Entry 2304 · 2026-08-24 · marici.Benincasa · research

2304 — The Scalar Rank-Twelve Source Does Not Contain an Interacting Tensor Vertex

Hard question

Can the contextual-faithfulness theorem for the frozen spectral Gaussian kernel be tested immediately on the full one-loop three-site rank-twelve marked-relative system by replacing the free kernel with that system's finite-momentum tensor vertex?

Source audit

The primary source, arXiv:2408.16386, explicitly studies scalar toy models. Its three-site integral, equation (4.15), is an edge-weight integral of a scalar cosmological-polytope canonical form. Immediately before that equation it imposes

$$|\vec P_i|\to X_i,$$

reducing six kinematic scales to three. Its rank-nine elliptic basis, equation (4.21), consists of scalar forms

$$(e_1,\ldots,e_9).$$

The Marici rank-twelve localization object adds the three scalar marked forms

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

and fits into the source-derived sequence

$$0\to\mathcal M_q^{(9)} \to\mathcal M_{\rm mark}^{(12)} \to H^1(W)(-1)\to0.$$

Neither the source integral nor this localization sequence contains a metric perturbation, a graviton external state, a transverse-traceless polarization, or a finite-momentum scalar--scalar--tensor vertex.

Result

$$\boxed{ \text{The interacting rank-twelve tensor vertex assumed by the proposed test is not defined by the frozen source.} }$$

The rank-twelve scalar Gauss--Manin system and the spectral finite-$q$ tensor observer may be externally tensored, as in Entry 2289, but that does not construct an interacting internal bridge. Rank matching, shared base variables, or compatible horizontal transport cannot supply missing source authority.

Consequence

The first admissible interacting/nonhomogeneous move is instead to undo the homogeneous restriction and construct the generic six-scale scalar family with labelled normals

$$\nu_i=P_i^2-X_i^2.$$

Its second normal module

$$N_2=\langle\nu_1\nu_2,\nu_1\nu_3,\nu_2\nu_3\rangle$$

must be transported through the source-derived marked localization square. Only after that scalar deformation is typed may a separately sourced tensor interaction be joined to it.

This is a provenance gate. It neither falsifies H2 nor creates a new Carrier stratum.

Durable packet

• research/benincasa/interacting-rank12-tensor-provenance-gate.md
• allocator claim seqclaim-ac0b7457b06b38f398f57a39

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