THE MUMFORD DYNAMICAL SYSTEM AND HYPERELLIPTIC KLEINIAN FUNCTIONS

The paper proves, via coefficient/limit manipulations of the Mumford Lax system and the conserved polynomial H_ξ = v_ξ^2 + u_ξ w_ξ, that v_ξ = -(1/2)∂_1 u_ξ and w_ξ = (ξ + h_1 - 2u_1)u_ξ - (1/2)∂_1^2 u_ξ, derives u_k' = ∂_k u_1, defines the (P,Q)-recursion, and establishes the key results ∂_k u_1 = ∂_1 P_k (k=2,…,g) and the stationary ODE P_g'' = 2(h_1-2u_1)P_g + 2∑_{i+j=g+1}P_i Q_j + (1/2)∑_{i+j=g} P_i'' P_j'' - 2 h_{g+1} under Pg+1=0. The candidate solution reproduces the same chain of ideas: conservation of H_ξ, extraction of η^g and ξ^g coefficients to obtain the two basic identities and the mixed-flow identity, identification of a triangular system with the (P,Q)-recursion, and use of Pg+1=0 to obtain the ODE. Differences are stylistic (coefficient-extraction vs. ξ→∞ limit), not substantive. The arguments agree step-for-step with the paper’s Theorems 4.1, 4.4, 5.1, and 5.5.