First-order asymptotic expansion of the binomial Bayesian prior

Establish a first-order large-N asymptotic expansion for the log-odds of the equilibrium prior in binomial Bayesian games, namely prove that for any 0 < x_A < x_B < 1 there exists a phase-dependent quantity \;^\varphi(x_A,x_B) such that \vartheta^*_N(x_A,x_B) = \vartheta^{*,\varphi}_\loopedsquare(x_A,x_B) + \frac{1}{N} \;^\varphi(x_A,x_B) + \mathcal{O}(N^{-2}), where \vartheta^{*,\varphi}_\loopedsquare is the phase-dependent limiting log-odds and \;^\varphi(x_A,x_B) is explicitly given (as in equation \eqref{eq:sampiPhiGammaResult}).

Background

Building on the periodic limiting prior conjecture, the authors derive a candidate first-order correction term for the log-odds \vartheta^*_N using Poisson summation and asymptotic expansions. The proposed correction depends on the phase \varphi and is expressed in closed form via transformed coefficient series.

While numerical evidence supports this expansion, a complete proof that the remainder is \mathcal{O}(N^{-2}) and that the stated coefficient indeed governs the first-order term has not been provided.