Higher-Order Pseudohyperboloids with the Merge Operation: A Geometric Foundation for Programmable Wave Confinement

This paper introduces higher-order pseudohyperboloids with the Merge operation as a geometric scaffold for programmable wave confinement, and as a candidate foundation for a broader design approach termed geometric wave engineering. The construction is governed by seven independent parameters (a, b, R, R₁, h, m, n) and generates an interior volume Ω₍, ₌ through recursive interval operations followed by geometric union. Unlike conventional resonator classes — such as whispering-gallery resonators, coupled-resonator optical waveguides, photonic crystals, bound-states-in-the-continuum platforms, and hyperbolic-metamaterial architectures — the proposed scaffold treats the topology of the confining region itself as a controllable design variable. The paper makes four contributions. First, it provides a formal recursive construction of Ω₍, ₌ in terms of interval families and the Merge operator. Second, it establishes several geometric properties of the construction, including connectivity switching through the axial parameter h, third-order disk–annulus bifurcation through R₁, and the compression of formal recursive branching into physically merged regions. Third, it formulates explicit design principles that together define geometric wave engineering as a coherent design-oriented perspective. Fourth, it outlines a limited but necessary statement of wave-relevant validation in the scalar-wave approximation, leaving full electromagnetic and acoustic modeling as the immediate next steps. The construction is scale-invariant within linear wave theory and is compatible with RF, microwave, optical, and acoustic regimes subject to the physical constitutive constraints of each domain. The geometry is preferentially manufactured by additive techniques (3D printing, lithography). All geometric claims are accompanied by a fully reproducible computational script provided as supplementary material.