Identities for ln 2 from the equation 1 − b = 2b^{2m+2}

Elementary manipulations of the fixed-point equation 1 − b = 2b^{2m+2}, arising from the condition Σ(-b)^k = 1/2 (k = 0 to 2m+1), yield connections to the Todd function t/(e^t − 1), the Dirichlet beta function L(s, χ₄), and the Lambert W function. The substitution b = e^{-t} produces a family of transcendental equations whose roots satisfy an exact decomposition of ln 2 involving the Bernoulli numbers. The Dirichlet series over the parameter space recovers L(s, χ₄) with alternating weights and (1 − 2^{-s})ζ(s) with constant weights. All results are unconditional. No conjectures are formulated.