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Theorem and explicit separation

FPRD-T72

A fixed repaired canonical-height gap

Exact statement

There is one partial integer-valued target over {a,b,c,d}\{a,b,c,d\} and, for every k≥3k\ge3, a horizon Nk=k+1+2kN_k=k+1+2^k whose exact minimum state budget is nk=k+3⋅2k−1+2n_k=k+3\cdot2^{k-1}+2. Even after minimizing over every global length-and-constant shift, repaired raw height is one while repaired canonical height is 3⋅2k−2=(1/2−o(1))nk3\cdot2^{k-2}=(1/2-o(1))n_k.

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

Context

Two countdown carriers have incompatible slopes. A global length shift can cancel one carrier but necessarily amplifies or preserves the other.

Hypotheses and scope

  • The target is undefined away from its displayed sparse support and is integer-valued on one fixed four-letter alphabet.
  • The theorem family is stated for k≥3, when both alternating carrier families and the dominant boundary term are present.
  • Both heights are minimized over all correct presentations at the exact minimum state budget and over all integer length-and-constant shifts.

Proof or evidence

Finite-horizon residuals force the minimum-state comb graph. Its canonical access tree forces boundary coordinates of alternating sign and magnitude 3·2^(j-2), while a ±1 raw weighting realizes the target.

Parity-comb separation →

Verification notes

Two earlier reconstructions and the repository reconciliation agree. A fresh dependency-free audit checks 4,192 live-prefix signatures and 200 integer gauges for k=3,...,10; see /research-artifacts/result-review-2026-09-01-batch-23/review_weighted_height_output.json.

Limitations

  • No novelty, total-target, rational-series, or full Pareto-frontier claim is made.
  • An earlier one-carrier witness collapses under the allowed length shift and is retained only as a negative control.

Open work

Determine whether the incompatible-carrier mechanism persists for a natural total rational series or under richer admissible baselines.

Notes

Produced by the review lane and independently reconstructed twice there; captured with attribution and reconciled here against this repository's normalization. Establishes that the global length-and-constant repair does NOT generally collapse canonical height to raw height. Mechanism: the target forces two countdown carriers of incompatible slope, +1 and -1, and a single global length drift adds the same coefficient to both, so no integer choice cancels both at once. The earlier one-carrier sparse comb is retained as a NEGATIVE CONTROL and is not a repaired witness: its target equals word length on its support, so lambda = -1 collapses both repaired heights to zero. Reconciliation clarified which edges carry the boundary coordinates: every c-edge lies in the BFS tree, and the boundary edge for block j is the chain edge from index 2^(j-2)+1 to 2^(j-2). NOT merged with richer baseline quotients or mode-dependent gauges. Novelty not assessed; the target is sparse and is not shown rational or finitely ambiguous.