Quantum Phase-Scale Symmetry and Gravity

Can electromagnetism and gravity be the compact and noncompact gauge branches of one quantum symmetry? We introduce a local phase–scale transformation acting on the wavefunction and its internal radius. This transformation leaves invariant the gauge-covariant kinetic phase, the lock Rω = c, and the normalized clock share Q_G = (ω_ℂ/ω)². Universal clock response, reciprocal clock–ruler closure, and an availability-weighted Gaussian information cost then yield ΔI_ℂ = −ln Q_G, ∇̄²Q_G = 8πGε/c⁴, and a definite information-stress tensor. For two positive-energy sources, the associated cross stress is tensile in the intersource region, produces U_ab = −Gm_am_b/d, and yields the inverse-square force. The same invariant determines both the lapse and the spatial metric, while its static spherical vacuum reproduces the Schwarzschild geometry exactly. Electromagnetism and gravity thus emerge as the U(1) and ℝ_{>0} branches of a single complex quantum symmetry.