Critical Square-Root Factors in Finite-Budget Geometry: Projection Efficiency, Hyperbolic Ball Models, Legendre Reciprocity, and Relativistic Hamiltonian Dynamics

This record contains a preprint by Yanliang Ma entitled Critical Square-Root Factors in Finite-Budget Geometry: Projection Efficiency, Hyperbolic Ball Models, Legendre Reciprocity, and Relativistic Hamiltonian Dynamics. The manuscript develops a mathematical-physics framework for recurring critical square-root factors arising in projection density, finite-budget efficiency laws, Lorentz normalization, Legendre reciprocal Hamiltonian structures, symplectic phase-space dynamics, static gravitational lapse efficiency, oblique rotational connection terms, angular Noether charges, and DBI-type square-root determinants. The main structural chain of the paper is: projection measure -> process efficiency -> Legendre energy -> symplectic dynamics, or equivalently: Jₚroj -> chiₚroc = Jₚroj^ (-alpha) -> L = -E₀ chi -> H -> XH. The paper is a conceptual and geometric preprint. It does not claim to derive special relativity, general relativity, symplectic mechanics, or DBI dynamics from the spherical projection model; rather, it proposes a conservative framework for organizing related square-root normalization and determinant factors. This version corrects the author information on the title page by adding location, email, and the Zenodo concept DOI. The scientific content is unchanged.