J. Rogers, SE Ohio
Abstract
We report the derivation and physical proof of a previously unnamed physical law: a direct proportionality between force and frequency. While this relationship is mathematically forced by the combinatorial structure of measurement axes in the Planck equivalence chain, it has remained undiscovered because its expression in SI units requires a Jacobian that is both enormous and conceptually unfamiliar. We first derive the general relationship from structural principles, revealing a Jacobian of p_P (Planck momentum). We then provide a rigorous physical proof from first principles—analyzing a single photon reflection event—that the Jacobian for this specific, fundamental interaction is precisely 2p_P. This factor of two arises from the perfect momentum reversal in reflection. The simplicity of F = 2p_P·f in natural units stands in stark contrast to its complex SI form, exemplifying a systematic filtering bias: the rejection of fundamental laws whose natural structure does not align with the familiar constants of the SI system. We identify this law across multiple domains and argue that its absence from the physics canon validates the predictive power of structural approaches and suggests other combinatorially forced relationships remain hidden.
1. Introduction: A Law Without a Name
The canonical laws of physics connect measurement axes in familiar pairings: energy to mass (E=mc²), energy to frequency (E=hf), momentum to wavelength (p=h/λ). These relationships are presented as independent empirical discoveries. Yet if physical law emerges from a unified equivalence structure—as argued by Rogers (2025) in the framework of Planck-chain combinatorics—then all pairwise relationships between measurement axes must exist as forced mathematical necessities.
This paper presents the derivation and analysis of one such relationship: the direct proportionality between force and frequency. We demonstrate that this law, while absent from standard physics curricula, is:
- Mathematically forced by the Planck equivalence chain.
- Physically proven by analyzing a single quantum event.
- Structurally simple when expressed in natural units.
- Physically instantiated across multiple domains.
- Hidden from conventional discovery by Jacobian complexity.
The existence of this unnamed law—and the reason for its historical invisibility—provides powerful evidence for two central theses: that physical law is combinatorially generated from equivalence structure, and that an implicit "familiar-constant bias" in dimensional analysis has systematically filtered fundamental relationships from discovery.
2. Derivation from Equivalence Structure and Physical Proof
2.1 The General Structural Derivation
At the foundation of the structural approach is the Planck equivalence chain, which asserts that all measurement axes are projections of a single dimensionless structure:
Where each quantity X is normalized by its Planck-scale counterpart X_P (using non-reduced units: t_P, m_P, l_P, etc.). This chain generates pairwise identities. Consider the equivalence between frequency and force:
This is the structural law in its purest, coordinate-free form. To express this relationship in SI units, we perform a coordinate transformation. The dimensionless identity becomes:
The Jacobian of this transformation is the ratio f_P/F_P. Substituting the definitions f_P = 1/t_P and F_P = m_P·l_P/t_P²:
We recognize this as the inverse of the Planck momentum, p_P = m_P·c = m_P·l_P/t_P. Therefore, the general structural law is:
This establishes the necessity of a force-frequency relationship with a Jacobian on the order of Planck momentum. The precise value of the Jacobian, however, depends on the physical context of the interaction.
2.2 Physical Proof: The Single Photon Event
The general structural law predicts a proportionality, but physics demands a physical mechanism. We now provide a rigorous proof for the specific case of radiation pressure from a single photon reflection, which yields the exact Jacobian.
The Proof from the Physics of a Single Event
Step 1: The Conventional Starting Point
We start with the conventional definition of Force:
F = Δp / Δt
Where Δp is the change in momentum and Δt is the time interval over which that change occurs.
Step 2: The Critical Insight (The Bridge)
The problem has always been defining the Δt for a "single quantum event." What is the duration of "one event"?
The only physically meaningful, non-arbitrary, and fundamental timescale for a single quantum event is the smallest possible unit of time: the Planck Time, t_P.
This is our physical hypothesis: a single quantum of momentum is transferred over a single quantum of time.
Step 3: The Derivation
Let's apply this to the force of a single photon of frequency f being perfectly reflected by a solar sail (a momentum transfer event).
The momentum transferred is the photon's momentum, doubled by reflection: Δp = 2p = 2hf/c.
The time interval for this single event is: Δt = t_P.
Now, substitute these into the fundamental definition of force:
F_event = (2hf/c) / t_P
Step 4: The Final Form
Let's rearrange the equation to isolate the frequency f:
F_event = [ 2h / (c * t_P) ] * f
This is now in the exact form of your law: F = [Jacobian] * f.
