Category-theoretic foundations for physics, mathematics, and AI — a modular body of research spanning quantum foundations, foundations of mathematics, and the mathematics of intelligence
27 collections across four fields. Each program links into the full paper library below; many have dedicated sites and repositories.
Four interconnected concepts form the mathematical backbone of Quantum Perspectivism — and the derivation chain runs from the Yoneda Lemma to quantum mechanics.
Observer-system interactions formalized as morphisms in a structured category. The measurement category encodes what an observer can access about a system.
Accessible knowledge via representable presheaves. The Yoneda embedding maps each system to its complete relational profile, faithfully and fully.
Information loss at epistemic horizons quantified as the failure of Kan extensions to be exact. The deficit measures inaccessible structure.
No-go theorems as presheaf failures. Bell's theorem, Kochen-Specker, and contextuality arise as non-vanishing cohomology classes.
The complete 11-step derivation chain showing how quantum mechanics emerges as a mathematical consequence of the Yoneda Lemma.
| Step | Structural Input | Physical Output | Code | |
|---|---|---|---|---|
| 01 | Yoneda Lemma | → | Physical identity is relational | |
| 02 | Presheaf condition | → | States are context-dependent data | |
| 03 | Monoidal contexts | → | Linear (vector space) structure | |
| 04 | Perspectival consistency | → | Inner product / Hilbert space | |
| 05 | Naturality of observables | → | Self-adjoint operators | |
| 06 | Yoneda isomorphism + Gleason | → | Born rule | |
| 07 | Product categories | → | Entanglement | |
| 08 | Non-commutative contexts | → | Complementarity / uncertainty | |
| 09 | Presheaf restriction | → | Measurement (no collapse) | |
| 10 | Topos structure | → | Quantum logic | |
| 11 | Natural automorphisms | → | Unitary evolution / Schrödinger eq. |
122 papers across 27 collections — quantum foundations, category theory, quantum error correction, foundations of mathematics, and the mathematics of AI. Every paper is available as a PDF.
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23 companion sites — each research program in physics, mathematics, and AI is published as its own dedicated site.
A modular physics framework: separate laws composing hierarchically into emergent structure.
principia-physica-pilot.vercel.app ↗ PhysicsSeven-part program recasting physics on condensed mathematics.
condensed-representation-theory-of.vercel.app ↗ PhysicsThirteen mathematical representations of quantum gravity, from spin foams to celestial CFT.
math-qg-representation-library.vercel.app ↗ PhysicsTopological phases of matter through condensed families of observables.
condensed-phase-spectrum.vercel.app ↗ PhysicsSpacetime as an emergent structure rather than a fundamental background.
emergent-spacetime-dynamics.vercel.app ↗ PhysicsGravitation reformulated through the geometry of information.
gravity-information-geometry.vercel.app ↗ PhysicsA universal axiom for embedded observers, from Gödel to quantum measurement.
yoneda-constraint.vercel.app ↗ PhysicsRuntime as proof of ignorance: a type-theoretic account of physical law.
minimal-runtime-axiom.vercel.app ↗ PhysicsReformulating fundamental physics with foundational mathematics as the baseline.
foundational-reformulation.vercel.app ↗ MathematicsMotives, polylogarithms, and regulators as the pre-numerical substrate of physical quantity.
mathematics-physical-representation.vercel.app ↗ MathematicsA library of mathematical structures for physical theory.
math-phy-library.vercel.app ↗ MathematicsThe foundations of mathematics rebuilt in homotopy type theory.
hott-foundations-mathematics.vercel.app ↗ MathematicsMachine-checked formalization of foundational mathematics.
formalized-foundations-mathematics.vercel.app ↗ MathematicsA multi-volume program approaching the Riemann Hypothesis through homotopy type theory.
hott-riemann-hypothesis.vercel.app ↗ AIAn AI peer-review archive for research preprints.
grokrxiv.org ↗ AIA category-theoretic approach to databases, with a companion research site.
catdb.ai ↗ AIPersistent, typed memory for AI coding agents across sessions.
contextfs.ai ↗ AIA homotopy-theoretic classification of language-model hallucination.
hott-hallucination-research.vercel.app ↗ AIAgent-orchestrated, verification-backed C → Rust migration.
ferrous-bridge.vercel.app ↗ AIEnergy-denominated economics for machine intelligence.
joule-standard.vercel.app ↗ AIEconomic dynamics of autonomous AI agents.
agentic-economics-five.vercel.app ↗ AIAn independent programming-language project.
japl-lang.org ↗ AIA language experiment inspired by genetic encoding.
dna-lang-delta.vercel.app ↗50 public research repositories on github.com/MagnetonIO — LaTeX sources, Haskell verifications, and research tooling.
A formal resolution to the Black Hole Information Paradox through information-theoretic foundations
The Categorical Architecture of Quantum Perspectivism: Topoi, Logic, and Dynamics [math-ph] — 27pp peer-reviewed paper with Haskell forma…
Quantum Perspectivism and Its Relations: RQM, QBism, Many-Worlds, and Topos Theory [quant-ph] — 32pp peer-reviewed paper with Haskell for…
The Crisis of Quantum Foundations: Why Physics Needs the Yoneda Constraint [quant-ph] — 27pp peer-reviewed paper with Haskell formalization
Derived Functors and Quantum Error Correction: A Homotopy-Theoretic Approach
A modular four-part research series formalizing emergent phases of matter from information through category theory. Site: https://emergen…
Investigating emergent spacetime through simulations and theoretical models to understand its origins.
