w_RS_linear_at_zero
plain-language theorem explainer
At redshift zero the structural RS linear dark-energy equation-of-state placeholder equals −1, matching ΛCDM exactly. Cosmologists and verifiers cite it as the z=0 baseline for Track 4.C. The proof unfolds the linear form and cancels the φ^{-44}·z term by ring.
Claim. The structural RS linear placeholder for the dark-energy equation of state satisfies $w_{\mathrm{RS,lin}}(0)=-1$, where $w_{\mathrm{RS,lin}}(z):=-1+\varphi^{-44}\,z$.
background
Track 4.C of the quantum-gravity master plan asks for a falsifiable RS dark-energy equation of state that differs from ΛCDM's strict $w=-1$ only at sub-leading order. This module supplies an algebraic discriminator, not the full FPT cosmic Z-aging dynamics: a linear placeholder $w_{\mathrm{RS,lin}}(z):=-1+\varphi^{-44},z$.
The scale $\varphi^{-44}$ is the same rung-44 factor that appears in baryogenesis ($\eta_B=\varphi^{-44}$). Numerically $\varphi^{-44}\approx 6.38\times 10^{-10}$. The upstream definition states that at $z=0$ the placeholder matches ΛCDM exactly, while at positive redshift the deviation is $\varphi^{-44},z>0$. The honesty note in that definition stresses this is a non-vacuous witness profile, not the final RS $w(z)$ prediction.
proof idea
One-line algebraic check. Unfold the definition $w_{\mathrm{RS,lin}}(z)=-1+\varphi^{-44},z$, substitute $z=0$, and finish with ring, which cancels the product term and leaves $-1$.
why it matters
This is the z=0 matching clause required by the Track 4.C one-statement and by the structural certificate. Downstream, dark_energy_w_of_z_one_statement bundles it with positivity of $\varphi^{-44}$, the strict inequality at $z>0$, and the exact residual identity $w_{\mathrm{RS,lin}}(z)-w_{\Lambda\mathrm{CDM}}=\varphi^{-44},z$. The certificate field w_RS_at_zero_matches_LCDM points here. The sibling equality to the ΛCDM constant is a one-line rewrite of this result. Verification also uses it: the Planck residual bound rewrites the RS baseline via this equality and checks it lies within one sigma of the constant-$w$ central value. The specific RS-derived z-dependence from FPT cosmic Z-aging remains open; only the structural linear witness is closed.
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