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 = -tan^2(1/2R); (ii) the volumes for d=2-5 are machine-computed by a unified exact integral; (iii) the small-angle volume coefficient is c_d = 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 d_max(R) ~ 4R^2 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).