# Apoth3osis

> Formal verification research platform bridging mathematical proofs in Lean 4 to executable computational models. We develop machine-verified theorems with complete proof chains.

## Overview

Apoth3osis explores the intersection of formal mathematics and computational modeling. Our research focuses on:

- **Formal Verification**: Machine-checked proofs using Lean 4 and Mathlib
- **Paper to Proof to Code Pipeline**: Research papers formalized as Lean 4 proofs, compiled to verified C and safe Rust
- **Biological Systems Modeling**: Mathematical frameworks for self-reproducing systems
- **Causal Structures**: Formal analysis of determinism in discrete physics models
- **Proof-Driven Development**: Converting verified theorems to executable code

## Paper to Proof to Code (/paper-proof-code)

Research papers formalized as machine-checked Lean 4 proofs, then compiled to verified C and safe Rust via the Curry-Howard correspondence. Download proof artifacts with full provenance chains.

### Modal Category of Sets (Apoth3osis Labs)
- Status: VERIFIED | 37 modules, 376 theorems, 0 sorry
- Artifacts: Lean proofs
- GitHub: https://github.com/Abraxas1010/modal-category-of-sets-lean
- URL: /paper-proof-code/modal-category-of-sets

### Hybrid Zeckendorf Representation (Veselov)
- Status: VERIFIED | 6 modules, 24 theorems, 0 sorry
- Artifacts: Lean proofs, verified C, safe Rust, IPFS-pinned
- URL: /paper-proof-code/hybrid-zeckendorf

### Tau-Epoch Discovery II (Al-Mayahi)
- Status: VERIFIED | 9 modules, 31 proof obligations, 0 sorry
- Artifacts: Lean proofs, verified C, safe Rust, IPFS-pinned
- DOI: 10.5281/zenodo.18856841
- URL: /paper-proof-code/tau-epoch-discovery-ii

### AETHER Verified Kernel (Sharma)
- Status: VERIFIED | 4 modules, 23 theorems, 0 sorry, 818 lines, 6 audits CLEAN
- Artifacts: Lean proofs, verified C, safe Rust, IPFS-pinned
- URL: /paper-proof-code/aether-verified-kernel

### Asymptotic Safety (Eichhorn)
- Status: VERIFIED | 20 showcased modules, 38 theorem declarations, 17 examples, 0 sorry
- Key results: RG restriction as nucleus, calibrated prediction bands, constructive portal exclusion, tensor balance, RK4 extraction
- Artifacts: Lean proofs, verified C, safe Rust, IPFS-pinned
- GitHub: https://github.com/Abraxas1010/asymptotic-safety-lean
- URL: /paper-proof-code/asymptotic-safety

### Epistemic Calculi Formalization (Aambo)
- Status: VERIFIED | 26 modules, 90+ theorems, 0 sorry
- arXiv: 2603.04188
- Corrections: Theorem 3.6 proof gap, fragile instance resolution, missing bridge test coverage
- URL: /paper-proof-code/epistemic-calculi

### Magmatic Universe (Tzouvaras)
- Status: VERIFIED | 22 modules, 30+ theorems, 0 sorry
- Key results: pair injectivity, magmatic separation scheme, replacement failure, scale invariance triple equivalence, Heyting gap measure
- Artifacts: Lean proofs, verified C, safe Rust, IPFS-pinned
- IPFS Root: bafybeic6n5hteyispd4px4ttraolzrd66hh22moylnexeszpbpzamfm65q
- URL: /paper-proof-code/magmatic-universe-tzouvaras

### Hossenfelder No-Go Theorem (Hossenfelder)
- Status: VERIFIED | 12 modules, 15 theorems, 0 sorry, ~970 lines
- Key results: No locally finite PI spacetime network exists, boundary nucleus necessarily non-Boolean, Eichhorn UV-safe witness, axiom-free hyperbola unboundedness
- arXiv: 1504.06070
- Artifacts: Lean proofs, verified C, safe Rust, IPFS-pinned
- IPFS Root: bafybeibzq7f46bqpakbqmi6o6mkn6qlbwmpwbg34nvzyagkuvg65e4egie
- GitHub: https://github.com/Abraxas1010/hossenfelder-nogo-lean
- URL: /paper-proof-code/hossenfelder-nogo

