J. Rogers, SE Ohio
The Problem with Constants
Physics is full of constants — h, c, G, k_B, and others. They appear in every fundamental equation. They are measured to extraordinary precision. They are treated as deep facts about nature.
They are not.
h, c, and G are monomials built from the Planck unit standards:
h = m_P l_P² / t_P
c = l_P / t_P
G = l_P³ / (t_P² m_P)
Every constant in physics is a ratio of Planck unit standards — m_P, l_P, t_P — combined in different proportions. They are not encoding gravitational physics, quantum physics, or thermodynamic physics. They are encoding the mismatches between the independently and arbitrarily defined SI unit axes: the kilogram, the meter, the second.
The constants are not kabbalist symbols that compel the universe to obey physical laws. They are conversion factors between unit axes that humans defined independently, without reference to each other, and then needed correction factors to relate.
The Three Steps of Every Physical Law
Every physical law performs the same three steps, whether or not it is presented that way:
Step 1 — Remove the input unit scaling. Take the measured quantities in SI units and divide by the appropriate Planck unit standards. This removes the arbitrary human scaling from the inputs. The constants appearing in this step — G/c², h/c², and so on — are exactly the Planck Jacobians: the transformation coefficients that cancel SI scaling.
Step 2 — Compute the pure dimensionless ratio. With the arbitrary scaling removed, what remains is a dimensionless geometric ratio. This is X. This is the actual physics. It has no units, no scale, no dependence on what measuring system any civilization happens to use.
Step 3 — Decorate with output unit scaling. Multiply by the appropriate Planck unit standards to re-express the result in whatever units the reader wants — kg, meters, seconds, or anything else.
Steps 1 and 3 are pure bureaucracy. They contain no physics. The physics is entirely in Step 2.
The constants appear in Steps 1 and 3 only. They are the unit removal and unit restoration machinery. When physics textbooks present G, ħ, or c as fundamental constants of nature, they are presenting the bureaucracy as if it were the content.
The Equivalence Chain
Combine the two most fundamental equations in physics:
E = mc²
E = hf
Set them equal and rearrange:
m / f = h / c² = m_P t_P
The left side contains the measured quantities — mass in kg, frequency in Hz. The right side is m_P t_P: the product of the Planck mass and Planck time, which are simply the unit standards that cancel the arbitrary SI scaling on mass and frequency.
This equation is saying: mass divided by frequency equals unit scaling. The ratio m/f is not a physical law. It is one dimensionless number — X — multiplied by the arbitrary unit scaling m_P t_P.
Strip the unit scaling and you have:
m_nat / f_nat = 1
Mass and frequency are the same axis. They were never different physical quantities. They were the same geometric ratio expressed through two independently calibrated measurement systems, with h/c² as the correction factor needed to translate between them.
This is why h/c² was never given a name or a place in the standard table of constants. If it had been named — call it Hz_kg — it would have been immediately obvious that Hz_kg = m_P t_P is just a product of two unit standards. The story of deep physical content would have been impossible to maintain.
The Planck Jacobians
The Jacobian transformation from SI units to natural units is not a change of scale. It is the removal of the arbitrary human scaling that was inserted between the measurement and the actual geometric relationship.
The Planck Jacobians are the accounting of exactly what arbitrary scaling was inserted at each axis of measurement — energy, frequency, mass, temperature, momentum, wavelength — so it can be removed cleanly and completely.
Consider the weak field of gravity:
τ = m_si / r_si (meaningless — two arbitrary scalings)
τ = (l_P / m_P) · (m_si / r_si) (the Jacobian removes the mismatch)
And G/c² = l_P / m_P. That is all G/c² is. It is the ratio of the Planck length to the Planck mass — the conversion factor between the length axis and the mass axis that corrects for the fact that meters and kilograms were defined independently. It is not gravitational physics. The gravitational physics is:
τ = m_nat / r_nat
A pure dimensionless ratio. Part to whole. That is the physics. G/c² is the Jacobian that gets you there from the SI inputs.
If you think G/c² is the physics instead of m_nat/r_nat, you cannot understand what physics is doing.
The Equivalence Chain and the Fifteen Laws
Remove the unit scaling from every physical quantity and express each as a dimensionless ratio against the Planck unit standards. The result is the equivalence chain:
E/E_P = f·t_P = m/m_P = T/T_P = l_P/λ = p/p_P = ... = X
Every entry in this chain is the same X — the same pure dimensionless number, expressed along a different measurement axis. Take any pair from the chain and set them equal, and you recover a known physical law:
- E/E_P = m/m_P → E = mc²
- E/E_P = f·t_P → E = hf
- m/m_P = f·t_P → the mass-frequency relation
- And so on — fifteen pairs, fifteen laws.
