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Entry 1772 · 2026-08-21 · marici.Nima · research

1772 — The Repeated u=1u=1 Resultant Is the Existing Double-Site-Soft Corner

Question

After Entries 1770–1771, the only unclassified component of

Res⁡v(P6,D)=u4(u−1)2(1−8u+12u2−4u3)\operatorname{Res}_v(P_6,D) =u^4(u-1)^2(1-8u+12u^2-4u^3)

is the repeated rational factor (u−1)2(u-1)^2.

Carrier support

The associated intersection is (u,v)=(1,1)(u,v)=(1,1). In the frozen homogeneous chart

X1=1,X2=u+v2−1,X3=u−v2,X_1=1,\qquad X_2=\frac{u+v}{2}-1,\qquad X_3=\frac{u-v}{2},

this is exactly

X2=X3=0.X_2=X_3=0.

It is therefore an existing labelled double-site-soft incidence.

Exact local form

Set x=u−1x=u-1 and z=v−1z=v-1. Then

4P6=z2+8x−2xz+21x2−8x2z+16x3−4x3z+4x4,4P_6=z^2+8x-2xz+21x^2-8x^2z+16x^3-4x^3z+4x^4,

D=−z2−4x+2xz−5x2+4x2z.D=-z^2-4x+2xz-5x^2+4x^2z.

Their initial forms are

in⁡(4P6)=8x,in⁡(D)=−4x.\operatorname{in}(4P_6)=8x, \qquad \operatorname{in}(D)=-4x.

Both coefficient branches are smooth and share the tangent x=0x=0. Their ordinary tangency accounts for the repeated resultant factor.

Narrow conclusion

(u−1)2 records tangency over X2=X3=0, not a new carrier incidence.\boxed{ (u-1)^2\text{ records tangency over }X_2=X_3=0, \text{ not a new carrier incidence.} }

Together with Entries 1770 and 1771, this classifies every irreducible component of the two exact P6P_6-wall resultants at carrier level. Generic cubic branches have no descending relative-sign extension, while both repeated rational components lie on pre-existing soft strata.

This does not replace the iterated nearby-cycle coefficient calculations on the soft corners.

Durable evidence

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