phi_pos

In Recognition Science, $\varphi$ is the unique self-similar fixed point forced by the T6 step of the unified forcing chain (the positive solution of $x=1+1/x$). It appears throughout the framework as the dilation factor of the $\varphi$-ladder and as the ratio that makes geometric sums close in closed form via $\varphi^2=\varphi+1$.

This module treats the Phase-9 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 the reach in epoch $m$ is $8\varphi^{-m}$. The infinite sum converges to the finite horizon $8\varphi^2$ precisely because $0<1/\varphi<1$.

The lemma is the elementary positivity fact needed before any comparison or summability argument involving $\varphi$ or $\varphi^{-1}$.

One-line wrapper: the statement is identical to Constants.phi_pos from the constants library, and the proof simply names that lemma.

Positivity of $\varphi$ is a prerequisite for the geometric-series construction of the recognition event horizon $8\varphi^2\approx 20.944$ comoving cells. Sibling facts in the same file (one_lt_phi, phi_inv_pos, phi_inv_lt_one, perEpochReach_pos, perEpochReach_summable) build the cascade that shows the cumulative reach is finite and strictly increasing toward that limit. That finite horizon is what freezes super-horizon structure at primordial amplitude and supplies the RS account of $\Omega_\Lambda$ freeze-out, with no fitted Hubble rate. The result sits under T6 (forced $\varphi$) and T7 (eight-tick octave) from the forcing chain.