CATEGORICAL CORRESPONDENCES BETWEEN MATHEMATICAL STRUCTURES AND COSMOLOGICAL SPACETIMES: Complete Formalization with Variational Dynamics and Operational Empirical Protocols

We present a rigorous mathematical-physical framework establishing structuralcorrespondences between a subcategory of number systems with metric completion(NumInf₀) and a physical category of Lorentzian spacetimes with operational informationmeasures (CosmTop). We formalize a covariant functor Φ: NumInf₀ → CosmTop, provepreservation of completion adjunction through natural transformations and commutativediagrams, and define CosmTop morphisms as causally continuous, information-preserving maps satisfying data-processing inequalities.Replacing non-operational uses of cardinalities, we introduce a scalar field φ withcovariant action, derive field equations, guarantee conservation via Bianchi identities,and provide explicit parametrizations connecting φ to observables: sound speed cₛ(k,φ) inthe radiation sector, phase corrections ΔΦ(f) in gravitational waves, equation of statew(z) for the scalar sector, and additional suppression term V(L) in optomechanicalinterferometry.We present contemporary experimental constraints on β, m_φ, and potentials V(φ),alongside realistic empirical protocols for CMB, gravitational waves, backgroundcosmology, and optomechanics, with sensitivity targets and noise/foreground modeling.Technical appendices include formal proofs of naturality, analysis templates, andsimulation specifications favoring reproducibility. Eigenvalue Spectrum of the Three-Dimensional Wave Operator in Cosmological Perturbation Theory: Rigorous Derivation and Observational Verification https://doi.org/10.5281/zenodo.18494333 MEASURABLE CHARACTERIZATION OF INTERMEDIATE CARDINALS: A CONDITIONAL FRAMEWORK VIA INFLUENCE FUNCTIONS ON P(N) https://doi.org/10.5281/zenodo.18074173 Intermediate Cardinal Structures via Measurable Influence Functions: A Conditional Framework on P(N) with Extension to Cosmological Scalar Field Interpretation https://doi.org/10.5281/zenodo.18077403 The Fabric Constituted by E = mc^2 : A Rigorous Framework for Spacetime as an Energetic Medium with Regional Vibrational Modes https://doi.org/10.5281/zenodo.18275005 The Continuum Informational Emergent Theory: A Novel Framework for Cosmology and Fundamental Physics Beyond the Big Bang Paradigm https://doi.org/10.5281/zenodo.18486919