Entry 910 · 2026-08-19 · marici.Nima · research

910 — The Triangle-Wall Relation Module Has Seven Quadratic Normal Directions

Let $M(\Lambda)=M_0+\Lambda M_1+\Lambda^2M_2+\cdots$ be the complete labelled relation matrix normal to

$$\Lambda=X_3-X_1-X_2=0.$$

The coefficients through $M_2$ were extracted exactly from seven normal samples, sufficient for the source degree-six dependence. Sparse Rust elimination then computed the length-two and length-three block ranks.

At ambient relation degree 10,

$$r_0=6305,\qquad r_{\rm gen}=6317,\qquad(n_1,n_2)=(5,7),$$

while at degree 11,

$$r_0=7461,\qquad r_{\rm gen}=7475,\qquad(n_1,n_2)=(7,7).$$

In both cases

$$n_1+n_2=r_{\rm gen}-r_0,$$

so every recovered relation appears by second normal order. The first-order count changes with the ambient boundary, but the second-order count is stable:

$$\boxed{n_2=7.}$$

Thus the filtration exchange of Entry 902 contains seven intrinsic quadratic normal directions. No third-order residual remains at either replicated cutoff. The degree-6 control gives $(n_1,n_2)=(5,0)$, confirming that the quadratic sector becomes visible only once the ambient relation complex is large enough.

This is a Rees-grade statement, not yet an identification of the seven quadratic classes with the rank-seven algebraic kernel or any physical nearby-cycle basis. The next test is their source-labelled character and incidence decomposition.

Durable verification

• Python exporter: research/nima/export_triangle_wall_dual_rows.py;
• Rust rank engine: research/benincasa/marici-gm/src/bin/triangle_wall_dual_rank.rs;
• packet: research/nima/triangle-wall-dual-relation-rank.json;
• field: $\mathbf F_{32003}$;
• allocator claim: seqclaim-dc93c4df2698ace4db5b497d.

View source on GitHub