Protocol Epistemology of Finiteness: Polyadic Algebra, Sheaf Cohomology and Topological Regularization in the Hyper-Rete Architecture

This is version 2.2.0 of the preprint "Протокольная эпистемология конечности: полиадическая алгебра, пучковая когомология и топологическая регуляризация в архитектуре Hyper-Rete" (Protocol Epistemology of Finiteness: Polyadic Algebra, Sheaf Cohomology and Topological Regularization in the Hyper-Rete Architecture). The work presents a rigorous mathematical framework synthesising enriched category theory, Sokolov's polyadic algebra (multidimensional matrices), Munerman's algebraic models of data processing, non-abelian Čech cohomology, Amari's information geometry, and Friston's free-energy principle. The central Correspondence Theorem states that an algebraic isomorphism of the rule-matrix model to its canonical acyclic form exists iff the first non-abelian Čech cohomology group Ĥ¹(X,𝒢) is trivial – providing an exact, computable criterion for logical consistency of a rule base. Major contributions: · Correspondence Theorem (both directions, complete proofs). · Hopf bifurcation theorem: non‑trivial Ĥ¹ induces a supercritical Hopf bifurcation in epistemic precision dynamics Π_t with explicit L₁ = -β₄/(2λ_c) < 0 (Poincaré normal form derivation). · ESS theorem (both Maynard‑Smith conditions): deontic recalibration strategy is evolutionarily stable; ESS-2 explicitly formulated as vacuous truth (strict Nash → ESS, Maynard Smith 1982). · Hyper‑Rete engine: AVX‑512 alpha‑net, polyadic Łukasiewicz‑JOIN over COO hypermatrices, bitwise Floyd‑Warshall closure (O(n³/64)), provably terminating recalibration module. · Worked example of Ĥ¹ computation for a 4-rule system. · Comparison table (Rete / TREAT / Drools / LTN / Hyper‑Rete). · Runnable Python benchmark: Python 3.14.5, NumPy 2.4.6, Apple-class CPU; 100% recalibration success; N=2: O(n^{1.66}), N=3: O(n^{5.16}) (log-log regression, R² = 0.998). Changes in v2.2.0 (compared to v2.1.0, DOI 10.5281/zenodo.20617316): · Terminal object theorem: corrected θ-correctness logic (violation = derivation of ⊥ with large weight). · Correspondence Theorem (⇐ direction): proved that c_{ij} ∈ 𝒢(U_i ∩ U_j) is an automorphism of matrix M (not merely of the protocol). · Hopf bifurcation: explicit derivation of c_{21} = -β₄ via Poincaré normal form; corrected coefficient β₄ = (1/4!) ∂⁴ F_total|_{W^c}. · ESS-2 condition: explicitly formulated as vacuous truth (ex vacuo verum) with reference to Maynard Smith (1982), p. 14. · Concavity of -λ𝒯: proved via negative semi-definiteness of the Hessian; separate treatment for continuous and discrete cases. · "Hallucination" definition: clarified the link C_{vv} = 1 ⇔ contradiction; established that C_{vv} = 1 is a necessary but not sufficient condition. · Worked example (Ex. 4.1): justification of c_{ij} assignment; clarification of c_{13} vs c_{31} direction in the non-abelian case. · GL_n(F)^p: corrected action to all p indices (was p-1); symmetrical action by all orientations. · Functor ℱ: resolved "immersion" vs "submersion" — only immersions are used. · Benchmark: full reproducible code in Appendix B; log-log regression methodology with R² = 0.998. · LaTeX: clean compilation without warnings (overfull/underfull hbox suppressed; \Join conflict resolved; silence package for unavoidable warnings). Previous versions: · v2.1.0: https://doi.org/10.5281/zenodo.20617316 · v1: https://zenodo.org/records/20600475 Keywords: protocol epistemology; polyadic algebra; Čech cohomology; topological regularization; Hyper‑Rete; information geometry; Hopf bifurcation; ESS; deontic recalibration; Łukasiewicz logic; Sokolov; Munerman. License: CC BY-NC-ND 4.0 (Attribution-NonCommercial-NoDerivatives)