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We build AI systems grounded in physics and mathematical first principles.
Hamiltonian dynamics. Statistical mechanics. Spectral theory. Quantum chaos. These aren't textbook abstractions -- they are the engine.
A Spectral Operator for the Riemann Hypothesis -- March 2026
Bryan Daugherty, Gregory Ward, Shawn Ryan
Unconditional proof of the Riemann Hypothesis. We construct a self-adjoint operator H_D on C²³ ⊗ L²([0,2π]) and prove D(s) = e^b · ξ(s) — every nontrivial zero of ζ(s) lies on Re(s) = 1/2. GUE pair correlation is derived as a theorem from the arithmetic trace formula and the Fundamental Theorem of Arithmetic (not assumed). Verified computationally: ‖R₂ − R₂^GUE‖₂ = 0.026 over 5,000,000 zeros. The universality constant Ω = 24 determines the fine-structure constant α_EM ≈ 1/137.03 with zero free parameters.
| Metric | Value |
|---|---|
| Verification | 140/140 automated checks pass |
| GUE Convergence | ‖R₂ − R₂^GUE‖₂ = 0.026 at N = 5M, power-law α = 0.2833 |
| Coupling Matrix | 23×23 J matrix, λ_max = 5.5232, κ = 23,015,945 |
| Topology | H₂ = 0 verified at 7 scales (N = 10³ to height ~10²²) |
| Monster Primes | 14/15 detected in spectral data |
| Lehmer Predictions | 28,160 testable predictions |
| Papers | Spectral Operator (v12.0) + Universality Constant (v1.3) -- On-Chain (BSV) |
| Data + Notebooks | 8 Jupyter notebooks, 20 data files, full reproduction scripts |
View Repository -- Open data (CC-BY-4.0). All claims independently verifiable for N ≤ 2,000 with standard Python.
- Featured: U₂₄ Spectral Operator -- Riemann Hypothesis, Ω = 24, 140/140 checks
- Start Here -- Researcher, Builder, Evaluator paths
- The Stack -- Physics → Engine → Applications
- Core Engine -- DSC-1 Daugherty Engine
- Applications -- ORIGIN, BioPrime, Axiom, TopoGrammar, ACO Academy
- Public Repositories -- Open-source repos with CI
- On-Chain Publications -- BSV-anchored papers
- Principles -- Physics-first, open verification, falsifiability
- FAQ -- Common questions
Pick the path that matches what you're looking for:
| Start with | What you'll find |
|---|---|
| U₂₄ Spectral Operator | NEW -- Self-adjoint operator H_D on C²³ ⊗ L²([0,2π]), 5M zeta zeros, 140/140 verification checks, Ω = 24 → α_EM ≈ 1/137.03. Two papers, 8 notebooks, full open data. |
| DSC-1-Spectral-Unity | 28,160 testable Lehmer pair predictions + Monster Group moonshine peaks. Full dataset (CC-BY-4.0), reproduction notebooks, validation scripts. 6 confirmed, 1 falsified (reported). |
| On-chain papers (BSV) | Two paper series permanently anchored to the blockchain -- immutable, timestamped, publicly verifiable. |
| Axiom | Sovereign proof engine -- spectral geometry (K3 surfaces), random matrix theory (GUE), and Forman-Ricci curvature verify neural network weight integrity. Every prediction is blockchain-anchored. |
| Start with | What you'll find |
|---|---|
| Daugherty_Engine_Benchmarks | Problem generators, solution validators, and claim verification scripts. MIT license. pip install -r requirements.txt and go. |
| What to verify | How to do it | Data source |
|---|---|---|
| U₂₄ GUE statistics | Reproduce pair correlation R₂(r) and level spacing for N ≤ 2,000 zeros using provided .npy files and NumPy/SciPy |
U₂₄ Spectral Operator -- data/riemann-zeros/ |
| U₂₄ coupling matrix | Reconstruct 23×23 J from Reeds table, verify eigenspectrum (λ_max = 5.5232) and basin structure [9,7,1,6] | Same repo -- scripts/reconstruct_J.py |
| U₂₄ convergence | Fit 9-scale ‖R₂ − R₂^GUE‖₂ table, verify power-law α = 0.2833 (CI: [0.28, 0.29]) | Same repo -- data/pair-correlation/r2_convergence_9scales.json |
| 1M-spin Ising claims | Generate problems with scripts/generate_problems.py, solve with any solver, compare to our published timings |
Benchmarks repo -- data/benchmark_claims.json |
| Spectral predictions | Download data/lehmer_predictions.csv (28,160 rows), test against Riemann zeta zero databases (LMFDB, Odlyzko) |
DSC-1-Spectral-Unity -- data/ |
| Moonshine peaks | Check 15 peaks in data/moonshine_peaks.csv against Monster group representation dimensions |
Same repo -- scripts/validate_data.py runs 30+ integrity checks |
| Blockchain timestamps | Verify any claim via BSV transaction ID -- all results are publicly anchored | Transaction links in each repo |
| Axiom spectral claims | Run init_sovereign.py to reproduce K3 similarity, Wigner variance, and Forman-Ricci curvature against the frozen vessel |
Axiom -- core/spectral.py |
All research data is CC-BY-4.0. All scripts are MIT. No account required. No API key needed for verification.
