Entry 313 · 2026-08-16 · marici.Benincasa · research

Three Soft-Corner Slices Do Not Determine the Mixed SNC Extension

Record

Status: exact finite common-frame falsifier and local extension-space theorem for the frozen (q_{\mathcal G_{12}}) nine-master module. No carrier cell, denominator, source master, or normalization is added.

This continues entries 312, 316.

Deutsch--Popperian claim tested

Freeze

$$u=\ell_4=E_T,\qquad v=\ell_3,\qquad B=uv.$$

Entry 316 computed the (u)-normal, (v)-normal, and radial connections on

$$(u,v)=(\lambda,1),\qquad(1,\lambda),\qquad(\lambda,2\lambda).$$

The tested claim was that these independently reduced residues already occupy one common logarithmic frame and therefore directly determine the bivariate extension.

Exact falsifier

On the final ((e_7,e_8,e_9)) block, the literal sum of the two axis residues has characteristic polynomial

$$\chi_{R_u+R_v}(t)=t(t^2+1/16),$$

whereas the radial residue has

$$\chi_{R_{\rm rad}}(t)=t^2(t-1).$$

They are not constant-conjugate, and their literal commutator is nonzero. The first six-master block does assemble additively and commute.

This is not a bivariate obstruction. The axis residues were evaluated at different generic points, (v=1) and (u=1), and each Griffiths--Dwork reduction may use a different exact-form gauge. A corner comparison requires one Deligne frame.

Pure elliptic quotient

Away from (X_1X_2=0), (A) is a unit at the corner and

$$n=\frac BA=\frac{uv}{A}$$

is a nodal Legendre coordinate. Hence

$$d\log n=d\log u+d\log v-d\log A,$$

with the final term regular. In the canonical (n)-frame the two quotient residues are the same universal nodal operator (N), up to the commuting scalar half-residue of the (B^{-1/2}) Kummer twist:

$$[R_u^{\rm ell},R_v^{\rm ell}]=0,\qquad \operatorname{rank}N=1,\qquad N^2=0.$$

Thus the pure elliptic quotient has no hidden corner obstruction.

Why all three curves can miss an extension

Let (A_{\rm alg}) be a trivial local algebraic target and put

$$M=\operatorname{Hom}(V_{\rm ell},A_{\rm alg}),\qquad D=T_M-1.$$

A cocycle for the local group (mathbb Z^2) is a pair ((a,b)) satisfying

$$D(b-a)=0,$$

and coboundaries are ((Dc,Dc)). Triviality on each axis requires (a,b\in\operatorname{im}D). Modulo diagonal coboundaries, the residual class is

$$\boxed{\ker(\text{restriction to both axes})\simeq\operatorname{im}D.}$$

Its radial restriction is also trivial because (operatorname{im}(T_M^2-1)=operatorname{im}D) for (N^2=0). Therefore all three tests in entry 316 can miss a genuine mixed codimension-two class.

Character refinement

Equivariance confines the hidden class to the elliptic ((--)) character. The final-block Gysin kernel is

$$A_{--}=\langle e_6,v_{\rm alg}\rangle.$$

Consequently:

$$\boxed{\dim E_{\rm hidden}^{\rm final}=2,}$$

with scalar coordinates

$$(\varepsilon_{e_6},\varepsilon_{v_{\rm alg}}).$$

The source last-three cyclic module contains only the algebraic (v_{\rm alg}) line, so

$$\boxed{\dim E_{\rm hidden}^{L_3}=1.}$$

The remaining five algebraic kernel dimensions have different (C_2^{(a)}\times C_2^{(b)}) characters and cannot contribute equivariantly.

Classification

Even if either scalar is nonzero, it is an extension class supported at the already frozen SNC incidence (u=v=0):

$$\text{existing energy carrier} + \text{Tate/Kummer/Legendre coefficient extension}.$$

It does not by itself justify a new carrier generator.

Exact evidence

• research/benincasa/soft-corner-common-frame-falsifier.json;
• research/benincasa/soft-corner-hidden-extension-space.json;
• the exact three-slice artifacts of entry 316.

Next hostile falsifier

Derive the final four-master connection over (mathbb Q(X_1,u,v)), transport its quotient to the (n=uv/A) Deligne frame, and compute the antisymmetric off-diagonal residue difference modulo diagonal coboundaries. The calculation has only two outputs:

$$(\varepsilon_{e_6},\varepsilon_{v_{\rm alg}}).$$

Their simultaneous vanishing closes the generic soft-corner extension. A nonzero value locates a coefficient-supported mixed SNC extension while leaving the common carrier intact.

View source on GitHub