diff --git a/README.adoc b/README.adoc deleted file mode 100644 index 183bf98..0000000 --- a/README.adoc +++ /dev/null @@ -1,412 +0,0 @@ -// SPDX-License-Identifier: CC-BY-SA-4.0 -// Copyright (c) Jonathan D.A. Jewell -# Absolute Zero - -**Formal Verification of Null Operations — two co-equal pillars: Certified Null _Effect_ (CNO) and Certified Null _Disclosure_ (OND)** - -image:https://img.shields.io/badge/License-MPL_2.0-blue.svg[MPL-2.0] -image:https://img.shields.io/badge/Sponsor-%E2%9D%A4-pink?logo=github[Sponsor,link="https://github.com/sponsors/hyperpolymath"] - -== License -image:https://img.shields.io/badge/OpenSSF-Best_Practices-green?logo=openssourcesecurity[OpenSSF Best Practices,link="https://www.bestpractices.dev/en/projects/new?repo_url=https://github.com/hyperpolymath/absolute-zero"] - - -**SPDX-License-Identifier: CC-BY-SA-4.0** - -Licensed under the Mozilla Public License 2.0. See `LICENSE` for full text. - -> *"The universe tends toward maximum entropy. A Certified Null Operation is a pocket of perfect computational order—a program that resists the Second Law. It does nothing, but in doing nothing, it says everything about the structure of computation itself."* - -## What is Absolute Zero? - -**Absolute Zero** is a research project exploring the formal verification of programs that provably compute nothing. We formalize **Certified Null Operations (CNOs)**—programs that, despite potentially complex implementations, can be mathematically proven to have zero net computational effect. - -**The Central Question**: Can we formally prove that a program does absolutely nothing? - -This seemingly trivial question leads to deep insights in: -- **Formal Verification**: Machine-checked proofs of program properties -- **Computational Complexity**: Understanding minimal computation -- **Reversible Computing**: Programs preserving thermodynamic reversibility -- **Esoteric Languages**: Using Malbolge as proof-of-concept - -## The Two Pillars - -Absolute Zero rests on **two co-equal pillars** — siblings of equal weight, not parent and extension: - -- **CNO — Certified Null _Effect_.** Certifies that an operation does *nothing to the world*: it terminates, maps input state to identical output state, is functionally pure, and is thermodynamically reversible (Landauer/Bennett). The conserved quantity is **state**. -- **OND — Observational Null Operations / Certified Null _Disclosure_.** Certifies that an operation *reveals nothing about its secret input* to a declared observer: its observable trace (timing, size, …) is constant over the secret, **relative to a declared observation model `O`**. The conserved quantity is the **secret→observable channel**. - -The two are **logically independent** (neither entails the other — a proved theorem, with witnesses), connected by a single *coupling dial* between a thing and the trace it casts to an observer: CNO witnesses the emptiness of the **effect** channel; OND certifies the emptiness of the **disclosure** channel (this dial is *framing, not theorem*). Unlike CNO — which lives entirely inside the formal model — every OND claim is **conditional on its declared `O`** and ships a **residue list** of out-of-scope observables: the honest, explicit boundary between the proof and the physical metal. - -See **`docs/TWO-PILLARS.adoc`** (narrative), **`docs/OND-ROADMAP.adoc`** (prioritised obligations), and **`docs/OND-PILLAR-STRUCTURE.adoc`** (module layout). The OND pillar is at the design/roadmap stage: its obligations are currently `Admitted` (specified, not yet proved) — the normal honest starting state, mirroring CNO's own open obligations. - -## Project Structure - -``` -absolute-zero/ -├── proofs/ # Formal proofs in multiple systems -│ ├── coq/ # Coq proofs (constructive) -│ │ ├── common/ # Core CNO framework (CNO.v) -│ │ ├── malbolge/ # Malbolge-specific proofs -│ │ ├── physics/ # Statistical mechanics (StatMech.v) -│ │ ├── category/ # Category theory (CNOCategory.v) -│ │ ├── lambda/ # Lambda calculus (LambdaCNO.v) -│ │ ├── quantum/ # Quantum computing (QuantumCNO.v) -│ │ └── filesystem/ # Filesystem CNOs (FilesystemCNO.v) -│ │ -│ ├── lean4/ # Lean 4 proofs (modern) -│ │ ├── CNO.lean # Core CNO framework -│ │ ├── StatMech.lean # Statistical mechanics -│ │ ├── CNOCategory.lean # Category theory -│ │ ├── LambdaCNO.lean # Lambda calculus -│ │ ├── QuantumCNO.lean # Quantum computing -│ │ └── FilesystemCNO.lean # Filesystem operations -│ │ -│ ├── z3/ # Z3 SMT verification (automated) -│ ├── agda/ # Agda proofs (dependent types) -│ ├── isabelle/ # Isabelle/HOL (production-grade) -│ ├── mizar/ # Mizar proofs (mathematical library) -│ ├── observation-models/ # OND: declared observation models O (proof inputs) -│ └── residue/ # OND: residue lists (the model-vs-metal gap) -│ # Each prover dir hosts CNO.* and (to author) co-equal OND.* modules -│ -├── interpreters/ # Language interpreters with CNO detection -│ ├── rescript/ # Malbolge (ReScript) -│ ├── brainfuck/ # Brainfuck (Python) -│ └── whitespace/ # Whitespace (Python) -│ -├── docs/ # Comprehensive documentation -│ ├── theory.md # Theoretical foundations -│ ├── examples.md # CNO examples across languages -│ ├── proofs-guide.md # How to write proofs -│ └── philosophy.md # Epistemology of nothingness -│ -├── examples/ # CNO example programs -│ ├── malbolge/ -│ ├── brainfuck/ -│ └── whitespace/ -│ -├── tests/ # Comprehensive test suite -│ ├── unit/ -│ └── proofs/ -│ -├── papers/ # Research paper drafts -│ └── .latex/ -│ -├── Justfile # Build automation -├── Containerfile # Containerized verification (Podman/Docker) -├── VERIFICATION.md # Detailed verification status -└── .gitlab-ci.yml # CI/CD pipeline -``` - -## Quick Start - -### Prerequisites - -**Fedora**: -```bash -sudo dnf install coq z3 nodejs opam just -npm install -g rescript@11.1 -``` - -**Ubuntu**: -```bash -sudo apt install coq z3 nodejs npm -npm install -g rescript@11.1 -cargo install just -``` - -### Build - -```bash -# Install dependencies -npm install - -# Build everything -just build-all - -# Verify all proofs -just verify-all - -# Run tests -just test-all -``` - -### Container (Podman/Docker) - -```bash -# Build image (Podman recommended) -podman build -t absolute-zero . - -# Run verification -podman run --rm absolute-zero just verify-all - -# Docker compatibility -docker build -t absolute-zero . -docker run --rm absolute-zero just verify-all -``` - -## What is a CNO? - -A **Certified Null Operation** is a program with the following formally proven properties: - -### Formal Definition - -``` -∀ σ : ProgramState, ∀ p : Program, - IsCNO(p) ↔ ( - Terminates(p, σ) ∧ // Always halts - FinalState(p, σ) = σ ∧ // Identity mapping - NoSideEffects(p) ∧ // Pure - ThermodynamicallyReversible(p) // Zero energy - ) -``` - -### Properties - -- ✅ **Termination**: Always halts -- ✅ **State Preservation**: Input state = Output state -- ✅ **Purity**: No I/O, no memory allocation -- ✅ **Reversibility**: Can be undone with zero energy cost (Landauer's principle) - -## Examples - -### Brainfuck CNOs - -```brainfuck - (empty program) ->< (move right then left) -+- (increment then decrement) ->><< (multiple balanced moves) -``` - -### Whitespace CNOs - -```whitespace - - -(three linefeeds = immediate halt) -``` - -### Malbolge CNOs - -```malbolge -(empty program - "Absolute Zero") -``` - -## Multi-Prover Verification - -For maximum confidence, we verify CNO properties in **six independent proof systems**: - -| Proof System | Foundation | Lines of Proof | Status | -|-------------|------------|----------------|--------| -| **Coq 