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Formally verified Formal Verification
Every project is a complete Lean 4 formalization with machine-checked proofs. The theorems compile, the invariants hold, and the code executes exactly as specified.
Foundational formalizations, extended research from published papers, and applied systems — all machine-checked in Lean 4.
Research papers and whitepapers on the agentic economy, formal verification, and autonomous systems.
Proof-driven computational visualizations. Each animation is parameterized by machine-checked invariants.
Complete pipeline: published paper to Lean 4 formalization to executable code. Every claim witnessed by a type-checked term.
10-layer machine-checked synthesis proving every non-Boolean nucleus produces an irreducible information gap. Instantiated across neural networks, spacetime, fluid dynamics, matter stability, dimensional emergence, causal invariance, spinors, control systems, and RG flow. 162 files, 1,486 declarations, zero sorry.
Formal verification framework for Neyman's CNCC using Heyting algebra nucleus operators as poly-time advice strings for reversible circuit structural explanations. 55 theorems, 20 modules, 1,037 lines, 0 sorry, twice adversarially audited.
Aambø’s categorical epistemic uncertainty framework formalized in Lean 4. Unifies probability, possibility, certainty factors, and Dempster–Shafer under a single monoidal-posetal spine with 8 axiom classes. 26 modules, 90+ theorems, 0 sorry. Includes corrections to Theorem 3.6 composition coherence.
Axiom-free Lean 4 formalization of Hossenfelder's no-go theorem: no locally finite network in Minkowski spacetime can be Poincaré-invariant. Constructive Heyting boundary interpretation with Eichhorn UV-safe witness proving boundary nucleus non-Booleanity. 12 modules, 15 theorems, 0 sorry, 0 axioms.
INT8 quantization lattices are Heyting algebras and ReLU gates are nucleus operators. Additive gap decomposition (quantization + gate + reconstruction), bitwidth monotonicity, and 74–96% gap optimization validated across MLP, CNN, and autoencoder architectures. 4 modules, 21 theorems, 298 lines, 0 sorry.
Machine-checked Lean 4 formalization of Eva Miranda’s computational dynamics program. Two pillars: (1) Billiard Turing completeness via TKFT/Cantor encoding with seismic validation (92.86%), (2) Navier-Stokes Turing completeness via cosymplectic geometry and Etnyre-Ghrist correspondence. 163 modules, 0 sorry.
Machine-checked Lean 4 formalization of Quni-Gudzinas' p-adic Hamiltonian decoupling framework. Ultrametric causal balls, tripartite energy conservation, bounded random walks, and the novel depth-truncation nucleus bridged to the Hossenfelder boundary gap.
SafEDMD-inspired residual-covariance error bounds and SDP Lyapunov controller synthesis for nucleus-bottleneck Koopman autoencoders, with 10 sorry-free Lean 4 modules.
Lean 4 formalization of Al-Mayahi’s PM-bounded τ-control framework: rational/tanh saturation operators as bounded contractions into a genuine set-level boundary nucleus, with risk functionals, τ-progression, soft gate blending, and multi-limit vector completion. 44 theorems, 0 sorry, 3 consecutive adversarial audit PASS.
Boltzmann-weighted beam search over tactic states with sheaf-consistent transport, Čech H¹ obstruction pruning, and proved geometric annealing convergence. 11 modules, 2,382 lines, 79 theorems, 0 sorry.
Eichhorn's asymptotic-safety RG slice formalized in Lean 4 as a genuine nucleus on UV-safe coupling regions, with calibrated mass-band theorems, constructive dark-matter exclusion, tensor balance, and RK4 extraction. Includes a Magma × AS bridge proving that nucleus-preservation on coupling regions mirrors scale-invariance on magmatic sets. 19 source modules, 38 theorem declarations, 17 examples, 0 sorry.
Machine-checked Lean 4 formalization of Deutsch & Marletto's Constructor Theory for cryptographic security. No-cloning implies eavesdropping impossibility. CHSH/Tsirelson bounds, BB84/E91 security. 92 modules, 3200+ theorems, 0 sorry.
