PDT-MC-1: Mathematical Closure Supplement

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: Verification and provenance.

An earlier mathematical closure supplement. Joint closure is addressed in R12 and verification and provenance in R13.

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.

This supplement provides the mathematical foundations for the PTG/PDT programme. It formalises the closure axioms, constructs the closure-compatible metric, proves the existence and uniqueness of solutions to the differential transport system, establishes the minimal winding and loading results, and derives the return algebras of the triplet, doublet, and singlet sectors. The central result is the Return Algebra Theorem, which shows that the triplet sector has return algebra End(

C^{3}

) of dimension 9, yielding the down-sector invariant

T_{3}

= 9 used in PTG-5 and PDT-II. PDT-MC-1 provides the mathematical backbone for the physical interpretation developed in PDT-I, PDT-II, and PDT-III.

Suggested citation

Graham Fincham, Daniel Hilton. “PDT-MC-1: Mathematical Closure Supplement.” Phase Differential Theory research programme. Version 1.1.1. 2026. https://www.phasedifferentialtheory.com/papers/pdt-mc-1

@misc{pdt_pdt_mc_1,
  title = {PDT-MC-1: Mathematical Closure Supplement},
  author = {Graham Fincham and Daniel Hilton},
  year = {2026},
  version = {1.1.1},
  url = {https://www.phasedifferentialtheory.com/papers/pdt-mc-1}
}

No DOI recorded. No repository record verified. Journal-review status not confirmed.