falsifierThreshold_pos
plain-language theorem explainer
At every positive redshift z the Track 4.C falsifier threshold φ^{-44}·z is strictly positive. Cosmologists citing the RS dark-energy w(z) discriminator use this to guarantee a nonempty precision band separating RS from ΛCDM. The proof unfolds the threshold definition and multiplies the two positivity facts φ^{-44}>0 and z>0.
Claim. For every real redshift $z>0$, the structural falsifier threshold $\varphi^{-44}\cdot z$ is strictly positive.
background
Track 4.C of the quantum-gravity master plan asks for a falsifiable dark-energy equation of state $w(z)$ that differs at sub-leading order from ΛCDM's constant $w=-1$. This module supplies the algebraic discriminator: the RS deviation is scaled by the rung-44 factor $\varphi^{-44}\approx 6.38\times 10^{-10}$, the same scale that appears in baryogenesis as $\eta_B=\varphi^{-44}$.
The structural placeholder is $w_{\mathrm{RS}}(z):=-1+\varphi^{-44}\cdot z$. The falsifier threshold at redshift $z$ is defined simply as $\varphi^{-44}\cdot z$. Upstream, $\varphi^{-44}$ is already known to be positive (phi_neg_44_pos, via positivity of $\varphi$ under negative integer powers). The present lemma lifts that fact to every positive redshift.
proof idea
One-line term proof after unfolding. Expand falsifierThreshold z to $\varphi^{-44}\cdot z$, then apply the real-multiplication positivity lemma to the pair of hypotheses $0<\varphi^{-44}$ (from phi_neg_44_pos) and $0<z$ (the local assumption). No further algebraic work.
why it matters
This positivity is the gate for the master-plan §7 falsifier band. Downstream, falsifier_band_at_redshift packages an existential precision equal to the threshold and proves that any measured $w$ within that band of $-1$ cannot equal the RS linear placeholder. The same fact is bundled into darkEnergyWofZStructuralCert, the module's closure certificate for Track 4.C.
Within Recognition Science the rung-44 scale ties cosmology to the $\varphi$-ladder forcing chain (T6 fixed point and the eight-tick octave). The specific FPT cosmic Z-aging dynamics that would replace the linear placeholder remain future work; this lemma only secures that the structural gap is open and positive wherever $z>0$.
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