Context
The allocator uses slots indexed by Fin s, while mathematical machines usually use named states. Reindexing removes that syntactic mismatch without adding computational power.
Proof or evidence
Lean transports initialization, readout, and all three visible commands, proves exact frontier transport, refinement transitivity, control-functionality invariance, and compilation of both named and original machines.
Verification notes
The audit checked that only control names move: input letters, action tags, stack symbols, full stack contents, and round timing remain unchanged.
Limitations
- A finite equivalence is supplied rather than synthesized from a generic Fintype instance.
- Reindexing neither constructs the refinement nor supplies a new allocator beyond T104.
Notes
A core-only ControlEquiv supplies forward and inverse control maps with two inverse laws. Reindexing changes only control names: the initial control, final-control readout, and each successful call, return, or local target are transported, while the observed letter, hidden action, stack symbol, and entire concrete stack remain exact in the same input round. Lean checks the resulting T105 refinement, exact frontier transport, epsilon-aware acceptance, refinement transitivity including local completeness, and control-functionality equivalence. A named start/pushed/kept machine is an exact reindexing of the genuine fork and compiles both itself and the original non-control-functional merged source on all words. The theorem does not synthesize an equivalence from Fintype, add finiteness fields to the base machine, decide whether a finite refinement exists, or construct an allocator independently of T104. External alphabet finiteness, positive scaffold degree, reversal, and LMR remain outside Lean. This opens no M34 and does not edit the paused EPAL publication.