Equilibrium Statistics as Conditional Laws and Conservation-Induced Correlations

We present a novel unified conditional-probability framework for relativistic systems in which conditioning on additive conservation laws simultaneously yields equilibrium occupation statistics and conservation-induced correlations. In this formulation, equilibrium arises as a conditional limit law of a closed system. The one-mode marginal gives Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac statistics at leading saddle order, with the conserved quantities fixing the exponential tilt and the microscopic occupation measure determining the statistics. Expanding the two-mode marginal to Gaussian order gives the leading finite-rank covariance between modes induced by exact conservation. When contracted with observables linear in mode occupations, this covariance gives their leading exact-conservation contribution. We use this structure to define projected observables orthogonal to selected conserved quantities. By construction, their covariance has no leading exact-conservation contribution. In small collision systems, where conservation effects are less suppressed by multiplicity and can survive standard nonflow suppressions, this provides a direct way to isolate conservation-aligned contributions to long-range correlations. We demonstrate this with PYTHIA8/Angantyr-generated p+Pb events at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=5.02\mathrm{T}\mathrm{e}\mathrm{V}$ by comparing ordinary and projected covariances, showing that the projection removes the conservation-aligned contribution while leaving the conservation-orthogonal covariance essentially unchanged.