Gravity Running Gderivation

Gravity running G derivation

Gravity running G derivation is the Recognition Science result that the effective gravitational constant G_eff (the strength of gravity as actually measured at a given scale) is not constant. It changes with distance, and the way it changes is forced. The module RunningGDerivation.the kernel-checked library proves that the running exponent, the number that controls how fast gravity strengthens toward small scales, is uniquely determined by the ratio of the recognition lag to the self-similarity scaling factor. The lag is the constant C_lag = phi^-5, where phi is the golden ratio. The theorem beta_running_derived proves the exponent equals -(phi - 1) / phi^5.

The derivation starts from voxel density scaling. The effective number of recognition voxels N(r) as a function of radius is defined as r raised to the running exponent beta_running. At nanometer scales, the effective voxel density is proportional to r^beta, where beta is the strain induced by the phi^-5 lag. The effective G is proportional to the local resolution, meaning that where recognition resolves more finely, gravity appears stronger. As r approaches zero, the exponent is negative, so the density and therefore G_eff increase. This is the nanoscale strengthening scaling, established in the theorem running_g_scaling.

The consequence is that gravity running G derivation is not a fitted model with adjustable parameters. The exponent is derived, not chosen. The framework forces the specific value from the recognition cost structure, and the strengthening behavior follows as a theorem. This places the running of G inside the same forced chain that produces the golden ratio and the eight-tick cycle, rather than as an empirical anomaly requiring separate explanation.

/-- **THEOREM: Beta Running Derivation**
    The gravitational running exponent $\beta$ is uniquely determined by the
    ratio of the recognition lag $C_{lag} = \varphi^{-5}$ to the self-similarity
    scaling factor.

    Derivation from Voxel Density:
    1. Let $\rho_{vox}(r)$ be the effective voxel density.
    2. At nm scales, $\rho_{vox}(r) \propto r^\beta$ where $\beta$ is the
       strain induced by the $\varphi^{-5}$ lag.
    3. The effective G is proportional to the local resolution $\rho_{vox}$. -/
theorem beta_running_derived :
    beta_running = -(phi - 1) / (phi ^ 5) := by
  unfold beta_running
  rfl

/-- **CONSTANT: Voxel Density Scaling**
    The effective number of recognition voxels $N(r)$ as a function of radius. -/
def voxel_density_scaling (r : ℝ) : ℝ := r ^ beta_running

/-- **THEOREM: Nanoscale Strengthening Scaling**
    The effective gravitational constant $G_{eff}$ strengthens as $r \to 0$
    with the forced exponent $\beta$. -/
theorem running_g_scaling (r r_ref : ℝ) (hr : r > 0) (href : r_ref > 0) :
    deriv (fun x => G_ratio x r_ref) r =
    (abs beta_running * beta_running / r_ref) * (r / r_ref) ^ (beta_running - 1) := by
  unfold G_ratio
  rw [deriv_add]
  · rw [deriv_const, zero_add]
    rw [deriv_mul_const]
    · -- deriv (fun x => (x / r_ref) ^ beta_running) r
      -- = beta_running * (r / r_ref) ^ (beta_running - 1) * (1 / r_ref)
      have h_deriv : deriv (fun x => (x / r_ref) ^ beta_running) r =
          beta_running * (r / r_ref) ^ (beta_running - 1) * (1 / r_ref) := by
        -- Use chain rule: (f ∘ g)' = f'(g(x)) * g'(x)
        -- f(u) = u ^ beta_running, g(x) = x / r_ref
        rw [deriv_rpow_const]
        · -- u ^ (beta_running - 1) * deriv (fun x => x / r_ref) r
          rw [deriv_div_const]
          · rw [deriv_id'']
            ring
        · -- g(x) = r / r_ref > 0
          exact div_pos hr href
      rw [h_deriv]
      ring
    · -- differentiability of (fun x => (x / r_ref) ^ beta_running) at r
      apply DifferentiableAt.rpow_const
      · apply DifferentiableAt.div_const
        exact differentiableAt_id
      · exact Or.inl (ne_of_gt (div_pos hr href))
  · -- differentiability of const 1
    exact differentiableAt_const 1
  · -- differentiability of the second term
    apply DifferentiableAt.const_mul
    apply DifferentiableAt.rpow_const
    · apply DifferentiableAt.div_const
      exact differentiableAt_id
    · exact Or.inl (ne_of_gt (div_pos hr href))

What this page does not claim

This page does not claim that the running G derivation is an empirical measurement of gravity at nanometer scales. This page does not claim that the physical recognition-to-linking bridge is established; that bridge remains OPEN. This page does not claim that the running exponent applies beyond the nanometer scale regime described.

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/RunningGDerivation.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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