Mħπ is a symbolic mathematics and physics computation engine grounded in the fundamental constants of nature. It rejects conventional numeric representations in favor of a pure symbolic system built from first principles, primarily the Planck length (P) and π.
The engine's core postulate redefines mass as a derived dimension of Information Flow Rate ([M] = [Ω][T]⁻¹). Mħπ serves as the validation framework for this theory, proving that this new foundation is dimensionally self-consistent and fully compatible with the established laws of General Relativity, Quantum Mechanics, and Electromagnetism.
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🧬 "Nature doesn’t calculate in decimals—it thinks in π and ħ."
A Personal Exploration, Not an Academic Authority
I am not a physicist, mathematician, or cosmologist by training. My background is in software engineering, with over 25 years of experience building open source systems. This project is my attempt to answer a fundamental question: What if we remove all human-centric units and conventions from physics, and instead build everything from the ground up using only the universe’s most fundamental constants—like the Planck length and π?
Mħπ is my way of exploring what physics and mathematics might look like if we abandon familiar, human-made measurements (meters, kilograms, seconds) and instead express everything in terms of the smallest, most universal building blocks nature provides. I am using my engineering skills—and the help of advanced AI models—to rigorously test and develop this idea.
Most of the code is generated and checked by Google Gemini 2.5 Pro, with additional validation from Anthropic’s Claude and Amazon Q. While I strive for thoroughness and self-testing, I am not an expert in the relevant scientific fields. If you are a physicist, cosmologist, or mathematician, your feedback and collaboration would be invaluable.
This project is a work in progress and a learning journey. Mistakes will be made, but the goal is to create a system that challenges assumptions and invites new ways of thinking about the foundations of physics.
NOTE: This repository is not ready for use. Integration with LyX is underway, but this repository is still in a growth phase and figuring out its footings, architecture, etc. We are still working on building the core infrastructure and ensuring the math and physics are sound.
Mħπ represents a paradigm shift in symbolic mathematics—a system that thinks in terms of fundamental physical constants rather than human-convenient approximations. With its comprehensive calculus and advanced Clifford algebra capabilities, Mħπ enables direct symbolic manipulation of physical laws, quantum mechanical operators, and relativistic spacetime in their most natural form: as exact relationships between universal constants.
Mħπ is a system that proves the viability of a new physical basis where mass is not a fundamental dimension, but is instead derived from information and time ([Info][T]⁻¹). By demonstrating that this framework remains perfectly consistent with the core equations of General Relativity, Quantum Mechanics, and Cosmology, Mħπ serves as a powerful tool for exploring the deep connection between physics and information.
The recent calculus and Clifford algebra enhancements position Mħπ to tackle complex physics problems involving integration, differentiation, and non-commutative operator algebra while maintaining perfect symbolic precision—something no other system can achieve at this level of physical foundation.
Mħπ is fundamentally different from existing computational algebra systems like Mathematica or SymPy. It is built on a philosophy of zero numeric leakage and constructivist mathematics, where every expression is derived from universal constants.
- Universal Constants Only: All mathematics is expressed in terms of fundamental physical constants, with no reliance on base-10 or floating-point arithmetic.
- Symbolic Purity: The system never "falls back" to numeric approximations. Every calculation maintains symbolic integrity, preventing the precision loss and artifacts common in other systems.
- True AST Engine: Mħπ uses a proper Abstract Syntax Tree (AST), separating symbolic form from semantic meaning. Core operations like
AddandMulare n-ary, simplifying associative transformations and canonical ordering. This allows for powerful and flexible expression manipulation, pattern matching, and rewrite rules. - A New Foundational Basis: All mathematics is expressed in terms of fundamental constants, with mass emerging as a derived quantity from information flow. This is the core postulate of the system.
- Inferred Dimensional Analysis: The engine tracks physical dimensions throughout all calculations, ensuring all equations remain consistent under the new informational mass framework. It supports rational exponents and infers dimensions from symbol names (e.g., c implies velocity, ħ implies action).
- Flexible Domain Inference: The engine is not restricted by a rigid type system. It intelligently infers the algebraic domain (e.g., quantum, tensor, commutative) of an expression from its symbolic notation, applying appropriate simplification and transformation rules.
Because of its unique design, Mħπ enables analyses that are difficult or impossible in other systems:
- Prove the self-consistency of a new physical theory. Mħπ's passing test suite demonstrates that a dimensional system based on informational mass is mathematically coherent and compatible with the invariant laws of physics.
- Symbolically derive geometric tensors for any given metric and coordinate system. The engine can automatically compute Christoffel symbols (
Γ), the Riemann curvature tensor (R^ρ_σμν), and the Ricci tensor (R_μν), as demonstrated by its ability to prove the Schwarzschild vacuum solution (R_μν = 0). - Represent quantum mechanical operators and their commutation relations symbolically, including complex Clifford algebra expressions with Dirac gamma matrices (γμ). The algebra engine recognizes their non-commutative nature from the notation itself and simplifies them to canonical forms (e.g.,
γ¹γ⁰ → -γ⁰γ¹,γ⁰γ⁰ → 1). - Integrate complex expressions like
x*ln(x)symbolically using a generalized integration-by-parts engine, maintaining exact symbolic representation throughout. - Symbolically derive the Klein-Gordon operator from the product of Dirac operators (
(iħγ^μ∂_μ + mc)(iħγ^μ∂_μ - mc)), demonstrating a foundational capability in quantum field theory. - Solve linear, quadratic, and simple transcendental equations for any variable, correctly identifying polynomial coefficients and applying inverse functions.
