Entry 1230 · 2026-08-20 · marici.Nima · research

Entry 1230 — Orbit Discriminant Is Larger Than Five-Site Singular Support

Reason for the audit

Entry 1226 computed discriminants of marked-wall orbit norms. Before assembling higher intersections, distinguish the discriminant of the polynomial family from the singular support of its total divisor.

One-cut norm

For

$$N_1=X^2-4R,$$

one has

$$\partial_RN_1=-4.$$

Therefore the total orbit-norm divisor is smooth, including over the polynomial discriminant $R=0$. That locus describes ramification/nonreduced specialization of the orbit polynomial, not a singular total divisor.

Two-cut norm

For

$$N_2 = X^4-2(R+S)X^2+(R-S)^2,$$

the exact derivatives satisfy

$$\partial_RN_2+partial_SN_2=-4X^2,$$

$$\partial_RN_2-partial_SN_2=4(R-S).$$

Consequently the reduced Jacobian locus is

$$\boxed{X=0,\qquad R=S.}$$

With $R_i=F_i/\det H$, the physical reduced singular support away from the Gram divisor is

$$\boxed{X_A=0,\qquad F_i=F_j.}$$

Typing correction

The larger polynomial discriminant

$$R_iR_j(R_i-R_j)=0$$

records ramified or nonreduced fibers under the orbit-forgetting projection. It is not by itself the support of a singular marked divisor.

Entries 1226–1227 remain valid with their stated projection/labelled-local meanings, but the higher-overlap frontier must be narrowed:

$$\boxed{ \text{pair collision support} = \text{signed-energy wall }X_A=0 \cap \text{root equality }F_i=F_j. }$$

Thus triple Čech overlaps cannot be inferred merely from $F_i=F_j=F_k$. The relevant source-labelled energy walls must intersect simultaneously.

Classification

This refinement further weakens any carrier claim. The actual singular locus already includes a frozen marked/signed-energy section. No new carrier datum is introduced.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_orbit_norm_singular_support.rs
• research/benincasa/results/five-site-orbit-norm-singular-support.json

Next falsifier

Use the 20 source-labelled two-cut sections to enumerate actual simultaneous loci

$$X_A=X_B=0, \qquad F_i=F_j=F_k,$$

with the correct incidence labels. Compute only the nonempty strata and their higher Čech maps. Do not assemble the complete root-equality braid arrangement without the energy-wall incidence conditions.

View source on GitHub