J. Rogers, SE Ohio
Every measurement you have ever made follows the same structure. Somewhere underneath the number on your instrument there is a pure, dimensionless ratio — the actual physical fact. To read it off a dial, you multiply it by an arbitrary scaling you chose in advance. The number you record is not the physics. It is the physics wearing your unit chart like a costume.
natural_ratio × unit_scaling = measured value
This is not a philosophical position. It is what measurement operationally does, every time, without exception. The natural ratio is invariant — every observer in every civilization finds it. The unit scaling is arbitrary — Babylonians, SI, CGS, it does not matter. The measured value changes with the unit chart. The natural ratio does not.
Physics is defined as the invariant. Therefore physics cannot carry units. If it has units, you are looking at a measurement — a natural ratio multiplied by an arbitrary scaling — not at the physics itself.
We Invented the Axes
Reality presents itself as unified. A falling rock, a warming gas, a vibrating string — one coherent substrate. We do not perceive unity. We perceive axes: mass, length, time, temperature, frequency. We experience these as independent because our senses are built that way. We measure them with different instruments. We assign them different units.
The independence is not in the physics. It is in the perception.
Once you impose multiple independent axes on a unified substrate, something is forced to appear: correction terms. Numbers that bridge the axes back together whenever a physical law tries to connect them. We gave those correction terms names. We called them constants of nature.
Newton understood this. He wrote his gravity law as F = MM/r² — a pure ratio, no correction term. The physics was in the ratio. He said explicitly that units have nothing to do with the physics.
After he died, we added G. We told ourselves we were completing his law. What we were actually doing was assuming that our SI units — kilograms, meters, seconds — were real, and patching the equation to work inside that assumption. G is the patch. Not a discovery. A patch for the coordinate origin we chose.
The Century of Constants
Newton's patch set the template. In rapid succession around 1900, three more constants appeared.
c — the speed of light. Appeared because we measured space in meters and time in seconds: two different arbitrary scalings for what turns out to be the same substrate axis. c is not a cosmic speed limit baked into the fabric of spacetime. c is l_P / t_P — the meter divided by the second.
h — Planck's constant. Appeared because we measured energy in joules and frequency in Hz: two different arbitrary scalings for the same substrate quantity. If you define E = f — one unit for both — h vanishes entirely. Not set to 1. Gone. It was never quantum mechanical content. It was a unit mismatch between two silos.
k_B — Boltzmann's constant. Appeared because we measured energy in joules and temperature in kelvin: two different arbitrary scalings for the same substrate quantity. k_B is not the bridge between mechanics and thermodynamics. It is the record that we gave thermal energy and mechanical energy different unit names.
Each one was celebrated as an independent discovery. Each one was seen as revealing something new about nature. Each one was recording that we had put a different arbitrary scaling on what was the same axis.
The dimensional formulas make this explicit:
c = L/T = l_P / t_P
h = ML²/T = m_P l_P² / t_P
k_B = ML²/(T²·θ) = m_P l_P² / (t_P² T_P)
G = L³/(MT²) = l_P³ / (m_P t_P²)
These are not abstract dimensional placeholders. L is l_P. T is t_P. M is m_P. θ is T_P. Every constant of nature is a specific combination of four numbers: the SI unit scalings for time, length, mass, and temperature. Those four numbers are the definitions of the second, the meter, the kilogram, and the kelvin — expressed in terms of the physics they are scaling away from.
Planck Found the Jacobians
In 1899, Max Planck derived a set of units by combining the constants in ways that produced pure numbers with no leftover SI factors. He found t_P, l_P, m_P, T_P. He looked at them and declared he had found God's units — nature's own fundamental scales, meaningful to any civilization in the universe.
He found the right numbers. He drew the wrong conclusion.
What Planck found were the Jacobians of a rotation — the components of the matrix that bridges SI coordinates to the natural ratios. They are not a destination. They are not a unit system. They are the numerical record of how far the second, the meter, the kilogram, and the kelvin missed the physics.
natural_ratio × t_P = time in seconds
natural_ratio × l_P = length in meters
natural_ratio × m_P = mass in kilograms
t_P is not a unit you switch to. t_P is already inside every SI time measurement ever made. You are already using it. The Planck scalings are implicit in SI. They were always there. Planck noticed them and thought he had found something beyond SI. He had found what was inside SI the whole time.
The natural ratios are not Planck units. They are the absence of any unit chart at all — the dimensionless physical reality that every unit chart is pointing at and none of them reach.
The 2019 Confession
In 2019, the SI system was redefined. The committee fixed the numerical values of c, h, k_B, and e to exact values by definition. They described this as anchoring the constants of nature.
