w_RS_is_CPL
plain-language theorem explainer
The RS dark-energy equation of state under the forced kernel equals the CPL form at parameters w₀ = −1 + δw₀ and wₐ = −δw₀. Cosmologists mapping Recognition thawing trajectories onto standard CPL fits cite this identity. The proof is a short algebraic reduction: positivity of 1+z, unfold both definitions, clear the denominator, and finish by ring.
Claim. For all real $\delta w_0$ and all redshifts $z > -1$, $w_{\mathrm{RS}}(\delta w_0,z) = w_{\mathrm{CPL}}(-1+\delta w_0,-\delta w_0,z)$, where $w_{\mathrm{RS}}(\delta w_0,z)=-1+\delta w_0/(1+z)$ and $w_{\mathrm{CPL}}(w_0,w_a,z)=w_0+w_a\,z/(1+z)$.
background
The module formalizes the paper "The Forced Redshift Kernel": dark energy is written $w(z)=-1+\delta w_0\cdot K(z)$, and the kernel $K$ is forced by two premises (rung factorization of aging-charge attenuation, and single-rung balance at the unique positive fixed point $\varphi^{-1}$ of $\rho=1/(1+\rho)$). The resulting RS equation of state is the one-parameter family $w_{\mathrm{RS}}(\delta w_0,z)=-1+\delta w_0/(1+z)$.
CPL is the standard phenomenological two-parameter form $w(z)=w_0+w_a,z/(1+z)$ used in supernova and BAO analyses. Matching the forced RS trajectory to a concrete CPL point $(w_0,w_a)$ is the bridge from the Recognition derivation to survey language.
The side condition $z>-1$ is the physical redshift domain; it guarantees the denominator $1+z$ is strictly positive.
proof idea
From $z>-1$, linear arithmetic gives $1+z>0$. Unfold the two definitions to the concrete equality
$$-1+\frac{\delta w_0}{1+z};\stackrel{?}{=};(-1+\delta w_0)+(-\delta w_0)\frac{z}{1+z}.$$
field_simp clears the common positive denominator; ring closes the resulting polynomial identity. No external lemmas beyond the two local definitions are required.
why it matters
Module consequences list this identity with the thawing-line relation $w_a=-(1+w_0)$, the sum rule $w_0+w_a=-1$, and the band $w_0\in(-1,-0.88)$. It places the forced kernel exactly on the CPL thawing line, so every RS prediction is a one-parameter CPL model rather than a free two-parameter fit.
That placement feeds the sibling results on the thawing line, the CPL sum rule, the $w_0$ band, and the no-phantom bound $w(z)\ge-1$. In the broader Recognition chain it is a cosmology-side readout of the rung law $\mathrm{occ},n=\varphi^{-n}$ (itself forced by T6 self-similarity and single-rung balance), not a new dynamical assumption.
The today-amplitude $\delta w_0\in(0,J(\varphi)]$ remains open; this theorem only identifies the shape.
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