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

This volume establishes the structural and geometric foundations of the research program Geometric Wave Engineering and introduces a class of recursively generated axisymmetric internal regions Ω_n and Ω_{n,m}. These regions are defined through families of radial intervals I_n(ξ), the recursion operator C_R, the canonizing Merge operator, an axial array layout, and dimensionless parameterization. The central object of study is neither a single branch of the generatrix nor an outer shell, but the entire internal volume, which is suitable for boundary-value, spectral, and computational problems.

The volume formalizes the interval-recursive apparatus for constructing higher-order geometries. It introduces the space of interval states, the basic seed profiles, the vertical and horizontal hyperbolic types, and the general recursive mechanism for the transition from order n to order n + 1. The fundamental geometric properties of the construction are established: well-definedness and idempotence of the Merge operator, the 1-Lipschitz property of the elementary mappings, non-negativity and boundedness of radial intervals, finiteness of the number of components, closedness of the interval-family graph, compactness of Ω_n and Ω_{n,m}, scale invariance, and completeness of the interval description.

The interest of this class of regions is motivated not only by their recursive geometric organization, but also by the focal structure of the hyperbolic generatrix. The foci of the generatrix are located outside the innermost region, yet they define geometrically distinguished directions associated with the seed family. Whether such focal structure has any non-trivial influence on internal ray or wave dynamics is treated in this volume as an open question rather than an established property; it is reserved for the falsifiable verification program of Chapter 12.

Volume 1 systematically separates proven geometric results from ray and wave effects that have not yet been established. Statements concerning energy localization, spectral windows, high Q-factor, radiation directivity, axial field concentration, and cross-domain applicability are not treated here as proven design properties. They are formulated as falsifiable hypotheses and are referred to the full-wave verification program developed in Chapter 12 for electromagnetic, acoustic, and quantum-wave models.

Volume 1 is therefore not a complete universal wave theory, but the first volume of a research program: it introduces a mathematical object, proves its basic geometric properties, fixes a reproducible computational standard, and provides a falsifiable bridge to subsequent volumes addressing physical applications.