Phase Differential Theory
Phase Differential
A Framework for Reality
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R01

Core axioms and consistency

The relational primitives, phase composition, retained history and the additional A6 realisation commitment.

Current foundationsPaper overview and reading guideManuscript PDF availableReview draft 1.0
Document version
Review draft 1.0
Date printed on the document
24 September 2026
Public release date
Not confirmed

This page is a reading guide written for the website. It is not the manuscript and does not reproduce its proofs. The manuscript PDF is linked below. It is a review draft and carries its own review notice. Authorship is not stated in the document and has not been confirmed. The public release date on this website has not been confirmed; the date shown is the date printed on the document. “Current foundations” does not by itself mean peer reviewed or experimentally confirmed.

Purpose

The question addressed

R01 states the axioms A0 to A6 and checks that they are consistent. It fixes what the theory begins with and what it deliberately leaves out.

Mathematical starting point

Objects and premises used

  • A set XX of distinctions, the circle group U(1)U(1) and a small groupoid GG of comparisons.
  • A phase functor Φ:G→BU(1)\Phi: G \to BU(1), a retained history hn=(r1,…,rn)h_n = (r_1, \dots, r_n) and a model-supplied admissibility law C(s)C(s).

Main results

What the paper establishes

  • A0 to A5 supply distinctions, circle phase, coherent comparison, the angle encoding, the minimal real carrier and retained history.
  • A6 adopts the first attained singleton s∗=min⁡{s:∣C(s)∣=1}s^* = \min\{ s : |C(s)| = 1 \} as the realisation event, when that minimum exists, is finite and precedes extinction.
  • Finite Canonical Selection (FCS) is kept as a separate, optional assumption that need not select one member.

Explanation

What each axiom supplies

A0 supplies distinctions but no distance, clock or embedding. A2 makes phase compose along comparisons: Φ(b∘a)=Φ(b)Φ(a)\Phi(b \circ a) = \Phi(b)\Phi(a). A5 lets the history grow without rewriting earlier records.
A6 does not guarantee that a process terminates, supply a probability law or establish universal determinism. The record count nn, episode parameter ss and calibrated time tt are distinct.

More on the mathematics page: Axioms →

Physical interpretation and scope

What connects this to physics

The axioms are structural. Connecting them to apparatus, clocks or spacetime requires the physical premises introduced in later papers.
R01’s A0 is the current distinction axiom. The earlier finite-capacity “Axiom 0” belongs to the historical paper PDT I and is not the same statement.

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Using this paper

View the R01 manuscript (PDF)Download PDF

R01. “Core axioms and consistency.” Phase Differential Theory current foundations. Review draft 1.0, document dated 24 September 2026. https://www.phasedifferentialtheory.com/foundations/r01

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Reading order is expository and is not a claim that each premise is derived in the preceding paper.