PTG VIII — PTG VIII: Structural Admissibility and Uniqueness of Transport Functionals

Reader summary

PTG VIII proves that PDT's transport functionals are not arbitrary choices. The focusing functional, the capacity density, and the entropy weights are uniquely determined, up to fixed equivalence classes, by the minimal data: a scalar field, a transport cone bundle, and a torsion-free connection. Focusing functionals must be quadratic forms in the generator from positive semi-definite tensors. Capacity densities must be linear forms from positive covectors. Entropy weights are positive triples, unique up to common positive scaling. Together these admissibility classes form a finite-dimensional space, completing PDT's structural closure.

Abstract

This paper establishes the structural admissibility and uniqueness properties of the transport functionals used throughout Phase Transport Geometry. We show that the focusing functional, the capacity density, and the entropy weights are not arbitrary choices but are uniquely determined, up to fixed equivalence classes, by the minimal geometric data: a scalar field, a transport cone bundle, and a torsion-free affine connection. We prove that any functional satisfying the required homogeneity, monotonicity, and compatibility conditions must lie in a finite-dimensional admissible class. These results complete the structural closure of the Phase Transport Geometry framework.

Falsifiable predictions

01
Every admissible focusing functional is a quadratic form arising from a positive semi-definite symmetric tensor on the transport cone.
02
Every admissible capacity density is a linear form arising from a covector field positive on the cone.
03
The PTG entropy weights are unique up to a single common positive scaling, leaving no extra tunable parameters in the framework.

Full paper

Introduction

The preceding papers introduced several scalar functionals: the focusing functional F(l^a) (PTG II), the capacity density rho(l^a) (PTG III), and the entropy weights (alpha, beta, gamma) (PTG VI). Each was required to satisfy specific homogeneity, monotonicity, and compatibility conditions.

The purpose of this paper is to show that these functionals are not arbitrary. They are uniquely determined, up to fixed equivalence classes, by the minimal data (M, Phi, L_p, grad). The main results are: a classification of admissible focusing functionals; a uniqueness theorem for capacity densities; a structural constraint on entropy weights; and a closure theorem showing that all admissible functionals form a finite-dimensional space.

Preliminaries

L_p is a closed convex phase-oriented cone at each p. A scalar functional G(l^a) is homogeneous of degree k if G(alpha l^a) = alpha^k G(l^a) for all alpha > 0.

Admissible focusing functionals

Definition

F : L -> R_{>=0} smooth, homogeneous of degree two, non-negative, compatible with the Raychaudhuri evolution, and invariant under admissible rescalings.

Classification

Any admissible F has the form F(l^a) = A_ab l^a l^b, where A_ab is a smooth symmetric positive semi-definite tensor field with A_ab l^a l^b >= 0 on L_p. Homogeneity of degree two implies a quadratic form; non-negativity and cone-compatibility imply positive semi-definiteness; rescaling invariance fixes the tensor structure.

Uniqueness class

F_1 and F_2 are equivalent iff they agree on L_p.

Admissible capacity densities

Definition

rho : L -> R_{>=0} smooth, non-negative, degree-one homogeneous, compatible with transport capacity evolution.

Uniqueness

Any admissible rho has the form rho(l^a) = B_a l^a, with B_a a smooth covector field positive on L_p. Degree-one homogeneity gives linearity; non-negativity and cone-compatibility give positivity.

Entropy weights

The entropy of PTG VI is H = integral over S_lambda of ( alpha h_ab h^ab + beta rho(l^a) + gamma sigma_ab sigma^ab ) dmu. The constants must satisfy alpha, beta, gamma > 0, and are unique up to a common positive scaling. Positivity ensures non-negativity of the entropy; monotonicity fixes the relative scaling; homogeneity fixes the overall scaling freedom.

Closure theorem

Structural closure

The admissible focusing functionals, capacity densities, and entropy weights form a finite-dimensional space determined entirely by the minimal geometric data (M, Phi, L_p, grad).

Discussion

The transport functionals used throughout PTG are uniquely determined, up to fixed equivalence classes, by the minimal data. This establishes the structural closure of the framework and prepares the ground for the constitutive closure analysis in PTG IX.

BibTeX

@techreport{pdt_ptg_viii_2026,
  author      = {Fincham, Graham and Hilton, Daniel},
  title       = {PTG VIII: Structural Admissibility and Uniqueness of Transport Functionals},
  institution = {Phase Differential Theory},
  year        = {2026},
  type        = {PDT working paper},
  number      = {PTG VIII}
}