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Theorem and robust separation

FPRD-T78

Canonical excess surviving every fixed-state correction

Exact statement

There is a fixed partial integer target on the regular support b+ca∗#Code Codeb^+ca^*\#\mathrm{Code}\,\mathrm{Code} for which hf1(N)≤1h_f^1(N)\le1 and Hf1(N)=Θ(N)H_f^1(N)=\Theta(N). For every fixed k≥1k\ge1, let Rf,khR^h_{f,k} and Rf,kHR^H_{f,k} take separate infima over all horizon-dependent kk-state deterministic sequential baselines with unrestricted integer weights and all exact residual presentations. Then Rf,kh(N)≤1R^h_{f,k}(N)\le1 and Rf,kH(N)=Θk(N)R^H_{f,k}(N)=\Theta_k(N).

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

Context

The value is floor(m/(i+1)) plus one exactly when the two binary-coded challenge parameters equal the unary address (i,m). Distinct slopes supply linear residual values; the equality bit prevents a bounded-state correction from enabling enough access-word collisions.

Hypotheses and scope

  • The correction is deterministic, sequential, defined on the target support, and has at most k states; its weights and topology may depend on the horizon.
  • The target itself is finitely described and has regular support, but is not max-plus rational at any ambiguity level.
  • Residual raw and canonical demand are independent infima; one baseline is not asserted to optimize both coordinates simultaneously.

Proof or evidence

Pigeonhole collisions in the baseline expose two unequal slopes. Independent challenge rows restrict possible presentation merges to O(k²) prefixes, leaving a large residual value on a forced BFS access word.

Fixed-state correction theorem →

Verification notes

The review reconstructed the asymptotic proof and added 32,832 raw-accumulation checks, 480 independent baselines, 92,640 shared-continuation identities, 46,656 exhaustive challenge configurations, and 11,690 cycle-mean checks. The archived 750-baseline audit remains supporting evidence.

Limitations

  • Presentation states are unbounded; the theorem trades baseline states against residual canonical height.
  • The equality challenge is deliberately engineered, and the target lies outside the max-plus rational class.
  • The zero baseline proves the optimized raw upper bound; the universal baseline quantifier proves the canonical lower bound.

Open work

Settle whether bounded raw height can coexist with unbounded corrected canonical height inside the finitely ambiguous rational class.

Notes

The target is not max-plus rational at any ambiguity level. The low-raw-height horizon machine uses Omega(N^2) states, so presentation states are free and the charged trade is baseline states against residual canonical height. The equality gadget is deliberately self-addressing. Novelty is not assessed.