darkEnergyWofZStructuralCert_inhabited
plain-language theorem explainer
The Track 4.C structural dark-energy certificate is inhabited: ΛCDM has constant w = −1, the RS linear placeholder matches at z = 0 and strictly exceeds −1 for z > 0 by φ^{−44}·z, with positive falsifier threshold. Cosmologists and the gravity master-theorem handoff cite it as the algebraic discriminator against ΛCDM. Proof is a one-line term packing the concrete certificate instance.
Claim. The structure bundling the Track 4.C structural dark-energy $w(z)$ facts is nonempty: $w_{\Lambda\mathrm{CDM}} = -1$, $\varphi^{-44} > 0$, the RS linear placeholder equals $w_{\Lambda\mathrm{CDM}}$ at $z = 0$, strictly exceeds it for every $z > 0$, the deviation equals $\varphi^{-44}\cdot z$, and the falsifier threshold is positive at every positive redshift.
background
Track 4.C of the quantum-gravity master plan asks for a falsifiable RS equation of state $w(z)$ that differs at sub-leading order from ΛCDM's strict $w = -1$, via the φ-rung dynamical history (FPT cosmic Z-aging). This module ships only the algebraic discriminator, not the full dynamics.
The RS scale is the rung-44 forcing factor $\varphi^{-44}\approx 6.38\times 10^{-10}$, the same scale that appears in baryogenesis as $\eta_B = \varphi^{-44}$. The structural placeholder is $w_{\mathrm{RS,lin}}(z) := -1 + \varphi^{-44}\cdot z$. ΛCDM is the constant $-1$. At $z = 0$ the two agree; at any $z > 0$ the RS value strictly exceeds $-1$ by $\varphi^{-44}\cdot z$.
The master certificate structure packages six facts: ΛCDM equals $-1$, positivity of $\varphi^{-44}$, match at zero, strict inequality at positive $z$, exact deviation magnitude, and positivity of the falsifier threshold $\varphi^{-44}\cdot z$.
proof idea
One-line term proof: inhabit Nonempty by the anonymous constructor applied to the concrete certificate value darkEnergyWofZStructuralCert. That value is itself a structure instance whose six fields are the already-proved sibling lemmas (w_LCDM_value_eq_neg_one, phi_neg_44_pos, w_RS_linear_at_zero, the positive-$z$ distinctness lemmas, the deviation-magnitude identity, and falsifierThreshold_pos). No further tactics.
why it matters
Closes the structural half of Track 4.C (master plan §4: Ω_Λ tension and dark-energy equation-of-state predictions). The certificate is the algebraic witness that RS $w(z)$ is suppressed by the rung-44 scale and is observationally distinguishable from ΛCDM once precision beats $\varphi^{-44}\cdot z$.
Downstream, track4_dark_energy_falsifier_endpoint_holds consumes this inhabitation as the fourth field of the Fork E endpoint theorem in the gravity master-theorem handoff integration lane. That endpoint packages concrete thresholds at $z = 1/2$ and $z = 1$ together with the near-ΛCDM exclusion rule.
Framework landmark: rung 44 is the same φ-ladder depth that forces baryon asymmetry $\eta_B = \varphi^{-44}$. Open question left explicit: the true FPT cosmic Z-aging $z$-dependence replaces the linear placeholder; only the structural discriminator is closed here.
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