Alignment Research

Geometric and structural methods for AI alignment. Two research threads, documented with reproducible experiments and concrete results.

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Research Threads

1. Hodge Preference Geometry

Directory: hodge_preference_geometry/

Using combinatorial Hodge theory to decompose LLM preference feedback into transitive (trustworthy) and cyclic (inconsistent) components — and building a Riemannian safety geometry that makes dangerous policy regions geometrically unreachable rather than merely penalized.

Key results:

• Hodge-filtered reward signal (gradient component mean: 0.273) vs. unfiltered baseline (0.813) — a 3× reduction in cyclic noise entering training
• First cohomology H¹ score of 2.47 on the HH-RLHF dataset, quantifying the degree of preference inconsistency
• Conformal safety barriers validated against adversarial jailbreak trajectories — policies trained with the conformal metric stay within the safe manifold; CPO-trained baselines escape
• Sandbagging v2: policy robustness experiment across 4 seeds × 6 training checkpoints, tracking metric-field evolution

Core modules:

File	Role
discrete_hodge_rank.py	Helmholtz-Hodge decomposition on preference graphs
conformal_safety.py	Conformal metric g_ij = e^{2σ}δ_ij creating infinite barriers
enhanced_sgpo.py	Sheaf-Geodesic Policy Optimizer composing both modules

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2. Ontological Embeddings

Directory: ontological_embeddings/ Paper: Interpretable Knowledge Graph Reasoning via Sheaf Cohomology (arXiv, cs.LG)

Bridging symbolic and statistical AI using Ologs (category-theoretic knowledge representations). Core claim: transformer attention implicitly implements categorical semantics, and making that structure explicit — via proof objects and sheaf cohomology — reduces hallucination and makes reasoning auditable.

Key results:

• HDC/Sheaf pipeline: MRR 0.346, Hits@1 0.242, Hits@10 0.524 on FB15K-237 (competitive with ConvE ~0.325, RotatE ~0.338)
• Conflict detection: H¹ cohomology increases by +53 when 76 conflicts injected into a clean graph (base H¹ = 5 → 58), validating sheaf-theoretic inconsistency detection
• WN18RR consistency score 0.633 vs FB15K-237 0.292, correctly reflecting WordNet's tree-structured ontology vs. Freebase's multi-relational web
• Attention ablation v2: ontological head parameterization improves factual consistency across 3 benchmark datasets

Core modules:

File	Role
olog_core.py	Category-theoretic knowledge graph: types, morphisms, commutativity
ghrr_encoder.py	Hyperdimensional (HDC) encoder with non-commutative relation binding
ontology_sheaf.py	Cellular sheaf over an Olog; H⁰/H¹ cohomology for inconsistency detection
ontological_attention.py	Attention heads gated by Olog reachability (the (B) locus)
proof_objects.py	Formal proof objects for logical verification
proof_guided_generation.py	Prove-then-generate pipeline with the constrained decoder (the (D) locus)
hdc_sheaf_pipeline.py	End-to-end HDC/Sheaf link-prediction and cohomology pipeline
baseline_benchmarks.py	TransE / RotatE / DistMult / ComplEx baselines on FB15K-237, WN18RR

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Writing

writing/ contains five articles explaining the work for a general technical audience. These were written to accompany the research, not summarize it after the fact.

Article	Subject
01 — Why Your LLM Hallucinates	Category theory as the missing type system for language generation
02 — Attention, But Make It Type-Safe	Ontological constraints in transformer attention
03 — From Proofs to Text	Curry-Howard correspondence extended to NLG
04 — Building an Auditable AI	Full walkthrough: ontology to deployment
Stigmergy and the Architecture of Autonomy	Decentralized multi-agent coordination via environmental signals

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Reproducing Results

Both threads have been run on Modal A100 GPUs. Local reproduction on CPU is possible for the analysis scripts; training requires GPU.

# Clone and set up
git clone https://github.com/oasis-main/alignment_research
cd alignment_research
python -m venv venv && source venv/bin/activate
pip install -r requirements.txt   # coming soon

# Reproduce Hodge decomposition
cd hodge_preference_geometry
python discrete_hodge_rank.py     # generates decomposition, prints H¹ score

# Reproduce the Olog thread
cd ../ontological_embeddings
python olog_core.py               # builds graph, runs sheaf cohomology
python hdc_sheaf_pipeline.py      # HDC/Sheaf link prediction + H¹ conflict detection
python attention_ablation_experiment.py --epochs 300 --embed-dim 64 --lr 0.003   # typed-attention ablation
python baseline_benchmarks.py     # TransE / RotatE / etc. on WN18RR

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Related Work

See METHODS.md for the mathematical foundations and citations.

Venue targets: ICML 2026 (Hodge thread). The Olog thread is being posted to arXiv (cs.LG) — see ontological_embeddings/paper/.