SIM-Resident Continuous-Field Execution as a Universal Operator–Boundary Compiler: Exact Operator Stacks, DPDE Shape Forcing, Robin/DtN Resonance Gates, and Lagrangian Solver Experiments

This paper consolidates a unified analytic framework developed across the project: a SIMresident control core driving a continuous-field screen, augmented by a Fourier-plane operator stage and programmable Robin boundary admittance, yields a universal operator-boundary compiler.The platform replaces raster buffers and discretized state tables with compact, equationdefined program states that generate spatiotemporal fields directly.Within this formalism we derive four interlocking result families.(i) An exact operator library implemented as compositions of real-space multiplication masks and Fourier-space spectral masks, enabling closed-form solvers and propagators for broad classes of constant-coefficient PDEs and producing specialfunction structure (e.g.Bessel/Hankel modes) as native radial identities.(ii) An exact DPDE shape-forcing compiler: for any analytic target geometry encoded as a field, the sustaining source terms required to make that geometry an exact steady solution are obtained by differentiation identities; articulated geometries follow from analytic implicit constructions.(iii) A Robin/DtN resonance-gate calculus: programmable boundary admittance yields explicit scattering and mode-selection laws, moving resonance poles and controlling Q-factors without altering the bulk operator family.(iv) A unified Lagrangian closure: bulk operator stacks and Robin boundaries arise from a single action principle, enabling a "Lagrangian solver" view that couples world dynamics to an observer subsystem written in explicit qualia coordinates, and supporting "physics-in-a-box" experiments where modified dispersions, fractional operators, heterogeneous couplings, and horizon-like boundaries are implemented as analytically specified masks and boundary terms.The result is a coherent end-to-end compilation chain from analytic specifications to realized fields, with applications spanning computation, control, sensing, experimental physics, and creative operator-defined media.