Constructing Certified Convergence: The Principle of Naturality, Part I

The Principle of Naturality (PN) is an explainable neuro-symbolic framework for detecting latent convergences among actors whose root premises are mutually exclusive. Its defining feature — and what sets it apart — is explainability: what the framework emits is not an opaque agreement score but a structured, auditable explanation of each convergence: the meeting point (a HUB), the per-actor inferential paths that reach it, the per-pair certificates that license it, and the normative justification for why that point was selected. The mechanism. Each actor's stance is expanded into an actor-relative inferential tree (P -> Q -> R -> ...), in which edges are defeasible inferential commitments — not formally valid deductions or strict causation — and each root-to-leaf branch reads as one possible line of reasoning. Cross-actor proposition pairs are validated by a three-stage cascade: an embedding pre-filter (S-BERT), bidirectional NLI with DeBERTa under pragmatic normalization with a contradiction veto, and an LLM role-inversion verification. Validated fusions are grouped with the Union-Find algorithm into connected components; within each, strict HUBs are closed multipartite cliques (pairwise certified across all actors) and lax HUBs are sets with complete cross-actor coverage. The selected HUB minimizes, lexicographically, the Total Topological Friction (TTF), a bicomposite path cost that balances displacement from each actor's root position against relevance to the dispute. The normative half. The selection rule is characterized as a Pareto-efficient utilitarian bargaining solution (4/4 cases), distinct from the Nash solution — which in GERD prefers a more balanced split — and from the egalitarian one. This makes the model's distributive commitment explicit and debatable, which is part of the explanation rather than a verdict. Evaluation. Four heterogeneous cases: the fictional control Selene (isolating the result from memorization), GERD (geopolitical, three states), Lia Lee (intercultural bioethics), and an intrapersonal conflict modeled from Bipolar I narratives (two antagonistic cognitive states within a single individual, with no architectural change). Against geometric baselines (cosine thresholding, single-linkage, k-means) none exceeds 7% precision; a stage ablation shows that the symbolic stages are what supply precision; and the convergences are shown not to be an artifact of depth-driven semantic collapse. Topological Data Analysis (Mapper, Vietoris-Rips persistence) is a post-hoc layer that visualizes the shape of the disagreement, with permutation tests confirming structured cross-actor separation (the strongest pair reaching z > 9). A descriptive mediation probe. An external proposal is tested against each actor's frozen tree — the trees are not re-expanded toward the proposal, since that would manufacture convergence by construction — which guarantees that a detected support is genuine. For each proposal PN reports the supporting node and the support level; the floor (minimum support) defines the binding actor and the convergence threshold; and a mediator iterates by rewording the proposal in a maximin loop (raising the floor) until it is supported by all parties or until no rewording lifts the binding actor. This connects to the egalitarian bargaining rule. What was left out, for honesty. Rough Set Theory / tolerance approximation was pruned: connected components and closed cliques do the work without that machinery. Several lines were refuted with data rather than opinion — the dialetheia/Phase-0 line (the NLI contradiction label reflects single-scene bias from SNLI, not true contradiction), Belnap logic as an empirical foundation, the cross-cluster account of geometric failure (which collapses under a null model), and the strong claim of "universal mediations" (which does not survive the full triple-control cascade and is retained only as a descriptive probe). The central, stated limit. PN's notion of convergence is model-relative: it is what the embedding, NLI, and LLM stages jointly certify. The only genuinely external reference is therefore human expert validation of HUB quality, which remains the principal open item. Contact Information: Author: Elio Javier Gogol Merletti Email: javier_gogol@hotmail.com LinkedIn: Elio Javier Gogol Merletti | LinkedIn Location: Mendoza, Argentina v17 (12 June 2026): Added §6 A Canonical Support Geometry for the Certificate: support-region axioms (reflexivity, substitution under certified equivalence, minimality) with a least-fixed-point uniqueness theorem (Knaster–Tarski), barycentric support coordinates λ derived solely from tree topology and certified pairs (deterministic, parameter-free), and the declared counting convention (A4) with measure-robustness check (ρ = 0.79–1.00). New results: strict-HUB closures collapse to single simplex points (within-closure λ-distance 0.000, 4/4 cases); corpus