Information Holarchies Categorical, Information-Geometric, and Field-Theoretic Foundations of Multiscale Adaptive Organization

A multiscale adaptive system—a cortex, an organism, an enterprise, a population of learningagents—is modelled here as an info-holarchy: a diagram of information-processing units (infoholons),indexed by a rooted tree of scales, in which each unit carries a belief about its ownenvironment and each edge carries a coarse-graining between environments. This second editionrebuilds the theory of [Sep26b] on a categorical foundation. Holarchies are functors from theposet of scales into a Markov category; consistent multiscale beliefs are precisely the cones,equivalently the elements of a limit, of the induced state functor; and the local-to-global problemof the first edition becomes a gluing problem whose solution on trees is guaranteed by conditionalproducts and whose failure on cyclic scale graphs is a Vorob’ev–Bell-type obstruction. In thelinear–Gaussian case the holarchy is a cellular sheaf, consistent beliefs are its zeroth cohomology,and the hierarchical free energy is a sheaf-Laplacian regularization.On this foundation we correct and sharpen the central results. The hierarchical free energyis re-derived from the generalized Pythagorean theorem of information geometry, separating anirreducible representational residual from a genuine consistency deficit; this repairs an inconsistencyin the first edition, where pushforwards of recognition densities generally left the parent’sfamily. The local–global consistency theorem is upgraded from an O(λ−1) bound on the penaltyto a sharp statement: the consistency violation decays as ∥y∥/λ with y the Lagrange multiplierof the strong-coupling problem, the penalty as 12∥y∥2λ−2, with a non-asymptotic bound2∥y∥/λ and a Γ-convergence statement for the general nonlinear case. Coarse-graining of dynamicsis identified with naturality squares (strong lumpability), informational closure follows, theeffective-information account of causal emergence is corrected (the supremal emergence gap overm macrostates is log2m, not attained), and the first edition’s conjectured inter-scale phase transitionis replaced by an exact theorem: the Kesten–Stigum / Evans–Kenyon–Peres–Schulmanreconstruction threshold for broadcasting on trees.The quantum part is rewritten in operator-algebraic terms (Umegaki and Araki relativeentropy, the Bogoliubov–Kubo–Mori metric selected by dual flatness among Petz’s monotonemetrics, Petz recovery as quantum Bayesian inversion), with holographic, noncommutativegeometricand M-theoretic material retained only at the level the literature supports. Theinfoton hierarchy of the companion papers [SJ26a, SJ26b] and the dually flat geometry of informationfield theory [Sep26a] are integrated: the Gaussian holarchy is an information fieldtheory on a tree, with the Legendre pair of cumulant generating function and effective actionas its dually flat structure and leaves-to-root Schur complementation as an exact renormalizationstep; and a holarchic Landauer theorem shows that the heat released by a holarchy thatupdates its holons locally is bounded below by the global entropy reduction plus the inter-holonmutual information it destroys. Self-reference is treated by Lawvere’s fixed-point theorem andKleene’s recursion theorem, yielding an impossibility theorem for complete self-models alongsidean existence theorem for finite self-describing holarchies. Every numerical claim is recomputed by scripts reproduced in the text. Throughout we mark what is proved, what is cited, what is conjectured, and what is a research program.