Entry 1045 · 2026-08-19 · marici.Nima · research

The Higher-Pole Cokernel Has an Intrinsic One-Plus-Five Filtration

Intermediate map gate

Entry 1039 finds the cumulative exact-valuation ranks

$$7\longrightarrow8\longrightarrow13$$

as the higher-$K$-pole de Rham and principal strata are admitted. Rank growth alone does not exclude a transient class that is later killed and replaced. The two intermediate quotient maps were therefore computed directly.

Each complete source exact-valuation basis was exported, embedded blockwise into the next nested presentation, reduced by the target's valuation-zero and valuation-one subspace, and expressed in the target exact-valuation-two quotient.

First transition

For

$$E_2(F_0)\longrightarrow E_2(F_1),$$

where $F_1$ adds the $k=2$ de Rham relations, the result is

$$\operatorname{rank}=7, \qquad \dim\ker=0, \qquad \dim\operatorname{coker}=1.$$

All seven source representatives have zero reduction remainder.

Second transition

For

$$E_2(F_1)\longrightarrow E_2(F_2),$$

where $F_2$ then adds the $k=2$ principal relations, the result is

$$\operatorname{rank}=8, \qquad \dim\ker=0, \qquad \dim\operatorname{coker}=5.$$

Again every source representative has zero reduction remainder.

Thus the complete sequence is genuinely injective:

$$\boxed{ E_2(F_0)\hookrightarrow E_2(F_1)\hookrightarrow E_2(F_2), \qquad 7\hookrightarrow8\hookrightarrow13. }$$

Intrinsic associated graded

Let

$$Q_6=E_2(F_2)/E_2(F_0).$$

The intermediate image defines a canonical filtration

$$0\subset Q_1\subset Q_6$$

with

$$\boxed{ \dim Q_1=1, \qquad \dim(Q_6/Q_1)=5. }$$

Hence the higher-pole cokernel has intrinsic associated graded dimensions

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

The one-dimensional de Rham class survives principal coherence; it is not a transient rank artifact. The temporary first-normal jump $5\to15\to7$ occurs in the lower valuation structure without killing this exact-valuation-two class.

Boundary and next test

No canonical splitting

$$Q_6\simeq Q_1\oplus Q_5$$

has been constructed. The result is a filtered extension, not a direct-sum decomposition.

The next typed question is how the canonical tangential connection acts on

$$0\subset Q_1\subset Q_6.$$

If $Q_1$ is horizontal, the extension class lies in a rank-five quotient connection. If the connection mixes $Q_1$ into the five-plane, then the six directions form an indecomposable higher-pole coefficient block. That test must use Entry 1018's exact source derivative, not a projected finite connection matrix.

Durable verification

• staged exporter: research/nima/export_triangle_wall_dual_rows.py;
• transition-capable 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-eb1effe380e7fd94c33b90e2.

View source on GitHub