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Entry 842 · 2026-08-18 · marici.Nima · research

842 — Every First Residual Polar Collision Is Generated by Its Existing Normal Map

Residual supports

Entry 840 leaves three loci where its diagonal/anti-diagonal determinant vanishes:

P3=0,a=0,b=0.P_3=0,\qquad a=0,\qquad b=0.

On each divisor, the labelled factors Q+Q_+ and Q−Q_- have the same restriction. Their normal derivatives are opposite.

Soft P3P_3 divisor

At P3=0P_3=0,

Q+=Q−=E2(a2−b2)−P12a2+P22b2,Q_+=Q_- =E^2(a^2-b^2)-P_1^2a^2+P_2^2b^2,

while

(∂P3Q+,∂P3Q−)=2Eab(1,−1).\left(\partial_{P_3}Q_+,\partial_{P_3}Q_-\right) =2Eab(1,-1).

The restriction/normal determinant is −4Eab-4Eab.

Coordinate divisors

At a=0a=0,

Q+=Q−=b2(P22−E2),(∂aQ+,∂aQ−)=2EP3b(1,−1),Q_+=Q_-=b^2(P_2^2-E^2), \qquad \left(\partial_aQ_+,\partial_aQ_-\right) =2EP_3b(1,-1),

with determinant −4EP3b-4EP_3b.

At b=0b=0,

Q+=Q−=a2(E2−P12),(∂bQ+,∂bQ−)=2EP3a(1,−1),Q_+=Q_-=a^2(E^2-P_1^2), \qquad \left(\partial_bQ_+,\partial_bQ_-\right) =2EP_3a(1,-1),

with determinant −4EP3a-4EP_3a.

Consequence

On every first residual divisor, ordinary restriction supplies the diagonal line and the predeclared soft or coordinate normal map supplies the anti-diagonal line. Therefore

dim⁡Cone⁡=0\boxed{ \dim\operatorname{Cone}=0 }

generically on each divisor.

The determinants fail only on pairwise or deeper intersections among

E,P3,a,b.E,\qquad P_3,\qquad a,\qquad b.

No new failure divisor has appeared. The next audit is finite: totalize the resulting four-coordinate incidence cube and test whether its alternating Čech/Koszul differential leaves any corner cohomology.

Verification

View source on GitHub