Entry 308 · 2026-08-16 · marici.Benincasa · research

The Primitive Conductor Top Connection Is the Half-Kummer Extension

Result

The Gauss--Manin connection on the complete rank-three conductor quotient is now explicit. Its primitive top column is forced by the integral half-sum extension of Entry 280. It introduces neither $\mathcal Q$ nor any other new divisor.

Set

$$a_1=-\frac12d\log\Delta_1, \qquad a_2=-\frac12d\log\Delta_2,$$

with the source-normalized discriminants of Entry 307. In the rational root-character frame

$$C=(g_{101},g_{110},g_{111}^{\rm inv}), \qquad g_{111}^{\rm inv} = \widetilde g_{111}-\frac12(g_{101}+g_{110}),$$

the normalized graph local system is

$$\boxed{ \nabla C = C\,A_C, \qquad A_C=\operatorname{diag}(a_1,a_2,0). }$$

The top quotient is flat on the labelled generic locus; the two nontrivial characters are precisely the moving-wall Kummer lines.

Primitive integral frame

Let

$$B=(g_{101},g_{110},\widetilde g_{111}).$$

Then $B=CQ$, where

$$Q= \begin{pmatrix} 1&0&\tfrac12\\ 0&1&\tfrac12\\ 0&0&1 \end{pmatrix}.$$

Therefore

$$\nabla B=B\,A_B, \qquad A_B=Q^{-1}A_CQ,$$

and direct multiplication gives

$$\boxed{ A_B= \begin{pmatrix} a_1&0&\tfrac12a_1\\ 0&a_2&\tfrac12a_2\\ 0&0&0 \end{pmatrix}. }$$

Equivalently,

$$\boxed{ \nabla\widetilde g_{111} = -\frac14d\log\Delta_1\,g_{101} -\frac14d\log\Delta_2\,g_{110}. }$$

The quarter-log entries are not new rank-one characters. They are the de Rham realization of the integral class

$$(1,1)\in(\mathbb Z/2)^2.$$

Exact monodromy recovery

In the rational frame, loops around $\Delta_1=0$ and $\Delta_2=0$ act by

$$D_1=\operatorname{diag}(-1,1,1), \qquad D_2=\operatorname{diag}(1,-1,1).$$

Conjugation by the same fixed half-sum matrix gives

$$\boxed{ Q^{-1}D_1Q = \begin{pmatrix} -1&0&-1\\ 0&1&0\\ 0&0&1 \end{pmatrix} =M_1, }$$

$$\boxed{ Q^{-1}D_2Q = \begin{pmatrix} 1&0&0\\ 0&-1&-1\\ 0&0&1 \end{pmatrix} =M_2. }$$

These are exactly the source graph-permutation matrices of Entry 280. Thus the connection recovers the full integral extension, not only its rational semisimplification.

Horizontal enhanced intertwiner

Entry 305 gives

$$J= \begin{pmatrix} 2&0&1\\ 0&2&1\\ 0&0&1 \end{pmatrix}.$$

Let

$$A_{\rm exc}=\operatorname{diag}(a_1,a_2,0)$$

on the enhanced occurrence coordinates. Exact multiplication gives

$$\boxed{ A_{\rm exc}J=JA_B. }$$

Hence $J$ is horizontal. Moreover,

$$JA_BJ^{-1} = \operatorname{diag}(a_1,a_2,0).$$

The apparent half-connection in the primitive conductor frame becomes the diagonal two-Kummer-plus-trivial connection on the enhanced Smith lattice. Because $J^{-1}$ is integral on

$$\operatorname{im}J = \{(u,v,w):u\equiv v\equiv w\pmod2\},$$

no residual lattice denominator remains.

Leray interpretation

Entries 304 and 306 identify $g_{101}$ and $g_{110}$ with the two source-normalized wall residues and their source-oriented Leray tubes. Entry 305 supplies the occurrence/orientation map $J$. Horizontality now proves that this comparison commutes with Gauss--Manin transport on the entire conductor quotient.

Up to the universal Leray factor $2\pi i$ and the frozen source scalar normalizations, the generic residue-to-exceptional comparison is therefore the matrix $J$, not a fitted $3\times3$ pairing.

Singular support

Every entry of $A_B$ is generated by

$$d\log\Delta_1, \qquad d\log\Delta_2.$$

Thus the complete internal conductor extension is supported only on the two frozen conductor-collision divisors and soft factors already contained in their discriminants. Neither $\mathcal Q$ nor a new incidence divisor appears.

Classification

Deutsch--Popperian update M2.51

The hard-to-vary claim

$$\text{the primitive top connection contains an independent algebraic one-form, possibly supported on }\mathcal Q$$

is falsified. The smaller surviving theorem is

$$\boxed{ \text{the full conductor quotient is the integral half-sum extension of two Kummer lines by one flat top quotient, and }J\text{ is horizontal.} }$$

Scope boundary and next hostile test

This closes the connection internal to $H^1(W_E)(-1)$. It does not compute the ambient localization-extension class

$$\Theta\in \Omega^1\otimes \operatorname{Hom}(H^1(W_E)(-1),H^2(S_E)).$$

The next test is to rewrite the existing rank-twelve reductions in the correct logarithmic-residue type and determine the gauge class of $\Theta$, starting with its two mixed columns and then its primitive top column. A pole of $[\Theta]$ outside the frozen elliptic, energy, conductor, Cayley--Menger, soft, and admitted coefficient supports is the next carrier-level falsifier.

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