A global plateau characterizes regularity

Exact statement

A language is regular exactly when the global quotient counts $q_L(N)=|Q_N(L)|$ are bounded, equivalently when one adjacent pair satisfies $q_L(N)=q_L(N+1)$ . At the first global plateau, the shifted letter actions descend to same-level actions and the quotient is the unique minimal residual DFA. Uniformly small individual fixed-length cuts do not suffice: nonregular length-only languages can have one row and zero one-way bits at every fixed cut.

StatusProved and internally audited

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

Context

The horizon quotients grow by surjective refinement. Once two adjacent global levels have equal finite size, the bond is bijective and no later observation can split a class.

Definitions

$q_L(N)$ counts residual behaviors on all future words of length at most $N$ , over all prefixes.
A global plateau uses the complete residual orbit, not a finite prefix sample and not one fixed-length cut.

Hypotheses and scope

The quotient counts are global over every prefix.
The minimal-DFA conclusion imports the classical residual theorem recorded as FPRD-T22.

Proof or evidence

An equal-size adjacent bond between finite quotients is bijective. Its kernel becomes a finite-index right congruence, so the shifted letter actions descend and the quotient recognizes the language. Conversely, a finite residual orbit bounds all horizons. The square-length language separates small nonuniform cut width from uniform recognizability.

Global plateau theorem →

Verification notes

The M26 audit checks plateau persistence, same-level action descent, unique factorization, conditional finite-window reconstruction, and the fixed-slice counterexample. It explicitly rejects false finite-prefix plateaus.

Limitations

A plateau observed only among prefixes up to a finite budget may split later.
Uniform bounds on separate exact cuts do not bound the number of global residual states.
Finite-window reconstruction is exact only when an independent residual-state bound is supplied.

Open work

Relate the global quotient sequence to established state-complexity and follower-set invariants without transferring incompatible constants.

Notes

The plateau is global over all prefixes. A finite-prefix plateau is not a regularity certificate. The square-length family supplies the direct fixed-slice versus uniform-recognition separation.