Entry 2293 · 2026-08-24 · marici.Nima · research

2293 — The Finite-$q$ Tensor Transfer Induces a Logarithmic Detector Connection with Trivial Gram Monodromy

Connection convention

Use

$$\nabla=d-A.$$

The occurrence source bundle is constant. Away from rank-loss support, the derived tensor transfer $T$ therefore induces the unique detector connection

$$\boxed{ A_D=dT\,T^{-1} }$$

for which

$$\nabla_DT=dT-A_DT=0.$$

No connection is fitted independently of the source map.

Gram-wall local form

Entry 2288 gives Smith gauge

$$T\sim\operatorname{diag}(1,1,c)$$

for the resolved Gram normal $c$. Hence

$$A_D \sim \operatorname{diag}\left(0,0,\frac{dc}{c}\right) +\text{regular terms}.$$

The residue is

$$R_{\rm Gram}=\operatorname{diag}(0,0,1).$$

Thus the tensor channel has one logarithmic transport direction, matching the single Cartier costalk of Entry 2288. Its local monodromy is

$$\exp(2\pi iR_{\rm Gram})=I_3.$$

The Gram class is supported but carries no nontrivial local-system character.

Gauss--Manin tensor compatibility

For either sector-specific Gauss--Manin system $(\mathbb V,A_{\mathbb V})$, put

$$A_{\mathbb V\otimes D} =A_{\mathbb V}\otimes I_3+I_{\mathbb V}\otimes A_D.$$

Then

$$1_{\mathbb V}\otimes T$$

is horizontal by the Leibniz rule. The direct Gaussian score block has constant connection and is horizontal by Entry 2289. Therefore the complete score-plus-tensor observer map is a typed morphism of connections, not merely an invertible map of generic fibers.

Classification

The Gram wall contributes:

• a simple logarithmic detector residue;
• identity monodromy;
• one recoverable Cartier costalk;
• no new elliptic or marked-relative coefficient character.

This completes the connection-level compatibility omitted from Entry 2289.

Verification

research/benincasa/checkers/finite_q_detector_connection.rs verifies the Smith-gauge connection, residue, monodromy character, and tensor-product rule.

View source on GitHub