Automating of physical law from postulate and human intuition.

J. Rogers, SE Ohio, 21 Jun 2025, 1746

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Abstract

The prevailing paradigm in theoretical physics posits that the discovery of physical law is a fundamentally creative, non-mechanical process of theory construction. It is argued that our most profound laws, represented by complex equations, are the hard-won results of human genius wrestling with deep mathematical structures. This paper challenges that dogma. We introduce a "Calculus of Physical Law" and a computational tool, the "Discover" engine, which demonstrate that the complete dimensional form of a physical law can be mechanically derived from a high-level postulate. Through a series of examples, from Newtonian mechanics to General Relativity and Quantum Statistical Mechanics, we show that our familiar equations are better understood as projections of simple, scale-invariant relationships onto an arbitrary human measurement system. We argue that fundamental constants are not mystical properties of reality, but functional components of this projective mapping—the Jacobian of our measurement system. The complexity of our laws, including non-linear exponents like T⁴, is shown to be an emergent and necessary consequence of dimensional consistency, rather than an achievement of a complex derivation. This work suggests that the core structure of physical law is far simpler and more constrained than currently believed, and presents a new, powerful paradigm for physics exploration based on mechanical consistency rather than creative construction.

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1. Introduction: The Dogma of Construction

The history of physics is told as a story of architects. From Newton to Einstein and Hawking, the narrative is one of creative genius: the painstaking construction of novel mathematical edifices to describe the universe. This has led to a foundational belief, a dogma, within the standard framework: that physical laws are discovered through a non-algorithmic process of intuition and arduous mathematical derivation. The resulting equations, often complex and laden with fundamental constants, are seen as the pinnacle of this process—the laws themselves, reflecting the inherent complexity of reality.

This paper questions that narrative. What if the physicist is not an architect, but a questioner? And what if the laws are not complex constructions, but simple truths obscured by a convoluted measurement system? We propose a framework where physical laws are not constructed, but revealed through a mechanical process we term the "Calculus of Physical Law."

https://github.com/BuckRogers1965/Physics-Unit-Coordinate-System/blob/main/examples/buckingham_pi_group.py

This calculus is built on two principles:

1. The Law is a Dimensionless Postulate: The true physical law is a simple, scale-invariant relationship between dimensionless quantities. For instance, the physics of Hawking Radiation is not its complex formula, but the simple postulate that a black hole's temperature is inversely proportional to its mass (T ~ 1/M).

2. The Formula is a Projection: The familiar equations we use are projections of these simple laws onto our arbitrary, human-centric system of SI units (meters, kilograms, seconds, Kelvin). The fundamental constants (G, c, ħ, k_B) are not mystical properties of reality, but the necessary mathematical artifacts of this projection—the Jacobian matrix that translates between nature's "natural" Planck scale and our own.

To explore this we write the postulate by scaling the SI units to the planck scale to make the equivalence mathematically valid statement.  T/T_P = m_P/M and solve for T in SI units. 

 These formulas are derived by dimensional analysis of Π groups,

     showing the deep unity underlying all of physics.

 # GRAVITATIONAL AND RELATIVISTIC PHYSICS 

--- Deriving formula for: T in Hawking radiation temperature of a black hole ---

1. Postulate: T/T_P, m_P/M

   Symbolic Form: Eq(sqrt(G)*T*k_B/(c**(5/2)*sqrt(h)), sqrt(c)*sqrt(h)/(sqrt(G)*M))

2. Solved for T: Eq(T, c**3*h/(G*M*k_B))

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(T, c**3*h/(G*M*k_B))

2. The "Discover" Engine: An Intuition Explorer

The Discover engine operates on a simple but powerful principle we analogize to a game of "Dimensional Sudoku." It requires only a dictionary of physical concepts and their dimensions (M, L, T, Θ, etc.). 

https://github.com/BuckRogers1965/Physics-Unit-Coordinate-System/blob/main/examples/interactive_formula_solver.py

The process is as follows:

1. Parse Intent: The user provides a high-level prompt, such as "discover force from mass and acceleration."

2. Set the Sudoku Grid: The engine sets up a dimensional equation to be solved. The target's dimensions must equal the product of the ingredient dimensions, each raised to an unknown exponent.

