Paper 23: Complex-Scale Quantum Mechanics: Geometric Neutrality and Hydrogenic Phase Shifts in a Gravitational Field

We derive a quantum-mechanical prediction from a framework in which the logarithmic scale coordinate of space is promoted to a single complex value zs ∈ C. The analysis has three main results. First, the requirement that the native Lagrangian be real forces the potential to be real-valued, automatically suppressing any imaginary classical force: the imaginary scale sector is classically neutral without any separate projection step. Second, be cause sR = ln(r/ℓ0) varies with position, projecting the hidden sI-sector onto the configurational wavefunction via a Born-Huang dedecomposition produces a Hermitian geometric correction to the Hamiltonian, ∆Hˆcs = ℏ^2/2me Gs(Φ) 1/r^2, where Gs(Φ) is the quantum metric of the sI -sector. Writing λ = Gs/2 (dimensionless), the first-order energy correction to hydrogenic state |nℓm⟩ in a gravitational potential Φ is ∆Enℓ(Φ) = λ Φ/c^2 EH/(n^3 (ℓ + 1/2)), ∆Enℓ ∈ R. Third, the state-dependence of ∆Enℓ produces a non-universal frequency residual between clock transitions, Rab,cd = λ EH (Fab/E^(0)ab − Fcd/E^(0)cd), where Fab=[n^3 a (ℓa + 1/2)]^−1 −[n^3 b (ℓb+1/2)]^−1, that cannot be mimicked by standard gravitational redshift and provides a clean experimental discriminator.Current optical clock data constrain λ ≲ 2 × 10^−9; the natural framework estimate λnat ≈ 4 × 10^−16 is safely consistent with all existing measurements.