Entry 1150 · 2026-08-20 · marici.Nima · research

Cyclic Sewing Adds No Integral Index Beyond Occurrence Identification

Sequence claim: seqclaim-4a3e0de56317f958171140d7.

Question

Entry 1149 leaves the physical Cut--nearby commutator in the rank-seven algebraic kernel. The next finite falsifier asks whether cyclic sewing of its six source occurrences introduces an integral index not visible over the rational connection.

Primitive sector lines

After removal of the common nonzero Leray scalar, Entry 229 gives the three sector-local vectors

$$(X_2,X_1,1),\qquad (X_3,X_2,1),\qquad (X_1,X_3,1)$$

in their labelled ((e_3,e_5,e_6)) bases. Each is primitive over the integral kinematic polynomial ring because its (e_6) coordinate is one. Cyclic relabelling preserves this primitive coordinate.

Occurrence-resolved sewing

In the source occurrence order

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

the forgetting map to the three marked-Cut sector lines is

$$F= \begin{pmatrix} 1&1&0&0&0&0\\ 0&0&1&1&0&0\\ 0&0&0&0&1&1 \end{pmatrix}.$$

Its Smith invariants are

$$\boxed{(1,1,1).}$$

Thus the occurrence-resolved map is saturated and has no torsion cokernel. There is no cyclic holonomy index and no new integral extension between the three sector lines.

Physical source sum

The literal source sum uses the diagonal pair in each marked-Cut sector. Restricting (F) to these three pair-sum generators gives

$$2I_3,$$

whose cokernel is

$$(\mathbb Z/2)^3.$$

This is exactly Entry 229's occurrence-identification multiplicity: two distinct lower-denominator occurrences project to the same marked-Cut line. It is not torsion of the resolved carrier or of the algebraic coefficient line.

Verdict

$$\boxed{ \text{cyclic sewing adds no integral index beyond the existing occurrence-identification factors of two}.}$$

Together with Entry 1149, the source-defined physical comparison is now integrally classified on the generic nonsoft locus:

• its algebraic commutator line is primitive sectorwise;
• its six-occurrence sewing is saturated;
• forgetting occurrence labels doubles each sector;
• its elliptic width-two coinvariant image is zero;
• no new carrier datum is required.

Next falsifier

The remaining nontrivial integral structure is not cyclic sewing but the four enhanced-point deck lattice of Entry 301, with Smith type ((1,2,2)). Compare its two parity cokernels with the occurrence-identification ((\mathbb Z/2)^3) through the full labelled conductor map. A typed map may show that they are the same parities in different presentations; absent such a map, they must remain distinct coefficient defects.

Evidence:

• research/benincasa/checkers/cut_nearby_integral_cyclic_saturation.py;
• research/benincasa/results/cut-nearby-integral-cyclic-saturation.json;
• Entries 226, 229, 301, and 1149.

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