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Theorem · exact resource bound

FPRD-T139

Linear-bit demand for exact seam profiles

Exact statement

For every ℓ≥3\ell\ge3, the prefixes us=1su_s=1^s induce 2ℓ−22^{\ell-2} distinct exact horizon-ℓ\ell seam maps. Thus an interface naming the complete map on every suffix guard needs at least ℓ−2\ell-2 bits, while slope and intercept use O(ℓ)O(\ell) bits; exact seam-profile demand is Θ(ℓ)\Theta(\ell).

StatusProved for the explicitly defined exact-profile interface; internally audited
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The lower bound counts distinct complete seam functions, not accepted traces. Recognition may merge prefixes whose full routing actions differ.

Definitions

  • For us=1su_s=1^s, Fus(ℓ)(q)=3−s(q+1)−1(mod2ℓ)F_{u_s}^{(\ell)}(q)=3^{-s}(q+1)-1\pmod{2^\ell}.
  • Two exact-profile states may merge only when these functions agree on every suffix guard.

Hypotheses and scope

  • ℓ≥3\ell\ge3.
  • The interface exposes the complete horizon map on hypothetical suffix guards; it is not merely an accept/reject state.

Proof or evidence

The multiplicative order ord⁡2ℓ(3)=2ℓ−2\operatorname{ord}_{2^\ell}(3)=2^{\ell-2} yields the distinct slopes. The guide proves that order by the self-contained induction v2(32m−1)=m+2v_2(3^{2^m}-1)=m+2. Recording slope and intercept modulo 2ℓ2^\ell gives the matching linear-bit upper bound.

Exact seam-profile demand proof →

Verification notes

This run checked the valuation induction, the state-count-to-bit conversion, and the trace-budget caveat. The source correctly limits the displayed family to horizons satisfying the stated exponential prefix-length budget.

Limitations

  • This is not a lower bound for paradoxical-trace recognition, finite automata for the accepted language, or Collatz computation.
  • Within a trace budget KK, this family alone yields only ℓ≤⌊log⁡2(K+1)⌋+2\ell\le\lfloor\log_2(K+1)\rfloor+2.

Open work

Keep this exact-interface bound separate from recognition lower bounds; seek a closer comparator for the profile-count formulation.

Notes

Distinctness uses ord_(2^ell)(3)=2^(ell-2), proved in the guide by a self-contained 2-adic valuation induction. This is a lower bound for exact routing profiles, not for paradoxical-trace recognition, finite automata for the accepted language, or general Collatz computation. Acceptance may merge prefixes whose complete seam actions differ. The displayed calibration uses prefix lengths up to 2^(ell-2)-1, so within a trace budget K it yields only ell<=floor(log_2(K+1))+2; no linear-in-K memory lower bound is claimed. This budgets the prefix only because the interface acts on hypothetical suffix guards; a convention requiring the suffix to occur inside the same trace would also charge ell symbols.