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Representation and closure theorem

FPRD-T125

Occurrence contexts and typed receipt ropes close under input

Exact statement

Binary even-palindrome dynamics factor exactly into one longest even-palindromic suffix occurrence and one reversed left-context rope. In the abstract persistent-sequence interface, typed mirror receipts, bridge-shadow transport, and common direct/reset rope recurrences close every online pop, prepend, reset, fresh-record, and retained-tail update with a fixed number of sequence operations. This is receipt semantics, not a certificate that a concrete sequence implementation meets the native per-symbol resource contract. A raw historical endpoint, immutable type-birth table, raw anchor-independent chunk, or bounded bridge chase does not supply the same interface.

StatusProved in Draft 2 and internally audited
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The representation stores the current longest even-palindromic suffix occurrence and the reversed context to its left. Typed receipts name the exact historical occurrence needed by a later transition, rather than only its word or endpoint.

Hypotheses and scope

  • The alphabet is fixed and finite; the published construction specializes to the binary alphabet.
  • Receipt closure is an abstract persistent-sequence statement. Its native implementation premise is separate and is stated by the complete Draft 5 Section 11.1 contract in T127/T128.

Proof or evidence

The proof derives the exact three-case word recurrence, mirror and selector receipt recurrences, bridge-shadow transport, and a fixed-operation closure theorem. Finite audits separately check 262,142 exhaustive word updates, 38,758 guarded shadow instances, and the stored selector against unbounded bridge-chase families.

Receipt dynamics and proof →

Verification notes

Draft 1 was found defective: it used bridge shadow outside one hypothesis and did not store a bounded pointer to the selected occurrence D_a(P). Draft 2 adds a mirror-prefix lemma and one transition receipt per direct table entry. A subsequent cold read confirmed these repaired interfaces; finite checks corroborate but do not prove them.

Limitations

  • Raw endpoints, immutable type-birth records, anchor-independent chunks, and bounded bridge chasing fail only as named representations; they are not general scaffold lower bounds.
  • The proof depends on distinguishing palindrome types from historical occurrences.

Open work

Obtain an external check of the bridge-shadow, mirror-prefix, and selector-receipt arguments.

Notes

The semantic receipt proof is retained, including Draft 2's mirror-prefix and typed-transition repairs. The native implementation obligation is separate and is stated in T127/T128. Representation obstructions are scoped to their named interfaces. Finite word audits are corroboration, not uniform proof; external specialist and novelty review remain open. See docs/integration/reconciliation/T125-T128.md for the current source edition, evidence boundary and seven-surface disposition.