transmittedFraction_pos

The module models the near-horizon recognition structure as a φ-self-similar potential barrier. At each rung boundary, energy splits by the golden-ratio partition $1 = \varphi^{-1} + \varphi^{-2}$, which is equivalent to $\varphi^2 = \varphi + 1$. The single-rung reflection coefficient is $|R|^2 = \varphi^{-2}$ and the reflected amplitude is $|R| = \varphi^{-1}$.

The transmitted fraction is defined as $\varphi^{-1}$ (the complementary share of the partition). The reflected fraction is $\varphi^{-2}$. Their sum is identically one by the defining equation of φ. Positivity of each share is the elementary arithmetic needed before one can compare either fraction to 1 or build geometric echo amplitudes $\varphi^{-n}$.

One-line term proof. The definition is $\mathrm{transmittedFraction} := \varphi^{-1}$. Apply Mathlib's inv_pos.mpr to the known fact phi_pos ($0 < \varphi$), which yields $0 < \varphi^{-1}$.

Feeds reflectedFraction_lt_one, which shows both partition shares are strictly less than one by combining this positivity with the complete-partition identity and linarith. That bound keeps the single-rung reflection coefficient inside $(0,1)$, so the geometric echo train (amplitude $\varphi^{-n}$, delay $n\cdot\Delta t_{\mathrm{echo}}$) is a genuine decaying series rather than a formal power of a number outside the unit interval.

In the Recognition framework this is structural, not fitted: the barrier scattering matrix is forced by $\varphi^2 = \varphi + 1$ (T6 self-similar fixed point). The QG-paper echo prediction is thereby a consequence of the substrate's φ-spacing, not a dimensional estimate. No open scaffold remains; the module is zero-sorry.