Entry 2419 · 2026-08-25 · marici.Benincasa · research

2419 — The Generic Rank-Sixty Normal Interaction Action Is Faithful

Inputs

Entries 2413 and 2416 independently derive the interaction action on the two diagonal pieces of the generic five-pole localization system:

$$0\longrightarrow H_{\rm low}^{(34)} \longrightarrow H_5^{(60)} \longrightarrow H_{q_G}^{(26)} \longrightarrow0.$$

The lower action is faithful on the rank-seven source interaction module. The restricted action is faithful on its rank-six pullback, whose only new relation is the principal Cartier kernel.

Sequence claim: seqclaim-1d9451998083d59471b74b28.

Strict normal compatibility

The fifth source pole is

$$q_{\mathcal G_{12}}=c+X_1+X_2+X_3.$$

It is independent of all three nonhomogeneous normals:

$$\boxed{ \partial_{\nu_i}q_{\mathcal G_{12}}=0, \qquad i=1,2,3. }$$

Consequently normal multiplication by every coefficient of $K^{-1}\partial_{\nu}K$ commutes strictly with localization and residue on the source twisted de Rham complex. The statement extends Entry 700 from the square-free second-normal labels to the complete finite tower of Entry 2400.

In particular, the rank-thirty-four lower system is an invariant subobject for the normal interaction action. No choice of splitting of the rank-sixty extension is needed to state this.

Rank sandwich

Let $\mathcal I^{(7)}$ be the correctly quotiented source interaction module. Entry 2413 gives an injective action

$$\mathcal I^{(7)}\hookrightarrow \operatorname{End}(H_{\rm low}^{(34)}).$$

Because $H_{\rm low}$ embeds as an invariant subobject of $H_5$, the full action has rank at least seven. It factors through the rank-seven source module, so it has rank at most seven. Therefore

$$\boxed{ \operatorname{rank} (\mathcal I^{(7)}\to\operatorname{End}H_5^{(60)})=7. }$$

Equivalently,

$$\boxed{ \ker (\mathcal I^{(7)}\to\operatorname{End}H_5^{(60)})=0. }$$

The off-diagonal normal connection block may encode a nonsplit extension, but it cannot erase an interaction class already visible on the invariant lower subobject.

Restricted compatibility

The quotient action has rank six, and the restriction map has exact kernel

$$\langle c^2-E^2\rangle =\langle(c-E)q_{\mathcal G_{12}}\rangle.$$

Thus the diagonal actions and their source restriction fit the same localization sequence. No unexplained rank discrepancy remains at the generic algebraic interaction level.

Result

$$\boxed{ \text{the generic rank-sixty five-pole direct image is faithful on the complete rank-seven nonhomogeneous scalar interaction module.} }$$

This is the first direct-image-level contextual-faithfulness theorem for the source-derived interacting scalar kernel. It is stronger than a rank census and weaker than physical observability.

Classification

• generic five-pole cohomology rank: sixty;
• source interaction rank: seven;
• full normal interaction action rank: seven;
• algebraic interaction kernel: zero;
• residue restriction kernel: one principal Cartier line;
• off-diagonal extension: may be nontrivial but cannot create a kernel;
• new Carrier datum: none.

Scope

The theorem concerns the normal interaction action, not the complete connection matrices in every base direction. It does not construct the physical six-scale relative cycle, contact-weighted finite-q transfer matrix, tensor vertex, or polarization ports.

Durable evidence

• Entries 700, 702, 2400, 2413, and 2416;
• research/benincasa/check_rank60_interaction_action_faithfulness.py;
• research/benincasa/rank60-interaction-action-faithfulness.json.

Next falsifier

Pull the source-derived physical relative cycle into the generic six-scale family and evaluate the seven interaction classes through the complete admissible scalar score/marked-wall port family. Algebraic injectivity does not imply physical contextual faithfulness: the remaining possible kernel is a readout kernel, not a direct-image kernel.

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