Chronotopic Metric Theory

We develop Resolution Geometry (RG), a protocol-explicit geometry of observability derived from a declared statistical experiment and an admissibility protocol. The primitive object is not a covariance decomposition or a physical model, but a complete observation system together with the rules that determine which distinctions are operationally admissible. Equality of observation laws induces a universal observational quotient, establishing the canonical domain on which all experiment-internal statements must be defined.

Within regular finite classical statistical models, the Fisher-Rao metric represents local distinguishability through the standard information-geometric characterization under sufficient-Markov invariance. Given a protocol metric and admissibility threshold, resolved directions are selected by a uniquely characterized spectral projector satisfying idempotence, metric self-adjointness, information compatibility, and threshold consistency. Observable sectors obtained in this way transport naturally under experiment isomorphisms and glue into observable atlases through identity and cocycle compatibility.

The resulting structure separates three logically distinct layers. The quotient layer identifies what an experiment can and cannot distinguish. The metric layer quantifies local distinguishability. The admissibility layer determines which distinguishable directions survive nuisance closure, stability, conditioning, and transport requirements. Entropy appears only downstream: observational quotients induce a decomposition of total uncertainty into observable and unresolved components, recovering Boltzmann multiplicity entropy under conditional equiprobability as a special case rather than a primitive assumption.

A covariance realization provides a concrete and computationally useful model of the general framework. In this setting, Fisher geometry is represented through Schur-complement identities, resolved-null coupling controls estimator improvement, canonical correlations provide observable-sector diagnostics, and transport structure admits gauge-theoretic and globalization interpretations. These constructions motivate applications to inverse problems, estimation theory, information geometry, and automated sector identification, but they are not taken as axioms of the framework itself.

The geometric picture that emerges is not a single smooth manifold. The resolved sector carries Fisher-Riemannian path geometry, while coupling and null structure naturally inhabit stratified orbit spaces assembled from eigenvalue chambers, Grassmannian angle strata, and cones of positive-semidefinite forms. Resolution Geometry therefore lives most naturally as a Whitney-stratified orbit-space bundle whose smooth base supports transport, holonomy, and monodromy, while its fibres encode admissible distinguishability, degeneracy structure, and unresolved directions.

The constituent mathematical ingredients$---$statistical experiment theory, observational quotients, Fisher geometry, spectral projectors, Schur complements, canonical correlations, cocycle descent, and stratified geometry$---$are classical. The contribution is not the introduction of a new primitive object, but their modular characterization and assembly into a single geometry of partial observability. Scope and non-claims are stated explicitly: the framework does not derive admissibility protocols from first principles, does not identify observability with ontology, and does not reduce physics to information. Its role is to characterize, for a declared experiment, the geometry of what can be distinguished, transported, and consistently inferred.

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The static, inferential, classificatory, and dynamical results are facets of a single object and are linked by exact identities; a result from any one layer rests on the invariants of the others. Readers building on any component are asked to cite the framework as a whole. Released under CC BY-NC-ND 4.0, [DOI]; the results are offered in the ordinary scientific manner, with the single request that their common origin be acknowledged.