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Theorem and exact calibration

FPRD-T62

Exact finite-horizon cost for the max-count series

Exact statement

For F(w)=max⁡(∣w∣a,∣w∣b)F(w)=\max(|w|_a,|w|_b), two one-state integer max-plus sequential components are exact, while no single finite sequential component is exact. Through horizon NN, the minimum width-one fit has 2⌊N/2⌋+12\lfloor N/2\rfloor+1 states and minimum canonical height ⌊N/2⌋\lfloor N/2\rfloor. Hence width-one state and canonical-height demand grow linearly, whereas an exact width-two cover has constant resource (2,2,1)(2,2,1).

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

Context

The example shows a sharp resource escape when cover width is fixed at one, together with a constant exact presentation when width two is allowed.

Definitions

  • Canonical height is measured after trim, access-push normalization, and initial-weight absorption; transition and final weights both count.
  • The clamped signed-count tracker at height h has 2h+12h+1 states and agrees through horizon 2h+12h+1.

Hypotheses and scope

  • Sequential components are finite integer max-plus machines.
  • The fine resource records cover width, total states, and canonical height.

Proof or evidence

A suffix-difference fooling argument gives the state lower bound. Gauge-normalized path differences give the canonical-height lower bound. The clamped tracker attains both bounds, and the two coordinate counters give the exact width-two cover.

Max-count calibration →

Verification notes

The proof, normalization theorem, calibration note, source comparison, and independent reconstruction were reconciled. The earlier provisional raw-height coordinate was replaced by the proved canonical-height bound.

Limitations

  • Cover width is not automaton ambiguity; no tropical 2-ambiguisability claim follows.
  • The fixed-width fine resources escape, but this alone is not a fixed-resource sewing defect.
  • The no-width-one baseline uses Klimann–Lombardy–Mairesse–Prieur (2004); the finite-horizon formulas are FPRD proofs.

Open work

Compare the finite-horizon state and canonical-height frontiers with standard weighted-automata approximation and sequentiality measures.

Notes

The exact two-cover and no-width-one baseline is imported from the direct max-count example in Klimann--Lombardy--Mairesse--Prieur (2004, Section 3.2, printed page 7), which proves the series is not unambiguous. The finite-horizon formulas were proved and corrected at Task 8 under the Task 6 gauge-canonical normalization theorem (docs/notes/m33-tropical-normalization.md): the calibration note docs/notes/m33-max-count-calibration.md establishes the exact state minimum, the tight canonical-height bound, and the corrected display replacing the spec's provisional (1, Theta(N), 1). Earlier language in this entry calling these "opening theorem targets" with a "provisional" height coordinate was superseded by that proof; statement text reconciled 2026-07-27 (validation-sprint claim table, row R4) with no change of status. Evidence class includes an isolated internal AI review (not external human review). This concerns finite sequential covers, not k-ambiguous weighted automata or tropical 2-ambiguisability. Raw state-count escape cannot by itself earn M34.