Abstract
In 2002, Duff, Okun and Veneziano engaged in a “trialogue” on the number of fundamental constants. Duff argued constants are not fundamental; Okun and Veneziano rejected that claim and offered different counts, but none operationalized the mechanism. While their philosophical claims were sound, the trialogue couldn’t stick the landing: it lacked the mathematical machinery to prove how physical law works, where constants come from, or why setting them to 1 is more than convention.
This paper introduces a three-step operational framework that walks through the existing math of every physical law to prove that natural ratios are already present in every formula right now. Constants are not fundamental entities; they are Planck Jacobians—unit scaling factors that are canceled in Step 1, revealed as pure geometry in Step 2, and multiplied back in Step 3. The framework is mathematically rigorous, computable (via the LawForge compiler), and cross-silo (deriving GR, quantum mechanics, and thermodynamics from the same postulate structure).
This is the rigor physics was missing. And it sticks the landing.
1. Introduction: The Trialogue That Couldn’t Stick the Landing
In 2002, Duff, Okun and Veneziano published “Trialogue on the number of fundamental constants” and engaged in a trialogue on the nature of fundamental constants. Duff's central thesis was clear:
“Constants like c, h, G, kB are not fundamental. Physics should be written in dimensionless form.”
Okun and Veneziano both claimed a different number. Duff's claim resonates with a deep intuition in physics: that the universe is unit-free, and our constants are artifacts of human measurement conventions. However, the trialogue failed to operationalize this claim. Duff argued that constants should disappear, but couldn’t show how they disappear, where they come back, or why the math works the way it does. Okun and Veneziano were able to reject that claim because setting constants to 1 lacks rigor. But they could not prove their claims either. This operationally showed a huge gap in the foundations of physics.
What Was Missing
| Duff’s Claim | What Was Missing |
|---|---|
| “Constants are not fundamental” | No rigorous framework for why |
| “Use natural units” | No operational method to go natural → SI |
| “Set constants to 1” | No derivation of where they reappear |
| “Physics is dimensionless” | No bridge between dimensionless → measured |
Duff had the philosophy but not the mathematical machinery. He couldn’t prove that natural ratios are already in every formula—he could only assert that they should be.
2. Newton’s Principia: The Original Rigor
To understand what Duff was missing, we must return to Newton’s Principia Mathematica (1687). Newton did not work in meters, seconds, or kilograms. He worked in natural proportions:
- “The square of the period is proportional to the cube of the distance.”
- “Force is proportional to mass times acceleration.”
- “Gravitation is proportional to the product of the masses and inversely proportional to the square of the distance.”
Newton wrote in ratios, not units. His laws were dimensionless by construction. The constants (like G) were added later, when metrology became the foundation of physics in the late 1800s. His proofs work in all unit charts.
The Historical Shift
| Newton (1687) | Modern Physics (Post-1800s) |
|---|---|
| Worked in proportions | Works in units |
| Laws are dimensionless | Laws are unit-decorated |
| Constants are conversion factors | Constants are fundamental entities |
| Natural ratios are primary | SI units are primary |
This shift from natural proportions to reified units is the root of the problem Duff faced. He was trying to argue for Newton’s original rigor, but without the operational machinery to recover it.
3. The Three-Step Framework: Operationalizing Natural Ratios
This paper introduces a three-step operational framework that walks through the existing math of every physical law to prove that natural ratios are already present:
Step 1: Remove Input Units
Action: Cancel arbitrary human unit standards by dividing measured quantities by non reduced Planck Jacobians (e.g., lP, tP, mP, EP, kP). Reduced planck units here are a category error because 1/2pi belongs to step 2, the invariant physics.
What it proves: Constants are just unit scaling. When you put a constant through Step 1, it goes to 1.
Step 1(c) = c/c = 1
Step 1(h) = h/h = 1
Step 1(G) = G/G = 1But measurement just cancels the unit scaling and leaves the natural ratio of that physical state behind. This data is the invariant to unit scaling—physics. With an observed measurement it removes the arbitrary unit scaling in that chart and leaves a ratio that is invariant to any unit scaling.