Machine-checked Lean 4 formalization of Walker & Cronin's Assembly Theory. Tight assembly index bounds (log₂(size) ≤ AI ≤ size-1), DAG join count equivalence, quotient monotonicity, molecular space semantics. 10 modules, 0 sorry.
Machine-checked Lean 4 formalization of Finzi et al.'s Epiplexity framework. Time-bounded MDL compression: S_T(X) + H_T(X). CSPRNG → high epiplexity, OWP factorization hardness. 23 modules, 0 sorry.
Machine-checked Lean 4 formalization of Pickhardt's payment channel network feasibility. Wealth constraints, cut capacity, Hall completeness, rebalancing invariance, EVM seam theorem. 42 modules, 0 sorry.
Laws of Form distinction calculus bootstrapped through boolean gates, λ-calculus, SKY combinators, eigenforms, and Heyting algebra to a verified dependent type checker. 25 phases, 0 sorry.
Datalog over F₂ as a computational model for chain complexes. Three evaluation modes (Boolean, Fuzzy, F₂), Gaussian elimination over GF(2), and Betti number computation — all formally verified in Lean 4. 0 sorry.
F₂ canonical representatives via deterministic RREF carry a genuine Mathlib Nucleus structure. Proves computable normal forms equal the algebraic Zₖ/Bₖ quotient. 0 sorry.
Wolfram’s ruliological analysis applied to λ-calculus. β-reduction multiway graphs with Church-Rosser confluence, branchial space analysis, and verified SKY combinator bridge. 0 sorry.
Ehrhard & Regnier’s Differential Linear Logic applied to SKY combinators. Exact chain rule, codereliction = derivative of exponential, and executable gradient descent synthesis loop. 0 sorry.
First Lean 4 formalization of Kleene’s K₁ realizability model with verified combinatory completeness. K and S combinators, Curry-Howard ↔ recursion theory bridge. 0 sorry.
UC composable hybrid key establishment combining quantum key distribution and post-quantum KEM. “Either breaks” property: security holds if QKD OR PQ-KEM is unbroken. References X-Wing, NIST FIPS 203. 0 sorry.
Post-quantum cryptographic primitives for Ethereum: XMSS/LeanSig signatures, Sumcheck polynomial IOPs, Fiat-Shamir transform, PQ HTLCs. References Drake/Khovratovich. 0 sorry.
Algebraic (not cryptographic) security proofs for Payment Channel Networks on Coinbase’s Base blockchain. Capacity conservation, flow balance, settlement correctness. 0 sorry.
Verified constraint solving for quantifier-free real polynomial systems. Tarski decidability, cylindrical algebraic decomposition, Positivstellensatz certificate generation. 0 sorry.
Information-theoretic side-channel security formalized via finite probability theory. Mutual information leakage bounds for timing, power, and EM channels. Data processing inequality. 0 sorry.
Futamura projections + adelic multi-scale coherence for organic AI alignment. Partial evaluation ↔ compilation equivalence, restricted product condition. 0 sorry.
Bidirectional lossless translation between Wolfram Physics hypergraph evolution and Lean 4 proof terms. Compact binary protocol, witness verification, and proved round-trip fidelity. 0 sorry.
Privacy-preserving regulatory compliance via zero-knowledge proofs. ZK compliance predicates, shielded transfers, selective disclosure, and credential revocation — all formally verified. 0 sorry.
On-chain verification oracle, ERC-721 audit certificates, and ZK credentials for formal mathematics — deployed on Base Sepolia. Contract invariants formally verified in Lean 4. 0 sorry.
CPP weakness relation carries meet-quantale structure: complete lattice with associative tensor distributing over joins. Frame distributivity and Mathlib locale correspondence. 0 sorry.
Neural network properties (robustness, fairness, monotonicity) expressed as algebraic constraints over weight matrices and activation functions. Machine-checkable in Lean 4 with CAB v2 certificates. 0 sorry.
Multi-agent algebraic-topological invariant discovery from triangulated surfaces. Lean 4 certified provability oracle for Betti numbers and Euler characteristic over GF(2) chain complexes. 3 theorems, standalone Python implementation.
