FPRD LabShare an approach

← Claims

Theorem

FPRD-T52

Presentation-independent access certificates

Exact statement

Only controller states that actually begin balanced segments of legal prefixes are needed in the replacement proof. Their entry balanced diameter cannot increase under a surjective reachable DFA morphism. Minimizing this diameter over regular controllers with the same trace on a fixed Dyck language defines βD(L)\beta_D(L) and gives aL(N)≤max⁡{1,(N+1)βD(L)+N}a_L(N)\le\max\{1,(N+1)\beta_D(L)+N\}.

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

Context

A controller can preserve costly distinctions on malformed words that the Dyck intersection never observes. The invariant removes that presentation artifact while keeping the envelope fixed.

Definitions

  • EM={q0}∪{δ(p,o):p follows a legal Dyck prefix, o is an opener}E_M=\{q_0\}\cup\{\delta(p,o):p\text{ follows a legal Dyck prefix},\ o\text{ is an opener}\}.
  • βD(L)=min⁡D∩R=LBmin⁡(R)ent\beta_D(L)=\min_{D\cap R=L}B_{\min(R)}^{\mathrm{ent}}.

Hypotheses and scope

  • The generalized-Dyck alphabet is fixed.
  • L has at least one finite regular controller presentation on that envelope.

Proof or evidence

Legal-prefix saturation computes the exact entry sources. DFA quotients preserve their images and can only shorten balanced witnesses. The nonempty integer set of trace-equivalent certificate values has an attained minimum, to which the sharpened access bound applies.

Dyck-trace minimax →

Verification notes

The proof and independent challenge checked entry-source exactness, quotient images, trace equivalence, and access bounds, including mutations that omit the start entry or reverse monotonicity.

Limitations

  • The theorem gives limit approximation, not a general terminating algorithm for beta_D.
  • The invariant is relative to one Dyck envelope and is not claimed invariant under recoding.

Open work

Investigate computable subclasses of the trace-class minimization problem and dependence on the chosen Dyck envelope.

Notes

This claim is relative to one fixed Dyck envelope and does not assert effective global minimization or invariance under recoding.