Exact starred-word shuffle quotient dichotomy

Exact statement

For nonempty words $x,y$ , $A_{\mathrm{Pre}}(x^*\amalg y^*)$ is an existential state quotient of $A_{\mathrm{POS}}(x^*\amalg y^*)$ if and only if $x=a^m$ and $y=a^n$ for one letter $a$ and integers $m,n\ge1$ .

StatusSelf-contained proof with a closed-form quotient certificate and executable audit

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

Context

The preceding bounded census left exactly the common-letter unary family as possible ordinary quotients.

Hypotheses and scope

Both words are finite and nonempty.
The quotient uses existential block transitions.

Proof or evidence

Necessity: any mismatched position pair creates a grid location with two incoming labels. Sufficiency: an explicit projection sends the APOS boundary and interior grid onto the cyclic mn-state prefix core, with the initial and final states preserved.

Exact dichotomy and constructive proof →

Verification notes

Independent code checks all 4,923 ordered word pairs over {a,b,c} with total length at most six and verifies the quotient formula for all 1,024 pairs 1≤m,n≤32, including the unit boundaries.

Limitations

The theorem classifies ordinary existential quotients; FPRD-SH-T06 separately classifies left-invariant quotients.
Specialist review and literature comparison remain open; no priority claim is made.

Open work

Use FPRD-SH-T06 for the stronger left-invariant classification.