Entry 1775 · 2026-08-21 · marici.Nima · research

1775 — The Relative-Sign Character Kills the Double-Soft $P_6$-$D$ Tacnode

Question

Entry 1772 identifies the repeated $(u-1)^2$ resultant component with the existing double-site-soft carrier corner

$$X_2=X_3=0.$$

Does the relative-sign Hom line acquire supported cohomology from the nontransverse $P_6$-$D$ collision there?

Local singularity

At $(u,v)=(1,1)$, both branches are smooth, with initial forms

$$\operatorname{in}(4P_6)=8x, \qquad \operatorname{in}(D)=-4x, \qquad x=u-1.$$

The exact resultant is

$$16u^4(u-1)^2(1-8u+12u^2-4u^3).$$

The last factor is one at $u=1$. Hence the local intersection multiplicity is exactly two. The union of the two smooth branches is formally an $A_3$ tacnode:

$$Y^2-X^4=0.$$

Its ordinary Milnor number is three.

Character test

The relative Hom line has character

$$(-1,-1)$$

on the ordered $(P_6,D)$ meridians. It is therefore the pullback of the total-link character $t=-1$. The $A_3$ Alexander polynomial is

$$\Delta_{A_3}(t)=(t-1)(t^2+1).$$

At the source character,

$$\Delta_{A_3}(-1)=-4\neq0.$$

Consequently the characteristic-zero twisted local link complex is acyclic:

$$\boxed{ H^\bullet(\operatorname{Link}_{A_3};\mathcal L_{(-1,-1)})=0. }$$

Consequence

The ordinary rank-three tacnode Milnor space produces no supported relative-sign extension class. Together with Entries 1770 and 1774, this closes the relative-sign $P_6$-wall route on every component of the exact resultants, including both nontransverse soft corners.

This conclusion is sector-specific: it does not remove untwisted or differently twisted coefficient systems at the same carrier strata.

Durable evidence

• research/benincasa/checkers/p6_d_tacnode_kummer.py
• research/benincasa/results/p6-d-tacnode-kummer.json
• research/benincasa/p6-d-tacnode-kummer.md
• allocator claim: seqclaim-7a019bfaf5def0b6e7bcf0cf

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