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.

[fjfaase]

Where does the 64 come from and what do you mean with 'proton mass stability'? The proton is believed to be stable because it is the lowest mass baryon. GUT theories say it might be unstable with a half time of at least 10^34. How does that relate to your number 64? Does the number have a unit?

[albert_roca]

64 is dimensionless. It comes from the model's holographic scaling law, where mass scales with surface complexity (m ∼ 4^i). The proton appears at i = 32.

  4^32= (2^2)^32 = 2^64

2^64 seems to be the minimum information density required to geometrically define a stable volume. The proton stability implies that nothing simpler can sustain a 3D topology. This limit defines the object's topological complexity, not its lifespan.

Please note that the model is being developed with IA assistance, and I realize that the onthological base needs further refinement.

The proton mass (m_p) is derived as:

  m_p = ((√2 · m_P) / 4^32) · (1 + α / 3)
  m_p = ((√2 · m_P) / √4^64) · (1 + α / 3)
  m_p ≈ 1.67260849206 × 10^-27 kg
  Experimental value: 1.67262192595(52) × 10^-27 kg
  ∆: 8 ppm.

G is derived as:

  G = (ħ · c · 2 · (1 + α / 3)^2) / (mp^2 · 4^64)
  G ≈ 6.6742439706 × 10^-11
  Experimental value: 6.67430(15) × 10^-11 m^3 · kg^-1 · s^-2
  ∆: 8 ppm.

α_G is derived as:

  α_G = (2 · (1 + α / 3)^2) / 4^64
  α_G ≈ 5.9061 · 10^–39
  Experimental value: ≈ 5.906 · 10^-39
  ∆: 8 ppm

The terms (1 + α / 3) and 4^64 appear in the three derivations. All of them show the same discrepancy from the experimental value (8 ppm). (Note: There is a typo in the expected output of the previous Python script; it should yield a discrepancy of 8.39 ppm, not 6 ppm.)

The model also derives α as:

  α^-1 = (4 · π^3 + π^2 + π) - (α / 24)
  α^-1 = 137.0359996
  Experimental value: 137.0359991.
  ∆: < 0.005 ppm.

Is it statistically plausible that this happens by chance? Are there any hidden tricks? AI will find a possible conceptualization for (almost) anything, but I'm trying to get an informed human point of view.