Entry 1228 · 2026-08-20 · marici.Nima · research

Entry 1228 — The Five-Site Double-Soft Corner Is a Uniform Four-Wall Arrangement

Frozen corner

Resolve

$$F_i=F_j=0$$

away from $\det H=0$ and the remaining Kummer branches. On the labelled cover use

$$R_i=y_i^2, \qquad R_j=y_j^2.$$

The four signed versions of a two-cut marked wall meet at

$$X=y_i=y_j=0.$$

Their equations are

$$X\pm y_i\pm y_j=0.$$

Exact local matroid

The four normal vectors are

$$(1,1,1), (1,1,-1), (1,-1,1), (1,-1,-1).$$

The four triple determinants are

$$4, 4, -4, -4.$$

Hence every triple is transverse and the four-set is the unique circuit:

$$\boxed{ M_{ m local}=U(3,4). }$$

The central Orlik–Solomon Betti vector is

$$\boxed{(1,4,6,3)}.$$

Blowing up the existing fourfold marked-intersection center produces an exceptional $\mathbf P^2$ containing four lines with no triple concurrence. This is the standard resolution of a frozen incidence stratum, not a new primitive carrier wall.

Deck inertia

The local deck group $(\mathbb Z_2)^2$ translates the four sign labels. Writing its nontrivial rational characters as $a,b,ab$, the cohomology decomposes as

$$H^0=1,$$

$$H^1=1+a+b+ab,$$

$$H^2=2a+2b+2ab,$$

$$H^3=a+b+ab.$$

Equivalently, the character values by degree are

$$(1,4,6,3)$$

at the identity and

$$(1,0,-2,-1)$$

at each nonidentity element.

Thus the three top classes are precisely the three nontrivial Kummer characters already compiled from the two branch directions.

Verdict

$$\boxed{ \text{double-soft collision} = \text{existing marked OS/Čech incidence} + \text{two-branch Kummer inertia}. }$$

No coefficient excess and no new carrier datum occur at the generic double-soft corner.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_double_soft_wall_arrangement.rs
• research/benincasa/results/five-site-double-soft-wall-arrangement.json

Next falsifier

Analyze the mixed Gram corner

$$F_i=F_j, \qquad \det H=0,$$

without dividing by $\det H$. Use the integral equations $\det(H)y_k^2=F_k$, derive the weighted local model, and test whether its exceptional coefficient object is generated by the existing Gram Kummer density and marked-incidence complex.

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