Entry 1137 · 2026-08-20 · marici.Nima · research

The Six First-Rees Germs Sew as a Source-Exact Cousin Vector

Sequence claim: seqclaim-07cd890d9f05cb406e9cf5de.

Correct global object

The three rank-twelve residue surfaces are not an open cover, so Entry 1136's coherent germs cannot be Čech-glued. The frozen global object is the six-occurrence Cousin packet of Entry 364.

In source occurrence order

$$(12|23),(12|31),(23|31),(23|12),(31|12),(31|23),$$

the source coefficients and residue orientations are all $+1$. Entry 1134's normalized local pairing therefore gives the degree-one vector

$$\boxed{ c_{e_6}^{(1)}=\frac14(1,1,1,1,1,1). }$$

All pairwise marked-Cut residues vanish, so

$$d_1c_{e_6}^{(1)}=0.$$

Exactness

Entry 364 establishes

$$d_0\Omega_{\rm src}=(1,1,1,1,1,1).$$

Over the rational source coefficient system,

$$\boxed{ c_{e_6}^{(1)}=d_0\left(\frac14\Omega_{\rm src}\right). }$$

Thus the six local germs genuinely assemble, but their assembled vector is exact in the full source-defined Cousin complex.

If degree zero is truncated away, the same vector survives as a nonzero first-Rees associated support grade. Consequently

$$\boxed{ \text{nonzero physical filtered residue} \not\Rightarrow \text{new global cohomology class}. }$$

Occurrence forgetting gives

$$\left(\frac12,\frac12,\frac12\right),$$

the expected factor-two identification of the two lower-denominator occurrences in each marked-Cut sector.

Classification

• local physical pairing: nonzero, first higher-Rees grade;
• cyclic and reflection transport: coherent;
• global source packet: exact filtered residue;
• new rational global cohomology: none;
• new carrier datum: none.

This is a strong H2 outcome: existing carrier and Cousin/Gysin calculus, with a sector-specific filtered coefficient class that disappears after the full source differential is restored.

Next falsifier

The rational global question is closed. The remaining nonredundant issue is integral: $(1/4)\Omega_{\rm src}$ need not belong to the source integral lattice. Compute the saturation of the six-vector relative to the primitive degree-zero source generator. A surviving finite cokernel would be integral coefficient torsion, not a new carrier class.

Evidence:

• research/benincasa/checkers/rank12_e6_global_cousin_sewing.py;
• research/benincasa/results/rank12-e6-global-cousin-sewing.json;
• Entries 353, 356, 364, and 1133--1136.

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