Higher-order noise statistics restore Heisenberg scaling under collective dephasing
Noisy-metrology theory characterizes decoherence by its two-point correlation function, equivalently the single-atom coherence time or noise spectrum. We show this is insufficient for entangled probes: two collective baths with identical single-atom $T_2$ but different higher-order statistics yield opposite entanglement-enhanced scaling. Under Gaussian Markovian collective dephasing a Greenberger--Horne--Zeilinger (GHZ) probe reaches an atom-number-independent sensitivity floor. For a fully Markovian compound-Poisson bath, in which collective dephasing is generated by a finite-rate sequence of unitary phase kicks, a Dicke coherence of order $q$ (a difference of $J_z$ eigenvalues) decays at $\Gamma_q=\Gamma[1-\mathrm{Re}\,\varphi(q)]$, with $\varphi$ the kick characteristic function; for any absolutely continuous kick law this rate saturates at large $q$ instead of growing as $q^2$, and a GHZ probe recovers Heisenberg scaling $\delta\omega\propto1/N$ over the window in which collective finite-rate noise dominates residual independent decoherence. We prove that the Gaussian floor is the exact worst case: at fixed single-atom coherence time every finite-rate kick statistics strictly beats it, and for arbitrary L\'evy phase noise the asymptotic entangled-probe sensitivity is set exclusively by the diffusive component. A converse bound shows that no input state, ancilla, or measurement improves on the GHZ scaling. The mechanism is purely exponential and CP-divisible, distinct from the Zeno, non-Markovian, nonlinear-generator, and error-correction routes. A dissipative analogue caps the Dicke superradiant burst. The full counting statistics of common noise thus emerge as a control axis for noisy quantum metrology, beyond the spectrum.