Geometric Capacity Constraints, Relative Entropy Bounds, and Covariant Operator Inequalities: A Four-Part Series

We develop a conditional mathematical framework linking covariant geometric capacity constraints in general relativity to operator level relative entropy bounds in quantum information theory. Part I derives sharp distinguishability budget theorems under finite control and finite influence assumptions, establishing area reach squared scaling and a structural separation between state counting (tiling reach) and rate constrained (flow reach) regimes. Part II introduces a covariant scalar capacity field coupled to the Einstein Hilbert action, derives the modified Einstein equation, proves on-shell conservation, and formulates enforcement dynamics via Karush–Kuhn–Tucker closure and exact cap storage release identities. Part III provides an operator-algebraic formulation of Relative Entropy Accumulation (REA), proving telescoping identities, CPTP monotonicity, and deriving spectral bounds for linear response operators. Part IV establishes an explicit operator inequality derivation bounding relative entropy increments by geometric capacity functionals under stated Stinespring dilated channel assumptions, incorporating symmetric cone structure and Euclidean Jordan algebra results. No claim of unification is made. The results establish conditional geometric information bounds under explicitly enumerated assumptions.