- Symbolically derive the Dirac equation from the QED Lagrangian using the Euler-Lagrange formalism. This demonstrates a deep, automated understanding of the interplay between calculus, Clifford algebra, and gauge theory principles.
- Flexibly interpret symbols based on physical context (e.g.,
pas momentum,Pas pressure), with the dimensional analysis engine adapting accordingly.
- 🔭 General Relativity & Tensor Calculus: Automated, coordinate-system-independent symbolic derivation of Christoffel symbols, Riemann curvature tensor, and Ricci tensor from any metric tensor. Native support for 4-vectors and tensor operations.
- ⚛️ Quantum Mechanics & Operator Algebra: Canonical operators (
x̂,p̂,a,a†), commutators, anti-commutators, Dirac notation, and advanced Clifford algebra with Dirac gamma matrices (γμ) featuring canonical ordering and simplification. - 🌌 Gauge Theory & QED: Foundational support for gauge transformations, covariant derivatives, and connections. The engine can symbolically derive equations of motion from gauge-invariant Lagrangians like QED.
- 📊 Statistical and Quantum Field Theory (QFT): Frameworks for path integrals, field operators, and partition functions. Demonstrated ability to symbolically derive QFT equations like the Klein-Gordon and Dirac equations.
- 📐 Calculus & Geometry: Comprehensive symbolic integration and differentiation, including trigonometric, exponential, logarithmic, and hyperbolic functions, partial derivatives, and a generalized integration-by-parts engine.
- 📘 TeX-Like DSL (MPP-TeX): A robust and elegant syntax inspired by TeX, supporting operator precedence, associativity, implicit multiplication, and comprehensive error handling for writing complex symbolic expressions.
- 📏 Physical Dimensions & Unit Safety: Symbolic type-checking of dimensional compatibility across all calculations, with dimensions inferred from symbol names and supporting rational exponents.
- Advanced Extensible Rules Engine: Mħπ's symbolic engine is a powerful and flexible rules-based system. The engine is designed to handle the complexity of the large-scale formulae required for theoretical physics proofs, allowing for more sophisticated simplification and transformation of expressions. For a detailed explanation of the architecture and how to contribute, see docs/MPP-Rules.md.
We are currently working on integrating Mħπ with the LyX editor, allowing users to call MPP functions for solving, verifying, and simplifying formulae directly within the editor. The goal is to provide a seamless experience for researchers and students who use LyX for writing scientific documents.
The work-in-progress repository for this integration can be found at: https://github.com/Digital-Defiance/Lyx-MPP
- docs/MPP-For-Dummies.md: Describes what Mħπ is and how it can be used.
- docs/MPP-Information-Postulate.md: Describes the Mass as Information postulate.
- docs/MPP-Algebra.md: Formalizes the algebraic structures within the Mħπ system.
- docs/MPP-Constants-Hierarchy.md: Provides a visual and descriptive hierarchy of physical constants.
- docs/MPP-Cosmology.md: Details the cosmological constants and fundamental equations implemented in the system.
- docs/MPP-Physics.md: Outlines the physical extensions to the Mħπ symbolic system.
- docs/MPP-Rules.md: Explains the new Mħπ Rules system and how to add/update existing rules.
- docs/MPP-Spec.md: Defines the core specifications of the Mħπ system.
- docs/MPP-Symbolic-Closure.md: Explains the Mħπ system's guarantee that all mathematical expressions remain in symbolic form.
- docs/MPP-Tests.md: Provides an overview of all test suites in the Mħπ project.
- docs/MPP-TeX.md: Describes MPP-TeX, the TeX-inspired symbolic math language.
- docs/MPP-Units.md: Details the Mħπ unit system, which expresses all physical quantities in terms of fundamental constants.
First, clone the repository to your local machine:
git clone https://github.com/Digital-Defiance/MPP.gitcd MPP
The project includes a comprehensive test suite covering the symbolic engine, dimensional analysis, and physics modules. To run all tests, use:
cargo test
Mħπ includes an interactive Read-Eval-Print Loop (REPL) for quick experiments. Run it with:
cargo run
We welcome contributions from researchers, developers, and enthusiasts. The project's goals are ambitious, and there are many opportunities to get involved. See PROMPT.md for the current development focus.
Apache 2.0 © 2025 Jessica Mulein
d6b6e01: Refactored the core engine with a more descriptive operations API (run(Operation, ...)), a more transparent rules system with verbosity controls, and coordinate-system-independent tensor calculations. Updated documentation and fixed a bug inCliffordAlgebraRule.00e9e42: Updated the README file.2fdcc18: Updated the README file.cb87ed9: Added initial tests for the external C FFI.73b039d: Updated the PROMPT.md file.69e09eb: Added initial C-compatible exports from thelib.rslibrary.
