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Theorem and promise-complexity separation

FPRD-T85

Deterministic archives versus mismatch certificates

Exact statement

Let (K=n+3), (W_x=\#\#x\#), and Dn={WxWy:x,y∈{0,1}n}D_n=\{W_xW_y:x,y\in\{0,1\}^n\}. Relative to (D_n), (operatorname{SmallPer}_K) is payload equality and its complement is inequality. Equality needs at least (2^n) NFA states; either deterministic polarity needs at least (2^n) states; inequality has an NFA with (2n+2) states and every such NFA has at least the least (r) satisfying (inom r{\lfloor r/2\rfloor}\ge2^n).

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

Context

Changing acceptance polarity leaves deterministic storage exponential but turns equality into an existential mismatch certificate for nondeterministic recognition.

Hypotheses and scope

  • Recognition is relative to the promise D_n; behavior on malformed words is unrestricted.
  • The small-period/equality correspondence is supplied by the fresh-marker construction of FPRD-T84.

Proof or evidence

Equality has a (2^n)-element NFA fooling set. Deterministic machines must distinguish all first payloads. For inequality, an NFA guesses one coordinate; reachable state sets form a Sperner antichain, giving the matching linear order of growth.

Promise-state separation →

Verification notes

The promise matrix, exact coordinate delay, linear NFA, fooling set, deterministic rows, and Sperner lower bound were independently checked through n=6 and proved for arbitrary n.

Limitations

  • The linear NFA is a promise monitor, not an exact recognizer over all input words.
  • The result does not expose successful runs or inversions and gives no lower bound for the full transducer decision language.

Open work

Identify a certificate-cover measure for the full language where run, anchor, and mismatch obligations must be guessed together.

Notes

Recognition is promise-relative: behavior outside D_n is unrestricted. The linear inequality NFA is therefore a promise monitor, not an exact standalone recognizer for the framed language. This theorem explains why a deterministic contextual archive lower bound need not transfer to the existential bad-inversion polarity. It does not expose successful runs or all inversions, improve the published complexity bound, or solve Automata Exchange 22.08. The automata lower-bound machinery is standard; no external novelty claim is made.