named_redshift_falsifier_bands
plain-language theorem explainer
At the master-plan redshifts z=1/2 and z=1, the RS structural falsifier threshold equals φ^{-44}/2 and φ^{-44}, and the linear RS equation-of-state equals -1 plus those same values. Cosmologists citing Track 4.C use this as the named numerical bands against which w(z) measurements are compared. The proof is a four-conjunct term packaging the specialized evaluations already proved at those two redshifts.
Claim. Let $\varphi^{-44}$ be the rung-44 scale and let the structural RS equation of state be $w_{\mathrm{RS}}(z)=-1+\varphi^{-44}\,z$, with falsifier threshold $T(z)=\varphi^{-44}\,z$. Then $T(1/2)=\varphi^{-44}/2$, $T(1)=\varphi^{-44}$, $w_{\mathrm{RS}}(1/2)=-1+\varphi^{-44}/2$, and $w_{\mathrm{RS}}(1)=-1+\varphi^{-44}$.
background
Track 4.C of the quantum-gravity master plan asks for a falsifiable dark-energy equation of state $w(z)$ that differs from $\Lambda$CDM's strict $w=-1$ at sub-leading order. This module supplies the algebraic discriminator: the deviation is suppressed 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 the linear form $w_{\mathrm{RS}}(z):=-1+\varphi^{-44}\cdot z$. The falsifier threshold at redshift $z$ is defined by $T(z):=\varphi^{-44}\cdot z$; a measurement of $w(z)=-1$ tighter than $T(z)$ would falsify the RS prediction that $w(z)-(-1)>0$ for $z>0$. The master plan names the concrete bands at $z=1/2$ and $z=1$.
Upstream lemmas already evaluate $T$ and $w_{\mathrm{RS}}$ at those two redshifts by unfolding the definitions and simplifying.
proof idea
Term-mode four-conjunct packaging. The proof is the anonymous constructor of a four-fold conjunction, feeding in the four specialized lemmas:
falsifierThreshold_at_redshift_halfandfalsifierThreshold_at_redshift_one(each unfolds $T(z)=\varphi^{-44}z$ at $z=1/2$ and $z=1$ and closes by ring),w_RS_linear_at_redshift_halfandw_RS_linear_at_redshift_one(the matching evaluations of $w_{\mathrm{RS}}$).
No new algebra is performed here; the declaration only bundles the named bands into one statement.
why it matters
This is the concrete numerical face of Track 4.C's falsifier: the two master-plan redshifts at which observers are told to compare $w(z)$ against $\Lambda$CDM. Downstream it is consumed by the module certificate darkEnergyWofZStructuralCert, which packages the constant $\Lambda$CDM value, positivity of $\varphi^{-44}$, matching at $z=0$, and the strict discriminator at positive $z$.
In the broader Recognition framework the scale $\varphi^{-44}$ is not free: it is the same rung that forces the baryon asymmetry on the $\varphi$-ladder. The module is closed (0 sorry); what remains open is the true FPT cosmic Z-aging dynamics that would replace the linear-in-$z$ placeholder with a derived functional form. Until that dynamics is shipped, these bands are structural witnesses, not dynamical predictions.
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