Entry 2426 · 2026-08-25 · marici.Benincasa · research

2426 — Gradient-Pivot Normal Lifts Glue as One de Rham--Čech Class

Local issue left by Entry 2424

Entry 2424 represents the physical normal Gauss--Manin derivative on a chart $U_p=\{\partial_pK\ne0\}$ by

$$V_i^{(p)}=-\frac{\partial_{\nu_i}K}{\partial_pK}\,\partial_p.$$

Its denominator is not a new physical pole, but this requires an overlap calculation: different pivot choices must represent one global class rather than unrelated local responses.

Pairwise overlap identity

On $U_p\cap U_r$, set

$$W_i^{(p,r)}=V_i^{(p)}-V_i^{(r)}.$$

Both lifts act on the kernel by $-\partial_{\nu_i}K$. Therefore

$$\boxed{W_i^{(p,r)}(K)=0.}$$

The overlap difference is a vertical vector field tangent to the Cayley--Menger boundary. For the fiber top form $\omega$, Cartan's identity gives

$$\boxed{ \mathcal L_{W_i^{(p,r)}}\omega =d\bigl(\iota_{W_i^{(p,r)}}\omega\bigr). }$$

Hence the two contact-weighted response forms differ by an exact meromorphic form on the frozen marked complement.

Triple coherence

For the three pivot charts $(U_c,U_a,U_b)$, the differences satisfy

$$\boxed{ W_i^{(c,a)}+W_i^{(a,b)}+W_i^{(b,c)}=0 }$$

componentwise. This was checked for all three labelled normals. The complete audit therefore contains nine pairwise tangency identities and three triple cocycle identities.

Result

$$\boxed{ \text{the local contact-weighted normal responses define one global de Rham--Čech class.} }$$

Gradient-pivot denominators are atlas data. They cannot be promoted to Carrier support. The only intrinsic failure of the gradient cover on the moving boundary is the already declared Cayley--Menger/Landau critical locus.

The contraction primitives are meromorphic along the frozen marked walls. Their residue compatibility belongs to the existing moving-wall relative complex; no ordinary-holomorphic comparison is asserted.

Scope

This theorem globalizes the scalar normal adapter for the main $K=0$ boundary at de Rham--Čech level. It does not establish tangency to every signed-minor boundary or evaluate its pairing with the Bunch--Davies cycle, and therefore does not yet establish rank-seven physical observability.

Durable evidence

• research/benincasa/check_physical_normal_lift_cech_coherence.py;
• research/benincasa/physical-normal-lift-cech-coherence.json;
• sequence claim seqclaim-bfaa012dd708c365bc6506d1.

Next falsifier

Push this class through the already typed moving-wall residue totalization. Verify that the Cartan primitives supply every required boundary homotopy, then pair the resulting global class with the source Bunch--Davies relative cycle. A residual overlap or residue class would be a coefficient extension; it would still lie on existing marked/Landau support unless a new divisor is independently produced.

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