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Theorem · elementary quotient invariant

FPRD-SH-T01

Loop labels survive existential quotients

Exact statement

For every existential state quotient π:A↠B\pi:A\twoheadrightarrow B, ΛA(q)⊆ΛB(π(q))\Lambda_A(q)\subseteq\Lambda_B(\pi(q)) for every state qq.

StatusSelf-contained proof
External reviewNo documented external or specialist review of these FPRD results is recorded.

Context

A state quotient can merge transitions but cannot delete a self-loop already carried by one member of a block.

Hypotheses and scope

  • The quotient uses existential block transitions.

Proof or evidence

If q has a σ-loop, the source and target of that edge both lie in π(q), so the quotient definition creates the same labelled loop on π(q).

Loop-spectrum monotonicity proof →

Verification notes

The one-line proof and the quotient convention were checked against every computational benchmark.

Limitations

  • The statement does not cover quotient notions that are allowed to delete transitions.
  • No novelty is claimed for loop preservation itself.

Open work

Use the invariant as a constant-time rejection test before exhaustive quotient search.