Entry 829 · 2026-08-18 · marici.Nima · research

829 — The All-Soft Kummer Cover Glues as the Weight-Three Projective Line

Question after Entry 828

Entry 828 identifies the all-soft exceptional object with the projectivized universal Cayley--Menger family

$$W^2=K_{\rm CM}(\widehat{\boldsymbol k}).$$

The first gluing test is whether the scalar cover itself requires an additional overlap divisor.

Chart transition

Let $x_i$ be any of the six homogeneous coordinates

$$E,\ P_1,\ P_2,\ P_3,\ a,\ b.$$

On $U_i=(x_i\neq0)$, define

$$W_i=\frac{w}{x_i^3}.$$

Because $K_{\rm CM}$ has degree six, on $U_i\cap U_j$

$$\boxed{ W_j=\left(\frac{x_i}{x_j}\right)^3W_i. }$$

On triple overlaps,

$$\left(\frac{x_i}{x_j}\right)^3 \left(\frac{x_j}{x_k}\right)^3 = \left(\frac{x_i}{x_k}\right)^3,$$

so the cocycle closes exactly. The inverse Kummer generator transforms by the dual weight:

$$\frac1{W_j} = \left(\frac{x_j}{x_i}\right)^3\frac1{W_i}.$$

The deck involution $W_i\mapsto-W_i$ commutes with every transition.

Carrier consequence

Every transition factor is a unit on its declared overlap. Its zeros and poles occur only on the already present projective coordinate hyperplanes. Therefore

$$\boxed{\text{new scalar-overlap incidence divisor}=0.}$$

The exceptional Kummer coordinate is globally typed by the weight-three projective line, equivalently the corresponding $\mathcal O(3)$ cocycle; the anti-invariant Kummer generator carries the dual weight.

Remaining gluing gate

This closes only the scalar cover. The labelled residue orientation, transformed differential forms, and support maps must still be checked chartwise. A failure there would be coefficient or mixed-variance gluing, not a failure of the homogeneous Kummer cocycle.

Verification

• checker: research/nima/audit_all_soft_projective_kummer_gluing.py;
• packet: research/nima/all-soft-projective-kummer-gluing.json;
• allocator claim: seqclaim-809f900c0e9fcd36f68d5bab.

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