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Counterexample and limitation theorem

FPRD-T109

No fixed action window is a universal coordinate doctrine

Exact statement

There is a one-letter, one-control partial deterministic visible machine whose forgotten-action projection is the universal unary language, yet for every fixed finite action-window depth two successful traces over the same word and control have the same retained window and different stacks, [7] and [].

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

Context

One trace calls once and then performs k local actions; another performs k+1 local actions. Their differing first action lies just beyond the retained suffix.

Proof or evidence

Lean constructs both traces for arbitrary k, proves equal windows including k=0, places the distinct stacks in one window-frontier fibre, derives failure of separation for every k, and separately proves universal-language acceptance.

Action-window escape →

Verification notes

The counterexample was checked against the original traced semantics rather than inferred from the refined machine. Its language is intentionally regular, preventing the coordinate failure from being misread as a PEG or scaffold lower bound.

Limitations

  • This refutes fixed action-window coordinates, not stationary allocation in general.
  • It supplies no language-class separation and does not imply that any context-free language lies outside PEG.

Open work

Search for another stationary representation that succeeds on the escape machine and separates representation failure from language complexity.

Notes

For arbitrary k, including zero, Lean constructs an exact call-then-k-locals trace and an exact all-locals trace on the same word, proves that their newest-first last-k action windows agree, places stacks [7] and [] in the same WindowFrontier fibre, and derives failure of WindowSeparated. A local-only trace proves all-word acceptance, including epsilon, so the projected language is universal and regular. This is a counterexample to one coordinate doctrine, not a PEG or scaffold lower bound: another stationary coordinate system may succeed. It does not prove CFL is outside PEL, opens no M34, and does not edit the paused EPAL publication.