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Theorem · replay invariance and path coherence

FPRD-T133

Stable causal support and garbage-fibre coherence

Exact statement

The least replay-closed causal endpoint support of a scaffold history is stable: preserving its supported descriptors and final-reachable labels preserves the final rooted concrete scaffold and the support itself. Every resulting causal-garbage fibre is a finite Cartesian product; single-coordinate triangles and distinct-coordinate commuting squares present its path groupoid and make its filled two-complex simply connected.

StatusProved and independently audited; no dedicated standalone checker
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

A node may be absent from the final reachable graph yet have supplied port cells queried while a later, still-visible edge was evaluated. Ordinary final reachability therefore under-approximates the data on which the final concrete scaffold depends.

Definitions

  • Start with every label and port cell of a final-reachable node. Whenever a supported old-top path was evaluated, add every older port cell it queried, including the cell where an invalid path failed; the least backward closure is CHC_H.
  • A garbage fibre fixes the final-reachable labels and every descriptor in this replay-closed support while varying all remaining finite coordinates.

Hypotheses and scope

  • Equal-length scaffold histories over the same finite row alphabet.
  • The comparison preserves every final-reachable label and every descriptor cell in the replay-closed support.

Proof or evidence

Chronological induction replays every supported descriptor through the same physical endpoints, preserving the rooted concrete graph. Recomputing the backward closure gives the same support. Unsupported coordinates are consequently independent finite factors; triangles compose changes of one coordinate and squares commute distinct coordinates, reducing every path to coordinate order and contracting every loop.

Replay invariance →

Verification notes

The replay induction, support-stability step, Cartesian-product consequence, and path-normalization argument were independently rechecked against the paper and proof audit. The source records finite support audits, but no dedicated standalone checker for T133 was identified.

Limitations

  • Causal support is an execution-specific dependency slice, not ordinary final reachability.
  • It is not asserted to be semantically minimal modulo bisimulation.
  • The triangle-and-square presentation applies inside one stable garbage fibre, not to mixed strong, garbage, and temporal derivations.

Open work

Study how causal support changes under strong and temporal moves; do not extend the garbage-cube theorem beyond one stable fibre without a transport proof.

Notes

Causal support is an execution-specific dependency slice, not ordinary final graph reachability and not asserted to be semantically minimal modulo bisimulation. The path-coherence result is internal to one stable garbage fibre and does not present mixed strong, garbage, and temporal derivations. No novelty claim is made.