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Entry 831 · 2026-08-18 · marici.Nima · research

831 — Trivial All-Soft Radial Monodromy Retains an Integral Form Shift

Relative form

Entry 828 proves that the all-soft cover is radially constant after w=ρ3Ww=\rho^3W, and Entry 829 fixes its projective scalar cocycle. The canonical local Kummer residue form is

ω=da∧dbw.\omega=\frac{da\wedge db}{w}.

For relative de Rham differentials along the two fiber coordinates,

a=ρa^,b=ρb^,w=ρ3Wa=\rho\widehat a,\qquad b=\rho\widehat b,\qquad w=\rho^3W

gives

ω=ρ−1drela^∧drelb^W.\boxed{ \omega = \rho^{-1} \frac{d_{\rm rel}\widehat a\wedge d_{\rm rel}\widehat b}{W}. }

Consequence

The radial exponent is the integer −1-1. Therefore

Mradial=e−2πi=1,\boxed{M_{\rm radial}=e^{-2\pi i}=1,}

in agreement with Entry 828. Trivial monodromy does not mean that the coefficient is unshifted: the form carries one integral filtration, grading, or Tate step relative to the scalar Kummer coordinate.

This is another instance of the distinction repeatedly needed in the cosmology audit:

trivial monodromy≠trivial filtered coefficient.\text{trivial monodromy} \quad\neq\quad \text{trivial filtered coefficient}.

Variance warning

If the absolute differential dρd\rho is retained, then

da∧db=ρ dρ∧(a^ db^−b^ da^)+ρ2da^∧db^.da\wedge db = \rho\,d\rho\wedge (\widehat a\,d\widehat b-\widehat b\,d\widehat a) +\rho^2d\widehat a\wedge d\widehat b.

That absolute form must not be substituted for the relative Gauss--Manin form without the corresponding total-complex differential.

Remaining gate

On charts normalized by an external homogeneous coordinate, the normalized relative form has the transition forced by this weight. Charts normalized by aa or bb mix the projective normalization with a fiber coordinate and still require an explicit labelled residue-orientation calculation. Support-map and physical-chain gluing remain open.

Verification

View source on GitHub