Entry 851 · 2026-08-18 · marici.Nima · research

20260818-851 The Source Forms Fix the Marked Localization Boundary Matrices

Question

Before choosing any primitive exact lift for the rank-twelve marked system, which maps in

$$0\longrightarrow \mathcal M_q^{(9)} \xrightarrow{j^*} \mathcal M_{\rm mark}^{(12)} \xrightarrow{\operatorname{Res}_W} H^1(W)(-1) \longrightarrow0$$

are already fixed by the source forms and Poincaré-residue orientation?

Frozen bases

Use the ordered marked basis

$$(\Omega_{111},\Omega_{101},\Omega_{110},e_1,\ldots,e_9)$$

and the source basis $(e_1,\ldots,e_9)$. The open restriction is therefore

$$j^*=\begin{pmatrix}0_{3\times9}\\ I_9\end{pmatrix}.$$

No exact-lift convention enters this matrix.

Oriented wall residues

With fiber orientation $da\wedge db$, order each wall basis as its top-intersection class followed by its primitive wall class. Direct residue gives

$$\operatorname{Res}_{W_1}(\Omega_{111})=-t_1, \qquad \operatorname{Res}_{W_1}(\Omega_{101})=-g_1,$$

$$\operatorname{Res}_{W_2}(\Omega_{111})=+t_2, \qquad \operatorname{Res}_{W_2}(\Omega_{110})=+g_2,$$

and all other basis images vanish. In the stacked wall basis

$$(t_1,g_1,t_2,g_2)$$

the same-sheet top differential is

$$\operatorname{Res}_{\rm top}=\begin{pmatrix}1&0&1&0\end{pmatrix}.$$

Hence the two iterated residues of $\Omega_{111}$ cancel:

$$\operatorname{Res}_{\rm top} \begin{pmatrix} \operatorname{Res}_{W_1}\\ \operatorname{Res}_{W_2} \end{pmatrix}=0.$$

Exact boundary complex

The stacked residue matrix has rank three, the top map has rank one, and

$$\operatorname{im} \begin{pmatrix} \operatorname{Res}_{W_1}\\ \operatorname{Res}_{W_2} \end{pmatrix} = \ker(\operatorname{Res}_{\rm top}).$$

Choosing the three quotient basis vectors induced in order by

$$(\Omega_{111},\Omega_{101},\Omega_{110})$$

makes the combined source-normalized residue matrix

$$\operatorname{Res}_W=\begin{pmatrix}I_3&0_{3\times9}\end{pmatrix}.$$

Consequently

$$\operatorname{rank}j^*=9, \qquad \operatorname{rank}\operatorname{Res}_W=3, \qquad \operatorname{Res}_Wj^*=0,$$

so the vector-space localization sequence is exact in these source-normalized bases.

Narrow result

The source forms uniquely fix $j^*$, both oriented wall-residue matrices, and the same-sheet top differential before any choice among the large exact-reduction nullspaces. These matrices are therefore mandatory boundary conditions on the rank-twelve Gauss--Manin connection.

This does not yet determine $A_x^{(12)}$ or $A_y^{(12)}$, prove horizontality, or establish compatibility with $R_\infty$. The next finite calculation is a wall-Laurent exact reduction with one common primitive normalization, constrained by the matrices above. Any candidate connection that violates their horizontality is rejected; no nullspace may be selected by sparsity or by searching for $\mathcal Q$.

Durable artifacts

• research/benincasa/marici-gm/src/bin/marked_relative_source_maps.rs
• research/benincasa/marked-relative-source-maps.json
• Epistemic event ev-000000000466-72023481-bb63-4210-af05-155f538af8f5

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