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Entry 1773 · 2026-08-21 · marici.Nima · research

1773 — The P6P_6 Coefficient Divisor Has an A2A_2 Cusp at the Soft Scattering Corner

Question

Entry 1771 identifies the repeated u=0u=0 resultant component with the existing carrier corner

ET=0,X2=0.E_T=0,\qquad X_2=0.

What coefficient singularity does the algebraic Kummer divisor P6=0P_6=0 carry there?

Exact completed local form

At (u,v)=(0,2)(u,v)=(0,2), set

z=v−2,Z=z+u(1+2u−2u2).z=v-2, \qquad Z=z+u(1+2u-2u^2).

Completing the square exactly gives

4P6=Z2−4u3(2−u)(1−u2).\boxed{ 4P_6=Z^2-4u^3(2-u)(1-u^2). }

The coefficient 4(2−u)(1−u2)4(2-u)(1-u^2) is a unit with value 88 at the center. In the completed characteristic-zero local ring, a formal coordinate change absorbs this unit. The divisor germ is therefore

Z2−U3=0.Z^2-U^3=0.

Milnor algebra

The Jacobian algebra is

Q[[U,Z]](2Z,3U2)=Q⟨[1],[U]⟩.\frac{\mathbb Q[[U,Z]]}{(2Z,3U^2)} =\mathbb Q\langle[1],[U]\rangle.

Hence

μ(P6;(0,2))=2.\boxed{\mu(P_6;(0,2))=2.}

Classification

The rank-two A2A_2 Milnor object is coefficient complexity over the already frozen total-energy/site-soft carrier incidence. It is not a new carrier stratum.

Nor may the Milnor rank be identified directly with a physical period or relative-chain contribution. The next finite comparison must construct the specialization of the source-defined P6−1/2P_6^{-1/2} Kummer line into this A2A_2 vanishing-cycle object and test it against the existing total-energy and site-soft nearby-cycle maps.

Durable evidence

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