8.19** | Constructive type theory | ~4000+ | ✅ 81 Qed, 19 Admitted (81% complete) | -| **Z3 4.13** | SMT solving | ~400 | ✅ Complete (10 theorems encoded, awaiting `z3`) | -| **Lean 4** | Dependent type theory | ~2500+ | ✅ Phases 1-4 complete | -| **Agda 2.6** | Dependent types | ~400 | ✅ Phase 1 complete (awaiting `agda`) | -| **Isabelle/HOL** | Higher-order logic | ~350 | ✅ Phase 1 complete (awaiting `isabelle`) | -| **Mizar** | Set theory | ~300 | ⚠️ Requires complex installation | - -### Verification Phases - -**🎉 Phase 1 Complete**: All core composition theorems proven, all proofs syntax-complete. - -**🎉 Phase 2-4 Complete**: Advanced theoretical foundations implemented (5 new modules, 10 proof files, ~3500 lines): - -| Module | Description | Coq | Lean 4 | -|--------|-------------|-----|--------| -| **Statistical Mechanics** | Landauer's Principle, thermodynamic reversibility | ✅ 9 Qed | ✅ | -| **Category Theory** | Universal CNO definition, model independence | ✅ 8 Qed | ✅ | -| **Lambda Calculus** | Functional programming CNOs | ✅ 9 Qed, 1 Admitted | ✅ | -| **Quantum Computing** | Quantum gates, unitary operations | 🟡 12 Qed, 5 Admitted | ✅ | -| **Filesystem Operations** | Valence Shell integration, practical CNOs | 🟡 8 Qed, 6 Admitted | ✅ | - -See [VERIFICATION.md](VERIFICATION.md) for detailed status and [PROOF-INSIGHTS.md](PROOF-INSIGHTS.md) for proof engineering knowledge. - -**Coq Proof Status** (2026-02-05): 81 Qed / 19 Admitted / 6 Defined / 63 Axioms across 10 files. 4 files fully complete (CNO.v, CNOCategory.v, StatMech.v, StatMech_helpers.v). - -**Next Step**: Complete remaining 19 Admitted proofs, then build container for machine verification. - -## Research Contributions - -### Core Formalization (Phase 1) -1. **Formal Definitions**: Define CNOs in 6 proof systems -2. **Multi-Prover Verification**: Cross-validate results across Coq, Z3, Lean 4, Agda, Isabelle -3. **Composition Theorems**: Prove CNOs compose under sequential execution -4. **Malbolge Verification**: Prove esoteric language CNOs -5. **Complexity Analysis**: Show CNO verification is undecidable in general - -### Advanced Foundations (Phases 2-4) -6. **Thermodynamic Formalization**: Rigorous Landauer's Principle and Bennett's reversible computing -7. **Category Theory**: Universal CNO definition as identity morphisms, model independence proofs -8. **Lambda Calculus**: Prove identity function (λx.x) is canonical functional CNO -9. **Quantum Computing**: Extend CNO theory to quantum gates and circuits -10. **Filesystem Operations**: Integrate Valence Shell reversibility, prove real-world CNOs - -### Practical Applications -11. **Secure Sandboxing**: Run untrusted code proven inert -12. **Compiler Optimization**: Detect and eliminate dead code -13. **Energy-Efficient Computing**: Baseline for zero-energy computation -14. **Transaction Systems**: Prove database rollback operations are CNOs - -## Theoretical Foundations - -### Landauer's Principle - -**Landauer's Principle** (1961): Erasing one bit of information dissipates at least `kT ln 2` of energy. - -At room temperature (300K): -``` -E_min = kT ln 2 ≈ 2.87 × 10⁻²¹ Joules per bit -``` - -**Implication**: A CNO dissipates **zero energy** because it erases no information. - -### Computational Complexity - -**Theorem**: The problem "Is program p a CNO?" is **undecidable** in general. - -**Proof**: Reduction from the halting problem. - -For finite-state programs with bounded execution, CNO verification is decidable. - -## Applications - -### 1. Secure Sandboxing - -Run untrusted code proven to be inert: -```python -if verify_cno(untrusted_program): - allow_execution(untrusted_program) -``` - -### 2. Compiler Optimization - -Detect and eliminate dead code: -```c -x = x + 1; -x = x - 1; // CNO: both lines can be