beta_running_derived
plain-language theorem explainer
The declaration fixes the gravitational running exponent beta to equal -(phi-1)/phi^5, where phi is the golden ratio self-similar fixed point. Researchers modeling nanoscale gravitational strengthening via voxel density scaling would reference this result. The proof reduces directly to unfolding the prior definition of beta_running followed by reflexivity.
Claim. $beta = - (varphi - 1) / varphi^5$, where $varphi$ is the self-similarity fixed point and the exponent arises from the recognition lag $varphi^{-5}$ in the voxel density scaling $rho_{vox}(r) propto r^beta$.
background
The module treats voxel density scaling as the effective number of recognition voxels N(r) as a function of radius. Recognition Science sets the lag constant C_lag equal to varphi^{-5}, which induces strain in the density at small r. Upstream, the definition beta_running supplies the explicit running exponent for gravitational strengthening as -(phi-1)/phi^5, with a numerical bound proved from phi in (1.61,1.62). The present theorem equates the derived form to this definition.
proof idea
The proof is a one-line wrapper that unfolds the definition of beta_running and applies reflexivity.
why it matters
This supplies the explicit value for the running exponent beta that enters nanoscale strengthening of G_eff as r approaches zero. It closes the derivation step from voxel density to the forced beta, linking directly to the self-similarity fixed point phi from the unified forcing chain and the recognition lag varphi^{-5}. No downstream uses appear yet, leaving open its insertion into full running-G calculations.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.