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Theorem and dichotomy

FPRD-T73

Height dichotomy for unambiguous max-plus series

Exact statement

If an integer max-plus series has a trim unambiguous realization, then it is either sequential, with bounded finite-horizon state, raw-height, and canonical-height demand, or it is nonsequential, in which case both height demands grow as Θ(N)\Theta(N). No intermediate raw- or canonical-height rate occurs inside this unambiguous class.

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

Context

Klimann, Lombardy, Mairesse, and Prieur import the exact twins/sequentiality/Lipschitz equivalence for trim unambiguous max-plus automata. The FPRD deduction turns a violating common-access/common-loop pair into a finite-horizon height lower bound.

Hypotheses and scope

  • The carrier is the integer max-plus semiring.
  • A trim unambiguous realization exists.

Proof or evidence

Unambiguity makes each violating arm the unique accepting run of its witness word. Repeating unequal-weight circuits produces a value gap linear in N, while any bounded-height sequential approximant can create only a bounded gap across the fixed exit suffixes.

Unambiguous height dichotomy →

Verification notes

The proof obligations were reconciled with the source's precise twin definition rather than the former shorthand "co-reachable circuits." A fresh audit checks 63 unary linear-gap controls and 32,767 length-shift identities; see /research-artifacts/result-review-2026-09-01-batch-23/review_weighted_height_output.json.

Limitations

  • The classical twins/sequentiality/Lipschitz equivalence is imported and not claimed as new.
  • Unambiguity is load-bearing; the dichotomy does not extend to all finitely ambiguous series.

Open work

Compare the quantitative finite-horizon formulation with the determinization and degree-of-sequentiality literature before making any positioning claim.

Notes

Scope is trim unambiguous integer max-plus automata. The result does not extend to finite ambiguity in general. Its mechanism is close to classical twinning and delay arguments, and no novelty claim is made. Reviewer-only sharpened constants are not part of this claim.