w_RS_linear_at_redshift_one
plain-language theorem explainer
At redshift z = 1 the structural RS linear dark-energy placeholder equals −1 + φ^{−44}. Cosmologists auditing Track 4.C falsifier bands cite this as the named z = 1 evaluation. The proof unfolds the linear form and the constant redshift, then closes by ring.
Claim. At redshift $z = 1$, the structural linear placeholder $w_{\mathrm{RS,lin}}(z) := -1 + \varphi^{-44}\, z$ evaluates to $-1 + \varphi^{-44}$.
background
Track 4.C of the quantum-gravity master plan asks for a falsifiable dark-energy equation of state that differs at sub-leading order from ΛCDM's strict $w = -1$. This module supplies only the algebraic discriminator: a linear-in-$z$ placeholder whose slope is the rung-44 scale $\varphi^{-44}$ (the same scale that appears in baryogenesis $\eta_B = \varphi^{-44}$).
The placeholder is $w_{\mathrm{RS,lin}}(z) := -1 + \varphi^{-44}, z$. Here $\varphi^{-44}$ is the positive real phi_neg_44, and redshift_one is the master-plan point $z = 1$. At $z = 0$ the form matches ΛCDM exactly; at positive redshift the deviation is $\varphi^{-44}, z > 0$. The honesty note in the definition is explicit: this is a non-vacuous witness, not the full FPT cosmic Z-aging dynamics.
proof idea
One-line algebraic evaluation. Unfold the definitions of the linear placeholder and of redshift one (so $z$ becomes the literal $1$), then apply ring to obtain $-1 + \varphi^{-44}\cdot 1 = -1 + \varphi^{-44}$. No external lemmas beyond those three local defs.
why it matters
Feeds the conjunction named_redshift_falsifier_bands, which packages Track 4.C's two named falsifier bands at $z = 0.5$ and $z = 1.0$. That parent theorem records both the threshold values and the corresponding $w$ evaluations, so this identity is the $z = 1$ half of the observational checklist.
In the broader Recognition framework the slope $\varphi^{-44}$ ties dark-energy structure to the same phi-ladder rung that forces baryon asymmetry. The module status is structural closure (0 sorry): the specific functional $z$-dependence from FPT cosmic Z-aging remains future work; this theorem only pins the linear witness at the named redshift.
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