Factorizations as relative Nielsen slices

Exact statement

Appending the inverse target identifies a fixed-target factorization space with a relative product-one tuple slice, equivariantly for Hurwitz moves and quotient maps.

StatusProved and independently checked

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

All 125 nonidentity length-three tuples in $S_3$ pass the comparison identities.

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

Classical Nielsen classes commonly impose conjugacy-class and generation conditions in addition to product one; this FPRD slice asserts only the displayed relative tuple conditions.
The finite audit checks the comparison, not the B5 application.

Open work

Use relative Nielsen stabilizers to organize complete lift fibres.