The Geography Is the Fisher Metric: An Information-Geometric Reading of Context, Interface, and Resolution

The preceding notes refused to treat arrays as tensors, contextual vectors as awareness, context windows as subjects, and domain interfaces as inventions. Each refusal asked for a named invariant. This note names one. On a regular parametric family of distributions the Fisher–Rao metric is the unique Riemannian metric monotone under Markov morphisms, and it is a tensor of type (0, 2) in the sense the teaching note required (Amari and Nagaoka, 2000; Mamun, 2026a). An embedding in R^d is a coordinate representation of a point on that manifold only after a statistical model has been specified. Cosine similarity after layer normalisation is not that metric.

Read this way, a context window is a restriction of the conditioning sigma-algebra, not a field of experience. A sparse interface is a large divergence from the conventional mixture, not a concatenated list. A resolution of an immediate problem is a projection, or the endpoint of a dual geodesic, under a divergence fixed by the problem. Situational scores and inventive mixtures remain informative functionals. They become geographic only where the Fisher structure, or a stated divergence that generates it, is the quantity held fixed under redescription. The geography is not the map of activations. The geography is the metric.

KSCCN article, 10 October 2026. Fifth note in the KSCCN sequence, following Tensors Beyond the Array (https://doi.org/10.5281/zenodo.23269680), Contextual Vectors Are Not Awareness (https://doi.org/10.5281/zenodo.23269878), The Window Is Not the Subject (https://doi.org/10.5281/zenodo.23269948) and The Interface Is Not the Invention (https://doi.org/10.5281/zenodo.23275777). Dr. Syed Muntasir Mamun, ORCID 0000-0001-6845-2853. The views expressed are those of the author.