Protocol Epistemology of Finiteness: Polyadic Algebra, Sheaf Cohomology, Topological Regularization in Hyper-Rete Architecture, and Seven Rigorous Extensions (v7.0.0 — Protocol Epistemology of Finiteness as a discipline, axiomatic PEF, Tarski–PEF theorem, Feferman reflection, HoTT connection, Munerman comparison)

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 7.0.0 introduces Protocol Epistemology of Finiteness (PEF) as a new formal discipline with:- Axioms of Immanence, Finite Closure, Incompleteness, Recalibration, and Limit (Axioms 1–5).- The Impossibility of Internal Semantic Criticism (Tarski–PEF Theorem): no protocol containing arithmetic can define its own truth predicate. This formally correlates the limit of global consistency with Tarski's undefinability theorem.- Feferman reflection: the hierarchy of reflective closures corresponds to the dimensional hierarchy of the nerve (0-simplices → 1-simplices → 2-simplices …).- Homotopy Type Theory (HoTT) connection: global consistency of a protocol family is equivalent to inhabitedness of certain identity types, with higher coherences corresponding to 2-simplex obstructions.- A structured comparison with Munerman (2024): strengths and complementarity of the polyadic matrix model and the cohomological control of PEF.- The PEF Manifesto: five operational principles for the new discipline. All results from previous versions are preserved and integrated:v6.0.0 additions: (i) analytic proof of β₄ > 0 via MP ≅ ℍ² with explicit condition γ < 1/5 (Theorem 3.7); (ii) derivation of the antisymmetric drift λJx from holonomy on ℍ² via the Gauss–Bonnet theorem (Lemma 3.8); (iii) Leapfrog Triejoin for 2-simplices, achieving O(|V|) complexity on sparse nerves and 109× speedup at n = 100 (Section 8.X); (iv) constructive bridge between the discrete Hodge–Laplacian penalty and the continuous Fisher–Rao determinant penalty (Remark unified-penalty). v5.0.0 components: discrete Hodge–Laplacian penalty, AVX2-accelerated Łukasiewicz JOIN (>10³× speedup), Python–Nauty bridge for tensor automorphisms, unified Hyper-Rete v7.0.0 engine. Every mathematical result is proved or experimentally benchmarked on physical hardware (AMD Zen 3, Intel Xeon with AVX-512). 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.