Atom stabilizer and tuple stabilizer are different

Exact statement

The colored atom stabilizer generates local atom lifts; the braid stabilizer of a complete quotient tuple governs path equivalence and must not be quotient out again in product existence.

StatusProved distinction

External reviewNo documented external or specialist review of this Quotient Arsenal result is recorded.

Context

Finite quotient words do not lift one factor at a time independently. Local affine lift fibres must be transported through the ordered product; the target defect then lives in a product-image cokernel that is invariant under quotient Hurwitz moves.

Proof or evidence

The fibre construction and Hurwitz action use the two groups in different domains.

Hurwitz fibre theorem core and independent audit →

Verification notes

The ordered-product proof was reconstructed with its square-zero and mod-eight domains explicit. An independent implementation checked 69,632 affine/quadratic matrix identities and 1,625 relative Nielsen and split-central identities, including an order-reversal hostile control.

Limitations

No novelty or specialist-review claim is made; the statement is limited to the displayed presentation and hypotheses.

Open work

Name both stabilizers explicitly in future quotient-path theorems.