Geometric Unification of Physical Interactions

Abstract This paper is presented in two parts. The first part (Geometric framework) is strictly derivational and establishes the unified principles. The second part (Heuristic topology and phenomenology) is exploratory by nature. It is intended to test the coherence and predictive power of the model within the broader geometric constraint without introducing free parameters or extra dimensions. I. Geometric framework The geometric unification of physical interactions is based on the topology of a three-dimensional space constrained by a fundamental geometric cost. The model defines intrinsic acceleration (aᵢ) as the superposition of the topological volumetric (m¹/3) and surface (m^−5/3) terms. Newton’s and Coulomb’s laws are integrated into a single equation (aᵢ = (U / r²) ⋅ (m / z + ∣n∣ / m) ), revealing the gravitational constant (G) not as a fundamental constant, but as an emergent and scale-dependent hybrid artifact. General relativity is integrated by imposing a physical limit on the geometric curvature of space (1. 03 × 10⁴3 N), resolving the main mathematical conflict with quantum mechanics. The surface term is identified as the exact geometric description of the spherical Casimir effect. The structural constant w = 2 is identified as a necessary component to explain the displacement between static geometry and dynamics (δ = √5). Time is described not as a fundamental dimension, but as the accumulated geometric distance required to resolve the structural cost of motion in a three-dimensional space. Electric charge is reinterpreted not as a fundamental entity but as the geometric quantization of the object’s structural radius, predicting discrete spatial steps in the acceleration of mesoscopic particles. The theory predicts the existence of a resonance mass (m_ϕ ≈ 4. 157 × 10^–9 kg) and a non-linear divergence from the classical geodesic trajectory at the mesoscopic scale, suggesting a revision of the strong equivalence principle. While the identity between inertial and gravitational mass is preserved, non-linear dynamics become a mathematical requisite for mesoscopic objects which are not strictly neutral. Experimental confirmation of this anomaly would fix the value of G with quantum precision. II. Heuristic topology and phenomenology The fractional quark charges (1/3, 2/3) are derived as geometric projection ratios, reinterpreting the Standard Model as an archaeological record of proton disintegration artifacts. The structural geometry is defined by the proton as a tetrahedron and the electron as an icosahedron. This icosahedral topology is further falsifiable through specific spectral shifts predicted for positronium (α / 24). The model proposes a discrete cubic spatial lattice, predicting a directional dependence of the decoherence time of spatially extended quantum states, such as entangled photon pairs, with a 90° periodicity. The neutron-proton mass difference is derived as a geometric cost of compression (∆: 10 ppm). The proton radius is derived as rₚ = 4 · ƛₚ · (1 – (α / (4 · π) ) ), matching experimental data (∆: 3 ppm). The geometric hierarchy is extended to nuclear structure, predicting the α particle charge radius and the proton magnetic moment. A finite structural radius for the electron (rₑ ≈ 5. 636 × 10^–15 m) is defined, eliminating QED singularities and suggesting that the electron cannot resolve the measurement of the proton due to its own size. The muon anomaly (g – 2) is derived without quantum corrections as the topological friction against the 12 vertices of the lepton structure (a_μ = (α / (2 · π) ) + (α² / 12) ) (∆: 4 ppm). The scaling symmetry between a particle and its orbital dynamics is defined, where α acts as the scaling factor and w determines the topological boundaries. The hierarchy ratio between Fs and Fg (10³8) is derived as Rₛ = 1 / αG = (mP / mₚ) ². Fundamental quantum phenomena, including the Rydberg formula and the uncertainty principle, are described in geometric terms. The Rydberg constant is derived as R_∞ = (mₑ · c · α²) / (2 · h) (∆: 0. 001 ppm). α is derived as an ordered polynomial of π across the three spatial dimensions defined by the 12-vertex spinor loop (α^–1 = (4 · π³ + π² + π) – (α / 24) ) (∆: 0. 005 ppm). The electron’s mass is derived as the inverse projection of a geometric shell defined by the binary metric limit (∆: 0. 01 ppm). The gravitational constant is derived as G = (ħ · c · 2 · (1 + α / 3) ²) / (mₚ² · 4⁶4) ≈ 6. 6742439706 × 10^–11 (∆: 8 ppm). Galactic rotation curves are explained as a transition to planar geometry without hypothetical matter. The dark energy to dark matter ratio (Ω_Λ / Ωₘ) is identified as the dynamic constant δ = √5 (∆: < 1%). The Hubble tension is reconciled as a local topological parallax (1 + 1/12) inherent to the icosahedral geometry of electromagnetic measurement, unifying CMB and SH0ES data (∆: 39 ppm). The vacuum catastrophe is resolved by deriving the cosmic density (ρ ≈ 10^–29 g/cm³) as the geometric dilution of the Planck density across the binary metric hierarchy. The model proposes a unified binary scaling law where proton mass (2⁶4), gravitational coupling (2¹28), and cosmic matter (2²56) emerge as informational bit-depth horizons. The hierarchy problem is resolved as a strict informational scaling artifact, where the gravitational coupling constant is identified at the binary metric limit (2¹28), emerging from the squared geometric harmonic (2⁶4) (∆: 8 ppm). In this context, the proton mass is derived geometrically from the Planck mass (∆: 8 ppm), and it is identified as the precise geometric projection of the fundamental spatial pixel (2 · lP) at the 64th binary harmonic of the metric, corrected by interaction costs. The model also predicts the stellar mass limit at the third geometric harmonic (i = 96), unifying nuclear and astrophysical scales. Sagittarius A* nodal symmetry is identified as an icosahedral projection, unifying micro and macro topologies. v21: Conceptual refinements, new sections, typographical corrections.