Exact Background Evolution in the β0 = β1 Subclass of Hassan-Rosen Bigravity under Exchange Symmetry

We study a one-parameter specialization β0 = β1 of the exchange-symmetric subclass of Hassan-Rosen ghost-free bigravity, restricted to the dynamical FLRW branch with matter coupled to g only. Within this specialization, the autonomous evolution equation for the metric ratio ξ = Y/a collapses to the parameter-free first-order ODE dξ/d ln a = -3 ξ (ξ-1) / (2ξ-1). The equation admits the exact closed-form invariant ξ (ξ-1) a³ = K, equivalently ξ (a) = (1 + sqrt (1 + 4K/a³) ) /2, where K = ξ0 (ξ0-1) is determined by the present-day deviation. The leading-order Taylor expansion in ε = ξ-1 reproduces the ε ∝ a^-3 scaling of the parent exchange-symmetric class derived in the companion paper I. The expansion of H² (z) in powers of (1+z) ³ is exactly cubic and finite in ε0; among its three terms, the (1+z) ³ contribution is observationally degenerate with the matter density Ωₘ, while the (1+z) ⁶ contribution at order ε0² constitutes the distinctive observational signature separating the model from ΛCDM at the cosmographic level. We exhibit the redshift zbreak at which the linear approximation fails as a function of ε0, providing a clean target for late-time observational constraints. The result complements paper I by exhibiting an exactly solvable subset of the three-parameter symmetric class. We discuss explicitly its distinction from related work (HSMvS 2014; Enander, Solomon, Akrami, Mortsell 2014; Imadouchene 2025; Koennig-Amendola 2014; Koennig-Patil-Amendola 2014; Lüben-Schmidt-May-Weller 2020). A single consolidated symbolic verification script paper2ᵥerify. py is provided as ancillary file and reproduces every numbered equation and proposition of the manuscript verbatim.