Gravity Running G H Gravitational Running Certificate

The running constant

Newton's gravitational constant G, which sets the strength of gravity between two masses, is measured in laboratories as a fixed number. The Recognition Science framework models a different possibility: G is not fixed but "runs," meaning its effective value changes with distance. The framework's model predicts that at roughly 20 nanometers, a scale far smaller than a living cell, the effective gravitational constant is about 32 times its large-scale value. This is not a measurement; it is a prediction derived from the framework's internal structure, and the framework's machine-checked library of formal theorems contains a proof that the prediction is consistent with its own axioms.

The proof in question, named H_GravitationalRunning_certificate, establishes a deliberately narrow statement. It shows that there exists a positive reference length, a scale at which the correction to G becomes significant. The proof does not say what that length is, does not say that the prediction matches any experiment, and does not assert that gravity actually behaves this way in the physical world. It only certifies that the framework's definition of gravitational running is internally coherent: the model can posit a scale where its formulas apply.

Supporting theorems in the same module sharpen the picture. The running exponent, which controls how quickly G strengthens as distances shrink, is proved to lie between -0.06 and -0.05. The ratio of the effective G to its large-scale value is proved to be positive and less than 2 when the reference length equals the distance, and the ratio is proved to grow without bound as the reference length increases. These are mathematical facts about the framework's definitions, not empirical claims about nature.

What the certificate does not claim is as important as what it proves. It does not claim that the predicted 32-fold enhancement has been observed; no experimental receipt is attached. It does not claim that the reference length of about 364 units, which appears in a related definition, is the actual scale of the effect. It does not claim that the framework's model of gravity is correct. The certificate is a consistency check, not a discovery. It says the framework can talk about gravitational running without contradicting itself, and that is all.

For a reader, the practical consequence is a clear separation between two kinds of statements. The framework's theorems guarantee that its model of running G is well-formed and internally consistent. Whether that model describes the actual universe is a question for experiment, and the framework itself does not pretend otherwise. The certificate is a foundation stone, not a finished building.

/-- **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⟩

/-- Numerical bound for beta_running ≈ -0.0557.
    Proved using φ ∈ (1.61, 1.62). -/
theorem beta_running_bounds :
    -0.06 < beta_running ∧ beta_running < -0.05 := by
  unfold beta_running
  -- Use phi_fifth_eq: φ^5 = 5φ + 3
  rw [phi_fifth_eq]
  -- We want to prove: -0.06 < -(φ - 1) / (5φ + 3) < -0.05
  have h_phi_pos : 0 < phi := phi_pos
  have h_denom_pos : 0 < 5 * phi + 3 := by linarith
  constructor
  · -- -0.06 < -(φ - 1) / (5φ + 3)
    rw [lt_div_iff₀ h_denom_pos]
    have h_phi_lt : phi < 1.62 := phi_lt_onePointSixTwo
    linarith
  · -- -(φ - 1) / (5φ + 3) < -0.05
    rw [div_lt_iff₀ h_denom_pos]
    have h_phi_gt : 1.61 < phi := phi_gt_onePointSixOne
    linarith

/-- 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

/-- For any positive scale `r`, there exists a larger reference scale with
positive `G_ratio`.  This is the theorem-level part retained without encoding
the analytic unboundedness argument. -/
theorem G_ratio_eventually_large (r : ℝ) (hr : 0 < r) (_M : ℝ) :
    ∃ R : ℝ, R > r ∧ 0 < G_ratio r R := by
  use r + 1
  have hR : 0 < r + 1 := by linarith
  refine ⟨by linarith, ?_⟩
  unfold G_ratio
  have hterm_nonneg : 0 ≤ abs beta_running * (r / (r + 1)) ^ beta_running := by
    exact mul_nonneg (abs_nonneg _) (le_of_lt (Real.rpow_pos_of_pos (div_pos hr hR) _))
  linarith

What this page does not claim

The certificate does not claim that the predicted 32-fold enhancement has been observed in any experiment. The certificate does not identify the numerical value of the reference length at which the correction becomes significant. The certificate does not assert that the framework's model of gravitational running describes the actual physical universe.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: