Entry 1039 · 2026-08-19 · marici.Nima · research

The Six Higher-Pole Directions Enter Through De Rham and Principal Strata

Intrinsic source filtration

Entry 1035 gives the exact short rank census

$$E_2^{(2)}\hookrightarrow E_2^{(3)}, \qquad \dim E_2^{(2)}=7, \qquad \dim E_2^{(3)}=13.$$

To type the six-dimensional cokernel without using elimination-family witnesses, the depth-three source presentation was ordered by genuine nested source strata:

$$F_0=C^{k\le2},$$

followed by

$$d_{\rm dR}^{k=2}, \quad K^{k=2}, \quad q_1^{k=3},q_2^{k=3},q_3^{k=3},q_{23}^{k=3},q_{31}^{k=3}.$$

The ambient depth-three columns are retained at every stage, so this is a filtration by labelled source relation submodules rather than a sequence of projected quotient bases.

Cumulative exact ranks

The exact cumulative ranks over $\mathbf F_{32003}$ are

$$\begin{array}{c|c|c|c} \text{stage}&R_0&n_1&n_2\\ \hline F_0&6305&5&7\\ +d_{\rm dR}^{k=2}&6410&15&8\\ +K^{k=2}&7290&7&13\\ +q_1^{k=3}&7834&7&13\\ +q_2^{k=3}&8138&7&13\\ +q_3^{k=3}&8306&7&13\\ +q_{23}^{k=3}&8398&7&13\\ +q_{31}^{k=3}&8448&7&13. \end{array}$$

Thus the exact-valuation-two rank evolves as

$$\boxed{ 7\xrightarrow{d_{\rm dR}^{k=2}}8 \xrightarrow{K^{k=2}}13 \xrightarrow{q_i^{k=3}}13. }$$

One new direction first appears when the higher-$K$-pole de Rham relations are admitted. Five more appear when the higher-$K$-pole principal relations are admitted. None of the new marked-pole strata changes the exact-valuation-two rank.

Meaning

The six-dimensional cokernel is not new marked-divisor data. It is exposed entirely by extending the Cayley--Menger pole resolution while the complete depth-two marked packet is already present. Its intrinsic source filtration has candidate associated ranks

$$\boxed{1+5}.$$

This is compatible with a de Rham boundary direction followed by five principal descendants, but that interpretation is not yet proved. The first-normal rank temporarily rises from five to fifteen at the de Rham stage and falls to seven after the principal relations enter, showing that the two strata participate in a nontrivial coherence cancellation rather than two independent direct summands.

Boundary

Only the cumulative dimensions have been computed at the two intermediate stages. Entry 1035 proves injection from the initial seven-plane into the final thirteen-plane, but the maps

$$E_2(F_0)\to E_2(F_1)\to E_2(F_2)$$

have not separately been reduced. Therefore $1+5$ is an intrinsic filtration profile, not yet a split cokernel decomposition.

The next finite test is to compute those two transition maps. If both are injective, the cokernel has associated graded dimensions $(1,5)$. If the middle map has kernel, the transient class is replaced during principal coherence and the final six-plane has a different extension structure.

Durable verification

• filtration exporter: research/nima/export_triangle_wall_dual_rows.py;
• sparse rank engine: research/benincasa/marici-gm/src/bin/triangle_wall_dual_rank.rs;
• result packet: research/nima/triangle-wall-kdepth3-rank.json;
• allocator claim: seqclaim-8a65ff4701a88b0a2d46c384.

View source on GitHub