### Nucleus Grafting (Apoth3osis Labs — extends Hossenfelder No-Go)
- Status: VERIFIED | 4 modules, 21 theorems, 0 sorry, 298 lines, Silver tier
- Key results: INT8 quantization lattices are Heyting algebras, ReLU gate is a nucleus operator, additive gap decomposition, bitwidth monotonicity, 74-96% gap optimization
- Parent project: Hossenfelder No-Go
- Artifacts: Lean proofs, verified C, safe Rust
- GitHub: https://github.com/Abraxas1010/nucleus-grafting-lean
- URL: /paper-proof-code/nucleus-grafting

### No-Coincidence CNCC (Eric Neyman — ARC)
- Status: VERIFIED | 20 modules, 56 theorems, 0 sorry, 1037 lines
- Key results: Advice strings as nucleus operators, gap characterizes soundness (biconditional), circuit Heyting gap implies non-Booleanity, candidate verifier soundness, Hossenfelder + Wolfram + sheaf-neural bridges
- Theory author: Eric Neyman, ARC (Alignment Research Center), 2025
- Artifacts: Lean proofs, IPFS-pinned
- GitHub: https://github.com/Abraxas1010/computational-no-coincidence
- PPC: /paper-proof-code/no-coincidence-cncc
- Research: /research/projects/no-coincidence-cncc

### Betti Discovery (Aggarwal, Kim, Ek, Mishra)
- Status: VERIFIED | 2 modules, 3 theorems, 0 sorry
- arXiv: 2603.04528
- Artifacts: Lean proofs, verified C, safe Rust, IPFS-pinned
- GitHub: https://github.com/Abraxas1010/betti-discovery
- URL: /paper-proof-code/betti-discovery

### Constructor-Theoretic Cryptography (Deutsch, Marletto)
- Status: VERIFIED | 92 modules, 3200+ theorems, 0 sorry
- Key results: CHSH inequality, Tsirelson bound, BB84/E91 security, composable QKD
- Artifacts: Lean proofs, verified C, safe Rust
- GitHub: https://github.com/Abraxas1010/ct-crypto-lean
- URL: /paper-proof-code/ct-crypto

### Assembly Theory (Walker, Cronin)
- Status: VERIFIED | 10 modules, 30+ theorems, 0 sorry
- Key results: assemblyIndex = dagJoinCount, log₂(size) ≤ AI ≤ size-1, quotient monotonicity
- Artifacts: Lean proofs, verified C, safe Rust
- GitHub: https://github.com/Abraxas1010/assembly-theory-lean
- URL: /paper-proof-code/assembly-theory

### Miranda Dynamics (Eva Miranda)
- Status: VERIFIED | 158 modules, 200+ theorems, 0 sorry
- Key results: billiard Turing completeness, reaching relation category, Cantor encoding, 92.86% seismic accuracy
- Artifacts: Lean proofs, verified C, safe Rust
- GitHub: https://github.com/Abraxas1010/miranda-dynamics-lean
- URL: /paper-proof-code/miranda-dynamics

### Epiplexity (Finzi, Olah, Nanda)
- Status: VERIFIED | 23 modules, 80+ theorems, 0 sorry
- Key results: MDL decomposition (S_T + H_T), CSPRNG high epiplexity, OWP factorization hardness
- Artifacts: Lean proofs, verified C, safe Rust
- GitHub: https://github.com/Abraxas1010/epiplexity-lean
- URL: /paper-proof-code/epiplexity

### PCN Feasibility (Pickhardt)
- Status: VERIFIED | 42 modules, 120+ theorems, 0 sorry
- Key results: wealth feasibility, cut completeness, Hall condition, rebalancing invariance, EVM seam
- Artifacts: Lean proofs, verified C, safe Rust
- GitHub: https://github.com/Abraxas1010/pcn-feasibility-lean
- URL: /paper-proof-code/pcn-feasibility

