Entry 1210 · 2026-08-20 · marici.Nima · research

Endpoint Discriminant Normalization Produces a Canonical Odd Line

Sequence claim: seqclaim-5d798378ca4aee9857d2edb9.

Resolved endpoint cover

Entry 1209 gives the doubled collision section $K_2^2=0$ on either radial endpoint. Retain the discriminant square root $\delta$ before forgetting its deck sheet. The endpoint cover is

$$R = \mathbb Q[K_2,\delta]/(\delta^2-K_2^2).$$

Its normalization is

$$N=\mathbb Q[K_2]\oplus\mathbb Q[K_2],$$

with

$$f(K_2,\delta) \longmapsto \bigl(f(K_2,K_2),f(K_2,-K_2)\bigr).$$

The two summands are the two labelled discriminant sheets; deck exchange swaps them.

Canonical normalization cokernel

The conductor is $K_2N$. The normalization sequence has cokernel

$$\boxed{ N/R\simeq\mathbb Q_{K_2=0}. }$$

It is generated projectively by the difference of the two sheet values. Deck exchange acts as

$$\boxed{\tau=-1}.$$

Thus the endpoint coefficient cell is canonical without choosing either lift $\delta=+K_2$ or $\delta=-K_2$. Choosing one branch would be noncanonical; retaining their anti-diagonal difference is intrinsic.

Finite cutoff calculations through degree 32 give normalization/image ranks

$$(6,5),\ (10,9),\ (18,17),\ (34,33),\ (66,65),$$

so the cokernel remains rank one.

Finite/infinity covariance

The finite and infinity endpoint equations are exchanged by

$$K_0\leftrightarrow K_4, \qquad z\leftrightarrow w.$$

Both therefore carry the same canonical odd normalization cokernel. An ordered residue orientation is still required to determine the sign of a Čech differential between the two endpoint occurrences, but not to define either local coefficient line.

Classification

$$\boxed{ \text{existing endpoint support} + \text{canonical rank-one odd coefficient costalk}. }$$

No new carrier datum appears. This is a normalization/conductor object of the sector-specific discriminant cover.

Remaining physical gate

This calculation does not produce a physical relative-chain map. The next test must derive the ordered endpoint residue/Gysin maps from the source measure and compare their images in the two odd normalization costalks. A zero or nonzero physical pairing may be stated only after that map exists.

Artifact

• research/benincasa/marici-gm/src/bin/five_site_endpoint_discriminant_normalization.rs
• research/benincasa/results/five-site-endpoint-discriminant-normalization.json

View source on GitHub