recognitionEventHorizon_eq

In Recognition Science cosmology, a recognition signal covers eight comoving cells per epoch (the T-7 eight-tick cadence) while the comoving scale dilates by $\varphi$ each epoch (T-6). The per-epoch reach is therefore $8/\varphi^m$, and the infinite cumulative reach is the geometric series summing to $8\varphi^2$.

The module defines that horizon as $8\varphi^2$. The golden-ratio identity $\varphi^2=\varphi+1$ (from the minimal polynomial $x^2-x-1=0$, re-exported locally from Constants) converts the expression into the linear form $8(\varphi+1)\approx 20.944$ comoving cells. No fitted Hubble rate or coupling enters: only the forced cadence 8 and the self-similar fixed point $\varphi$.

Unfold the definition of the horizon ($8\varphi^2$), then rewrite with the local lemma equating $\varphi^2$ to $\varphi+1$ (a re-export of Constants.phi_sq_eq). The result is immediate algebraic substitution; no series summation or analysis is required at this step.

This closed form is the algebraic handle used by the sandwich bounds that place the horizon strictly between the dyadic rungs $2^4=16$ and $2^5=32$, and by the one-statement packaging of the freeze-out law. Those results establish that structure at scale 16 is sub-horizon (homogenized) while structure at scale 32 is super-horizon (frozen at primordial amplitude). The identity therefore sits on the T-6/T-7 forcing chain that produces the RS $\Omega_\Lambda$ freeze-out without tuned parameters, matching the numeric cosmogenesis simulation.