Building Physics from Unity: A Constructive Approach

 J. Rogers, SE Ohio

Abstract

We demonstrate that fundamental physical laws can be constructed from pure unity (the number 1) through the mathematical structure of measurement itself. Rather than viewing physical constants and relationships as discovered properties of nature, we show they emerge necessarily from how we fragment and scale a unified reality into observable quantities.

1. The Foundation: Unity and Tautology

Begin with the ultimate tautology:

X = X

This is completely content-free: pure identity, demanding nothing of reality. Multiply both sides by unity:

X · (1) = X · (1)

Still tautological. Now express unity as a ratio of dimensional quantities:

X · (energy scale / energy scale) = X · (time scale / time scale)

This remains a double tautology—we have said nothing yet. But rearrange:

E_si / (energy scale) = f_si · (time scale)

If we interpret X as having dimensions and choose our scales carefully, we can (with non-reduced Planck units) write:

E / E_P = (1/t) · t_P

Where E_P and t_P are characteristic scales (non reduced Planck energy and Planck time). Solving for E:

E = (E_P · t_P) · f

Define the product of these scales as a constant:

h ≡ E_P · t_P

And we obtain:

E = hf

Planck's relation emerges from pure tautology plus dimensional scaling.

2. The Unified Scale

Physical reality has one unified scale, not separate Planck units for each quantity. There exists a single fundamental scale S, and what we call the non-reduced Planck Units E_P, m_P, l_P, t_P are merely the scaling of that one physical scale expressed in different human measurement conventions.

The dimensionless quantity at each point in reality:

X = E/S = m/S = l/S = t/S = T/S = p/S = ...

All physical quantities, when properly scaled, point to the same dimensionless value X. The apparent differences between "energy," "mass," "length," and "time" are artifacts of our measurement conventions, not fundamental distinctions in nature.

3. The Periodic Table of Physical Law

Take the unified expression pairwise to generate all fundamental relationships:

X = T/T_P = f·t_P = m/m_P = l_P/l = E/E_P = p/p_P=...

Each pair yields a physical law:

Energy-Frequency (Planck Relation)

E/E_P = f·t_P → E = hf

Where h = E_P·t_P

Mass-Energy (Einstein Relation)

E/E_P = m/m_P → E = mc²

Where c² = E_P/m_P

Momentum-Wavelength (de Broglie Relation)

p/p_P = l_P/λ → p = h/λ

Energy-Temperature (Boltzmann Relation)

E/E_P = T/T_P → E = k_B T

Where k_B = E_P/T_P

Every fundamental relationship in physics can be derived by equating different expressions for X. The "laws" are consistency conditions—ensuring our fragmented measurements of the unified reality remain coherent.

4. The Constants as Conversion Factors

What we call "fundamental constants" are unit conversion factors between arbitrary human measurement schemes:

• c converts between space and time measurements
• ℏ converts between energy and frequency measurements
• G converts between mass and spatial curvature measurements
• k_B converts between energy and temperature measurements

These aren't properties discovered in nature—they're exchange rates we created when we chose to measure energy in joules, mass in kilograms, length in meters, time in seconds, rather than recognizing these as the same thing measured along different conceptual axes.

5. The Necessity of Fragmentation

Why We Cannot See Unity

X is defined relationally: X = (this thing) / (that thing)

To perceive X requires:

1. A reference frame (the denominator—our scale, our "1")
2. Something measured against it (the numerator—the phenomenon)
3. A split between observer and observed

Before fragmentation: No contrast, no difference, no observable structure After fragmentation: Relationships, laws, phenomena—but not unity itself

The act of observation necessarily creates fragmentation.

What We Actually Observe

Not X itself, but:

• Differences in X (ΔX)
• Ratios between measurements
• Gradients and rates of change
• Relationships between quantities

We only ever access relationships, never absolutes.

The Measurement Trap

Suppose you devise a new way to "directly measure X." What you've actually done:

1. Invented a new conceptual axis (call it Q)
2. Established a scale for it (Q_P)
3. Measured the ratio: Q/Q_P = X

You've added another spoke to the wheel—another projection, another fragmentation, another way to not see X directly.

Every measurement has this structure:

• Choose a conceptual axis (what am I measuring?)
• Establish a reference scale (what's my unit?)
• Obtain a ratio (the "measurement")

There is no escape from this structure. No view from nowhere. No direct access to unity.

6. Physical Law as Constructed, Not Discovered

The Construction Recipe

To create any physical law:

1. Start with unity: X = X
2. Choose two conceptual axes (e.g., energy and frequency)
3. Put scales on them (E_P and t_P)
4. Demand they point to the same X: E/E_P = f·t_P
5. Rearrange: E = (E_P·t_P)·f
6. Name the constant: h ≡ E_P·t_P
7. Declare the law: E = hf

What Seemed Like Discovery

Physical laws are:

• The geometry of how our measurement conventions relate
• Tautologies expressed in dimensional clothing
• The self-consistency requirements of our conceptual fragmentation
• Bookkeeping that ensures our measurements remain coherent

Why This Works

Because there is only one X. Every way we measure it must give compatible results. Physical law is the mathematical structure that ensures our fragmented observations of unified reality stay consistent with each other.

7. Implications

The Nature of Physics

Physics is not the study of nature's fundamental properties—it is the study of how our conceptual fragmentations relate to each other. The laws describe our interface with reality, not reality itself.

Why Physics is Mathematical

Physics is mathematical because it is literally made of measurement structure. Mathematics is the language of relationships, ratios, and transformations—precisely what remains accessible when unity itself cannot be observed.

The Universe from 1

The entire edifice of physical law can be constructed from:

• The number 1 (unity)
• The act of establishing scales (measurement conventions)
• The requirement of self-consistency (mathematics)

No additional physical principles required. The universe emerges from unity and the structure of how we fragment it into observables.

8. Conclusion

We do not see unity—we see what our senses fragment it into. Physical law is not discovered in nature but constructed from the mathematics of measurement itself. The constants are conversion factors. The laws are consistency conditions. And X, the unified reality, remains forever inaccessible except through the fragmented relationships we call physics.

The territory is inaccessible. We only ever get better and better maps. But now we understand: the maps themselves are constructed from unity and the unavoidable structure of observation.