OmegaLambdaCert
plain-language theorem explainer
Certificate structure packaging the RS dark-energy fraction: raw saturation equals 11/16, the EM correction α/π lies in (0.002, 0.004), final Ω_Λ sits in (0.683, 0.686), and that value is within 2σ of Planck 2018, using one measured CODATA α. Cosmologists citing the phase-saturation derivation of Ω_Λ point here as the single obligation bundle. Instantiation assembles sibling equalities and interval lemmas; no new algebra lives in the structure itself.
Claim. A certificate for the dark-energy density parameter is a record of five obligations: the raw saturated fraction equals $11/16$; the electromagnetic correction $\alpha/\pi$ (CODATA $\alpha$) lies in $(0.002, 0.004)$; the corrected value $\Omega_\Lambda = 11/16 - \alpha/\pi$ lies in $(0.683, 0.686)$; $|\Omega_\Lambda - 0.6889| < 2 \times 0.0056$ (Planck 2018 within $2\sigma$); and $\Omega_\Lambda$ equals exactly $11/16$ minus the measured CODATA fine-structure constant over $\pi$.
background
The module derives the cosmological constant fraction from phase saturation on the eight-tick cycle. The 8-tick DFT carries a forced 44-mode budget; of these, 11 modes are Q₃-symmetric (3 spatial axes plus gauge sector, equivalently the [4,2,2] Gray-code asymmetry under S₃ breaking). Addressing uses 2⁴ = 16 bits (2 bits per epoch half), so the raw saturated fraction is 11/16.
The electromagnetic coupling supplies a one-loop correction: the fraction of EM-active modes is of order α/(2π), which reduces effective saturation by α/π. In this module the correction uses the external CODATA anchor α ≈ 7.297×10⁻³ (one measured input), not a constructed RS α. Thus Ω_Λ := omega_raw − em_correction = 11/16 − α_CODATA/π.
Upstream, EarlyUniverse carries a parallel definition with α_lock; here the certificate deliberately pins the measured anchor so the numerical interval and Planck comparison are auditably external. Sibling defs fix N_modes_saturated = 11, tick_addressing = 16, and the interval lemmas on the correction and final value.
proof idea
No proof body: this is a structure (Prop bundle). Each field is a named obligation whose witnesses are supplied by the downstream inhabitant omegaLambdaCert. That theorem fills raw_fraction from omega_raw_eq, correction_bounds from the pair of EM-correction inequalities, final_interval from the combined Ω_Λ interval lemma, planck_consistent from the RS-vs-Planck comparison, and one_measured_input from the canonical-form rewrite that unfolds the mode counts and normalizes the arithmetic. The structure itself only declares the interface.
why it matters
This is the canonical certificate for the cosmology claim Ω_Λ = 11/16 − α/π. It is the type inhabited by omegaLambdaCert, the module's top-level theorem that packages raw combinatorics, the small EM shift, the predicted band (0.683, 0.686), and 2σ consistency with Planck 2018 (0.6889 ± 0.0056).
Framework landmarks: the eight-tick octave (T7) supplies the 2⁴ addressing and the mode budget; the Q₃ / [4,2,2] count fixes the numerator 11. The deliberate choice of CODATA α (reverted 2026-07-06) keeps the certificate honest about one external input, after an earlier zero-parameter attempt was rejected because the constructed α seed is an identification, not a derived coupling.
Anyone auditing whether RS predicts dark energy without a free Ω_Λ parameter cites this structure and its inhabitant.
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