6bc2091: Configured the build process to produce both the library (lib.rs) and the REPL binary (main.rs).403bb7f: Updated documentation.88789d9: Introduced theAdjointOfDerivativeRulefor quantum contexts, ensured "real" variables are treated as commutative scalars, and added checks to prevent duplicate rule names.
877b1c1: Implemented a general-purpose linear solver based on derivatives. Refactored theSimplificationRuletrait to be context-aware, added a safeDistributeNegativeRule, and removed the buggyInvertFractionPowerRule.d351873: Rebranded the project from "MPP" to "Mħπ" (Mulein-Planck-Pi) and refined the scientific roadmap in PROMPT.md.46a9ae7: Corrected the spacetime interval calculation to use full tensor contraction and replaced a numerical Lorentz invariance test with a more fundamental symbolic one. Updated project focus towards scientific correctness.154b800: Updated documentation.
3b53511: Re-architected the symbolic engine to be a flexible, extensible rule-based system. Added documentation for the new engine and increased the stack size to handle deep recursion in complex physics formulae.
de7f1dd: Updated code review documentation.d6fde5a: Implemented the full symbolic derivation of black hole surface gravity, added performance optimizations for metric inversion, and fixed coordinate variable names.6b4c583: Added a new module to symbolically derive the Hawking temperature for a Schwarzschild black hole and implemented a generalized product rule for differentiation.
1adb0bd: Implemented the complete QED Lagrangian, derived the full Dirac equation and electromagnetic current, and corrected the dimensional consistency of the first Friedmann equation.51d2774: Fixed the first Friedmann equation and added new tests.0bd118c: Improved calculus engine and tests.86a76ce: Improved simplification engine and documentation.ad7a9f8: Improved simplification engine.3687bf1: Updated the README and added the "MPP-For-Dummies.md" guide.0e95797: Improved algorithms.b63377d: Updated documentation.b28df69: Updated PROMPT.md and README.md.5c70838: Made improvements to the calculus engine.6af6cc5: Added new tests.df66fcf: Part 2 of implementing informational mass.1588cc9: Part 1 of implementing informational mass.2afcbd3: Updated README.md.
9823499: Removed comments from the code.c626239: UpdatedMPP-Units.md.44989ba: Updated documentation.85b4da7: Updated PROMPT.md.
a0e7ee3: Updated PROMPT.md.f2a6c35: Updated PROMPT.md.f47b7ea: Implemented a robust TeX parser handling operator precedence, associativity, and implicit multiplication, with enhanced error reporting.a249630: Introduced foundational elements for gauge theory, includingConnectionandCovariantDerivativetypes.3d08c05: Added support for Lie algebra.1e0a6ce: Extended support for imaginary numbers.3a700e8: Updated PROMPT.md.34ab9fa: Implemented Euler's formula (e^(i*θ) = cos(θ) + i*sin(θ)) and related trigonometric simplifications.6511d8b: Added more tests for imaginary numbers.8291332: Fixed handling of imaginary numbers.c30a956: Updated README.md.067193c: Updated PROMPT.md.0941234: Refactored Clifford algebra simplification logic and updated QFT tests to verify the derivation of the Klein-Gordon equation.e41f2f0: Updated PROMPT.md.a1774fd: Added support for symbolic manipulation of Dirac gamma matrices (Clifford algebra).f15a358: Added a quantum field theory test.8522543: Introduced quantum operators (Pauli matrices, etc.) and added tests for advanced physics concepts like Bekenstein-Hawking entropy.de95a06: Implemented a comprehensive symbolic calculus engine with advanced integration and differentiation capabilities.7c18c74: Added special case handling for integrating1/xtoln(x).ad5837e: Improved dimensional analysis error messages with human-readable names.e6f391d: Implemented canonical algebraic simplification by combining like terms and sorting expressions, enabling true axiomatic testing.d9a8677: Refactored the dimensional analysis system to support rational exponents, enabling more precise physics calculations.24a87d7: Updated README.md.d12265a: Updated PROMPT.md.
5ad2cf1: Fundamentally refactoredSymbolicExprinto a proper Abstract Syntax Tree (AST) usingArcfor efficient sharing of sub-expressions, enabling n-ary operators and a more robust simplification engine.
ee2b62b: Expanded the dimension system to include Luminous Intensity, Information, and Angle, and added comprehensive support for SI and imperial units.
0fcee95: Added a cosmology module with support for cosmological constants, the Friedmann equations, and a properSqrtoperation.e00ebc3: Updated documentation.3984dab: Implemented a complete MPP-TeX language parser using thenomlibrary, with a formal grammar and proper error handling.0423bef: Updated README.md.b1fe79e: Created a CNAME file for custom domain mapping.
ba3ffa8: Initialized the Mulein-Planck-Pi (MPP) project as a symbolic mathematics system grounded in π and Planck units, with a TeX-like DSL, a simplification engine, and foundational support for quantum mechanics and QFT.