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 = ℏ²/2me Gs (Φ) 1/r², 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/c² EH/ (n³ (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=n³ a (ℓa + 1/2) ^−1 −n³ 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.
Donald G Palmer (Thu,) studied this question.