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module module moderate

IndisputableMonolith.Verification.OmegaLambdaPlanckLikelihood

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Attaches the Planck 2018 Ω_Λ measurement to the RS-derived dark-energy fraction Ω_Λ = 11/16 − α/π. Certifies that the residual sits inside the published two-sigma band and packages the check as an inhabited likelihood certificate. Cosmology auditors of the quantum-gravity falsifier register would cite this module. The work is arithmetic comparison of fixed constants against the derived bound, plus a small certificate wrapper.

claimLet $\Omega_\Lambda = 11/16 - \alpha/\pi$ be the RS dark-energy fraction. With Planck 2018 central value $\Omega_\Lambda^{\mathrm{P18}}$ and uncertainty $\sigma>0$, the residual $|\Omega_\Lambda - \Omega_\Lambda^{\mathrm{P18}}|$ is strictly less than $2\sigma$, so $\Omega_\Lambda$ lies in the Planck two-sigma interval. The module records an active dataset attachment and an inhabited likelihood certificate for that comparison.

background

Recognition Science derives the cosmological constant fraction from phase saturation as $\Omega_\Lambda = 11/16 - \alpha/\pi$, and proves the numerical band $\Omega_\Lambda \in (0.680, 0.700)$ in the upstream cosmology module. That closed-form value is the object under test here.

This verification module sits in the falsifier-register stack. Upstream dataset attachments supply named observational records and sensitivity metadata for every row of the quantum-gravity master plan §7 register. The present file specializes that pattern to the Planck 2018 dark-energy fraction: it fixes the published central value and sigma, forms the residual against the RS prediction, and exposes positivity and two-sigma comparison lemmas.

Notation is ordinary: central value, one-sigma width, residual, and the two-sigma threshold $2\sigma$. Certificates are Prop-level wrappers that package the inequality as a reusable status token for the likelihood layer.

proof idea

Definition block first: Planck central value, sigma, residual, and the two-sigma width as concrete constants, with positivity lemmas for $\sigma$ and $2\sigma$.

Comparison lemmas then show the residual is strictly below $2\sigma$ and therefore that RS $\Omega_\Lambda$ lies in the Planck two-sigma interval. These are direct numerical inequalities against the already-derived RS band and the fixed Planck numbers.

A boolean/status flag marks the dataset attachment as active. The certificate type and its inhabited instance wrap the two-sigma success into the register's likelihood token; inhabitation is a one-line constructor application once the inequality is in hand.

why it matters in Recognition Science

Feeds the Falsifier Likelihood Register, which aggregates Sessions 107--115 into the dataset-specific likelihood and status layer over the quantum-gravity master plan §7 falsifier register. Without this attachment, the Ω_Λ row would lack a concrete observational likelihood token.

Upstream, it consumes the Ω_Λ derivation (core claim: $\Omega_\Lambda = 11/16 - \alpha/\pi$ in $(0.680, 0.700)$) and the structural dataset-attachment module. Together they turn a pure RS cosmology identity into a falsifiable, register-visible check against Planck 2018.

In the broader framework this is a verification hinge, not a forcing-chain step: it does not re-derive $\phi$, the eight-tick octave, or $D=3$, but it does lock the dark-energy fraction prediction to a named external dataset so the falsifier stack can report pass/fail with a certificate.

scope and limits

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declarations in this module (13)