Field-Theoretic Residual Clock Response and the Criterion for a Common Operational Time

Version v1.0. This work develops an explicit field-theoretic realization of a residual-induced clock-rate response and separates it from the stronger requirement of a common operational time. Starting from a broken-$U(1)$ parent theory with gauge-invariant phase mismatch $$\\alpha_\\mu = \\partial_\\mu \\Phi - A_\\mu,$$ we condition on a quasistatic spacelike residual background with supported spatial strength $$X = \\vert{}\\alpha\\vert{}_h^2.$$ The broken-phase radial amplitude responds as $$\\rho_X^2 = v^2 - \\frac{X}{\\lambda_\\Psi},$$ and a neutral portal-coupled phase clock acquires $$\\omega_C^2(X) = \\omega_{C0}^2 - \\frac{g_C}{\\lambda_\\Psi} X.$$ In the probe and adiabatic regime this yields the normalized clock-rate factor $$L_C(X) = \\frac{\\omega_C(X)}{\\omega_{C0}} = \\sqrt{1 - \\eta_C X},$$ with the calculable leading coefficient $$\\lambda_{\\mathrm{eff}} = \\frac{g_C}{2\\lambda_\\Psi \\omega_{C0}^2}.$$ This provides an explicit parent-model realization of the leading weak residual-clock response considered phenomenologically in the companion work Residual Clock Dynamics: Operational-Time Rescaling and Effective Relaxation in a Minimal $U(1)$ Model (Version v1.0, Zenodo DOI: 10.5281/zenodo.22735204). The exact nonlinear exponential completion of that phenomenological response is not derived here and remains model dependent. For multiple directly coupled clocks, the analysis shows that common normalized rates require $$\\frac{g_a}{\\omega_{a0}^2} = \\frac{g_b}{\\omega_{b0}^2},$$ revealing a generic universality obstruction: a residual-induced clock-rate shift does not by itself define a common operational time. A universal residual-dependent matter metric, $$\\widetilde{g}_{\\mu\\nu} = L^2(X) g_{\\mu\\nu},$$ is then introduced as a sufficient effective-field-theory mechanism for common clock scaling. This common-metric coupling is not derived from the parent theory, and matching its clock factor to the parent phase-clock response is an EFT matching assumption. Under exact common conformal scaling, the common factor cancels from local dimensionless ratios of co-located ideal clocks, producing a complementary common-mode observability obstruction. The paper also distinguishes pure time reparameterization from metric dynamics, isolates the Lorentzian obstruction to constructing a globally positive quadratic residual norm from a real one-form and the spacetime metric alone, and analyzes linear response about nonzero supported residual backgrounds. The resulting hierarchy is $$\\text{response} \\longrightarrow \\text{universality} \\longrightarrow \\text{observability}.$$ The work is intended as a field-theoretic and effective-theory analysis of these logically distinct requirements, not as a UV-complete theory of time or a claim of experimental detection.