Shadow Path Geometry: From the Constant of Incomplete Closure (0.353) Through Λs (0.396) to the Resolution Gap (0.420)

This paper introduces a geometric framework for understanding scale-dependent structure in physical constants through a hierarchy of recursive phase thresholds. Three dimensionless invariants are identified and studied: • the Constant of Incomplete Closure L≈0.353L \approx 0.353L≈0.353• the equilibrium threshold Λs≈0.396\Lambda_s \approx 0.396Λs≈0.396• the Resolution Gap RG≈0.420RG \approx 0.420RG≈0.420 These values emerge from the interaction of circular geometry (π\piπ), recursive growth (φ\varphiφ), and exponential damping (eee) within a compactified phase space. A Möbius-type transformation is used to model orientation-reversing recursion, providing a formal mechanism for phase inversion and threshold formation. The framework is developed using fixed-point analysis, fractional linear maps, and eigenmode structure on a closed phase manifold. A statistical null model is included to test whether observed physical constants cluster non-randomly around the predicted thresholds. The resulting model provides a unified geometric interpretation of time, vacuum structure, and scale quantization, linking recursive phase dynamics to experimentally measured constants and particle mass hierarchies. All mathematical definitions, operators, and validation procedures are provided in the appendix. For further information about the ENSO Framework, please contact Eric Needham:ensotheory1@gmail.com