Entry 849 · 2026-08-18 · marici.Nima · research

849 — The Generic Marked-Relative Map Exists Abstractly, but Its Source-Normalized Matrix Does Not

Correct geometric model

The generic rank-nine source system is already geometrically identified as

$$\mathcal M_q^{(9)}=H^2(S_E;\mathcal K).$$

For the two finite source walls

$$W=W_1\cup W_2,$$

Entry 279 proves the canonical localization sequence

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

where

$$\mathcal M_{\rm mark}^{(12)} =H^2(S_E\setminus W;\mathcal K)$$

and the ranks are

$$0\longrightarrow9\longrightarrow12\longrightarrow3\longrightarrow0.$$

Thus the source-to-relative map exists canonically at object level. It is the localization pullback $j^*$; no splitting is required.

Nested infinity localization

The rank-nine absolute object itself fits into

$$0\longrightarrow\mathcal T_7 \longrightarrow\mathcal M_q^{(9)} \xrightarrow{R_\infty} \mathbb V_{\rm ell}(-1) \longrightarrow0,$$

with ranks $7,9,2$. The marked-relative problem is therefore a nested extension:

$$\text{elliptic quotient} \leftarrow \mathcal M_q^{(9)} \hookrightarrow \mathcal M_{\rm mark}^{(12)} \twoheadrightarrow H^1(W)(-1).$$

This is the correct home for an elliptic–Tate/marked extension class.

Missing datum

The canonical object-level arrow is not yet represented in a source-normalized master basis. Nor is the full generic bivariate rank-twelve Gauss–Manin connection known.

The existing local residue packets cannot determine it uniquely. The exact reconstruction audit leaves nullities

$$\boxed{95,\qquad346,\qquad664}$$

in the one-wall, two-wall, and top six-order systems. These are exact-lift gauge freedoms, not evidence for additional coefficient rank. Choosing a solution from them after inspecting $\mathcal Q$ would fit the answer.

Correct first construction

The required object is a source-normalized relative de Rham reduction engine retaining four strata simultaneously:

1. the absolute $q$-only rank-nine sector;
2. the $W_1$ residue sector;
3. the $W_2$ residue sector;
4. the same-sheet top intersection sector.

It must output the inclusion $j^*$, all residue matrices, and the full rank-twelve connection in one common exact-lift convention. Only then is the horizontality equation

$$\nabla_{\rm mark}j^*=j^*\nabla_{\rm src}$$

and the off-diagonal extension support well-defined.

Consequence for $\mathcal Q$

Previous calculations exclude $\mathcal Q$ from the pure elliptic quotient, the generic absolute rank-nine connection, and the canonical local marked top coefficient. The remaining typed possibility is narrower:

$$\boxed{ \mathcal Q\text{ may occur only in the canonically reconstructed off-diagonal marked extension class.} }$$

No conclusion about that class is available until the four-stratum engine exists.

Verification

• checker: research/nima/audit_marked_relative_source_map_inventory.py;
• packet: research/nima/marked-relative-source-map-inventory.json;
• allocator claim: seqclaim-ffd2788fceceec82bb864a20.

View source on GitHub