support demography (pure / F-endpoint / carrier populations, with carriers systematically shallower); two-dimensional equity reading (construction balance β_λ vs cost balance β_TTF) extending the bargaining analysis — in all four cases the utilitarian-selected HUB is not the most equitable available, with the equity premium quantified per case. Reported a pre-registered negative control: interior simplex pairs do not certify above random (0/300 vs 0/300) — the geometry is a canonical representation of the certificate, not an additional detector. Corrected the strict-HUB inventories to match the released artifacts (GERD 3, Selene 2 strict + 1 lax, Jamison 11 equivalence classes over 38 clique instances, Lia Lee 37) and bounded the Sudan-pivot claim to the utilitarian-selected HUB. Added previously missing citations (Knaster; Tarski; Nash; Kalai; RoBERTa; MiniLM; Mapper/Singh et al.) and minor textual repairs. v16 (3-6-26): Second Place in the 2026 GCSP Prize for Transformative Futuresin Peace and Security (Geneva Centre for Security Policy, Geneva, Switzerland), among 337 submissions from 95 countries,evaluated by an international jury including representatives of ETH Zurich, the University of Oxford, and the Swiss Federal Administration. Topological Detail per Case. v 15 (31-05-2026): Repositioned the framework around explainability (X-NeSy): the emitted object is now a structured, auditable explanation — convergence point, per-actor inferential paths, per-pair certificates, and the normative justification of selection — rather than a score. Removed Rough Set Theory / tolerance approximation (connected components via Union-Find plus closed multipartite cliques now subsume it; strict/lax HUBs redefined as cliques and complete cross-actor coverage). Added a normative characterization of HUB selection as a Pareto-efficient utilitarian bargaining solution, distinguished from the Nash and egalitarian solutions. Made the three-stage validation cascade explicit (embedding pre-filter; bidirectional averaged NLI under pragmatic normalization with contradiction veto; LLM role-inversion verification). Added geometric baselines (cosine, single-linkage, spherical k-means; ≤7% precision against the validated relation), a stage ablation isolating the symbolic stages, and a control showing convergences are not an artifact of depth-driven semantic collapse. Introduced a descriptive mediation-design probe (external proposals tested against frozen actor trees; support level, convergence threshold τ*, binding actor, and a maximin reformulation loop). Retired previously explored lines now ruled out with data (paraconsistent/dialetheia reading of the NLI contradiction label, Belnap logic as empirical foundation, the cross-cluster account of geometric failure) and dropped the LLM-as-judge baseline as non-independent; corrected reporting figures (embedding pre-filter elimination 14–81%; GERD 54%). v 14.1 (13 may2026): Major conceptual and formal refinement of the PN framework: clarified the distinction between inferential compatibility and formal logical equivalence, separated semantic validation from operational closure, restructured graph architecture (GC/GF/GO), refined strict/lax HUB definitions under tolerance-based rough sets, and reduced overly strong causal and ontological claims to align the framework with its neuro-symbolic and representation-dependent nature. v 13.1 (12 may2026): The document is reorganized, graph types are defined, and redundant content is removed. v 12.1 (10 may2026): Writing errors, missing references, and lack of TDA analysis of cases are corrected v 10.3 (9 may2026): Formal integration of Pawlak’s Rough Sets and Dung’s Argumentation Framework to refine consensus cores and node acceptability; removed the theta-Pi dimensional artifact, centering metrics on the 768d Delta-sim-Pi gradient; includes full analytical results for Case Study 4 and externalizes raw JSON reports, wind map outputs, and permutation logs to GitHub for academic replicability. v10.2 (4 -may2026): fixes structural errors in algorithmic normalization, case ordering, and the resolution of equivalence classes to optimize the framework’s core execution, including precise definitions of HUB (strict and lax variants), Topological Friction restricted to validated cores, and the Internal Pairwise Validation Rate (IPVR). v10 (19 Abril2026): Added Case Study 4 — Selene Station controlled fictional scenario (§4.4) eliminating corpus, cross-actor, and parametric contamination; bicomposite field deviation analysis (§4.4.6, Table 4.22) demonstrating domain-invariant δ ≈ 0.40 across 6,003 nodes in four cases; Semantic Divergence Analysis formalization (§2.5) with divergence gradient ΔsimΠ and actor divergence coefficient D(aᵢ); Topological Dissociation Analysis (§2.6) with Mapper-based definitions for Cognitive Dissociation Degree, Topological Bridges, and Semantic Rumination Loops; consolidated cross-case comparison table (Table 4.23); section numbering correcti