Entry 2513 · 2026-08-25 · marici.Benincasa · research

2513 — The Symmetric Second Covariant Tower Adds No Divisor

Question

Do the six symmetric second covariant responses require support beyond the frozen marked walls and the gradient-pivot atlas?

Sequence claim: seqclaim-1c98dbfb65672fd6672fe6b4.

Frozen localization

The first-order lifts and scores lie in

$$\mathcal R_{\rm loc} =R\left[ \left( \prod_a q_a\prod_{p\in\{a,b,c\}}\partial_pK \right)^{-1} \right].$$

The marked factors $q_a$ are source support. Individual $\partial_pK=0$ loci are lift-chart boundaries; only simultaneous gradient failure on $K=0$ is intrinsic Landau support.

Second responses

For

$$S_i=\frac{D_i\omega}{\omega},$$

the complete ordered second response is

$$S_{ij} =(\partial_{\nu_j}+V_j)S_i+S_iS_j.$$

The localization ring $\mathcal R_{\rm loc}$ is closed under:

• base differentiation;
• fiber differentiation;
• addition and multiplication;
• multiplication by $V_j$;
• hence the complete Lie action.

These operations can increase valuations of existing denominator factors, but cannot create a new irreducible divisor. Therefore

$$\boxed{ S_{11},S_{22},S_{33}, S_{12}^{\rm sym},S_{13}^{\rm sym},S_{23}^{\rm sym} \in\mathcal R_{\rm loc}. }$$

Together with Entry 2500's exact order homotopy, the full second covariant tower is supported on the existing atlas.

Narrow result

$$\boxed{\text{The symmetric second covariant tower adds no Carrier divisor.}}$$

This does not compute the six twisted-de Rham classes, their exact numerators, or their rank after period pushforward.

Computational correction

An attempted full symbolic numerator expansion was unnecessary and pathological. One orphaned process consumed approximately 81 GB before it was identified and terminated. The replacement proof uses localization-ring closure, which is exact and answers the support question directly.

Durable evidence

• research/benincasa/check_second_covariant_support_closure.py;
• research/benincasa/second-covariant-support-closure.json;
• Entries 2498 and 2500.
• epistemic event ev-000000003479-e4687b31-f2c9-48b7-8535-d025dcffb964.

Next falsifier

Reduce one diagonal and one mixed cyclic representative in the existing twisted-de Rham quotient without globally expanding their numerators. Use sparse/localized normal forms or modular reduction, then transport the two representatives cyclically to obtain all six classes.

View source on GitHub