One Geometry, Three Frameworks

Three independently developed frameworks — the Quarter Circle (QC), the Zeta-Filter (ZF), and the Harmonic Framework (HF) — are unified here. All three arrive at the same boundary σ = ½, the same constant L (1, χ₄) = π/4, and the same structural role for the number 2. The central result is the Master Equation: ζ (r*) / ζ (1−r*) = r* / (2π (1−π/4) ) which determines K = Ωᵥisible/Ωdark from the Riemann zeta function and π alone, with no free parameters and no cosmological measurement as input. The equation arises from the Timeless-Temporal Crossing: the −0 side of the Zeta Line is atemporal (outside time entirely), while the +1 side acquires time at the moment of crossing. At t = 0 — the Big Bang — the timeless −0 transmission amplitude χ (r*) = ζ (r*) /ζ (1−r*) is converted to the temporal +1 crossing fraction FGUE = r*/ (1−π/4) by the factor 2π, the circumference of the full circle. The equation predicts K = 0. 18800, Ωᵥis = 0. 05038, consistent with Planck 2018 at 1. 4σ. The gap δp = 0. 060 between the two frameworks decomposes exactly into a pre-existing quantum asymmetry (2−π) /4 and the post-Big-Bang order parameter δ = ½−r*. Eight precision predictions follow, all within 1σ of observation. Three mathematical results — the existence and uniqueness proof for r*, the 80-digit numerical verification, and the analytic proof that 2π arises exactly from the structure of the functional equation — are established in Addenda A and B.