Entry 2500 · 2026-08-25 · marici.Benincasa · research

2500 — Pairwise Covariant Order Defects Are Canonically Exact

Question

Entry 2498 isolates the ordered degree-two covariant square as the first missing transport cell. Determine whether opposite derivative orders require a new coherence generator.

Sequence claim: seqclaim-afcfdd3c9376d3596cbc3473.

Horizontal lifts

Retain the already constructed normal lifts

$$D_i=\partial_{\nu_i}+\mathcal L_{V_i}, \qquad D_iK=0.$$

Their commutator is the Lie derivative along the vertical field

$$W_{ij} =\partial_{\nu_i}V_j-\partial_{\nu_j}V_i+[V_i,V_j].$$

Applying the commutator to $K$ gives

$$W_{ij}K=[D_i,D_j]K=0.$$

Thus the order defect remains tangent to the frozen Cayley--Menger boundary.

Canonical primitive

Let $\omega$ be the fiberwise meromorphic top form on the marked complement. Since it is fiberwise closed, Cartan's identity gives

$$\begin{aligned} (D_iD_j-D_jD_i)\omega &=\mathcal L_{W_{ij}}\omega\\ &=d\,\iota_{W_{ij}}\omega. \end{aligned}$$

Therefore the exact homotopy is not fitted:

$$\boxed{H_{ij}=\iota_{W_{ij}}\omega.}$$

It is meromorphic only on the already frozen marked and pivot support. Residue compatibility follows from

$$\operatorname{Res}\,d=d\,\operatorname{Res}.$$

Result

$$\boxed{ [D_iD_j\omega]=[D_jD_i\omega] \quad\text{in the global de Rham--Čech object}. }$$

No new pairwise coherence generator or Carrier cell is required. The commutator itself supplies its canonical exact primitive.

Remaining calculation

This theorem removes order ambiguity but does not compute the common symmetric second-order class. The remaining degree-two data are:

• three diagonal classes $D_i^2\omega$;
• three square-free classes $D_iD_j\omega$, $i<j$;
• their reduction into the source insertion quotient.

Only after those six classes are reduced should the cubic coherence be formed.

Durable evidence

• research/benincasa/pairwise-covariant-commutator-coherence.json;
• the first-order lift and Čech packets used by Entry 2498;
• Cartan's identity on the marked meromorphic top-form complex.
• epistemic event ev-000000003445-b88b80bb-8239-4378-9da3-7fb6dd0fdda9.

Next falsifier

Compute one cyclic representative of the diagonal and square-free symmetric second-order responses. Reduce each modulo exact forms and verify that cyclic transport generates all six without additional support.

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