Gravity coupling matches the 128-bit integer limit to 6 ppm

[albert_roca]

I'm asking for a sanity check on the hierarchy problem, or the ≈ 10^38 gap between gravity and the strong force.

The results of this test suggest the universe might use a 128-bit integer architecture.

1. HYPOTHESIS

• The proton is defined at the 64-bit integer limit (2^64).

• Gravity is defined at the 128-bit integer limit (2^128).

If this is true, the gravitational coupling constant (alpha_G) should be the least significant bit (LSB) of this structure (2^-127) (1), modulated by the same geometric cost factor found in the proton mass (1 + alpha / 3) (2).

(1) The model derives G from the proton mass scaling, resulting in a factor of 2 / 2^128, which simplifies to 2^-127.

(2) The term (1 + alpha / 3) is the geometric interaction cost derived for the proton mass in the holographic model.

2. FORMULA

  alpha_G_model = (1 + alpha / 3)^2 / 2^127

3. DATA (CODATA 2022)

  mp    = 1.67262192e-27 kg   (proton mass)
  hbar  = 1.05457181e-34 J s
  c     = 299792458 m/s
  G_exp = 6.67430e-11         (experimental G)
  alpha = 7.29735256e-3       (fine-structure constant)

4. VERIFICATION

Standard experimental coupling from G_exp:

  alpha_G_exp = (G_exp * mp^2) / (hbar * c)
  Value: 5.906063e-39

Prediction using ONLY alpha and powers of 2 (no G, no mass):

  alpha_G_model = (1 + 0.00729735256 / 3)^2 * 2^-127

  2^-127 (raw LSB)   ≈ 5.877471e-39
  Correction factor  ≈ 1.004870
  alpha_G_model      ≈ 5.906099e-39

5. RESULTS

  Experimental: 5.906063e-39
  Predicted:    5.906099e-39
  Delta:        3.6e-44
  Discrepancy:  6 ppm

6. PYTHON

  # Constants (CODATA 2022)
  mp    = 1.67262192e-27
  hbar  = 1.05457181e-34
  c     = 299792458
  G     = 6.67430e-11
  alpha = 7.29735256e-3

  # 1. Experimental coupling
  alpha_G_exp = (G * mp**2) / (hbar * c)

  # 2. 128-bit Model Prediction
  alpha_G_model = ((1 + alpha/3)**2) * (2**-127)

  # Comparison
  ppm = abs(alpha_G_model - alpha_G_exp) / alpha_G_exp * 1e6

  print(f"Exp:   {alpha_G_exp:.6e}")
  print(f"Model: {alpha_G_model:.6e}")
  print(f"Diff:  {ppm:.2f} ppm")

7. QUESTION

In your opinion, is this a numerological coincidence, or is it rather a structural feature?

Preprint: https://doi.org/10.5281/zenodo.17847770

[fjfaase]

What is so special about 2^127?

[albert_roca]

The model identifies the proton mass stability at the 64-bit limit (2^64). Since gravitational interaction scales with m_p^2 , the hierarchy gap corresponds to the square of that limit:

  (2^64)^2 = 2^128

The geometric derivation involves a factor of 2, linked to the holographic pixel diagonal (√2 )^2:

  2 / 2^128 = 2^−127

2^−127 represents the least significant bit (LSB) of a 128-bit integer.