EXACT MINIMAL AND MAXIMAL TOROIDAL CLOSURES OF A BRACHISTOCHRONE CURVE V1.0

This paper presents a metrologically explicit closed-form geometric model that assigns exact geometric bounds to the closure of a brachistochrone curve constrained to a toroidal surface. The construction is purely algebraic: all fundamental length scales are defined by exact rational coefficients, π, and SI units, with no physical interpretation assumed. At the microscopic end, a horn-torus degeneration introduces the minimal closure circumference c0 = 29 27 × 10 −35 m ≈ 1.074 × 10 −35 m. At the macroscopic end, equating the toroidal surface area to the de Sitter horizon area Λ = 45927 42050 × 10 −52 m −2 ≈ 1.092 × 10 −52 m −2 , yields the maximal major-cycle closure length Lmax = 23200 567 π × 10 87 m ≈ 1.285 × 10 89 m. These three quantities-the minimal closure c0, the cosmological constant Λ, and the maximal closure Lmax-constitute the metrological constraints of the model. They are presented in both exact and decimal form to emphasise their dual roles as algebraic invariants and usable numerical benchmarks. The resulting framework provides a closed-form reference scale suitable for calibration, comparison, and discrete-geometric investigations, including Regge-calculus contexts, without invoking dynamical assumptions. arXiv Endorsement Request: Endorse via code QUDWJF.