track4_dark_energy_falsifier_endpoint_holds
plain-language theorem explainer
Fork E packages Track 4.C's dark-energy w(z) falsifier: thresholds at z=0.5 and z=1 equal φ^{-44}/2 and φ^{-44}, any measured w closer to ΛCDM than the threshold cannot match the RS linear prediction, and the structural certificate is inhabited. Gravity-track auditors and cosmologists cite it when wiring the handoff integration. The proof is a four-field term constructor from the named threshold lemmas, the separation theorem, and certificate inhabitation.
Claim. The Track 4 dark-energy falsifier endpoint holds: the falsifier threshold at redshift $z=1/2$ equals $\varphi^{-44}/2$, the threshold at $z=1$ equals $\varphi^{-44}$, and for every $z>0$ and every measured equation-of-state $w$, if $|w-w_{\Lambda\mathrm{CDM}}|$ is strictly below the threshold at $z$ then $w\neq w_{\mathrm{RS\text{-}linear}}(z)$. Moreover the structural $w(z)$ certificate is nonempty.
background
This module is the Track 7 integration-lane receipt for parallel fork handoffs. Fork E is the Track 4.C dark-energy $w(z)$ falsifier-band refinement: it records named redshift thresholds and a formal separation between ΛCDM-near measurements and the RS structural linear prediction, without upgrading the discovery claim.
The falsifier threshold is a positive band width on the equation-of-state axis, scaled from $\varphi^{-44}$ (half that width at $z=1/2$, full width at $z=1$). The separation theorem states that any measured $w$ closer to the ΛCDM value than this band cannot equal the RS linear $w(z)$. Upstream, the structural certificate is already inhabited, and the two named threshold identities are proved by unfolding and ring.
RS-native units fix $c=1$ and $\hbar=\varphi^{-5}$; the $\varphi$-ladder supplies the small scale $\varphi^{-44}$ that sets the band. The endpoint is a pure packaging Prop: four conjuncts the integration lane can consume as a single handoff.
proof idea
Term-mode four-tuple constructor. The first two fields are the named lemmas equating the threshold at redshift half to $\varphi^{-44}/2$ and at redshift one to $\varphi^{-44}$ (each an unfold-and-ring proof). The third field is a lambda that applies the separation theorem: if $|w_{\mathrm{measured}}-w_{\Lambda\mathrm{CDM}}|$ is below the threshold at $z>0$, then $w_{\mathrm{measured}}$ differs from the RS linear prediction. The fourth field is inhabitation of the structural $w(z)$ certificate. No extra tactics; the endpoint is exactly the conjunction of those four upstream facts.
why it matters
Fork E is the dark-energy handoff in Track 7. Downstream, the one-statement integration theorem consumes this endpoint together with the many-body channel lift, Schläfli-to-stationarity reductions, physical residual/Bianchi interface, Page-capacity tick layer, and Track 6 falsifier-sensitivity package, while deliberately not asserting the fully unconditional discovery theorem. The integration certificate instance wires the same endpoint as a named field.
In the Recognition framework this is the concrete falsifier band for structural $w(z)$: measurements that hug ΛCDM inside the $\varphi^{-44}$-scaled window rule out the RS linear branch at that redshift. It closes the Track 4.C packaging leaf so the gravity master handoff can treat dark energy as a theorem-grade discriminator rather than an open structural gap. Remaining open work sits on Track 1 displacement-class leaves, not on this band.
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