HoTT Hallucination Research

Hallucination is not a bug -- it's a theorem.

A three-paper trilogy establishing the Hallucination--Homotopy Correspondence (HHC): a complete, functorial classification of LLM hallucination types via homotopy type theory and algebraic topology.

Website: hott-hallucination-research.vercel.app

Papers

Paper 1: Topological Hallucination Detection (23 pages)

A Homotopy-Theoretic Classification of LLM Failure Modes

Proves five cases of the HHC, mapping each hallucination type to a topological invariant:

Hallucination Type	Invariant	Obstruction
Unjustified inference	pi_0	Disconnection
Circular reasoning	pi_1 != 0	Non-contractible loop
Inconsistent justifications	pi_2 != 0	Incoherent 2-cell
Fabricated entity chain	H_n != 0	Homological hole
Compositional drift	Holonomy	Transport anomaly

• PDF | LaTeX | Haskell

Paper 2: Type-Theoretic Generation (24 pages)

Hallucination-Free Language Generation via Dependent Type Theory

Five generation principles replacing statistical sampling with proof search:

1. Generation as type inhabitation -- arg min C(a) subject to Gamma |- a : A
2. Certified derivations -- every term comes with a checkable proof
3. Abstention on empty types -- return bottom when no valid claim exists
4. Context as fibration -- dependent telescope, not flat sequence
5. Attention as limit -- weighted limit in Sem, not weighted sum in Vect

• PDF | LaTeX | Haskell

Paper 3: HoTT Synthesis (28 pages)

A Unified Theory of Semantic Correctness for Language Models via Homotopy Type Theory

Unifies detection and generation via:

• Detection-Generation Adjunction -- Generate -| Detect as adjoint functors

• Semantic Monad -- T = Detect . Generate with unit, multiplication, monad laws

• Completeness Theorem -- every hallucination is detected; generation never produces undetected hallucinations

• POC Architecture -- Generate-Check-Abstain loop with persistent homology monitoring

• PDF | LaTeX | Haskell

Key Results

• 21 theorems proven across 75 pages
• 5x5 duality: each hallucination type pairs precisely with one generation principle (emergent property visible only across the trilogy)
• Fundamental Obstruction Theorem: any language model with a contractible representation space will necessarily hallucinate
• Completeness Theorem: the HHC + type-theoretic generation system is sound and complete

Haskell Code

Three standalone modules, verified clean under ghc -Wall -Wextra -Werror:

# Run any module
runghc code/topological-detection/TopologicalDetection.hs
runghc code/type-theoretic-gen/TypeTheoreticGen.hs
runghc code/hott-synthesis/HoTTSynthesis.hs

Dependencies: base only (GHC boot libraries). No cabal/stack required.

Project Structure

papers/
  topological-hallucination-detection/   # Paper 1: detection via homotopy invariants
  type-theoretic-generation/             # Paper 2: generation via type inhabitation
  hott-synthesis/                        # Paper 3: unified framework
  synthesis.md                           # Cross-paper analysis
code/
  topological-detection/                 # Haskell: 5 obstruction detectors
  type-theoretic-gen/                    # Haskell: type inhabitation + derivations
  hott-synthesis/                        # Haskell: full HoTT-LM POC
sources/                                 # Original source papers
images/og/                               # Open Graph images (1200x630)
website/                                 # Next.js site (deployed to Vercel)
social-posts.md                          # Posts for Twitter/X, LinkedIn, Facebook, Bluesky

Author

Matthew Long YonedaAI Research Collective | Magneton Labs LLC Chicago, IL | matthew@yonedaai.com

License

Copyright 2026 Magneton Labs LLC. All rights reserved.