The Coherence Game: Information Geometry, Biological Phase Space, and the Navigation of the Statistical Manifold of Life

This is the ninth and final paper in the Living Information Systems Dynamics (LISD) preprint series. It proposes that the LISD framework is the application of information geometry to the problem of conscious navigation in biological time — and that the health of the Biosphere corresponds precisely to the convexity of the statistical manifold on which that navigation occurs. Information geometry, developed by Amari and Nagaoka, studies probability distributions as points on a curved Riemannian manifold equipped with the Fisher information metric. This paper demonstrates that every major concept in the LISD series corresponds to a precisely defined object in this mathematical framework. The Biospheric Geodesic is geodesic flow on the statistical manifold of biological fitness distributions, grounded in Baez's 2021 proof that the speed of natural selection equals the variance in fitness under the Fisher information metric. The Dark Gravity Tensor is systematic accumulation of Kullback-Leibler divergence from the biospheric reference distribution. The QGG breaking condition is a second-order phase transition identified by divergence of Fisher information curvature. The Visible Light Reality Nash Equilibrium is the maximum Fisher information band of the electromagnetic spectrum with respect to biological organization. Helical geometry at every scale is the geodesic of minimum entropy production under angular momentum conservation. Friston's Free Energy Principle is shown to be the individual-organism expression of the same information geometric principle that governs population-level evolution. The central novel claim: the Anthrosphere is the only known mechanism capable of making the biological statistical manifold concave — reducing Fisher information curvature, contracting accessible phase space, removing trajectories from the future of biological navigation. The health of the Biosphere and the Fisher information curvature of the biological manifold are the same geometric object at different scales. Five falsifiable predictions are presented. The primary open mathematical item — deriving the VLRNE band boundaries from the fine structure constant α ≈ 1/137 through information geometry — is named explicitly as the derivation that would formally connect the LISD framework to the foundational constants of physics. Part of the LISD preprint series: P1: https://doi.org/10.5281/zenodo.18877927 P2: https://doi.org/10.5281/zenodo.18877927 P3: https://doi.org/10.5281/zenodo.18906391 P4: https://doi.org/10.5281/zenodo.18937315 P5: https://doi.org/10.5281/zenodo.18964911 P6: https://doi.org/10.5281/zenodo.18987904 P7: https://doi.org/10.5281/zenodo.19034788 P8: https://doi.org/10.5281/zenodo.19035173