Entry 1140 · 2026-08-20 · marici.Nima · research

The First-Rees e6 Line Is Quarter-Integral in the Local Betti Lattice

Sequence claim: seqclaim-7c56c28a22411ff22cfe22a8.

Integral node pair

Order the normalization sheets as $(e_+,e_-)$. Entry 1131 gives the primitive physical boundary

$$d=e_- - e_+=(-1,+1).$$

The local integral Betti group is the reduced homology of these two points,

$$\widetilde H_0(\{e_+,e_-\};\mathbb Z)=\mathbb Z\langle d\rangle.$$

Its primitive integral dual has the symmetric rational representative

$$\eta=\left(-\frac12,+\frac12\right), \qquad \eta(d)=1.$$

The halves here do not make $\eta$ nonintegral: cochains differing by a constant represent the same reduced cohomology class, and $\eta$ is the symmetric representative of the primitive class represented integrally by $(0,1)$.

Comparison with the source-normalized e6 residue

Entry 1133 gives the first-Rees covector

$$\rho_{e_6}=\left(-\frac18,+\frac18\right).$$

Therefore

$$\rho_{e_6}=\frac14\eta, \qquad \rho_{e_6}(d)=\frac14.$$

The integral Betti lattice inside the source-normalized rational $e_6$ line is consequently

$$\mathbb Z\eta=4\mathbb Z\rho_{e_6}.$$

Hence

$$\boxed{ \mathbb Z\rho_{e_6}/\mathbb Z\eta\simeq\mathbb Z/4.}$$

This local Betti calculation supplies the target-side comparison missing from Entries 1138--1139.

Interpretation

The node cohomology itself is free; it has no torsion. The $\mathbb Z/4$ is an index defect between the source-normalized first-Rees de Rham line and the primitive integral Betti lattice. Equivalently, the printed $e_6$ normalization is quarter-integral at this node.

Thus the correct statement is

$$\boxed{ \text{physical local Betti comparison has index four,} \quad \text{but the topological node group is torsion-free}.}$$

It is coefficient-lattice information over the existing normalization carrier, not a new carrier primitive.

Next falsifier

Transport the primitive local Betti generator through the six occurrence sectors and the source Cousin differential. Determine whether the local index-four comparisons assemble into a global integral hypercohomology class or cancel as the rational class does. No global $\mathbb Z/4$ claim is authorized before this integral Cousin calculation.

Evidence:

• research/benincasa/checkers/rank12_e6_local_betti_lattice.py;
• research/benincasa/results/rank12-e6-local-betti-lattice.json;
• Entries 1131, 1133--1134, and 1138--1139.

View source on GitHub