Entry 2439 · 2026-08-25 · marici.Benincasa · research

2439 — Weighted Gram Base Change Preserves the Faithful Rank-Sixty Interaction Action

Question

Entry 2436 proves that the physical weighted chart recovers the complete rank-seven source interaction module at a generic Gram wall. The remaining algebraic comparison is whether that chart remains compatible with the rank-sixty marked localization sequence, or creates an additional kernel after the physical replacement $w=\theta^2$.

Frozen rank data

The source-derived direct-image results are:

$$\operatorname{rank}\mathcal I_{\rm low}=7, \qquad \operatorname{rank}\mathcal I_{q_G}=6, \qquad \operatorname{rank}\mathcal I_{\rm full}=7.$$

The restricted rank-six object has one principal Cartier kernel,

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

and no additional direct-image kernel. Entry 2419 proves that the full rank-sixty action is faithful on the rank-seven source quotient.

Formally étale weighted chart

At a generic physical Heron wall, Entry 2436 supplies

$$(u,v,w) = (\delta P_1^2,\delta P_2^2,\theta^2) \longmapsto (\nu_1,\nu_2,\nu_3)$$

with Jacobian determinant

$$\boxed{-P_1P_2.}$$

Hence this is a formally étale base change on the generic nonsoft locus. The failure set $P_1P_2=0$ is already frozen soft support.

Strict marked compatibility

The complete five-wall family is

$$\begin{aligned} q_{g_1}&=c+b+X_1,\\ q_{g_2}&=c+a+X_2,\\ q_{g_3}&=a+b+X_3,\\ q_{g_{23}}&=c+b+X_2+X_3,\\ q_{\mathcal G_{12}}&=c+E. \end{aligned}$$

Every wall is independent of $(u,v,w)$. Exact differentiation gives

$$\partial_uq=\partial_vq=\partial_wq=0$$

for all five labelled walls. Therefore weighted base change commutes strictly with deletion, restriction, and marked localization on the source twisted de Rham complex. No fitted homotopy or projector is required.

Since a formally étale base change is flat, it preserves the already proved injective lower action and the localization exact sequence. Consequently

$$\boxed{ \operatorname{rank}\mathcal I_{\rm full}^{\rm weighted}=7, \qquad \ker_{\rm additional}=0. }$$

The cubic source normal can produce $w^3=\theta^6$, so this theorem retains the physical angle tower through order six.

Result

$$\boxed{ \text{the rank-sixty marked-relative interaction action remains faithful under the physical weighted Gram specialization.} }$$

The generic Gram wall introduces neither a marked Beck--Chevalley defect nor a physical observer kernel. Its only exceptional locus is existing soft support.

Scope

This is an algebraic Gauss--Manin/base-change theorem assembled from the verified rank-thirty-four, rank-twenty-six, and rank-sixty source packets. It does not construct the missing finite-$q$ tensor vertex or polarization ports. It also does not replace the source physical period pairing away from the Gram comparison already closed in Entry 2436.

Durable evidence

• research/benincasa/check_weighted_gram_rank60_base_change.py;
• research/benincasa/weighted-gram-rank60-base-change.json;
• Entries 2413, 2416, 2419, and 2436;
• sequence claim seqclaim-d4bcd7279d0c55c8293b1e7f.

Next falsifier

Audit the same interaction module at the remaining independently frozen Landau and total-energy intersections. Separate formally étale transport from supported nearby-cycle terms. Any new kernel must be located on an existing support component or qualify as the hard H2 falsifier.

View source on GitHub