The Unified Master Equation: A Dimensionless Constant α = 1.5 as the Foundation for a Theory of Everything

We present the Unified Master Equation (UME), a symmetry-based framework that unifies gravitational, quantum, and cosmological dynamics through a single structural constant: α = 1.5. This asymmetry between expansion and contraction emerges from a pre-geometric Δ–Σ vacuum and is treated not as a free parameter, but as a measurable, RG-stabilized quantity—anchored empirically in the range 1.47–1.53.

From this foundation, we derive structurally a wide set of observables across scales and sectors: the cosmological expansion function H(z), the fine-structure constant α_EM, the electron–proton mass ratio, the proton radius anomaly, the anomalous magnetic moments (g−2) of e⁻ and µ, neutrino masses, and the strong CP suppression—with α=1.5 as the sole structural input, complemented by known physical scales (e.g., M_Pl, m_e) for RG flow. The framework also reproduces the deceleration parameter q₀ ≈ –0.40 and the ΛCDM-compatible expansion history without requiring Ω_m or Ω_Λ as input.

A novel contribution is the hierarchical mapping of α = 1.5 into ubiquitous 3/2 scalings in quantum and statistical systems—from partition functions and fermion gases to QCD plasma and gravitational dynamics. This “dimensional echo” is formalized as a causal tree from the Δ–Σ stem to RG-stable branches and observable leaves.

UME also makes falsifiable predictions: a Δ-boson mediator, deviations in short-range gravity, and signatures in gravitational wave spectra.

Together, these results position UME as a mathematically explicit and testable unification scheme—with roots in symmetry, branches in known physics, and leaves in observable precision.

At the RG-stable point α = 1.5, Ward-balanced Δ–Σ dynamics enforce Zt = Zx, deriving the universal light cone (v* = c) as a structural prediction rather than a postulate. This provides a fundamental conceptual bridge between pre-geometric vacuum structure and Lorentz invariance.