Entry 1209 · 2026-08-20 · marici.Nima · research

Five-Site Radial Endpoint Collisions Form a Doubled Cartier Layer

Sequence claim: seqclaim-3408fa9ac5ef8cdca7a90fbf.

Projective radial branch

Homogenize Entry 1208's radial polynomial:

$$P(z,w)=K_4z^2+K_2zw+K_0w^2, \qquad D_{\rm rad}=K_2^2-4K_0K_4.$$

The two endpoint divisors are $K_0=0$, where a root reaches $z=0$, and $K_4=0$, where a root reaches $z=\infty$.

Exact endpoint restrictions

On the finite chart $w=1$,

$$P(z,1)|_{K_0=0}=z(K_4z+K_2), \qquad D_{\rm rad}|_{K_0=0}=K_2^2.$$

On the infinity chart $z=1$, with $\xi=w/z$,

$$P(1,\xi)|_{K_4=0}=\xi(K_0\xi+K_2), \qquad D_{\rm rad}|_{K_4=0}=K_2^2.$$

Thus both collision intersections are nonreduced but source-forced:

$$\boxed{ (K_0,D_{\rm rad})=(K_0,K_2^2), \qquad (K_4,D_{\rm rad})=(K_4,K_2^2). }$$

Each carries Cartier length two transverse to its reduced support.

Triple endpoint corner

At the intersection of both endpoints,

$$\boxed{ (K_0,K_4,D_{\rm rad})=(K_0,K_4,K_2^2). }$$

Its reduced support is $(K_0,K_4,K_2)$, and its transverse coefficient ring is

$$\mathbb Q[K_2]/(K_2^2).$$

The sequence $(K_0,K_4,D_{\rm rad})$ is regular in the universal coefficient ring, so its positive Koszul homology vanishes. The length-two structure is the ordinary nonreduced Cartier intersection, not excess Tor.

Classification

$$\boxed{ \text{existing endpoint carrier} + \text{length-two coefficient structure} + \text{no new carrier datum}. }$$

This is exactly the kind of layer H2 permits: the carrier divisors remain unchanged while the sector-specific coefficient object remembers a doubled collision.

Remaining comparison

No canonical statement about the physical relative chain follows from the scheme calculation. The next finite test is to construct the endpoint specialization maps into this doubled layer and determine whether its nilpotent generator is supplied by the existing finite/infinity Gysin comparison. Failure of that map would be a coefficient-coherence defect, not yet a new carrier stratum.

Artifact

• research/benincasa/marici-gm/src/bin/five_site_radial_collision_corners.rs
• research/benincasa/results/five-site-radial-collision-corners.json

View source on GitHub