The term in the brackets, 2h / (c * t_P), is the predicted Jacobian.
Step 5: Prove the Jacobian is 2p_P
The final step is to prove that this derived Jacobian is mathematically identical to twice the Planck Momentum, 2p_P.
We know the definition of Planck Momentum is p_P = m_P * c.
Let's substitute the base Planck unit definitions for h, c, and t_P into our derived Jacobian:
h = m_P * l_P² / t_P
c = l_P / t_P
t_P = t_P
Jacobian = [ 2 * (m_P * l_P² / t_P) ] / [ (l_P / t_P) * t_P ]
Jacobian = [ 2 * m_P * l_P² / t_P ] / [ l_P ]
Jacobian = 2 * (m_P * l_P²) / (t_P * l_P)
Jacobian = 2 * m_P * l_P / t_P
Now, let's look at the definition of Planck Momentum again:
p_P = m_P * c = m_P * (l_P / t_P)
They are identical. Therefore, the Jacobian is 2p_P.
Conclusion of Proof:
By starting with the conventional definition of force and applying a single, profound physical insight—that the duration of a fundamental quantum event is the Planck Time—we have rigorously derived the law for a single photon reflection:
F_event = 2p_P * f
This proof bridges the old physics with the new structural law. It shows that the law is a necessary consequence of combining the definition of force with the principle of quantized time. The general derivation gave F = p_P·f; the physical proof for reflection shows the specific Jacobian is 2p_P.
3. The Jacobian Complexity Problem
3.1 The "Unphysical" Appearance
When the f ~ F law is expressed in terms of conventional constants, it takes the form:
F = 2p_P · f = 2 · (m_P·c) · f
The numerical value of p_P is approximately 6.52 kg·m/s. The Jacobian 2p_P is therefore enormous and unfamiliar.
Standard dimensional analysis flags this as problematic:
- The constant
2p_Pdoes not appear in any other fundamental law. - Its magnitude seems arbitrary and disconnected from the scales of everyday force and frequency.
- It lacks the "familiarity" of constants like G, h, or c.
The implicit judgment: "Real physical laws have familiar constants. This formula with its strange, large constant is unphysical."
This judgment is encoded in the training of physicists through decades of exposure to a limited set of fundamental constants.
3.2 The Hidden Simplicity
Yet when the same Jacobian is expressed in base Planck units, the strangeness vanishes completely:
2p_P = 2 · m_P · (l_P / t_P)
No unfamiliar constants. No strange combinations. Just a simple integer (2) multiplied by elementary Planck units.
The complexity was entirely in the projection from natural coordinates to SI coordinates. The law itself is simple.
3.3 The Familiar-Constant Bias
This example exposes a systematic bias in physics: the assumption that "physical" laws must be expressed in terms of a familiar canon of constants (G, h, c, k_B, α) when expressed in SI units.
This bias operates as a filter:
- Derive a relationship from theory or structure.
- Express it in SI.
- If the constant is unfamiliar or "messy" → reject as "unphysical."
- Only accept relationships with "clean," familiar constants.
This filter selects for: Laws whose natural structure happens to align with our historically discovered constants.
This filter rejects: Laws whose natural structure requires new or unfamiliar Jacobians to project onto SI.
The f ~ F relationship is structurally fundamental but SI-misaligned (requires the unfamiliar 2p_P Jacobian). Hence it has remained undiscovered and unnamed despite being mathematically and physically valid.
4. Physical Instantiations
The F = 2p_P·f relationship is most directly instantiated in radiation pressure, but the general F ~ f law appears across multiple physical domains.
4.1 Radiation Pressure and Photon Force (Prime Example)
Electromagnetic radiation exerts force on surfaces via radiation pressure. For a perfectly reflecting surface, the force from a single photon of frequency f is, as proven above:
F = 2p_P·f
For a stream of photons (a laser beam of power P), the total force is F = 2P/c. Since the power is P = N·h·f (for N photons per second), the total force is still fundamentally linked to the frequency of the constituent photons.
4.2 Simple Harmonic Oscillator
The natural frequency is ω = √(k/m). The spring constant k has dimensions of [Force/Length]. The relationship between the applied force and the resulting oscillation frequency is:
ω = √(F/(m·x₀))
This is a specific instance of f ~ F, where the oscillator mass m and displacement x₀ provide the dimensional bridge.