Reusable witness-chain proof pattern: symbolic PDE → Madelung derivation → trust bridge → runtime export → oracle verification. A 43-line type-level kernel instantiates across quantum physics, trust engineering, and smart contract audit. 11 modules, 1,348 lines, 0 sorry.
6-phase formalization of Leinster–Shulman magnitude homology in Lean 4. From chain complexes through Künneth, Mayer–Vietoris, diagonality, persistent/VR bridges, to iterated spectral convergence. 90+ sorry-free theorems.
Formally-verified Lyapunov governors, Chebyshev eviction guards, and Betti fingerprinting from Teerth Sharma’s AETHER paper, integrated into AgentHALO production runtime. 20+ theorems, 710 tests, 2 hostile audits clean.
Al-Mayahi’s internal time (τ) across 6 scientific domains, formalized in Lean 4. Two-Clock Projection Law, structural class exclusion, and τ-Coordination bridge to multi-agent consensus. 19 modules, 46+ theorems, 0 sorry.
Veselov’s hybrid hierarchical number system, formalized in Lean 4 with rigorous Rust benchmarks. First formally-verified number representation to outperform GMP: 55x on sparse addition at 1M bits. 68 theorems, 7 experiments, 0 sorry.
Quality-diversity search over Lenia-based autonomous agents with Markov blanket shells, eigenform crystallization, and game-theoretic strategies. Agents grow in continuous 3D flow fields, develop concentric shells, crystallize toroidal eigenform gates, and bridge when algebraic depth reaches geometric morphism threshold.
Proof-carrying Koopman autoencoder with nucleus-constrained latent space and Lean-certified ReLU/threshold operators.
Machine-verified formalization of semantic closure theory from López-Díaz & Gershenson. Mechanizes how self-reference and meaning emergence arise through fixed points of closure operators on (M,R) systems.
Lean 4 proof that confluence and causal invariance are independent in Wolfram Physics rewriting systems. Mechanizes Piskunov’s 2020 counterexamples with WM148 case study under fresh-vertex semantics.
Machine-checked proof that multiway rewriting systems form ∞-groupoids (Kan complexes). SKY combinators with full confluence, Wolfram Physics case studies, and topos-theoretic gluing with sieves as complete Heyting algebras. Builds on Goertzel’s reflective metagraph rewriting.
10-layer machine-checked synthesis proving every non-Boolean nucleus produces an irreducible information gap. Instantiated across neural networks, spacetime, fluid dynamics, matter stability, dimensional emergence, causal invariance, spinors, control systems, and RG flow. 162 files, 1,486 declarations, zero sorry.
Formal verification framework for Neyman's CNCC using Heyting algebra nucleus operators as poly-time advice strings for reversible circuit structural explanations. 55 theorems, 20 modules, 1,037 lines, 0 sorry, twice adversarially audited.
Aambø’s categorical epistemic uncertainty framework formalized in Lean 4. Unifies probability, possibility, certainty factors, and Dempster–Shafer under a single monoidal-posetal spine with 8 axiom classes. 26 modules, 90+ theorems, 0 sorry. Includes corrections to Theorem 3.6 composition coherence.
Axiom-free Lean 4 formalization of Hossenfelder's no-go theorem: no locally finite network in Minkowski spacetime can be Poincaré-invariant. Constructive Heyting boundary interpretation with Eichhorn UV-safe witness proving boundary nucleus non-Booleanity. 12 modules, 15 theorems, 0 sorry, 0 axioms.
INT8 quantization lattices are Heyting algebras and ReLU gates are nucleus operators. Additive gap decomposition (quantization + gate + reconstruction), bitwidth monotonicity, and 74–96% gap optimization validated across MLP, CNN, and autoencoder architectures. 4 modules, 21 theorems, 298 lines, 0 sorry.
Machine-checked Lean 4 formalization of Eva Miranda’s computational dynamics program. Two pillars: (1) Billiard Turing completeness via TKFT/Cantor encoding with seismic validation (92.86%), (2) Navier-Stokes Turing completeness via cosymplectic geometry and Etnyre-Ghrist correspondence. 163 modules, 0 sorry.
Machine-checked Lean 4 formalization of Quni-Gudzinas' p-adic Hamiltonian decoupling framework. Ultrametric causal balls, tripartite energy conservation, bounded random walks, and the novel depth-truncation nucleus bridged to the Hossenfelder boundary gap.
