Paper 0: Distortion of the Geodesic Unit Cell in Positively-Curved Constant-Curvature Space — Exact Evaluation of Edge, Angle, Area, and Volume

Foundations volume (Paper 0) of the series "The Dual Geometry of Wavelength Space and Frequency Space. " This paper defines, as an elementary differential-geometry computation, the geometric distortion of a unit cell placed in a positively-curved constant-curvature space of curvature radius R, preserving geodesic length. Main results: (i) the vertex angle is fixed in closed form cos theta = -tan² (1/2R) ; (ii) the volumes for d=2-5 are machine-computed by a unified exact integral; (iii) the small-angle volume coefficient is cd = d (d-1) /12 = C (d, 2) /6; (iv) an existence threshold R*d = 1/ (2 arcsin (1/sqrt d) ) for each dimension; (v) edge length, vertex angle and 2-face area are dimension-independent (intrinsic 2D quantities) while only the volume is dimension-dependent; (vi) a curvature meter inverts sign and magnitude of curvature from a single local angle (dimension-universal) ; (vii) a dimensional ceiling dₘax (R) ~ 4R² with a censorship ceiling fixed at 4 yields a staircase in which dimension emerges from zero and locks at d=4. The 1-dimensional logic wave (d=1) has exactly zero distortion for any R, giving the geometric ground for the curvature-exactness used throughout the series. No physical identification is made. All numbers are reproducible with the bundled verification script; all verification items pass (two-party verification protocol).