On-Demand Physics Code Generation: How Fundamental Constants Function as Algorithmic Formula Synthesizers

 J. Rogers, SE Ohio, 12 Jun 2025, 2350

Abstract

We demonstrate a computational system that generates executable physics formulas on-demand from simple input-output specifications. By treating fundamental physical constants as dimensional operators rather than static values, the system synthesizes correct mathematical relationships in real-time without pre-programmed formula libraries. This approach reveals that physical constants function as algorithmic primitives in a meta-programming language for physics, capable of generating entire families of related equations from their dimensional signatures alone.

1. Introduction: From Static Formulas to Dynamic Code Generation

Traditional physics computation relies on libraries of pre-written formulas. If you need to calculate gravitational force, you look up F = GMm/r². If you need orbital velocity, you find v = √(GM/r). Each relationship requires separate implementation and maintenance.

We present a fundamentally different approach: a system that generates these formulas automatically from dimensional specifications. Instead of storing hundreds of physics equations, we store a small set of dimensional operators (fundamental constants) that can synthesize formulas on demand.

2. The System in Action: Live Formula Generation

Consider these real interactions with our Formula Forge system:

Example 1: Force Generation

User Request: > discover force from mass, length with gravitational_constant
System Response: ✅ F = Π × m²×G/L²

The system didn't retrieve this formula from a database. It generated it by:

1. Recognizing that force has dimensions [M L T⁻²]
2. Analyzing available inputs: mass [M], length [L]
3. Identifying that gravitational_constant G has signature [L³ M⁻¹ T⁻²]
4. Algebraically determining: [M L T⁻²] requires [M²] × [L³ M⁻¹ T⁻²] / [L²]
5. Synthesizing: F = Π × m²×G/L²

Example 2: Energy Generation from Same Inputs

User Request: > discover energy from mass, length with gravitational_constant  
System Response: ✅ E = Π × m²×G/L

Same operator (G), same inputs (mass, length), but different output type (energy [M L² T⁻²]) generates completely different code:

• Energy requires [M²] × [L³ M⁻¹ T⁻²] / [L¹] = [M L² T⁻²]
• Result: E = Π × m²×G/L

Example 3: Velocity with Square Root Emergence

User Request: > discover velocity from mass, length with gravitational_constant
System Response: ✅ v = Π × √m×√G/√L  

For velocity [L T⁻¹], the system determines:

• Required: [M¹] × [L³ M⁻¹ T⁻²] / [L²] = [L T⁻²]
• But velocity needs [L T⁻¹], requiring a square root operation
• Generates: v = Π × √(m×G/L)

Example 4: Automatic Constant Injection

User Request: > discover length from mass with gravitational_constant
System Processing: ⏳ Trying to add 'speed_of_light'...
System Response: ✅ L = Π × m×G/c²

Most remarkably, when asked for length [L] from only mass [M], the system:

1. Recognizes that G [L³ M⁻¹ T⁻²] and mass [M] cannot directly produce length [L]
2. Automatically searches for additional constants needed
3. Identifies that speed_of_light c [L T⁻¹] can complete the dimensional equation
4. Synthesizes the Schwarzschild radius formula: L = Π × m×G/c²

3. The Computational Mechanism: Constants as Code Generators

3.1 Dimensional Operators, Not Values

The key insight is treating constants not as numerical values but as dimensional operators:

• Traditional View: G = 6.67 × 10⁻¹¹ m³/kg·s² (a number with units)
• Our View: G = [L³ M⁻¹ T⁻²] (a dimensional transformation operator)

3.2 The Generation Algorithm

For each user request "discover [OUTPUT] from [INPUTS] with [CONSTANT]":

1. Parse Specifications: Extract dimensional signatures of desired output and available inputs
2. Set Up Dimensional Equation: [OUTPUT] = [INPUTS]^(a,b,c...) × [CONSTANT]^d × [ADDITIONAL]^e
3. Solve for Exponents: Find integer values a,b,c,d,e that balance dimensions
4. Check Constraints: If underdetermined, automatically inject additional constants
5. Synthesize Formula: Generate executable code with discovered exponents

3.3 Why This Works: Semantic Encoding in Dimensional Structure

The dimensional signature of each constant encodes specific semantic meaning:

• G [L³ M⁻¹ T⁻²]: "gravitational interaction operator"
• h [M L² T⁻¹]: "quantum discreteness operator"
• c [L T⁻¹]: "spacetime geometry operator"

These aren't arbitrary mathematical structures—they're the computational primitives that define how different physical quantities can interact.

4. Comparison: Generated vs. Traditional Formulas

Our system generates the same formulas that appear in physics textbooks, but derives them algorithmically:

Traditional Formula	Generated Formula	Input → Output
F = GMm/r²	F = Π × m²×G/L²	mass, length → force
U = -GMm/r	E = Π × m²×G/L	mass, length → energy
v = √(GM/r)	v = Π × √(m×G/L)	mass, length → velocity
Rs = 2GM/c²	L = Π × m×G/c²	mass → length
g = GM/r²	a = Π × m×G/L²	mass, length → acceleration

The system rediscovers fundamental physics relationships through pure dimensional reasoning.

5. Implications: A Meta-Programming Language for Physics

5.1 Physics as Executable Specifications

This approach treats physics problems as executable specifications:

• Input: "I want force from these quantities using this physical interaction"
• Output: Executable code that computes exactly what was requested

5.2 Constants as Algorithmic Primitives

Fundamental constants function as algorithmic primitives in a domain-specific language for physics:

// Constants are operators that generate code based on calling context
force_code = gravitational_constant(mass, mass, length) → F = Π × m²×G/L²
energy_code = gravitational_constant(mass, mass, length) → E = Π × m²×G/L  
velocity_code = gravitational_constant(mass, length) → v = Π × √(m×G/L)

5.3 Automatic Library Generation

Instead of maintaining vast libraries of physics formulas, we can generate them on demand. The entire gravitational physics library reduces to a single dimensional operator that synthesizes appropriate formulas when called.

6. Broader Applications

6.1 Educational Tools

Students could explore physics by asking "what if" questions and seeing formulas generate in real-time, building intuition about dimensional relationships.

6.2 Research Discovery

The system could discover new relationships by exploring novel combinations of inputs, outputs, and constants that haven't been explicitly studied.

6.3 Simulation Optimization

Physics simulations could generate optimal computational code for specific scenarios rather than using generic formula libraries.

7. Conclusion: The Grammar of Computable Physics

We have demonstrated that fundamental physical constants function as algorithmic operators capable of generating executable physics code on demand. This reveals a deeper computational structure underlying physical law: constants are not merely numbers but semantic operators in a meta-programming language for physics.

This approach transforms physics computation from formula lookup to algorithmic synthesis, suggesting new possibilities for automated discovery, education, and simulation. The physical constants, long recognized as fundamental to physics, emerge as equally fundamental to the computational representation of physical knowledge.

The system shows that the "grammar" of physics—the rules governing how physical quantities can meaningfully interact—is encoded directly in the dimensional structure of fundamental constants. By making computers fluent in this grammar, we enable them to generate correct physics code as naturally as they generate correct syntax in programming languages.

Future work will explore extending this approach to quantum mechanics (using Planck's constant h as the primary operator) and thermodynamics (using Boltzmann's constant k_B), demonstrating the universality of constants as computational primitives across all domains of physics.