omega_lambda_from_phi_proven
plain-language theorem explainer
The dark-energy density is fixed as Ω_Λ = 11/16 − α/π, sits inside the Planck 2σ band, and the baryon-asymmetry rung is forced by D = 3. Cosmologists and RS auditors cite this as the closed Track 4.A clause inside the quantum-gravity master statement. The proof is a pure term packaging of the existing Track4ACert witness and its Nonempty proof.
Claim. The Recognition Science dark-energy prediction holds: $\Omega_\Lambda = 11/16 - \alpha/\pi$, this value lies inside the Planck $2\sigma$ observational band, and the baryon-asymmetry rung integer is forced by spatial dimension $D=3$ (via the Track 4.A certificate).
background
Gravity.MasterTheorem authors the twelve-clause quantum-gravity master statement (Track 7.A). Eight clauses are CLOSED and discharged from existing Lean certificates; five remain as hypothesis inputs. This declaration is the CLOSED cosmology clause: dark energy and the baryon asymmetry from the phi-ladder.
Track 4.A supplies the certificate structure with five clauses. The load-bearing ones here are: the exact formula $\Omega_\Lambda = 11/16 - \alpha/\pi$ (raw eight-tick share $11/16$ minus the electromagnetic correction $\alpha/\pi$), the observational band check against Planck, the $2\sigma$ consistency flag, and the $\eta_B$ rung forced by $D=3$ along three convergent routes (gap-from-dimension, chirality×torsion, fermionic DOF). Upstream, track4ACert inhabits that structure by rfl on the formula unfold and by the existing rung and band certificates.
In RS-native units the same ladder that forces $\phi$ (T6) and $D=3$ (T8) therefore fixes the cosmological constant density without a free dark-energy parameter.
proof idea
Term-mode constructor, not a tactic script. The goal type omega_lambda_from_phi is inhabited by pairing (i) a four-field record drawn fieldwise from Cosmology.Track4ACert.track4ACert — namely omegaLambda_formula, omegaLambda_band, planck_2sigma, and etaB_dimension_route — with (ii) the upstream theorem track4ACert_inhabited : Nonempty Track4ACert. No new algebra is performed; the proof is a pure packaging of the already-verified Track 4.A certificate into the master-theorem clause shape.
why it matters
This is one of the eight CLOSED clauses of the Track 7.A master statement. Downstream, rs_quantum_gravity_master_conditional discharges it inside the conjunction that yields RSQuantumGravityMaster once the five still-open hypothesis inputs are supplied. The non-circularity audit closed_certs_hold likewise lists omega_lambda_from_phi among the six closed certificate conjuncts that must hold independently of the open tracks.
Framework landmarks: T7 (eight-tick octave) supplies the raw $11/16$ share; T8 ($D=3$) forces the $\eta_B$ rung; the fine-structure band on $\alpha$ enters only as the small EM correction $\alpha/\pi$. The clause therefore ties cosmology to the same forcing chain that fixes $c$, $\hbar$, and $G$ in RS-native units, with zero free dark-energy parameter. It does not by itself close the unconditional master theorem; Tracks 1.B/1.C, 2.C/2.D, 3.C, 6.B, and 6.C remain open.
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