Saturation Geometry and the Structural Origin of the Born Rule: Measurement from Born/KKT No-Excess Selection

This paper develops an operator-theoretic account of quantum measurement and the Born rule within a constrained dynamical framework given by a coupled Dirac–Lambda 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, a fixed ultraviolet anchor, and a Karush–Kuhn–Tucker (KKT) variational structure. The spectral filters, determinant prescriptions, and normalization conventions are fixed globally and are not tuned per background. The main result is a Category C internal framework resolution candidate via Born/KKT No-Excess Coimage Selection (NECS) active-record selection. The paper replaces the earlier scalar strong-convexity hypothesis with a KKT-loaded active readout quadratic form, written in the paper as Q_mu = J_mu^* M_mu J_mu. Here J_mu is the active residual readout map, and M_mu is the canonical L2(mu) multiplier metric induced by the KKT multiplier measure. The observed record is not identified with the full raw persistent source algebra. Instead, it is selected by a Born/KKT instance of the NECS parent law as the no-excess coimage of the persistent source algebra under the active readout map. In plain terms, the observed pointer record is the part of the persistent Born-side source structure that remains visible through the active readout map, with any readout-invisible surplus quotiented away. Persistent source-side surplus invisible to J_mu is therefore quotient-silent for the observed record. Under this selected no-excess record algebra, KKT stationarity and positivity of the active readout quadratic form force record-diagonality. Collapse in the pointer sector is therefore enforced by saturation and active-record selection rather than postulated. The paper also derives Hilbertization of the active response geometry on the finite active spectral spine. The irreversible Lambda-channel defines a faithful active state, and the linearized response Hessian acts as a Schur multiplier on the same spectral spine. This intertwines the response flow with the modular flow, fixes the Schatten exponent to p = 2, and yields the trace rule P(k) = Tr(rho P_k), reducing to the usual squared-amplitude Born weight for pure states and rank-one pointer events. The framework is conditional and falsifiable. It depends on active residual coimage generation, Spectral-Spine Response Closure, the capacity inequality, the ultraviolet anchor, stability of the saturated configuration, quadratic Born weights, and the predicted infrared suppression behavior. Failure of these structures would disconfirm the model. The paper does not invoke collapse, a preferred basis, or Born probabilities as independent interpretational postulates. Measurement and Born weights emerge structurally from saturation geometry through KKT-loaded active readout, no-excess coimage selection, and Hilbertized response geometry within the coupled Dirac–Lambda system.