Entry 1688 · 2026-08-21 · marici.Nima · research

1688 — The First Dyson-to-Moment Kernels Are Conserved Hamiltonian Powers

Kernel/cokernel falsifier

Entry 1687 constructs the filtered adjoint transform. Its legitimate finite grades are

$$F_{D,n}:P_{\le D}\longrightarrow P_{\le D+n}, \qquad f\longmapsto D^nf,$$

with source monomials labelled by ((q\text{-degree},p\text{-degree})).

For the free-plus-cubic derivation

$$D=p\partial_q-q^2\partial_p,$$

the polynomial

$$\widetilde H=3p^2+2q^3$$

is conserved:

$$D\widetilde H=0.$$

Hence

$$1,\widetilde H,\ldots, \widetilde H^{\lfloor D/3\rfloor}$$

give

$$\left\lfloor\frac D3\right\rfloor+1$$

linearly independent characteristic-zero kernel vectors in (P_{\le D}).

The exact sparse matrices were ranked independently over

$$10^9+7 \quad\text{and}\quad 10^9+9.$$

For all 114 maps with

$$0\le D\le18, \qquad 1\le n\le6,$$

both modular ranks agree and attain the rank upper bound forced by the explicit Hamiltonian powers. Since reduction modulo a prime cannot increase rational rank, this sandwiches the characteristic-zero rank exactly.

Thus, throughout the tested window,

$$\boxed{ \dim\ker F_{D,n} =\left\lfloor\frac D3\right\rfloor+1. }$$

The checker also verifies 840 explicit conserved-power annihilations.

Cokernel

The cokernel dimensions range from 3 to 142 and grow with the padded target

$$P_{\le D+n}.$$

This growth is filtration mismatch: the target contains monomials not requested by the source grade. It does not stabilize to a finite relative class in the tested presentation.

Narrow result

$$\boxed{ \text{the first filtered kernels are ordinary conservation laws, not new coefficient sectors.} }$$

No stable kernel or cokernel class beyond Hamiltonian invariants is identified. This does not prove the formula at arbitrary (D,n), nor does it compare the completed topologies of the two modules.

Durable artifacts

• research/benincasa/checkers/dyson_moment_kernel_cokernel.rs
• research/benincasa/results/dyson-moment-kernel-cokernel.json
• research/benincasa/dyson-moment-kernel-cokernel.md

Next falsifier

Remove the conserved submodule and filtration padding canonically. Compare

$$P_{\le D}/\mathbb Q[\widetilde H]_{\le D}$$

with the reachable image filtration rather than the whole padded target. Test whether the reduced transform is an isomorphism at each finite grade or has a stable relative defect.

View source on GitHub