MPP

Mħπ Symbolic Closure Guarantee

Overview

The Mħπ system provides a symbolic closure guarantee, which ensures that all mathematical expressions remain in symbolic form throughout all operations, unless explicit numeric evaluation is requested. This document outlines the principles, implementation, and verification of this guarantee.

Principles

  1. No Numeric Leakage: The system never introduces floating-point literals or numeric approximations during symbolic operations.
  2. Symbolic Representation: All mathematical expressions are represented using the SymbolicExpr enum, which provides a complete symbolic representation for all supported mathematical concepts.
  3. Constructivist Approach: All expressions derive from fundamental constants (π, P, τ, ħ, G, k) and remain in terms of these constants throughout all operations.
  4. Explicit Evaluation: Numeric evaluation is only performed when explicitly requested by the user, and is clearly separated from symbolic operations.

Implementation

The symbolic closure guarantee is implemented through several mechanisms:

  1. Symbolic Expression Type: The SymbolicExpr enum provides variants for all supported mathematical concepts, ensuring that all expressions can be represented symbolically.

  2. Simplification Rules: The simplifier in simplify.rs applies rewrite rules that preserve symbolic form. For example:

    • Rational numbers are simplified to their lowest terms but remain as Rational(a, b) or Const(n) (for integer results).
    • Operations with special values (0, 1, etc.) are simplified according to algebraic rules, but remain symbolic.
    • Derivatives and integrals are computed symbolically, with results expressed in terms of the original symbolic variables and constants.

  3. Algebraic Structure Preservation: The system respects the algebraic structures of different domains:

    • The commutative ring structure of basic arithmetic operations (Add, Mul, Pow).
    • The non-commutative algebra of quantum mechanical operations.
    • The tensor algebra of relativistic physics.

  4. Dimensional Analysis: Physical dimensions are tracked throughout all calculations, ensuring that expressions maintain dimensional consistency and remain in terms of fundamental constants.

Verification

The symbolic closure guarantee is verified through comprehensive testing:

  1. Unit Tests: The tests/symbolic_closure.rs file contains unit tests that verify symbolic closure for all operations:

    • Basic arithmetic operations (Add, Mul, Pow)
    • Calculus operations (Derive, Integral)
    • Quantum mechanical operations (Ket, Bra, BraKet)
    • Tensor operations (Tensor, Contract)
    • Spacetime operations (Four)
    • Dimensional analysis (Dimension)

  2. Property-Based Tests: The test suite includes property-based tests that verify symbolic closure for various combinations of expressions:

    • Addition of various symbolic expressions
    • Multiplication of various symbolic expressions
    • Exponentiation with various bases and exponents
    • Differentiation of various expressions
    • Integration of various expressions

  3. Comprehensive Path Coverage: The test suite exercises all code paths in the simplifier to ensure that symbolic closure is maintained regardless of the simplification path taken.

  4. Helper Function: The is_symbolic function in tests/symbolic_closure.rs provides a way to check if an expression contains any non-symbolic elements, and is used throughout the test suite to verify symbolic closure.

Examples

Here are some examples of how the symbolic closure guarantee works in practice:

  1. Rational Simplification:

    1/3 * 3 * π → π

    The simplifier recognizes that 1/3 * 3 = 1 and simplifies the expression to π, maintaining symbolic form.

  2. Derivative Calculation:

    d/dr (π * r^2) → 2 * π * r

    The derivative is computed symbolically, with the result expressed in terms of the original symbolic variables and constants.

  3. Quantum Mechanical Operations:

    |ψ⟩⟨φ|

    Quantum mechanical operations are represented symbolically, with no numeric approximations.

  4. Tensor Operations:

    T^μν_ρσ

    Tensor operations are represented symbolically, with indices and components maintained in symbolic form.

  5. Constant Relationships:

    c * τ → P
c * Δ → P

    Relationships between fundamental constants are recognized and simplified, but remain in symbolic form.

Conclusion

The symbolic closure guarantee is a fundamental aspect of the Mħπ system, ensuring that all mathematical expressions remain in symbolic form throughout all operations. This guarantee is essential for the system’s ability to provide exact, constructivist mathematics based on fundamental constants, without relying on base-10, binary, or floating-point arithmetic.

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