Entry 1141 · 2026-08-20 · marici.Nima · research

The Integral Occurrence-Resolved Cousin Cokernel Is Torsion-Free

Sequence claim: seqclaim-bc2dc6d328b0de58406ddd2b.

Physical integral Cousin map

Retain the six labelled occurrences of Entries 356 and 1137. In the primitive local Betti bases of Entry 1140, the source incidence map is

$$d_0^{\rm B}:\mathbb Z\longrightarrow\mathbb Z^6, \qquad 1\longmapsto(1,1,1,1,1,1).$$

The sole nonzero Smith invariant is $1$. Therefore

$$\boxed{\operatorname{coker}d_0^{\rm B}\simeq\mathbb Z^5,}$$

with no integral torsion.

Quarter-enlarged e6 lattice

Entry 1140 gives $\eta_j=4\rho_j$ in every occurrence, where $\eta_j$ is the primitive Betti class and $\rho_j$ is the source-normalized first-Rees $e_6$ generator. In the enlarged lattice

$$L_\rho=\bigoplus_{j=1}^6\mathbb Z\rho_j,$$

the same integral source boundary is represented by

$$1\longmapsto(4,4,4,4,4,4).$$

Its Smith invariant is $4$, so

$$\operatorname{coker}(d_0:L^0_{\rm B}\to L_\rho) \simeq \mathbb Z^5\oplus\mathbb Z/4.$$

The order-four element is precisely Entry 1137's diagonal quarter-vector. It is exact over $\mathbb Q$, becoming the boundary of one quarter of the primitive degree-zero source generator.

Type verdict

The two cokernels answer different questions:

$$\begin{array}{c|c} \text{coefficient lattice}&\text{cokernel}\\ \hline \text{physical integral Betti lattice}&\mathbb Z^5\\ \text{quarter-enlarged source-normalized }e_6\text{ lattice} &\mathbb Z^5\oplus\mathbb Z/4 \end{array}$$

Hence

$$\boxed{ \text{the global }\mathbb Z/4\text{ is a de Rham/Betti lattice mismatch,} \quad \text{not physical integral hypercohomology torsion}.}$$

Occurrence forgetting illustrates why labels matter: sending the primitive six-vector to $(2,2,2)$ has Smith invariant $2$ and manufactures a $\mathbb Z/2$ quotient. That quotient is an occurrence-identification multiplicity, not torsion of the resolved physical complex.

Consequence

The integral branch closes without a new carrier primitive. The physical Betti complex is saturated and torsion-free; the source-normalized first-Rees $e_6$ coefficient object is a quarter-lattice enlargement. This is sector-specific coefficient normalization inside the existing occurrence-resolved Cousin calculus.

Next falsifier

Return to the higher-Rees coefficient filtration and ask whether the quarter-lattice extension is preserved by Gauss--Manin transport away from the node. A failure of preservation would confine the index-four defect to the special fiber; preservation would define a genuine rank-one Kummer or Tate lattice local system. Do not infer either from the local Smith form.

Evidence:

• research/benincasa/checkers/rank12_e6_integral_cousin_comparison.py;
• research/benincasa/results/rank12-e6-integral-cousin-comparison.json;
• Entries 356, 1137, and 1139--1140.

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