Two-layer sewing is affine then quadratic

Exact statement

For corrections $1+2x_i+4y_i$ , a fixed first-layer word lifts by affine equations in $y_i$ , while variation across the complete $x_i$ -solution fibre is quadratic and order-sensitive.

StatusProved and algebraically replayed

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

69,632 exhaustive matrix-algebra checks verify the expansion and its ordered cross terms.

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 formula is pinned to the declared mod-eight matrix layer.

Open work

Use the polar form to detect when the quadratic obstruction collapses.