FPRD LabShare an approach

← Claims

Theorem · arbitrary-arity quotient transfer

FPRD-SH-T11

Boolean products preserve componentwise left quotients

Exact statement

Let ∥ψ\Vert_\psi be an arbitrary-arity memoryless Boolean product of NFAs, and let EiE_i be a left-invariant equivalence on component AiA_i. The coordinatewise product equivalence E=∏iEiE=\prod_iE_i is left-invariant on ∥ψ(A1,…,An)\Vert_\psi(A_1,\ldots,A_n), and ∥ψ(A1,…,An)/E≅∥ψ(A1/E1,…,An/En)\Vert_\psi(A_1,\ldots,A_n)/E\cong\Vert_\psi(A_1/E_1,\ldots,A_n/E_n).

StatusSelf-contained predecessor proof with strengthened independent boundary audit
External reviewNo documented external or specialist review of these FPRD results is recorded.

Context

The 2026 Boolean-product framework permits arbitrary arity and a separate allowed participant-set policy for every letter. Its public abstract emphasizes right-invariant reductions; reversal and a direct predecessor proof supply the left-handed transfer used here.

Hypotheses and scope

  • Event support is memoryless: admissibility depends only on the current letter and participant set.
  • Each component equivalence is left-invariant, including saturation of its initial set.

Proof or evidence

For one fixed participant set, advancing coordinates use the component predecessor-profile equalities; stuttering coordinates remain inside the same component blocks. This proves product left invariance. The same support witness proves that quotienting commutes exactly with the product construction.

Boolean-product transfer theorem →

Verification notes

The source checker covers 8,192 transition/policy/partition cases. An independent implementation reconfirms all 5,408 predecessor-compatible cases and additionally checks 56,448 saturated initial-set pairs and 86,528 final-set pairs.

Limitations

  • This is likely a classical reversal consequence or framework synthesis; no novelty or priority claim is made.
  • It concerns the product of component automata, not automatically the natural prefix automaton of a Boolean-product expression.

Open work

Compare the canonical product of component prefix quotients with the natural prefix automaton of one Boolean-product expression.