Constants Gravitational Constant G Rs Pos

A positive constant

Newton's gravitational constant G is the number that sets the strength of gravity in Newton's law of universal gravitation: the force between two masses is proportional to the product of the masses divided by the square of their distance, and G is the proportionality factor. In the SI system, G is notoriously hard to measure, with a value of approximately 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻² and a relative uncertainty of about 2.2 × 10⁻⁵. It is one of the least precisely known fundamental constants of physics.

The Recognition Science framework treats G not as a measured input but as a derived quantity. In its native units, the framework sets the fundamental recognition wavelength λ_rec = 1 and the speed of light c = 1, and it defines the Planck constant as hbar = φ⁻⁵, where φ is the golden ratio (1 + √5)/2. The framework's derivation gives G = λ_rec² · c³ / (π · hbar), which reduces to G = φ⁵ / π. Numerically, φ⁵ ≈ 11.09 and π ≈ 3.14, so this gives G ≈ 3.53 in these native units.

Within the framework, the declaration G_rs_pos is a formal theorem in the machine-checked library of formal theorems. It states that the derived value G_rs = φ⁵ / π is strictly greater than zero. The proof is direct: φ is positive, so its fifth power is positive; π is positive; and the quotient of two positive numbers is positive. The theorem is a small but necessary piece of the framework's structure, since many later statements about gravity require knowing that G is a positive quantity.

The framework's library also records a stronger combined statement, gravitational_constant_derived, which asserts both that G_rs is positive and that it equals φ⁵ / π. This is the formal resolution of the framework's registry item C-002: what determines the gravitational constant. The framework's claim is that G has no free parameters; it is fixed by the golden ratio and pi through the ledger geometry of recognition events.

What G_rs_pos does not claim is just as important. It does not assert that the framework's derived numerical value matches the measured SI value of G. The framework's value is in its own native units, and converting it to SI requires a scale for length, time, and mass that the framework does not provide in this declaration. The theorem also does not claim that the derivation is complete: it establishes positivity and the algebraic form, but the physical bridge from recognition events to Newtonian gravity is a separate, open question in the framework.

/-- Newton's gravitational constant G in RS-native units.
    G = λ²_rec · c³ / (π · ℏ) with λ_rec = c = 1, ℏ = φ⁻⁵.
    Thus G = φ⁵ / π. -/
noncomputable def G_rs : ℝ := phi ^ 5 / Real.pi

/-- G > 0. -/
theorem G_rs_pos : 0 < G_rs := by
  unfold G_rs
  apply div_pos
  · exact pow_pos phi_pos 5
  · exact Real.pi_pos

/-- **C-002 Resolution**: The gravitational constant is determined by φ and π.

    G = φ⁵/π has no free parameters. It arises from the ledger geometry:
    - λ_rec: the fundamental recognition wavelength (ℓ₀ = 1 in RS units)
    - c: speed of light (1 in RS units)
    - ℏ: Planck constant (E_coh = φ⁻⁵ in RS units)

    The "least precisely known" constant in SI becomes a derived quantity. -/
theorem gravitational_constant_derived :
    0 < G_rs ∧ G_rs = phi ^ 5 / Real.pi :=
  ⟨G_rs_pos, rfl⟩

What this page does not claim

The derived value of G matches the measured SI value of G. The derivation of G is complete and physically justified. The framework provides a unit conversion from its native units to SI.

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/Constants/GravitationalConstant.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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