PTG-1: Closure Architecture and the Minimal Phase Manifold

Daniel Hilton

Earlier research paper

This paper remains available as part of the development of Phase Differential Theory. The R00–R13 foundational suite provides the current formulation of the theory. Read this paper in its original model and publication context.

Current foundations Research history

Current foundational treatment

Historical backgroundPrincipal topic: Phase transport and representations.

An earlier closure architecture and phase manifold. The current account of phase composition and the minimal phase carrier is in R01 and R02.

A topic connection does not mean that every claim in this paper is retained or that a current paper reproduces its proof.

Original abstract

Preserved from the original manuscript.

Phase Transport Geometry (PTG) begins from a primitive structural axiom: the existence of an irreducibly collective triplet closure relation. This axiom generates the closure hierarchy (1,2,3) and determines the minimal compact phase manifold capable of supporting independent singlet, doublet, and triplet sectors. Under explicit structural assumptions, we show that this manifold is diffeomorphic to (

S^{1}

)^6 and prove its conditional minimality and uniqueness. PTG-1 provides the geometric foundation for compact transport (PTG-2), the closure-compatible metric (PTG-3), sector structure (PTG-4), defect mass scaling (PTG-5), and the physical interpretation developed in the PDT series.

Suggested citation

Graham Fincham, Daniel Hilton. “PTG-1: Closure Architecture and the Minimal Phase Manifold.” Phase Differential Theory research programme. Version 1.1.1. 2026. https://www.phasedifferentialtheory.com/papers/ptg-1-2026

@misc{pdt_ptg_1_2026,
  title = {PTG-1: Closure Architecture and the Minimal Phase Manifold},
  author = {Graham Fincham and Daniel Hilton},
  year = {2026},
  version = {1.1.1},
  url = {https://www.phasedifferentialtheory.com/papers/ptg-1-2026}
}

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