There Are No Units in Nature: Why the Natural Ratios Are Inevitable

J. Rogers, SE Ohio

Physicists like to say we “set c=h=G=kB=1c=h=G=kB=1” in natural units, and then everything becomes magically simple. But they never say what that actually means. Because it is never don with any rigor. If they did then they would se it is ababout harmonizing axis of measurement to a single physical scale.

My revised paper, “The Structure of Physical Law as a Grothendieck Fibration,” makes a stronger claim:

• The simplicity of natural/Planck units is not a convenient trick.
• It is a proof that there is only one real “thing” underneath all of physics.

And the constants c,h,G,kB are the unavoidable price we pay for insisting there are separate things like “mass,” “length,” and “time.”

This isn’t just aesthetics. It says: any universe with coherent physics and observers like us must have something like the Planck scale and something like our constants. There is no other way to do measurement on a single underlying reality.

1. The core move: reality is already unified, axes are in your head

The usual story:

• There are independent fundamental quantities: mass, length, time, energy, temperature…
• We discover “mysterious” conversion factors between them: c between space and time, h between energy and frequency, G between mass and geometry, k_B between energy and temperature.

My story:

• There is one coherent, dimensionless physically real substrate Su.
• Our “fundamental quantities” are just axes we made up to describe that one thing from different perspectives.
• The independence of Mass, Length, Time, etc. is a perceptual error, not a feature of the world.

Once you fracture a unity into separate axes and put arbitrary scles on each concept, you create gaps between them. The constants are not little bridges out there in nature; they are the measured size of the distortion you introduced by splitting something that was never separate.

2. The categorical skeleton: a fibration of measurement

The paper formalizes this with category theory, but the picture is simple:

• There’s a base layer of pure concepts (mass‑like, time‑like, energy‑like directions).
• There’s a measurement layer where you attach numbers and units (kg, m, s, J, K…).
• There’s a projection from measurements back to concepts.

This projection has a specific structure (a Grothendieck fibration), and physical laws are special arrows (“Cartesian liftings”) that sit above simple, unit‑free relations.

Example:

At the concept level: “energy is equivalent to mass” (no units, just E∼m).

At the measurement level: “E=mc^2” — which is that same relation, expressed in a weird, skewed coordinate system we call SI.

In this language, c^2 is not a mysterious physical ingredient. It’s a Jacobian: the conversion factor that appears because we chose axes that are misaligned with the underlying physical real unity.

3. The new part: Planck scale as a structural inversion point

The biggest addition in this update is an explicit inversion argument that explains why the Planck jacobians align in the unique place where our distortions cancel out.

Look at three types of ratios:

Mass: m/m_P – direct.

Length: l_P/λ – inverted.

Frequency: f⋅t_P – inverted.

You can see the pattern:

Mass goes up → m/m_P goes up.

Wavelength goes up → l_P/λ goes down.

Frequency goes up → f t_P goes up, but period goes down. 

The substrate ties these together m/m_P = l_P/λ = f t_P=⋯=X.

That equation is the Equivalence Chain. It says: once you normalize everything correctly, all of these “different” quantities are just different names for the same dimensionless number X.

Now, where do all these ratios equal 1?

m=1*m_P

λ=1*l_P

f=1/t_P

That special point is what we mistakingly call the "Planck scale". Geometrically, it’s the unique “inversion point” where all reciprocal relationships balance and the log‑space curves cross. At that point, our axes are as aligned with reality as they can possibly be, and all the Jacobians collapse to 1.

The key punchline:

You don’t pick the Planck scale; the Planck scale is what you get when you stop lying to yourself about the independence of your axes.

4. Why the constants are structurally unavoidable

The paper proves a “structural necessity” theorem in plain language:

Start with one coherent substrate where everything can interact.

Let an observer describe it using multiple axes (mass, length, time, etc.).

Assume there are real laws – nontrivial relations between those axes.

Then:

There must exist some distinguished system of scales where all laws look simple (the Planck‑like system).

In any other system, you are forced to introduce constants like c,h,G,k_B as correction factors. They are the Jacobians of your choice of description.

So in this view:

• A universe with coherent physics and no constants is impossible, if you insist on talking in fractured axes like kg, m, s, K.
• A universe with “many unrelated fundamental scales” is also impossible, if those scales are supposed to interact.
• The Planck scale and the constants are not oddities of our universe. They are what any universe looks like when a fragmented observer measures a unified substrate.

5. Why this matters (and to whom)

For working physicists

• It explains why “natural units” feel natural: they are the coordinates where your axes finally line up with the substrate.
• It demotes constants from “deep mysteries to be explained by more physics” to geometric bookkeeping for your unit choices.
• It unifies dimensional analysis (Buckingham ππ) and Noether/Lie symmetries as operations on the same underlying fibration, not two separate tricks in different textbooks.

If you’ve ever felt like c,h,G,k_B are more about our description than about reality, this gives you a rigorous way to say that.
For philosophers of physics

• It attacks the idea that “mass,” “length,” and “time” are ontologically primitive.
• It treats the observer’s concept‑splitting as the source of constants and complexity.
• It gives a precise sense in which “laws are coordinate artifacts” of a deeper, unit‑free structure.

This is not just wordplay; it’s backed by categorical structure and an explicit Equivalence Chain that reproduces ~15 major laws (Einstein, Planck, de Broglie, Stefan–Boltzmann, Newtonian gravity, etc.) as projections of a single tautology.
For mathematically inclined readers

• You get a clean Grothendieck fibration π:E→B where laws are Cartesian liftings.
• Constants become cocycles / connection coefficients in a “measurement bundle.”
• There’s a clear path to higher‑category generalizations (constants as 2‑morphisms, measurement as a stack, etc.).

If you like the idea that “physics is a bad coordinate chart on something simple,” this paper is almost literally that sentence turned into math.

6. The real philosophical punchline

The old question: “Why do c,h,G,kBc,h,G,kB have these specific values, and why do Planck units seem so special?”

The new answer:

• Because you chose to describe a single thing with multiple conceptual axes.
• Because you insisted on having “mass” over here and “length” over there and “time” somewhere else.

The constants are the cost of that insistence. The Planck scale is the unique point where the bill sums to zero.

Or in one line:

There are no units in nature. There is only a unified substrate, and the constants you are forced to invent when you split it apart.