Emergent Fisher Halos from Information Geometry

Dark matter phenomenology is usually introduced at the level of microscopic models or parametric density profiles. Rotation curves, cluster dynamics and lensing maps are then treated as constraints on an a priori unknown mass distribution. Modified gravity proposals instead change the relation between baryonic matter and force law, but struggle to reproduce clustered halo behaviour and cosmological constraints in a unified way. In the universal information hydrodynamics (UIH) framework developed in earlier works, which already support a weak field scalar sector that reproduces the Newtonian Poisson law in a coarse grained setting. In this paper we apply the same Fisher geometry to self gravitating discs and show that it naturally generates an extended scalar halo. We study Gaussian density fields on a two dimensional disc with a broad class of local covariance or precision operators with finite correlation length, and prove that, for discs large compared to this scale, the Fisher information in the constant mode grows extensively with the disc area, with boundary corrections that scale only with the perimeter. Interpreted as a scale dependent kinetic prefactor for a scalar field that tracks a common mode shift of the disc density, this yields a radial Fisher coupling that grows quadratically with disc radius. When this scalar is coupled to the Newtonian potential in the weak field limit, the associated gradient energy density behaves like an extended halo with density profile approximately proportional to one over radius squared, producing flat rotation curves without introducing new particle species or modifying Newtonian gravity. The mechanism is geometric rather than phenomenological: once the Fisher geometry and scalar channel are fixed by the UIH axioms, the halo follows from an area law for Fisher curvature on discs and a simple Newtonian coupling. We develop the construction analytically and test it with numerical experiments. Two lattice models realise the Fisher area law and its boundary corrections in concrete finite difference operators. A Newtonian scalar halo model with a running Fisher coupling reproduces flat outer rotation curves and links the halo amplitude to disc compactness and a single coupling scale. An information theoretic renormalisation group picture, in which the Fisher coupling saturates beyond disc scales, produces a finite total halo mass and a Keplerian decline at very large radii. A suite of numerical tests explores sensitivity to the baryonic profile, compactness and correlation length, and shows that the Fisher scalar halo can reproduce the qualitative structure of observed rotation curves while remaining anchored to the same information geometry that organises the UIH programme.