A No-Meta Theory of Relativity of Theories

This preprint develops a “no-meta” version of the Theory of Relativity of Theories (TRoT) on top of a persistence-first, value-anchored law-space geometry. Instead of treating evaluators, meta-learners and model selectors as external oracles, the paper embeds them as internal natural laws in the same entropy–transport background as the systems they assess. On the geometric side, each theory TTT in a TRoT category ThThTh is mapped into a fibered Bures–HK entropy–transport (FBHK) law-space LawLawLaw, yielding a theory-space metric and an induced Hellinger–Kantorovich (HK) distance on probability measures over theories. Under standard assumptions (discrete or countable theory catalogues, log-entropy–transport costs, λλ–geodesic convexity), value-anchored functionals on (P (Th), d) (P (Th), d_^) (P (Th), d) admit unique EVI gradient flows that describe theory-selection dynamics. These flows are shown to be invariant under HK-compatible TRoT equivalences, providing a dynamic analogue of categorical equivalence of scientific theories. On the evaluative side, the paper introduces a hierarchy of evaluators: (H1) arbitrary external protocols; (H2) persistence-compatible protocols whose losses are monotone transforms of a common persistence functional; and (H3) fully PFHS/value-anchored evaluators that also satisfy geometric and small-gain conditions. The main theorem establishes a two-way correspondence between persistence-compatible external protocols and a subclass ₏ₕ of internal evaluation theories whose observables embed into the same persistence-first natural-law field. In this precise sense, no-meta is treated as a design restriction: admissible evaluators are exactly those that can be realised as internal laws sharing the same PFHS/FBHK background. Finally, the framework is connected to practical model selection and meta-learning. MDL- and PAC-Bayes-type criteria and gradient-based meta-learning schemes are represented as evaluation theories and value-anchored functionals on theory space, rather than as external principles. A discrete three-theory example in the Fisher–Rao (HK) limit illustrates the induced replicator-type ODE for theory weights and clarifies how different monotone persistence profiles change the dynamics without changing the induced order on theories. The joint contraction analysis extends prior PFHS/value-anchored work by adding the theory-distribution component to a coupled law–agent–defect system.