Information Geometry of PAO

This paper proves an existence theorem: any regular statistical channel family equipped with an immersive self-referential write-back policy induces a genuine Fisher information metric on persistent-state space, under conditions broad and easy to check. This resolves, at the level of existence, PAO-Foundation’s central open problem – the construction of an information metric governing the accessibility of neighboring futures – for the entire class of sufficiently regular Persistent Adaptive Organizations, independent of any one state space or noise family. PAO-Binary, PAO-Smooth, and PAO-Sphere – together the bacterium trilogy – serve as explicit examples confirming this general result: each exhibits one concrete, exactly solvable channel family – multiplicative correlation on two points, von Mises noise on the circle, von Mises–Fisher noise on the sphere – for which the metric this paper proves must exist can also be computed outright, and, for PAO-Sphere, its curvature found exactly. Whether that geometry is homogeneous, curved, or exhibits sign-changing curvature are logically separate questions, addressed at progressively stronger levels of generality: we prove that full symmetry of the policy forces the induced metric to be the unique invariant metric the symmetry permits – for the sphere, the confirming example on which this pattern is checked exactly, that invariant metric is the familiar round one, giving constant curvature immediately – and we prove that breaking this symmetry removes the group-theoretic obstruction responsible for that triviality. What fills the space the obstruction vacates is examined at two further, weaker levels: PAO-Sphere’s own theorem stands as an exact existence witness that genuine, sign-changing curvature is achievable by this architecture, and we state, explicitly as a conjecture rather than a theorem, that non-trivial curvature should be expected generically once symmetry is broken on a manifold of dimension at least two. We do not derive this metric in closed form for any new example beyond the three already confirming it; that is deliberately left to future work. This paper establishes that Fisher information geometry arises naturally and generally from regular self-referential write-back policies. The spherical example confirms that such geometries can exhibit genuine, sign-changing curvature, suggesting that the phenomenon is architectural rather than accidental. Determining the precise genericity of non-trivial curvature across arbitrary channel families remains an important open problem.