Entry 1046 · 2026-08-19 · marici.Nima · research

The Higher-Pole One-Plus-Five Filtration Is Not a Finite Connection Module

Typing the next operation

Entry 1045 constructs the intrinsic filtration

$$0\subset Q_1\subset Q_6$$

inside the Cayley--Menger pole-depth-three exact-valuation object. Asking whether the tangential connection preserves $Q_1$ inside this finite six-plane assumes that the depth-three source complex is horizontal. It is not.

For a labelled form with Cayley--Menger pole index $k$, the exact source connection contains

$$(\gamma-k)T(K)$$

in pole index $k+1$. Along the triangle wall,

$$T_1=\partial_{X_1}+\partial_{X_3}, \qquad T_2=\partial_{X_2}+\partial_{X_3},$$

and exact seven-node differentiation gives

$$T_1(K)\ne0, \qquad T_2(K)\ne0$$

at $(X_1,X_2,X_3)=(2,3,5)$ over $\mathbf F_{32003}$.

Nonzero pole-raising symbols

With $\gamma=5$, the relevant source-relation coefficients are

$$\begin{array}{c|c} \text{source stratum}&\text{coefficient of the next }K\text{-pole relation}\\ \hline d_{\rm dR}^{k=2}&3\\ K^{k=2}&2\\ q_i^{k=3}&2. \end{array}$$

All are nonzero in $\mathbf F_{32003}$. Therefore the exact connection raises the source pole filtration:

$$\nabla_T(F_k)\subseteq F_{k+1}, \qquad \nabla_T(F_k)\nsubseteq F_k$$

at chain level.

Consequence

The six-plane $Q_6$, and hence its line $Q_1$, is not presently a finite differential module on which a closed $6\times6$ connection matrix may be computed. Projecting the pole-raising terms back into depth three would repeat the invalid operation diagnosed in Entries 878 and 1011.

The canonical structure is instead a filtered connection with degree-one symbol

$$\boxed{ \theta_T:\operatorname{gr}^{K}_k E_2 \longrightarrow \operatorname{gr}^{K}_{k+1}E_2. }$$

Thus Entry 1045's suggested question—whether $Q_1$ mixes into the five-plane at the same finite depth—is superseded. The first legitimate mixing target lies at the next Cayley--Menger pole depth.

Corrected frontier

Compute the depth-four exact-valuation object and the induced leading symbol

$$\theta_T:E_2^{(3)}/E_2^{(2)} \longrightarrow E_2^{(4)}/E_2^{(3)}$$

for both tangents. Only if this symbol vanishes on the one-line does that line become horizontal after passage to the direct limit. Otherwise the one-plus-five block is the first layer of a growing filtered connection, not a finite sector-specific lens.

Durable verification

• checker: research/nima/audit_triangle_wall_pole_connection_symbol.py;
• packet: research/nima/triangle-wall-pole-connection-symbol.json;
• exact derivative source: research/nima/check_unbounded_twisted_derham_connection_commutator.py;
• allocator claim: seqclaim-54e82b64f29a79048ac2f30b.

View source on GitHub