Research Documents
The complete WILL Relational Geometry research in three parts. All results derive from the single principle SPACETIME ≡ ENERGY and the closure condition \(\kappa^{2}=2\beta^{2}\), without free parameters.
Part I: Relativistic Foundations
A relational rediscovery of Special and General Relativity from the SPACETIME ≡ ENERGY principle. Derives Lorentz factors, energy-momentum relations, Schwarzschild geometry, and Einstein field equations using only two dimensionless projections: \(\beta\) (kinematic) and \(\kappa\) (potential) — without metrics, tensors, or free parameters.
- Singularity-free gravity via bounded density \(\rho_{\max}=c^{2}/(8\pi G r^{2})\)
- Equivalence of inertial and gravitational mass derived, not postulated
- All GR critical surfaces (photon sphere, ISCO, horizons) emerge as simple \((\kappa,\beta)\) fractions
- Closed-form Relational Orbital Mechanics without \(G\), mass, or differential formalism
Part II: Cosmological Dark Sector
Applies Relational Geometry to cosmology and galactic dynamics. With zero free parameters, builds an unbroken derivation chain from first principles to observational data across 20 orders of magnitude, replacing the “dark sector” paradigm with transparent geometric ontology.
- Hubble parameter \(H_0\approx 68.15\) km/s/Mpc derived from CMB temperature and \(\alpha\) — within 1% of Planck 2018
- Supernova flux residuals < 0.015 mag across full Pantheon+ range
- CMB acoustic spectrum and low-quadrupole anomaly resolved via \(S^2\) topology
- Galactic rotation curves for 175 SPARC galaxies without fitting
- Wide binary anomaly and dark lensing explained geometrically
Part III: Relational Quantum Mechanics
Derives the complete hydrogen atom structure from geometric and topological closure alone. Quantization emerges as an inevitable consequence of relational self-consistency — not as a separate postulate. No classical force analogues, probabilistic wavefunctions, or differential equations are used.
- Bohr radius and quantized energy levels from first principles
- Fine-structure constant \(\alpha\) identified as the ground-state kinematic projection \(\beta_1\)
- Sommerfeld–Dirac fine-structure formula derived geometrically
- Spin and Pauli exclusion as topological chirality of \(S^1\) winding
Supplementary Documents
Appendix I — Empirical Validation
Complete list of empirical tests and observational comparisons