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Entry 833 · 2026-08-18 · marici.Nima · research

833 — The Two All-Soft Fiber Gysin Forms Agree Exactly on Their Overlap

Mixed-variance construction

Entry 832 shows that the aa- and bb-normalized charts cannot be glued to the external subcover by an ordinary relative Jacobian. Their canonical form-level comparison is obtained by contracting the relative area form with the fiber Euler field:

η=ιa∂a+b∂b(da∧dbw)=a db−b daw.\eta = \iota_{a\partial_a+b\partial_b} \left(\frac{da\wedge db}{w}\right) = \frac{a\,db-b\,da}{w}.

This is the radial Gysin form derived from the source orientation.

The two charts

On UaU_a, set

z=ba,Wa=wa3.z=\frac ba,\qquad W_a=\frac w{a^3}.

Then

ηa=a−1dzWa.\eta_a=a^{-1}\frac{dz}{W_a}.

On UbU_b, set

z′=ab,Wb=wb3.z'=\frac ab,\qquad W_b=\frac w{b^3}.

The ordered orientation gives

ηb=−b−1dz′Wb.\eta_b=-b^{-1}\frac{dz'}{W_b}.

Exact overlap

On Ua∩UbU_a\cap U_b,

z′=z−1,dz′=−z−2dz,b−1=a−1z−1,Wb=z−3Wa.z'=z^{-1},\qquad dz'=-z^{-2}dz,\qquad b^{-1}=a^{-1}z^{-1},\qquad W_b=z^{-3}W_a.

Substitution yields

ηb=ηa.\boxed{\eta_b=\eta_a.}

The coordinate-inversion sign and the ordered-residue sign cancel exactly. No additional overlap homotopy or incidence divisor is needed for the canonical Kummer form.

Consequence

The scalar cover, external relative forms, and the two fiber-chart radial Gysin forms now all glue. The remaining question is support-sensitive: every labelled signed-energy, triangle, soft, coordinate-boundary, and Cayley--Menger restriction must commute with this Gysin comparison.

Form-level coherence does not by itself prove those support-map squares. Physical-chain activation also remains separate.

Verification

View source on GitHub