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Theorem and exact counterexample

FPRD-T74

Min-count has constant height and linear state demand

Exact statement

For fmin⁡(w)=−min⁡(∣w∣a,∣w∣b)f_{\min}(w)=-\min(|w|_a,|w|_b), intrinsic max-plus ambiguity degree and exact sequential cover width are both two. For every N≥2N\ge2, the minimum width-one state demand is 2⌊N/2⌋+12\lfloor N/2\rfloor+1, while both minimum raw height and minimum canonical height equal one. Thus the unambiguous height dichotomy cannot extend to ambiguity degree two.

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

Context

Min-count is a sharp boundary example: state demand grows linearly, yet an optimal clamped difference tracker keeps canonical height constant.

Definitions

  • fmin⁡=max⁡(−∣w∣a,−∣w∣b)f_{\min}=\max(-|w|_a,-|w|_b), so two one-state sequential components give an exact width-two cover.

Proof or evidence

Finite-horizon residual differences force one state for each signed count in [−floor(N/2),floor(N/2)]. The clamped tracker attains that state count and is already in canonical height-one form. At N=1 the target is identically zero, so one state of height zero suffices and the N≥2 hypothesis is sharp.

Min-count counterexample →

Verification notes

A fresh exhaustive audit checks all 65,515 binary words through horizons 2,...,14, 511 pairwise state lower-bound witnesses, and the BFS normalization; see /research-artifacts/result-review-2026-09-01-batch-23/review_weighted_height_output.json.

Limitations

  • The result is a counterexample to extending T73 beyond unambiguous realizations, not a classification of finitely ambiguous series.
  • The relation to max-count is an additive length shift, not a negation duality.

Open work

Seek a rational series with a genuinely sloped state-versus-canonical-height frontier rather than another flat corner example.

Notes

The relation to max-count is addition of a length gauge, not negation duality. All three pinned demand coordinates are invariant under negation; the withdrawn orientation-sensitive account is not revived.