eliminated -``` - -### 3. Reversible Computing - -CNOs establish baseline for zero-energy computation. - -### 4. Formal Methods Education - -CNOs provide pedagogical examples for learning proof assistants. - -## Development - -### Build Commands - -```bash -just build-all # Build everything -just verify-all # Verify all proofs -just test-all # Run all tests -just clean # Clean build artifacts -just stats # Project statistics -``` - -### Testing - -```bash -# Python tests -python3 tests/unit/test_cno_properties.py - -# Interpreter tests -python3 interpreters/brainfuck/brainfuck.py -python3 interpreters/whitespace/whitespace.py - -# Proof verification -just verify-coq -just verify-z3 -``` - -### CI/CD - -GitLab CI automatically: -- Builds all proofs in multiple systems -- Runs interpreter tests -- Verifies CNO properties -- Deploys documentation - -## Documentation - -- **[Theory](docs/theory.md)**: Mathematical foundations -- **[Examples](docs/examples.md)**: CNO examples across languages -- **[Proofs Guide](docs/proofs-guide.md)**: How to write proofs -- **[Philosophy](docs/philosophy.md)**: Epistemology of nothingness -- **[CLAUDE.md](CLAUDE.md)**: AI assistant guide - -## License - -**SPDX-License-Identifier: CC-BY-SA-4.0** - -This project is licensed under the **Palimpsest-MPL License 1.0 or later** (MPL-2.0). - -The Palimpsest-MPL extends Mozilla Public License 2.0 with provisions for ethical use, post-quantum cryptographic provenance, and emotional lineage protection. - -See [LICENSE](LICENSE) for full license text. - -**Fallback**: Where platform requirements mandate OSI-approved licenses, MPL-2.0 may be used with appropriate documentation. - -## Citation - -If you use Absolute Zero in research, please cite: - -```bibtex -@misc{jewell2025absolute, - title={Absolute Zero: Formal Verification of Certified Null Operations}, - author={Jewell, Jonathan D. A.}, - year={2025}, - publisher={GitLab}, - howpublished={\url{https://gitlab.com/maa-framework/6-the-foundation/absolute-zero}}, - note={Coq and Z3 verification of computationally inert programs} -} -``` - -## Related Work - -- **CompCert**: Formally verified C compiler (Isabelle/HOL) -- **seL4**: Formally verified microkernel -- **Landauer, R. (1961)**: "Irreversibility and Heat Generation" -- **Bennett, C. (1973)**: "Logical Reversibility of Computation" - -## Research Questions - -1. **Classification**: Can we classify all CNOs up to equivalence? -2. **Complexity**: What is the computational complexity of CNO verification? -3. **Obfuscation**: What's the most complex program provable as a CNO? -4. **Language-Independence**: Do CNO properties hold across languages? -5. **Quantum CNOs**: What does "null operation" mean for quantum programs? - -## Contributing - -Contributions welcome! See [CONTRIBUTING.md](CONTRIBUTING.md). - -Areas of interest: -- Proof engineering (porting to other assistants) -- PL theory (new CNO classifications) -- Esoteric languages (more test cases) -- Applications (practical use cases) - -## Contact - -**Jonathan D. A. Jewell** -- GitLab: [@hyperpolymath](https://gitlab.com/hyperpolymath) -- GitHub: [@Hyperpolymath](https://github.com/Hyperpolymath) -- Email: jonathan@metadatastician.art - -## Acknowledgments - -- **Ben Olmstead**: Creator of Malbolge -- **Coq Development Team**: Excellent proof assistant -- **Microsoft Research**: Z3 SMT solver -- **Proof Assistant Communities**: Lean, Agda, Isabelle, Mizar - ---- - -**Status**: Proofs verified ✓ | Theorems established ✓ | More work to do ✓ - -*"Absolute Zero does nothing. But in doing nothing, it does everything."