Projection Invariants in Rotating Curve Families: Universality of the Diametral Abel Transform and Variational Characterization of the Circular Family

We study a one-parameter family of circles, each of radius n/4, whose centers rotate on a circle of radius n/4, all passing through a common point. Using the co-area formula, the theory of pushforward measures, and classical Abel inversion, we establish four results. First, the envelope of the family is the circle of diameter n, and the radial superposition density rho (r) = 1/sqrt ( (n/2) ² - r²) is the Radon-Nikodym derivative of the incidence pushforward measure with respect to dr, where F is the incidence map and K is the normalized kinematic (Haar) measure on the family. Second, the surface integral over the limit disk gives M (n) = n * pi, which equals the perimeter of the envelope circle and constitutes the circular analogue of the Cauchy-Crofton identity. Third, we prove a universality theorem: for any admissible incidence function g: 0, pi/2 -> 0, R (continuous, strictly decreasing, g (0) =R, g (pi/2) =0), the diametral Abel projection yields Pg (0) = pi independently of g. This constitutes a closed-form analytic identity with identically zero residual error; it is structurally distinct from numerical or iterative approximation algorithms. The universal value pi arises as the total Haar measure of the half-orbit 0, pi/2 of SO (2) ; the factorization Mg = ng * pi, with ng = 2 * integral of g (u) du, separates topological and geometric information. Fourth, g (u) = R * cos (u) is the unique stationary point of the harmonic-oscillator action; we show it is in fact the strict global minimum of the action on the admissible class, confirmed by the positive definiteness of the second variation. Open problems on envelope structure, singularity classification, and reconstruction of g from the Abel projection via a Volterra equation of the first kind are stated precisely.