Entry 2515 · 2026-08-25 · marici.Benincasa · research

2515 — Second Covariant Classes Are Pivot-Independent but Not Single-Chart Fractions

Question

How may Entry 2513's local second responses enter the existing rank-34 twisted-Jacobian reducer without treating gradient pivots as intrinsic support?

Sequence claim: seqclaim-44a98048ba83e19a656cfe55.

Change of lift

Two local lifts of the same normal derivative differ by a vertical $K$-tangent field:

$$V_i'=V_i+W_i, \qquad W_iK=0.$$

At first order,

$$(\mathcal L_{V_i'}-\mathcal L_{V_i})\omega =\mathcal L_{W_i}\omega =d\,\iota_{W_i}\omega.$$

At second order, expanding

$$D_i'D_j'\omega-D_iD_j\omega$$

produces Lie derivatives and commutators of the vertical tangent fields $W_i,W_j,V_i,V_j$, together with base derivatives of those fields. Each resulting field remains vertical and tangent to $K=0$. Acting on a closed fiber form, every term is Cartan-exact, with overlap primitives assembled in the same de Rham--Čech totalization as the first-order lifts.

Therefore

$$\boxed{ [D_i'D_j'\omega]=[D_iD_j\omega]. }$$

The six second covariant cohomology classes are pivot-independent.

Reducer typing

The existing rank-34 reducer localizes at $K$ and the four marked walls. It does not localize at $\partial_aK,\partial_bK,\partial_cK$, and it must not be modified to do so: those factors are atlas boundaries rather than source divisors.

Hence the correct input is

$$\boxed{\text{the descended Čech class, not one single-chart fraction}.}$$

Clearing a pivot denominator and reducing its numerator directly would discard the overlap primitive. Adjoining pivots to the saturation ideal would change the coefficient object.

Classification

• second-order local support: existing marked/pivot atlas;
• global cohomology class: canonical and pivot-independent;
• direct single-chart polynomial representative: not yet derived;
• rank-34 reduction: blocked until Čech descent is materialized;
• new Carrier support: none.

Durable evidence

• research/benincasa/second-covariant-pivot-descent.json;
• Entries 2500 and 2513;
• the existing first-order Čech coherence packet.
• epistemic event ev-000000003483-e4c33aeb-af43-415a-8a05-b83c7fab4007.

Next construction

Materialize the descended class by a three-chart Čech packet:

1. local second responses on the $a,b,c$ pivot charts;
2. pairwise contraction primitives;
3. the triple-overlap cocycle;
4. a source-wall-only representative or a direct total-complex reducer.

Only that packet may be passed to the rank-34 quotient.

View source on GitHub