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Refinement transfer theorem

FPRD-T105

Finite control refinements transfer canonical allocation

Exact statement

If a partial deterministic visible machine has a finite control refinement whose refined frontier is control-functional, then a stationary scaffold decides the original forgotten-action projection. The refinement preserves the same input, hidden action, and concrete stack in both directions after every prefix.

StatusMechanically checked and internally audited
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

A refined control acts as a finite causal color: histories that merge at one original control can be separated before the control-functional allocator is applied.

Hypotheses and scope

  • The refinement preserves initialization, acceptance, and every successful call, return, and local step in both directions with the same concrete stack.
  • The refined machine is control-functional and has finitely indexable controls.

Proof or evidence

Reverse-list induction proves exact frontier projection and acceptance for all prefixes, including epsilon. The calibration separates pushed and empty branches that collide at one original control and compiles the refined language through T104.

Finite causal refinement →

Verification notes

The soundness and completeness directions were checked for same-round preservation; no output lag or change of hidden action is permitted.

Limitations

  • The theorem does not prove that every bounded frontier admits a finite refinement.
  • It does not decide refinement existence or handle histories requiring unboundedly many colors.

Open work

Characterize when a bounded frontier admits a finite causal refinement, rather than merely assuming one is supplied.

Notes

The refinement laws preserve the initial control, final-control readout, and every successful call, return, and local command in both directions with the same input round, hidden action, and concrete successor stack. Reverse-list induction establishes exact frontier projection for every prefix and exact projected acceptance, including epsilon. The Fin-2 calibration merges a call stack [7] and local stack [] at one accepting original control and proves directly that the original frontier is not control-functional. Its Fin-3 refinement separates the branches, is control-functional by FPRD-T104's existing proof, and its generic compiled scaffold accepts the original language exactly on all words. The theorem assumes only a finite control refinement and derives its allocator through FPRD-T104; it does not assume an allocator. It does not prove that every bounded frontier admits a finite causal coloring, decide refinement existence, or handle frontiers requiring unboundedly many refined controls. Finiteness of the external input and stack alphabets, positive scaffold degree, reversal, and LMR remain outside Lean. This opens no M34 and does not edit the paused EPAL publication.