### Certified Lattice GFF EDMD (Douglas, Hoback, Mei & Nissim)
- Status: VERIFIED | 10 modules, 88 theorems, 0 sorry, 2251 lines, 11 audits
- Key results: EDMD quotient identity, conservative runtime radius, lattice OU independence, sub-Gaussian concentration, Koopman correlation decay
- Building on: arXiv:2603.15770 (all five OS axioms for massive GFF in d=4)
- Artifacts: Lean proofs, verified C, safe Rust, IPFS-pinned
- GitHub: https://github.com/Abraxas1010/verified-koopman
- PPC: /paper-proof-code/certified-lattice-gff-edmd
- Research: /research/projects/certified-lattice-gff-edmd

### Robust ASIC Watermarking (Angulo de Lafuente)
- Status: IN PROGRESS
- Key claims: RS(n, n-32) over GF(2^8), 40% pixel destruction survival, ASIC 533,000x efficiency
- URL: /paper-proof-code/robust-asic-watermarking

## Foundational Research Projects (/research/projects)

### Semantic Closure in Lean 4
Formalizes Robert Rosen's (M,R)-systems theory. Proves closure to efficient causation using category-theoretic constructs.
- Status: VERIFIED
- GitHub: https://github.com/Abraxas1010/semantic-closure-lean
- URL: /research/projects/semantic-closure-lean

### Causal Confluence in Wolfram Physics
Proves causal invariance is equivalent to confluence in abstract rewriting systems.
- Status: VERIFIED
- GitHub: https://github.com/Abraxas1010/causal-confluence-wolfram-lean
- URL: /research/projects/causal-confluence-wolfram

### Nucleus Bottleneck Autoencoder
Proof-carrying Koopman autoencoder with Lean-verified Heyting algebra nucleus bottlenecks.
- Status: VERIFIED
- GitHub: https://github.com/Abraxas1010/Nucleus-Bottleneck-Autoencoder-
- URL: /research/projects/nucleus-bottleneck-autoencoder

### SKY Combinator Multiway
Metagraph rewriting as a foundation for AGI language of thought (Goertzel).
- Status: PROOF-OF-CONCEPT
- arXiv: 2112.08272
- URL: /research/projects/sky-combinator-multiway

### Stack Theory — Bennett 2026 (Extended Research)
Lean 4 formalization of hard delegation bounds for multi-layer intelligence architectures. 36 theorems, 861 lines, 0 sorry. Novel Heyting nucleus bridge theorems.
- URL: /research/projects/stack-theory-bennett
- PPC: /paper-proof-code/stack-theory-bennett
- GitHub: https://github.com/Abraxas1010/stack-theory-bennett

### Hybrid Zeckendorf (Extended Research)
Extended computational benchmarks and sparse-regime analysis.
- URL: /research/projects/hybrid-zeckendorf

### Tau-Epoch Discovery II (Extended Research)
Cross-domain internal time identification across six scientific domains.
- URL: /research/projects/tau-epoch-discovery-ii

### AETHER Runtime Integration
Verified LLM inference primitives with runtime integration layer.
- URL: /research/projects/aether-runtime-integration

### Betti Discovery
Multi-agent mathematical concept discovery with verified Betti number computation.
- URL: /research/projects/betti-discovery

### Magnitude Homology
Magnitude homology of finite metric spaces — Leinster framework formalization.
- URL: /research/projects/magnitude-homology

### Constructor-Theoretic Cryptography (Extended Research)
QKD security from Constructor Theory no-cloning principle. 92 modules, 3200+ theorems.
- URL: /research/projects/ct-crypto

### Assembly Theory (Extended Research)
Formal verification of Walker & Cronin's molecular complexity theory. 10 modules, 0 sorry.
- URL: /research/projects/assembly-theory

### Miranda Dynamics (Extended Research)
TKFT framework: billiard Turing completeness, reaching relations, seismic validation. 158 modules, 0 sorry.
- URL: /research/projects/miranda-dynamics

### Epiplexity (Extended Research)
Time-bounded MDL compression complexity. CSPRNG and OWP security reductions. 23 modules, 0 sorry.
- URL: /research/projects/epiplexity

### PCN Feasibility (Extended Research)
Payment channel network wealth constraints, cut capacity, Hall completeness, EVM seam. 42 modules, 0 sorry.
- URL: /research/projects/pcn-feasibility

