track4A_headline
plain-language theorem explainer
Bundles the RS dark-energy fraction Ω_Λ = 11/16 − α_CODATA/π with its proved band (0.683, 0.686), 2σ Planck 2018 overlap, and three independent forcings of the baryon-to-photon φ-rung exponent −44 at D = 3. Cosmologists citing Track 4.A structural predictions use this single certificate. The proof is a seven-conjunct refine that discharges each clause by an already-proved lemma or definitional unfold.
Claim. One has $\Omega_\Lambda = 11/16 - \alpha_{\mathrm{CODATA}}/\pi$, with $0.683 < \Omega_\Lambda < 0.686$ and $|\Omega_\Lambda - \Omega_\Lambda^{\mathrm{Planck\,2018}}| < 2\,\sigma_{\mathrm{Planck}}$. Simultaneously the three RS rung integers for the baryon-to-photon ratio equal $-44$: the gap-from-dimension route at spatial dimension $D=3$, the chirality$\times$torsion route, and the fermionic degrees-of-freedom route.
background
Track 4.A is the master certificate that packages three already-closed cosmology results into one structural theorem: the exact η_B rung integer, the Ω_Λ formula and numerical band, and Planck 2018 consistency. Status is theorem-grade (zero sorry, no RS-internal axiom).
The dark-energy density parameter is defined by $\Omega_\Lambda = 11/16 - \alpha/\pi$. The seed $11/16$ is the vacuum-mode fraction of the eight-tick D=3 ledger; the electromagnetic correction uses the external CODATA anchor $\alpha = 7.2973525643\times 10^{-3}$. Spatial dimension is the forced constant $D := 3$ from the forcing chain (T8).
The baryon-to-photon rung is obtained three ways. Gap-from-dimension: $\eta_B(d) = A - \mathrm{dimensionGap}(d) = 1 - d^2(d+2)$, which at $d=3$ is $-44$. Chirality$\times$torsion: $-(4\times 11) = -44$. Fermionic DOF route likewise yields $-44$. Upstream equalities eta_B_rung_from_dimension_at_D3, eta_B_rung_from_chirality_eq, and eta_B_rung_from_fermionic_eq already pin each integer.
proof idea
Seven-way refine ⟨?_, …⟩ assembling pre-proved facts.
- Unfold
omega_lambda,omega_raw,em_correctionand close byrfl(definitional identity with CODATA α). 2–3. Left and right bounds ofomega_lambda_intervalgive $0.683 < \Omega_\Lambda < 0.686$. rs_consistent_with_plancksupplies the 2σ Planck 2018 overlap. 5–7.eta_B_rung_from_dimension_at_D3,eta_B_rung_from_chirality_eq, andeta_B_rung_from_fermionic_eqeach return the integer equality to $-44$.
No new algebra is performed here; the certificate is pure conjunction of upstream closures.
why it matters
This is the headline discharge of master-plan Track 4.A (sub-tasks 1–3) and the audit row “Ω_Λ structurally derived,” upgrading those items from open/conditional to theorem. It sits at the junction of T8 ($D=3$), the eight-tick ledger seed $11/16$, and the single external α input, producing a narrow structural band that overlaps Planck 2018.
The triple convergence of the −44 rung (dimension gap, chirality×torsion, fermionic DOF) is the structural backbone for the φ-ladder placement of η_B. Downstream consumers are not yet wired in-repo (used_by empty); the certificate is the published endpoint of Track 4.A itself.
Explicitly left open: Track 4.B (why the cost-minimum vacuum mode-sum is exactly the $\phi^{-44}$ rung rather than $10^{120}$ times larger) and Track 4.C (Ω_Λ tension and dark-energy equation-of-state predictions).
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