Saturation Geometry and the Structural Emergence of Measurement and the Born Rule in a Capacity-Constrained Dirac–Λ System

This paper develops a fully operator-theoretic account of quantum measurement and the Born rule within a constrained dynamical framework given by a coupled Dirac–Λ system. In this Tier-1 setting, a reversible Dirac carrier is coupled to an independent irreversible (record) channel through a scale-by-scale capacity inequality enforced via a Karush–Kuhn–Tucker (KKT) structure and a fixed ultraviolet anchor. All spectral schemes and determinant prescriptions are globally fixed and non-tunable. The main results are: • A canonical record (pointer) algebra defined intrinsically as the fixed-point algebra of the modular automorphism group of the saturated stationary state. No external measurement basis is inserted. • A quantitative convexity gap showing that persistent record-coherence carries strictly positive Dirac-side capacity cost. At any active (saturated) scale, complementary slackness forces record-diagonality. Collapse is therefore enforced as a feasibility condition, not postulated. • A rigidity theorem showing that modular implementability of the linearized saturation dynamics forces Hilbert (GNS/L²) geometry, excluding non-Hilbert Schatten p-structures and fixing the quadratic Born exponent. • A derivation of the Born rule as the unique positive, normalized, unitarily covariant bounded linear functional on pointer projections once L² geometry is enforced. • A quantitative irreversible activation cost at measurement, including ultraviolet and infrared asymptotics and a capacity-limited decoherence rate bound with 1/T suppression in the infrared regime. The framework is falsifiable: violations of the capacity inequality, failure of the ultraviolet anchor, instability of the saturated configuration, deviation from quadratic Born weights, or absence of the predicted infrared suppression regime would disconfirm the model. This work does not rely on interpretational postulates and does not invoke collapse as an axiom. Measurement and Born probabilities emerge structurally from saturation geometry within a capacity-limited reversible/irreversible coupling.