When Does a Clock Become Physical? Reparameterization No-Go and Residual Clock–Generator Coupling

This paper asks a basic structural question: when does a state-dependent clock represent physical structure rather than a mere reparameterization of time? The starting point is deliberately restrictive. If a dynamical vector field is multiplied everywhere by a strictly positive scalar lapse, then the oriented state-space trajectories are unchanged; only their parametrization changes. A state-dependent clock rate therefore does not become physical by declaration. This positive-lapse reparameterization result is used as a no-go gate for subsequent clock models. The paper then identifies three distinct structures that can survive this test: Relational nonuniversality: Different clock sectors can respond differently to the same residual datum, so normalized clock ratios retain observable differential information even when a universal common mode cancels. Covariant lapse-response stress: When the lapse-dependent sector is embedded in a generally covariant action, a nonzero Hilbert-tensor contribution provides a sufficient tensor witness that the lapse dependence is not removable by coordinate-time relabeling alone. Operator nonscalarity: For $$K_T = L^{-1/2} H L^{-1/2},$$ a strongly commuting clock operator $\widehat L$ is equivalent to a single scalar time rescaling only when it is spectrally scalar on the nonzero-energy support of $\widehat H$. A minimal gauge-invariant realization is built from the residual one-form $$\alpha = d\Phi - A,$$ which sources a positive operational lapse and a positive residual generator-cost form. Under the stated admissibility, density, boundary, and coercivity hypotheses, the zero-cost sector is the aligned vacuum almost everywhere. The same framework yields a lapse-dependent Hilbert stress response and a regulated operator realization near clock-degenerate sectors. For a network of clock comparisons, the paper also derives an exact relational structure. If $\kappa_i$ denotes the logarithmic clock response at vertex $i$ and $D$ is the graph incidence matrix, then pairwise residuals satisfy $$r = D\kappa.$$ On a connected graph, the global common mode is unidentifiable, while the cycle-space component provides an exact noiseless compatibility test for the scalar vertex-clock model. Loss of a positive lower bound is treated separately as a boundary flag $B_{\partial}$. It signals that the global positive-lapse equivalence argument has left its domain of applicability, but it is not taken by itself as evidence of physical inequivalence. The resulting framework is intended as a theorem-controlled physicality criterion for residual clock–energy models: a physical clock is not identified merely by state dependence, but by relational, covariant-stress, or operator structure that survives the reparameterization test.