A Variational Functional Equivalent Framework for the Riemann Hypothesis: Hilbert Kernel Spectral Measure Model (Chinese‑English Dual‑Document)

This paper constructs a quadratic energy functional defined on the space of signed Borel measures. Using a Hilbert integral kernel, the Riemann Hypothesis is equivalently trans formed into a non-negativity condition of the steady-state measure of gradient flow. The full framework includes kernel symmetry, strict positivity, coercivity, Gateaux differentia bility, existence and long-time weak convergence of gradient flow. Finally, an equivalent proposition is established between the non-negativity of steady-state measure and the zero distribution of Riemann ζ function. The framework can be formally verified in Lean4, with only the core equivalence proposition left as an unproven gap. 本文构造定义在带符号 Borel 测度空间上的二次能量泛函,以 Hilbert 积分核将黎曼猜想 等价转化为梯度流稳态测度的非负性条件。整套框架包含:核对称性、泛函严格正定性、强 制性、Gateaux 可微性、梯度流存在性与长时间弱收敛,最后建立稳态测度非负与黎曼 ζ 函 数零点分布的等价命题。框架可使用 Lean4 形式化验证,仅核心等价命题保留待证缺口。