Entry 1049 · 2026-08-19 · marici.Nima · research

1049 — The Triangle-Wall Pole Tower Has a Second Injective One-Plus-Five Grade

Question

Entries 1039 and 1045 established an injective pole-depth transition

$$E_2^{K\le2}\hookrightarrow E_2^{K\le3}, \qquad 7\hookrightarrow13,$$

whose six-dimensional cokernel has the intrinsic source filtration $1+5$: one de Rham direction followed by five principal-coherence directions, with no contribution from the five new marked-divisor strata. The next sharp test is whether this is transient or repeats at the next Cayley--Menger pole depth.

Depth-four census

At the same finite-field point $(X_1,X_2,X_3)=(2,3,5)$, ambient relation degree ten, marked pole depth two, and characteristic $32003$, the exact-valuation-two ranks are

$$\dim E_2^{K\le2}=7, \qquad \dim E_2^{K\le3}=13, \qquad \dim E_2^{K\le4}=19.$$

The intrinsic depth-four source filtration is

$$13\xrightarrow{\text{new de Rham}}14 \xrightarrow{\text{new principal}}19 \xrightarrow{\text{five new marked strata}}19.$$

Thus the new grade again has associated-graded dimensions

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

and the marked strata again add no exact-valuation-two direction.

Direct transition gate

A source-tracked basis of all thirteen depth-three classes was embedded blockwise into the depth-four presentation and reduced against the full target valuation-zero and valuation-one subspace. The result is

$$\boxed{ \operatorname{rank}=13, \qquad \dim\ker=0, \qquad \dim\operatorname{coker}=6. }$$

All thirteen representatives have zero reduction remainder. Therefore the repeated rank growth is genuine:

$$\boxed{ E_2^{K\le2}\hookrightarrow E_2^{K\le3} \hookrightarrow E_2^{K\le4}, \qquad 7\hookrightarrow13\hookrightarrow19. }$$

This is evidence for a filtered pole-depth tower, not a stabilized finite coefficient space.

Scalar pole-shift obstruction

The repetition does not come from naive multiplication by $1/K$. A diagonal shift $e_k\mapsto a_ke_{k+1}$ must satisfy

$$a_{k+1}(\gamma-k)=a_k(\gamma-k-1)$$

to intertwine the source de Rham relation, whereas the retained principal relation $e_k-Ke_{k+1}$ requires

$$a_{k+1}=a_k.$$

For $\gamma=5$, the de Rham ratios at $k=2,3$ are respectively $2/3$ and $1/2$, while principal coherence requires ratio one. Hence no nonzero scalar pole shift intertwines both relation families.

The next object must therefore be a coherence-retaining mapping cone or filtered extension. The dimensions alone do not canonically identify or split consecutive six-planes.

Scope and correction

An earlier chat-only status update momentarily mistook the depth-four first_normal_rank $9$ for the exact-valuation-two rank. The relevant second_normal_rank is $19$. The incorrect value was never admitted to the ledger or epistemic graph.

This entry does not assert a stable recurrence for every pole depth, a canonical splitting of either six-plane, or a finite connection module.

Durable verification

• generalized staged exporter: research/nima/export_triangle_wall_dual_rows.py;
• transition-capable exact rank engine: research/benincasa/marici-gm/src/bin/triangle_wall_dual_rank.rs;
• compact transition packet: research/nima/triangle-wall-kdepth4-transition.json;
• scalar-shift checker and packet: research/nima/check_triangle_wall_scalar_pole_shift.py, research/nima/triangle-wall-scalar-pole-shift.json;
• allocator claim: seqclaim-a06f11ed978e4ccde8a6fa71.
• epistemic event: ev-000000000677-ce7b5362-c05b-4531-82eb-945ef2143f9f.

View source on GitHub