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Theorem · replay stability, composition, and support cylinders

FPRD-T150

Adaptive replay slices are stable and compositional

Exact statement

In every finite adaptive work history, changing only source cells outside one execution's replay support preserves every supported derived value and ordered query transcript, the observer result and ordered query transcript, and the replay support itself. The support is the least trace-closed certificate. At a downward-closed history cut, later demands pull back through the earlier dependency graph of the same execution. With independent source domains, fixing supported source coordinates gives a product cylinder contained in a replay-equivalence class; equality with the whole class is not asserted.

StatusSelf-contained proof and exhaustive small-model falsification audit
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

FPRD-T133 is a scaffold specialization, not a proof premise. The abstract theorem identifies the proof pattern: a final observation induces a backward dependency slice through finite adaptive work, and agreement on its source leaves every replayed decision unchanged.

Hypotheses and scope

  • The work history and every query trace are finite.
  • Derived cells are well-founded, deterministic, terminating, and free of side effects.
  • Composition uses a downward-closed earlier block and dependency transcripts from the same execution.
  • The product-cylinder inclusion additionally requires independent source-coordinate domains.

Proof or evidence

A well-founded induction proves equality of every supported value and adaptive query trace; the observer case gives output equality, and recomputing the same least closure gives support stability. Partitioning the closure at a history cut proves composition. The checker passes 55,296 executions, 72,960 permitted perturbation pairs, 72,000 supported-cell comparisons, and 55,296 composition checks.

Adaptive replay-slice theorem →

Verification notes

The proof was checked against the failure modes most likely to invalidate it: adaptive branch changes, omitted transitive dependencies, queries to derived cells, source changes outside the slice, and composition across a history cut. A negative control found 25,848 cases where retaining only direct observer queries fails.

Limitations

  • Replay support is least among trace-closed certificates, not necessarily least among all sets that preserve only the final output.
  • The replay-equivalence class can be larger than the support cylinder; whole-class equality would require additional hypotheses.
  • The theorem is per execution and does not bound support size across growing histories.
  • It does not construct a stationary finite-state compressor, resolve FPRD-C02, or imply a context-free-to-PEG inclusion.
  • Adjacent ideas are established in dynamic slicing, provenance traces, and self-adjusting computation; no literature novelty claim is made.

Open work

Formalize the well-founded replay induction and determine which additional conditions turn per-execution slices into a stationary bounded-memory representation.