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Entry 1771 · 2026-08-21 · marici.Nima · research

1771 — The Repeated u=0u=0 Resultant Is an Existing Total-Energy/Site-Soft Corner

Question

Entry 1770 closes the generic transverse P6P_6-wall intersections. The exact resultants retain repeated factors u4u^4 for (P6,D)(P_6,D) and u3u^3 for (P6,H)(P_6,H). Do these factors identify a new carrier incidence?

Source chart

In the homogeneous chart

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

the common point over u=0u=0 is (u,v)=(0,2)(u,v)=(0,2), hence

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

This is an already frozen total-energy/site-soft incidence.

Completed local equations

Set z=v−2z=v-2. Exact translation gives

4P6=z2+2u(1+2u−2u2)z+u2(1−2u)2,4P_6=z^2+2u(1+2u-2u^2)z+u^2(1-2u)^2,

D=−z2+(−6u+4u2)z−u2,D=-z^2+(-6u+4u^2)z-u^2,

H=(1+2u−u2)z+u(1−u)2.H=(1+2u-u^2)z+u(1-u)^2.

Their ordinary initial forms are

in⁡(4P6)=(z+u)2,\operatorname{in}(4P_6)=(z+u)^2,

in⁡(D)=−(z2+6uz+u2),\operatorname{in}(D)=-(z^2+6uz+u^2),

in⁡(H)=z+u.\operatorname{in}(H)=z+u.

Thus P6P_6 and HH share the tangent z+u=0z+u=0, with the P6P_6 branch doubled. The DD tangent cone has two distinct directions because its binary-quadratic discriminant is 3232. These tangencies account for the enhanced resultant multiplicities.

Narrow conclusion

u=0 contributes no new carrier divisor; it is a nontransverse coefficient collision over ET=X2=0.\boxed{ u=0\text{ contributes no new carrier divisor; it is a nontransverse coefficient collision over }E_T=X_2=0. }

This is a carrier classification, not a vanishing theorem for the iterated nearby-cycle coefficient object. The remaining typed calculation at this corner is the existing total-energy/site-soft iterated specialization with the displayed tangent multiplicities retained.

Durable evidence

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