From FLRW to Generalized Amari: Recognized Cosmological Modifications, the Inverse Problem, and the Origin of Global Attention

We establish a downward chain of isomorphisms linking recognized modifications of the Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology to the generalized non-local Amari equation that was recently introduced in the companion work [9]. The chain has three steps. First, we review three recognized modifications ofFLRW: bulk viscosity (Israel–Stewart), anisotropic Bianchi I (Ellis–MacCallum), and non-local gravity (Deser–Woodard). Second, we prove that each of these modifications is isomorphic, under the same hydrodynamic reduction that was used in the companion work [11] to identify the local Amari equation with the two-dimensionalNavier–Stokes equation, to a corresponding modification of the Navier–Stokes equation: bulk viscosity corresponds to an additional Laplacian term with coefficient νQ, Bianchi I anisotropy corresponds to an asymmetric drift uA ·∇Ω, and non-local gravitycorresponds to an additional source SA. Third, we solve the inverse problem: given a modified Navier–Stokes equation with a Laplacian coefficient νQ, a drift uA, and a source SA, we construct the generalized non-local Amari kernel Kglobal = Wlocal +A·Q whose hydrodynamic limit reproduces the given modification. The inverse problem is solved explicitly for each of the three cases, with the minimal regularity assumptions on the kernel.The downward chain is then applied to cosmology: the observed accelerated expansion of the Universe is interpreted as the combined effect of bulk viscosity (the isotropic contribution νQ), Bianchi I anisotropy (the anisotropic contribution uA), and non-local gravity (the source SA), without introducing ad hoc scalar fields or adjustable parameters. We state the constructive hypothesis (H-cosmo), which makesthis interpretation precise and falsifiable. The chain is finally applied to cognitive systems: the same generalized kernel describes the global-attention regime of large language models and the default mode network of the human brain, recovering theresults of [9] as a consequence of the cosmological reduction.The paper contains no adjustable parameters: every constant is fixed by the recognized cosmological modification that is taken as input. The inverse problem isolates the precise conditions under which the generalized Amari kernel is uniquely determinedby the modified Navier–Stokes equation, and it identifies the remaining freedom (the choice of the asymmetry tensor A) as a new direction for future work.