The Gravitational Constant as a Uniqueness Constraint in Algorithmic Formula Discovery

J. Rogers, SE Ohio, 13 Jun 2025, 1205

(Building upon "A Categorical Theory of Measurement")

Abstract

We investigate the algorithmic role of the gravitational constant, G, within a categorical framework for physics. While our previous work established that G is a composite scaling morphism, not a fundamental entity, here we demonstrate its crucial function as a uniqueness constraint that prunes the solution space in automated formula discovery. By analyzing discovery queries with and without the G constraint, we show that G's unique dimensional signature ([M⁻¹ L³ T⁻²]) acts as a "key" that fits a specific "lock" in unit space, collapsing a multitude of possible physical laws into the single, correct gravitational relationship. This reveals that constants, while not always fundamental, serve an essential algorithmic purpose by providing the necessary context to resolve ambiguity and guide discovery engines to a unique, physically relevant solution.

1. Introduction: The Ambiguity of Under-Constrained Systems

In our "Categorical Theory of Measurement," we model physical law discovery as a functorial mapping from a set of input measurement axes to a target output axis. An algorithmic implementation of this, the "Formula Forge," seeks to find a composition of morphisms that connects these axes. A common challenge in this process is ambiguity: a simple query, such as discovering Force from Mass and Length, is under-constrained. The dimensional space allows for multiple valid solutions corresponding to different physical contexts (e.g., kinematic force, centripetal force). The system can generate multiple valid "Pi groups," but cannot determine which one is appropriate without further information.

This paper explores how physical constants, specifically the gravitational constant G, resolve this ambiguity. We will demonstrate that G's role is not merely to provide numerical scaling, but to introduce a powerful topological constraint in the dimensional search space, effectively isolating the unique solution corresponding to gravitational phenomena.

2. The Algebraic "Lock": Defining the Problem Space

Let us consider the query discover Force from Mass, Length. Our engine seeks to solve for the exponents a, b, ... in the general equation:

F = Π ⋅ mᵃ ⋅ Lᵇ ⋅ (other quantities...)

In the language of dimensional analysis, this is a system of linear equations in the vector space of exponents for the base dimensions [M, L, T]:

[M¹ L¹ T⁻²] = [M]ᵃ [L]ᵇ ⋅ ...

Without additional constraints, the engine finds multiple valid solutions by introducing other fundamental quantities from its library (e.g., time t or velocity c):

1. Kinematic Solution (F = ma): F = Π ⋅ m¹ ⋅ L¹ ⋅ t⁻²

2. Centripetal Solution (F = mv²/r): F = Π ⋅ m¹ ⋅ c² ⋅ L⁻¹

The engine correctly identifies that multiple "Pi groups" can be formed, but lacks the context to select one. The problem space contains a "lock"—a dimensional gap that must be filled—but there are multiple "keys" that could potentially fit.

3. The Uniqueness of the Key: The Dimensional Signature of G

The command "with gravitational_constant" fundamentally alters the query. It instructs the engine to limit its search to solutions that incorporate the specific "key" of G. The defining characteristic of this key is its unique dimensional "shape":

[G] = [M⁻¹ L³ T⁻²]

This signature is not arbitrary. It is the precise dimensional ratio required to bridge the gap between a static mass configuration (m² / L²) and the dynamic output of Force. Let us define the "lock shape" as the required transformation:

[Lock Shape] = [Target Output] / [Chosen Inputs]
[Lock Shape] = [Force] / [Mass² / Length²] = [M¹ L¹ T⁻²] / [M² L⁻²] = [M⁻¹ L³ T⁻²]

The engine performs a simple but powerful pattern-matching operation:

[Lock Shape] == [Key Shape]
[M⁻¹ L³ T⁻²] == [M⁻¹ L³ T⁻²]

The perfect match between the problem's "lock" and the constant's "key" immediately prunes the search tree. All other potential solutions (like the kinematic or centripetal forms) are discarded because they do not utilize the G key. The search space collapses from multiple possibilities to a single, unique solution.

4. G as a Composite Morphism: A Dynamic Key

Our framework reveals that G is a composite morphism: G = c³ ⋅ t_P² / Hz_kg. This internal structure explains its dynamic behavior in different contexts. For example, in the derivation of the Schwarzschild Radius (L_s = mG/c²), the c² in the denominator interacts with G's internal structure:

L_s = m ⋅ [c³ ⋅ t_P² / Hz_kg] / c² = m ⋅ [c¹ ⋅ t_P² / Hz_kg]

Here, the context of the problem (1/c²) has dynamically morphed the key before its use. This demonstrates that the constraint is not static; it is an operator that can be modified by the other inputs, creating a new, context-specific "key" that fits a different "lock."

5. Conclusion: Constants as Contextual Constraints

While G is not a fundamental entity in the same way as t_P or c, it serves an equally vital, albeit different, purpose. It is a contextual constraint object. Its unique dimensional shape, though emergent from the composite nature of SI units, provides an unambiguous guide for algorithmic discovery.

Our findings show:

1. Ambiguity is a Lack of Context: Multiple valid formulas can exist for the same set of base quantities. The "correct" formula is dependent on the physical context.

2. Constants Provide Context: Introducing a constant like G is the algorithmic equivalent of specifying the context—in this case, "the domain of Newtonian gravity."

3. The "Shape" of a Constant is a Uniqueness Filter: The dimensional signature of a constant is a powerful filter that drastically limits the search space, enabling efficient and unambiguous discovery of the relevant physical law.

Therefore, the role of constants in physics is twofold. They are, as we have previously shown, scaling factors that bridge our measurement systems to a natural reality. But they are also, as demonstrated here, essential navigational beacons in the vast dimensional space of physical law. They are the keys that unlock unique solutions from a landscape of infinite possibilities, making the algorithmic discovery of physics not just possible, but elegant and efficient.