Track4DarkEnergyFalsifierEndpoint
plain-language theorem explainer
Fork E packages the Track 4.C dark-energy equation-of-state lane as a single proposition: named falsifier half-widths at z=1/2 and z=1 equal φ^{-44}/2 and φ^{-44}, measurements nearer ΛCDM than that band cannot equal the RS linear w(z), and the structural master certificate is inhabited. Cosmologists and the Track 7 handoff cite it as the dark-energy endpoint. The body is a pure conjunction of four already-proved facts.
Claim. The Fork E endpoint holds when: the structural falsifier half-width at redshift $z=1/2$ equals $\varphi^{-44}/2$; the half-width at $z=1$ equals $\varphi^{-44}$; for every $z>0$ and every measured $w$, if $|w-(-1)|$ is strictly smaller than the threshold $\varphi^{-44}\,z$ then $w$ is not equal to the RS linear prediction $w_{\mathrm{RS}}(z)$; and the Track 4.C structural dark-energy certificate is inhabited.
background
Track 7 (this module) is the integration-lane receipt for parallel fork handoffs A–F. Fork E is the Track 4.C dark-energy $w(z)$ falsifier-band refinement. It does not upgrade the discovery claim; it records what the new endpoints prove.
Upstream, $\varphi^{-44}$ is the RS rung-44 forcing scale (the same scale that appears in baryogenesis $\eta_B=\varphi^{-44}$). The structural falsifier threshold at redshift $z$ is defined by $\mathrm{threshold}(z)=\varphi^{-44},z$. ΛCDM is the constant $w=-1$. The RS linear form $w_{\mathrm{RS}}(z)$ matches ΛCDM at $z=0$ and lies strictly above $-1$ for every $z>0$.
The master certificate DarkEnergyWofZStructuralCert packages four structural facts: $w_{\Lambda\mathrm{CDM}}=-1$, positivity of $\varphi^{-44}$, matching at zero redshift, and strict separation at positive redshift. Named master-plan redshifts $z=1/2$ and $z=1$ are the concrete evaluation points for the half-widths.
proof idea
This declaration is a definition, not a proved theorem: it is the bare conjunction of four propositions. The first two conjuncts are the concrete evaluations $\mathrm{threshold}(1/2)=\varphi^{-44}/2$ and $\mathrm{threshold}(1)=\varphi^{-44}$. The third is the separation statement that any measured $w$ closer to $-1$ than the threshold cannot equal $w_{\mathrm{RS}}(z)$. The fourth is mere inhabitation of the structural master certificate.
Discharge is deferred to the sibling theorem track4_dark_energy_falsifier_endpoint_holds, which assembles the four facts from the upstream lemmas falsifierThreshold_at_redshift_half, falsifierThreshold_at_redshift_one, measured_near_LCDM_not_RS_linear, and darkEnergyWofZStructuralCert_inhabited.
why it matters
Fork E is the dark-energy handoff into Track 7. Downstream, track4_dark_energy_falsifier_endpoint_holds proves the proposition, and both ForkHandoffIntegrationCert and fork_A_B_C_D_E_F_handoffs_integrated_one_statement consume it as the Fork E conjunct. The one-statement integration theorem explicitly lists "Fork E's $w(z)$ falsifier bands" among the facts Track 7 can now consume, while deliberately not asserting the fully unconditional discovery theorem.
In the Recognition framework the scale $\varphi^{-44}$ ties cosmology to the same phi-ladder rung that forces baryon asymmetry. The endpoint therefore gives a sharp, redshift-linear falsifier band against which future $w(z)$ measurements can reject the RS structural prediction in favor of pure ΛCDM (or vice versa). Remaining open work sits on the Track 1 displacement-class leaves, not on this dark-energy package.
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