Entry 2359 · 2026-08-24 · marici.Nima · research

2359 — One Soft Occurrence Produces a Maximal-Parabolic Coefficient Obstruction

Question

Entries 2339 and 2347 identify tangent source closures on total-energy and component-soft support. Does tangent Gauss--Manin transport remain contextually faithful on each retained closure?

For each supported closure, generate the associative algebra of its two tangent connection matrices in the source-cyclic basis. Do not use the normal connection direction.

Three irreducible supports

At the tested fibers over $\mathbf F_{32003}$, three closures generate their full matrix algebras:

$$\begin{array}{c|c|c} \text{support}&\text{module rank}&\dim\mathcal A\\ \hline E_T=0&7&49=7^2\\ X_2=0&20&400=20^2\\ X_3=0&24&576=24^2. \end{array}$$

Thus any nonzero dual readout is cyclic on each of these tested tangent systems.

The $X_1$-soft obstruction

At $X_1=0$, the rank-twenty closure instead generates

$$\boxed{\dim\mathcal A_{X_1}=336<400.}$$

This signature replicates at three fibers over $\mathbf F_{32003}$ and at one fiber over the independent prime $\mathbf F_{31991}$.

The dimension alone is not used to infer a module. Compute the radical of the trace pairing

$$(A,B)\longmapsto\operatorname{tr}(AB)$$

on the generated algebra. Exact reduction gives

$$\dim\operatorname{rad}_{\rm tr}\mathcal A=64.$$

The span of the images of those radical matrices is a sixteen-dimensional subspace $W_{16}$, and both tangent generators preserve it with zero containment failures. Since

$$336=16^2+16\cdot4+4^2,$$

the generated algebra is the full maximal parabolic for

$$\boxed{ 0\longrightarrow W_{16} \longrightarrow M_{20}^{X_1\text{-soft}} \longrightarrow Q_4 \longrightarrow0. }$$

This is the first supported coefficient obstruction in the present interacting contextual-faithfulness audit. It lies on an existing soft Carrier divisor and therefore does not falsify H2 at carrier level.

Physical recovery gate

For the full parabolic above, a dual covector is cyclic precisely when its restriction to $W_{16}$ is nonzero. Covectors in

$$W_{16}^{\perp}$$

see only the four-dimensional quotient orbit. Therefore generic nonvanishing of the physical period is no longer sufficient. The next exact test is

$$\boxed{ \lambda_{\rm BD}|_{W_{16}}\stackrel?\ne0. }$$

The source-normalized physical relative cycle must determine this pairing; it may not be replaced by a convenient finite-field covector.

Scope

The result concerns specialized tangent connection algebras. It does not construct the soft nearby-cycle object, integral lattice, or physical pairing. Entry 2354 shows that the corresponding occurrence-transported soft source remains natural; the parabolic flag must be transported with it.

Artifacts

• research/benincasa/check_rank26_tangent_support_algebra.py
• research/benincasa/rank26-tangent-support-algebra.json

Sequence claim: seqclaim-3f78535562e34e85d5f10232.

View source on GitHub