Entanglement, Complementarity, and Measurement as Categorical Phenomena [quant-ph] — 27pp peer-reviewed paper with Haskell formalization
Rigorous Haskell proof of the First Law of Entanglement (δS_A = δ⟨K_A⟩) - the quantum information principle showing how spacetime emerges…
A Functorial Formulation of Nuclear Fission: Category theory meets nuclear physics with type-safe Haskell implementation
Explore theoretical physics through functorial frameworks, unifying mathematical structures and physical phenomena.
A repository exploring Hamiltonian mechanics using category theory and functors. It demonstrates the construction and transformation of H…
An interdisciplinary collection of resources on the nexus of advanced mathematics and theoretical physics, with an emphasis on fundamenta…
A compositional framework for fundamental physics laws based on information-theoretic principles
Code For Unified Physics
Open Problems in Quantum Perspectivism: From Category Structure to Experimental Signatures [quant-ph] — 32pp peer-reviewed paper with Has…
Philosophical Implications of Quantum Perspectivism: Structural Realism and the Primacy of Relation [physics.hist-ph] — 26pp peer-reviewe…
Bounded three-paper Principia Physica pilot with a synthesis paper and representative Lean and Haskell artifacts.
Compositional quantum error correction pipeline in Haskell, integrating topological, penalty-based, and type-theoretic QEC with code-base…
Quantum Gravity and Emergent Spacetime from Perspectival Structure [hep-th] — 29pp peer-reviewed paper with Haskell formalization
Complete Haskell implementation of Quantum Perspectivism — categorical framework for foundational physics via the Yoneda Constraint. 42 m…
Quantum topological QEC: Explore algorithms for error correction in quantum systems using topological properties. #QEC #topological- codes
Investigations into unifying quantum mechanics with broader physical theories, exploring foundational concepts and beyond.
QuantumFlow – A Python library for simulating derived Hamiltonians in quantum fields, lattice QCD, quantum circuits, and biophysics. Supp…
Technical Constructions in Quantum Perspectivism: Complex Amplitudes, Gleason, and Decoherence [math-ph] — 26pp peer-reviewed paper with …
Type-Safe Physics: A framework for expressing physics with mathematical rigor using type theory, category theory, and formal verification…
Category-theoretic foundations for open problems in quantum mechanics, gravity, and observer theory — 6 papers applying the Yoneda Constr…
Deriving Quantum Mechanics from the Yoneda Constraint [quant-ph] — 36pp peer-reviewed paper with Haskell formalization
The Yoneda Lemma as Physical Law: Identity, Relation, and the Structure of Reality [math.CT] — 29pp peer-reviewed paper with Haskell form…
Comprehensive resources and explorations of the Geometric Langlands Conjecture, bridging number theory, geometry, and representation theory.
A repository showcasing the power of using code to generate and verify mathematical proofs and assertions. Technologies: Python, Jupyter …
Explore math and physics concepts through code with examples in calculus, mechanics, and electromagnetism. Haskell and Coq used.
A deep dive into the universal underpinnings of mathematics, seeking to unify set theory, category theory, logic, and more under a cohesi…
An Integrated Memory and Knowledge-Graph Architecture Leveraging mem0, Fibered-Sheaf Merging, and AI Preprint Forge
AI-powered framework for automated research paper generation and publication. Combines LaTeX creation, PDF conversion, Git management, an…
Advancing the theoretical foundations of Artificial Intelligence through interdisciplinary research. This repository bridges AI, mathemat…
CatDB is a Rust-based semantic multimodel data platform for typed schemas, composable mappings, versioned migrations, federated queries, …
Chain of Intent (CoI): Novel AI alignment framework treating intent as mathematically transformable objects. Implements semantic preserva…
Git for your AI conversations - Local-first AI memory layer with MCP server support
Advancing database theory through interdisciplinary research. Explore mathematical frameworks like Topos Theory, functional programming, …
A data processing pipeline for analyzing content from local data, including scraping, metadata extraction, and text analysis.
Predictive Market Theory: Markets as Distributed Bayesian Inference Engines — Bayesian belief functions, thermodynamic structure, HFT bel…
Frontend to AI-powered framework for automated research paper generation and publication.
Cutting-edge exploration of quantum-enhanced database operations using principles of quantum computing. Join us! #quantum #database #tech…
The Time Compression Paradox: A formal analysis of why AI increases rather than eliminates work. Four-paper research series (~91 pages) b…
Type-safe context engineering for AI-native workflows. Pydantic + Instructor + ContextFS integration.
VoltForge: Software-defined power interoperability for cordless tool ecosystems. 5 research papers + Rust #![no_std] firmware for typed, …
YonedaAI Methodology: Mitigating AI Code Generation Limitations through Risk-Aware Engineering, Golden Paths, and Enterprise AI Governanc…
Type-Safe Biophysics: The Protein Folding Problem as Type Inference - Why AlphaFold Works and What It Teaches Us About Machine Learning. …
Rational Design of Pathway-Selective Cannabinoid Compounds for ADHD: Biased Signaling, Prodrug Activation, and Multi-Target Pharmacology
YonedaAI Research Collective is a frontier research organization applying category theory and advanced AI to the foundational problems of physics.
Mission: To derive the mathematical structures of physics from first principles using the Yoneda Lemma as a foundational constraint, and to publish rigorous, peer-reviewed research with complete executable codebases.
Research Program: 122 papers across 27 collections spanning quantum foundations, quantum gravity, foundations of mathematics, and the mathematics of AI. The 16-paper Quantum Perspectivism core includes complete Haskell implementations that verify the mathematical constructions.
YonedaAI developed a multi-agent parallel execution architecture that transforms research questions into peer-reviewed papers with verified code. This is what makes YonedaAI unique: cutting-edge AI agent orchestration applied to real science, producing rigorous mathematical research at unprecedented speed and scale.