omegaLambdaCert
plain-language theorem explainer
Packages the dark-energy fraction certificate: raw mode saturation is exactly 11/16, the EM correction α/π lies in (0.002, 0.004), and the corrected Ω_Λ sits in (0.683, 0.686), within 2σ of Planck 2018, using only CODATA α as measured input. Cosmologists citing the RS Ω_Λ derivation use this as the single entry point. Proof is a structure constructor that wires five already-proved lemmas.
Claim. There exists a certificate that the RS dark-energy fraction satisfies: the raw saturated fraction equals $11/16$ exactly; the electromagnetic correction $\alpha/\pi$ lies in $(0.002, 0.004)$; the corrected value $\Omega_\Lambda = 11/16 - \alpha/\pi$ lies in $(0.683, 0.686)$; $|\Omega_\Lambda - 0.6889| < 2\cdot 0.0056$ (Planck 2018 within $2\sigma$); and the only measured input is the CODATA fine-structure constant.
background
The module derives the cosmological constant density fraction from eight-tick phase-mode saturation. The 8-tick DFT carries 44 frequency modes; 11 of them are $Q_3$-symmetric (three spatial axes plus gauge sector, equivalently the $[4,2,2]$ Gray-code asymmetry under $S_3$ breaking). The addressing denominator is $16 = 2^4$ (4-bit labels on the eight-tick cycle), so the raw saturated fraction is $11/16$.
Electromagnetism supplies a one-loop vacuum correction: the EM-active mode fraction is $\alpha/(2\pi)$, which subtracts $\alpha/\pi$ from the raw budget. With CODATA $\alpha$, this correction is a few parts in a thousand. The certificate structure OmegaLambdaCert records five obligations: exact raw fraction, correction bounds, final numerical interval, Planck consistency, and the one-measured-input canonical form $\Omega_\Lambda = 11/16 - \alpha_{\mathrm{CODATA}}/\pi$.
Upstream lemmas already establish each piece: omega_raw_eq for $11/16$; em_correction_pos2 and em_correction_small for the $(0.002,0.004)$ window via $\pi$ decimal bounds; omega_lambda_interval for $(0.683,0.686)$; and omega_lambda_canonical_form identifying the CODATA input.
proof idea
Structure constructor, not a deep argument. Each field of OmegaLambdaCert is filled by a named upstream lemma:
raw_fraction:=omega_raw_eq(exact $11/16$).correction_bounds:= pair ofem_correction_pos2andem_correction_small(lower/upper bounds on $\alpha/\pi$).final_interval:=omega_lambda_interval(itself the pair of the $0.683$ and $0.686$ inequalities).planck_consistentunfolds the Planck 2018 central value and $1\sigma$ error, then applies the already-provedrs_consistent_with_planckcomparison.one_measured_input:=omega_lambda_canonical_form, which rewrites $\Omega_\Lambda$ as $11/16 - \alpha_{\mathrm{CODATA}}/\pi$ by unfolding the saturated-mode and tick-addressing constants and normalizing.
No new arithmetic is performed at this site.
why it matters
This is the module-level certificate for the RS dark-energy claim: $\Omega_\Lambda = 11/16 - \alpha/\pi \in (0.683, 0.686)$, consistent with Planck 2018 at $2\sigma$, with a single measured input. It closes the derivation chain that begins from the eight-tick octave (forcing-chain T7) and the three spatial dimensions (T8), which fix the $2^4$ addressing and the 11 $Q_3$-modes.
The module doc states the core theorem explicitly and records the 2026-07-06 policy that CODATA $\alpha$ remains an external anchor rather than a constructed constant (the constructed seed $4\pi\cdot 11$ is excluded by measurement at $>30{,}000\sigma$). Downstream use count is currently zero; the certificate is the public face of the cosmology derivation rather than an intermediate lemma.
Framework contact: eight-tick period $2^3$, $D=3$, and the $\alpha$ band all appear as structural inputs to the mode budget and the EM correction.
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