SafEDMD-inspired residual-covariance error bounds and SDP Lyapunov controller synthesis for nucleus-bottleneck Koopman autoencoders, with 10 sorry-free Lean 4 modules.
Lean 4 formalization of Al-Mayahi’s PM-bounded τ-control framework: rational/tanh saturation operators as bounded contractions into a genuine set-level boundary nucleus, with risk functionals, τ-progression, soft gate blending, and multi-limit vector completion. 44 theorems, 0 sorry, 3 consecutive adversarial audit PASS.
Boltzmann-weighted beam search over tactic states with sheaf-consistent transport, Čech H¹ obstruction pruning, and proved geometric annealing convergence. 11 modules, 2,382 lines, 79 theorems, 0 sorry.
Eichhorn's asymptotic-safety RG slice formalized in Lean 4 as a genuine nucleus on UV-safe coupling regions, with calibrated mass-band theorems, constructive dark-matter exclusion, tensor balance, and RK4 extraction. Includes a Magma × AS bridge proving that nucleus-preservation on coupling regions mirrors scale-invariance on magmatic sets. 19 source modules, 38 theorem declarations, 17 examples, 0 sorry.
Machine-checked Lean 4 formalization of Deutsch & Marletto's Constructor Theory for cryptographic security. No-cloning implies eavesdropping impossibility. CHSH/Tsirelson bounds, BB84/E91 security. 92 modules, 3200+ theorems, 0 sorry.
Machine-checked Lean 4 formalization of Walker & Cronin's Assembly Theory. Tight assembly index bounds (log₂(size) ≤ AI ≤ size-1), DAG join count equivalence, quotient monotonicity, molecular space semantics. 10 modules, 0 sorry.
Machine-checked Lean 4 formalization of Finzi et al.'s Epiplexity framework. Time-bounded MDL compression: S_T(X) + H_T(X). CSPRNG → high epiplexity, OWP factorization hardness. 23 modules, 0 sorry.
Machine-checked Lean 4 formalization of Pickhardt's payment channel network feasibility. Wealth constraints, cut capacity, Hall completeness, rebalancing invariance, EVM seam theorem. 42 modules, 0 sorry.
Laws of Form distinction calculus bootstrapped through boolean gates, λ-calculus, SKY combinators, eigenforms, and Heyting algebra to a verified dependent type checker. 25 phases, 0 sorry.
Datalog over F₂ as a computational model for chain complexes. Three evaluation modes (Boolean, Fuzzy, F₂), Gaussian elimination over GF(2), and Betti number computation — all formally verified in Lean 4. 0 sorry.
F₂ canonical representatives via deterministic RREF carry a genuine Mathlib Nucleus structure. Proves computable normal forms equal the algebraic Zₖ/Bₖ quotient. 0 sorry.
Wolfram’s ruliological analysis applied to λ-calculus. β-reduction multiway graphs with Church-Rosser confluence, branchial space analysis, and verified SKY combinator bridge. 0 sorry.
Ehrhard & Regnier’s Differential Linear Logic applied to SKY combinators. Exact chain rule, codereliction = derivative of exponential, and executable gradient descent synthesis loop. 0 sorry.
First Lean 4 formalization of Kleene’s K₁ realizability model with verified combinatory completeness. K and S combinators, Curry-Howard ↔ recursion theory bridge. 0 sorry.
UC composable hybrid key establishment combining quantum key distribution and post-quantum KEM. “Either breaks” property: security holds if QKD OR PQ-KEM is unbroken. References X-Wing, NIST FIPS 203. 0 sorry.
Post-quantum cryptographic primitives for Ethereum: XMSS/LeanSig signatures, Sumcheck polynomial IOPs, Fiat-Shamir transform, PQ HTLCs. References Drake/Khovratovich. 0 sorry.
Algebraic (not cryptographic) security proofs for Payment Channel Networks on Coinbase’s Base blockchain. Capacity conservation, flow balance, settlement correctness. 0 sorry.
