Gravity as Logarithmic Frequency-Potential: A Mathematical Framework for Lapse-Gradient Control and External-Field Surfing

This paper develops a mathematical framework in which gravitational potential is represented by the logarithm of the lapse function in static and asymptotically flat relativistic spacetimes. In Newtonian gravity, a scalar potential controls acceleration through its gradient. In general relativity, the equivalent scalar structure is not globally available in arbitrary spacetimes; however, in static geometries the lapse N encodes the local rate of proper time relative to a chosen asymptotic time. We define the dimensionless frequency-potential ϕ = ln(N) and treat its spatial derivative as the lapse-gradient field governing redshift, static-observer acceleration, and weak-field recovery of the Newtonian potential. The paper makes three theorem-level contributions to the mathematical interpretation of static relativistic gravity. First, it formalizes ϕ as an additive potential for multiplicative frequency ratios. Second, it separates curvature, clock-rate potential, and control work in a way suitable for theorem-level development. Third, it formulates external-field surfing as a constrained variational and optimal-control problem over worldlines in Einstein-constrained external lapse fields, rather than as an uncontrolled or reactionless force law. The completed manuscript establishes the conceptual framework, proves the static-field identities, develops the Schwarzschild and weak-field examples, states the variational problem with explicit assumptions, and records the domain and conservation-law limitations.