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Uniform bounded-window theorem

FPRD-T108

Any fixed hidden-action window gives a conditional allocator tier

Exact statement

For any fixed natural number k, if the newest-first window of the last at most k successful hidden actions separates stacks at each observed word and target control, then pairing control with that window gives a finite control-functional refinement and a stationary scaffold. The theorem includes k=0 and epsilon.

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

Context

Rather than adding separate depth-two and depth-three claims, one theorem handles every fixed finite causal window.

Definitions

  • A chronological successful trace records exactly one hidden action per observed input round; its k-window retains the newest at most k actions.

Proof or evidence

Lean proves window update laws, original-to-refined and refined-to-original trace translations, exact window-frontier semantics, derived control functionality, and the compiler theorem. A depth-two example is separated although its last actions agree.

Bounded action-window refinement →

Verification notes

The zero-depth, empty-word, same-round, truncation-order, and supplied finite-enumeration boundaries were checked explicitly.

Limitations

  • The theorem does not say that some finite k works for every projected visibly pushdown language.
  • It neither computes a sufficient depth nor handles genuinely unbounded causal memory.

Open work

Find structural conditions that bound a sufficient action-window depth, or move to a different stationary coordinate system when no fixed depth works.

Notes

Chronological successful traces are defined on the original partial machine and record exactly one hidden action in the same input round. Their newest-first last-k window is bounded by construction; failed commands remain absent. Lean checks the update and append laws, k = 0, both original-to-refined and refined-to-original trace translations, the exact original traced WindowFrontier at every refined control, and the empty word. Thus the separation hypothesis is not circular. A two-round calibration merges call stack [7] and local stack [] at one final control after the same last local action, so it is formally not last-action separated; the preceding action makes the depth-two windows distinct, and the original traced frontier is proved two-window separated. The general compiler theorem retains a supplied enumeration of the full product carrier rather than pretending reachable-state finiteness is enough. This does not prove that some fixed finite depth works for every projected VPL, compute or bound a sufficient depth, synthesize the enumeration, or handle histories requiring unbounded memory or another stationary representation. External alphabet finiteness, positive scaffold degree, reversal, and LMR remain outside Lean. This opens no M34 and does not edit the paused EPAL publication.