DarkEnergyWofZStructuralCert
plain-language theorem explainer
A certificate packing the Track 4.C structural dark-energy equation-of-state discriminator: ΛCDM has constant w = −1, while a linear-in-z RS placeholder deviates by φ^{−44}·z and matches ΛCDM only at z = 0. Cosmologists and master-plan auditors cite it for named z = 0.5 and z = 1 falsifier bands plus a measurement-separation clause. As a structure, it is inhabited by assembling the module’s proved field lemmas; no separate proof body.
Claim. A certificate records: (i) the ΛCDM equation of state equals $-1$; (ii) $\varphi^{-44}>0$; (iii) the structural RS profile $w_{\mathrm{RS}}(z):=-1+\varphi^{-44}z$ equals $-1$ at $z=0$ and strictly exceeds $-1$ for every $z>0$; (iv) the deviation is exactly $\varphi^{-44}z$; (v) the falsifier threshold $T(z):=\varphi^{-44}z$ is positive for $z>0$, with $T(1/2)=\varphi^{-44}/2$ and $T(1)=\varphi^{-44}$; (vi) any measured $w$ closer to $-1$ than $T(z)$ cannot equal $w_{\mathrm{RS}}(z)$; (vii) the linear form is an explicit structural placeholder.
background
Track 4.C of the quantum-gravity master plan asks for a falsifiable dark-energy equation of state $w(z)$ that differs from ΛCDM’s strict $w=-1$ at sub-leading order, driven by the φ-rung dynamical history (FPT cosmic Z-aging). This module closes only the algebraic discriminator, not the full dynamics.
ΛCDM is encoded as the constant $w_{\mathrm{LCDM}}=-1$. The RS scale is $\varphi^{-44}$ (same rung that appears in baryogenesis $\eta_B=\varphi^{-44}$). The structural witness is the linear placeholder $w_{\mathrm{RS}}(z)=-1+\varphi^{-44}z$, which matches ΛCDM at $z=0$ and rises by $\varphi^{-44}z$ at positive redshift. The falsifier threshold is $T(z)=\varphi^{-44}z$, with named master-plan redshifts $z=1/2$ and $z=1$.
The module status is structural theorem: zero sorry, zero RS-internal axiom. The doc-comment on the linear profile states explicitly that it is “NOT the RS dark-energy prediction”; the true FPT $z$-dependence remains future work.
proof idea
This declaration is a structure type, not a proved theorem: it lists the Prop fields a Track 4.C cert must carry. Inhabitation is separate. The canonical witness darkEnergyWofZStructuralCert fills each field by a named lemma from the same module: $w_{\mathrm{LCDM}}=-1$, positivity of $\varphi^{-44}$, match at $z=0$, strict inequality for $z>0$, exact deviation identity, positivity of $T(z)$, the four named-band equalities at $z=1/2$ and $z=1$, the measurement-separation implication, and the honest-scope identity restating the linear placeholder. No tactic script lives on the structure itself.
why it matters
This is the master cert type for Cosmology Track 4.C (“Ω_Λ tension and dark-energy-equation-of-state predictions”). Downstream, darkEnergyWofZStructuralCert inhabits it and darkEnergyWofZStructuralCert_inhabited records non-emptiness. Gravity handoff integration consumes the same content as Track4DarkEnergyFalsifierEndpoint: named $z=0.5$ and $z=1.0$ thresholds plus the formal separation theorem for measurements closer to ΛCDM than the RS structural band.
Framework link: the deviation amplitude is the rung-44 factor $\varphi^{-44}$ on the φ-ladder, tying dark-energy discrimination to the same scale as baryogenesis. The honest-scope field keeps the claim non-overstated: linear-in-$z$ is a non-vacuous placeholder until FPT cosmic Z-aging supplies the true functional form. That open dynamical question is the natural next closure target after this structural bundle.
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