w_RS_linear_at_redshift_half
plain-language theorem explainer
At redshift z = 1/2 the structural RS dark-energy placeholder equals −1 + φ^{−44}/2. Cosmologists citing Track 4.C falsifier bands use this exact evaluation. The proof unfolds the linear form and the constant 1/2, then closes by ring algebra.
Claim. The structural RS equation-of-state placeholder satisfies $w_{\mathrm{RS}}^{\mathrm{lin}}(1/2) = -1 + \varphi^{-44}/2$, where $\varphi^{-44}$ is the rung-44 forcing scale.
background
Track 4.C of the quantum-gravity master plan asks for a falsifiable dark-energy equation of state that differs from ΛCDM’s strict $w = -1$ at sub-leading order. This module supplies only the structural discriminator: a linear-in-$z$ placeholder $w_{\mathrm{RS}}^{\mathrm{lin}}(z) := -1 + \varphi^{-44}, z$, not the full FPT cosmic Z-aging dynamics.
The scale $\varphi^{-44}$ is the same rung-44 factor that appears in baryogenesis ($\eta_B = \varphi^{-44}$). It is positive and tiny ($\approx 6.38\times 10^{-10}$). The named master-plan redshift $z = 1/2$ is the constant $1/2$. At $z = 0$ the placeholder matches ΛCDM exactly; at positive redshift the deviation is $\varphi^{-44}, z$.
proof idea
Term-mode one-liner. Unfold the definitions $w_{\mathrm{RS}}^{\mathrm{lin}}(z) = -1 + \varphi^{-44}, z$ and $\mathrm{redshift_half} = 1/2$, then apply ring to obtain $-1 + \varphi^{-44}/2$. No external lemmas are required beyond the three local definitions.
why it matters
Feeds the parent theorem named_redshift_falsifier_bands, which packages Track 4.C’s two named falsifier bands at $z = 0.5$ and $z = 1.0$. The master plan requires a concrete, non-vacuous witness that RS $w(z)$ differs from ΛCDM by a $\varphi^{-44}$-suppressed amount; this evaluation pins the $z = 0.5$ half of that witness.
The rung-44 scale links dark-energy structure to the same $\varphi$-ladder that forces baryon asymmetry. The true functional $z$-dependence (FPT cosmic Z-aging) remains open; the linear placeholder is only the algebraic discriminator.
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