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

The axiom. Irreducible statistical dependence implies a connecting structure. This is the foundational principle of causal inference, applied without exception—including to quantum entanglement. The variational formulation. The axiom is given mathematical form as a variational principle on bipartite quantum systems. The action S = ∫i⟨Φ̇|Φ⟩ − ⟨Φ| (KA + KB) |Φ⟩dτ has modular flow as its Euler–Lagrange equation, the local-unitary group U (dA) × U (dB) as its symmetry, and the entanglement spectrum as its conserved Noether charges. The local form of this conservation law is the spectral equivalence of reduced modular Hamiltonians, spec (KA) = spec (KB), which is the informational analog of Newton's third law; Newton's third law itself is recovered as a special case in the thermal limit. The action is shown to be uniquely determined—not chosen—by six published uniqueness theorems (Schmidt decomposition, Stone–Tomita–Takesaki, Aharonov–Anandan, bipartite splitting, internal-consistency state-preservation via Noether, Ostrogradsky stability). The dependence axiom is therefore not a philosophical commitment but a variational principle of the same kind as those that organize the rest of physics. The central new theorem. For any bipartite pure state, the Holevo capacity of the measurement-induced channel exactly equals the entanglement entropy, the maximum is uniquely achieved by Schmidt-basis measurement, and the excess capacity is identically zero (χ = I = SE). This Capacity Saturation Theorem is the framework's quantitative foundation and follows from the variational structure. Derivations from saturation: The Born rule—derived as the unique probability rule compatible with capacity saturation, with an independent derivation from channel thermalization. Two distinct paths to the same result. The Tsirelson bound SCHSH ≤ 2√2 derived as the matching point where simulation cost equals channel capacity. Einstein's field equations via Bisognano–Wichmann and Jacobson's thermodynamic construction, with quantum corrections of order O (exp (−A/16Għ) ) from finite thermalization precision. The cosmological constant Λ = 12/ (N·Għ·t²ₐge) ≈ 10⁻¹²¹ in Planck units—within one order of magnitude of the observed value ~10⁻¹²². Uses one observed input (the age of the universe) and the number of massless species; no free parameters. A 120-order-of-magnitude improvement over the QFT prediction. Additional theorems (50+ total): the Bekenstein–Hawking entropy, Hawking temperature, holographic bound, Bekenstein bound, generalized second law, classical area theorem, no-hair theorem, Page curve, firewall paradox resolution, prohibition of closed timelike curves, arrow of time, singularity resolution, two-phase inflation, flatness and horizon problem resolutions, Hubble parameter range, monogamy from information budget, fine-tuning resolution, Reeh–Schlieder, Kochen–Specker contextuality, quantum Zeno effect, no-signaling, CPT theorem, measurement without collapse, second law of entanglement thermodynamics, Landauer bound for entangled systems, vacuum channel capacity, entanglement–energy equivalence. Predictions: SCHSH temporal decay; decoherence rate ≤ MSS bound; gravitational entanglement bound for BMV-class experiments; entanglement-triggered gravitational collapse; entanglement content of gravitational waves; exponential corrections to black hole entropy. What this paper claims. The paper introduces a variational principle that formalizes the dependence axiom, proves the action is forced (not chosen) by six published uniqueness theorems, derives the entanglement spectrum as a Noether charge, recovers Newton's third law as a special case in the thermal limit, and shows that the resulting framework unifies quantum foundations, quantum information, general relativity, and cosmology under a single organizing principle. The contribution is both new mathematics—most notably the Capacity Saturation Theorem, the variational formulation with its uniqueness chain, and the cosmological constant prediction—and structural unification across results previously developed in disconnected subfields. The framework is presented as a compression of known physics with one falsifiable prediction already passed (Λ to within an order of magnitude, zero free parameters) and several further predictions proposed. No established result in quantum mechanics, general relativity, or quantum information theory is contradicted. Correspondence: studentₜᵥsbohr@proton. me