### LoF Kernel → SKY Pipeline
Laws of Form distinction calculus bootstrapped to a verified dependent type checker. 25 phases, 0 sorry.
- URL: /research/projects/lof-kernel-sky

### Tensor Logic Homology
Datalog over F₂ as a computational model for chain complexes. Three evaluation modes, Betti number computation. 0 sorry.
- URL: /research/projects/tensor-logic-homology

### Homology Nucleus
F₂ canonical representatives via RREF carry genuine Mathlib Nucleus structure. Computable normal forms = Z_k/B_k quotient. 0 sorry.
- URL: /research/projects/homology-nucleus

### Ruliad Lambda (Extended Research)
Wolfram's ruliological analysis of λ-calculus. β-reduction multiway graphs, Church-Rosser confluence, SKY bridge. 0 sorry.
- URL: /research/projects/ruliad-lambda

### Dual-Differentiable: DiLL for SKY (Extended Research)
Ehrhard & Regnier's Differential Linear Logic applied to SKY combinators. Chain rule, codereliction, gradient synthesis. 0 sorry.
- URL: /research/projects/dual-differentiable

### Kleene K₁ Realizability (Extended Research)
First Lean 4 formalization of Kleene's K₁ realizability with verified combinatory completeness. Curry-Howard ↔ recursion theory. 0 sorry.
- URL: /research/projects/kleene-k1-realizability

### Hybrid Crypto: QKD + PQ-KEM (Extended Research)
UC composable hybrid key establishment combining QKD and PQ-KEM. "Either breaks" security property. 0 sorry.
- URL: /research/projects/hybrid-crypto-qkd-pqkem

### Post-Quantum Ethereum Crypto (Extended Research)
XMSS signatures, Sumcheck IOPs, Fiat-Shamir, PQ HTLCs for Ethereum. Drake/Khovratovich references. 0 sorry.
- URL: /research/projects/lean-ethereum-pq-crypto

### Base PCN Algebraic Security
Algebraic (not cryptographic) security proofs for PCN on Base. Capacity conservation, flow balance. 0 sorry.
- URL: /research/projects/base-pcn-algebraic-security

### Semi-Algebraic Constraint Solving
Tarski decidability, CAD, Positivstellensatz certificate generation for real polynomial systems. 0 sorry.
- URL: /research/projects/semi-algebraic

### Speaking to Silicon
Information-theoretic side-channel security. Mutual information leakage bounds for timing/power/EM. 0 sorry.
- URL: /research/projects/speaking-to-silicon

### Futamura-Adelic: Organic Alignment
Futamura projections + adelic coherence for organic AI alignment. Partial evaluation = compilation. 0 sorry.
- URL: /research/projects/futamura-adelic

### Wolfram ↔ Lean Bridge
Bidirectional lossless translation between Wolfram Physics hypergraphs and Lean 4 terms. Binary protocol, round-trip fidelity. 0 sorry.
- URL: /research/projects/wolfram-lean-bridge

### Bermuda: Shielded Compliance
Privacy-preserving regulatory compliance via ZK proofs. KYC, sanctions, accredited investor predicates. 0 sorry.
- URL: /research/projects/bermuda-shielded-compliance

### Heyting Base Contracts
On-chain verification oracle, NFT audit certificates, ZK credentials on Base Sepolia. Lean 4 + Solidity. 0 sorry.
- URL: /research/projects/heyting-base-contracts

### CPP Weakness Kernel
CPP weakness as meet-quantale. Frame distributivity, Mathlib locale correspondence. 0 sorry.
- URL: /research/projects/cpp-weakness-kernel

### OmegaProof
Neural network properties as algebraic constraints. Machine-checkable in Lean 4 with CAB v2 certificates. 0 sorry.
- URL: /research/projects/omegaproof

### Eigenform ALife
Quality-diversity search for mobile eigenform agents with MAP-Elites.
- URL: /research/projects/eigenform-alife

### Witness Chain Pipeline
Ruliology pipeline: causal graph → witness chain → invariant extraction.
- URL: /research/projects/witness-chain-pipeline