Step 2: Do the Physics as Pure Ratio
Action: Work with dimensionless ratios only.
What it proves: This is the only physics. The universe is pure geometry (X), unit-free, no axes, no measurement, no calculation, no perception.
E/E_P = m/m_P = f/f_P = XStep 3: Decorate with Output Units
Action: Multiply back by Planck Jacobians to express in SI.
What it proves: Constants reappear as unit scaling, not as fundamental entities.
E = X · E_P = X · (m_P · c²) = m · c²4. The Proof: Every Formula Already Uses Natural Ratios
You can now walk through the existing math of any physical law to show that natural ratios are already there:
Example 1: E = mc²
Take the formula: E = mc²
Apply Step 1: Divide by Planck Jacobians
E/E_P = m/m_PYou get: X = X (natural ratio)
Apply Step 3: Multiply back by Jacobians
E = X · E_P = X · (m_P · c²) = m · c²
The formula is the same. But now you know: - The natural ratio X is already there. - The constant c is just unit scaling.
Example 2: Hawking Temperature
Take the formula: T = c³h / (16π²GMkB)
Apply Step 1: Divide by Planck Jacobians
T/T_P = m_P/MYou get: T/T_P = m_P/M = X (natural ratio)
Apply Step 3: Multiply back by Jacobians
T = X · T_P = X · (c³h / 16π²Gm_Pk_B) = c³h / (16π²GMk_B)
The formula is the same. But now you know: - The natural ratio X is already there. - The constants c, h, G, kB are just unit scaling. The geometry of 1/(16pi^2) is a separate concern from the unit scaling.
All 15+ Laws from the Equivalence Chain
This works for every law:
| Law | Step 1 (Natural Ratio) | Step 3 (SI + Constants) |
|---|---|---|
| E = mc² | E/E_P = m/m_P = X | E = mc² |
| F = Gm₁m₂/r² | F/F_P = (m₁/m_P)(m₂/m_P)/(r/r_P)² | F = Gm₁m₂/r² |
| E = hf | E/E_P = f/f_P = X | E = hf |
| T = c³h/(16π²GMk_B) | T/T_P = m_P/M = X | T = c³h/(16π²GMk_B) |
Every formula uses natural ratios right now. Constants are just unit decoration.
5. Why This Sticks the Landing
| Duff 2002 | This Framework |
|---|---|
| “Constants are not fundamental” (philosophy) | “Constants are Jacobians” (math) |
| “Use natural units” (suggestion) | “Steps 1–3” (procedure) |
| “Set constants to 1” (convention) | “Step 1 cancels them to 1” (derivation) |
| No bridge from dimensionless → SI | Explicit bridge (Step 3) |
| Couldn’t stick the landing | Sticks the landing |
The Mathematical Rigor
This framework is not just philosophy — it’s operational:
- It’s executable: LawForge implements Steps 1–3 as a compiler.
- It’s reproducible: Code is public on GitHub.
- It’s derivable: You can prove the steps for any law.
- It’s cross-silo: Works for GR, quantum mechanics, thermodynamics, all from one postulate.
6. Conclusion: The Rigor Duff Was Missing
Duff couldn’t stick the landing because he had no machinery to make it rigorous.
This framework has the machinery: - Steps 1–3. -
Planck Jacobians. - LawForge compiler. - X as terminal
object.
And this is not a suggestion about how we should to physics. The 3 steps are operationally how physics works right now in every formula. This is why dimensional analysis works.
You can now walk through the existing math of every physical law to prove:
This is what the 3 steps do. Every formula uses natural ratios in step 2 right now. Constants are just unit scaling.
That’s the rigor Duff was missing.
That’s why this framework sticks the landing.
References
Duff, M. J., Okun, L. B., & Veneziano, G. (2002). Trialogue on the number of fundamental constants. Journal of High Energy Physics, 2002(03), 023. https://doi.org/10.1088/1126-6708/2002/03/023
Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica.
Rogers, J. (2026). LawForge: The Physics Law Discovery Engine. GitHub. https://buckrogers1965.github.io/LawForge/
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