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Theorem and exact lower bound

FPRD-T80

A max-plus series of exact ambiguity degree three

Exact statement

The integer max-plus series f△(w)=max⁡{−3∣w∣a,−∣w∣,−3∣w∣b}f_{\triangle}(w)=\max\{-3|w|_a,-|w|,-3|w|_b\} has intrinsic ambiguity degree exactly three.

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

Context

On words a^x b^y, the series has three open cones with gradients (-3,0), (-1,-1), and (0,-3).

Proof or evidence

A two-ambiguous automaton would yield at most two affine forms on each eventual product cell, but two affine zero sets cannot cover a full-dimensional lattice subset in all three cones. Three one-state components give the matching upper bound.

Three-gradient example →

Verification notes

The review checked all 131,071 binary words through length 16, excluded 31,878 pairs of small affine forms on an 11-by-11 hostile grid, and rechecked every residue-cell/cone rank. The proof remains the cut-state and affine-cover argument, not finite testing.

Limitations

  • The result establishes one exact intrinsic ambiguity value; it is not a classification theorem.
  • No novelty or external-review claim is made.

Open work

Use the gradient criterion on less symmetric examples and compare the resulting lower bounds with existing ambiguity hierarchies.

Notes

The proof restricts to a*b*, factors at the block boundary, and uses three persistent gradients. The earlier path-scheme proof is withdrawn. The primary unary source is identified but not held in repository custody. Novelty is not assessed. The separate all-k baseline-robustness theorem is not promoted because its full proof packet is absent from the public repository.