H_rref_sync_period
plain-language theorem explainer
The definition asserts existence of a reference scale r_ref equal to the fundamental length ell0 multiplied by phi to the power 360. This scale is chosen so that the running gravitational ratio at 20 nm is within 2 of 32. Researchers testing gravity at nanoscale distances would reference this to link the enhancement directly to the D=3 forcing chain. The construction is an explicit definition without additional lemmas.
Claim. There exists a positive real number $r_0$ such that $r_0 = ell0 * phi^{360}$ and $|G_ratio(20*10^{-9}, r_0) - 32| < 2$, where $G_ratio$ is the enhancement factor of Newton's constant at the given distance relative to its macroscopic value.
background
The RunningG module formalizes the prediction that Newton's constant runs at nanometer scales, with $G_eff(r) = G_∞ * (1 + |β| * (r / r_ref)^β)$ and β derived from the φ-ladder as approximately -0.056. ell0 is the fundamental RS length unit (voxel size), equal to 1 in native units. G_ratio(r, r_ref) computes the numerical enhancement at distance r when the reference scale is r_ref, drawing on the J-cost reparametrization G(t) = cosh(t) - 1 from the functional equation.
proof idea
This is a direct definition that unfolds the proposition by exhibiting the explicit form r_ref = ell0 * phi^360 together with the positivity condition and the numerical bound on the ratio at 20 nm. No lemmas are applied; the body simply packages the sync-period scale and the target enhancement of 32.
why it matters
This anchors the nanometer-scale running-G prediction to the forcing chain, specifically the sync period arising from D=3 (T8) and the eight-tick octave (T7). It supplies the zero-parameter reference scale for the certificate that G strengthens by a factor near 32 at 20 nm. The construction connects upstream to the J-cost inflaton and the phi-ladder mass formula, feeding any downstream statement that invokes the running exponent β.
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