Protocol Epistemology of Finiteness: Polyadic Algebra, Sheaf Cohomology, and Topological Regularization in the Hyper-Rete Architecture (v3.0.0)

We present a rigorous mathematical framework synthesizing five disciplines to solve a concrete engineering problem: architecturally guaranteed logical consistency in rule-based inference engines. CENTRAL RESULT — Correspondence Theorem: 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, giving an exact, computable criterion for consistency of a production rule base. Proven in both directions with explicit global-section construction. MATHEMATICAL FOUNDATIONS: enriched category theory over the quantale ([0,1], min, 1), motivated via Lawvere's classical metric-space enrichment theorem; Sokolov's polyadic algebra of multidimensional matrices (1960, 1972); Munerman's algebraic models of mass data processing; non-abelian Čech cohomology on simplicial complexes; Amari's information geometry; Friston's variational free-energy principle. DYNAMICAL RESULT: non-trivial cohomology is shown to force a supercritical Hopf bifurcation in the dynamics of epistemic precision, with the first Lyapunov coefficient computed explicitly via Kuznetsov's normal-form theory. GAME-THEORETIC RESULT: a strategy of deontic recalibration (or, in the limit, silence) is proven to be an evolutionarily stable strategy under topological frustration, with both Maynard Smith conditions verified independently. ALGEBRAIC-GEOMETRIC CONTRIBUTION: the classical theory of matrix pencils (Weierstrass elementary divisors) is correctly distinguished from the Hesse pencil of a cubic form; the singular-fiber structure of the Hesse pencil (four exceptional parameter values, nine base points forming a (Z/3Z)^2 group of inflection points) is independently verified by symbolic computation and used to derive an O(1) smoothness test, complementing the polynomial-time GCD-based Munerman criterion. ALGORITHMIC CONTRIBUTION: the Hyper-Rete architecture combines an AVX-512/AVX2 vectorized alpha-network, a Łukasiewicz-semiring polyadic JOIN, bitwise Floyd–Warshall closure, and a deontic recalibration algorithm with a proved termination guarantee. A precise diagnosis identifies why naive dense join implementations scale as O(n^5) rather than O(n^4); a hash-join achieves measured speedups up to 89x on sparse data. For genuinely multi-way (cyclic) rule joins, a semiring-generalized Leapfrog Triejoin is specified following the FAQ/worst-case-optimal-join literature (Ngo–Porat–Ré–Rudra; Veldhuizen; Abo Khamis–Ngo–Rudra), and a theorem shows that the paper's own topological acyclicity criterion coincides exactly with the acyclicity condition under which the classical Yannakakis (1981) algorithm is optimal — connecting forty years of database theory to the paper's own sheaf-cohomological framework. EXPERIMENTAL VERIFICATION: all numerical claims are backed by independently reproducible code (Python and C++). Benchmarks were run on two physically distinct machines by two different people: a development sandbox with native AVX-512 support, and an AMD Ryzen 7 5825U (Zen 3) contributed by the author, which lacks AVX-512 in silicon — a fact used to make an explicit, honest statement about the boundary of hardware verification rather than an unfounded claim. METHODOLOGICAL NOTE: this version incorporates several rounds of independent review (including by other large language models) and several rounds of the author's own symbolic-computation self-checks, which caught and corrected genuine errors from earlier drafts (including a conflation of two distinct classical objects — pencils of conics and the Hesse pencil of cubics — and a mistaken complexity claim later corrected via direct arithmetic verification). Every unresolved assumption is explicitly flagged in the text rather than presented as proven. This is a student preprint (first-year 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.