H_GPrecision
plain-language theorem explainer
H_GPrecision encodes the empirical hypothesis that the Recognition Science-derived gravitational constant agrees with the CODATA 2018 value to within 22 ppm. Metrologists and theorists comparing first-principles constant derivations to experiment would cite it when testing the Planck gate identity. The body is a direct existential statement that bounds the absolute difference by an error smaller than 10^{-15}.
Claim. Let $G_{RS}$ be the gravitational constant obtained from the Recognition Science Planck gate identity $G = λ_{rec}^2 c^3 / (π ħ)$. The hypothesis asserts that there exists a real number $ε$ such that $|G_{RS} - 6.67430 × 10^{-11}| < ε$ and $ε < 10^{-15}$.
background
The module Phase 12.3 evaluates the gravitational constant to six or more significant figures from the Planck gate identity. Upstream Constants.G supplies the RS-native form $G = λ_{rec}^2 c^3 / (π ħ)$, while Constants.Codata.G hard-codes the benchmark value 6.67430e-11. JCostInflaton.G re-expresses the underlying J-cost in log coordinates as $G(t) = cosh(t) - 1$, and LogicAsFunctionalEquation.Identity guarantees that self-comparison costs zero, anchoring the functional equation derivations.
proof idea
This is a definition whose body is the existential proposition itself. No lemmas are invoked; the statement directly encodes the error-bound hypothesis. The single downstream use is the theorem gravitational_constant_precision, which simply unwraps the definition to assert the precision claim.
why it matters
H_GPrecision supplies the hypothesis for the theorem gravitational_constant_precision, closing the loop from the derived constants (T5 J-uniqueness, T6 phi fixed point, eight-tick octave) to the observable value of G. It tests the RS prediction $G = φ^5 / π$ in native units against the 22 ppm tolerance, with the alpha band and D = 3 as related landmarks. The open question is whether future measurements remain inside the stated error.
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