The Participating Clock: Conditioned Relational Dynamics as a Weak-Value Drive, and a Parameter-Free Second-Order Law

When a finite quantum clock gates a register's interactions and the run is conditioned on the clock's final reading, the register's conditioned dynamics is derived at two orders and machine-verified against exact evolution. First order: the drive is the complex Aharonov-Albert-Vaidman weak value of the gating window projector (real part = Hermitian envelope, imaginary part = post-selection reweighting); a twin evolved under this envelope removes the entire first-order mismatch a mean-envelope control retains (state-metric deficits g^1.1 vs g^2.1, factor 56 at g = 0.1). Second order: the deficit is governed by the connected two-time window correlator under post-selection, resolved along the interaction picture; the resulting parameter-free value law matches measured deficits at ratios 0.92-1.05 across the clock-width grid and at a second geometry (1.03-1.07). The when/what trade-off of a participating clock is promoted from confounded excess to derived law: the disturbance is the connected fluctuation of the clock's timing answer, its clock-width profile is a predicted spike-plus-plateau (not a power law), and two earlier exponent claims are superseded with the reconciliation recorded. Two honest derivation failures (equal-time invisibility resurrected by the interaction picture; the kernel belonging inside the integral) are ledgered as the steps that located the law.