TITLE:
The Generalized HKR Theory: Derived Epistemology and Topological Information Dynamics
AUTHORS:
Fabian Matthias Zimber
KEYWORDS:
Derived Epistemology, Information Geometry, Quantum Darwinism, Renormalization Group, Percolation Universality
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.10,
October
16,
2026
ABSTRACT: This article develops the generalized HKR theory as a unified geometric and information-theoretic model of the transition from context-dependent representations to path-independent objective facts. The construction combines a fact-state bundle over a Riemannian context manifold with the Hochschild-Kostant-Rosenberg filtration, Hopkins-Kuhn-Ravenel character-theoretic depth, Fisher-Rao information geometry, quantum Darwinism, and renormalization-group dynamics. In the reported simulations, the context-space scalar curvature has a hyper-stable minimum of approximately −2.25, while consensus concentrates the normalized Wilson-loop invariant at approximately 1.000. Fisher-Rao natural-gradient revision reaches tolerance in 15 iterations, compared with 179 ordinary-gradient iterations, giving an acceleration factor of 11.9. The quantum-Darwinist partial-information curve saturates at a system entropy of 1.000 bits for an environment fraction below 0.05 and yields redundancy 117.6. Finite-size scaling for lattice sizes from 64 to 1024 gives
ν≈1.398
,
γ/ν
≈1.731
, and
β/ν
≈0.109
, establishing the reported second-order transition in the two-dimensional site-percolation universality class near the critical threshold 0.593. Active-inference, social-channel, biomimetic, and recursive tests further produce a self-referential score of 0.637 and a 16-claim composite score of 0.881. Together these results establish a scale-invariant, topologically coherent mechanism for emergent objectivity.