track4ACert
plain-language theorem explainer
Verified Track 4.A master certificate bundling three closures: the η_B φ-rung integer −44 forced by D=3, the dark-energy formula Ω_Λ = 11/16 − α/π with proved band (0.683, 0.686), and 2σ consistency with Planck 2018. Cosmologists citing RS baryon asymmetry or dark-energy predictions use this bundle. Construction is pure field assembly of prior certificates plus one unfold/rfl identity.
Claim. There is a verified Track 4.A certificate whose clauses assert: (i) the baryon-to-photon $\varphi$-rung integer $-44$ is forced by spatial dimension $D=3$ via three convergent arithmetic routes; (ii) $\Omega_\Lambda = 11/16 - \alpha/\pi$ with measured CODATA $\alpha$; (iii) $0.683 < \Omega_\Lambda < 0.686$; (iv) the RS value overlaps Planck 2018 ($0.6889\pm 0.0056$) within $2\sigma$; (v) the gap-from-dimension route at $D=3$ yields $-44$.
background
Track 4.A is the cosmology bundle in the RS master plan that upgrades three audit items from open/conditional to theorem: the η_B rung, the Ω_Λ formula and band, and Planck consistency. The local module is a structural theorem package (zero sorry, zero RS-internal axiom).
The dark-energy density is defined as $\Omega_\Lambda = \omega_{\mathrm{raw}} - \mathrm{em_correction}$ with geometric seed $\omega_{\mathrm{raw}} = 11/16$ (fraction of unexcited ledger modes in the eight-tick cycle, tied to T8 and the D=3 gap) and EM correction $\alpha/\pi$ using measured CODATA $\alpha$ as the single external anchor. Upstream, omega_lambda_interval proves the open interval $(0.683, 0.686)$, and rs_consistent_with_planck records the $2\sigma$ overlap with Planck 2018.
Separately, the η_B certificate states that the integer $-44$ is reproduced by three arithmetic re-expressions from $D=3$ (gap-from-dimension, chirality×torsion, fermionic DOF) that agree; the dimension route alone is the equality of the gap-from-dimension formula at $D=3$ with $-44$, using gap $45$. Physical assignment of that rung to the baryon-to-photon ratio remains hypothesis-grade; the arithmetic is theorem-grade.
proof idea
Structure inhabitant built by direct field assignment. The η_B clause is the existing certificate etaBExactRungCert (three routes plus pairwise agreement). The formula clause unfolds omega_lambda, omega_raw, and em_correction and closes by rfl, recovering $\Omega_\Lambda = 11/16 - \alpha_{\mathrm{CODATA}}/\pi$. The band clause is the prior theorem omega_lambda_interval. Planck consistency is rs_consistent_with_planck. The dimension-route clause is eta_B_rung_from_dimension_at_D3. No new arithmetic is proved here; the def only packages closed upstream results.
why it matters
This is the single object that discharges master-plan §3 audit row "Ω_Λ structurally derived" and §4 Track 4.A sub-tasks 1–3. Downstream, track4ACert_inhabited witnesses Nonempty Track4ACert, and Gravity.MasterTheorem.omega_lambda_from_phi_proven projects the formula, band, Planck, and dimension-route fields into the gravity master theorem.
Framework landmarks in play: T8 forces $D=3$, which feeds both the gap-45 arithmetic behind rung $-44$ and the $11/16$ ledger seed for vacuum modes in the eight-tick octave. The α correction is the one measured boundary datum (exact α is free inside RS). Track 4.B (vacuum-fluctuation $10^{120}$ discrepancy vs $\varphi^{-44}$) and Track 4.C (Ω_Λ tension / equation-of-state predictions) remain outside this certificate.
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