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Allocator theorem

FPRD-T104

Control-functional frontiers have a canonical stationary allocator

Exact statement

If every reachable visible control carries at most one live concrete stack, then one fixed slot per control is a canonical stationary allocator. Its transition depends only on the active-control mask, the current input, and current stack tops, and it compiles through FPRD-T102 and FPRD-T101.

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

Context

When a control names at most one stack after every prefix, the control itself is a persistent coordinate and collisions at that control are semantically harmless.

Definitions

  • Control-functional means that two frontier configurations with the same word and control necessarily have the same concrete stack.

Proof or evidence

Lean checks candidate soundness and completeness, stationary candidate selection, collision safety, the reachable exactness invariant, both directions of the allocator step law, exact readout, and a three-control branching calibration.

Control-functional allocation →

Verification notes

The load-bearing collision step was inspected separately: every candidate reaches the same target control at the same extended prefix, so control functionality makes their target stacks equal before one representative is chosen.

Limitations

  • Control functionality is sufficient, not necessary, and is stronger than a frontier-cardinality bound.
  • The theorem does not decide control functionality or handle several distinct stacks at one control.
  • Classical choice denotes a finite transition table only after the relevant carriers are known finite.

Open work

Determine useful structural tests or sufficient conditions for control functionality in natural projected machines.

Notes

Control-functional means that after every observed prefix each visible control carries at most one live concrete stack. The control is therefore a canonical slot address. Lean checks exact call, return, and local candidate soundness and completeness, stationary candidate choice, fixed active-mask coordinates, singleton initialization, final-control readout, reachable exactness, and both directions of the one-letter allocator law. When several derivations reach one target control, the proof uses control functionality at that same extended prefix to show that an arbitrary selected candidate has the common target stack. The Fin-3 calibration branches on one input into a call stack [7] and a local empty stack at distinct controls, then proves exact compiled language equality for all words. Classical choice denotes a finite transition table when the input, stack alphabet, and controls are finite; those finiteness facts, positive degree, reversal, and LMR remain external. This does not prove that arbitrary bounded frontiers are control-functional, does not construct an allocator when one control has multiple distinct stacks, and does not prove the universal FPRD-C02 proposition. This opens no M34 and does not edit the paused publication.