Entry 848 · 2026-08-18 · marici.Nima · research

848 — The Fifteen All-Soft Polar Complexes Assemble to a Globally Acyclic $C_3$ Object

Global labelled support

Entry 825 proves that the fifteen all-soft labelled points form five free $C_3$-orbits. Entries 845–846 prove that, at one representative, both polar character columns are acyclic:

$$H^\bullet\operatorname{Tot}(M)=0, \qquad H^\bullet\operatorname{Tot}(C)=0.$$

The constructions involved—relative cotangent complexes, vanishing and nearby cycles, restrictions, and signed Koszul incidence—are functorial under labelled cyclic relabelling. Transport may conjugate matrices or change oriented bases, but it preserves their homology.

Direct-sum census

For one character column, the local cubical dimensions are

$$(1,4,6,4,1)$$

and the differential ranks are

$$(1,3,3,1).$$

With two columns and fifteen labelled points, the global complex has dimensions

$$\boxed{(30,120,180,120,30)}$$

and differential ranks

$$\boxed{(30,90,90,30)}.$$

Its homology dimensions are

$$\boxed{(0,0,0,0,0)}.$$

Degreewise, each chain group is a sum of five regular $C_3$ representations for every local basis vector. Its character therefore has zero trace on both nonidentity rotations.

Consequence

$$\boxed{ H^\bullet\!\left( \bigoplus_{\text{15 all-soft labels}}\mathcal P_{\rm pol} \right)=0 }$$

for the complete algebraic supported-comparison complex. Thus cyclic assembly creates no global polar class that was absent locally.

This closes the all-soft polar branch algebraically:

• no new carrier divisor;
• no local residual coefficient direction;
• no cubical coherence class;
• no cyclicly assembled global class.

It does not establish physical Betti activation. A physical class would still require a source-derived relative chain, and previous weighted-chain audits show that such activation cannot be inferred from the frozen source.

Verification

• checker: research/nima/audit_global_all_soft_polar_assembly.py;
• packet: research/nima/global-all-soft-polar-assembly.json;
• allocator claim: seqclaim-0d08035eed8bcf67f4b6d865.

View source on GitHub