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Theorem · canonical residual completion and stabilization criterion

FPRD-T156

Residual towers complete the language and detect regularity

Exact statement

The inverse limit of the finite-horizon residual tower is a compact totally disconnected deterministic response machine with continuous input transitions. The full Myhill--Nerode residual space embeds equivariantly and densely. A language is regular exactly when the tower stabilizes, equivalently when its quotient sizes are bounded or one restriction map Q_(h+1) to Q_h is bijective.

StatusSelf-contained proof with exhaustive finite audit
External reviewThe theorem is aligned with established profinite and topological recognition theory; no external review of this FPRD formulation is recorded.

Context

FPRD-T155 isolates the static quotient memory at one cut. T156 supplies the canonical dynamics of those quotients across all future horizons.

Hypotheses and scope

  • The input alphabet is finite.
  • Responses are exact Boolean language membership values.
  • Each finite quotient carries the discrete topology and the inverse limit carries the limit topology.

Proof or evidence

Restriction and derivative maps commute, so input letters act continuously on the inverse limit. Prefix residuals are dense because every finite coordinate is realized. Stabilization is equivalent to a finite-index right congruence. The audit covers 32,768 finite language trees, 5,898 binary DFAs, 2,195,456 transport identities, and twelve ideal-residual witnesses.

Residual completion theorem →

Verification notes

The proof was checked for surjectivity of restrictions, derivative well-definedness, arbitrary-word transport, density without assumed surjectivity, ideal completion points, and exact DFA partition stabilization.

Limitations

  • The compact completion can add coherent ideal residuals that no finite prefix realizes.
  • The theorem supplies canonical semantic transport, not an encoding-independent numeric cost for physically realizing it.
  • Profinite and topological recognition are established theories; the claimed contribution is the explicit response-horizon organization for FPRD.
  • No independent specialist review is recorded.

Open work

Use the stationary framework D45/T157 and later compilation results; the remaining frontier is structural lower bounds, spectrum emptiness, or explicit compilation for consequential presentations.