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Entry 1774 · 2026-08-21 · marici.Nima · research

1774 — The P6P_6 Kummer Character Kills the Soft A2A_2 Milnor Plane

Question

Entry 1773 finds an ordinary rank-two A2A_2 Milnor object at

ET=X2=0.E_T=X_2=0.

Does the source-defined algebraic coefficient line P6−1/2P_6^{-1/2} specialize to a nonzero supported class there?

Link presentation

The link of

Z2−U3=0Z^2-U^3=0

is the trefoil. With source-compatible orientation, take

V=(10−11).V=\begin{pmatrix}1&0\\-1&1\end{pmatrix}.

The Alexander presentation and polynomial are

tV−VT,Δ(t)=det⁡(tV−VT)=t2−t+1.tV-V^T, \qquad \Delta(t)=\det(tV-V^T)=t^2-t+1.

The Kummer line P6−1/2P_6^{-1/2} assigns character t=−1t=-1 to a positive meridian of the coefficient divisor. Therefore

(−V−VT)=(−211−2),det⁡=3.(-V-V^T)= \begin{pmatrix}-2&1\\1&-2\end{pmatrix}, \qquad \det=3.

This matrix is invertible over characteristic zero. Degree-zero invariants also vanish because the character is nontrivial. Hence the twisted local link cohomology is acyclic:

H∙(Link⁡A2;L−1)=0.\boxed{H^\bullet(\operatorname{Link}_{A_2};\mathcal L_{-1})=0.}

Consequence

The source Kummer line selects neither one nor both of the ordinary A2A_2 Milnor directions. The rank-two Milnor algebra in Entry 1773 is real coefficient-divisor geometry, but it does not produce a supported class in the P6−1/2P_6^{-1/2} sector.

This closes the algebraic Kummer contribution at the generic total-energy/site-soft cusp. It does not rule out contributions from other coefficient blocks or from a separately supplied physical relative chain.

Durable evidence

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