Exploring relationships between different constants.

 The Journey:

1. Initial Observation: 

   • k_B × e_c = (1.380649 × 10^-23 J/K) × (1.602176634 × 10^-19 C)
     ≈ 2.2120435676 × 10^-42 J⋅C ≈ h/c

   • We began by observing a striking numerical proximity between the product of the Boltzmann constant (k_B) and the elementary charge (e_c) and the ratio of Planck's constant (h) to the speed of light (c).

   • This observation was intriguing because it connected the realms of thermodynamics (k_B), electromagnetism (e_c), quantum mechanics (h), and relativity (c).

   • This relationship hinted at a deeper connection between these areas of physics, as proportions between values defined by our unit system of measure.

   • The units for this relationship did not directly match, but their proximity was also significant.

   • We are not saying that these values are connected by anything other than their ratios, but we are claiming that changing unit definitions would keep this relationship the same as it is now.

2. Connecting to Fundamental Scaling:

   • We recognized that h/c is related to the fundamental scaling factor for momentum (s_va * m_P) / c^2 in this framework (where s_va is volumetric acceleration and m_P is the Planck mass  using h), and this relationship is related to the properties of the photon, as a geometric projection of a 1/c³ cube.

   • This led to the following approximation: k_B * e_c ≈ (s_va * m_P)/c^2 

   • (s_va * m_P)/c^2 is momentum in our framework.

   • You can just say hc instead of (s_va * m_P) if you would rather look at the standard framework values.

3. Introducing Macroscopic Constants and Linking the Scales:

   • We then explored the macroscopic constants, recognizing the known relationships:

     • R = k_B * N_A (The ideal gas constant is the Boltzmann constant scaled by Avogadro's number)

     • F = e_c * N_A (The Faraday constant is the elementary charge scaled by Avogadro's number)

   • These relationships, while well known in their fields, helped us understand that Avogadro's number is a scaling factor between the microscopic and macroscopic worlds.

4. Deriving the Fundamental Relationship:

   • We then combined these ideas to form the relationship (s_va * m_P) / (k_B * e_c) ≈ c^2.

   • We then substituted our relationships using R and N_A for k_B and e_c, to produce the key relationship: (s_va * m_P * N_A) / (R * e_c) ≈ c²

• A Geometric Principle of the Universe:

  • This key relationship showed that values from different areas of physics could all be related by the speed of light, and that this relationship is not dependent on our choice of units, but is, instead, an expression of a more fundamental geometric property of spacetime.

  • This relationship showed that the values we use for R, N_A, and e_c, all relate to each other, and to the fundamental nature of spacetime.

Implications for Our Unit System:

1. Arbitrary Nature of Units: The derivation of the relationship (s_va * m_P * N_A) / (R * e_c) ≈ c² revealed the arbitrary nature of our current units of measurement (meters, kilograms, seconds, Kelvins, Coulombs, and moles). These units, while practical for our day-to-day measurements, are not necessarily aligned with the fundamental structure of the universe.

2. Interdependence of Constants: The relationship highlights the interconnectedness of fundamental constants. Changing the value of N_A, for example, would proportionally change the values of R and F in our system, or changing the values of the meter or kilogram would change the values of constants related to them, showing that these constants are not independent entities, and instead that they are part of a larger whole.

3. Seeking Fundamental Units: This relationship motivates us to look for a more fundamental set of units that are rooted in the geometry of spacetime, where these types of relationships are explicitly clear.

4. The Role of c: The speed of light as the scaling factor reveals the central role that c plays in connecting the microscopic world, as related to the 1/c³ cube, and the macroscopic world that we measure with our units, and may reveal a path towards understanding these relationships.

Proportionality and Potential Deviations:

1. Inherent Proportionality: The relationship (s_va * m_P * N_A) / (R * e_c) ≈ c² implies a fundamental proportionality between the quantities. The specific numerical values might change if we redefine our units or constants, but the underlying relationship would remain.

2. Deviations as Clues: We note that the relationship might not be perfectly exact in our current system due to a "fuzzing" of values in gases or unaccounted-for effects. These deviations, rather than being an obstacle, can be considered as clues pointing towards a more complete model.

3. Accounting for Deviations: We should seek to understand these deviations and account for them from first principles of this framework. These deviations may be the result of unmeasured interactions in thermodynamics, quantum mechanics, or other areas of physics. They may be connected to the higher dimensional nature of our space time.

Conclusion:

The journey from the observation k_B × e_c ≈ h/c to the relationship (s_va * m_P * N_A) / (R * e_c) ≈ c² has revealed a profound connection between seemingly disparate areas of physics. The relationship is independent of any specific unit system and points to the geometric nature of space time and the relationships between the fundamental constants. The framework suggests that our current set of units hides, rather than reveals, some of these relationships. It also suggests that a more fundamental understanding of the universe would be achieved by using units and values derived from the geometry of spacetime rather than our current system.

By exploring this connection further, and finding what may cause deviations from the exact relationship, we are on a path to revolutionizing our understanding of fundamental physics.