Top-layer product-cokernel sewing theorem

Exact statement

In a square-zero top layer, local lifts sew exactly when the target defect lies in the sum of their prefix/suffix-transported direction spaces; the residual class in the cokernel is the obstruction.

StatusProved

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 ordered product expansion gives the affine image and target-membership criterion directly.

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

The theorem assumes the declared square-zero layer and complete local direction spaces.

Open work

Use the cokernel before escalating to a higher quotient layer.