w_RS_linear_deviation_magnitude
plain-language theorem explainer
The RS linear placeholder for the dark-energy equation of state differs from ΛCDM's constant w = −1 by exactly φ^{−44}·z at every redshift z. Cosmologists citing Track 4.C use this identity as the algebraic size of the discriminator. The proof is a one-line unfold-and-ring reduction from the definitions.
Claim. For every real redshift $z$, the structural RS profile satisfies $w_{\mathrm{RS}}(z) - w_{\Lambda\mathrm{CDM}} = \varphi^{-44}\, z$, where $w_{\mathrm{RS}}(z) := -1 + \varphi^{-44}\, z$ 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 at sub-leading order. This module supplies the structural algebraic discriminator, not the full FPT cosmic Z-aging dynamics: a linear-in-z placeholder suppressed by the rung-44 scale.
ΛCDM is encoded as the constant $w_{\Lambda\mathrm{CDM}} = -1$. The RS scale $\varphi^{-44}$ is the same forcing factor that appears in baryogenesis ($\eta_B = \varphi^{-44}$ via the phi-rung ladder with rung value $-44$). The placeholder profile is $w_{\mathrm{RS}}(z) := -1 + \varphi^{-44}, z$, so it matches ΛCDM at $z = 0$ and deviates linearly thereafter.
The honesty note in the definition is explicit: this linear form is a non-vacuous witness for the discriminator inequality, not the derived RS $w(z)$.
proof idea
Term-mode proof by definitional reduction. Unfold the two sides: $w_{\mathrm{RS}}(z)$ expands to $-1 + \varphi^{-44}, z$ and $w_{\Lambda\mathrm{CDM}}$ expands to $-1$. The difference is then $\varphi^{-44}, z$ by ring. No external lemmas are required beyond the three local definitions.
why it matters
This identity is the quantitative core of the Track 4.C structural package. It is conjoined into the one-statement theorem dark_energy_w_of_z_one_statement, which packages ΛCDM constancy, positivity of $\varphi^{-44}$, agreement at $z=0$, strict separation at positive $z$, and the exact deviation formula. Downstream, w_RS_linear_abs_deviation_eq_threshold rewrites the absolute separation as the falsifier threshold by applying this identity and nonnegativity of $\varphi^{-44}, z$ for $z \ge 0$. The master certificate darkEnergyWofZStructuralCert bundles the related positivity and distinctness facts.
Framework landmark: the suppression scale is the same rung-44 factor that forces baryon asymmetry, tying dark-energy discrimination to the phi-ladder. The open question left explicit by the module is the true FPT Z-aging $z$-dependence; only the linear placeholder is closed here.
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