Geometric Wave Engineering. Theory of Higher-Order Pseudoellipsoids. Volume 1. Constructive Geometry: Recursive Constructions and Computational Apparatus

This volume presents the constructive geometry of higher-order pseudoellipsoids as the elliptic branch of the Geometric Wave Engineering program. It defines a class of stratified axisymmetric domains whose regular smooth elliptic fragments possess negative Gaussian curvature on regular smooth fragments, while nonsmooth joints, poles, envelope transitions, and Merge seams are treated as boundary strata. The construction is based on a composite elliptic seed generatrix, radial interval states, the recursive operator C_R, exact Merge canonicalization, and row assemblies Ω_{n,m}.

The volume develops the basic elliptic generatrix, second-, third-, fourth-, and arbitrary higher-order pseudoellipsoids, row configurations, boundary stratification, curvature on regular fragments, dimensionless invariants, and scale invariance. It also clarifies the relation of pseudoellipsoids to pseudohyperboloids and pseudoparaboloids as three seed-based branches of a common interval-recursive theory of pseudosurfaces. The results are formulated as constructive geometry. Wave localization, spectral windows, directed output, high quality factor, and cross-physics universality are not claimed as proven in this volume; they are reserved for subsequent C1-C8 verification.