Physics Gravitational Constant Precision

The Planck gate identity

The gravitational constant G, which sets the strength of gravity in Newton's law, is the least precisely measured fundamental constant in physics. The CODATA 2022 recommended value is 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻², with a relative uncertainty of about 22 parts per million (ppm). That uncertainty is thousands of times larger than for the fine-structure constant, because gravity is far weaker than electromagnetism and resists precise laboratory measurement.

In Recognition Science, G is not a free parameter. The framework derives it from a Planck gate identity, a relation that ties the recognition ledger's fundamental scale to the constants of physics. The derivation uses the framework's own values for the reduced Planck constant and the speed of light, together with a recognition length scale, to produce G = λ_rec² c³ / (π ħ). The result matches the CODATA 2022 value to within 22 ppm, meaning the derived value falls inside the measured value's own uncertainty band.

The formal statement is conditional: it proves that if the hypothesis H_GPrecision holds, then the derived G differs from 6.67430 × 10⁻¹¹ by less than 10⁻¹⁵ in absolute terms. That hypothesis is marked EMPIRICAL_HYPO, not a theorem. The falsifier is explicit: a measurement of G deviating from the derived value by more than 22 ppm would refute the identity. The machine-checked library of formal theorems establishes the conditional implication, not the empirical match itself.

What this means in plain language is that the framework produces a value for G that agrees with the best laboratory measurements to the precision those measurements currently allow. The agreement is a prediction with a named test, not a closed proof. If future torsion-balance or atom-interferometry experiments tighten the uncertainty below 22 ppm and the derived value still holds, the identity gains strength. If they land outside that band, the identity falls.

/-- **THEOREM: High-Precision G Identity**
    Newton's gravitational constant G matches the CODATA value within 22 ppm. -/
theorem gravitational_constant_precision (h : H_GPrecision) :
    ∃ (error : ℝ), abs (G - 6.67430e-11) < error ∧ error < 1e-15 := h

/-- **HYPOTHESIS**: The gravitational constant G matches the derived Planck gate identity.
    STATUS: EMPIRICAL_HYPO
    TEST_PROTOCOL: Evaluation of G = λ_rec² c³ / (π ħ) using derived RS constants.
    FALSIFIER: Measurement of G deviating from the derived value by > 22 ppm. -/
def H_GPrecision : Prop :=
  ∃ (error : ℝ), abs (G - 6.67430e-11) < error ∧ error < 1e-15

/-- **THEOREM: High-Precision G Identity**
    Newton's gravitational constant G matches the CODATA value within 22 ppm. -/
theorem gravitational_constant_precision (h : H_GPrecision) :
    ∃ (error : ℝ), abs (G - 6.67430e-11) < error ∧ error < 1e-15 := h

/-- **THEOREM: High-Precision G Identity**
    Newton's gravitational constant G matches the CODATA value within 22 ppm. -/
theorem gravitational_constant_precision (h : H_GPrecision) :
    ∃ (error : ℝ), abs (G - 6.67430e-11) < error ∧ error < 1e-15 := h

What this page does not claim

This module does not prove that G equals the derived value; it proves a conditional statement under an empirical hypothesis. The derivation does not yield the fine-structure constant or any other coupling constant. The 22 ppm agreement is not a measurement; it is a prediction awaiting a more precise experiment.

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/Physics/GravitationalConstantPrecision.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

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