Plithogenic Metric Equivalence and Metric Transport in Evolving Epidemiological State Spaces

Abstract

We develop an operator-theoretic theory of metric equivalence and metric transport for evolving epidemiological states represented within a plithogenic Hilbert space. Let Hepi = HT ⊕ HI ⊕ HF represent truth, indeterminacy, and falsity components associated with epidemiological information, including disease occurrence, exposure, immunity, diagnostic status, environmental risk, and intervention uncertainty. For each temporal frame t, a contradiction structure is represented by a self-adjoint operator D̂t, from which a strictly positive metric operator Gt is constructed. Two epidemiological state operators A, B ∈ B(Hepi) are said to be Gt metrically equivalent when A*GtA = B*GtB. We establish a reduction to classical metric equivalence, characterize the relation through positive square roots and partial isometries, and identify the associated metric invariants. We then introduce epidemiological metric transport operators satisfying A*tGt+1At = Gt and prove that local metric preservation at every temporal transition implies global metric preservation across every finite sequence of epidemiological frames. An approximate metric defect is introduced together with perturbation estimates. The epidemiological component is formulated theoretically: epidemiological variables are represented as components of an evolving uncertain state, while no patient-level or population-level dataset is required. Finite dimensional examples illustrate metrically equivalent and non-equivalent epidemiological transition operators. The resulting theory provides an operator-theoretic basis for studying the preservation of epidemiological geometry under evolving uncertainty, contradiction, and state  transition.

DOI: 10.5281/zenodo.22782528