When two reasoning systems are compared — the parties to a conflict, two rival theories, two successive states of one mind — no single mathematical structure exists of which the usual measurements (similarity, entailment, agreement) are partial views. This work introduces that structure: certified convergence spaces. Given a family of consequence trees and a certified relation between propositions of different actors — symmetric, non-transitive — the family of support regions generated by a transfer operation is a canonical object, identifiable with the definable sets of a tolerance approximation space; the lattice of agreement, the rough set of its uncertainty, and the geometry of its distribution are three derived views that mutually forget one another (Structure Theorem; zero violations across four heterogeneous conflicts, two independent implementations, and a brute-force oracle). The object can be measured. Its boundary — apparent agreement without a certificate; formally, the illusion of agreement — covers about half of the corpus in all four cases, no similarity threshold eliminates it, and it is enriched in similar-yet-opposed pairs (confirmed out of sample): the illusion is not half-formed agreement but opposition masked as similarity. When an external proposition is read against the object, its certified consequences belong almost exclusively (95.5%, versus 18% expected by chance at matched similarity) to the already-certified core: what is viable connects to what is real. The computational realization that produces the certificate is measured separately: 71–84% plausibility under blind judges, verified rejection of irrelevant material, a characterized and mitigated dominant error mode, a baseline against a state-of-the-art LLM — equally plausible, impossible to audit — and nine refuted hypotheses reported in full. Validation by human domain experts is declared as the pending external limit. Working draft (R2, 16 July 2026), written in Spanish. Companion to "The Principle of Naturality" (Part I), currently under open review at the journal Neurosymbolic Artificial Intelligence (Tracking #990-2023). This preprint establishes the public record of the introduced object (certified convergence spaces), the Structure Theorem, and its measurements.
ELIO JAVIER GOGOL MERLETTI (Sat,) studied this question.
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