Entry 2399 · 2026-08-25 · marici.Benincasa · research

2399 — The Ambient Marked Extension Splits Rationally at a Physical Conductor Pinch

Question

Entry 2397 proves that each positive physical conductor collision carries a separately observed rank-one wall Kummer class. The remaining local gate is whether the ambient rank-nine absolute block can attach a hidden supported class through the rank-twelve localization extension

$$0\longrightarrow\mathcal M_9 \longrightarrow\mathcal M_{12} \longrightarrow\mathcal W_3 \longrightarrow0.$$

This can be decided at generic conductor support without reconstructing a particular $9\times3$ primitive extension block.

Sequence claim: seqclaim-ee4fbc3c2ef0fdfa68b67b76.

Exact physical witness

Use Entry 2397's point

$$(x,y,z;E) =\left(\frac5{14},\frac35,\frac3{70};1\right).$$

At this point,

$$\Delta_1=0, \qquad \Delta_2=\frac{561}{1225}\ne0.$$

The signed-energy letters are

$$(\ell_1,\ell_2,\ell_3,\ell_4) =\left(-\frac27,-\frac15,\frac{32}{35},1\right),$$

so the pure elliptic discriminant is nonzero. The algebraic quartic is also a unit:

$$\mathcal Q=\frac{51}{245}\ne0.$$

Thus this is a generic point of the first conductor divisor, not an elliptic, soft, total-energy, or quartic intersection. The absolute surface family is smooth there.

Local monodromy types

Around the punctured normal disk to $\Delta_1=0$, the rank-nine absolute Gauss--Manin system is locally unramified:

$$T_{\mathcal M_9}=I_9.$$

In the oriented wall basis $(q_0,q_1,q_2)$, only $q_1=g_{101}$ is ramified:

$$T_{\mathcal W_3}=\operatorname{diag}(1,-1,1).$$

Therefore the local extension problem for the supported line is governed by

$$V=\operatorname{Hom}(\mathcal K_-,\mathcal M_9), \qquad T_V=-I_9.$$

For a local system on a punctured disk,

$$H^1(\mathbf Z,V)=V/(T_V-I)V.$$

Over $\mathbf Q$,

$$T_V-I=-2I_9$$

is invertible. Hence

$$\boxed{ \operatorname{Ext}^1_{\mathrm{Loc}_{\mathbf Q}} (\mathcal K_-,\mathcal M_9)=0. }$$

Any logarithmic off-diagonal representative at this generic conductor point is removable by a boundary-preserving rational triangular gauge. No choice of primitive solver section is required for this conclusion.

Supported cone and observer

The rational supported nearby object therefore contains exactly the already identified wall line:

$$\psi_{\Delta_1}^{\rm supp}\mathcal M_{12} \simeq\mathcal K_-.$$

Entry 308's horizontal Leray matrix sends its generator to

$$g_{101}\longmapsto(2,0,0)^T.$$

Thus the ambient localization extension adds no rational class hidden from the physical route observer. By exchanging labels, the same theorem holds at a generic positive point of $\Delta_2=0$.

Integral qualification

The argument is intentionally rational/de Rham. Over $\mathbf Z$, the cokernel of $-2I$ is $(\mathbf Z/2)^9$. The calculation therefore does not prove an integral splitting of the ambient extension. This is a specific remaining lattice question, not a rational rank or monodromy class and not new support.

Result

$$\boxed{ \begin{gathered} \text{the ambient rank-twelve marked extension splits rationally at a}\ \text{generic positive conductor pinch, and its only supported rational}\ \text{class is the already faithful wall Kummer line.} \end{gathered}}$$

Classification

• Carrier support: existing conductor/Landau divisor;
• absolute coefficient block: smooth and unramified;
• wall coefficient block: one semisimple Kummer line;
• rational ambient extension class: zero;
• physical route kernel: zero;
• possible integral refinement: two-torsion, uncomputed;
• new Carrier datum: none.

Scope

This theorem is local at a generic smooth point of either conductor divisor. It does not cover intersections with soft, elliptic, Gram, or marked endpoint support, nor the nonhomogeneous six-scale direct image or tensor polarization system.

Durable evidence

• research/benincasa/check_conductor_ambient_extension_local_splitting.py;
• research/benincasa/conductor-ambient-extension-local-splitting.json;
• Entries 307, 308, 851--855, 869, and 2397.

Next falsifier

Test the same extension integrally. Use the source Smith lattice and the fixed occurrence/Leray map to determine whether any of the possible $(\mathbf Z/2)^9$ ambient extension classes is actually realized. A realized torsion class must then be paired with the integral physical cycle; it is coefficient/lattice data unless it forces additional support.

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