A new radial, natural, higher order intermediary of the main problem four decades after the elimination of the parallax.

The paper shows that selecting H0,1 as µ/p·(1/4)C2,0(R⊕^2/r^2)(2−3s^2) (Deprit’s elimination of the parallax) makes θ cyclic at first order and yields a first-order generating function W1 that contains no equation of the center φ; moreover, the φ–f coupling is postponed to third order, easing the computation of second-order short-period terms . The paper also presents an alternative “neutral” first-order choice that keeps the 1/r^3 structure (H0,1 = (µ/r^3)(1/2)C2,0R⊕^2(1−3s^2/2)), derives the corresponding W1 in Delaunay and compact polar forms, and shows how to remove θ from K2 and K3 while maintaining factorization by 1/r^3; with small eccentricity, K2 and K3 lose dependence on e and ω, yielding a radial higher-order intermediary . The model solution faithfully reproduces these constructions and conclusions (including the small‑e truncation argument and the central point that the true simplification comes from making θ cyclic at first order rather than literally removing parallactic factors). One minor caveat: the model proposes an explicit closed-form W1 for the parallax-elimination case in polar variables that is not printed in the paper; while likely derivable, it should be checked algebraically against the homological equation. Overall, there is substantial agreement in method and results.