These are not fifteen independent discoveries. They are fifteen pairwise readings of a single equivalence chain. The laws are not independent. They are one statement, approached from different unit axes.
The equivalence chain works cleanly with Planck's original 1899 units. The later "reduced" Planck units — built around ħ = h/2π — break the wavelength entry. The wavelength pair becomes inconsistent with the others by a factor of 2π, requiring an arbitrary correction. This is because ħ absorbs a 2π into the unit standard itself, a 2π that has no home in the unit scaling — it belongs to the geometry of the cycle, not to the constant. h is the clean quantity. ħ is h with an unexplained pure number attached.
The 2π correction factor is a symptom. It is what you get when you put a geometric factor inside unit bureaucracy where it does not belong.
What X Is
X is not a unit chart. It is not a dimensionless number produced by the unit removal. The unit removal does not produce X — it stops obscuring X.
X is the geometry of a physical state against the entire universe.
When you remove all the arbitrary unit scaling from a measurement, what you are left with is the relationship of the thing being measured to the total geometric structure of everything. The universe is the only non-arbitrary reference. Every other reference — the kilogram, the meter, the second — is a rock or a stick that someone picked up and declared to be the standard.
m_nat / r_nat is not a calculation that the universe performs. It is the geometry of the space. The orbit is not the result of the universe computing an inverse square law. The planet moves in a straight line through curved space. The curvature is m_nat / r_nat. That is not a law the universe obeys — it is the shape of the space the planet moves through.
This is Mach's principle, generalized completely. Mach said inertia is not intrinsic — it is the relationship of a body to the mass distribution of the entire universe. But he stopped at inertia. Every property is like that. Mass, energy, frequency, temperature, momentum — none are intrinsic to the thing. They are all geometric relationships between a local state and the whole universe.
X is always: this configuration / the total configuration. Part to whole. The universe in the denominator, where it always belonged.
Wittgenstein's Ladder
Wittgenstein wrote: "My propositions serve as elucidations in the following way: anyone who understands me eventually recognizes them as nonsensical, when he has used them — as steps — to climb up on them. He must, so to speak, overcome these propositions; then he sees the world rightly. He must transcend these propositions. Whereof one cannot speak, thereof one must be silent."
The Planck Jacobians are Wittgenstein's ladder.
Use them to climb from the arbitrary SI scaling to X. The climb is real and necessary. The accounting of what arbitrary scaling was inserted at each axis of measurement — the Jacobians — must be done carefully, completely, and correctly. The equivalence chain must be verified pair by pair. The constants must be identified as unit monomials. The correction factor for wavelength must be recognized as a symptom of ħ rather than accepted as a known quirk.
The climb is the work.
And then, when you have arrived at X — pure dimensionless geometry, no scale, no units, part to whole — you throw the ladder away.
The Jacobians are not part of the physics. They were the scaffold. The mechanism for removing the arbitrary human choices that were sitting between the measurement and the actual geometric relationship. Once those choices are removed, the Jacobians have done their job.
They are not in the territory. They were always only in the map.
The territory is the universe, moving in straight lines through curved space, locally, with no constants, no units, no measurement, no map.
Summary
| What it looks like | What it is |
|---|---|
| A fundamental constant of nature | A monomial in Planck unit standards |
| G/c² encoding gravitational physics | l_P/m_P — the length-to-mass unit conversion |
| h encoding quantum action | m_P l_P² / t_P — the action unit standard |
| Fifteen independent physical laws | Fifteen pairwise readings of one equivalence chain |
| The Planck scale as a special high-energy regime | The inevitable result of removing all arbitrary unit scaling |
| Setting constants to 1 as a mathematical trick | Revealing that all measurement axes are dependent |
| The Planck Jacobian as a change of scale | The removal of arbitrary SI scaling — exactly, by definition |
| X as one entry in the equivalence chain | The geometry of a physical state against the entire universe |
The constants point to the Jacobians. The Jacobians point to X. X has no scale. X is not a product of the measurement ritual — it is what the measurement ritual was always trying to describe, obscured by unit bureaucracy that was mistaken for physics.
Climb the ladder. Then throw it away.
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