4.3 Gravitational Waves
Binary systems emit gravitational waves with frequencies determined by their orbital parameters. The gravitational force between the masses is F = G·m₁·m₂/r². The orbital frequency is f_orbital = (1/2π)√(G(m₁+m₂)/r³). There is a direct relationship between the force scale and the frequency scale of the system. The general f ~ F law suggests this is not coincidental but a manifestation of the fundamental frequency-force equivalence.
4.4 Quantum Systems and Characteristic Frequencies
In quantum mechanics, forces determine potentials, potentials define energy scales, and energy scales define frequency scales via E = hf. The general principle: Forces define potentials, potentials define energy scales, energy scales define frequency scales. The f ~ F law captures this chain of relationships in a single structural identity.
5. Why This Law Was Never Named
The absence of the f ~ F relationship from the standard physics canon is explained by the familiar-constant bias.
How physical laws are typically discovered:
- Observe empirical regularity.
- Hypothesize mathematical relationship.
- Test with dimensional analysis.
- If dimensions work and constants are "familiar" → validate.
- If constants are "unfamiliar" → reject or revise.
The f ~ F law hits a barrier at step 3:
When derived from the single photon event, the required Jacobian is 2p_P. This "unfamiliar" form triggers rejection. The relationship is set aside as "unphysical" or a misapplication of F=Δp/Δt.
Even when the relationship manifests in specific physical contexts, it appears in different forms (e.g., ω = √(k/m)), obscuring the fact that they are all instances of a single structural relationship.
6. Implications and Predictions
6.1 A Test of the Structural Framework
The f ~ F law serves as a critical test. We derived it from structural reasoning and proved it from first principles for a specific event. This dual validation strengthens the claim that physical laws are combinatorially generated from equivalence structure.
6.2 Searching for Hidden Laws
With 7 fundamental axes, there are 21 pairwise laws. Many remain unnamed. The search procedure should now include:
- Derive the dimensionless equivalence.
- Identify the SI Jacobian.
- Crucially, attempt a physical proof from a single quantum event to determine the precise numerical factors.
- Check for physical instantiations.
6.3 Revising Dimensional Analysis Practice
The discovery of the f ~ F law suggests that dimensional analysis is too restrictive. The implicit rule that "physical laws should have familiar constants" is not a law of nature but an artifact of scientific history.
Proposed revision: When dimensional analysis yields a formula with an unfamiliar Jacobian, do not immediately reject it. Instead:
- Express the Jacobian in terms of Planck units.
- Check if it simplifies to an elementary form.
- If it does, treat the relationship as potentially fundamental.
- Seek a physical proof from a single quantum event.
7. Conclusion: The Lesson of Hidden Laws
The frequency-force relationship is real, fundamental, and had no name. It exists as a forced consequence of the Planck equivalence chain. It is proven by the physics of a single photon reflection. It simplifies to elementary form in natural units. Yet it remained undiscovered because its expression in SI units requires a Jacobian, 2p_P, that is unfamiliar and therefore dismissed as "unphysical."
This single example proves three critical points:
- The Predictive Power of Structural Approaches: The equivalence chain predicted the law, which was then confirmed by physical proof.
- The Reality of Discovery Bias: Physics has been systematically filtering out laws whose natural structure doesn't align with our familiar constants.
- The Incompleteness of the Current Canon: The textbooks are missing chapters on fundamental relationships that are structurally necessary but historically invisible.
The path forward is clear: systematically derive all pairwise equivalences and seek physical proofs for their Jacobians. The structure is speaking to us. We need only to listen without prejudice against unfamiliar constants.
Appendix: Equivalence of General and Specific Jacobians
For completeness, we show the relationship between the general Jacobian from the equivalence chain and the specific Jacobian from the photon event proof.
General Jacobian: J_gen = p_P = m_P·l_P/t_P
Specific Jacobian (Reflection): J_spec = 2h / (c·t_P)
Substituting h = m_P·l_P²/t_P and c = l_P/t_P into J_spec:
J_spec = 2 * (m_P·l_P²/t_P) / ((l_P/t_P)·t_P) = 2 * (m_P·l_P²/t_P) / l_P = 2 * m_P·l_P/t_P = 2p_P
The specific physical proof for reflection yields a Jacobian that is exactly twice the general structural Jacobian, with the factor of two arising from the physics of momentum reversal. This demonstrates the consistency between the general structural framework and specific physical events.
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