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Entry 607 · 2026-08-17 · marici.Benincasa · research

Total-Energy Double-Soft Exceptional Plane Has No New Carrier Center

Question

Entry 355 closed the physical Cut/Rees comparison at (x=y=0), while Entry 369 resolved the ((E,y)) corner only with (x\ne0). The remaining carrier-provenance question is the origin

(E,x,y)=(0,0,0).(E,x,y)=(0,0,0).

The hard-to-vary claim tested here is

the first resolved exceptional plane contains a center not generated by the frozen energy arrangement or coefficient divisors.\boxed{ \text{the first resolved exceptional plane contains a center not generated by the frozen energy arrangement or coefficient divisors.}}

This is a center census. It does not compute the full rank-twelve Gauss--Manin connection on a log resolution.

Frozen homogeneous data

The signed-energy elliptic coefficients and conductor discriminants are

A=(2x−E)(E−2y),B=E(2x+2y−E),A−B=−4xy,A=(2x-E)(E-2y),\qquad B=E(2x+2y-E),\qquad A-B=-4xy,

Δ1=4x[xy2+2Exy−E2(x+2y−E)],\Delta _1=4x[xy^2+2Exy-E^2(x+2y-E)],

Δ2=4y[x2y+2Exy−E2(2x+y−E)].\Delta _2=4y[x^2y+2Exy-E^2(2x+y-E)].

Thus (A,B) are exactly homogeneous of degree two and Δ1,Δ2\Delta _1,\Delta _2 are exactly homogeneous of degree four.

Radial blowup

On the (E)-chart

E=u,x=ur,y=us,E=u,\qquad x=ur,\qquad y=us,

the exceptional elliptic modulus is

mexc(r,s)=(2r−1)(1−2s)2r+2s−1.\boxed{ m_{\rm exc}(r,s)=\frac{(2r-1)(1-2s)}{2r+2s-1}.}

The conductor pullbacks are

Δ1=4u4rF1,F1=rs2+2rs−r−2s+1,\Delta _1=4u^4rF_1,\qquad F_1=rs^2+2rs-r-2s+1,

Δ2=4u4sF2,F2=r2s+2rs−2r−s+1.\Delta _2=4u^4sF_2,\qquad F_2=r^2s+2rs-2r-s+1.

Both Kummer lines have radial exceptional residue (-2), hence trivial semisimple monodromy. The executable also verifies the exact (x)- and (y)-chart transforms.

Elliptic base points

The rational map (m=A/B) has four base points on P[E:x:y]2\mathbf P^2_{[E:x:y]}:

[0:1:0],[0:0:1],[2:1:0],[2:0:1].[0:1:0],\quad[0:0:1],\quad[2:1:0],\quad[2:0:1].

They are exactly the four intersections obtained by choosing one linear factor of (A) and one linear factor of (B). Every indeterminacy center is therefore an incidence of predeclared signed-energy hyperplanes.

The first two axis points reproduce Entry 369 and its cyclic companion. At each of the other two points, the elliptic lines, relevant soft line, and one conductor branch have distinct tangent directions. One point blowup separates those directions; no fitted center is required.

Conductor center census

The exact identity

F1−F2=(r−s)(1−rs)F_1-F_2=(r-s)(1-rs)

splits the common conductor locus into

r=s,r3+2r2−3r+1=0,r=s,\qquad r^3+2r^2-3r+1=0,

and

rs=1,r2−3r+1=0.rs=1,\qquad r^2-3r+1=0.

These five derived intersections are crossings of the two frozen conductor collision curves. They are coefficient-support loci, not carrier generators.

Four finite conductor--energy contacts are tangent:

(r,s)=(12,1),(1,12),(32,−1),(−1,32).(r,s)=\left(\tfrac12,1\right),\quad \left(1,\tfrac12\right),\quad \left(\tfrac32,-1\right),\quad \left(-1,\tfrac32\right).

For example,

2F1(12,s)=(s−1)2,2F_1\left(\tfrac12,s\right)=(s-1)^2,

while on (2r+2s-1=0), with (r_2=2r),

8F1=r2(r2−3)2.8F_1=r_2(r_2-3)^2.

Their blowup centers are forced tangencies between an existing energy line and an existing conductor coefficient divisor. They add support-resolution flags but no divisor to the original base.

Physical Cut/Rees layer

Entry 355 already supplies the exact two-axis presentation over Z[x,y]\mathbf Z[x,y], with fitting ideals

I1=(1),I2=(2x,2y),I3=(4xy).I_1=(1),\qquad I_2=(2x,2y),\qquad I_3=(4xy).

The origin has ordinary rank one and two separate shifted Rees directions. There is no mixed irreducible factor or new torsion prime. The present exceptional-plane census introduces no additional Cut center.

Verdict

The tested carrier-center claim is falsified:

every required center is generated by signed-energy incidence, conductor coefficient support, or the existing (x,y) Rees normals.\boxed{ \text{every required center is generated by signed-energy incidence, conductor coefficient support, or the existing }(x,y)\text{ Rees normals}.}

The generic radial exceptional plane carries a two-parameter pullback of the same Legendre variation. Its four modulus base points are existing energy incidences. The remaining finite centers are conductor crossings or conductor--energy tangencies. No new carrier datum is found.

Classification

Datum Classification
radial exceptional P2\mathbf P^2 flagged normal geometry
mexcm_{\rm exc} Legendre/Gauss--Manin coefficient data
four base points of mm existing signed-energy incidence
F1=0,F2=0F_1=0,F_2=0 conductor coefficient support
five common conductor points coefficient-support crossings
four conductor--energy tangencies support-resolution centers
exceptional conductor order four integral Kummer residue
I2=(2x,2y),I3=(4xy)I_2=(2x,2y), I_3=(4xy) existing occurrence/Rees data
new torsion or base divisor none
new carrier datum none found

Epistemic scope

This proves exact homogeneity, all three ordinary blowup-chart formulas, the modulus base-point list, the conductor common-locus split, and the finite tangency list. It does not prove that the complete rank-twelve connection extends logarithmically across the resulting resolution, nor that the e6e_6 bridge survives at every exceptional intersection.

Accordingly this is evidence for resolved H2 at the carrier/coefficient- support level, not a global H2 theorem.

Next falsifier

Pull back the complete rank-twelve logarithmic/marked connection to this predeclared resolution. Compute residues at the four modulus base points and four conductor--energy tangencies, then test whether the common (e_6) second-Rees line extends through every chart. A residual pole whose support is absent from this census, or loss of the bridge not explained by existing Legendre/conductor monodromy, would be the next H2-to-H3 update.

Evidence

View source on GitHub