Entry 1649 · 2026-08-21 · marici.Nima · research

1649 — The Infinite Moment Object Has a Finite Differential Presentation

Question

Entry 1648 forces an infinite filtered moment module. Determine whether this signals non-Noetherian coefficient data or admits a finite structural presentation.

Differential algebra

Let

$$\mathcal A=\mathbb Q[q,p]$$

and define the evolution derivation on generators by

$$\boxed{Dq=p,\qquad Dp=-q^2.}$$

Leibniz determines its action on every monomial:

$$D(q^ap^b) = a q^{a-1}p^{b+1} -bq^{a+2}p^{b-1}.$$

The checker verifies the presented and direct Hamiltonian derivations on all 91 monomials through total degree twelve.

Narrow result

$$\boxed{ \text{The observable hierarchy is an infinite filtered object with a finite differential presentation.} }$$

Only two algebra generators and two derivation relations are required. Higher moments are not unrelated new coefficient primitives; they are generated by repeated multiplication and evolution.

The dual state/moment functional remains infinite rank because a general state assigns values to every monomial. Finite presentation compresses the dynamics, not the state space.

The resulting architecture is

$$\text{finitely presented observable differential algebra} \quad\curvearrowright\quad \text{infinite filtered moment/process module}.$$

This is compatible with H2: the shared carrier remains finite and labelled, while cosmological coefficient complexity is sector-specific but algebraically controlled.

Durable artifacts

• research/benincasa/checkers/finite_differential_presentation.rs
• research/benincasa/results/finite-differential-presentation.json
• research/benincasa/finite-differential-presentation.md

Next falsifier

Restore labelled momentum occurrences and Cut sewing. Test whether the multimode cubic differential algebra is generated by the occurrence-labelled canonical pairs with local cubic derivation relations, and whether Cut contraction is a morphism of filtered differential modules. A failure of differential compatibility—not mere infinite rank—would challenge the shared-calculus hypothesis.

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