Admissible Boundary Projection

This work presents a structural discovery about how physical systems governed by standard laws (e.g., electromagnetism, spacetime geometry, signal propagation) manifest observable boundary behavior. The paper demonstrates that constraint-selected continuation necessarily projects as boundary geometry. That is, when admissible trajectories, those capable of persisting under perturbation, remaining orderable, and staying within bounded constraints, encounter geometric, material, or representational limits, their continuation does not fail stochastically or dynamically. Instead, it projects deterministically onto boundary structures already observed in physical systems (e.g., shells, horizons, saturation fronts, wakes, caustics). This work does not introduce new physical laws, constants, forces, or dynamics. Rather, it clarifies a necessary structural relation that is already implicit in observed boundary phenomena: boundary behavior is the geometric projection of admissible continuation under constraint. The analysis is grounded in the admissibility-first framework developed in earlier works, which establishes admissible continuation as the minimal condition for empirical persistence. Within that framework, this paper identifies boundary projection as the inevitable geometric consequence when admissibility is exhausted or pinched by constraints. Key contributions include: A formal description of boundary projection as a deterministic geometric outcome, not a failure mode. A dimensionless, scale-free diagnostic invariant expressing the fraction of admissible continuation remaining. A translation of boundary behavior into standard mathematical language using viability kernels, feasibility sets, and projector decompositions. Empirical grounding through optical, electromagnetic, and representational systems, demonstrating that collapse and irreversibility can be directly observed as boundary projection without invoking probabilistic or metaphysical assumptions. Because the result is structural rather than dynamical, it applies across domains without analogy: from optics and electromagnetism to biological, cognitive, and informational systems. The paper reframes collapse, irreversibility, and loss of meaning as geometric facts of constrained continuation, not emergent mysteries or entropy-driven events. This work should be read as a clarifying and consolidating contribution: it makes explicit a selection rule that physical systems already obey, renders it measurable, and shows how boundary phenomena become legible as projections of admissibility loss rather than unexplained breakdowns. Keywords by Domain Core Framework admissible continuation boundary projection constraint-selected persistence viability kernel feasibility geometry structural invariants dimensionless diagnostics Physics & Electromagnetism electromagnetic propagation boundary conditions waveguides plasma shells photon spheres horizon structure Maxwell equations (structural interpretation) Mathematics & Systems Theory viability theory constraint geometry feasible sets projection onto boundaries irreversibility without entropy deterministic collapse Optics & Signal Systems diffraction boundaries saturation effects caustics sensor clipping representational limits signal persistence Cross-Domain / General scale-independent structure persistence under constraint non-probabilistic collapse structural explanation empirical resolvability