The Geometric Coupling Constant: Defining α/2π as the Invariant Ratio Between Charge Count and Time Experience

J. Rogers, SE Ohio

Abstract

The fine structure constant, α, has long been regarded as the most profound mystery in modern physics—a dimensionless number that dictates the strength of all electromagnetic interactions, yet lacks a clear physical origin. This paper argues that the mystery is an artifact of a flawed ontology. We demonstrate that the more fundamental, cycle-based constant κ_e = α/2π is not a number, but a fundamental ratio with a clear, mechanical meaning. By re-framing physics through the primacy of time experience and a mechanical model of the particle, we show that κ_e is the invariant ratio between a dimensionless cause (the Charge Count of an interaction) and its physical effect (the Radiated Time Experience, measured as the frequency of an emitted photon). This framework demystifies the fine structure constant, revealing it as the "gear ratio" or "transducer efficiency" of the fundamental interaction mechanism of the universe. It quantitatively measures the fundamental asymmetry between the baseline unity of gravity and the "offset" coupling of the electromagnetic interaction.

1. Introduction: The Mystery of α

The fine structure constant, α ≈ 1/137, is the cornerstone of quantum electrodynamics (QED). It is a pure number that appears ubiquitously, setting the scale for atomic energy levels, the probability of photon emission, and the strength of the electromagnetic force. Richard Feynman famously remarked, "It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man."

This mystery persists because the standard model treats α as a fundamental, empirically-determined input. Its value is known to extraordinary precision, but its physical meaning remains opaque. This paper posits that the mystery of α is a direct result of a category error: we have been treating a ratio as a number. By establishing a more physically intuitive, mechanically-grounded ontology, we can derive α/2π not as a given, but as a necessary consequence of the universe's internal architecture.

2. Foundational Principles: A Mechanistic Ontology

To understand α/2π, we must first state the axioms of the system in which it operates. This framework is built upon four principles:

The Primacy of Time Experience (τ): The fundamental reality is not spacetime, but a dynamic, geometric structure of local time rates. All physical quantities—mass, energy, momentum—are different manifestations of this "time experience" and the gradients (Δτ) between different states.

The Particle as a Two-Handle Engine: A fundamental particle (e.g., an electron) is not a point. It is a composite entity with two distinct, orthogonal interfaces or "handles"—mechanically analogous to a rigid body with two perpendicular lever arms:

• The Mass Handle: Couples directly to the geometry of the time field (gravity). It is the seat of inertia and defines the particle's global time experience relative to the cosmos (Σ(M/R)). Forces applied through this handle produce motion with no internal stress—the coupling efficiency is unity.

• The Charge Handle: Couples to the electromagnetic field through a lever arm mechanism. It is the interface for local, targeted forces. Forces applied through this handle create internal stress on the particle's structure, which must be transduced into motion. This transduction has a measured efficiency of k_e = α/2π.

Charge as a Dimensionless Count (q_count): Electric charge is not a substance or a separate dimension. It is a dimensionless integer (e=1) that "counts" the number of fundamental charge interfaces involved in an interaction.

Radiation as Jettisoned Time Experience: A photon is not a fundamental particle. It is a quantized, propagating record of a change in a particle's time experience (Δτ). It is emitted when an interaction is too violent for the particle's internal structure to absorb mechanically. The frequency (f) of the photon is the direct, physical measure of this jettisoned Δτ.

3. The Lever Arm Mechanism: Understanding the Bidirectional Transduction

The charge handle operates as a lever arm mechanism. This single mechanical structure explains both how electromagnetic forces produce motion AND how motion produces radiation. Understanding this bidirectional transduction is key to grasping the physical meaning of κ_e.

3.1. Forward Direction: Charge Interaction → Motion

Consider the fundamental electromagnetic force between two charges. The standard Coulomb force law can be rewritten to expose its local, multiplicative structure:

F = k_e × (q1/r) × (q2/r)

This is not mere algebraic manipulation. It reveals that each charge creates a local geometric stress field with intensity (q/r), and the force emerges from the product of these two local field intensities at the point of interaction, transduced by the constant k_e.

