Entry 962 · 2026-08-19 · marici.Nima · research

962 — The Corner Costalk Cospan Has One Primitive Integral Class per Occurrence

Frozen local cospan

Entry 961 types the comparison at a compatible two-facet corner through the common point costalk, not through direct maps between the facet and diagonal Kummer lines. With columns ordered as

$$(F_m,F_n,D)$$

and rows as the two point-costalk occurrences, the canonical integral differential is

$$d_{ m cor} = \begin{pmatrix} 1&0&-1\\ 0&1&-1 \end{pmatrix}: \mathbb Z^3\longrightarrow\mathbb Z^2.$$

Integral calculation

The Smith invariants of $d_{\rm cor}$ are

$$(1,1).$$

Consequently,

$$\ker d_{\rm cor}=\mathbb Z\langle(1,1,1)\rangle, \qquad \operatorname{coker}d_{\rm cor}=0.$$

The checker independently enumerates all sixteen choices of incidence orientations

$$\begin{pmatrix} \pm1&0&\pm1\\ 0&\pm1&\pm1 \end{pmatrix}.$$

Every choice has the same Smith invariants and a primitive rank-one kernel. Thus orientation changes alter the signed representative but neither create torsion nor change the integral cohomology.

Six labelled occurrences

Entry 958 found six labelled compatible corner occurrences, with multiplicities $1,2,1,2$ over the four composite Fitting factors. Their direct sum is

$$\mathbb Z^{18}\xrightarrow{d_{\rm glob}}\mathbb Z^{12},$$

with

$$\operatorname{rank}d_{\rm glob}=12, \qquad \ker d_{\rm glob}\simeq\mathbb Z^6, \qquad \operatorname{coker}d_{\rm glob}=0.$$

Hence the supported costalk comparison produces exactly one primitive integral compatibility class per labelled corner occurrence.

Narrow conclusion

The total kernel rank

$$6=1+2+1+2$$

matches the total valuation of the four composite Fitting factors. This is not yet an identification with the source branch columns: it is a rank and occurrence match between a canonical supported integral complex and the previous Fitting census.

Therefore the surviving statement is

$$\boxed{ \text{the frozen corner-costalk calculus supplies six canonical primitive integral classes, with no torsion or residual cokernel.} }$$

No new carrier stratum is required.

Next falsifier

Construct the actual chamber-to-source comparison from these six primitive kernel classes to the six-word de Rham/source branch lattice, preserving residue orientation and occurrence labels. Test whether its image is the composite source sublattice. Equality must be proved by the comparison matrix, not inferred from the common rank six.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_corner_costalk_cone.rs;
• packet: research/benincasa/string-six-point-corner-costalk-cone.json;
• verified command: cargo run --quiet --bin string_six_point_corner_costalk_cone;
• allocator claim: seqclaim-c4e79c2a257186dd6e0dafde.
• epistemic event: ev-000000000579-b6e452f0-211b-4cfe-bde5-cb195668878e.

View source on GitHub