The Gate Equation: A Cantor-Spectral Mapping of the Lineweaver–Patel Mass–Radius Diagram

Background Lineweaver and Patel [1] demonstrated that every known object in the universe occupies a triangular region on a log(mass)–log(radius) diagram, bounded by the Schwarzschild line (r = 2Gm/c²) from above and the Compton line (λ = ħ/mc) from below, intersecting at the Planck mass–length point. Separately, the Aubry–André–Harper (AAH) Hamiltonian at its self-dual critical point produces a Cantor-set energy spectrum whose gap structure has been shown to predict atomic radius ratios, with a mean error of 6.7% across 54 elements [2]. Whether these two frameworks—one cosmological and one atomic—share a common spectral architecture has not been investigated. Methods We re-expressed the Lineweaver–Patel boundaries in Planck units as the dimensionless inequality 1/µ ≤ ρ ≤ 2µ and mapped both the radial coordinate and the density-isoline slope onto the φ-bracket address system (bz = round[log(r/l _P )/log(φ)]) used in the AAH spectral framework. The gate angle Θ from the companion atomic-radius formula was connected to the density slope Γ via the continuous interpolator Γ = (2Θ − 1)/(2Θ + 1). Results The Schwarzschild and Compton boundaries correspond to the two largest spectral gaps in the five-band AAH Cantor spectrum: the gold gate (σ₂) and the silver gate (σ₁), respectively. The total span of the allowed triangle, measured in φ-brackets from the Planck vertex (bz = 0) to the Hubble radius (bz = 294), satisfies N × W = 137.3 ≈ α ⁻¹ , where W is the AAH wall fraction and α is the fine-structure constant. The probability of an object traversing all four Cantor gates is W ⁴  ≈ 0.048, matching the observed cosmic baryon density Ω _b  ≈ 0.049 to within 2%. The gate angle Θ from the atomic-radius formula maps continuously to the density slope: Θ = 1 yields Γ = +1/3 (atomic-density objects), whereas the limits Θ → 0 and Θ → ∞ reproduce the Compton and Schwarzschild boundaries exactly. Conclusions These correspondences suggest that the Lineweaver–Patel mass–radius diagram and the AAH Cantor spectrum share a common underlying structure, with the same gate transmission constant L = 1/φ ⁴ governing boundaries at both the atomic and cosmological scales. The framework provides a single dimensionless inequality that locates every known object from fundamental particles to the observable universe.