Measurement as Ratio: The Invariant Beyond Units and Constants

J. Rogers, SE Ohio

This paper is at: https://github.com/BuckRogers1965/Physics-Unit-Coordinate-System/tree/main/docs

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Abstract

Measurement is the comparison of an object to a standard, yielding a dimensionless ratio. The subsequent attachment of a unit label is a human convention, not a discovery of an intrinsic property. The so‑called fundamental constants of physics—$c,h,G,k_B$—appear only because we have already fixed our unit system. When we divide a measured quantity by its corresponding Planck value (a combination of these constants expressed in the same units), both the arbitrary human unit and the Planck “scale” cancel completely, leaving a pure dimensionless number $X$ that depends on nothing but the object and the unified substrate. This number is the only physically meaningful invariant. The equality of $X$ across all conceptual axes (mass, length, time, temperature, etc.) is the algebraic expression of the universe’s unity. In this view, constants are not part of the invariant; they are merely the scaffolding we use to strip away our own conventions, and they vanish entirely when we do so.

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1. What Measurement Is

Consider a balance. On one pan sits a standard mass stamped “1 kg”. On the other pan sits an apple. A rider is moved along a graduated bar until equilibrium; the rider’s position reads “0.2”. What has been discovered?

The physical fact is the equilibrium condition. That condition yields a dimensionless number: the ratio of the apple’s mass to the standard mass. Because the bar is linear, the reading means

$$\frac{m_{\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{l}\mathrm{e}}}{m_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}}}=0.2$$.

The apple does not possess the label “kg”. The label belongs to the standard. The act of measurement compares the apple to that standard, and the result is a pure number—a ratio.

This is not philosophy; it is the operational definition of measurement. Every measurement—a ruler, a clock, a thermometer—follows the same pattern: compare to a standard, obtain a dimensionless ratio, then by convention attach the standard’s unit to the object, multiplying the ratio by that unit. The unit label is transferred, not discovered.

2. The Arbitrariness of the Standard

The standard itself is arbitrary. The kilogram was once a platinum‑iridium cylinder; today it is defined by fixing Planck’s constant. Regardless, the choice of unit is a human convention. Any other choice (gram, pound, solar mass) would serve equally well; the numerical ratio $m_{\mathrm{\text{apple}}}/m_{\mathrm{\text{standard}}}$ would change accordingly, but the physical relation between apple and standard remains invariant.

When we write $m_{\mathrm{\text{apple}}}=0.2\mathrm{\text{kg}}$, we perform a conventional act: we take the pure ratio $0.2$ and attach the unit “kg” that belongs to the standard. We then speak as if the apple has a property “0.2 kg”. This reification obscures the relational nature of measurement.

3. The Unified Substrate and the Invariant $X$

The universe does not come pre‑divided into “mass”, “length”, “time”, etc. Those are conceptual axes we impose. Beneath them lies a single, coherent substrate of interacting phenomena. Every object participates in that substrate, and every interaction involves all aspects of reality simultaneously.

If the substrate is truly unified, then for any object there exists a single dimensionless number—call it $X$—that captures its relation to the whole. This number does not depend on any unit, any constant, or any axis. It is the raw, unit‑free coordinate of the object in the substrate. All measurements, regardless of which axis we use, are attempts to determine $X$.

4. Constants as Scaffolding: Canceling the Standard

The constants $c,h,G,k_B$ are not fundamental parameters that appear in $X$. Instead, they are conversion factors that we have defined within our unit system. Their numerical values are determined by that system. They serve as a bridge: given a measurement in kilograms, we can use the constants to completely eliminate the arbitrary standard.

Take the apple’s mass measured in kilograms: $m=0.2\mathrm{\text{kg}}$. The Planck mass is a combination of the constants:

$m_P=\sqrt{\frac{hc}{G}}$.

But note: $h$, $c$, and $G$ are themselves expressed in the same unit system (SI). Thus $m_P$ is simply a fixed number of kilograms: $m_P\approx 5.456\times {10}^{-8}\mathrm{\text{kg}}$.

Now form the ratio:

$\frac{m}{m_P}=\frac{0.2\mathrm{\text{kg}}}{m_P\mathrm{\text{kg}}}$.

