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Normalization theorem

FPRD-T103

Absorbing rejecting sinks can be pruned before allocation

Exact statement

A total deterministic visible machine with a designated absorbing rejecting sink has exactly the same forgotten-action projection as the partial machine obtained by discarding every command whose target is that sink. Equality holds after the same prefix, including epsilon, and any allocator for the pruned frontier 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

A totalized rejecting sink preserves language acceptance but can pollute a frontier with irrelevant sink stacks. Pruning removes those failed branches in the same input round.

Hypotheses and scope

  • The sink is nonaccepting and every call, return, and local command from it remains in it.
  • Acceptance depends only on control; no assumption that the initial control differs from the sink is needed.

Proof or evidence

Lean proves step projection, preservation of successful pruned runs, the absorption argument for accepted total runs, two-way all-word language equality, epsilon behavior, and direct allocator/compiler composition.

Rejecting-sink pruning →

Verification notes

The audit checked that pruning happens on the same observed letter, that sink stack changes do not matter once absorption is established, and that the result does not smuggle in allocator existence.

Limitations

  • Pruning normalizes the semantic interface; it does not construct an allocator.
  • The theorem requires a genuinely absorbing nonaccepting sink.

Open work

Check the pruning theorem against alternative totalization conventions used in the visibly pushdown literature.

Notes

Lean 4.30.0 checks exact equality between the existing total one-stack evaluator and a direct tagged visible evaluator, syntactic pruning of sink-entering commands, preservation of every successful pruned run, the absorption argument that a total accepted run never entered the sink, two-directional all-words projected-language equality, same-prefix timing, empty-word behavior even when the initial state is the sink, and composition with the existing T102 allocator and T101 compiler. The finite calibration retains a call that pushes 7, prunes local and false call commands in the same round, and checks epsilon plus both one-letter outcomes. The theorem does not construct a stationary allocator for the pruned frontier. Carrier finiteness, positive degree, reversal, and LMR remain external. This opens no M34 and does not edit the paused EPAL publication.