Certified Convergence Spaces: The Principle of Naturality, Part II

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 builds one, and with it a way to operate on reasoning: 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). An axiomatic uniqueness result shows the region operator is the only one compatible with three axioms — the object is characterized, not chosen. Once reasonings are objects, they can be operated on: ordered, intersected, projected, and perturbed — an external proposition can be injected and read, with a per-pair certificate, for where it attaches and whom it bridges. 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. Certified agreement itself has fine structure: its wiring is more modular than chance with its degrees, and small crossings select balance across actors while large shared cores tilt toward one. 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, and a baseline against a state-of-the-art LLM — equally plausible, impossible to audit. Ten refuted hypotheses are reported in full — including one previously published as confirmed (anchoring in the core), which died on re-audit when its candidate universe was corrected. Validation by human domain experts is declared as the pending external limit. Revision 3 (R3), bilingual: Spanish original and English translation included. Companion to "The Principle of Naturality" (Part I), under open review at Neurosymbolic Artificial Intelligence (Tracking #990-2023). This preprint establishes the public record of the object (certified convergence spaces), the Structure Theorem and its uniqueness, and the measurements.