Λ-2PS Toroidal Field Framework

The Λ-2PS Toroidal Field Framework introduces a physics-informed digital-twin architecture for magnetic-field reconstruction, charged-particle tracking, anomaly detection, and adaptive control in systems with toroidal geometry. The framework defines an effective scalar field, Λ(t, θ, φ), over a toroidal domain to organize a reduced and interpretable representation of magnetic deviations, structural deformation, thermal drift, uncertainty, and control inputs. The Λ field is not identified with the magnetic field B, the electromagnetic potential Aμ, or the cosmological constant. Maxwell’s equations and Lorentz dynamics remain the fundamental physical layer. Volumetric magnetic-field corrections are represented through divergence-free vector bases derived from high-fidelity electromagnetic simulations. The proposed implementation combines proper orthogonal decomposition of vector-potential snapshots, toroidal harmonic descriptors, topology-aware global components, and dynamic state estimation. This separation preserves the distinction between the interpretable geometric field Λ, the reduced computational state, the physically admissible electromagnetic basis, and the observable magnetic field. Two competing reduced-order dynamics are considered: a first-order diffusive model for slow thermal drift and relaxation, and a second-order hyperbolic model for inertial, oscillatory, or resonant behavior. Model selection is determined empirically through out-of-sample prediction, uncertainty calibration, stability analysis, and computational performance. The framework develops two principal operating regimes. The first is an inverse-reconstruction problem inspired by the ATLAS muon spectrometer, in which magnetic-field deviations, detector geometry, and particle momentum may be estimated jointly. The second is an adaptive beam-control problem inspired by the GaToroid concept, using state estimation and model predictive control to support non-rotating toroidal particle-therapy systems. A Kalman-based data-assimilation formulation, observability analysis, safety constraints, falsifiable hypotheses, validation metrics, baseline comparisons, and a staged experimental program are presented. The architecture also introduces the Sophia Factor, 𝒮Λ, as an operational coherence index, and the Δ-Witness as an auditable, hash-linked record of sensor data, inferred states, uncertainty, operating modes, control decisions, verified constraints, and observed outcomes.