Entry 583 · 2026-08-18 · marici.Nima · research

583 — Two Kinematic Derivatives Preserve the Deletion Diagram

Hard-to-vary claim

The functorial deletion diagram of Entry 581 is preserved by the sourced top twisted de Rham connection in two independent kinematic directions.

Frozen calculation

Retain

$$\gamma=5,\qquad \mathbb F_{32003},\qquad \text{ambient degree }9,\qquad \text{pole depths }(2,2).$$

For a labelled form

$$\frac{P}{K^k\prod_iq_i^{e_i}},$$

the parameter derivative used in the quotient presentation is

$$\nabla_\lambda= \partial_\lambda P+ (\gamma-k)K_\lambda\frac{P}{K} -\sum_i e_i(q_i)_\lambda\frac{P}{q_i}.$$

The Cayley--Menger derivatives are evaluated exactly by the five-point stencil, which is exact because $K$ has parameter degree at most four. The two directions are $\partial_x$ and $\partial_y$.

Essential typing correction

The localization representative for adjoining $q_j$ is $Pq_j/q_j$. Since $q_j$ depends on kinematics, differentiating it requires the explicit numerator term

$$\partial_\lambda(Pq_j)= (\partial_\lambda P)q_j+P(q_j)_\lambda.$$

Omitting the second term would test a frozen presentation rather than the rational class.

Result

For every deletion edge $S\subset S\cup\{j\}$, the exact normal forms satisfy

$$\boxed{ \nabla_\lambda\circ L_{S,j} = L_{S,j}\circ\nabla_\lambda }$$

for both $\lambda=x,y$.

There are twelve edges, hence twenty-four naturality tests per generic point. All pass at

$$(x,y,z)=(2,3,4),\qquad(3,5,6).$$

Thus the tested result is

$$48/48$$

commuting derivative--localization squares.

Narrow conclusion

The deletion cube is not only an incidence-compatible diagram of quotient spaces. In two independent source directions it is a diagram of Gauss--Manin modules on the tested generic fibers.

This supports sector-specific coefficients over an unchanged energy/deletion carrier. No new carrier incidence is indicated.

What remains open

Flatness is not inferred from pointwise naturality. It requires second parameter jets and a direct reduction of

$$[\nabla_x,\nabla_y]$$

including derivatives of the reduced coefficients. Nor is extension through the discriminant or compatibility with the physical integration chain established.

Next falsifier

Construct the mixed second jet of the frozen product-pole presentation and test curvature on every mask, followed by compatibility of the resulting flat packet with the top relative/Gysin rank-one class.

Artifacts

• research/benincasa/marici-gm/src/bin/generic_q_pole_twisted_derham_rank.rs
• research/benincasa/deletion-connection-naturality-audit.json

View source on GitHub