J. Rogers, SE Ohio
When you remove every human unit standard from measurement, every physical quantity reduces to the same ratio against the universe. That ratio is what Mach was pointing at — and he didn't realize it was universal.
Physics has long known a set of equivalences. Each major field owns one link in a chain:
E/E_P = f·t_P = m/m_P = λ_P/λ = p/p_P = T/T_P = X
Relativists knew E ~ m. Quantum theorists knew E ~ f. Thermodynamicists knew E ~ T. Wave mechanicists knew f ~ 1/λ. Each community published their link and stayed inside their silo. Nobody applied transitivity across all of them simultaneously.
When you do apply transitivity — when you express every quantity in non-reduced Planck units and let the unit standards cancel — every quantity in the chain collapses to the same dimensionless ratio X. Not similar ratios. Not related ratios. The identical ratio.
What X actually is
The Planck Jacobian is the mathematical operation that performs this collapse. It cancels the human unit standards against the Planck unit chart — and crucially, both cancel completely. The human units disappear. But so do the Planck units. Becue the Planck units are SI unit scaling inverted. Neither survives the operation.
What remains is not a ratio against the Planck scale. The Planck scale is the solvent, not the residue — it cancels with the human unit standards and leaves nothing behind. No kilograms, no meters, no seconds, no Planck units. Just X: a pure, unitless ratio whose only reference is the entire universe itself.
The Planck Jacobian and the human unit standards annihilate each other. What survives is the raw relationship each quantity has to the whole — the universe as the sole remaining standard, carrying no units of any kind.
Mach saw it for inertia. It's true for everything.
Ernst Mach argued that inertia is not an intrinsic property of matter. It is a relationship — specifically, a relationship between a body and the rest of the universe. Mass doesn't resist acceleration because of something it contains. It resists acceleration because of its relational situation within the whole.
Mach was right. But he didn't realize he had found a universal principle, not a special one. Because when the Planck Jacobian cancels every unit standard simultaneously, every quantity — energy, frequency, temperature, momentum, length, mass — becomes the identical unitless ratio X against the same sole reference: the entire universe.
Mach's principle is not a special fact about inertia. It is a universal fact about measurement itself. Every measurable quantity is a relationship to the universe. Strip away every unit standard — human and Planck alike — and the only thing left is that relationship. Every axis collapses to the same X.
What this means for the constants
If every quantity is the same ratio X, then the equations connecting them — E=hf, E=mc², E=k_BT — are not deep discoveries about nature. They are tautologies. X = X, stated fifteen different ways for the fifteen pairwise combinations of six quantities.
The constants h, c, k_B, G are not profound facts about the universe. They are the conversion factors between the human unit systems that each silo developed independently — the fingerprints of our measurement conventions, not properties of nature. Einstein said as much about c²: mass and energy are the same thing measured two ways, and c² is unit scaling between them. The same is true of every constant in the chain.
Since 2019, the SI system has fixed h, c, e, and k_B by definition. If you change h, you have not discovered something new about nature. You have redefined the kilogram. The international metrology community enacted this conclusion without drawing its philosophical implications. The constants are conventions. Hume's guillotine applies: no amount of physics can prove that a mile has 5280 feet, and no amount of physics can give the constants values that are anything other than artifacts of our unit choices.
The natural ratios are what the universe looks like without us in the way
Strip away the kilogram, the meter, the second, the kelvin — every measurement standard humanity invented to navigate the world at human scales — and what remains is pure relation. The Planck Jacobian is the operation that performs this stripping. It cancels the human standards, cancels itself, and leaves only the universe relating to itself, with no intermediary.
This is what Mach was reaching for when he said inertia was relational. It is what Einstein was reaching for in his last thirty years when he searched for a unit-free description of the universe. None of them applied transitivity across every silo simultaneously. That step — seeing that every axis of measurement reduces to the identical X, that Mach's relational principle is universal and not special, that the Planck Jacobian cancels everything including itself leaving only the universe — is what the journal structure made impossible to publish and what self-publishing in March 2026 has now put on record.
The natural ratios are not a mathematical curiosity. They are what the universe looks like when you stop measuring it against a standard you invented and use the universe itself as your yardstick.
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