iota_O$7.
• Sharpness at identifiability barrier: If two distinct signals yield identical window-sums but have different neutral status, no sound procedure can resolve the case conclusively (Washburn et al., 27 Feb 2026).

7. Structural and Geometric Properties

Key properties of the canonical cost $\iota_O$8:

• Symmetry: $\iota_O$9.
• Unique minimizer and nonnegativity: $s$0, with equality iff $s$1.
• Strict convexity in log-space: $s$2.
• Separable structure (d'Alembert law): $s$3.
• Bregman divergence form: $s$4 for $s$5.
• Minimality: No alternative $s$6 can satisfy the prescribed axioms (Washburn et al., 5 Feb 2026).

The associated geometry is intrinsically degenerate in logarithmic space and pseudo-Riemannian in the original variables, with explicit singularity loci. The metric structure and divergence relations provide a multifaceted characterization useful in statistical certification, optimization, and informational geometry contexts (Washburn et al., 8 Apr 2026).

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References:

(Washburn et al., 5 Feb 2026, Washburn et al., 27 Feb 2026, Washburn et al., 8 Apr 2026)