falsifierThreshold_at_redshift_half

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 strict $w = -1$. This module supplies the algebraic discriminator: the RS deviation is suppressed by the rung-44 factor $\varphi^{-44}$ (the same scale that appears in baryogenesis $\eta_B = \varphi^{-44}$), with a linear-in-$z$ placeholder $w_{\mathrm{RS}}(z) := -1 + \varphi^{-44}\cdot z$ as a non-vacuous witness.

The falsifier threshold at redshift $z$ is defined by $\mathrm{falsifierThreshold}(z) := \varphi^{-44}\cdot z$. The named master-plan redshift $z = 1/2$ is the constant $1/2$. Evaluating the threshold there is the content of this lemma; the companion evaluation at $z = 1$ is the full rung-44 scale itself.

One-line algebraic identity. Unfold the definitions $\mathrm{falsifierThreshold}(z) = \varphi^{-44}\cdot z$ and $\mathrm{redshift_half} = 1/2$, then apply ring to obtain $\varphi^{-44}\cdot(1/2) = \varphi^{-44}/2$. No external lemmas are required.

Feeds three parents. First, named_redshift_falsifier_bands packages the two master-plan bands: threshold at $z=0.5$ is $\varphi^{-44}/2$ and at $z=1$ is $\varphi^{-44}$, together with the matching $w_{\mathrm{RS}}$ values. Second, the Track 4.C one-statement theorem bundles the full structural claim (match at $z=0$, strict excess over $-1$ at positive $z$, deviation exactly $\varphi^{-44}\cdot z$). Third, the gravity handoff track4_dark_energy_falsifier_endpoint_holds consumes this equality as a Fork E endpoint for the integration lane.

Within the Recognition framework the rung-44 scale is the same $\varphi$-ladder factor that forces baryon asymmetry; here it sets the amplitude of the dark-energy discriminator. The specific FPT cosmic Z-aging $z$-dependence remains future work; this lemma only pins the linear placeholder at the named half-redshift band.