Certified Convergence Spaces: The Principle of Naturality, Part II

When several reasoning systems — the parties to a conflict, rival theories, distinct cognitive architectures — deploy the same problem from their own premises, we have mathematics to compare words, propositions and opinions, but not reasonings. This work presents a calculus that answers four questions about a family of consequence trees certified pairwise by mutual textual entailment: what each proposition commits to and whose support that is (the support region, with a backward query that exhibits every path); where the parties already coincide structurally and where they only seem to (the common ground and the boundary); where the reasoning of one depends on that of another (bridges and a cross-commitment matrix); and what each must abandon to adopt a proposal (injection, certified opposition, contraction cost). Each answer is a computed set, and each comes with its guarantee: the space is a closure system, so the common ground exists, is unique and is itself definable for any family of actors; the closure is additive, so the canonical base of implications is explicit, derivation is complete and the closure query costs O(|V|+|F|); and injecting a proposal is, under the declared conflict relation, a Hansson base revision satisfying five postulates with minimal contraction and four with dependency contraction. On two real conflicts the calculus exhibits commitments no party stated — the premises of Egypt and Ethiopia both commit, by different paths, to African Union mediation — a common ground of 370 and 551 propositions made of the consequences of breakdown rather than of solutions, and, with the opposition order pre-registered before certifying, recovers both orders and shows that the highest-cost actor is the one whose reasoning depends most on what the proposal denies, not the one that disagrees most. 70–96% of what a region says about other reasonings it says through chains of fusions; sensitivity to certificate error is measured by ablation and reported as a limit, and a tested propagation hypothesis that did not survive its controls is reported as negative. A negation bias in the inference judge is identified and corrected. Expert validation remains the declared external limit.