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Entry 1537 · 2026-08-21 · marici.Nima · research

1537 — Mean-Mass and Unequal-Mass Jet Transport Is Formally Flat

Status

Exact formal-connection calculation on the double-Gysin infinity-jet generating section.

Coordinates and section

Write

a=4yˉ2,τ=(y1−y2)2,z=X−1.a=4\bar y^2,\qquad \tau=(y_1-y_2)^2,\qquad z=X^{-1}.

The normalized corner section is

F(a,τ;z)=2(1−az2)(1−τz2).F(a,\tau;z) =\frac{2}{(1-az^2)(1-\tau z^2)}.

Its parameter derivatives close on the same line:

∂aF=z21−az2F,∂τF=z21−τz2F.\boxed{ \partial_aF =\frac{z^2}{1-az^2}F, \qquad \partial_\tau F =\frac{z^2}{1-\tau z^2}F. }

Mixed curvature

Define the scalar formal connection coefficients

Aa=z21−az2,Aτ=z21−τz2.\mathcal A_a=\frac{z^2}{1-az^2}, \qquad \mathcal A_\tau=\frac{z^2}{1-\tau z^2}.

They have no cross-dependence and commute. Therefore

∂aAτ−∂τAa+[Aa,Aτ]=0.\boxed{ \partial_a\mathcal A_\tau -\partial_\tau\mathcal A_a +[\mathcal A_a,\mathcal A_\tau] =0. }

Equivalently,

∂a∂τF=∂τ∂aF.\boxed{\partial_a\partial_\tau F =\partial_\tau\partial_aF.}

The exact checker verifies both the first-order equations and the mixed identity.

Physical fiber

At τ=0\tau=0,

Aτ=z2,\mathcal A_\tau=z^2,

recovering Entry 1534's even-grade shift. Mean-mass transport remains

Aa=z21−az2,\mathcal A_a=\frac{z^2}{1-az^2},

which generates the one-channel recurrence along the physical locus.

Meaning

The source-derived coefficient family has no mixed coherence defect between tangential mean-mass transport and invariant transverse transport. Its formal two-parameter totalization is already flat; no additional homotopy cell is required at this level.

This is a formal coefficient-connection statement. It is not yet an identification with a physical Gauss–Manin connection on relative cycles. That stronger claim would require a source pairing or comparison morphism.

Durable evidence

View source on GitHub