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

Paper 13 3 conjectured that the scale coordinate of the (x, y, z, s) framework is a single complex value zs ∈ C. Paper 17 5 established that classical field equations are neutral with respect to the imaginary part sI, placing the conjecture correctly in the quantum sector. This paper derives the first concrete, testable quantum-mechanical prediction of the complex-scale framework, resolved through four steps. The native Lagrangian is manifestly real; this requires the potential V (x^µ, zs) ∈ R, which automatically forbids imaginary energy corrections and ensures classical neutrality without any separate projection step. The complex-scale sector enters quantum mechanics through a Born-Huang-type geometric projection of the hidden sI -sector onto the configurational wavefunction. Because sR = ln (r/ℓ0) varies with position, the geometric scalar potential of the projection is ∆Hˆcs = ℏ²/2me Gs (Φ) 1/r², where Gs (Φ) is the quantum metric of the sI -sector. Writing λ = ℏ² Gs/ (2mea²0 EH) (dimensionless), the complex-scale energy correction to hydrogenic state |nℓm⟩ is ∆Enℓ (Φ) = λ Φ/c² EH/ (n³ (ℓ + 1/2) ), ∆Enℓ ∈ R. For a clock transition |a⟩ → |b⟩ at two heights: ∆ϕ^ (12) ab (T) = −λEH/ℏ Fab (Φ1 − Φ2) /c² T, Fab = 1/ (n³a (ℓa + 1/2) ) − 1/ (n³b (ℓb + 1/2) ). The level-structure factor Fab makes the correction non-universal: it differs between transitions and cannot be mimicked by standard gravitational redshift. The dimensionless coupling is bounded by current optical clock precision, λ ≲ 2 × 10^−9, while the natural framework estimate λnat ≈ 4 × 10^−16 is safely consistent with all existing measurements.