Physical Constants as Inherent Semantic Operators for Constraining Automated Discovery

J. Rogers, SE Ohio, 12 Jun 2025, 2347

Abstract
We present a new framework for understanding the role of fundamental physical constants (G, h, k_B, c) in the structure of physical law. This framework reinterprets these constants not as passive numerical values, but as active mathematical operators whose dimensional signatures inherently define and constrain specific domains of physical phenomena. Within a generative model of physics where laws are derived by navigating a graph of dimensional nodes, a constant's unique dimensional structure functions as a unique "address" or "key." When a dimensional problem requires the specific transformation represented by a constant's signature, that constant is naturally and logically selected by the system. This process acts as an emergent semantic pruning mechanism, transforming a computationally intractable search for formulas into a highly efficient, context-aware pathfinding problem. This reveals that the semantic context of "gravity" or "quantum mechanics" is not information to be added to a system, but is already fully contained within the mathematical structure of G and h themselves.

1. Introduction: The Grammar of Discovery

The pursuit of an automated physics discovery engine confronts a primary challenge: the boundless space of possible mathematical relationships. A system that attempts to find new laws by naively combining base Jacobeans unit scalings between base axis of measurements and the time axis faces a combinatorial explosion. The key to tractable discovery lies in understanding the inherent grammar that governs the language of physics—a grammar that provides the rules and context to guide a search for meaningful "sentences" (i.e., physical laws).

This paper proposes that fundamental physical constants are the cornerstone of this grammar. We will demonstrate that their role transcends that of mere conversion factors. The unique dimensional signature of each constant is, in itself, a complete semantic operator that defines a domain of physical interaction. By building a system fluent in this dimensional grammar, we can achieve a highly efficient form of automated discovery where physical context emerges directly from the mathematics.

2. The Unit Space: A Graph of Physical Dimensions

We begin by modeling the landscape of physics as a graph of nodes and potential connections, the "Unit Space."

• Nodes: The fundamental axes of measurement, representing dimensions such as Mass [M], Length [L], Time [T], and Temperature [Θ].

• The Discovery Problem: The derivation of a physical formula is equivalent to finding a valid path or algebraic composition of nodes that connects a set of inputs to a target output. For example, deriving a formula for Force [M L T⁻²] from Mass [M] and Acceleration [L T⁻²].

In its raw form, this graph is noisy and its connections are computationally expensive to explore. The structure needed to navigate it efficiently is provided by the constants.

3. Constants as Unique Mathematical Operators

Our central thesis is that a fundamental constant is best understood as a unique mathematical operator defined entirely by its dimensional signature. This signature acts as a "key" that fits a specific "lock" in a dimensional equation. The constant doesn't represent a physical context; its mathematical form is the context.

Consider the primary constants:

•  (The Gravitational Operator): The dimensional signature [L³ M⁻¹ T⁻²] is a unique mathematical structure. It is precisely the operator required to transform a [M²/L²] term into a [Force] term ([M L T⁻²]). When a physical problem demands this specific transformation, G is not just the best choice; it is the only correct operator in the lexicon of fundamental constants. Its selection is a matter of mathematical necessity.

•  (The Quantum Operator): The signature [M L² T⁻¹] (Energy × Time) is the unique operator that bridges the Energy and Frequency [T⁻¹] domains. Any problem requiring a conversion between particle-like energy and wave-like frequency will naturally select h as the necessary mathematical bridge.

•  (The Thermodynamic Operator): The signature [M L² T⁻² Θ⁻¹] (Energy / Temperature) is the unique operator that relates the statistical energy of ensembles to the macroscopic property of temperature. It is the necessary term to solve any equation connecting these two domains.

4. Emergent Semantic Pruning: How the Grammar Works

The "semantic pruning" of the search space is not a programmed-in heuristic; it is an emergent property of this grammatical system. An automated "Formula Forge" engine operates as follows:

1. Problem Definition: The engine is tasked with finding a formula for a target quantity from a set of inputs (e.g., discover Force from Mass, Length).

2. Algebraic Requirement Analysis: The engine sets up a dimensional equation and determines the required dimensions of an unknown operator X that would solve the equation. For Force ~ X * Mass^a * Length^b, it searches for simple integer exponents a and b.

3. Operator Identification: Upon finding a simple combination (e.g., a=2, b=-2), it calculates the required dimensional signature for X: [X] = [Force] / [M² L⁻²] = [L³ M⁻¹ T⁻²].

4. Lexicon Matching: The engine searches its library of fundamental operators for one matching this exact signature. It finds G.

5. Formula Construction: The engine concludes that the simplest valid law involves the operator G and constructs the result: F = Π * G * M² / L².

The search was not "told" to look in the domain of gravity. The mathematical requirements of the problem itself led it inevitably to the gravitational operator G. The context was not an input to the search; it was the output of the dimensional algebra.

5. Conclusion: From Brute Force to Grammatical Fluency

Viewing physical constants as inherent semantic operators resolves the challenge of combinatorial explosion in automated discovery. It reframes the problem from a "brute-force" search to one of grammatical parsing.

This understanding provides a foundational principle for building the next generation of AI tools for physics. Such a tool does not need to be programmed with high-level human intuition about "gravity" or "quantum mechanics." It only needs to be made fluent in the language of dimensional algebra. By doing so, it will naturally deduce the correct context and select the appropriate physical constant, not because it has been taught the meaning, but because it understands the grammar.

The physical constants, therefore, are the pillars of this grammar. Their dimensional signatures are not arbitrary; they are the definitions of the fundamental verbs of physical law, and by understanding them as such, we can build systems that can learn to speak, and perhaps one day write, the language of the universe.