phiL_pos

LogicReal is the Cauchy completion of the recovered rationals, wrapped so Mathlib's completed reals can be reused without polluting global instances. Transport toReal : LogicReal → ℝ is an order embedding: $x < y$ if and only if $\mathrm{toReal},x < \mathrm{toReal},y$, and it sends zero to zero.

This module mirrors Recognition Science constants on that line. Each definition is written in LogicReal; companion theorems show that toReal recovers the classical real constant from IndisputableMonolith.Constants. The recovered golden ratio is $$\varphi_L = \frac{1 + \sqrt{5}}{2}$$ built from fromReal and the logic-side square root, and toReal_phiL states $\mathrm{toReal},\varphi_L = \mathrm{Constants.phi}$.

One short tactic block. Rewrite the goal with the order-transport equivalence lt_iff_toReal_lt, then simplify the endpoints by toReal_zero and toReal_phiL. The residual goal is the ordinary real inequality $0 < \mathrm{Constants.phi}$, discharged by Constants.phi_pos.

φ is the self-similar fixed point forced at T6 in the Unified Forcing Chain, and it appears throughout the RS ladder (mass formula, $\hbar = \varphi^{-5}$, $G = \varphi^5/\pi$, Berry threshold $\varphi^{-1}$). Placing a strict positivity fact on the recovered line is the first ordered property needed before inequalities such as $\varphi_L > 1$ or rung comparisons can be stated inside LogicReal rather than after transport.

The module's charter is exactly this: every RS constant mirrored in LogicReal must recover its classical real value under toReal. No downstream consumers are wired yet (used_by is empty), so the lemma is infrastructure for later LogicReal inequalities on tick, octave, $\hbar$, and $\alpha^{-1}$.