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Conditional compilation theorem with Lean verification

FPRD-T94

Stationary derivative covers compile exactly

Exact statement

Suppose a fixed synchronous multistack executor represents a forgotten-action frontier through a fixed derivative basis and fixed slots, with exact initialization, a preserved validity invariant, exact one-letter updates, and exact final-state readout. Then after every word its represented relation equals the pushdown frontier. The executor and its compiled scaffold therefore recognize exactly the source language, including at the empty word.

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

Context

The theorem makes a proposed frontier representation prove four local obligations; the global recognition result then follows mechanically.

Hypotheses and scope

  • One horizon-independent derivative basis, slot assignment, transition, invariant, and readout work for every prefix.
  • The validity invariant holds initially and is preserved by every input transition.

Proof or evidence

Induction on the input combines initial exactness, invariant preservation, the exact stationary step, and FPRD-T92. The checked compiler then transfers executor recognition to a scaffold. One-slot visible-stack and two-slot parallel examples instantiate the interface.

Stationary-cover transfer →

Verification notes

The Lean theorems cover_run_exact, cover_decides_iff, and compiled_cover_decides_iff were matched to the written proof and calibration examples. The finite-carrier PEG corollary remains an external source-based composition.

Limitations

  • The theorem assumes a stationary cover; it does not prove that every projected visible frontier has one.
  • Finite-horizon or prefix-dependent covers do not satisfy the hypothesis.
  • The PEG corollary requires finite carriers and an explicit reversal orientation.

Open work

Construct a stationary bounded-slot cover for broader projected frontiers, or prove a representation-independent resource obstruction.

Notes

Lean 4.30.0 proves the principal derivative laws, preservation of the invariant along every run, cover_run_exact, cover_decides_iff, and compiled_cover_decides_iff. The one-slot visible-stack and two-slot finite parallel calibrations both instantiate the same interface. The theorem is conditional on a supplied stationary exact step and readout; Lean does not prove that every forgotten-action frontier has such a cover, nor does this claim invoke the external LMR correspondence.