The Full Channel: Deriving the Born Rule, the Tsirelson Bound, and Einstein's Equations from Capacity Saturation

We begin from an axiom of statistical inference — that irreducible statistical dependence implies a connecting structure — and apply it without exception to quantum entanglement. The resulting framework yields a capacity saturation theorem: for any bipartite pure state, the Holevo capacity of the measurement-induced channel exactly equals the entanglement entropy, and the correlation exactly consumes this capacity. From saturation alone, we derive three results that standard quantum mechanics postulates or proves only algebraically: (i) the Tsirelson bound SCHSH ≤ 2√2 follows from the requirement that the communication cost of simulating the correlation cannot exceed the channel capacity; (ii) the Born rule is the unique probability rule compatible with exact capacity saturation; and (iii) entanglement monogamy follows from a quantitative information budget constraint. We further show that the self-sizing property of the channel resolves the fine-tuning problem identified by Wood and Spekkens for causal models of quantum correlations, derive Born rule emergence as thermodynamic equilibrium, establish a second law of entanglement thermodynamics, and — via the Bisognano–Wichmann theorem and Jacobson's thermodynamic construction — derive Einstein's field equations from the Clausius relation applied to the channel's thermal structure. The complete chain runs from a single statistical axiom to both quantum mechanics (Born rule, Tsirelson bound) and general relativity (Einstein's equations). Several speculative extensions were proposed during development and killed by the paper's own theorems; these failures are reported as informative.