Phase Differential Theory
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R03

Selection and persistent records

Conditions for a unique surviving candidate, symmetry obstructions and conditional record stability.

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

R03 determines when a shrinking family of candidates reaches a unique survivor, when symmetry prevents it, and under what conditions a record remains stable.

Mathematical starting point

Objects and premises used

  • A finite, exhaustive family of mutually exclusive candidates and a nested admissibility law C(s2)⊆C(s1)C(s_2) \subseteq C(s_1) for s2≥s1s_2 \ge s_1, from R01.
  • Positive finite loss times TrT_r supplied by the model.

Main results

What the paper establishes

  • The last-survivor criterion (Theorem R03.1).
  • Ties at the largest loss time pass straight from two or more candidates to none, so no A6 event occurs.
  • Record stability is conditional on the dynamics of the recording system.

Theorem R03.1: The last-survivor criterion

Result statement (summary)
For a finite candidate set with at least two members, each candidate rr having a positive finite loss time TrT_r and C(s)={r:s<Tr}C(s) = \{ r : s < T_r \}, a singleton occurs before extinction exactly when the largest loss time is unique. The first singleton occurs at the second-largest loss time.
Assumptions and scope: The loss times are supplied by the model. The theorem identifies a singleton; it does not explain the physical origin of the loss law.
Link to this statementShort proof on the mathematics page

The full proof is given in the R03 manuscript (PDF).

Explanation

Worked example: three candidates

With TA=2T_A = 2, TB=1T_B = 1 and TC=3T_C = 3, B leaves at stage 1 and A at stage 2. C is the first unique survivor at s=2s = 2, the second-largest loss time, and is recorded.
If TA=TBT_A = T_B were the largest loss time, A and B would leave together and no record would be made.

More on the mathematics page: Last-survivor selection →

Physical interpretation and scope

What connects this to physics

The theorem identifies a singleton in a supplied process. A6 adopts that singleton as realised. Explaining the physical loss law is a further task for the model.

Read, download and cite

Using this paper

View the R03 manuscript (PDF)Download PDF

R03. “Selection and persistent records.” Phase Differential Theory current foundations. Review draft 1.0, document dated 24 September 2026. https://www.phasedifferentialtheory.com/foundations/r03

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