* diff --git a/README.md b/README.md index e74a0d6..bf47767 100644 --- a/README.md +++ b/README.md @@ -1,5 +1,484 @@ -Canonical README: [README.adoc](README.adoc) — this pointer file replaces a near-duplicate copy (deduplicated 2026-06-12; unique content, the sponsor badge, was folded into README.adoc). + +**Formal Verification of Null Operations — two co-equal pillars: +Certified Null *Effect* (CNO) and Certified Null *Disclosure* (OND)** + +![MPL-2](https://img.shields.io/badge/License-MPL_2.0-blue.svg) [![Sponsor](https://img.shields.io/badge/Sponsor-%E2%9D%A4-pink?logo=github)](https://github.com/sponsors/hyperpolymath) + +# License + +[![OpenSSF Best Practices](https://img.shields.io/badge/OpenSSF-Best_Practices-green?logo=openssourcesecurity)](https://www.bestpractices.dev/en/projects/new?repo_url=https://github.com/hyperpolymath/absolute-zero) + +**SPDX-License-Identifier: CC-BY-SA-4.0** + +Licensed under the Mozilla Public License 2.0. See `LICENSE` for full +text. + +\> **"The universe tends toward maximum entropy. A Certified Null +Operation is a pocket of perfect computational order—a program that +resists the Second Law. It does nothing, but in doing nothing, it says +everything about the structure of computation itself."** + +# What is Absolute Zero? + +**Absolute Zero** is a research project exploring the formal +verification of programs that provably compute nothing. We formalize +**Certified Null Operations (CNOs)**—programs that, despite potentially +complex implementations, can be mathematically proven to have zero net +computational effect. + +**The Central Question**: Can we formally prove that a program does +absolutely nothing? + +This seemingly trivial question leads to deep insights in: - **Formal +Verification**: Machine-checked proofs of program properties - +**Computational Complexity**: Understanding minimal computation - +**Reversible Computing**: Programs preserving thermodynamic +reversibility - **Esoteric Languages**: Using Malbolge as +proof-of-concept + +# The Two Pillars + +Absolute Zero rests on **two co-equal pillars** — siblings of equal +weight, not parent and extension: + +- **CNO — Certified Null *Effect*.** Certifies that an operation does + **nothing to the world**: it terminates, maps input state to identical + output state, is functionally pure, and is thermodynamically + reversible (Landauer/Bennett). The conserved quantity is **state**. + +- **OND — Observational Null Operations / Certified Null *Disclosure*.** + Certifies that an operation **reveals nothing about its secret input** + to a declared observer: its observable trace (timing, size, …) is + constant over the secret, **relative to a declared observation model + `O`**. The conserved quantity is the **secret→observable channel**. + +The two are **logically independent** (neither entails the other — a +proved theorem, with witnesses), connected by a single **coupling dial** +between a thing and the trace it casts to an observer: CNO witnesses the +emptiness of the **effect** channel; OND certifies the emptiness of the +**disclosure** channel (this dial is **framing, not theorem**). Unlike +CNO — which lives entirely inside the formal model — every OND claim is +**conditional on its declared `O`** and ships a **residue list** of +out-of-scope observables: the honest, explicit boundary between the +proof and the physical metal. + +See **`docs/TWO-PILLARS.adoc`** (narrative), **`docs/OND-ROADMAP.adoc`** +(prioritised obligations), and **`docs/OND-PILLAR-STRUCTURE.adoc`** +(module layout). The OND pillar is at the design/roadmap stage: its +obligations are currently `Admitted` (specified, not yet proved) — the +normal honest starting state, mirroring CNO’s own open obligations. + +# Project Structure + + absolute-zero/ + ├── proofs/ # Formal proofs in multiple systems + │ ├── coq/ # Coq proofs (constructive) + │ │ ├── common/ # Core CNO framework (CNO.v) + │ │ ├── malbolge/ # Malbolge-specific proofs + │ │ ├── physics/ # Statistical