The Common-Time Pocket: Time Dilation as Two Hyperbolic Tilts on One Plane

The series reads structure through a winding e^((λ+i)θ) whose modulus is a rate and whose argument is a phase. Applied to relativistic time, the invariant rate is c and the two familiar dilations become two hyperbolic tilts of a worldline: the special-relativistic factor is γ = cosh φ with φ the rapidity (tanh φ = v/c), and the static gravitational lapse is dt/dτ = cosh χ with cosh χ = (1 − 2GM/rc²)^(−1/2). Both are exact. On the single plane (φ, χ) the shared "now" of a bound system is a coherence pocket near the origin: to first order the fractional desync is ε = ½(φ² + χ²), so a tolerance boundary is a circle of radius ρ = √(2ε), circular orbits lie on the ray χ = √2 φ, and the whole solar system sits deep in the corner. Exactly, a clock moving at local speed v at radius r has dt/dτ = cosh φ cosh χ, so ε = 1 − 1/(cosh φ cosh χ); the plane is bounded by two catastrophes, the light cone (φ → ∞) and the event horizon (χ → ∞), and radial free-fall from rest at infinity is exactly the diagonal φ = χ, reaching both at once at the horizon. Throughout, "common time" is the classical coordination of proper times among worldlines; the i is a classical oscillation, and no quantum-coherence claim is made. All mechanism is imported; the paper adds a re-description and no new number.