What they operationally did:
fix t_P exactly → defines the second
fix c = l_P/t_P exactly → defines the meter via l_P = c·t_P
fix h = m_P l_P²/t_P → defines the kg via m_P = h·t_P/l_P²
The SI base units are now defined by the Planck scalings. We did not say that is what we were doing. But operationally, that is what we did.
The 2019 redefinition is the institutional proof that the constants were always unit scalings. We locked them to exact values without admitting why those values were the right ones to lock — because admitting that would require admitting that G, c, h, k_B were never constants of nature in the first place.
The Gasket
Here is the contradiction that sat in plain sight for a century.
Accepted fact one: unit scaling is arbitrary. Every physicist agrees. Taught in the first lecture of every physics course.
Accepted fact two: physics is invariant to unit scaling. Every physicist agrees. Also taught in the first lecture.
What follows: if you write physics inside a unit chart, something must absorb the arbitrariness while keeping the physics invariant. That something is the constants. The constants are the gasket between the two accepted facts.
A gasket between two facts that both deny the constants are fundamental cannot itself be fundamental. The logic is direct. We accepted both facts. We refused to follow the implication one step further. We stopped exactly there and called what we found a deep mystery — the fine-tuning problem, the anthropic principle, the multiverse conjecture. Entire research programs built on the question of why the constants have the values they do.
The fine-tuning problem is asking why the Babylonians chose that particular time interval. That is the entire mystery. The values of the constants are the values of the second, the meter, the kilogram, the kelvin — set by Roman marching cadences, human body parts, a platinum cylinder in Paris, and the freezing point of water. The universe did not tune them. History did.
The Laws Were Never Independent
Once you see the constants as unit scalings, something else dissolves: the independence of the laws.
The celebrated unification formula — Hawking temperature — contains c, G, h, k_B all together. It is called the great meeting point of quantum mechanics, relativity, gravity, and thermodynamics. In natural ratios it says:
T_H / T_P = m_P / M × 1/(4π)²
A hot thing cools as it gets heavier. Scaled by the geometry of a sphere. That is the entire content. The four constants are the unit scalings for temperature and mass dressed up as a unification of four frameworks. Strip the unit chart and there is nothing left to unify, because the substrate was never fragmented.
Every fundamental law is one leg of a single tautology:
T/T_P = f·t_P = m/m_P = l_P/λ = E/E_P = p/p_P = X
One dimensionless scalar. Six different unit-scaled projections. Fifteen pairwise laws of physics. All of them are X = X.
E = mc² is X = X with a mass/energy Jacobian. E = hf is X = X with an energy/frequency Jacobian. λ = h/p is X = X with a momentum/wavelength Jacobian. de Broglie did not discover a new law. He found another angle on the same object.
The probability that fifteen independently discovered laws would form the complete combinatorial closure of six quantities — every pair connecting, every Jacobian consistent with every other — by accident is less than 10⁻²². It is not a coincidence. It is one thing.
What Newton Knew
We did not progress past Newton. We regressed.
Newton wrote ratios. He knew the physics was in the ratios. He knew the units were scaffolding. He left G out of his gravity law because G had no place in the physics — only in the unit chart.
We put G in. We reified the SI coordinate origin as the origin of physical reality. We added c, h, k_B in rapid succession, each time mistaking the gasket for the phenomenon. We built a century of physics on the assumption that the measured value — natural_ratio × unit_scaling — was the real thing, not a combination of the real thing and our arbitrary scaffolding.
We could not see it because we had no concept of how unit charts work as an operational fact. We did not know that every measurement is a tautology. We did not know that every constant is a Jacobian. We did not know that the Planck scalings were already inside SI, implicit, waiting to be noticed.
Planck noticed them in 1899 and thought he found God's rulers.
We fixed their values by international vote in 2019 and called it anchoring the constants of nature.
Both times we were looking directly at the unit scaling and seeing something else.
The Core Statement
Physics is invariant to unit scaling. Units are arbitrary. Therefore the constants — which change when units change — are unit scalings, not physics.
The natural ratios are what remains when every unit scaling is divided out. They are not a unit chart. Not Planck units. Not natural units. They are the absence of any measurement convention at all — the single dimensionless physical reality that every unit chart in every civilization is pointing at from a different arbitrary angle.
The laws of physics are not independent discoveries. They are projections of one tautology through different pairs of unit scalings. The constants are the projection coefficients. The physics is X = X.
We mistook perception for reality. The axes are ours. The substrate is not.
This argument is developed formally as a Grothendieck fibration in The Structural Necessity of Physical Law (Rogers, 2025), where the terminal object S_u, the forced emergence of Planck-like scales, and the derivation of all fundamental laws from a single equivalence chain are proved as categorical necessities, not empirical coincidences.