The kilograms cancel. What remains is a pure number—the quotient of two numbers expressed in the same arbitrary unit. That number does not depend on the kilogram. It does not even depend on the constants, because the constants were used only to compute $m_P$ in kilograms, and that computation is exactly what cancels the unit.

The result is simply a number. It has no memory of the standard, no memory of the Planck mass, no memory of the constants. It is the invariant $X$.

We can write this directly as:

$X=\frac{m}{m_P}\mathrm{\text{(a pure number)}}$.

Because both numerator and denominator share the same unit, the unit vanishes. The constants that went into defining $m_P$ vanish as well—they were only a ladder, and once we climb it, we leave it behind.

5. The Same $X$ Across All Axes

Because the substrate is unified, the same invariant $X$ must be obtained regardless of which axis we use to measure the object. For length:

$X=\frac{l}{l_P}$,

where $l_P=\sqrt{Gh/c^3}$ (again expressed in meters). For frequency:

$X=\frac{f}{f_P}$,

with $f_P=\sqrt{c^5/(Gh)}$ (expressed in hertz). For temperature:

$X=\frac{T}{T_P}$,

with $T_P=\sqrt{c^{5}h/(G{k}_B^2)}$ (expressed in kelvin).

In each case, the Planck value is a fixed number in the corresponding human unit. Dividing by it cancels that unit, yielding the same dimensionless $X$. The equalities among these ratios are not accidental; they are the algebraic shadow of the substrate’s unity. They also explain why the constants take the values they do relative to our unit system: they are precisely the conversion factors that make all these normalized ratios equal to the same $X$.

6. Why the Constants Are Not Fundamental

A common misconception is that the constants are “fundamental parameters” that set the scale of nature. The present analysis shows the opposite: the constants are derived from the combination of our arbitrary unit system and the invariant $X$. If we chose different units, the numerical values of $c,h,G,k_B$ would change, but $X$ would remain the same. In fact, if we choose units where the constants become 1 (cancelling the unit stanards with Planck jacobians), then $X$ is simply the measured quantity itself—no constants remain. This reveals that the constants were never intrinsic; they were merely the conversion factors needed to express the invariant $X$ in terms of our chosen human units.

The reduced Planck constant $\hslash =h/(2\pi )$ does not appear in this story because the factor $2\pi$ is irrelevant to unit scaling. It is a mathematical convenience that just creates cleaner notation in formulas, not to the structure of measurement.

7. Implications

• Measurement reveals ratios, not properties. What we call “mass”, “length”, “time” are labels we attach to ratios.
• The only invariant is the dimensionless number $X$. It depends on the object and the substrate, not on any human convention.
• Constants are scaffolding. Their numerical values are artifacts of our unit system; they disappear entirely when we form the invariant $X$.
• Natural units are simply the choice to measure in units where the scaffolding becomes 1, making the invariants directly visible.
• Physical laws (e.g., $E=m{c}^2$, $E=hf$) are not independent; they are all expressions of the single relation $X=X$, projected onto different axes.

8. Conclusion

We began with a simple balance, an apple, and a standard. We saw that measurement yields only a dimensionless ratio, and that attaching a unit to the object is a convention. We then used the constants not as fundamental properties of nature, but as scaffolding that allows us to cancel our arbitrary units and reveal the true invariant $X$—a pure number that relates the object directly to the unified substrate. The same $X$ emerges from measurements of length, frequency, temperature, and every other axis, because the substrate is one.

The constants are the ladder; $X$ is the destination. When we finally look at the world without the ladder, we see only dimensionless numbers—and the unity that makes them equal across all axes.

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Appendix A: Local Equivalences Without Global Transitivity

This appendix documents how standard physics already equates the major “silo” quantities pairwise—space with time, energy with mass, energy with frequency, energy with temperature, mass with inverse length, and so on—yet typically stops short of enforcing global transitivity across all silos. The result is a patchwork of local identifications that, if taken seriously and closed under transitivity, collapse into the single invariant $X$ described in the main text.

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A.1 Local identifications inside each silo

1. Relativity: space $\leftrightarrow$ time $\leftrightarrow$ spacetime

   • Special relativity introduces Minkowski spacetime, where space and time are components of a single four‑vector.
     
