Odyssey: Constructing Verifiable Local Truth-Preserving Foundation Models
06:00 · June 29, 2026 · arXiv cs.AI RSS
We introduce a categorical framework called ODYSSEY for constructing verifiable, local truth-preserving foundation models as compositions of foundries: building-block architectural components that specify a cover of local contexts, local representation families, restriction maps, gluing rules, obstruction policies, update obligations, and human-facing views. A foundry is an organized sheaf of knowledge that carries within it an argumentation component. Concrete foundries are built from generic foundries such as evidence/argument, operational decision, institutional/financial, market meaning, scientific challenge, research-program, assistant-build, and evaluation-harness foundries. Universal Foundry Learning (UFL) formalizes foundry construction as a composition of left and right Kan extensions, with left Kan extension rolling local artifacts into candidate foundries and right Kan extension enforcing the restriction, gluing, obstruction, and argumentation conditions required for promotion. Foundry SQL (FSQL) is a small typed query surface for slicing maintained foundry artifacts that uses TICKET (Topos Integration using Causal Kan Extension Transformers) certification for admitting external or pre-built models into durable ODYSSEY state. ODYSSEY is fully implemented and tested across a wide spectrum of concrete foundries, showing that the same categorical machinery supports domain construction, artifact replay, sheaf diagnostics, grounded Toulmin/local-LLM scrutiny, residual-obstruction ledgers, and optimized TICKET-compatible causal-claim extraction across heterogeneous sources. This paper is to be presented as a 2.5 hour tutorial at ICML 2026. The tutorial home page is at https://bit.ly/4ajS0nA.
Summary
Odyssey presents a categorical framework for assembling verifiable foundation models from modular components called foundries. Each foundry functions as an organized sheaf that encodes a cover of local contexts, families of local representations, restriction maps, gluing rules, obstruction policies, update obligations, and human-facing views. Rather than collapsing documents or domains into single embeddings, the approach treats representation as the explicit combination of covering and gluing, so that local predictive and logical models remain inspectable and their consistency or failure to agree is preserved as durable artifacts.
Concrete foundries are assembled from generic building blocks such as evidence/argument, operational decision, institutional/financial, market meaning, scientific challenge, research-program, assistant-build, and evaluation-harness foundries. These are orchestrated by five specialized agents: Scylla for human-facing design briefs and explanations, Homer for workflow skeletons, Athena for representational semantics and cross-sheaf bridges, Prometheus for materializing Topos World Models and auditing gluing, and Toulmin for turning maintained state into warranted claims with explicit grounds, qualifiers, and rebuttals.
Universal Foundry Learning formalizes the construction process through left and right Kan extensions. The left extension aggregates local artifacts into candidate foundries, while the right extension enforces the required restriction, gluing, obstruction, and argumentation conditions before promotion. Foundry SQL supplies a typed query surface over these artifacts, and TICKET certification enables external pretrained models, including GPT-style systems, to be admitted into durable Odyssey state while preserving causal-claim extraction.
The system has been implemented and exercised across multiple domains, including retailer, brand, corporate, and research-program foundries, demonstrating support for artifact replay, sheaf diagnostics, grounded Toulmin scrutiny, residual-obstruction ledgers, and optimized causal-claim extraction. The work is scheduled for presentation as a 2.5-hour tutorial at ICML 2026.
Why it matters
This research is highly relevant to Dutch AI researchers focusing on transparent, ethical, and verifiable AI, aligning strongly with EU AI Act requirements. The rigorous mathematical framework for truth-preserving foundation models offers significant theoretical advancements for advanced AI practitioners.
