Certified Convergence Spaces: The Principle of Naturality, Part II

When several reasoning systems — the parties to a conflict, rival theories, distinct cognitive architectures — develop the same problem from their own premises, we have mathematics to compare words, propositions and opinions, but not reasonings. This work builds an object on which that comparison is computed. Each party's reasoning (its premise and what it derives) and the judgment of natural-language inference models on every pair of claims from different parties are represented as two tables, with five primitive operations defined on them: reduction, thresholding, Boolean product, closure and per-actor counting. Every other quantity is a derived construction: who is committed to a claim without having formulated it, and by what path; what all parties hold at once and how much each contributes; which claims cross from one reasoning to another; what accepting a proposal would cost each party; and, with no proposal at all, which conflicts do not dissolve. Proved. The region of a claim is the least fixed point of a monotone operator on a finite set: it exists, is unique and does not depend on traversal order. Its extension is an additive closure operator whose closed sets are the down-sets of a preorder and form a distributive lattice. The implication system has a unique canonical basis, obtained at no extra asymptotic cost, and each query is linear-time reachability. The contraction for accepting a proposal satisfies inclusion, consistency with respect to the proposal and structural retention. Measured on a documented dispute (the Nile/GERD case) and a fictional control corpus, with the same models and thresholds. Reciprocity of implication decides over ninety percent of the certification frontier. Pairs chosen by vector similarity alone coincide with the certified ones four and sixteen times in a hundred. Derived quantities hold for the declared threshold, not for a neighborhood of it. The base inference model assigns high contradiction to three of four unrelated pairs, and that measurement fixes the order in which the models are consulted. A third corpus was excluded by a pre-declared admission test. Did not survive. A proposal foreign to the problem costs as much as the legitimate one, so cost magnitude does not measure relevance. And the certificate, run across the two corpora, marks more pairs than within the first: it recognizes shared courses of action, not identical propositions. Both are reported as limits. The calculus does not claim that a conclusion is true, nor that the parties agree with it: it claims that, holding what they hold, they are committed to it. It returns the options and their cost; the choice stays with people.