   • In units where $c=1$, spatial distance and time interval share the same unit: a meter and the time it takes light to travel a meter are numerically identical.
     
   • Operationally, “one second” is defined by light travel over a fixed distance; conceptually, time and length are already unified in that frame.

2. Relativistic energy: energy $\leftrightarrow$ mass

   • The relation $E=m{c}^2$ makes energy and mass proportional.
     
   • In units where $c=1$, mass and energy carry the same dimension; a particle can be labeled interchangeably by its “mass” or its “rest energy.”

3. Quantum mechanics: energy $\leftrightarrow$ frequency, time $\leftrightarrow$ energy

   • The Planck relation $E=hf$ identifies energy with frequency; choosing units with $h=1$ makes them numerically identical.
     
   • The energy–time uncertainty relation ties energy scales to time scales, reinforcing the link between temporal structure and energetic structure.

4. Statistical mechanics: energy $\leftrightarrow$ temperature

   • The relation $E\sim k_{B}T$ identifies characteristic energies with temperatures.
     
   • In units where $k_B=1$, “temperature” is literally just energy per degree of freedom; the numerical distinction disappears.

5. Quantum field theory: mass $\leftrightarrow$ inverse length $\leftrightarrow$ inverse time

   • A particle’s Compton wavelength satisfies ${\lambda }_C\sim 1/m$ (with $\hslash =c=1$), so mass and inverse length are interchangeable.
     
   • Frequencies and time scales also enter via dispersion relations; in natural units, mass, energy, inverse length, and inverse time all share the same dimension.

Within each theoretical silo, then:

• Relativity collapses space and time.
  
• Relativity plus mass–energy equivalence collapses mass and energy.
  
• Quantum theory collapses energy and frequency, and ties energy to time scales.
  
• Statistical mechanics collapses energy and temperature.
  
• Quantum field theory collapses mass, inverse length, inverse time, and energy.

Each community implicitly says, “in the right units, these two are the same thing,” but usually only within its own conceptual neighborhood.

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A.2 The transitivity that is not enforced

Taken together, these equivalences form a connected graph:

• Nodes: $\{\mathrm{\text{space}},\mathrm{\text{time}},\mathrm{\text{mass}},\mathrm{\text{energy}},\mathrm{\text{frequency}},\mathrm{\text{temperature}},\mathrm{\text{inverse length}},\dots \}$.
  
• Edges: proportionalities such as $c$, $h$, $k_B$, and $\hslash$ that become 1 in suitable units.

By ordinary reasoning:

• If space $\leftrightarrow$ time (via $c$), and time $\leftrightarrow$ energy scales (via uncertainty relations), and energy $\leftrightarrow$ mass (via $E=m{c}^2$), and mass $\leftrightarrow$ inverse length (via Compton wavelength), then space is connected to inverse length and mass and temperature and frequency through a chain of identifications.
  
• Once all the conversion factors are set to 1 (Planck or natural units), every edge becomes an equality of numerical variables.

However, in practice:

• Each field uses the equivalence it needs and then stops.
  
• Relativists talk about spacetime, but do not typically say “length is just inverse temperature” in the same breath, even though the chain of known equivalences leads there.
  
• Stat mech texts treat $k_{B}T$ as an “energy scale” but rarely connect that directly to, say, an inverse length scale via the full transitive closure of all constants.
  
• QFT works happily with mass $\sim$ inverse length $\sim$ inverse time, but still labels these as different “kinds” of quantity.

The result is a locally unified, globally segregated picture: each silo is internally consistent and uses some subset of equivalences, yet the discipline as a whole does not promote the full connected graph to a single equivalence class under transitivity.

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A.3 From local equivalences to a single invariant $X$

The main text of the paper takes the next step:

1. Start with the network of identifications already accepted in each silo.
   
2. Strip away human units by normalizing to Planck (or equivalent natural) scales.
   
3. Take transitivity seriously across the entire network.

Under this view:

• Once $c=h=G=k_B=1$, all the proportionality constants become identity maps.
  
