Constants Gravitational Constant Gravitational Constant Derived

The derived constant

Newton's gravitational constant G measures the strength of gravity: it appears in Newton's law of universal gravitation, F = G m₁m₂/r², and sets the force between two masses at a given distance. In SI units, G ≈ 6.674 × 10⁻¹¹ N·m²/kg², and it is notoriously difficult to measure precisely because gravity is so weak compared to other forces. The CODATA 2022 recommended value has a relative uncertainty of about 2.2 × 10⁻⁵, making G the least precisely known fundamental constant of the SI system.

Recognition Science (RS) proposes a different origin for G. In RS-native units, the framework derives G = φ⁵/π, where φ = (1 + √5)/2 ≈ 1.618 is the golden ratio and π ≈ 3.14159 is the circle constant. This expression has no free parameters: it is a pure ratio of two mathematical constants. The derivation starts from the framework's central result, the cost function J(x) = (x + 1/x)/2 − 1, which is forced by five plain conditions including reciprocal symmetry and a composition law. From J, the framework derives the golden ratio as the unique self-similar scaling, and from that scaling it derives the constants of nature.

The specific declaration named gravitational_constant_derived in the framework's machine-checked library of formal theorems establishes two facts about G_rs, the RS-defined gravitational constant. First, it proves that G_rs is positive: G_rs > 0. Second, it proves that G_rs equals φ⁵/π by definition. The declaration is a theorem in the library, meaning it is checked by a computer proof assistant and rests on no framework-specific axioms beyond the standard logical postulates of the ambient type theory.

What this declaration does not claim is equally important. It does not claim that the numerical value φ⁵/π matches the measured SI value of G to within experimental uncertainty. That comparison is an empirical check, not a theorem. The declaration establishes the formal definition and its positivity, nothing more. It also does not claim that the derivation of G from the cost function is complete in the sense of being fully formalized; the chain from J to φ to G is partly documented in the library, but the physical bridge from recognition events to the gravitational constant remains a target for further formalization.

In plain terms, the declaration says: within the framework, the gravitational constant is defined as the ratio φ⁵/π, and this ratio is positive. It is a clean formal statement, but it does not by itself prove that this ratio equals the measured G. That is a separate, empirical question that the framework addresses elsewhere, not in this declaration.

/-- 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

/-- **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 declaration does not claim that φ⁵/π numerically equals the measured SI value of G. The declaration does not claim that the full physical derivation from recognition events to gravity is formalized; that bridge remains open. The declaration does not claim that G is a free parameter; it is defined, not fitted.

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)

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: