Foundational research in Category Theory, Topos Theory, and Topological Deep Learning.
This repository contains the six research papers that form the mathematical foundation of the Dynamis Agentic Operating System. These papers are the culmination of over seven years of research intersecting the formalization of learning through category and topos theory along the design constraints of the models by Xavier Zubiri, Bernard Lonergan, Lev Vygotsky, and Jean Piaget. Aside from Paper I, that is currently available on Zenodo, the other five papers are to be treated as research drafts and notes. Papers 2-4 were recompiled and revised systematically whereas Paper V and VI are to be seen as recent developments. When published, an updated entry will be placed here with their official dating.
This is only one instance of the Topos Semantic AI approach. Other avenues will be explored in time.
Establishes the categorical framework for multi-agent consensus using Topos Theory and Sheaf Cohomology.
Defines how autonomous agents can reach consensus on complex, semantic data without a central coordinator. Introduces the Holonomy Defect as a measure of agent disagreement.
Published: Zenodo (DOI: 10.5281/zenodo.18445406)
Formalizes time-ordering guarantees using temporal logic and a BLAKE3-based cryptographic clock.
Proves that the 1kHz Proof-of-History tick provides deterministic ordering for agent coordination across distributed nodes.
Defines the constructive type theory underlying Topos Agent architecture.
Introduces "Inventio" (discovery) as the process by which agents synthesize new logical modalities in response to unresolved topological defects.
Bridges the gap between categorical abstractions and performant Java 25 implementation.
Documents the engineering decisions behind Panama FFM native cryptography, Virtual Threads for consensus, and AVX-512 vectorized Sheaf Laplacian computation.
Describes Sheaf Neural Networks for topology-aware machine learning on agent interaction graphs.
Details how trust signals propagate via Laplacian diffusion on the cellular Sheaf, replacing naive GNN message-passing with mathematically grounded topology-aware learning.
Introduces topological invariants as a measure of computational difficulty for adversarial attacks.
Proves that certain classes of attacks against the Sheaf consensus require solving problems that are hard in a topological sense, providing security guarantees beyond standard cryptographic assumptions.
@article{mendoza2026topos,
title={Topos Semantics for Agentic Reasoning},
author={Mendoza, Marcos},
year={2026},
publisher={Zenodo},
doi={10.5281/zenodo.18445406}
}CC BY 4.0
Copyright 2026 Marcos Mendoza / Cryptozoa.