• Every “axis”—mass, length, time, temperature, frequency, etc.—is just a different coordinate chart on a single substrate.
  
• For a given object, the normalized quantities $m/m_P$, $l/l_P$, $t/t_P$, $T/T_P$, $f/f_P$, and so on are not merely dimensionless; they are equal to the same invariant number $X$, because they are all descriptions of the same underlying relation to the unified substrate.

In other words, the discipline has already built the ladder:

• It has identified all the rungs pairwise (inside each silo) and even adopted unit systems where the constants become 1.
  
• What it has not done is declare the ladder collapsed: enforce global transitivity and announce that there is only one invariant coordinate left after all equivalences and normalizations are applied.

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A.4 Why the global closure is typically avoided

This appendix does not claim that global transitivity is logically impossible within current physics; rather, it notes that it is not standard practice to enforce it. Reasons include:

• Conceptual convenience: Keeping “mass,” “length,” and “temperature” as distinct labels is pedagogically and practically useful, even if they become numerically equivalent in certain units.
  
• Multiple dimensionless parameters: The Standard Model and cosmology appear to involve many independent dimensionless couplings and ratios; this encourages a view with many invariants instead of a single $X$.
  
• Disciplinary silos: Each subfield optimizes its own language and rarely insists on a fully unified ontology across all others.

The paper’s proposal is precisely to close the loop: to recognize that the community has already accepted enough pairwise identifications that, when taken together and normalized by Planck jacobians that cancel the unit standards, they naturally define a single dimensionless invariant $X$ that survives the collapse of all silos.

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References

1. Annenberg Learner – “Learning Math: Measurement – Part B: The Role of Ratio (Fundamentals of Measurement)”. learner This resource explicitly frames measurement as comparing an unknown to a standard and emphasizes that the result is a ratio, not an intrinsic property, which underpins your Section 1 claim that measurement yields a dimensionless number before any unit label is attached.

2. “Measurement in Science,” Stanford Encyclopedia of Philosophy (SEP). plato.stanford The SEP article provides a rigorous philosophical and operational account of measurement as the assignment of numbers via comparison procedures and standardized instruments, supporting your treatment of units and standards as conventions layered on top of a more primitive comparison process.

3. “Measurement,” Wikipedia. en.wikipedia This overview describes measurement as the process of associating numbers with physical quantities according to rules and standards, reinforcing your argument that the act itself is a structured comparison to a conventional unit rather than the discovery of a built‑in label like “kg” in the object.

4. “Dimensionless Physical Constant,” Wikipedia. en.wikipedia This entry defines dimensionless constants as pure numbers whose values are independent of any unit system, which dovetails with your invariant $X$ as a unit‑free quantity and supports your claim that only such dimensionless combinations are truly universal.

5. J.‑P. Uzan, “Dimensionless constants and cosmological measurements,” arXiv:1304.0577. arxiv Uzan argues that only dimensionless combinations of constants are operationally meaningful in cosmology and fundamental physics, directly resonating with your thesis that $c,h,G,k_B$ are scaffolding for constructing invariant, dimensionless ratios rather than themselves being part of the invariant.

6. “Planck Units,” Wikipedia. en.wikipedia This article defines Planck units by setting $c,\hslash ,G,k_B$ to 1 and shows how quantities like the Planck mass, length, and temperature arise from these constants, supporting your construction of $m_P,l_P,T_P$ as unit‑dependent scales used only to cancel human units and expose the pure number $m/m_P$, $l/l_P$, etc.

7. “Planck units,” TCS Wiki. tcs.nju.edu The TCS Wiki summary highlights that Planck units arise from combining constants associated with relativity, quantum theory, gravitation, and thermodynamics, which aligns with your view that these constants are conversion bridges between conceptual axes rather than independent “fundamental properties” appearing in $X$.

8. “UNIT 1: Philosophy of Measurement – Calibration,” engineering metrology notes. scribd These metrology notes stress that a measurement result is a number representing the ratio of the quantity to a chosen unit, and they discuss standards and calibration as conventional but necessary structures, lending technical support to your Sections 2 and 4, where the kilogram and other standards are treated as arbitrary scaffolding used to reveal the underlying dimensionless invariant.