Planck Scale Crossover and Elastic–Gravity Mapping in USP Field Theory (msf:48510)

This document develops a structured interpretation of the Planck scale within USP Field Theory by examining the geometric crossover between quantum localization and relativistic gravitational radius.Planck units arise from the fundamental constants (ħ, c, G):ℓₚ = √(ħG / c³)tₚ = √(ħG / c⁵)mₚ = √(ħc / G)Eₚ = mₚc² = √(ħc⁵ / G)The crossover condition follows from equating the Compton wavelength and the Schwarzschild radius:λ_C = ħ / (Mc)r_s = 2GM / c²At the Planck scale (scale-level relation):λ_C ≈ r_sThis geometric equality identifies the Planck mass and Planck length as the boundary where intrinsic quantum localization curvature balances relativistic gravitational curvature.A second structural mapping is introduced: gravitational coupling may be re-expressed dimensionally as an effective elasticity parameter of the oscillatory continuum:G ~ c⁴ / T_fT_f ~ c⁴ / GThe corresponding scale:F_P = c⁴ / Gmatches the standard Planck force magnitude (~10⁴³ N), showing that the elasticity interpretation remains consistent with conventional Planck-unit dimensional analysis.In this framework:• No modification of General Relativity is proposed.• No alteration of Quantum Field Theory is implied.• No spacetime discretization is assumed.The analysis operates strictly at the scale (dimensional) level. Order-unity geometric factors are not treated as physically decisive for the crossover identification.The Planck scale is therefore interpreted not as a fundamental particle scale, but as a geometric curvature boundary emerging from the intersection of:Quantum localization: λ_C ∝ 1/MRelativistic gravitational radius: r_s ∝ MThis reinterpretation provides a structural lens for Planck-scale geometry while remaining fully aligned with established physics at accessible energies.