mechanics (StatMech.v) + │ │ ├── category/ # Category theory (CNOCategory.v) + │ │ ├── lambda/ # Lambda calculus (LambdaCNO.v) + │ │ ├── quantum/ # Quantum computing (QuantumCNO.v) + │ │ └── filesystem/ # Filesystem CNOs (FilesystemCNO.v) + │ │ + │ ├── lean4/ # Lean 4 proofs (modern) + │ │ ├── CNO.lean # Core CNO framework + │ │ ├── StatMech.lean # Statistical mechanics + │ │ ├── CNOCategory.lean # Category theory + │ │ ├── LambdaCNO.lean # Lambda calculus + │ │ ├── QuantumCNO.lean # Quantum computing + │ │ └── FilesystemCNO.lean # Filesystem operations + │ │ + │ ├── z3/ # Z3 SMT verification (automated) + │ ├── agda/ # Agda proofs (dependent types) + │ ├── isabelle/ # Isabelle/HOL (production-grade) + │ ├── mizar/ # Mizar proofs (mathematical library) + │ ├── observation-models/ # OND: declared observation models O (proof inputs) + │ └── residue/ # OND: residue lists (the model-vs-metal gap) + │ # Each prover dir hosts CNO.* and (to author) co-equal OND.* modules + │ + ├── interpreters/ # Language interpreters with CNO detection + │ ├── rescript/ # Malbolge (ReScript) + │ ├── brainfuck/ # Brainfuck (Python) + │ └── whitespace/ # Whitespace (Python) + │ + ├── docs/ # Comprehensive documentation + │ ├── theory.md # Theoretical foundations + │ ├── examples.md # CNO examples across languages + │ ├── proofs-guide.md # How to write proofs + │ └── philosophy.md # Epistemology of nothingness + │ + ├── examples/ # CNO example programs + │ ├── malbolge/ + │ ├── brainfuck/ + │ └── whitespace/ + │ + ├── tests/ # Comprehensive test suite + │ ├── unit/ + │ └── proofs/ + │ + ├── papers/ # Research paper drafts + │ └── .latex/ + │ + ├── Justfile # Build automation + ├── Containerfile # Containerized verification (Podman/Docker) + ├── VERIFICATION.md # Detailed verification status + └── .gitlab-ci.yml # CI/CD pipeline + +# Quick Start + +## Prerequisites + +**Fedora**: `` `bash `` `sudo` `dnf` `install` `coq` `z3` `nodejs` +`opam` `just` `npm` `install` `-g` `rescript@11.1`\` + +**Ubuntu**: `` `bash `` `sudo` `apt` `install` `coq` `z3` `nodejs` `npm` +`npm` `install` `-g` `rescript@11.1` `cargo` `install` `just`\` + +## Build + +```bash +# Install dependencies +npm install + +# Build everything +just build-all + +# Verify all proofs +just verify-all + +# Run tests +just test-all +``` + +## Container (Podman/Docker) + +```bash +# Build image (Podman recommended) +podman build -t absolute-zero . + +# Run verification +podman run --rm absolute-zero just verify-all + +# Docker compatibility +docker build -t absolute-zero . +docker run --rm absolute-zero just verify-all +``` + +# What is a CNO? + +A **Certified Null Operation** is a program with the following formally +proven properties: + +## Formal Definition + + ∀ σ : ProgramState, ∀ p : Program, + IsCNO(p) ↔ ( + Terminates(p, σ) ∧ // Always halts + FinalState(p, σ) = σ ∧ // Identity mapping + NoSideEffects(p) ∧ // Pure + ThermodynamicallyReversible(p) // Zero energy + ) + +## Properties + +- ✅ **Termination**: Always halts + +- ✅ **State Preservation**: Input state = Output state + +- ✅ **Purity**: No I/O, no memory allocation + +- ✅ **Reversibility**: Can be undone with zero energy cost (Landauer’s + principle) + +# Examples + +## Brainfuck CNOs + +```brainfuck + (empty program) +>< (move right then left) ++- (increment then decrement) +>><< (multiple balanced moves) +``` + +## Whitespace CNOs + +```whitespace + +(three linefeeds = immediate halt) +``` + +## Malbolge CNOs + +```malbolge +(empty program - "Absolute Zero") +``` + +# Multi-Prover Verification + +For maximum confidence, we verify CNO properties in **six independent +proof systems**: + +\| Proof System \| Foundation \| Lines of Proof \| Status \| +\|-------------\|------------\|----------------\|--------\| \| **Coq +8.19** \| Constructive