Six Birds Verified XII: The Geometry of Access

Abstract

An observer reads parts of a system through linear probes and knows how the parts overlap. It asks how much it can read, whether its readings fit together, and where they fail to. For finite-dimensional rational state spaces, declared access answers all three through linear algebra and cohomology. What can be read at one place has a dimension: that of the span of the probes, not their number. A global picture reproduces the compatible local readings exactly when it hides nothing and has the right dimension. Misfits that no pairwise check can see are classes in a first cohomology group. On graphs joined by transports they are governed by holonomy, which moves global sections and obstructions together. Compatible readings come from compatible states exactly when a class built from the invisible directions vanishes. In a signed threshold filtration, the rank of signed probes is the number of points minus the number of components exactly below a critical scale: the first scale, if any, with an unbalanced cycle. Rank is not continuous; two explicit gates bound it. Growth of access always stops, but without a certificate a plateau need not be final. All results are formalized in Lean 4.