w_RS_linear
plain-language theorem explainer
Linear-in-redshift placeholder for a dark-energy equation of state: w(z) = -1 + φ^{-44} z. Track 4.C structural work cites it as a non-vacuous witness that some upward-deviating profile exists and differs from exact ΛCDM. At z = 0 it matches w = -1; the slope is the baryogenesis rung-44 scale. The body is a one-line algebraic combination of -1 and φ^{-44}.
Claim. For redshift $z \in \mathbb{R}$, define the structural linear profile $w_{\mathrm{lin}}(z) := -1 + \varphi^{-44}\, z$, where $\varphi^{-44}$ is the rung-44 forcing scale.
background
Track 4.C of the quantum-gravity master plan asks for a falsifiable dark-energy equation of state w(z) that differs at sub-leading order from ΛCDM's strict w = -1. This module ships only the algebraic discriminator: a structural linear placeholder, not the full FPT cosmic Z-aging dynamics.
The rung-44 scale φ^{-44} (about 6.38 × 10^{-10}) is the same factor that appears in baryogenesis as η_B = φ^{-44}. ΛCDM is the constant value -1. The linear profile is -1 plus that tiny slope times z, so it equals -1 today and rises for z > 0.
Honesty note from the source: this is not the physical RS dark-energy prediction. The correct RS form is the antitone cosmic-aging kernel w(z) = -1 + J(φ)/(1+z) with amplitude J(φ) ≈ 0.118 today. The linear profile is kept only as a generic upward-deviating witness for carrier-independence and discriminator arguments.
proof idea
Pure definition: expand as the real number -1 plus the product of the upstream constant φ^{-44} with the redshift argument z. No lemmas, no tactics.
why it matters
Feeds the Track 4.C one-statement theorem, which packages ΛCDM = -1, positivity of φ^{-44}, match at z = 0, strict excess at positive z, and exact deviation φ^{-44} z. The master cert structure records the same four structural facts.
Downstream falsifier machinery uses it directly: exact ΛCDM at positive z is not the linear profile; absolute separation equals the falsifier threshold; measurements closer to ΛCDM than that threshold cannot equal the profile; named redshift bands and the dataset-facing falsifier band all quote it.
In the Recognition framework this is scaffolding for the Ω_Λ / w(z) prediction line, not a closed physical claim. The open question remains the true z-dependence from FPT cosmic Z-aging (canonical Δw kernel), which this placeholder deliberately does not supply.
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