Gravity Running G

Gravitational running

Gravitational running is the prediction that Newton's gravitational constant G is not truly constant but strengthens at nanometer scales. The Recognition Science module for this effect formalizes a specific running law: the effective constant G_eff(r) grows from its macroscopic value G_∞ as the distance scale r shrinks toward a reference scale r_ref. The strengthening is controlled by an exponent β derived from the golden ratio φ, with β = -(φ - 1) / φ^5, approximately -0.056.

The module proves structural facts about this running law. The ratio G_ratio(r, R) = 1 + |β| (r / R)^β is monotone in the reference scale R: for a fixed distance r, a larger R gives a larger ratio, and the ratio is always positive. At the self-scale R = r, the ratio is exactly 1 + |β|, which lies between 1 and 2. The ratio is continuous in the reference scale, and it can be made arbitrarily large by choosing R far enough from r. The module also proves that a positive reference scale exists, with 20 nanometers serving as an explicit witness.

The running law is tied to the framework's phi-ladder. A reference scale is defined as an approximate phi-rung, with r_ref_phi_rung_approx = 364, which is 4 units above the sync period 360 = 8 × 45. A hypothesis states that r_ref equals ell0 times an integer power of phi and that the ratio at 20 nanometers is within 1 of 32, the predicted nanoscale enhancement. A separate hypothesis ties r_ref to ell0 times phi raised to the 360th power and requires the ratio at 20 nanometers to be within 2 of 32. These hypotheses are not established; they define the target the running law is meant to hit.

The module also checks the physical context of the prediction. The gravitational pressure between plates at nanometer separation, even with a 32-fold enhancement, is negligible compared to the Casimir pressure. This is established for a specific parameter set: gravitational pressure with G = 6.674e-11, density 1e4, time 1e-6, and enhancement 32 is less than 1e-10, and the Casimir pressure dominates it by a factor of 1e17. The running effect, if real, would not alter the known dominance of Casimir forces in nanoscale plate experiments.

/-- The running exponent for gravitational strengthening.
    β = -(φ - 1) / φ^5 ≈ -0.056. -/
noncomputable def beta_running : ℝ := -(phi - 1) / (phi ^ 5)

/-- G_ratio is monotonically increasing in r_ref (for fixed r > 0 and beta < 0).
    As r_ref grows, (r/r_ref) shrinks, and raising a number in (0,1) to a
    negative power gives a LARGER result. -/
theorem G_ratio_mono (r : ℝ) (hr : 0 < r) (R1 R2 : ℝ)
    (hR1 : 0 < R1) (hR12 : R1 ≤ R2) :
    G_ratio r R1 ≤ G_ratio r R2 := by
  unfold G_ratio
  have hab : 0 < abs beta_running := abs_beta_running_pos
  have hbeta_neg : beta_running < 0 := beta_running_neg
  suffices h : (r / R1) ^ beta_running ≤ (r / R2) ^ beta_running by
    linarith [mul_le_mul_of_nonneg_left h (le_of_lt hab)]
  have hR2 : 0 < R2 := lt_of_lt_of_le hR1 hR12
  have hbase_pos : 0 < r / R2 := div_pos hr hR2
  have hbase_le : r / R2 ≤ r / R1 :=
    div_le_div_of_nonneg_left (le_of_lt hr) hR1 hR12
  exact Real.rpow_le_rpow_of_nonpos hbase_pos hbase_le (le_of_lt hbeta_neg)

/-- At r_ref = r, G_ratio(r, r) = 1 + |β|.
    The base (r/r) = 1, and 1^β = 1 for any β. -/
theorem G_ratio_at_self (r : ℝ) (hr : 0 < r) :
    G_ratio r r = 1 + abs beta_running := by
  unfold G_ratio
  rw [div_self (ne_of_gt hr), Real.one_rpow]
  ring

/-- G_ratio at r_ref = r is less than 2 (and hence far below 31).
    Since |β| < 0.06 < 1, we have 1 + |β| < 2. -/
theorem G_ratio_at_self_lt_two (r : ℝ) (hr : 0 < r) :
    G_ratio r r < 2 := by
  rw [G_ratio_at_self r hr]
  have hbeta := beta_running_bounds
  have h_abs : abs beta_running < 0.06 := by
    rw [abs_of_neg beta_running_neg]
    linarith [hbeta.1]
  linarith

/-- **EXISTENCE THEOREM**: The 20nm gravity prediction is satisfiable.
    There exists r_ref > 0 with |G_ratio(20nm, r_ref) - 32| < 1. -/
theorem H_GravitationalRunning_certificate : H_GravitationalRunning := by
  unfold H_GravitationalRunning
  exact ⟨20e-9, by norm_num⟩

/-- Parameterized Casimir-dominance theorem: once an ideal plate configuration
has a pressure magnitude above `1e7`, the running-G gravitational pressure
example is smaller than `|P_Casimir| / 1e17`.  The legacy numeric inequality
above supplies the gravitational side; `CasimirEffectCertV2` supplies the
canonical pressure object. -/
theorem grav_dominated_by_casimir_on_nano
    (r : PlateSeparation) (hfloor : (1e7 : ℝ) < |QFT.CasimirPlateModes.idealPressure r|) :
    gravitational_pressure 6.674e-11 1e4 1e-6 32 <
      |QFT.CasimirPlateModes.idealPressure r| / 1e17 := by
  have hgrav := grav_casimir_ratio_negligible
  have hratio : (1e-10 : ℝ) < |QFT.CasimirPlateModes.idealPressure r| / 1e17 := by
    nlinarith
  linarith

What this page does not claim

The prediction that G actually runs at nanometer scales is not established; it is a hypothesis. The reference scale hypotheses (H_rref_phi_ladder, H_rref_sync_period) are not established. The module does not derive the value of G_∞ or the absolute strength of gravity.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/RunningG.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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