A finitary structural account of subsystem attribution in closed two-subsystem comparison worlds. The primitive data are a set of joint states equipped only with binary comparison predicates; no topology, metric, dynamics, background geometry, or gauge structure is assumed. Rectangular completeness is shown to be equivalent to profile-maximality, and the resulting quotient semantics yields a precise characterization of when subsystem attribution is and is not invariant.
A mathematically closed physical theory must be written without external scaffolding. If observers, clocks, laboratories, and comparison protocols are all internal to the same world being modeled, then no background manifold, state space linearity, causal order, or field content can be accepted as primitive by convention alone. This manuscript develops the closure principle as a classification program, showing that the admissible architecture is highly rigid across geometric, field-theoretic, and scalar sectors.
We prove that a genuinely closed three-dimensional system is necessarily diffeomorphic to S³. The argument does not start with a sphere and verify that it is closed; it starts from the closure constraint already in force at the manifold stage and derives that the only consistent geometry is S³. The mechanism uses the orthonormal frame bundle and the action groupoid of admissible comparisons.
This paper introduces a formal class of structural framework data and four independent axes of structural input: externalization, artificial factorization, premature globalization, and reification of repair. A theorem establishes their logical independence. The taxonomy is applied to recurring tensions in classical mechanics, quantum mechanics, quantum field theory, general relativity, and statistical mechanics.
Standard cosmology classifies the universe as "open," "flat," or "closed" by the curvature parameter of FLRW solutions. These labels describe the geometry of solutions within a pre-assumed framework and do not address the prior question of what structure is admissible. This paper proves that cosmological and structural uses of "open" and "closed" operate at different logical levels and cannot be identified.
We prove that closure as rectangular completeness stands in a precise relation to standard physical notions of closed two-subsystem systems. For classical Hamiltonian systems it is exactly equivalent to the full-product component of standard closure; for quantum systems it captures the product-state sector while entangled states furnish a concrete obstruction. Rectangular completeness strictly extends standard closure notions on every sector admitting a genuine product-state description.