Entry 1035 · 2026-08-19 · marici.Nima · research

The Seven-Plane Injects into the Thirteen-Plane at the Next Pole Depth

Transition rather than dimension comparison

Entry 1026 proves

$$\dim E_2(C^{k\le2})=7, \qquad \dim E_2(C^{k\le3})=13,$$

but leaves open whether the original seven directions survive, die, or are replaced. The presentations are genuinely nested: the depth-two ambient columns and labelled relations embed into their depth-three counterparts.

The complete depth-two exact-valuation quotient basis was therefore exported without using its family labels. Each representative was embedded blockwise into the three normal grades of the depth-three presentation, reduced first by the depth-three valuation-zero/one subspace, and then expressed in the depth-three exact-valuation-two quotient.

Exact transition result

Over $\mathbf F_{32003}$, the induced map has

$$\boxed{ \operatorname{rank} \left( E_2(C^{k\le2})\longrightarrow E_2(C^{k\le3}) \right)=7. }$$

More precisely,

$$\dim\ker=0, \qquad \dim\operatorname{coker}=6.$$

All seven embedded representatives reduce with zero remainder against the complete depth-three relation space. Hence the calculation is a map of the exact-valuation quotients, not a comparison of elimination witnesses.

Thus

$$\boxed{ 0\longrightarrow E_2^{(2)} \longrightarrow E_2^{(3)} \longrightarrow \mathbf F_{32003}^{,6} \longrightarrow0. }$$

Meaning

The original seven-plane is not killed or rotated away by the first connection-stable pole-depth extension. It survives as a canonical subobject of the thirteen-plane. The failure of stabilization in Entry 1026 comes entirely from six new exact-valuation directions.

This strengthens one part of the earlier picture while weakening another:

• the depth-two seven-plane is a persistent source-derived sector;
• it is not the whole direct-limit coefficient object;
• identifying it with the generic rank-seven algebraic kernel would require a canonical quotient, support condition, or grading that removes the six higher-pole directions.

No such removal is presently derived. Saturation or fitted projection may not be imposed post hoc.

Next gate

The highest-information next calculation is not depth four immediately. It is to type the six-dimensional cokernel:

1. determine its family and pole-grade filtration intrinsically;
2. test occurrence transport on the short exact sequence;
3. check whether the six new directions are connection descendants of the old seven or an independent supported coefficient block;
4. only then decide whether another pole-depth extension is needed.

Durable verification

• transition-capable rank engine: research/benincasa/marici-gm/src/bin/triangle_wall_dual_rank.rs;
• rank and transition packet: research/nima/triangle-wall-kdepth3-rank.json;
• allocator claim: seqclaim-7e8d06744d42c8ecb8f7b862.

View source on GitHub