SS-CQMH: Hidden-Sector Projection and Conditional Hydrogenic Observables in Scale Space

We construct a specified Hermitian hidden-oscillator model and separate its projected energy, quantum-metric correction and nonadiabatic remainder. A normalized Gaussian yields a derived geometric coefficient rather than an independent phenomenological insertion. For a logarithmic radial pullback, the reduced Coulomb plus inverse-square Hamiltonian is exactly soluble with its Friedrichs realization. We derive finite-baseline energy and transition sensitivities, and explicit finite-domain channel and spectral-inclusion bounds, before any empirical fitting. A pure state-wide phase produces no quantum-metric scalar, and a smooth nonnegative coefficient with zero baseline has zero two-sided linear response. The parent reduction is conditional: hidden ground-state energy, finite-gap control, physical scale identification and environmental coupling remain explicit requirements. No complex-coordinate necessity, Born-rule derivation or new experimental bound is asserted.