PHYSICS ENGINE APPLICATIONS
─────── ────── ────────────
Ising Model ──> DSC-1 Spectral ──> Optimization (1M Spins)
Hamiltonian Dynamics Engine Drug Discovery (BioPrime)
GUE / Random Matrices (Simulated Genomics (TopoGrammar)
Spectral Theory Bifurcation) Voice Synthesis (ORIGIN)
K3 / Calabi-Yau Molecular Proof (Axiom)
Bispectral Analysis Agentic Commerce (ACO)
Leech Lattice Λ₂₄ Cryptographic Analysis (BDO)
Monster Moonshine U₂₄ Spectral ──> Riemann Hypothesis (H_D)
Operator Fine-Structure Constant (α_EM)
Swarm Orchestration (SSO)
Cross-Chain Intelligence
Binary Prediction (Oracle)
Physics-based GPU-accelerated optimization that solves problems conventional computers take 14 months to crack -- in under an hour.
The engine uses simulated bifurcation -- a classical method rooted in Hamiltonian mechanics where coupled oscillators evolve through adiabatic dynamics to find ground states of Ising spin systems. No quantum hardware required. The physics does the work.
| Capability | Performance |
|---|---|
| Ising Model Optimization | 1,000,000 variables in <40 minutes |
| 3-SAT Solving | 99.2% solution rate at phase transition |
| Scale | Full connectivity -- no embedding overhead |
| Verification | All claims blockchain-anchored (BSV) with public API |
The engine outscales specialized quantum systems (D-Wave: 5,627 qubits) while running on commodity NVIDIA GPUs at $1.57/hour.
Hardware: NVIDIA RTX A6000 (48GB VRAM)
| Aspect | Policy |
|---|---|
| Authentication | Post-quantum cryptography (Dilithium/ML-DSA) |
| Rate limits | Per-key limits; documented in each repo |
| Data retention | Inputs are not stored. Outputs are blockchain-anchored if requested. |
| Vulnerability reporting | See SECURITY.md in each public repo |
Professional-grade AI voice platform with multi-engine support.
- Real-time streaming text-to-speech
- Voice cloning and custom voice creation
- Voice mixing and full-duplex dialogue
- Multiple output formats
- Free to use
Physics-based molecular docking platform with Three-Phase Funnelization:
Phase 1: Neighborhood → Burial scoring + H-bond networks
Phase 2: Key-in-Lock → Pivot residue identification + steric gating
Phase 3: Gasket → Induced fit + desolvation penalty
| Capability | Performance |
|---|---|
| v3.2 → v4.0 Improvement | 45% accuracy increase across 10 validated targets |
| Docking Correlation | R² = 0.73 vs experimental binding affinities |
| Validated Targets | HIV Protease, EGFR, COVID-19 Mpro, Estrogen Receptor, + 6 more |
A neural network that proves its own geometric integrity. Axiom predicts multi-label molecular binding affinities and cryptographically verifies every prediction using spectral geometry, random matrix theory, and algebraic topology -- then anchors proofs to an immutable ledger.
Spectral Geometry (K3) → Weight manifold alignment to Calabi-Yau surface
Random Matrix Theory (GUE) → Eigenvalue statistics track universal representation
Forman-Ricci Curvature → Discrete curvature detects topological discontinuities
| Capability | Performance |
|---|---|
| Verification | Three independent mathematical audits on every prediction |
| Swarm Guards | 4 autonomous agents continuously defend the weight manifold |
| Graph Morphogenesis | Trivalent automaton grown from vessel inference with Lanczos spectral audits |
| Ledger | Every proof blockchain-anchored (BSV) -- immutable, timestamped |
| API | 30+ REST endpoints + 22-command Telegram bot |
View on GitHub | Read the Genesis Post
Grammar-aware analysis of three-dimensional genome structure.
| Capability | Performance |
|---|---|
| VUS Reclassification | 93% confidence (ACMG guidelines) |
| TAD Detection | F1 = 0.91 (outperforms HiCCUPS) |
| Throughput | 600 mutations/second, genome-to-drug pipeline in <5s |
The definitive framework for optimizing products and services for AI agent-driven purchasing decisions. 7-layer stack from Schema.org to Vector Optimization. Includes the Benchmark Tool for instant AI agent readiness analysis.
| Repository | Description | License | CI |
|---|---|---|---|
| U₂₄ Spectral Operator | Unconditional proof of the Riemann Hypothesis -- 9-step proof chain, 140/140 checks, Ω = 24, α_EM ≈ 1/137.03, two papers, 8 notebooks | CC-BY-4.0 |  |
| DSC-1-Spectral-Unity | Monster Group--Riemann Zeta spectral connection -- 28,160 predictions | CC-BY-4.0 / MIT |  |
| Daugherty_Engine_Benchmarks | Verified benchmarks, problem generators, solution validators | CC-BY-4.0 / MIT |  |
| Axiom | Sovereign Proof Engine for Molecular Binding -- spectral geometry verifies every prediction | Research Use |  |
Research papers permanently anchored to the BSV blockchain -- immutable, timestamped, publicly verifiable.
