Resolving the last Forbidden "Why" Questions: A Geometric Derivation of Lightspeed c and Energy Constancy within the Reciprocal Z + 1/Z Single-Particle Framework

Resolving the last Forbidden “Why” Questions: A Geometric Derivation of Lightspeed c and EnergyConstancy within the Reciprocal Z +1/Z Single-Particle Framework This paper provides the complete mathematical and topological resolution to the foundationalopen questions left by - The Forbidden "Why" Question- - From a Single ParticleUniverse to Space and Time - de Leeuw (2026) - (http://doi.org/10.5281/zenodo.23036173)regarding the physical constant c andcosmological energy balance within a single-particle universe. Operating strictly within the relational Z +1/Z framework, we demonstrate that the speed of light c is not an axiomaticvelocity limit, but the precise geometric boundary of causal reach. This limit is derived fromthe ϵ-compatibility classes between a continuous, timeless circular manifold and its reciprocalEuclidean projection. Under iterative scaling, the phase-space volume of interacting U(1)circle groups imposes a strict information capacity threshold (Nmax = e^2^π^2 ), which, whenmodified by a three-dimensional isotropic spatial efficiency factor (f3D ≈ 0.802), analyticallyconverges to the exact value of c ≡ 299, 792, 458. Furthermore, the paradox of cosmicenergy balance is resolved: the non-closing path integral of the continuous oscillator naturallygenerates open Riemann phase sheets. When normalized to build a three-dimensionalspace-time, this topological evolution manifests as a metric stretching that perfectly drivesthe observed Hubble expansion without violating global invariance. Finally, we show thatthe Planck scale and its fundamental constants emerge directly from the system’s unit radius,establishing W = Z + 1/Z as the complete, self-generating function for the physicalroots of the universe. Required predecessor: 1. The Forbidden "Why" Question- - From a Single Particle Universe to Space and Time - de Leeuw (2026) - Zenodo DOI: 10.5281/zenodo.23036173 For refence of introduction to TSO . Time–Space Oscillations: A Geometric and Deterministic Approach to Relativistic and Quantum PhenomenaZenodo DOI: 10.5281/zenodo.17534734The foundational article introducing the TSO framework and its applications to both relativistic and quantum phenomena. Time Dilation and the Nature of Gravitational and Inertial ForcesZenodo DOI: 10.5281/zenodo.17543059Explores the TSO perspective on time dilation and the interplay of inertial and gravitational forces. Time–Space Oscillations and ElectromagneticsZenodo DOI: 10.5281/zenodo.17591325Applies the TSO model to electromagnetic phenomena, revealing novel interpretations of wave propagation and field interactions. Time–Space Oscillations and Quantum Mechanics Zenodo DOI: 10.5281/zenodo.17670668Extends TSO to encompass quantum mechanical principles, including Heisenberg uncertainty, Schrödinger dynamics, path integrals, entanglement, zero-point energy, and the Quantum Zeno effect.