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 $\sqrt{s_{\mathrm{NN}}}=5.02~\mathrm{TeV}$ by comparing ordinary and projected covariances, showing that the projection removes the conservation-aligned contribution while leaving the conservation-orthogonal covariance essentially unchanged.