Entry 1163 · 2026-08-20 · marici.Nima · research

Branch-Enhanced Four-Cycle Triple Points Have Order-Six Kummer Inertia

Sequence claim: seqclaim-ec80e0a7ae58cc9816fd3e32.

Retracted by Entry 1164. This entry incorrectly retained the edge-degree-two term $4\det(G)z_1$ in the projective infinity branch. The correct degree-four branch is $4B_4=-\Delta^T\operatorname{adj}(G)\Delta$. Consequently the five forced incidence points are branch-singular when activated, and the smooth-branch order-six calculation below does not apply to them. It is retained only as a record of the falsified intermediate calculation.

Finite support reduction

Entry 1162's 296 labelled forced triple incidences occupy only five projective points in the four edge variables:

$$[1:0:0:0],\ [0:1:0:0],\ [0:0:1:0],\ [0:0:0:1], \ [1:-1:1:-1].$$

Their occurrence multiplicities are respectively

$$38,38,38,38,144.$$

Thus the branch-enhancement problem is a finite five-point evaluation, not an unbounded search through the 28 terms.

Source quartic

Let $G$ be the Gram matrix of three independent external momentum increments and set

$$z_i=y_i^2, \qquad \Delta=(z_2-z_1,z_3-z_2,z_4-z_3)^T.$$

The homogeneous one-loop Cayley--Menger/Baikov branch quartic is

$$\boxed{ 4B_4 =4\det(G)z_1-\Delta^T\operatorname{adj}(G)\Delta. }$$

Evaluation at the five incidence points produces five explicit external Gram-minor equations. None vanishes identically: the identity Gram matrix gives values

$$3,2,2,1,1.$$

At the alternating point, the support is exactly

$$\det G=0.$$

Hence every activation divisor is a pullback/fiber product of the existing Cayley--Menger branch with a frozen marked-incidence section. No new carrier equation is introduced.

Generic branch-hit normal form

At a smooth branch point choose base coordinates $(u,v)$ with branch $u=0$. On the double cover,

$$u=w^2.$$

Three distinct base lines through the point pull back to three smooth branches

$$v+c_iw^2=0, \qquad c_i\ne c_j.$$

Their union is

$$\boxed{ f(w,v)=\prod_{i=1}^3(v+c_iw^2)=0. }$$

It is weighted homogeneous of degree six for weights

$$\operatorname{wt}(w)=1, \qquad \operatorname{wt}(v)=2.$$

Each pair has contact order two. The delta invariant is six and

$$\boxed{\mu(f)=10.}$$

Monodromy and coefficient type

The Milnor-algebra Hilbert coefficients are

$$(1,1,2,2,2,1,1),$$

and the spectral multiplicities are

$$\begin{array}{c|rrrrrrr} \alpha&-\frac12&-\frac13&-\frac16&0&\frac16&\frac13&\frac12\\ \hline m_\alpha&1&1&2&2&2&1&1. \end{array}$$

The semisimple monodromy orders occur with ranks

$$\boxed{ 1:2, \qquad2:2, \qquad3:2, \qquad6:4. }$$

Two ordinary blowups, equivalently the $(1,2)$-weighted blowup, resolve the divisor using only rational exceptional strata. Therefore the generic enhanced coefficient object is mixed Tate with finite Kummer/Artin inertia of order dividing six. It is not a new elliptic or higher-genus block.

Architectural result

The strongest four-site hostile incidence test currently closes as

$$\boxed{ \text{existing Cayley--Menger/marked carrier} +\text{ weighted nearby-cycle calculus} +\text{ finite Kummer/Artin coefficients}. }$$

This is positive evidence for H2 and against both a new four-site carrier primitive and a universal coefficient architecture.

Scope and next falsifier

The calculation assumes the quartic branch is smooth and transverse in the normal direction at the selected incidence point. Deeper loci where the external Gram matrix degenerates further, or where the branch itself is singular/tangent, remain separate supported tests.

The next step is to organize the five activation equations into cyclic orbits and test their intersections with soft, signed-energy, and total- energy support. Only a residual local system not generated by the current Gram, Kummer, and elliptic objects would update the carrier hypothesis.

Evidence:

• research/benincasa/checkers/audit_four_cycle_branch_enhanced_triples.py;
• research/benincasa/results/four-cycle-branch-enhanced-triples.json;
• Entries 1154 and 1159--1162.

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