Mechanically, this can be understood as follows:

• The charge interaction creates stress on the lever arm
• This stress must be transduced through the particle's internal structure into actual motion of the mass center
• The transduction efficiency is k_e
• Motion through the mass handle (gravity) requires no such transduction—its efficiency is unity

3.2. Reverse Direction: Motion → Radiation

The same lever arm that transduces charge interactions into motion serves as a relief valve when internal stress cannot be absorbed mechanically. Consider an electron undergoing bremsstrahlung (braking radiation):

• The moving electron has a time experience τ_moving
• It interacts via its charge handle with a stationary target (τ_static)
• The mismatch in time experiences (Δτ) creates stress on the lever arm
• When this stress exceeds what the internal structure can absorb, the excess is jettisoned as a photon
• The photon's frequency f is the direct measure of this released Δτ
• The photon carries momentum, causing the electron to recoil

3.3. The Fundamental Ratio

From the reverse direction (radiation), we can identify the fundamental ratio. When a charge count q_count undergoes an interaction violent enough to produce radiation, the radiated photon has frequency f. The invariant ratio is:

κ_e = f / q_count

This is the central thesis of this paper. The constant κ_e is not a "magic number." It is the measured transduction efficiency of the charge lever arm mechanism—the same mechanism operating in both directions.

By comparing this with the established formulas of physics, we can identify κ_e as related to the fine structure constant by the geometric factor of a cycle: α = 2πκ_e. In natural units where the mass handle's coupling is unity, κ_e ≈ 0.00116 is simply the measured coupling efficiency of the charge handle.

4. Physical Interpretation: The Measured Coupling Ratio

This definition transforms α/2π from a mystery into a precise technical specification of the universe's fundamental interaction architecture.

4.1. A Universal Transduction Ratio

κ_e represents the measured coupling efficiency of the charge lever arm. Its value, κ_e ≈ 0.00116, is not "weak" or "strong"—it is simply the empirical ratio. Just as a gear ratio of 3:1 is neither good nor bad but simply describes the mechanism, κ_e is the measured fact of how charge interactions transduce into physical effects.

4.2. The Measure of a Fundamental Asymmetry

In the natural, geometric units of this framework, the coupling through the mass handle (gravity) is 1—it is direct, co-linear, producing no internal stress. The coupling through the charge handle (electromagnetism) is κ_e—it operates through a lever arm, creating internal stress that must be transduced.

The fact that κ_e ≠ 1 is the quantitative measure of the fundamental asymmetry between these two handles. It is not that one force is "stronger" or "weaker," but that they operate through different geometric mechanisms with different measured efficiencies:

F_gravity = 1 × (geometry of mass distribution)

F_EM = κ_e × (geometry of charge distribution)

4.3. The Bidirectional Nature

The same lever arm mechanism that appears in Coulomb's law (charge interaction → motion) appears in radiation formulas (motion → photon emission). This is not coincidental—it is the same physical structure operating in both directions:

Forward: (q1/r) × (q2/r) → [lever arm stressed] → F = κ_e × (local interactions)

Reverse: Δτ → [lever arm stressed] → [excess jettisoned] → photon with f = κ_e × q_count

The appearance of κ_e in both contexts confirms it as the fundamental transduction coefficient of the charge handle mechanism.

5. Conclusion: The Demystification of α

The fine structure constant α is not a fundamental number. The more fundamental physical constant is the Geometric Coupling Constant, κ_e = α/2π. This constant is not an arbitrary parameter, but a ratio with a clear, mechanical meaning:

κ_e is the measured transduction efficiency of the charge lever arm—the invariant ratio between charge count interactions and their physical effects, whether those effects manifest as forces or as radiated photons.

This insight dissolves a century of mystery:

• It provides a physical mechanism: the charge handle operates as a lever arm that transduces interactions with measured efficiency κ_e

• It explains the bidirectional nature: the same mechanism appears in both Coulomb's law and radiation formulas because it is the same physical structure

• It demystifies the hierarchy of forces: gravity operates directly through the mass handle (efficiency = 1), while electromagnetism operates through the charge lever arm (efficiency = κ_e ≈ 0.00116). These are not "strengths" but measured geometric coupling efficiencies

• It reveals α as a technical specification: κ_e is the gear ratio of the fundamental transduction mechanism, with the factor of 2π arising from the cyclic geometry of the interaction

The universe is not built with magic numbers. It is a rational, mechanical system built on measured geometric ratios. The "mystery" of the fine structure constant was never in the number itself, but in our failure to see the simple lever arm mechanism it describes. We were measuring a transduction efficiency and wondering why it had the value it did, without realizing we were looking at the measured coupling ratio of two fundamentally different geometric mechanisms—one direct (mass), one levered (charge).