Geometric Unification of Physical Interactions

[Superseded version. Please refer to the latest one.] Newtonian, electrostatic, and relativistic regimes unified in a single equation. Empirically falsifiable inertial anomaly at mesoscopic scale (m_ϕ ≈ 4.157 × 10^-9 kg). Proton radius derived as r_p = 4 · ƛ_p · (1 – (α / (4 · π))) (3 ppm discrepancy). Muon anomaly derived without quantum corrections (4 ppm discrepancy). Neutron-proton mass difference derived geometrically (10 ppm discrepancy). Finite radius of the electron derived geometrically at r_e ≈ 5.636 × 10^-15 m. Hierarchy problem (10^38 gap) resolved as a geometric scaling artifact (R_s = 1 / α_G = (m_P / m_p)^2). α revealed as a geometric polynomial of π with icosahedral correction (0.005 ppm discrepancy). Galactic rotation curves resolved as a topological transition to planar geometry. CMB and SH0ES data reconciled as a local topological parallax (39 ppm discrepancy). Proton mass derived holographically from Planck mass (8 ppm discrepancy). Ω_Λ / Ω_m ratio identified as the dynamic constant √5 (< 1% discrepancy). Sagittarius A* nodal symmetry identified as an icosahedral projection (ξ ≈ 1.25). Gravitational constant derived at G ≈ 6.6742439706 × 10^-11 (8 ppm discrepancy). 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 (a_i) as the superposition of the topological volumetric (m^1/3) and surface (m^-5/3) terms. Newton’s and Coulomb’s laws are integrated into a single equation (a_i = (U / r^2) ⋅ (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 × 10^43 N), resolving the main mathematical conflict with quantum mechanics. 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. The structural constant w = 2 is identified as a necessary component to explain the displacement between static geometry and dynamics in a three-dimensional space. 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. The neutron-proton mass difference is derived as a geometric cost of compression (∆: 10 ppm). Time is described not as a fundamental dimension, but as the accumulated geometric distance required to resolve structural cost of motion in a three-dimensional topology. The model predicts that the decoherence time of spatially extended quantum states, such as entangled photon pairs, should exhibit a subtle directional dependence, with a 90° periodicity. The unification is validated through precise retro-predictions without free parameters or extra dimensions. The proton radius is derived as r_p = 4 · ƛ_p · (1 – (α / (4 · π))), matching experimental data (∆: 3 ppm). The geometric hierarchy is extended to nuclear structure, accurately predicting the α particle charge radius (w^3) and the proton magnetic moment (g_p). A finite structural radius for the electron (r_e ≈ 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)) + (α^2 / 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 F_s and F_g (10^38) is derived as R_s = 1 / α_G = (m_P / m_p)^2. Fundamental quantum phenomena, including the Rydberg formula and the uncertainty principle, are described in geometric terms. α is derived as an ordered polynomial of π across the three spatial dimensions defined by the 12-vertex spinor loop (α^-1 = (4 · π^3 + π^2 + π) – (α / 24)) (∆: 0.005 ppm). Galactic rotation curves are explained as a transition to planar geometry without hypothetical matter. The dark energy to dark matter ratio (Ω_Λ / Ω_m) is identified as the dynamic constant δ = √5, matching experimental data with a discrepancy < 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). This icosahedral topology is further falsifiable through specific spectral shifts predicted for Positronium (α / 24). The vacuum catastrophe is resolved by deriving the cosmic density (ρ ≈ 10^-29 g/cm^3) as the geometric dilution of the Planck density across the binary metric hierarchy. The model proposes a unified binary scaling law where proton mass (2^64), gravitational coupling (2^128), and cosmic matter (2^256) 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 (2^128), emerging from the squared geometric harmonic (2^64) (∆: 8 ppm). In this context, the proton mass is derived geometrically from the Planck mass (∆: 8 ppm), validating the holographic scaling law across 19 orders of magnitude, and it is identified as the precise geometric projection of the fundamental spatial pixel (2 · l_P) 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 (ξ ≈ 1.25), unifying micro and macro topologies. 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) / (m_p^2 · 4^64) ≈ 6.6742439706 × 10^-11 (∆: 8 ppm). v18: Conceptual refinements, new sections, typographical corrections.