Protocol Epistemology of Finiteness: A Cohomological Consistency Criterion for Rule-Matrix Systems (CohoRete, v12.0.0)

Version 12.0.0 adds an algebraic theory of probability on unital W*-algebras (§11) as a strict generalisation of the Kolmogorov model. A commutative W*-algebra is shown to be isomorphic to L^∞(Σ,μ), correcting a draft formulation that conflated this with the Gelfand–Naimark theorem for general compact Hausdorff spaces — the correct statement requires a hyperstonean spectrum (Dixmier, 1951). The Correspondence Theorem is extended to the non-commutative case via the sheaf of state-preserving automorphism groups Aut_ω. The Tomita–Takesaki modular automorphism group is proved to be a canonical subgroup of Aut_ω, uniquely characterized by the KMS condition (Takesaki, 1970). The Kochen–Specker/Peres–Mermin contextuality example is reformulated as a non-trivial first Čech cohomology class. Five questions are left explicitly open: a modular analogue of the Bockstein theorem, von Neumann entropy as an algebraic analogue of Shannon entropy, the descent problem for W*-algebras (existence of a global model from compatible local data without prior embedding), a quantum Fisher metric compatible with the Amari connection of §4, and an explicit cocycle computation for the Peres–Mermin example. 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.