The Web of Equivalence: A Structural Unification of Physical Law through the Planck Chain

J. Rogers, SE Ohio

Abstract
This paper demonstrates that the canonical equations of physics are not independent empirical discoveries but the inevitable consequences of projecting a single equivalence chain at the Planck unit chart. By encoding seven fundamental measurement axes—frequency, length, mass, energy, momentum, force, and temperature—into a unified dimensionless chain, we obtain 21 pairwise equivalences that reproduce the familiar laws of mechanics, relativity, thermodynamics, and quantum theory. This framework reveals physical constants (

        $c, h, k_B, G$  

) not as fundamental mysteries but as Jacobian factors ensuring coherence across unit projections. Extensions with additional axes (pressure, density, acceleration, charge) expand the combinatorial law space, suggesting a generative rather than empirical origin of physical law. Unlike conventional unification programs that attempt to merge the fundamental forces, this approach provides a higher-order structural unification: the unification of law itself across all physical domains.

---

1. Introduction

Physics has long presented its laws as distinct empirical generalizations: Newton’s (

        $F=ma$

), Einstein’s (

        $E=m{c}^2$      

), Planck’s (

        $E=hf$

), and Boltzmann’s (

        $E = k_B T$

). Yet each law invokes physical constants whose numerical values shift across unit systems. This dependence on unit charts implies that these constants are not metaphysical but structural conversion factors. The deeper invariant lies in the Planck unit system, where all quantities reduce to dimensionless ratios.

This paper argues that at the Planck chart (NOT the reduced Plank chart), all physical measurement axes collapse into a single equivalence chain. Each axis—frequency, mass, length, energy, momentum, force, temperature—can be expressed as a ratio against its Planck unit counterpart. Linking them together yields:

        $$\frac{T}{T_P}=f\cdot t_P=\frac{m}{m_P}=\frac{l_P}{l}=\frac{E}{E_P}=\frac{p}{p_P}=\frac{F}{F_P}$$      

This chain's structure, incorporating both direct (m/m_P) and inverse (l_P/l) proportionalities, is the key. From this chain, 21 pairwise equivalences emerge, each corresponding to a canonical physical law. Thus, physics is revealed not as a collection of unrelated discoveries but as the projection of a single invariant web.

---

2. The Equivalence Chain

At the Planck chart, each axis of measurement has a natural unit: (

        $t_P, l_P, m_P, E_P, p_P, F_P, T_P$
      

). By normalizing measurements to these units, all axes become dimensionless. Equating them in a chain asserts that the universe does not privilege one axis over another—all are slices of a single invariant. The inclusion of inverse relationships like that of length is critical, reflecting the duality inherent at the quantum scale where greater energy implies shorter wavelength.

---

3. Generating Physical Laws

Pairwise comparisons of the seven equivalences yield 21 formulas. The structure of the chain now correctly generates the foundational laws of both relativity and quantum mechanics:

• Mass-Energy Equivalence:

  $\frac{E}{E_P} = \frac{m}{m_P} \implies E = m \left(\frac{E_P}{m_P}\right) \implies E = mc^2$      

• Planck-Einstein Relation:

  $\frac{E}{E_P} = f \cdot t_P \implies E = f (E_P \cdot t_P) \implies E = hf$ 

• de Broglie Relation:

  $\frac{p}{p_P} = \frac{l_P}{l} \implies pl = p_P l_P \implies p = \frac{h}{l}$     

• Compton Wavelength:

  $\frac{m}{m_P} = \frac{l_P}{l} \implies ml = m_P l_P \implies l = \frac{h}{mc}$    

• Boltzmann's Law:

  $\frac{E}{E_P} = \frac{T}{T_P} \implies E = T \left(\frac{E_P}{T_P}\right) \implies E = k_B T$
        

What appear as independent “laws of nature” are simply projections of dimensionless equivalence at the Planck scale, with constants acting as Jacobian factors in SI or Imperial charts.

---

4. Historical Context: Laws as Independent Discoveries

Before this framework, physical laws were generally seen as unrelated discoveries, each emerging independently through observation, experimentation, and theoretical insight:

• Classical mechanics was developed to describe planetary motion and terrestrial phenomena, leading to Newton’s

          $F=m\mathrm{a.}$

• Relativity arose from puzzles in electromagnetism and the constancy of light speed, producing

          $E=mc^2$
  

  and Lorentz invariance.

• Quantum mechanics emerged from blackbody radiation and atomic spectra, introducing

          $E=hf$

  .

• Thermodynamics was formulated independently through studies of heat, energy, and entropy.

Historically, these laws were treated as isolated peaks, with constants like

        $c,h,k_B,G$

acting as empirical bridges rather than structural connectors. Each law had to be “found” separately, and their interrelations were often discovered retrospectively or piecemeal.

The Planck equivalence chain reframes this view: all these laws are now seen as inevitable pairwise projections of a single underlying equivalence. Rather than being independent, each law is a necessary consequence of the dimensionless structure at the Planck scale. This transforms physics from a patchwork of empirical findings into a predictable combinatorial lattice, where the web itself dictates the form of fundamental law.

---

5. Structural Unification vs. Force Unification

Conventional physics seeks to unify the fundamental forces (electromagnetism, weak, strong, gravitation) by reducing them to a single interaction. These programs treat the laws themselves as separate but attempt to merge their underlying fields.

By contrast, the equivalence web offers a higher-order structural unification. It shows that classical mechanics, relativity, quantum theory, and thermodynamics all arise as inevitable pairwise projections of a single equivalence chain. The constants that distinguish these domains—(

        $c, h, k_B, G$
      

)—are not gateways to separate regimes, but conversion factors ensuring that the same invariant ratio web projects coherently across all measurement axes.

Thus, this is not a unification of forces, but a unification of law itself: the recognition that mechanics, relativity, thermodynamics, and quantum theory are not independent domains but are structurally entailed by the same generative principle.

---

6. Extension of the Web

Adding further axes—pressure, density, acceleration, charge—expands the combinatorial law space. With

        $n$

axes, the number of pairwise laws is

        $\binom{n}{2}$
      

. For seven axes we obtain 21 laws; with ten, 45 laws. The apparent proliferation of physical law is an artifact of projection from a compact invariant structure.

---

7. Implications

1. Constants Reinterpreted: The constants

           $c,h,k_B,G$      

   are unit harmonization factors, not metaphysical properties.

2. Laws as Projections: Physical laws are not discovered independently; they are forced pairwise identities in the equivalence web.

3. Structural Unification: Physics reduces to a generative algebra of equivalences. Classical, quantum, relativistic, and thermodynamic laws share the same source.

4. Beyond Force Unification: The equivalence web offers a more radical unity than merging forces: it shows that law itself has a single invariant origin.

---

8. Conclusion

The discovery that canonical physical laws collapse into a single equivalence chain reframes the nature of physics itself. Law is not a patchwork of empirical generalizations but a geometric consequence of the Planck equivalence chain. Constants lose their mystery, and physics is revealed as the projection of a single invariant dimensionless structure. Unlike force unification, which merges interactions while keeping domains of law separate, this framework unifies the very form of law across all domains.

---

Future Work

Future work will catalog the full combinatorial expansion with additional axes, formalize the Jacobian role of constants within categorical frameworks, and explore whether all known physical equations can be derived as pairwise projections of a minimal equivalence chain. The long-term goal is to establish the Planck equivalence web as the structural foundation for a unified physics of law itself.