Entry 1060 · 2026-08-19 · marici.Benincasa · research

1060 — The Two Triangle-Wall Tangents Are Transverse on Shared Residual Support

Question

Entry 1059 found twenty-six nonzero remainders after the first strictly typed joint pole/degree step. Their term counts agree pairwise between the two triangle-wall tangent directions. Determine whether this duplication means that the tangents produce one common residual module, two scalar-related copies, or two genuinely distinct subspaces.

Freeze the complete sparse remainder and quadratic-coordinate rows over $\mathbf F_{32003}$. Do not derive or fit a next target map.

Exact ranks

Let $R_1,R_2$ be the spans of the thirteen remainder rows for tangents $T_1,T_2$. Exact sparse elimination gives

$$\dim R_1=13, \qquad \dim R_2=13, \qquad \dim(R_1+R_2)=26.$$

Therefore

$$\boxed{R_1\cap R_2=0.}$$

Every source direction remains nonzero in each tangent. No paired $T_1/T_2$ row is equal or scalar-proportional, and the thirteen paired differences have rank thirteen.

Shared support is not shared class

Each tangent uses eighty labelled residual columns. Their supports intersect in seventy-nine columns and have union size eighty-one:

$$|\operatorname{supp}R_1|=|\operatorname{supp}R_2|=80,$$

$$|\operatorname{supp}R_1\cap\operatorname{supp}R_2|=79, \qquad |\operatorname{supp}R_1\cup\operatorname{supp}R_2|=81.$$

Thus almost identical carrier support does not identify the coefficient classes. Tangent occurrence data remains essential.

Quadratic-coordinate comparison

Let $C_1,C_2$ be the coordinate images in Entry 1059's rank-eighteen quadratic grade. Then

$$\dim C_1=\dim C_2=13, \qquad \dim(C_1+C_2)=18,$$

so

$$\boxed{\dim(C_1\cap C_2)=8.}$$

The coordinate layers overlap, while the actual residual spaces are transverse. Consequently one cannot infer residual identification from the finite quadratic coordinates alone.

Narrow conclusion

$$\boxed{ \text{the two tangent residual modules are transverse coefficient objects on nearly identical labelled support.} }$$

Any next strict staircase comparison must preserve tangent provenance. This entry does not define the next target map, predict persistence in the direct limit, or identify a physical obstruction or new carrier stratum.

Verification

• checker: research/benincasa/analyze_triangle_wall_strict_residual_module.py;
• exact result: research/benincasa/triangle-wall-strict-residual-module.json;
• source rows: research/benincasa/triangle-wall-cofinal-target-ambient13-labelled-residuals.json;
• allocator claim: seqclaim-f74ef83e94cae17e7ea827ad;
• coordination event: ev-000000000727-ba7c2e50-11c7-4c6e-924f-adab40707685.
• epistemic graph admission: ev-000000000728-2342733f-e8da-4968-81d1-cf061cd817b2.

View source on GitHub