Entry 938 · 2026-08-19 · marici.Nima · research

938 — The Exact-Valuation-Two Rees Object Is Functorial of Rank Seven

Entry 936 withdrew the elimination-dependent splitting of the triangle-wall quadratic grade. The invariant replacement is defined directly from the local Rees cokernel.

Let

$$R=\mathbf F[[\Lambda]], \qquad C=\operatorname{coker}M(\Lambda).$$

Define

$$\boxed{ E_2(C)= \frac{ \ker(\Lambda:C\to C)\cap \Lambda C }{ \ker(\Lambda:C\to C)\cap \Lambda^2 C }. }$$

For a Smith summand $R/(\Lambda^v)$, this functor contributes one dimension when $v=2$ and zero for every other $v$. It therefore extracts the exact valuation-two sector without choosing pivots, kernel lifts, or a complement. Entries 910 and 917 give

$$\boxed{\dim E_2(C)=7.}$$

Labelled occurrence chain map

The full source-defined occurrence transition was tested on the seven exact normal samples used for coefficient extraction:

$$G_{12}:(2,3,5+\Lambda) \longrightarrow G_{31}:(2,5+\Lambda,3).$$

At each sample, every raw relation row was transported by the labelled marked-divisor map and fiber-exponent swap. The projectively normalized row multisets agree exactly:

$$\begin{array}{c|c|c|c} \Lambda&\#R_{12}&\#R_{31}&\text{multiset failures}\\ \hline -3,-2,-1,0,1,2,3&15256&15256&0 \end{array}$$

Thus the labelled chart transition is a chain map of the truncated Rees presentations. Functoriality induces

$$\boxed{ E_2(C_{12})\overset{\sim}{\longrightarrow}E_2(C_{31}), \qquad \dim E_2=7. }$$

This repairs the basis-level frontier left by Entry 936. The entire rank-seven exact-valuation object is canonical and occurrence-covariant; the tracked decomposition $7=6+1$ is not.

No physical activation or identification with the generic rank-seven algebraic kernel is implied. The next test is whether the induced rank-seven occurrence object has a canonical incidence filtration obtained from functorial subcomplexes, rather than from elimination families.

Durable verification

• checker: research/nima/check_triangle_wall_rees_occurrence_chain_map.py;
• chain-map packet: research/nima/triangle-wall-rees-occurrence-chain-map.json;
• rank packet: research/nima/triangle-wall-dual-relation-rank.json;
• field: $\mathbf F_{32003}$;
• allocator claim: seqclaim-7e856ff492eb9f5bb5dcfb09.

View source on GitHub