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Conditional direct recognition consequence

FPRD-T128

A direct scaffold route for binary even palindromes

Exact statement

Conditional on a persistent sequence implementation satisfying the complete direct Draft 5 Section 11.1 contract, the receipt construction, whole-client normalization, opaque-atom staging and simultaneous record compiler yield a fixed finite letter-synchronous native scaffolding automaton for EPAL_2={ww^R:w in {0,1}*}, with symbolic finite resource bounds. Loff--Moreira--Reis then gives an ordinary complete-match PEG for reversal(EPAL_2)=EPAL_2. This conditional direct route uses neither a strict real-time multitape machine nor Slisenko's palindrome premise; the separate T89/T90 transfer route remains independent.

StatusConditional construction with an explicit sequence contract; symbolic resource bounds
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

An earlier FPRD paper already proved EPAL_2 in PEG through strict real-time multitape machines. This theorem gives a second route through a direct persistent representation whose state exposes the historical occurrence needed by the next comparison.

Hypotheses and scope

  • A persistent sequence implementation satisfies the complete Draft 5 Section 11.1 contract; this premise is not discharged by the concrete implementation evidence.
  • FPRD-T125 supplies abstract receipt closure; T127 supplies conditional staging and whole-client normalization supplies a bounded plan under its own hypotheses; T126 compiles that plan.
  • The Loff-Moreira-Reis theorem is used in the scaffold-to-PEG direction, and EPAL_2 is reversal-invariant.

Proof or evidence

Given the complete sequence contract, receipt closure, opaque-atom staging and whole-client normalization produce a bounded simultaneous plan whose finite-control acceptance tests emptiness of the live reversed context. T126 compiles it letter-for-letter; Loff-Moreira-Reis gives the PEG consequence for its reversal. This is a checked conditional consequence, not a concrete implementation certificate.

Direct theorem and dependency chain →

Verification notes

The current Draft 5 and reconciled root ledger govern: the direct consequence is conditional on the complete Section 11.1 contract, Appendix D remains informal implementation evidence, and numerical radius/distance remain withdrawn. The following dated source-audit trail does not discharge that premise. Draft 3 corrected the compiler's initial-heap case. A 10 September 2026 source audit then found that Kaplan–Tarjan does not itself establish the stronger finite-record/opacity interface used by T127. The exact client audit corrected the call schedule to eight/eight/two, and later semantic and assertion reviews closed the source-level list premise. The typed call certificate checks the corrected endpoint recurrence through public uncons=118. A fresh mutation attack then showed that the numerical radius checker accepts meaningless repeated selector lists, so 55/874, R*=1,750, and distance 1,751 are withdrawn rather than described as checked. The exact OCaml 4.14.1 interprocedural IR supplies complete child edges, qualified formals, case and nullary-pattern topology, ordered calls, return leaves, and build slots. Its callback-specialized cat closure has 81 bodies and no unresolved ordinary direct call. The strengthened admission gate binds the exact archived source and CMT bytes, checks slot-to-edge identity, and partitions all 952 source sites: 405 fixed structural, 73 fixed control, and 474 context-sensitive across 42 bindings. Eighteen admission mutations reject. A new checked contract rejects treating those 474 generic sites as a flat semantic table: Packet permits arbitrary symbolic Pair^k(alpha) depth, so the type proof must be context-indexed and symbolic. Its 21 mutations reject. This is still not old/fresh interpretation: those type schemes, 42 callback substitutions, and two assertion obligations remain. Allocation A*=8,838, q=7, degree 61,868, whole-client normalization, and the word, receipt, assembly, and acceptance arguments are unchanged. Under the complete sequence contract the direct theorem uses symbolic finite source-model bounds, not the withdrawn numerical corollary. The direct route has no Galil, Slisenko, real-time multitape, or general Turing-machine premise. The reviews share model dependencies and are not external validation.

Limitations

  • The conclusion was already established by the earlier FPRD transfer paper; this record claims a different direct route, not the first refutation of the Loff–Moreira–Reis conjecture.
  • The direct theorem is scoped to the abstract immutable source-record model, not literal OCaml heap allocation, GC, exceptions, or machine addresses.
  • The former A=6K+4 and r=6K+8 formulas remain withdrawn. A*=8,838 and q=7 remain supported in the abstract source-record model, but R*=1,750 and compiler distance 1,751 are also withdrawn pending a real provenance certificate.
  • No novelty determination or documented external specialist review is claimed.

Open work

Extract and check symbolic type schemes for the 474 contextual cat source sites, preserving arbitrary Pair^k(alpha) depth, finish branch-correlated callback substitution, then extend the fail-closed interpreter separately to uncons and seek external review.

Notes

Status proof applies to the checked conditional consequence of the Section 11.1 sequence contract. It does not assert that the concrete implementation has been certified. The 21 September Draft 5 is the current aligned manuscript; root Drafts 1/2 and previous implementation restorations are historical evidence. Numerical radius and distance remain withdrawn. T90's independent classical-transfer membership conclusion is not downgraded. Causal Example 9.6 needs assurance-level and edition alignment in a future coordinated publication. No fresh formal verification, unconditional direct membership proof, novelty claim, research count or paper revision results from this migration. See docs/integration/reconciliation/T125-T128.md.