Formal Proof: Physical Constants are Composite Jacobians

 J. Rogers, SE Ohio

Theorem

All fundamental physical constants are composite Jacobians that scale between arbitrary measurement systems and the natural ratios of the Planck scale.

Proof

Part 1: Establishing the Nature of Measurement Systems

Lemma 1.1: Any measurement system consists of arbitrarily scaled base units.

Proof: By definition, measurement systems like SI establish base units (meter, kilogram, second) through:

• Historical conventions (e.g., meter originally defined as fraction of Earth's circumference)
• Anthropocentric scaling (e.g., units sized for human use)
• Independent treatment of dimensions (treating length, mass, time as orthogonal)

The specific numerical values of these units are arbitrary choices, not determined by physical necessity. ∎

Lemma 1.2: Physical laws express relationships between quantities that exist independently of measurement systems.

Proof: Physical phenomena (e.g., gravitational attraction, electromagnetic interactions) occur in nature regardless of how humans choose to measure them. The relationships between physical quantities must therefore be independent of any specific measurement system. ∎

Part 2: The Dimensional Structure of Physical Laws

Lemma 2.1: Physical laws require dimensional consistency.

Proof: Any meaningful physical equation must maintain dimensional homogeneity. For an equation A = B to be valid, [A] = [B] where [X] denotes the dimensions of X. This is a fundamental requirement of dimensional analysis. ∎

Lemma 2.2: When expressed in arbitrary measurement systems, physical laws often require constants to maintain dimensional consistency.

Proof: Consider Newton's law of gravitation F ∝ m₁m₂/r². In dimensional terms: [F] = [M][L][T]⁻² [m₁m₂/r²] = [M]²[L]⁻²

Since [F] ≠ [m₁m₂/r²], a constant k with dimensions [k] = [L]³[M]⁻¹[T]⁻² is required to maintain dimensional consistency. Similar dimensional mismatches requiring constants exist throughout physics. ∎

Part 3: The Planck Unit System as Natural Reference

Lemma 3.1: Planck units form a complete system of natural units.

Proof: Planck units are defined as:

• l_P = √(hG/c³) (Planck length)
• m_P = √(hc/G) (Planck mass)
• t_P = √(hG/c⁵) (Planck time)
• T_P = √(hc⁵/Gk_B²) (Planck temperature)

These form a complete system where all physical quantities can be expressed as dimensionless ratios. ∎

Lemma 3.2: At the Planck scale, physical laws reduce to simple relationships between dimensionless ratios.

Proof: When all quantities are expressed in Planck units, they become dimensionless ratios. For example:

• E/E_P = m/m_P (energy-mass equivalence)
• F/F_P = (m₁/m_P)(m₂/m_P)(r_P/r)² (gravitational force)

These relationships contain no physical constants, only pure numerical ratios. ∎

Part 4: Constants as Composite Jacobians

Lemma 4.1: Physical constants can be factored into products of Planck units.

Proof: By direct computation:

• c = l_P/t_P
• h = m_P × l_P²/t_P
• G = l_P³/(m_P × t_P²)
• k_B = m_P × l_P²/(t_P² × T_P)

Each constant is a specific combination of Planck units raised to appropriate powers. ∎

Lemma 4.2: The dimensional structure of each constant corresponds to the Jacobian matrix needed to transform between arbitrary units and Planck units.

Proof: In mathematics, a Jacobian matrix represents the scaling factors needed to transform between coordinate systems. For physical constants:

• c = l_P/t_P represents the scaling between length and time dimensions
• G = l_P³/(m_P × t_P²) represents the scaling between mass, length, and time dimensions
• The dimensional structure of each constant precisely matches the transformation needed to convert from arbitrary units to natural Planck-scale ratios ∎

Lemma 4.3: Constants can be algorithmically constructed from first principles.

Proof: Given any physical law with dimensional mismatch:

1. Identify the dimensional signature required: [k] = [Output]/[Input]
2. Construct the constant by replacing abstract dimensions with Planck units: k = l_P^a × m_P^b × t_P^c × T_P^d
3. The resulting expression will match the empirically determined constant

This algorithm works for all physical constants, demonstrating they are constructed rather than discovered. ∎

Part 5: The Function of Constants as Scaling Operators

Lemma 5.1: Constants function as scaling operators that transform measurements from arbitrary units to natural ratios.

Proof: Consider any physical quantity Q measured in arbitrary units. Its natural ratio is Q/Q_P. To express this in arbitrary units: Q/Q_P = Q_arbitrary/Q_P

Rearranging: Q_arbitrary = (Q/Q_P) × Q_P

The term Q_P represents the scaling factor (Jacobian) needed to convert from natural ratios to arbitrary units. For composite quantities, these scaling factors combine to form the physical constants. ∎

Lemma 5.2: The numerical value of any constant depends only on the choice of measurement system, not on fundamental properties of nature.

Proof: Since constants are composites of Planck units, and Planck units have fixed values in nature, the numerical value of a constant in any system depends solely on how that system defines its base units relative to Planck units. Different measurement systems will assign different numerical values to the same physical constant, demonstrating that constants are properties of measurement systems, not of nature itself. ∎

Conclusion

By the above lemmas, we have proven that:

1. Physical constants are composite quantities constructed from Planck units
2. Planck units function as fundamental Jacobians (scaling factors)
3. Constants therefore function as composite Jacobians
4. Their mathematical role is to scale between arbitrary measurement systems and the natural ratios of the Planck scale
5. Physics fundamentally operates through these natural ratios at the Planck scale
6. The apparent complexity of physical laws emerges from our choice to use arbitrarily scaled, fragmented measurement systems

Therefore, all fundamental physical constants are composite Jacobians that scale between arbitrary measurement systems and the natural ratios of the Planck scale. ∎