Protocol Epistemology of Finiteness: Polyadic Algebra, Sheaf Cohomology, Topological Regularization in Hyper-Rete Architecture, and Six Rigorous Extensions (v6.0.0 — analytic β₄>0, holonomic rotation, Leapfrog Triejoin for 2‑simplices, unified penalty bridge)

Abstract

We present a rigorous mathematical framework for protocol epistemology of finiteness — a synthesis of enriched category theory, Sokolov's polyadic algebra, Munerman's algebraic data-processing models, non-abelian Čech cohomology, and Amari's information geometry with Friston's free-energy principle. CENTRAL RESULT (Correspondence Theorem, v3.0.0–v3.1.0):An algebraic isomorphism of a rule-matrix model to its canonical acyclic form exists if and only if the first non-abelian Čech cohomology group is trivial. This gives an exact, computable criterion for consistency of production rule bases, proven in both directions with explicit global-section construction. VERSION 6.0.0 introduces four rigorous extensions that close previous assumptions and scale the framework further: 1. Analytic proof of β₄ > 0 (Theorem 3.7)Using the isometry M_P ≅ ℍ² of the statistical manifold of univariate normal distributions, we derive an explicit condition γ < 1/5 for the topological penalty parameter that guarantees β₄ > 0, thereby strictly proving supercritical Hopf bifurcation. Previously this was only numerically observed. 2. Holonomy → antisymmetric drift (Lemma 3.8)The long-standing assumption of a rotational term λJx in the stochastic differential equation is now derived from the holonomy of the hyperbolic plane via the Gauss–Bonnet theorem. The holonomy angle equals the hyperbolic area enclosed by the centre manifold boundary. 3. Leapfrog Triejoin for 2‑simplices (Section 8.X)The check of the 2‑cocycle condition on nerves is reformulated as a triangle query and solved with a semiring-aware Leapfrog Triejoin. On sparse nerves (|E| = O(|V|)) it scales as O(|V|) instead of the previous cubic, delivering a measured 109× speedup at n = 100. 4. Constructive penalty bridge (Remark unified-penalty)A canonical map Φ : M_P → C¹(X, G) links the discrete Hodge–Laplacian penalty (on the nerve) and the continuous Fisher–Rao determinant penalty (on the parameter manifold) via T_cont(θ) = f(T_disc(Φ(θ))) + O(‖θ−θ*‖³). This unifies the two earlier penalty definitions. All v5.0.0 components are preserved and integrated: discrete Hodge–Laplacian penalty, AVX2‑accelerated Łukasiewicz JOIN (>10³× speedup), Python–Nauty bridge for tensor automorphisms, and the unified Hyper‑Rete v6.0.0 engine. Every mathematical result is proved or experimentally benchmarked on physical hardware (AMD Zen 3, Intel Xeon with AVX‑512). Explicit references include Boxler (1989), Arnold (1998), Kolda & Bader (2009), Ngo et al. (2012), Abo Khamis et al. (2016), Veldhuizen (2014), Grochow–Qiao (2021). All unresolved assumptions are explicitly flagged. This is a student preprint (Applied Mathematics and Computer Science, Smolensk State University), intended for eventual development into a Bachelor's thesis. Shared under CC BY‑NC‑ND 4.0; any commercial use of the algorithms, mathematical models (Hyper‑Rete and derivatives), or source code requires a separate written license agreement with the author.