We propose tying the fundamental scale of gravitationally induced space-time blurring to proper time rather than to spatial length, and postulate that, under an ensemble reading of the quantum state in which each branch carries its own world line, proper-time fluctuations are statistically independent across branches. Interference is sensitive to the difference of branch phase contributions; with independent clocks this difference accumulates at the frequency ω_eff = E_br/ℏ. For a composite system we define branch-sensitive energy operationally after removing the Hamiltonian sector common to both branches and measuring the remaining split-sector energy from its physical vacuum or ground state. This prescription is invariant under a global shift of the total Hamiltonian. It yields V/V_0 = exp(−χ) with χ ∝ m² t^(2/3) for a centre-of-mass interferometer. Confronted with sodium-nanoparticle interferometry of the Vienna group, this gives an upper bound ξ ≲ 1.1. A decisive test is near: at m ~ 1 MDa and v ≈ 25 m/s the model predicts practically complete loss of interference, whereas standard quantum mechanics expects measurable fringes.