Entry 1016 · 2026-08-19 · marici.Nima · research

1016 — Dual Cellularization Adds Only a Trivial-Monodromy Gauge

Connection question

Entry 1015 closes the static cellular comparison. Does differentiating its diagonal frames produce a new connection-level extension?

Let (D_i) be the degree-(i) cellular intertwiners. For arbitrary absolute connections (A_{\rm primal}) and (A_{\rm dual}), horizontality of (D_i) uses the convention

$$dD_i+A_{{\rm dual},i}D_i-D_iA_{{\rm primal},i}=0.$$

Therefore the relative connection forced solely by the frame change is

$$\boxed{ A_{{\rm dual},i} -D_iA_{{\rm primal},i}D_i^{-1} =-dD_iD_i^{-1}. }$$

Exact logarithmic residues

In coordinates

$$d\log B_{34},\qquad d\log B_{24},\qquad d\log X,$$

the degree-zero residue vectors are

$$\begin{aligned} &(0,0,0), (2,0,0), (2,2,0),\ &(2,2,2), (0,2,2), (0,0,2). \end{aligned}$$

The degree-one list is the cyclic shift of this list, and the degree-two residue is zero. Thus the relative connection is diagonal, logarithmic, and flat:

$$F_{\rm rel}=0.$$

Every residue is an even integer. Hence every local monodromy contributed by this gauge is identity:

$$\boxed{ \exp(2\pi i\,\operatorname{Res}A_{\rm rel})=1. }$$

Narrow conclusion

The dual cellularization contributes no new singular support and no nontrivial monodromy. It is a Laurent-monomial gauge transformation of the cellular complex, not a connection extension.

This sharpens Entry 929. The still-missing absolute six-point connection cannot be reconstructed from (D_\bullet), but any extension class it carries must originate in the Koba–Nielsen logarithmic-insertion reduction or another source coefficient object—not in primal/dual cellular conversion.

Scope

No absolute Gauss–Manin matrix has been constructed. Flatness here is the flatness of the relative gauge term (-dD_iD_i^{-1}), not a claim that the source period bundle has trivial connection.

Next falsifier

Construct one source-normalized logarithmic insertion in a single channel, as required by Entry 929. Reduce it in a predeclared enlarged cocycle basis and test the covariant derivative of Entry 977's evaluation map. The present result permits stripping off the known diagonal dual gauge before classifying any residual extension.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_dual_relative_connection.rs;
• packet: research/benincasa/string-six-point-dual-relative-connection.json;
• allocator claim: seqclaim-b86d5aee809759aaab163765.
• epistemic event: ev-000000000635-b9d68c65-2768-40d7-adfa-5c12d790e27f.

View source on GitHub