Constants Codata G Pos

The positivity lemma

Newton's gravitational constant G, the number that sets the strength of gravity in Newton's law and in general relativity, is a positive quantity. In the International System of Units, its accepted value is about 6.67430 × 10⁻¹¹ cubic meters per kilogram per second squared. The Recognition Science framework stores this empirical value in a separate, quarantined module, clearly marked as CODATA 2018 reference data, so that the framework's derived constants never silently depend on measured numbers.

Within that module, the declaration G_pos is a machine-checked lemma stating that 0 < G. Its proof is direct: it unfolds the definition of G as the decimal number 6.67430e-11 and verifies by numeric computation that this value is greater than zero. The same pattern establishes positivity for the speed of light c and the reduced Planck constant hbar, and from each positivity result the framework derives the corresponding non-zero statement, such as G_ne_zero.

In Recognition Science, the framework's own derived value for G is not this CODATA number. The framework derives G = φ⁵/π from its forcing chain, where φ is the golden ratio, as a theorem rather than as a free parameter. The CODATA constants live in a separate namespace precisely to keep those two kinds of content apart: the derived constants form the certified surface, while the empirical values are available only for explicit comparison and reporting.

What G_pos does not claim is any physical content beyond the arithmetic fact. It does not assert that gravity is attractive, that the measured value is exact, or that the CODATA 2018 number is the framework's own derived constant. It is a bookkeeping lemma: a guarantee that the stored reference value is a positive real number, so that later code and proofs can rely on that fact without re-checking the decimal expansion.

lemma G_pos : 0 < G := by unfold G; norm_num

/-- RS-native gravitational coupling projection through the recognition/Planck
    bridge: \(G = \lambda_{\text{rec}}^2 c^3 / (\pi \hbar)\).

    This is not a prediction of the SI value of Newton's constant.  SI conversion
    requires the dimensional bridge in `Foundation/SIBridgeClosure.lean`. -/
noncomputable def G : ℝ := (lambda_rec^2) * (c^3) / (Real.pi * hbar)

What this page does not claim

G_pos does not assert that gravity is attractive or that the measured value is exact. G_pos does not claim that the CODATA 2018 number is the framework's own derived constant.

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/Codata.lean
$ lake env lean IndisputableMonolith/Constants.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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