type theory \| ~4000+ \| ✅ 81 Qed, 19 Admitted +(81% complete) \| \| **Z3 4.13** \| SMT solving \| ~400 \| ✅ Complete +(10 theorems encoded, awaiting `z3`) \| \| **Lean 4** \| Dependent type +theory \| ~2500+ \| ✅ Phases 1-4 complete \| \| **Agda 2.6** \| +Dependent types \| ~400 \| ✅ Phase 1 complete (awaiting `agda`) \| \| +**Isabelle/HOL** \| Higher-order logic \| ~350 \| ✅ Phase 1 complete +(awaiting `isabelle`) \| \| **Mizar** \| Set theory \| ~300 \| ⚠️ +Requires complex installation \| + +## Verification Phases + +**🎉 Phase 1 Complete**: All core composition theorems proven, all +proofs syntax-complete. + +**🎉 Phase 2-4 Complete**: Advanced theoretical foundations implemented +(5 new modules, 10 proof files, ~3500 lines): + +\| Module \| Description \| Coq \| Lean 4 \| +\|--------\|-------------\|-----\|--------\| \| **Statistical +Mechanics** \| Landauer’s Principle, thermodynamic reversibility \| ✅ 9 +Qed \| ✅ \| \| **Category Theory** \| Universal CNO definition, model +independence \| ✅ 8 Qed \| ✅ \| \| **Lambda Calculus** \| Functional +programming CNOs \| ✅ 9 Qed, 1 Admitted \| ✅ \| \| **Quantum +Computing** \| Quantum gates, unitary operations \| 🟡 12 Qed, 5 +Admitted \| ✅ \| \| **Filesystem Operations** \| Valence Shell +integration, practical CNOs \| 🟡 8 Qed, 6 Admitted \| ✅ \| + +See \[VERIFICATION.md\](VERIFICATION.md) for detailed status and +\[PROOF-INSIGHTS.md\](PROOF-INSIGHTS.md) for proof engineering +knowledge. + +**Coq Proof Status** (2026-02-05): 81 Qed / 19 Admitted / 6 Defined / 63 +Axioms across 10 files. 4 files fully complete (CNO.v, CNOCategory.v, +StatMech.v, StatMech_helpers.v). + +**Next Step**: Complete remaining 19 Admitted proofs, then build +container for machine verification. + +# Research Contributions + +## Core Formalization (Phase 1) + +1. **Formal Definitions**: Define CNOs in 6 proof systems + +2. **Multi-Prover Verification**: Cross-validate results across Coq, + Z3, Lean 4, Agda, Isabelle + +3. **Composition Theorems**: Prove CNOs compose under sequential + execution + +4. **Malbolge Verification**: Prove esoteric language CNOs + +5. **Complexity Analysis**: Show CNO verification is undecidable in + general + +## Advanced Foundations (Phases 2-4) + +6. **Thermodynamic Formalization**: Rigorous Landauer’s Principle and + Bennett’s reversible computing + +7. **Category Theory**: Universal CNO definition as identity morphisms, + model independence proofs + +8. **Lambda Calculus**: Prove identity function (λx.x) is canonical + functional CNO + +9. **Quantum Computing**: Extend CNO theory to quantum gates and + circuits + +10. **Filesystem Operations**: Integrate Valence Shell reversibility, + prove real-world CNOs + +## Practical Applications + +11. **Secure Sandboxing**: Run untrusted code proven inert + +12. **Compiler Optimization**: Detect and eliminate dead code + +13. **Energy-Efficient Computing**: Baseline for zero-energy computation + +14. **Transaction Systems**: Prove database rollback operations are CNOs + +# Theoretical Foundations + +## Landauer’s Principle + +**Landauer’s Principle** (1961): Erasing one bit of information +dissipates at least `kT` `ln` `2` of energy. + +At room temperature (300K): `` ` `` `E_min` `=` `kT` `ln` `2` `≈` `2.87` +`×` `10⁻²¹` `Joules` `per` `bit`\` + +**Implication**: A CNO dissipates **zero energy** because it erases no +information. + +## Computational Complexity + +**Theorem**: The problem "Is program p a CNO?" is **undecidable** in +general. + +**Proof**: Reduction from the halting problem. + +For finite-state programs with bounded execution, CNO verification is +decidable. + +# Applications + +## 1. Secure Sandboxing + +Run untrusted code proven to be inert: `` `python `` `if` +`verify_cno(untrusted_program):` `allow_execution(untrusted_program)`\` + +## 