Catenoid Bridge Geometry as a Geometry–Flow Vacuum: Topological Quantization, Integer Spectra, and Double-Barriers

We present a complete geometric reformulation of the catenoid bridge black hole metric as a Geometry–Flow vacuum: a solution of the harmonic flow field equations dω = 0, d†ω = 0,g_µν = F(ω), with no reference to Newton’s constant, stress–energy, or matter sources. In this formulation geometry emerges directly from the flow 1-form ω, and the catenoid bridge becomes a nontrivial harmonic representative of the throat topology S 1 × R. We derive the wave dynamics on this background, showing that the effective potential possesses a universal double-barrier structure, creating a resonant cavity whose quasinormal frequencies obey exact integer ratios. This integer spectrum arises from the winding modes of the throat topology, not from any quantum postulate, thereby exhibiting topological quantization without ℏ. The catenoid bridge metrically connects two asymptotically flat regions and produces characteristic gravitational-wave echoes with timescales of order 0.1 s for stellar-mass objects. We further show that the catenoid bridge sits at θ = 0 of the catenoid–helicoid associate family of minimal surfaces. The helicoid sector (θ = π/2) corresponds to purely oscillatory, unitary Geometry–Flow dynamics, while the catenoid sector corresponds to dissipative or resonant dynamics. A “stacked Wick rotation” in the associate parameter θ continuously interpolates between these regimes and naturally motivates a unified Geometry–Flow operator. An appendix provides an emergent Einstein-like formulation of the Geometry–Flow vacuum equations for readers accustomed to general relativity. This appendix derives the analogue of the Einstein tensor from the flow-functional action, still without Newton’s constant, and shows how the catenoid bridge satisfies the resulting vacuum equation.