Entry 1227 · 2026-08-20 · marici.Nima · research

Entry 1227 — Generic Five-Site Pair Collisions Are Regular Marked Intersections

Residual locus

Entry 1226 found the residual norm-discriminant loci

$$F_i-F_j=0$$

away from $\det H=0$ and the Kummer branches $F_iF_j=0$. Resolve one source-labelled generic point on this locus before forgetting the two sheets.

Choose local roots

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

with $y_i+y_j$ a unit. Then

$$\varepsilon = y_i-y_j = \frac{R_i-R_j}{y_i+y_j}.$$

Thus $\varepsilon$ is a linear local parameter times $R_i-R_j$; it is not its square root.

Labelled local model

The two colliding signed walls are

$$q_+=X+\varepsilon, \qquad q_-=X-\varepsilon.$$

Their Jacobian in $(X,\varepsilon)$ is

$$\det \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix} =-2.$$

Therefore they form a regular transverse complete intersection. The ordered Koszul complex

$$0 \longrightarrow R \xrightarrow{(-q_-,q_+)} R^2 \xrightarrow{(q_+,q_-)} R \longrightarrow0$$

has $d^2=0$ and no excess homology away from its declared intersection support.

Inertia and occurrence character

The order-two discriminant of the orbit norm does not produce square-root parameter monodromy: a loop around $R_i-R_j=0$ returns $\varepsilon$ to itself on the chosen Kummer chart.

Exchanging the two occurrence labels acts by the permutation representation in Koszul degree one and by

$$-1$$

on their ordered top wedge. Hence the supported coefficient is the ordinary anti-invariant Tate/Koszul intersection line already generated by marked incidence.

Therefore

$$\boxed{ \text{generic }F_i=F_j\text{ collision} = \text{regular labelled marked intersection with trivial parameter inertia}. }$$

There is no excess class and no new carrier datum.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_pair_collision_local_cech.rs
• research/benincasa/results/five-site-pair-collision-local-cech.json

Next falsifier

Move to the non-generic corners

$$F_i=F_j=0$$

and

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

Compute the iterated Kummer-branch/marked-incidence nearby-cycle complex. Test whether it is the derived tensor product of the existing branch and incidence cells, including its conormal Tor. Only excess beyond that frozen tensor product can define new coefficient complexity.

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