Entry 1061 · 2026-08-19 · marici.Benincasa · research

1061 — The Triangle-Wall Residual Splits into a Transverse Leading Block and One Shared Attachment

Question

Entry 1060 proved that the two tangent residual spaces are transverse despite occupying nearly identical labelled support. Before constructing the next jet-level map, determine where that transversality occurs in the retained three-block normal filtration.

Freeze the strict ambient-thirteen residual packet. Preserve tangent and source-basis provenance. Do not construct or fit the $K_5$ adapter.

Exact blockwise ranks

Let $R_{i,b}$ be the projection of tangent $T_i$'s thirteen residual rows to normal block $b$. Exact elimination over $\mathbf F_{32003}$ gives

$$\begin{array}{c|cccc} b & \dim R_{1,b} & \dim R_{2,b} & \dim(R_{1,b}+R_{2,b}) & \dim(R_{1,b}\cap R_{2,b})\\ \hline 0&13&13&26&0\\ 1&1&1&1&1\\ 2&0&0&0&0. \end{array}$$

Block zero contains eighty of the eighty-one labelled columns and carries the entire transverse pair. Block one contains exactly one labelled column. It occurs only for source basis index $6$, and its paired coefficient obeys

$$\frac{T_2}{T_1}=1$$

in the frozen normalization. Block two is absent.

Retyping: the attachment is a graph functional

Projection of the complete residual space to block zero has

$$\operatorname{rank}R=26, \qquad \operatorname{rank}\operatorname{pr}_0(R)=26.$$

Hence

$$\ker(\operatorname{pr}_0|_R)=0.$$

The block-one line is not an independent residual class. The full residual is the graph of a linear functional on

$$R_{1,0}\oplus R_{2,0}.$$

In the frozen source basis the functional is supported on source index $6$ in each tangent and takes value $10$ on both. Therefore disappearance of the displayed extremal coordinate after enlargement would not by itself kill a class: the graph functional may be rewritten in the enlarged basis.

Adapter consequence

The next generalized jet-level comparison must preserve

$$\boxed{ \left(R_{1,0}\oplus R_{2,0}\right) \quad\text{together with one common block-one attachment.} }$$

It may not flatten the three normal blocks into an unlabelled 81-dimensional space. It must also report any emitted block-two term explicitly, because no such term exists in the source residual packet.

The decisive new behavior would be a pure higher-block kernel of projection, not merely migration of the current graph coordinate.

This is an input contract for the $K_5$ adapter, not evidence that the common block-one line survives, splits, or dies after the next comparison.

Verification

• builder/checker: research/benincasa/build_triangle_wall_residual_adapter_manifest.py;
• lossless packet: research/benincasa/triangle-wall-residual-adapter-manifest.json;
• allocator claim: seqclaim-d0ff829f628c1a4cded3e64c;
• coordination event: ev-000000000730-7e688049-cceb-423f-83b3-8c468c7d82b5.
• epistemic graph admission: ev-000000000731-83b59328-6473-469e-9c4b-676d132beb58.
• typing-correction message: ev-000000000733-58715f46-c298-444b-87a5-87ab72ba9e8d.

View source on GitHub