PTG IX: PTG IX: Constitutive Closure and Autonomy of the Framework

Reader summary

PTG IX closes the programme. Every admissible focusing functional and capacity density arises from a single constitutive tensor; no other admissible scalar, vector, or tensor functionals exist. The membrane evolution equation is the unique first-order admissible flow on the horizon. The entire framework is autonomous: every admissible construction is determined by the minimal data (manifold, scalar field, transport cones, torsion-free connection). PTG is therefore a closed and complete geometric theory. No additional axioms, functionals, or degrees of freedom can be added without violating the minimal-data or admissibility constraints.

Abstract

This paper establishes the constitutive closure of Phase Transport Geometry. Building on the structural admissibility results of Phase Transport Geometry VIII, we show that the admissible focusing functionals, capacity densities, entropy weights, and membrane evolution equations form a closed and autonomous mathematical system. No additional geometric, analytic, or variational structures can be added without violating the minimal data assumptions or the admissibility constraints. We prove that all admissible transport functionals arise from a single constitutive tensor, and that the entire framework is uniquely determined by the minimal geometric data. This completes the Phase Transport Geometry programme.

Falsifiable predictions

01
Every admissible PTG functional, including F, $\rho$ , and the entropy integrand, derives from a single constitutive tensor.
02
The membrane evolution equation is the unique first-order admissible flow on the transport horizon.
03
PTG is autonomous: no admissible scalar, vector, or tensor structure can be added beyond those already present in PTG I to VIII.

Full paper

Introduction

Phase Transport Geometry is constructed from the minimal data (M,

\Phi

L_{p}

, grad). The preceding papers introduced several scalar and tensorial functionals: F(

l^{a}

) (PTG II),

\rho(l^{a})

(PTG III), A, C, K (PTG V), H(

\lambda

) (PTG VI),

V^{a}

(PTG VII), and the admissibility classes (PTG VIII).

The purpose of this paper is to show that these objects form a closed and autonomous system. No additional admissible functionals exist. No further degrees of freedom can be introduced. The framework is complete.

Constitutive tensors

Definition

A constitutive tensor is a smooth symmetric tensor field

C_{ab}

, positive semi-definite on

L_{p}

, and compatible with the transport cone.

Representation theorem

Every admissible focusing functional and every admissible capacity density arises from a constitutive tensor: F(

l^{a}

) =

C_{ab} l^{a} l^{b}

and

\rho(l^{a})

C_{ab} l^{a} n^{b}

for some smooth covector

n^{b}

compatible with

L_{p}

Constitutive closure

Definition

A set of admissible functionals is constitutively closed if no additional functional can be added without violating homogeneity, positivity, cone compatibility, or structural admissibility.

Closure theorem

The admissible focusing functionals, capacity densities, entropy weights, and membrane evolution equations form a constitutively closed set. No additional admissible scalar, vector, or tensor functionals exist. By the representation theorem any candidate must arise from a constitutive tensor, but PTG VIII shows all such tensors are already included.

Autonomy of the framework

Definition

PTG is autonomous if all admissible constructions are uniquely determined by the minimal data (M,

\Phi

L_{p}

, grad).

Autonomy theorem

PTG is autonomous. Every admissible functional, invariant, or evolution equation is uniquely determined by the minimal data. PTG VIII shows the admissibility classes are finite-dimensional and data-determined; PTG II to VII show all higher structures depend only on these functionals.

Uniqueness of the membrane evolution equation

The membrane evolution vector field

V^{a}

\alpha G^{a}

\beta \sigma_{\star b}^{a} k^{b}

is the unique admissible first-order evolution equation on the transport horizon. Any admissible evolution must be tangent to

S_{\star}

, homogeneous of degree zero, and compatible with entropy monotonicity; these force the given form.

No additional degrees of freedom

Rigidity

Any admissible scalar functional on

L_{p}

is a linear combination of F(

l^{a}

\rho(l^{a})

, and <<\sigma_{ab} \

\sigma

^{ab}>>. Homogeneity and positivity restrict it to quadratic or linear forms; cone and deformation-tensor compatibility restrict it to the stated basis.

Completion of the PTG programme

PTG I to IX form a complete and autonomous geometric theory. No additional axioms, functionals, or structures can be added without violating the minimal data assumptions or the admissibility constraints.

Discussion

PTG is a closed and autonomous geometric framework. All admissible functionals arise from a single constitutive tensor and no additional degrees of freedom exist. This completes the Phase Transport Geometry programme.

Suggested citation

@techreport{pdt_ptg_ix_2026,
  author      = {Fincham, Graham and Hilton, Daniel},
  title       = {PTG IX: Constitutive Closure and Autonomy of the Framework},
  institution = {Phase Differential Theory},
  year        = {2026},
  type        = {PDT working paper},
  number      = {PTG IX}
}