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Theorem and algorithm

FPRD-T50

Exact access calculus and a linear bound

Exact statement

For a finite controller (M) intersected with a generalized-Dyck language, shortest balanced-controller distances determine exact raw-coordinate arrivals. Minimizing those arrivals inside each bounded semantic row and then maximizing computes (a_{L_M}(N)). If (B_M) is the largest finite balanced distance from an underlying-DFA-reachable source, then aLM(N)≤max⁡{1,(N+1)BM+N}a_{L_M}(N)\le\max\{1,(N+1)B_M+N\}; the class-wide bound is pointwise optimal.

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

Context

Finite control rules out the arbitrary access delays seen in the sparse-marker calibration. The proof makes the constraint effective rather than merely asymptotic.

Definitions

  • dM(p,q)d_M(p,q) is the shortest length of a balanced word carrying controller state (p) to (q).
  • ρM(s,q)\rho_M(s,q) is the shortest prefix reaching raw stack/controller coordinate ((s,q)).
  • A semantic row receives the minimum arrival among all raw coordinates that induce it.

Hypotheses and scope

  • The generalized-Dyck alphabet has at least one matched bracket pair.
  • The controller is finite and total; the Dyck envelope and presentation are fixed.

Proof or evidence

The last unmatched opener gives an exact min-plus recurrence for raw arrivals. A height-h prefix has h+1 balanced segments; replacing each by a shortest witness yields the linear bound. A controller family attains equality at each chosen positive horizon.

Controlled-Dyck access calculus →

Verification notes

The proof was checked against production and independent implementations, 404,061 literal membership cells, mutation tests, and a separate sharpness construction. Review repaired the least-versus-greatest wording for the balanced-distance inequalities and sharpened the source set.

Limitations

  • The coefficient belongs to the chosen presentation, not intrinsically to the language.
  • The presentation attaining equality may depend on N.
  • The result does not cover arbitrary visibly-pushdown, context-free, or PEG presentations.

Open work

Compare the presentation certificate with established visibly-pushdown and weighted shortest-witness formulations.

Notes

The linear coefficient is presentation-dependent. The sharpness presentation may depend on N; no one fixed presentation is claimed to attain equality at every horizon. The claim neither minimizes over equivalent controllers nor extends to arbitrary visibly pushdown, context-free, or PEG presentations.