Entry 1534 · 2026-08-21 · marici.Nima · research

1534 — The Second-Normal Mass Deformation Is the Even-Grade Shift Operator

Status

Closed all-grade operator identity for the first invariant deformation away from the physical mass diagonal.

Generating function

Use the physical mean and invariant normal coordinates

$$a=4\bar y^2, \qquad \tau=\delta^2=(y_1-y_2)^2, \qquad z=X^{-1}.$$

The normalized double-Gysin jet is

$$\boxed{ F(z,\tau) =\frac{2}{(1-az^2)(1-\tau z^2)}. }$$

The physical equal-mass fiber is

$$F_{\rm phys}(z)=F(z,0)=\frac{2}{1-az^2}.$$

Transverse operator

Differentiating in the first source-visible normal coordinate gives

$$\boxed{ \left.\frac{\partial F}{\partial\tau}\right|_{\tau=0} =z^2F_{\rm phys}(z). }$$

Thus the second-normal deformation acts on the physical infinity jet by the even-grade shift operator.

Coefficientwise, for $m\ge1$,

$$\boxed{ \left. \frac{\partial C^{(2m)}}{\partial\tau} \right|_{\tau=0} =C_{\rm phys}^{(2m-2)} =2a^{m-1}. }$$

All odd grades remain zero.

Meaning

The first unequal-mass correction is not an independent coefficient family. It is canonically generated from the physical one-channel jet by shifting the filtration two steps:

$$\boxed{ N_{\rm mass}=z^2. }$$

This is the local extension operator carried by the ramified source map. The normal coefficient object therefore has the form

$$\text{physical jet} \xrightarrow{\;z^2\;} \text{second-normal jet},$$

rather than two unrelated towers.

The result provides a finite algebraic model for a nearby-cycle or Gauss–Manin extension in which the transverse deformation is determined by a filtration-shift operator.

Durable evidence

• research/nima/check_supercritical_infinity_jet.sage;
• Entries 1531–1532;
• allocator claim seqclaim-229fdc3c1f75a8ac6750410e.

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