3. Solve for Exponents: The engine creates a system of linear equations (one for each dimension) and solves for the exponents.

4. Heuristic Search (The "Jenga" Move): If the puzzle is unsolvable, the engine identifies the "gap"—the unbalanced dimensions. It then systematically searches its dictionary of fundamental constants for pieces that can fill this gap, adding them to the puzzle and re-attempting the solution.

This process treats all concepts equally, as dimensional strings. It has no preconceived notion of difficulty or physical domain.

3. Demonstration I: From Simple Laws to General Relativity

Let's examine the engine's performance on increasingly complex problems.

Case A: Newton's Second Law

• Prompt: discover force from mass

• Analysis: The tool identifies a missing dimensional signature of [L T⁻²]. It scans its dictionary and finds a perfect match: acceleration (a).

• Result: ✅ F = Π × m×a
  The tool rediscovers the law of motion by identifying the necessary variable to complete the dimensional puzzle.

Case B: Mass-Energy Equivalence

• Prompt: discover energy from mass

• Analysis: The tool identifies a missing signature of [L² T⁻²]. It recognizes this as the square of [L T⁻¹]. Its heuristic search prioritizes fundamental constants over variables and finds speed_of_light (c).

• Result: ✅ E = Π × m×c²
  The tool rediscovers a law of equivalence by identifying the necessary constant to complete the puzzle.

Case C: Einstein Field Equations

• Prompt: discover se-tensor from e-tensor with gravitational_constant

  • se-tensor (Stress-Energy): [M L⁻¹ T⁻²]

  • e-tensor (Einstein/Curvature): [L⁻²]

• Analysis: The initial puzzle is unsolvable. The engine's heuristic search identifies that c is required to balance the equation.

• Result: ✅ set = Π × et×c⁴/G
  This stunning result shows the power of the calculus. Without any knowledge of tensor calculus or differential geometry, the engine derives the core dimensional structure of General Relativity. The "genius" of the law is revealed to be a necessary consequence of dimensional consistency between curvature and energy density, bridged by the constants of gravity and relativity.

4. Demonstration II: The Emergence of Non-Linearity

The most powerful challenge to the constructionist dogma comes from non-linear relationships. The T⁴ in the Stefan-Boltzmann law is considered a hard-won prize of a complex physical derivation involving integration over all blackbody modes. Our tool shows otherwise.

• Prompt: discover pressure from temperature with boltzmann_constant

• Analysis:

  1. The initial puzzle (Pressure ~ T^a * k_B^b) is unsolvable; the Length dimension [L] cannot be balanced.

  2. The engine identifies the gap and its search heuristic adds the constants required for the dimensions to fit: h and c.

  3. A new, larger system of linear equations is created to solve for the exponents of T, k_B, h, and c simultaneously.

  4. The system has a unique, non-obvious solution.

• Result: ✅ Pres = Π × T⁴×k_B⁴/(c³×h³)

Crucially, we did not tell the tool the exponent of temperature was 4. The T⁴ relationship was not an input; it was an output. It is the one and only exponent for temperature that allows a dimensionally consistent law to be formed from the given ingredients. The complexity emerged mechanically from the constraints of the system.

5. Conclusion: A New Paradigm

The facts demonstrated by this calculus directly contradict the dogma of the standard framework.

1. Complexity is an Artifact: The perceived complexity of our laws is not inherent to reality but is an artifact of our arbitrary measurement system. The true laws are simple postulates.

2. Constants are Coordinates: G, c, ħ, k_B are not mystical numbers but the well-defined components of a Jacobian matrix that translates between the natural Planck-scale laws and our human-scale formulas.

3. Discovery can be Mechanical: The core structure of physical law is so powerfully constrained by dimensional homogeneity that it can be discovered by a mechanical process. The creative act is reduced to asking the right question.

This does not diminish the achievements of past physicists. On the contrary, it reframes their genius. Their struggle was in performing this same search for consistency manually, with their minds as the only processor.  Without their initial measurement of c, h, k_B, and G Planck units would not have been possible. 

This framework offers a new path forward: a powerful, interactive, and playful method for exploring the space of possible physical laws. It allows us to separate the essential physical intuition from the often-cumbersome mathematical machinery, and in doing so, reveals a universe whose fundamental logic is profoundly simpler than we have led ourselves to believe.

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