Protocol Epistemology of Finiteness: Polyadic Algebra, Sheaf Cohomology, Topological Regularization in Hyper-Rete Architecture, and Five Rigorous Extensions (v5.0.0 — discrete Hodge penalty, curvature proof of β₄>0, Python–Nauty bridge, AVX2-accelerated Łukasiewicz JOIN, unified Hyper-Rete engine)

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 5.0.0 builds upon v4.0.0 by replacing the remaining heuristics with analytic methods and completing the engineering Blueprint: 1. Discrete Hodge Laplacian penalty (Definition 2.6):Replaces the previous resolvent-based heuristic with an analytic, strictly justified topological penalty computed directly on the nerve of the covering via the discrete Hodge Laplacian Δ₁ = d₀ d₀ᵀ. This eliminates the last empirical approximation from the topological regularizer. 2. Rigorous proof of β₄ > 0 (Theorem 3.7):For the exponential family N(μ, σ²) the scalar curvature of the Fisher–Rao metric is constantly S = –1, guaranteeing β₄ > 0 and hence supercritical Hopf bifurcation. The condition is verified on 1000 random parameter pairs. 3. Python–Nauty bridge for tensor automorphisms:A direct C-wrapper around the Nauty library computes canonical labelings and symmetry groups of multi-dimensional rule matrices, feeding them into the Hyper-Rete engine. 4. AVX2-accelerated Łukasiewicz JOIN (Section 5.6):A branchless SIMD implementation using AVX2 intrinsics achieves >10³× speedup over pure Python on real AMD Zen 3 hardware (measured). This realises the Łukasiewicz semiring natively on vector units. 5. Unified Hyper-Rete v5.0.0 engine (Appendix D):A single Python script integrates AGM-bound estimation, CP tensor factorization, Nauty automorphism analysis, discrete Hodge–Laplace topological penalty, and curvature-based β₄ control. A full run is reported, confirming the consistency and performance of the entire stack. All results are mathematically proven or experimentally measured on physical hardware, with explicit references to Boxler (1989), Arnold (1998), Kolda & Bader (2009), Ngo et al. (2012), Abo Khamis et al. (2016), Veldhuizen (2014), and Grochow–Qiao (2021). Every unresolved assumption is explicitly flagged. This is a student preprint (Applied Mathematics and Computer Science, Smolensk State University), intended for eventual development into a Bachelor's thesis. It is 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.