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Theorem and exact state count

FPRD-T76

Exact two-arm state demand and height windows

Exact statement

For the exact partial domain a∗#∪a∗$a^*\#\cup a^*\$, with f(am#)=xmf(a^m\#)=x^m and f(am$)=ymf(a^m\$)=y^m, set dm=x−mymd_m=x^{-m}y^m, D(N)=max⁡m<N∣dm∣D(N)=\max_{m<N}|d_m|, and E(N)=max⁡m<Nmax⁡{∣xm∣,∣ym∣}E(N)=\max_{m<N}\max\{|x^m|,|y^m|\}. For N≥1N\ge1, if d0,…,dN−1d_0,\ldots,d_{N-1} are distinct, then Af1(N)=N+1A_f^1(N)=N+1, D(N)/4≤hf1(N)≤max⁡{min⁡(∣x∣,∣y∣),D(N)}D(N)/4\le h_f^1(N)\le\max\{\min(|x|,|y|),D(N)\}, and E(N)/2≤Hf1(N)≤E(N)E(N)/2\le H_f^1(N)\le E(N). The raw-height window does not require delay distinctness.

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

Context

The family isolates two geometries: branch delay controls raw height, whereas absolute power growth controls pinned canonical height. A fixed three-state unambiguous machine realizes the target, so the growing state formula is specifically deterministic.

Hypotheses and scope

  • Correctness includes agreement on undefined words through the horizon.
  • Delay distinctness is required for the exact state count and canonical lower bound, but not for the raw-height bounds.

Proof or evidence

Continuation ratios force distinct spine states. A path machine stores the common x-drift on the spine and the delay on one exit, while canonicalization pushes each access value onto the exits.

Two-arm transfer theorem →

Verification notes

The review checked 784 integer parameter cases, 5,712 distinct-delay indices, 136 exact canonical split indices, 112 equal-arm hostile controls, and 93 exact Heisenberg word-norm cases.

Limitations

  • The theorem concerns one two-arm family and deterministic finite-horizon realization only.
  • Its canonical-height bound is generally a factor-two window, not an exact formula; x=y is a hostile boundary where a two-state sequential machine suffices.

Open work

Complete the targeted comparison with quantitative delay, twinning, and cyclic-subgroup-distortion results.

Notes

These are pinned width-one deterministic finite-horizon demands. Over H_3(Z), x=X and y=XZ give state demand N+1, raw height Theta(sqrt N), and canonical height Theta(N). That Heisenberg corollary collapses under a global length gauge and is not itself a gauge-reduced witness.