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Theorem · exact classification

FPRD-SH-T06

No starred-word shuffle has a left quotient

Exact statement

For all nonempty words x,yx,y, APre(x∗⨿y∗)A_{\mathrm{Pre}}(x^*\amalg y^*) is not a left-invariant quotient of APOS(x∗⨿y∗)A_{\mathrm{POS}}(x^*\amalg y^*).

StatusSelf-contained all-length proof with exact boundary and parameter audits
External reviewNo documented external or specialist review of these FPRD results is recorded.

Context

The ordinary quotient classification leaves only common-letter unary pairs; the question is whether a different partition can recover the stronger predecessor compatibility.

Hypotheses and scope

  • Both words are finite and nonempty.
  • Left invariance is tested by equality of predecessor-block profiles.

Proof or evidence

The initial state forces the first row and column locations into the target state q00. Left invariance then gives that source block at most two predecessor blocks, while target q00 has three unless (m,n)=(1,1). In the exceptional case the interior location must join q00 but lacks its initial predecessor.

Complete left-quotient obstruction →

Verification notes

An independent closed-form constructor checks all 4,096 pairs with 1≤m,n≤64. Direct labelled-map enumeration tests 69,975 maps in six boundary cases, finds 29 ordinary quotient maps, and finds no left-invariant map.

Limitations

  • The result is specific to starred-word shuffles and the cited APOS/APre constructions.
  • The primary sources establish the automaton constructions and the general 2023 shuffle failure, not this family classification; no novelty or priority claim is made.
  • No external review or priority claim is recorded.

Open work

Test whether an analogous predecessor obstruction classifies broader non-starred shuffle syntax.