Entry 1226 · 2026-08-20 · marici.Nima · research

Entry 1226 — Residual Five-Site Norm Discriminants Are Projection Collisions

Frozen marked cover

Use the 26 physical marked sections on the rank-32 five-site Kummer cover. Their cut-depth profile is

$$1+5+20:$$

one section with no root, five sections with one root, and twenty connected-subgraph sections with two boundary roots.

The deck-orbit norm forgets the labelled sheet while retaining the marked divisor. This operation is not a new carrier construction.

One-cut orbit

For a one-cut wall $X+2y=0$, with $y^2=R$, the orbit norm is

$$N_1(X)=X^2-4R.$$

Its discriminant is

$$\operatorname{Disc}_X(N_1)=16R.$$

Thus its only collision is the existing Kummer branch $R=0$.

Two-cut orbit

For a two-cut wall $X+s_iy_i+s_jy_j=0$, the four-sheet norm is

$$N_2(X) = X^4-2(R_i+R_j)X^2+(R_i-R_j)^2.$$

Exact multiplication over the four sign choices and an independent root-difference calculation give

$$\boxed{ \operatorname{Disc}_X(N_2) = 4096R_i^2R_j^2(R_i-R_j)^2. }$$

On the physical cover

$$R_i=\frac{F_i}{\det H},$$

so away from $\det H=0$ the residual collision is

$$F_i-F_j=0.$$

The twenty connected-subgraph labels realize all ten unordered edge pairs exactly twice.

Type of the residual divisor

The loci $F_i=0$ and $F_j=0$ are existing edge-soft/Kummer branches. At

$$F_i-F_j=0$$

the Kummer cover is generically smooth. What collides are two labelled signed marked walls after projection to the base. Therefore

$$\boxed{ F_i-F_j=0 \text{ is marked relative-incidence/coefficient support, not a primitive carrier wall.} }$$

Keeping occurrence and sheet labels resolves the collision. Promoting the discriminant of the orbit norm to a new carrier generator would confuse an orbit-forgetting projection with source incidence.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_marked_orbit_norms.rs
• research/benincasa/results/five-site-marked-orbit-norms.json

Next falsifier

Resolve one labelled generic collision over $F_i=F_j\neq0$. Compute the local relative Čech/Kato complex for the two colliding signed walls and its inertia under the transposition of the two roots. Test whether the resulting coefficient class is generated by the existing marked-incidence comparison. A new carrier claim would require failure of that labelled resolution, not merely the base norm discriminant.

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