phi_sq_eq

In Recognition Science, $\varphi$ is the unique positive self-similar fixed point forced by T6 (the solution of $x^2 - x - 1 = 0$ greater than 1). The identity $\varphi^2 = \varphi + 1$ is the algebraic content of that fixed-point equation and is the workhorse for every closed-form reduction involving powers or inverses of $\varphi$.

This module studies the de Sitter recognition event horizon. A signal covers eight comoving cells per epoch (T7 eight-tick cadence) while the comoving scale dilates by $\varphi$ each epoch, so per-epoch reach is $8,\varphi^{-m}$. The infinite cumulative reach is the geometric series $\sum_m 8,\varphi^{-m} = 8/(1-\varphi^{-1})$. The identity $\varphi^2 = \varphi + 1$ converts that sum into the clean closed form $8\varphi^2 = 8(\varphi+1)$.

Upstream, Constants.phi_sq_eq records the same identity with the explicit radical definition of $\varphi$ and a short ring/linear-combination proof.

One-line term wrapper: the lemma is definitionally Constants.phi_sq_eq. That upstream proof unfolds $\varphi = (1+\sqrt{5})/2$, uses $(\sqrt{5})^2 = 5$, and finishes by ring_nf plus a linear combination clearing the remaining radical term. No local tactics are needed.

The horizon formula in this module is $8\varphi^2$; rewriting it as $8(\varphi+1)$ is exactly this identity, so every subsequent bound on partial reach and freeze-out cites it. Downstream it seeds $\varphi^3 = 2\varphi+1$ (tidal-locking resonance), $1/\varphi = \varphi-1$ (quasicrystal ratio and J-band evaluations), the numeric J($\varphi$) band $(0.11,0.13)$, and the condensed-matter critical-energy window used in the falsifiable $T_c\sim 80$–$120,$K prediction. Framework landmarks: T6 ($\varphi$ forced) and, via the horizon sum, T7 (eight-tick octave). It is pure algebra, but it is the single rewrite that turns the geometric series into the RS-native constant $8\varphi^2\approx 20.944$.