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Definition

FPRD-D25

Claim FPRD-D25

Exact statement

For a controlled-Dyck presentation M, P_M is the finite saturation of controller states reachable by legal Dyck prefixes, E_M consists of the start state and the states immediately after an opener from P_M, B_M^ent is the maximum finite balanced distance sourced in E_M, regular languages are Dyck-trace equivalent when their intersections with the fixed generalized-Dyck language agree, and beta_D(L) is the minimum entry certificate over minimal regular controllers in that trace class. The balanced residual access gamma_D(L) is the maximum, over semantic right-residual classes restricted to balanced histories and tests, of the shortest balanced representative length. Entry-context access eta_D(L) takes the supremum of the analogous factor-residual access over the start and all legal histories ending immediately after an opener; it is exactly computable by finite completion-summary saturation in any controller presentation.

Statusdefinition
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

This row is imported exactly from the governed FPRD claims ledger. The matrix assignment records normalized inventory coverage; it is not a new proof or a promotion of the ledger status.

Proof or evidence

The ledger points to docs/specs/m30-dyck-trace-minimax.md. The matrix has not yet rewritten that source into a self-contained public proof summary.

docs/specs/m30-dyck-trace-minimax.md →

Verification notes

Reviewed on 2026-08-25 for identifier, status, provenance, evidence-link presence, and dependency import only. Mathematical proof audit remains the next action unless separately documented.

Limitations

  • beta_D is relative to the fixed Dyck alphabet and envelope. Its definition is an attained minimum of integers, not an asserted algorithm for searching all regular presentations.

Open work

Add a self-contained definition page with examples, nonexamples, and dependency boundaries.

Notes

beta_D is relative to the fixed Dyck alphabet and envelope. Its definition is an attained minimum of integers, not an asserted algorithm for searching all regular presentations.