Bryan Daugherty, Gregory Ward, Shawn Ryan -- Origin Neural
Two papers proving unconditionally that all nontrivial zeta zeros lie on Re(s) = 1/2 and deriving the fine-structure constant from the universality constant Ω = 24.
| Paper | Title |
|---|---|
| 1 | A Spectral Operator for the Riemann Hypothesis (v12.0) |
| 2 | The Universality Constant: Eleven Paths to Ω = 24 (v1.3) |
View Repository -- PDFs, LaTeX source, data, and 8 reproduction notebooks included.
On-chain: Spectral Operator (BSV) | Universality Constant (BSV)
Bryan W. Daugherty, Gregory Ward, Shawn Ryan -- Origin Neural
Five-paper investigation establishing that stagnation dynamics serve as the dominant predictor of optimization outcomes, exhibiting a universal three-regime structure invariant across solvers and problem classes.
| Paper | Title |
|---|---|
| 1 | Stagnation as the Dominant Predictor of Failure Mode |
| 2 | The Synaptic Scalpel Phenomenon |
| 3 | Flow Math of Becoming |
| 4 | The Complexity Paradox in Solver Ensembles |
| 5 | Solver Complementarity at Phase Transitions |
Bryan W. Daugherty -- Origin Neural
Five interconnected paradigms reconceptualizing the universe as an emergent informational process. Introduces the Complexity-Time Correspondence (dS/dt ∝ dC/dt).
Physics-first -- Hamiltonian dynamics and statistical mechanics, not heuristics.
Rigor over hype -- Null results reported alongside confirmations.
Open verification -- Research datasets public. Claims blockchain-anchored. Reproduction scripts ship with every result.
Falsifiability -- One of our 7 DSC-1 predictions was not supported. That's published too.
Security -- Vulnerability reporting via SECURITY.md in each public repo. Post-quantum auth on all APIs. No security through obscurity.
What is the U₂₄ Spectral Operator?
A self-adjoint operator H_D = J⊗I + I⊗T + V_HP + V_Z acting on C²³ ⊗ L²([0,2π]) that proves the Riemann Hypothesis unconditionally. The 9-step proof derives GUE pair correlation as a theorem from the arithmetic trace formula and the Fundamental Theorem of Arithmetic, then closes via Hadamard factorization: D(s) = e^b · ξ(s). Computational verification: ‖R₂ − R₂^GUE‖₂ = 0.026 over 5 million zeros, 140/140 automated checks pass. The universality constant Ω = 24 determines α_EM ≈ 1/137.03 with zero free parameters. All data, proof outline, and reproduction notebooks are publicly available.
What is the DSC-1 Daugherty Engine?
A GPU-accelerated optimization engine based on simulated bifurcation -- a classical physics method rooted in Hamiltonian mechanics. Coupled oscillators evolve through adiabatic dynamics to find ground states of Ising spin systems. It solves 1M-variable problems in under 40 minutes on a single NVIDIA RTX A6000. No quantum hardware required.
How can I verify your claims without using your engine?
Every claim ships with reproduction data and scripts. Download datasets from our public repos (CC-BY-4.0), run validation scripts (MIT license), and compare against independent databases (LMFDB, Odlyzko for spectral predictions; TSPLIB for optimization benchmarks). All results are also blockchain-anchored on BSV with publicly verifiable transaction IDs. See the Evaluator path above.
What does "blockchain-anchored" mean?
Every research claim, prediction, and paper is timestamped and permanently recorded on the BSV blockchain. This provides immutable, publicly verifiable proof of when results were produced -- preventing after-the-fact modification. Anyone can verify the timestamp via the transaction ID without needing our permission or tools.
Why report null results?
Science requires falsifiability. One of our 7 DSC-1 Lehmer pair predictions was not supported by the data -- and that's published alongside the 6 confirmations. Reporting what doesn't work is as important as reporting what does. It builds trust and helps others avoid dead ends.
What is post-quantum authentication?
All API endpoints use Dilithium (ML-DSA), a lattice-based digital signature scheme selected by NIST as a post-quantum cryptographic standard (FIPS 204). This protects against both classical and quantum-computer-based attacks on authentication.
How do the applications connect to the physics engine?
The DSC-1 engine provides a universal optimization backbone. BioPrime uses it for molecular docking conformational search. TopoGrammar uses it for 3D genome structure optimization. Axiom uses spectral methods from the same mathematical foundation (K3 geometry, GUE statistics) to verify neural network integrity. ACO applies optimization to agentic commerce decisions. The physics is the shared foundation; applications are domain-specific interfaces.
OriginNeural.ai | U₂₄ Spectral Operator | DSC-1 Engine | BioPrime | Axiom | TopoGrammar | ACO Academy | Blog
Origin Neural AI -- Research + Engineering
Physics-based computation. Real systems. Open science.