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Theorem and sharp bound

FPRD-T70

State-conditioned raw/canonical height coupling

Exact statement

For a trim deterministic sequential integer-weighted machine with mm states, raw height hh, pinned BFS access tree TT, and canonical height HH, one has H≤max⁡{D+2,max⁡e∉TℓT(e)}h≤(m+1)hH\le\max\{D+2,\max_{e\notin T}\ell_T(e)\}h\le(m+1)h. After minimizing within a fixed state budget nn, hf,≤n≤Hf,≤n≤(n+1)hf,≤nh_{f,\le n}\le H_{f,\le n}\le(n+1)h_{f,\le n}, and the coefficient n+1n+1 is attained.

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

Context

Canonicalization can accumulate many raw edge weights on one non-tree edge or final coordinate. The theorem bounds that accumulation by the access-tree geometry and shows the coarse state-dependent factor is sharp.

Definitions

  • DD is the depth of the pinned BFS access tree.
  • ℓT(e)\ell_T(e) is the length of the fundamental undirected tree cycle closed by a non-tree edge ee.

Hypotheses and scope

  • The machine is trim, deterministic, sequential, and integer-weighted.
  • Canonical height includes transition and final weights after initial absorption.

Proof or evidence

Tree access weights contain at most D raw edge weights; a non-tree canonical edge is a signed sum around its fundamental cycle. A unary spike family forces n+1 states and canonical final value n+1 while raw height remains one.

Height coupling →

Verification notes

The proof was independently reconstructed against the repository normalization. The current audit checks the sharp family through 30 states and the one-state pointwise hostile example; see /research-artifacts/result-review-2026-09-01-batch-22/review_prequery_height_output.json.

Limitations

  • The inequality h≤H is not pointwise for a fixed machine; it holds after minimization because the canonical form competes in the same state-bounded section.
  • The theorem is a coupling envelope, not a conservation law or novelty claim.

Open work

Compare the coupling envelope with standard weight-pushing bounds and isolate useful graph classes with a smaller coefficient.

Notes

Produced by the review lane; captured with attribution and independently verified here against the repository's own normalization rather than the statement. Elementary weight pushing; novelty NOT assessed. The first inequality is NOT pointwise - a fixed machine can have canonical height below raw height - and holds only after minimizing, because a canonical form is itself a competing presentation in the same section. Recorded as an optimal worst-case coupling envelope inside one state-budget section, not as a conservation law and not as evidence that canonical height lacks independent content.