w_RS_linear_abs_deviation_eq_threshold
plain-language theorem explainer
At every nonnegative redshift the absolute gap between the structural RS linear w(z) placeholder and the ΛCDM constant w = −1 equals the Track 4.C falsifier threshold φ^{−44} z. Cosmologists citing the dark-energy discriminator or the formal falsifier band use this identity. The proof rewrites the signed deviation, unfolds the threshold, and drops the absolute value by nonnegativity of the product.
Claim. For every real redshift $z \ge 0$, $\bigl|w_{\mathrm{RS,lin}}(z) - w_{\Lambda\mathrm{CDM}}\bigr| = \varphi^{-44}\, z$, where $w_{\mathrm{RS,lin}}(z) = -1 + \varphi^{-44} z$ and $w_{\Lambda\mathrm{CDM}} = -1$.
background
Track 4.C of the quantum-gravity master plan asks for a falsifiable dark-energy equation of state that differs from ΛCDM’s strict $w = -1$ at sub-leading order, suppressed by the rung-44 scale $\varphi^{-44}$ (the same scale that appears in baryogenesis $\eta_B = \varphi^{-44}$). This module ships only the algebraic discriminator: a structural linear-in-$z$ placeholder, not the full FPT cosmic Z-aging dynamics.
ΛCDM is the constant $w_{\Lambda\mathrm{CDM}} = -1$. The RS structural witness is $w_{\mathrm{RS,lin}}(z) := -1 + \varphi^{-44} z$, so the two agree at $z = 0$ and separate linearly for $z > 0$. The falsifier threshold is defined as that same product $\varphi^{-44} z$. Upstream, the signed identity $w_{\mathrm{RS,lin}}(z) - w_{\Lambda\mathrm{CDM}} = \varphi^{-44} z$ is already proved by unfolding and ring, and $\varphi^{-44} > 0$ follows from positivity of $\varphi$.
proof idea
Term-mode, three steps. First rewrite the absolute deviation via the signed identity w_RS_linear_deviation_magnitude, which gives $\varphi^{-44} z$ without absolute value. Unfold the definition of the falsifier threshold so the goal is $|\varphi^{-44} z| = \varphi^{-44} z$. Discharge with abs_of_nonneg, using that the product is nonnegative: $\varphi^{-44} > 0$ (from phi_neg_44_pos) and $z \ge 0$ (hypothesis).
why it matters
This is the absolute-value form of the Track 4.C structural separation: the RS/ΛCDM gap is exactly the falsifier threshold at every $z \ge 0$. Downstream, the symmetric swap |w_{\Lambda\mathrm{CDM}} - w_{\mathrm{RS,lin}}(z)| = \mathrm{threshold}(z) is a one-line consequence, and the formal falsifier band measured_near_LCDM_not_RS_linear uses it to prove that any measurement closer to ΛCDM than the threshold cannot equal the RS structural prediction.
In the Recognition framework this closes the algebraic half of master-plan Track 4.C (Ω_Λ tension and dark-energy $w(z)$). The rung-44 factor ties the discriminator to the same $\varphi$-ladder scale as baryogenesis. The specific functional $z$-dependence from FPT cosmic Z-aging remains future work; the linear placeholder is only a non-vacuous witness for the inequality.
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