EVE-AUREOLITH Ω: Exact Observable-Kernel Realizability, Minimum Acoustic Drive, and Certified Broadband Floquet Amplification in a Causal Maxwell-Lorentz Model
Abstract
EVE-AUREOLITH Ω develops a constructive mathematical framework connecting a desired electromagnetic response to the acoustic actuation required to generate it. The research addresses a defined part of the physical realizability gap in programmable electromagnetic systems: determining whether a theoretically specified response can be produced by a fixed material–actuator model under explicit resource constraints, and identifying the obstruction when it cannot. The central contribution is a necessary-and-sufficient inverse-realizability theorem for a periodic, homogeneous, small-signal, single-resonance Maxwell-Lorentz material coupled to a finite-dimensional damped acoustic plant. Within this class, the framework connects observable electromagnetic memory, material stiffness, acoustic reachability, and minimum quadratic drive effort. It provides an explicit construction for feasible targets and a precise model-based obstruction for infeasible targets. The release contains a 91-page manuscript, complete conditional proofs, executable research software, exact-arithmetic certificates, synthetic numerical results, and reproducibility records. The principal results are: Recovery of material dynamics from the electromagnetic impulse kernel.For the fixed, calibrated single-resonance model, the complete electric impulse kernel determines the time-dependent material stiffness through a fourth diagonal time coefficient. A separate Volterra consistency identity tests whether the recovered stiffness explains the entire proposed electromagnetic response. This separates local parameter recovery from global response consistency. The theorem assumes an exact, phase-resolved kernel containing the required short-delay information across modulation phases. Recovering this information reliably from finite, noisy experimental measurements remains a separate inverse problem. An exact acoustic feasibility threshold and minimum-effort construction.The required material trajectory is decomposed into modulation harmonics and compared with the acoustic plant’s harmonic transfer matrices. Realizability requires the appropriate range conditions, admissible material bounds, and convergence of a weighted Fourier effort series. When these conditions hold, the series gives the exact minimum average quadratic actuator effort. A target is realizable within a prescribed budget precisely when this minimum does not exceed that budget. The corresponding minimum-effort drive is given explicitly through weighted pseudoinverses of the harmonic transfer operators. This creates a direct chain from a target electromagnetic kernel to a material trajectory, an actuator feasibility decision, and an executable mathematical drive waveform. A finite-force correction to ideal switching constructions.Bounded switching speed alone does not guarantee compatibility with an inertial actuator. A continuous piecewise-linear displacement can have bounded velocity while its corners require impulsive acceleration and force. The manuscript establishes this obstruction and develops finite-force alternatives. These include cubic transitions with continuous velocity and bounded acceleration, together with a mean-preserving triangular-average construction for bounded scalar targets in the fully actuated class. The resulting approximation statements include explicit response-error and resource estimates. Positive realizability gaps at fixed finite resources.Under the stated assumptions, fixing the acoustic plant, modulation period, and finite drive budget produces a compact reachable set of complete electromagnetic kernels. Consequently, an unattainable complete target has a strictly positive distance from that set in the specified uniform kernel norm, provided the feasible set is nonempty. This result gives mathematical content to a resource-limited realizability gap: an infeasible complete response cannot always be approached arbitrarily closely while the same finite resource limit is retained. The manuscript also derives quantitative estimates connecting material-trajectory separation to electromagnetic response error. A finite-force construction with a continuous-band certificate.The featured example uses a causal Lorentz model with positive material damping and an explicitly calculated inertial acoustic drive. Two cubic transitions connect material-stiffness plateaus between 0.4 and 9 over a dimensionless period of 4.98. Its exact resource calculations give an average squared actuator force of approximately 301.2196, average acoustic damping heat of approximately 7.1287, and maximum absolute force of approximately 47.4554. These are dimensionless model quantities; they are not measured electrical power ratings. An exact-integer, outward-rounded dyadic interval computation covers 100 contiguous closed cells spanning the entire wave-number interval [0.9, 1.1]. It establishes that every electromagnetic period matrix in this interval has a real Floquet multiplier below −2.5, and therefore spectral radius greater than 2.5. The certificate covers a continuous interval. It does not rely on interpolating sampled eigenvalues. Its meaning is broadband linearized Floquet amplification within the declared model; finite-port transmission performance and sustained nonlinear operation require additional analysis. Energy accounting and an explicit electromechanical interpretation.The coupled formulation includes external drive work, material dissipation, acoustic damping, and reciprocal mechanical backreaction. An ideal positive-element RLC ladder is derived whose Bloch equations reproduce the featured Maxwell–Lorentz dynamics. The mechanically variable capacitor reproduces the material energy and associated backreaction force. This supplies an explicit interpretation within lumped circuit theory and makes the source of amplification accountable through the external drive. Fabrication, parasitic effects, radiation, and measured actuator calibration remain further validation steps. The candidate research contribution is the integration of observable-kernel identification, whole-response consistency, exact acoustic resource geometry, finite-force synthesis, positive-gap analysis, and a certified broadband example into one construction-or-obstruction framework. Historical priority of this full combination remains unestablished. Its potential practical importance lies in making inverse electromagnetic design more physically discriminating. Within a calibrated model, the framework can identify targets that an actuator cannot realize, calculate the least drive effort for feasible targets, and expose hidden costs introduced by abrupt switching or high-frequency material modulation. These capabilities could support the development of programmable dispersive circuits, acoustically controlled electromagnetic media, and reproducible wave-control design methods. The implemented software includes an exact rational compiler for finite-harmonic scalar targets, interval propagation routines, synthetic examples, and a separate Lua resource checker. The finite-harmonic compiler operates on targets specified through their material-stiffness polynomial; a general experimental kernel-reconstruction pipeline is outside its implemented scope. Broader optimization statements in the manuscript retain their explicit oracle and regularity assumptions. During preparation of the repository edition, all 1,350 suite-reported internal checks passed. The Lua rational resource calculation also passed. The complete 100-cell interval certificate was regenerated, with every scientific JSON field matching the supplied certificate; only execution timing differed. All three exact harmonic compiler examples reproduced their supplied results. The archive includes the manuscript and LaTeX sources, Python implementation, Lua checker, interval certificates, synthetic data, figures, test logs, claim ledger, proof map, prior-art notes, experimental falsification protocol, and SHA-256 integrity manifests. The research establishes conditional results for a specified effective material and actuator class. Independent peer review, proof-assistant formalization, experimental realization, and a universal characterization of three-dimensional electromagnetic materials remain outstanding. The featured construction demonstrates feasibility within its model; its global performance optimality is not established. Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki.Research version: 1.0.0.Repository edition: 1.0.0-hf.1.
