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Theorem

FPRD-T40

Finite future quotients reconstruct controlled-Dyck semantics

Exact statement

For every finite controlled-Dyck system and horizon NN, finite balanced-word saturation computes all reachable configurations through stack height NN, including those reached by arbitrarily long balanced detours, and bounded future signatures compute a finite quotient SNS_N preserving readout, bonding, and named degree-lowering letter actions. The family (SN)(S_N) is the minimum reachable reduced graded summary in its declared category. With the exact semantic quotient given the topology of agreement on bounded futures, its inverse limit is action- and readout-preserving homeomorphic to that quotient.

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

Context

A finite controller permits arbitrarily long balanced detours, but their controller effect is summarized by a finite saturated relation. This makes every bounded-height reachable carrier and every bounded-future quotient effectively finite.

Definitions

  • WD⊆Q×QW_D\subseteq Q\times Q is the least controller relation containing identity and neutral transitions and closed under composition and matched wrapping.
  • SN=CN/ker⁡BNS_N=C_N/\ker B_N, where BNB_N is the Boolean response table for all suffixes of length at most NN.
  • Exact semantic equivalence ∼∞\sim_\infty identifies reachable states agreeing on every finite continuation; write XM=RM/∼∞X_M=R_M/{\sim_\infty}.

Hypotheses and scope

  • The controlled-Dyck system has a finite total controller and finite visible alphabet.
  • Carriers and quotients contain only start-reachable states.
  • The universal minimum is taken among surjective graded summaries preserving readout, bonding, and named degree-lowering actions.

Proof or evidence

Finite saturation computes the effect of every balanced detour on the controller, and a stack recurrence computes the reachable controller states over each saved stack. Partition refinement by current output and successor signatures computes each S_N. Every valid summary forces a unique surjection onto S_N, so reduced summaries are isomorphic. Bounded-height threads either stabilize on a raw semantic class or escape to unbounded stack height; the latter has the all-false future and is already represented by absorbing failure. This supplies the surjectivity needed for the homeomorphism and makes the topology explicit.

Bounded quotients, universal property, and inverse limit →

Verification notes

The M24 proof boundary and new independent checker agree on global reachability, finite carriers, tail behavior, quotient computation, bonding, named degree-lowering actions, and the deep-stack/failure limit. The checker covers all 64 two-state one-bracket controllers with one neutral letter, 30,392 signature cells, 1,572 bonds, and 11,760 action comparisons. The exact-length layered theorem is kept separate from this at-most-length quotient tower.

Limitations

  • No single finite level is claimed to recover every infinite-horizon semantic distinction.
  • The minimum is relative to the declared reachable graded-summary category, not arbitrary machine presentations.
  • The theorem does not provide an asymptotic classification of quotient growth for every controller.

Open work

Add a compact computed example of the quotient tower and clarify how its topology compares with standard visibly-pushdown semantics.

Notes

Category admission requires reachability, surjectivity, and preservation of readout, bonding, and named actions. Redundant refinements are valid; reducedness characterizes the canonical minimum. No global VPA control-state minimum is asserted.