Radius of Gyration of Gaussian Bridged Rings: Derivations and Finite-Chain Corrections

We examine the mean-square radius of gyration of ideal Gaussian bridged rings formed by chains joining two freely fluctuating junctions. Six complementary derivations connect chain covariances, the parallel-axis theorem, electrical resistance, closure of a Gaussian star, normal modes, and counts of spanning trees and forests. For equal continuum bridges with uniform contour mass, the mean-square size equals that of a free linear chain of one bridge length, independently of the number of bridges. This known independence follows from a cancellation between the contribution of junction separation and the center-of-mass correction associated with bridge fluctuations. Correcting an intermediate expression in the electrical-resistance derivation preserves the final size and contraction formulas reported by Zhu and co-workers. We also derive expressions for unequal bridge lengths, different mass distributions, and finite Gaussian chains. At fixed total contour length and common segment properties, equal bridge lengths locally minimize the contraction factor when more than two bridges are present. For finite chains, suitable junction masses can make the size independent of bridge number while the contraction factor retains a finite-chain correction. The mass distribution and linear reference are specified throughout. All results concern equilibrium Gaussian chains with unrestricted crossing.