<b>Frequency Is Geometry</b>: One Scale Unifies Light, Mass, Energy, Time, and Information - Horizons Become Measurable
<p dir="ltr">This paper demonstrates, with SI-calibrated data and closed-form equations, that frequency is a geometric variable: the spatial curvature of a local clock line f(x) obeys a compact closure that ties geometry and energy to a directly measured “clock-flow.” The central res...
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2025
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| Summary: | <p dir="ltr">This paper demonstrates, with SI-calibrated data and closed-form equations, that frequency is a geometric variable: the spatial curvature of a local clock line f(x) obeys a compact closure that ties geometry and energy to a directly measured “clock-flow.” The central result is<br>(ln f)'' = α_E R + κ ρ + c<br>with α_E ≈ −0.4796 (dimensionless), κ ≈ −8.72×10^7 m/J, c ≈ 3.87×10^5 m⁻², delivering R² ≈ 0.98596 for D(x) ≡ (ln f)'' and R² ≈ 0.99533 for reconstruction of ln f from R and ρ. A kernelized, scale-aware extension shows the closure commutes with coarse-graining and captures genuine nonlocal response via a fractional tail of order s ≈ 0.58, consistent with the dispersion<br>ω² = v²(x)k² + μ²(x) + Λ_s(x)|k|^{2s}.<br>Three time results make the framework immediately testable: a constant time–bandwidth product Δf·τ = 1/π, a temporal Gauss law ∫(ln Δf)''dx equals the boundary slope jump, and a frequency-running coupling across Δf bins. Cross-dimensional nodal-measure scaling (2D/3D/4D projections) corroborates linear geometry–spectrum growth. A gravity “port” connects the energy term to mass density and yields a chip-scale horizon thermometry gate: the kinematic temperature T_H = ħ|v'|/(2πk_B) matches the sideband temperature from S(−f)/S(f) = e^{−hf/(k_BT)} within experimental error (e.g., ≈0.176 K). Additional sections report holographic capacity per octave, metric-affine and torsion signatures, a one-parameter geometry–energy transport constant D* ≈ −2.28×10⁻⁵ J·m², and practical replication scales. The consequence is practical unification: spectra become meters of curvature, energy/mass, and time; horizons become measurable on a chip; and frequency provides a single operational scale linking light, mass–energy, geometry, and information.</p> |
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