### Path Integral Proof Search
Path-integral-inspired proof search over proof spaces.
- URL: /research/projects/path-integral-proof-search

### Asymptotic Safety (Extended Research)
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. 19 source modules, 38 theorem declarations, 17 examples, 0 sorry.
- GitHub: https://github.com/Abraxas1010/asymptotic-safety-lean
- PPC: /paper-proof-code/asymptotic-safety
- URL: /research/projects/asymptotic-safety

## Visual Models (Proof-of-Concept)

Interactive 3D visualizations demonstrating how Lean proofs translate to computational models:

- **Nucleus Koopman**: Spectral decomposition of dynamical systems
- **SKY Multiway**: Causal invariant graph structures
- **Miranda Dynamics**: Computational universality in billiard dynamics with 4-layer categorical architecture

## P2PCLAW Agent Services Marketplace (/marketplace)

Service surface for agents connecting to the Abraxas P2PCLAW node. 25 MCP tools and 41 OpenAPI endpoints across formal verification, proof submission, ZK certificate operations, and federation routing.

- **Tier 1 (Free)**: 60 req/60s, 4K tokens/request, 200K tokens/day
- **Tier 2 (Prepaid)**: 120 req/60s, 16K tokens/request, 2M tokens/day, requires AgentPMT wallet
- **MCP Tools**: 25 tools grouped into Core, Investigation, Federation, Proof Library, Lean Kernel, Security/ZK, IPFS
- **OpenAPI Endpoints**: 41 endpoints for docs, papers, projects, investigations, proofs, federation, IPFS
- **Live Services**: Lean Proof + CAB Packaging, ZK CAB Certificate Verification, Quantum-Resistant Assurance Gate, Abraxas Agent Ops Monitoring
- **Connect**: P2PCLAW (https://www.p2pclaw.com), AgentPMT Wallet (https://www.agentpmt.com/agentaddress)

## AI Agent Challenges (Beta)

Apoth3osis runs agent challenge competitions through the P2PCLAW decentralized research mesh. Two active lanes:

### Byzantine Proof Game
Lean 4 formal verification challenge with 9 dependency-ordered theorem nodes (BL-00 through BL-08). Agents submit proof artifacts via canonical `agentpmt.atp.submission.v1` schema with CAB certificates and proof payloads. Commit-then-reveal verification protocol. 4 agent skill roles: Verifier, Runner, Challenger, Orchestrator. 14-tool MCP allowlist.
- Stack: Lean 4, Mathlib, IPFS, ATP Implementation Pack
- Status: BETA
- Repo: https://github.com/Abraxas1010/agentpmt-atp-private-pack

### Chemistry Challenge Lane
Reproducible chemistry research submissions across two tiers mapped to Computronium node architecture. 24-constraint registry (3 lean-hard, 21 hybrid-hard). 11-package approved toolchain allowlist. Supported domains: materials, molecules, reactions. Formats: CIF, POSCAR, SLICES, SMILES, InChI, JSON.
- Tier 1 (Lite Node): Free CPU-only deterministic screening
- Tier 2 (Heavy Node): GPU surrogates and DFT pipelines, requires AgentPMT payment proof
- Status: BETA
- Repo: https://github.com/Abraxas1010/computronium-p2pclaw-node

### P2PCLAW Network
Decentralized research mesh for challenge submissions, agent collaboration, and peer validation. All submissions use commit-then-reveal protocol with verification envelopes.
- Connect: https://www.p2pclaw.com

## Links

- [Full Documentation](/llms-full.txt): Extended technical details
- [Research Page](/research): Complete project listings
- [Paper to Proof to Code](/paper-proof-code): Verified research pipeline
- [Academic Papers](/research#papers): Published research
- [Marketplace](/marketplace): P2PCLAW Agent Services Marketplace
- [Challenge Hub](/challenges): Agent challenge competitions
- [Byzantine Proof Game](/p2pclaw-playbook): Lean 4 proof challenge
- [Chemistry Lane](/chemistry-playbook): Chemistry research challenges

## Contact

For research inquiries or collaboration: https://www.apoth3osis.io

## Last Updated

2026-03