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Theorem and saturation witness

FPRD-T83

One rooted crossing type realizes the full period quotient

Exact statement

For every finite nonempty output alphabet there is one fixed deterministic three-sweep transducer and one fixed legal idempotent crossing-effect/component type ((E,C)) whose legal same-type two-sided output-context equivalence is exactly the generic congruence defining (M_B), for every (B). The witness uses silent outer sweeps and a copying reverse sweep, so every output word and every context (x[-]y) occurs within that one rooted type.

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

Context

Fixing an ordinary crossing effect and one rooted component does not necessarily remove any generic output context.

Hypotheses and scope

  • The transducer has three deterministic sweeps; only the middle right-to-left sweep emits output.
  • The equivalence ranges over legal adjacent intervals of the fixed rooted type.

Proof or evidence

Every interval has the same three-loop effect. Reversing the spatial block realizes an arbitrary trace word, and three adjacent reversed blocks realize an arbitrary two-sided context. Thus typed and generic equivalence coincide.

Rooted-type saturation →

Verification notes

The crossing sequences, component direction, trace order, idempotence, context construction, and equality of the two congruences were independently checked and finitely instantiated.

Limitations

  • The long productive intervals are not output-minimal anchors.
  • The external period cap varies while the witness transducer remains fixed; this is not a transducer-size lower bound.

Open work

Replace bare rooted typing by a carrier that marks two output-minimal anchor occurrences and their legal intervening factor.

Notes

The saturating intervals need not be output-minimal. This does not prove saturation for the two output-minimal anchor occurrences required by a sweeping inversion, does not remove the successful-run/inversion quantifier, and does not solve Automata Exchange 22.08. The period bound is externally fixed while the witness transducer is fixed. No external novelty claim is made.