Entry 2416 · 2026-08-25 · marici.Benincasa · research

2416 — The Restricted Interaction Module Has Only the Principal Localization Kernel

Question

Entry 2413 proves that the generic rank-thirty-four lower direct image is faithful on the rank-seven source interaction module. The complementary test is the rank-twenty-six residue system on

$$q_{\mathcal G_{12}}=c+E=0.$$

The restricted system must be derived independently; its rank cannot be inferred from the lower calculation.

Sequence claim: seqclaim-5224f2a3d02f6cb558004e18.

Source restriction

In Entry 2400's notation, the third diagonal coefficient and cubic unit are

$$D_3=c^2, \qquad U=1.$$

On the residue divisor $c=-E$, they obey the additional exact identity

$$\boxed{ D_3-E^2U =c^2-E^2 =(c-E)q_{\mathcal G_{12}} =0. }$$

This is a principal Cartier restriction relation. It exists before cohomological reduction and has source-fixed normalization.

Together with Entry 2413's three universal relations, it reduces the source interaction rank from seven to six:

$$\boxed{ \mathcal I_{\rm low}^{(7)} \longrightarrow \mathcal I_{q_G}^{(6)}, \qquad \ker=\langle D_3-E^2U\rangle. }$$

The map is surjective at the generic source-polynomial level.

Independent restricted quotient

The finite reducer imposes $c+E=0$, retains all four marked lines, and constructs the two tangential logarithmic critical equations in $(a,b)$. The inversion relation uses the complete restricted divisor. The resulting standard-monomial ranks are

$$\begin{array}{c|c|c|c} \text{point}&\text{prime}&\operatorname{rank}H_{q_G} &\operatorname{rank}\langle K_\alpha/K\rangle\\ \hline A&32003&26&6\\ B&32009&26&6. \end{array}$$

The four modular kernel vectors agree exactly with:

1. the three source interaction identities of Entry 2413 after restriction;
2. the principal identity $D_3-E^2U=0$.

No fifth relation appears.

Result

$$\boxed{ \ker(\mathcal I_{\rm raw}^{(10)}\to H_{q_G}^{(26)}) =\langle \text{three universal source relations}, D_3-E^2U \rangle. }$$

Hence, on the correctly restricted source quotient,

$$\boxed{ \mathcal I_{q_G}^{(6)} \hookrightarrow H_{q_G}^{(26)} }$$

is generically injective. The rank drop $7\to6$ is principal localization data, not coefficient loss.

Interaction localization sequence

At the source-coefficient level the two independently computed sides now assemble as

$$\boxed{ 0\longrightarrow \langle(c-E)q_{\mathcal G_{12}}\rangle \longrightarrow \mathcal I_{\rm low}^{(7)} \longrightarrow \mathcal I_{q_G}^{(6)} \longrightarrow0. }$$

This is the finite interaction analogue of deletion–restriction. The remaining issue is not rank but the off-diagonal extension/coherence map in the full rank-sixty direct image.

Classification

• lower source interaction rank: seven;
• restricted source interaction rank: six;
• restriction kernel: one principal Cartier line;
• restricted cohomology rank: twenty-six;
• restricted interaction class rank: six;
• additional cohomological kernel: zero;
• new Carrier datum: none.

Scope

The theorem does not yet construct the full rank-sixty Gauss–Manin extension or the physical relative-cycle pairing. It proves both diagonal interaction maps and their source-level restriction kernel.

Durable evidence

• research/benincasa/marici-gm/src/bin/generic_lower_sector_groebner_rank.rs with NORMAL_TOWER=restricted;
• research/benincasa/check_restricted_interaction_class_faithfulness.py;
• research/benincasa/restricted-interaction-class-faithfulness.json;
• independent runs at points A/B and primes 32003/32009.

Next falsifier

Construct the normal-derivative extension block between the rank-thirty-four lower and rank-twenty-six restricted systems. Test whether the principal kernel line supplies the complete coherence cell. Any additional off-diagonal interaction class must have source-labelled support and must be tested against the physical residue/Leray observer before interpretation.

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