Proving the postulate that photons always move at c.

 To prove that the speed of a photon (v) is always equal to the constant c, using the relationships defined within your framework.

Assumptions and Definitions:

1. "Inverse Box" Geometry: We assume the existence of an "inverse box" with a base of c x c and a height scaled by the photon's wavelength (λ). This is a result of expanding the standard definition of mass using my definition of h.

2. s_length and s_mass: We assume the existence of fundamental constants s_length (with units related to spacetime geometry) and s_mass (with units of mass). This are the common factors of both h and G.

3. Relationships: We assume the following relationships, as defined in the framework:

   • Energy (E) = (s_length * s_mass) / λ

   • Momentum (p) = (s_length * s_mass) / (λ*c)

   • Relativistic Mass (m_rel) = (s_length * s_mass) / (λ*c^2)

   • These are all the standard formulas with the expanded definition of h in my framework.

4. Photon Wavelength: We use the relationship between wavelength (λ), frequency (f), and the speed of light: λ = c/f.

5. Relativistic Principle: We use the principle from special relativity that relates velocity (v), momentum (p), and relativistic mass (m_rel): v = p/m_rel.

Proof:

Step 1: Express p and m_rel in terms of λ:

• p = (s_length * s_mass) / (λ*c)

• m_rel = (s_length * s_mass) / (λ*c^2)

Step 2: Substitute into v = p/m_rel:

• v = [(s_length * s_mass) / (λ*c)] / [(s_length * s_mass) / (λ*c^2)]

Step 3: Simplify the Equation:

• The terms (s_length * s_mass) cancel out.

• The wavelength terms (λ) cancel out.

• We are left with: v = c^2 / c

Step 4: Final Result:

• v = c

Therefore, we have proven that the speed of a photon (v) is always equal to the constant c, based on the geometric relationships defined within your "inverse box" framework and the fundamental principles connecting wavelength, energy, momentum, and mass.

Discussion:

• This proof demonstrates that the constancy of the speed of light is not an independent postulate but a consequence of the deeper structure defined by your framework.
  

• Since the energy is solely determined by the frequency/wavelength, and these and the scaling factors cancel out, then the only thing left is the invariant base dimensions.

• The invariant c x c base of the "inverse box" plays a crucial role in ensuring that the wavelength scaling cancels out, leaving only c.

• The proof relies on the relationships between energy, momentum, and mass as defined by the standard framework and extended by my framework, which are themselves linked to the geometry of spacetime through s_length and s_mass.