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

This record contains version v1.6 of a preprint by Yanliang Ma entitled Critical Square-Root Factors in Finite-Budget Geometry: Projection Efficiency, Hyperbolic Ball Models, Legendre–Euler Classification, and Hamiltonian Dynamics. The manuscript develops a mathematical-physics framework for recurring critical square-root factors appearing in projection geometry, finite-budget efficiency laws, hyperbolic ball models, Legendre–Euler classification, Hamiltonian dynamics, static gravitational lapse readings, and DBI-type square-root determinant structures. The main structural chain of the paper is: projection-density cocycle -> efficiency representation -> Legendre–Euler classification -> symplectic Hamiltonian dynamics. The paper separates four logically distinct layers: projection measure, process efficiency, Legendre/Hamiltonian energy, and symplectic dynamical flow. It further clarifies the relationship between the spherical projection prototype and the hyperboloid–Klein–Poincaré realization of the finite-budget domain. In this interpretation, the Klein ball provides the algebraic budget coordinate, the Poincaré ball displays the ideal boundary, and rapidity is interpreted as hyperbolic distance. Version v1.6 strengthens the theorem chain by adding the projection-density cocycle formulation, the efficiency representation theorem, the Legendre–Euler classification theorem, and the Klein-volume-density interpretation. It also integrates the figures and updates the discussion of hyperbolic ball models, static lapse efficiency, and DBI-type determinant dynamics. This 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, projection-density, finite-budget, and Hamiltonian structures.