Double-cover lifting needs a relative boundary-phase receipt

Exact statement

For a connected double cover, a cover-preserving mapping class has a boundary-pointwise lift exactly when its relative phase ε in reduced H⁰ of the boundary vanishes; for a planar twist, ε(t_c^m)=(m mod 2)·ω(c)·χ_c.

StatusProved by path lifting and mod-two intersection; internally audited

External reviewNo documented external or specialist review of this FPRD deduction is recorded.

Context

Odd/even branch parity detects whether a twist changes sheets locally, but not which boundary fibres acquire that change. Relative boundary phase supplies the missing spatial receipt.

Definitions

ω is the F₂-valued character classifying the connected double cover.
χ_c records the interior boundary components separated from the exterior boundary by c.

Hypotheses and scope

The mapping class preserves the cover character and fixes boundary components on the base.

Proof or evidence

For an arc from the exterior boundary to boundary j, ε_j is the sheet change along f(a_j)a_j⁻¹. Arc independence follows from preservation of ω; simultaneous boundary-fixed lifting is equivalent to all relative phases agreeing. A twist contributes intersection with c times ω(c).

Boundary-phase lift theorem →

Verification notes

Arc choices, homomorphism signs, even powers, and cut-vector independence were rechecked; 3,048 finite cut-formula cases provide transcription corroboration.

Limitations

The receipt decides boundary-pointwise liftability, not whether a zero-receipt block is one allowed positive Dehn-twist atom.

Open work

Label the actual Chapter 4 factor curves by their cut vectors and compare the resulting zero-receipt blocks with the thesis’s liftable packets.