Protocol Epistemology of Finiteness: Polyadic Algebra, Sheaf Cohomology, Topological Regularization in Hyper-Rete Architecture, and Four Rigorous Extensions (v4.0.0 — sparse JOIN, antisymmetric drift, convexity, analytic penalty)

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 4.0.0 adds FOUR rigorous extensions, replacing earlier hypotheses and heuristics with strict theorems: 1. Sparse Łukasiewicz-JOIN complexity (Theorem 5.1):Proven that the Łukasiewicz-JOIN in COO format has complexity O(nnz(A)·nnz(B)·k), where k is the number of common (Cayleyan) indices. This replaces the earlier O(n^5) discussion for dense data. Additionally, approximate JOIN via CP tensor factorization with controlled error (Theorem 5.3) and worst-case optimal Leapfrog Triejoin for Łukasiewicz semirings (Theorem 5.5) are included. 2. Antisymmetric drift term (λJx) in SDEs on the center manifold (Theorem 3.5):Strictly derived via stochastic center manifold theory (Boxler, 1989) and decomposition of the effective coupling matrix M = A·B⁻¹ into symmetric (dissipative) and antisymmetric (rotational) parts. This replaces the earlier heuristic assumption. 3. Convexity of the log-partition function for exponential families (Lemma 4.1):Proven via positive definiteness of the Fisher information matrix. Geodesic convexity on compact manifolds is analyzed (Lemma 4.3). This replaces the earlier regularity assumption in the ESS proof. 4. Analytic topological penalty (Definition 5.1):Sigmoidal heuristic replaced by a resolvent-based regularizer Φ(-ln det g(θ)) = (1/γ) ln(1 - γ ln det g(θ)), providing an infinite barrier, non-vanishing gradient, and explicit link to H¹(X,G) ≠ {*}. All four results are fully proven, with explicit references to Boxler (1989), Arnold (1998), Kolda & Bader (2009), and the worst-case optimal join literature (Ngo et al., 2012; Abo Khamis et al., 2016; Veldhuizen, 2014). EXPERIMENTAL VERIFICATION: all numerical claims are backed by independently reproducible code (Python and C++). Benchmarks run on two physically distinct machines: a development sandbox with native AVX-512 support and an AMD Ryzen 7 5825U (Zen 3), with an honest statement about the boundary of hardware verification. METHODOLOGICAL NOTE: this version incorporates multiple rounds of independent review (including by large language models) and symbolic-computation self-checks. 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.