Verified constraint solving for quantifier-free real polynomial systems. Tarski decidability, cylindrical algebraic decomposition, Positivstellensatz certificate generation. 0 sorry.
Information-theoretic side-channel security formalized via finite probability theory. Mutual information leakage bounds for timing, power, and EM channels. Data processing inequality. 0 sorry.
Futamura projections + adelic multi-scale coherence for organic AI alignment. Partial evaluation ↔ compilation equivalence, restricted product condition. 0 sorry.
Bidirectional lossless translation between Wolfram Physics hypergraph evolution and Lean 4 proof terms. Compact binary protocol, witness verification, and proved round-trip fidelity. 0 sorry.
Privacy-preserving regulatory compliance via zero-knowledge proofs. ZK compliance predicates, shielded transfers, selective disclosure, and credential revocation — all formally verified. 0 sorry.
On-chain verification oracle, ERC-721 audit certificates, and ZK credentials for formal mathematics — deployed on Base Sepolia. Contract invariants formally verified in Lean 4. 0 sorry.
CPP weakness relation carries meet-quantale structure: complete lattice with associative tensor distributing over joins. Frame distributivity and Mathlib locale correspondence. 0 sorry.
Neural network properties (robustness, fairness, monotonicity) expressed as algebraic constraints over weight matrices and activation functions. Machine-checkable in Lean 4 with CAB v2 certificates. 0 sorry.
Multi-agent algebraic-topological invariant discovery from triangulated surfaces. Lean 4 certified provability oracle for Betti numbers and Euler characteristic over GF(2) chain complexes. 3 theorems, standalone Python implementation.
Reusable witness-chain proof pattern: symbolic PDE → Madelung derivation → trust bridge → runtime export → oracle verification. A 43-line type-level kernel instantiates across quantum physics, trust engineering, and smart contract audit. 11 modules, 1,348 lines, 0 sorry.
6-phase formalization of Leinster–Shulman magnitude homology in Lean 4. From chain complexes through Künneth, Mayer–Vietoris, diagonality, persistent/VR bridges, to iterated spectral convergence. 90+ sorry-free theorems.
Formally-verified Lyapunov governors, Chebyshev eviction guards, and Betti fingerprinting from Teerth Sharma’s AETHER paper, integrated into AgentHALO production runtime. 20+ theorems, 710 tests, 2 hostile audits clean.
Al-Mayahi’s internal time (τ) across 6 scientific domains, formalized in Lean 4. Two-Clock Projection Law, structural class exclusion, and τ-Coordination bridge to multi-agent consensus. 19 modules, 46+ theorems, 0 sorry.
Veselov’s hybrid hierarchical number system, formalized in Lean 4 with rigorous Rust benchmarks. First formally-verified number representation to outperform GMP: 55x on sparse addition at 1M bits. 68 theorems, 7 experiments, 0 sorry.
Quality-diversity search over Lenia-based autonomous agents with Markov blanket shells, eigenform crystallization, and game-theoretic strategies. Agents grow in continuous 3D flow fields, develop concentric shells, crystallize toroidal eigenform gates, and bridge when algebraic depth reaches geometric morphism threshold.
Proof-carrying Koopman autoencoder with nucleus-constrained latent space and Lean-certified ReLU/threshold operators.
Machine-verified formalization of semantic closure theory from López-Díaz & Gershenson. Mechanizes how self-reference and meaning emergence arise through fixed points of closure operators on (M,R) systems.
Lean 4 proof that confluence and causal invariance are independent in Wolfram Physics rewriting systems. Mechanizes Piskunov’s 2020 counterexamples with WM148 case study under fresh-vertex semantics.
Machine-checked proof that multiway rewriting systems form ∞-groupoids (Kan complexes). SKY combinators with full confluence, Wolfram Physics case studies, and topos-theoretic gluing with sieves as complete Heyting algebras. Builds on Goertzel’s reflective metagraph rewriting.
Proof-driven computational visualizations. Each animation is parameterized by machine-checked invariants — what you see is the proof running.
Our pipeline does more than verify existing claims — it extends them. When we formalize a research paper in Lean 4, the type checker forces us to fill every gap, resolve every ambiguity, and prove every implicit assumption. The result is invariably richer than the original.
