Bounded census of starred-word shuffles

Exact statement

Among all 306 expressions $x^*\amalg y^*$ over $\{a,b,c\}$ with $|x|+|y|\le4$ , 288 have no ordinary prefix quotient; the 18 ordinary-quotient cases are all unary same-letter controls, and none has a left-invariant quotient.

StatusExhaustive within the stated finite domain; its pattern is now proved by FPRD-SH-T05

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

Context

The census tests how frequently the quotient failure occurs and identifies a narrow family of ordinary-quotient controls.

Hypotheses and scope

The alphabet is $\{a,b,c\}$ , both words are nonempty, and $|x|+|y|\le4$ .

Proof or evidence

The checker enumerates all words in the domain, constructs both automata, and exhausts compatible partitions for ordinary and left quotients.

Census table and scope →

Verification notes

The totals split as 9, 54, and 243 cases at total word lengths 2, 3, and 4; the corresponding no-quotient counts are 6, 48, and 234.

Limitations

A finite census is evidence, not a general theorem; the general proof is recorded separately as FPRD-SH-T05.

Open work

Retain this census as the discovery record and finite cross-check for FPRD-SH-T05.