w_RS
plain-language theorem explainer
The RS dark-energy equation of state under the forced BIT kernel is $w(z)=-1+\delta w_0/(1+z)$. Cosmologists matching RS predictions to CPL constraints (DESI, Euclid, Roman) cite this form. It is the direct specialization of the deviation ansatz $w=-1+\delta w_0\cdot K(z)$ once rung dilution forces $K(z)=1/(1+z)$.
Claim. For amplitude $\delta w_0\in\mathbb{R}$ and redshift $z\in\mathbb{R}$, the Recognition Science dark-energy equation of state is defined by $w_{\mathrm{RS}}(\delta w_0,z)=-1+\dfrac{\delta w_0}{1+z}$.
background
The module forces the BIT dark-energy kernel from two premises: rung factorization (attenuation across $m+n$ $\varphi$-rungs multiplies) and single-rung balance at the unique positive fixed point of $\rho=1/(1+\rho)$, namely $\varphi^{-1}$. The deviation ansatz is $w(z)=-1+\delta w_0\cdot K(z)$; the forced kernel is $K(z)=1/(1+z)$ (equivalently $\varphi^{-n}$ on the lattice $1+z=\varphi^n$).
Among the BIT kernel families, this is the inv_one_plus_z member. The Chevallier-Polarski-Linder (CPL) form $w_{\mathrm{CPL}}(w_0,w_a,z)=w_0+w_a,z/(1+z)$ then becomes the natural comparison target: the forced kernel sits exactly on the thawing line $w_a=-(1+w_0)$.
Local status: everything stated from the two premises is theorem-level; the BIT aging mechanism and single-channel selection remain hypotheses, and the today-amplitude band $\delta w_0\in(0,J(\varphi)]$ is open.
proof idea
Definitional one-liner. Specialize the deviation form $w=-1+\delta w_0\cdot K(z)$ at the forced kernel $K(z)=1/(1+z)$; no lemmas or tactics are required.
why it matters
This is the working equation of state for the forced-kernel cosmology. Downstream, w_RS_is_CPL identifies it with CPL at $w_0=-1+\delta w_0$, $w_a=-\delta w_0$; no_phantom proves $w(z)\ge-1$ for $\delta w_0\ge0$ and $z>-1$ (sign falsifier); deviation_ceiling caps the deviation by the Carnot bound $J(\varphi)$; and omega_gap_explanation_retired uses the pinned sign to kill the "BIT explains the Planck-RS $\Omega_\Lambda$ gap" hypothesis. All five clauses of the one-statement summary bit_kernel_shape_one_statement either define or quote this object. Framework landmarks: $\varphi$-rung dilution (T6 self-similarity) and the J-cost ceiling $J(\varphi)$. Open: the admissible today-amplitude interval.
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