2. Compiler Optimization + +Detect and eliminate dead code: `` `c `` `x` `=` `x` `+` `1;` `x` `=` +`x` `-` `1;` \` + +## 3. Reversible Computing + +CNOs establish baseline for zero-energy computation. + +## 4. Formal Methods Education + +CNOs provide pedagogical examples for learning proof assistants. + +# Development + +## Build Commands + +```bash +just build-all # Build everything +just verify-all # Verify all proofs +just test-all # Run all tests +just clean # Clean build artifacts +just stats # Project statistics +``` + +## Testing + +```bash +# Python tests +python3 tests/unit/test_cno_properties.py + +# Interpreter tests +python3 interpreters/brainfuck/brainfuck.py +python3 interpreters/whitespace/whitespace.py + +# Proof verification +just verify-coq +just verify-z3 +``` + +## CI/CD + +GitLab CI automatically: - Builds all proofs in multiple systems - Runs +interpreter tests - Verifies CNO properties - Deploys documentation + +# Documentation + +- **\[Theory\](docs/theory.md)**: Mathematical foundations + +- **\[Examples\](docs/examples.md)**: CNO examples across languages + +- **\[Proofs Guide\](docs/proofs-guide.md)**: How to write proofs + +- **\[Philosophy\](docs/philosophy.md)**: Epistemology of nothingness + +- **\[CLAUDE.md\](CLAUDE.md)**: AI assistant guide + +# License + +**SPDX-License-Identifier: CC-BY-SA-4.0** + +This project is licensed under the **Palimpsest-MPL License 1.0 or +later** (MPL-2.0). + +The Palimpsest-MPL extends Mozilla Public License 2.0 with provisions +for ethical use, post-quantum cryptographic provenance, and emotional +lineage protection. + +See \[LICENSE\](LICENSE) for full license text. + +**Fallback**: Where platform requirements mandate OSI-approved licenses, +MPL-2.0 may be used with appropriate documentation. + +# Citation + +If you use Absolute Zero in research, please cite: + +```bibtex +@misc{jewell2025absolute, + title={Absolute Zero: Formal Verification of Certified Null Operations}, + author={Jewell, Jonathan D. A.}, + year={2025}, + publisher={GitLab}, + howpublished={\url{https://gitlab.com/maa-framework/6-the-foundation/absolute-zero}}, + note={Coq and Z3 verification of computationally inert programs} +} +``` + +# Related Work + +- **CompCert**: Formally verified C compiler (Isabelle/HOL) + +- **seL4**: Formally verified microkernel + +- **Landauer, R. (1961)**: "Irreversibility and Heat Generation" + +- **Bennett, C. (1973)**: "Logical Reversibility of Computation" + +# Research Questions + +1. **Classification**: Can we classify all CNOs up to equivalence? + +2. **Complexity**: What is the computational complexity of CNO + verification? + +3. **Obfuscation**: What’s the most complex program provable as a CNO? + +4. **Language-Independence**: Do CNO properties hold across languages? + +5. **Quantum CNOs**: What does "null operation" mean for quantum + programs? + +# Contributing + +Contributions welcome! See \[CONTRIBUTING.md\](CONTRIBUTING.md). + +Areas of interest: - Proof engineering (porting to other assistants) - +PL theory (new CNO classifications) - Esoteric languages (more test +cases) - Applications (practical use cases) + +# Contact + +**Jonathan D. A. Jewell** - GitLab: +\[@hyperpolymath\]() - GitHub: +\[@Hyperpolymath\]() - Email: +[jonathan@metadatastician.art](jonathan@metadatastician.art) + +# Acknowledgments + +- **Ben Olmstead**: Creator of Malbolge + +- **Coq Development Team**: Excellent proof assistant + +- **Microsoft Research**: Z3 SMT solver + +- **Proof Assistant Communities**: Lean, Agda, Isabelle, Mizar + +------------------------------------------------------------------------ + +**Status**: Proofs verified ✓ \| Theorems established ✓ \| More work to +do ✓ + +**"Absolute Zero does nothing. But in doing nothing, it does +everything."**