Synchronized unary threshold audit

Exact statement

Exact construction checks 576 strong and 576 arbitrary synchronized pairs for $1\le m,n\le24$ ; exhaustive search over 10,534 compatible partitions independently verifies the ordinary- and left-quotient boundary in the six smallest arbitrary-synchronization instances.

StatusExact structural and exhaustive boundary checks pass

External reviewNo documented external or specialist review of these FPRD results is recorded.

Context

The computation audits the symbolic orbit and quotient formulas and independently exhausts the smallest candidate partitions.

Proof or evidence

Every strong pair is isomorphic on its accessible part. Arbitrary synchronization has an ordinary quotient exactly when one exponent is 1 in the parameter sweep and a left quotient only at (1,1); exhaustive search agrees in every boundary case.

Synchronized threshold audit →

Verification notes

The six exhaustive instances are (1,1), (1,2), (2,1), (1,3), (3,1), and (2,2). A separate constructor reproduces all 576 strong and 576 arbitrary parameter cases and checks their bounded word semantics against an independent oracle.

Limitations

Finite checks audit the implementation; the all-parameter conclusions come from FPRD-SH-T07 and FPRD-SH-T08.

Open work

Extend the exact checker to a non-unary synchronized family only after its prefix receipt is stated precisely.