Track4ACert
plain-language theorem explainer
Certificate structure packing five Track 4.A clauses: the baryon-to-photon rung −44 forced by D=3, the dark-energy identity Ω_Λ = 11/16 − α_CODATA/π, the proved band (0.683, 0.686), Planck 2018 consistency at 2σ, and an explicit gap-from-dimension witness. Cosmologists and auditors of the RS master plan cite it as the single discharge point for Track 4.A. It is a pure structure definition; content lives in the field proofs of its inhabitant.
Claim. A Track 4.A certificate is a record of five facts: (1) an exact-rung certificate that the baryon-to-photon rung integer is forced by spatial dimension $D=3$ along three convergent arithmetic routes; (2) $\Omega_\Lambda = 11/16 - \alpha_{\mathrm{CODATA}}/\pi$; (3) $0.683 < \Omega_\Lambda < 0.686$; (4) $|\Omega_\Lambda - \Omega_\Lambda^{\mathrm{Planck\,2018}}| < 2\,\sigma_{\mathrm{Planck}}$; (5) the gap-from-dimension formula at $D=3$ evaluates to $-44$, i.e. $1 - D^2(D+2) = -44$.
background
Track 4.A is the cosmology bundle in the RS master plan that closes three items at once: the integer rung governing the baryon-to-photon ratio η_B, a structural formula for the dark-energy fraction Ω_Λ, and a 2σ check against Planck 2018. The module status is structural theorem (zero sorry, zero RS-internal axiom); this structure is the packaging type.
Spatial dimension is the forced constant $D=3$ (T8/T9). The gap-from-dimension route defines the η_B rung as $1 - d^2(d+2)$; at $d=3$ this is $-44$. The exact-rung certificate EtaBExactRungCert records three arithmetic re-expressions of that integer (gap-from-dimension, chirality×torsion, fermionic DOF) that agree and do not take empirical η_B as input; the assignment of the rung to η_B itself remains hypothesis-grade upstream.
Dark energy is $\Omega_\Lambda = 11/16 - \alpha/\pi$. The seed $11/16$ is the vacuum-mode fraction of the eight-tick ledger at $D=3$; the correction uses the external CODATA anchor $\alpha = 7.2973525643\times 10^{-3}$ (one measured input). Upstream lemmas already prove the numerical band and the Planck overlap.
proof idea
No proof body: this is a structure declaration. Each field is a Prop (or a nested certificate structure) naming a pre-existing closure. The inhabitant track4ACert fills the fields by direct reference: the exact-rung certificate is etaBExactRungCert; the Ω_Λ identity is rfl after unfolding the raw seed $11/16$ and the EM correction $\alpha/\pi$; band and Planck clauses are the corresponding theorems from OmegaLambdaDerivation; the dimension-route witness is the evaluation of eta_B_rung_from_dimension at $D=3$. Nonemptiness is then ⟨track4ACert⟩.
why it matters
This is the typed discharge point for master-plan §4 Track 4.A and the §3 audit row "Ω_Λ structurally derived". Downstream, track4A_headline restates the five clauses as a single conjunction, and track4ACert_inhabited records nonemptiness. In the gravity stack, MasterTheorem.omega_lambda_from_phi is the proposition "carried Ω_Λ content ∧ Nonempty Track4ACert", and omega_lambda_from_phi_proven builds that witness from the five fields of track4ACert.
Framework landmarks in play: T8 forces $D=3$; the eight-tick octave supplies the ledger seed $11/16$; the φ-ladder places the η_B exponent at rung $-44$. The structure does not touch Track 4.B (why the vacuum mode-sum is $\varphi^{-44}$ rather than $10^{120}$ times larger) or Track 4.C (Ω_Λ tension and dark-energy equation of state).
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