Protocol Epistemology of Finiteness: A Cohomological Consistency Criterion for Rule-Matrix Systems (CohoRete, v8.1.0)

Abstract

This preprint develops a cohomological consistency criterion for rule-matrix systems: 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 Ȟ¹(X,G) is trivial (Correspondence Theorem, both directions, explicit global-section construction). Version 8.1.0 adds Giraud's exact sequence for non-abelian cohomology (Giraud, 1971), correctly distinguishing two senses of "extension" of the coefficient sheaf, and states — as an explicitly unproven research conjecture, not a result — a possible link between the second cohomology group H²(X,A) and the open O(n^5) dense-JOIN problem. Version 8.0.0 material is carried forward unchanged: a C++ implementation of Leapfrog Triejoin with correctness tests and benchmarks (up to 258× speedup on triangle queries; an order-sensitivity bug found and fixed during benchmarking is documented in the text), and a formalization of negation in production rules via balance of signed graphs (Ȟ¹ with ℤ₂ coefficients) with an explicit non-abelian extension for finite groups. The framework draws on Sokolov's polyadic matrix algebra, Munerman's algebraic data-processing models, and Amari's information geometry; where results are proved only for restricted cases (the normal family, finite groups) rather than in general, or remain conjectural, this is stated explicitly in Section "Limitations" rather than implied by the framing — e.g. β₄ > 0 is proved for the normal family only; the antisymmetric drift term is a geometrically motivated derivation with one open step; dense JOIN at N≥3 remains an open algorithmic problem; the cohomological-barrier conjecture for dense JOIN is explicitly unestablished. 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 requires a separate written license agreement with the author.