omega_lambda_lt_11_16
plain-language theorem explainer
The RS dark-energy density parameter sits strictly below the geometric seed 11/16. Anyone citing the C-010 upper bound or the registry certificate needs this inequality. The proof unfolds the definition and applies linear arithmetic once α/π > 0 is in hand.
Claim. Let $\Omega_\Lambda^{\mathrm{RS}} := 11/16 - \alpha/\pi$, with $\alpha > 0$ the measured fine-structure constant. Then $\Omega_\Lambda^{\mathrm{RS}} < 11/16$.
background
Module C-010 treats the cosmological-constant problem: QFT vacuum energy overshoots observation by ~10^120, while data give Ω_Λ ≈ 0.7. Recognition Science fixes the density parameter by the closed formula
Ω_Λ = 11/16 − α/π.
The seed 11/16 comes from the D = 3 ledger (eight-tick octave forced at T8, gap-45 synchronization, LCM structure). The single measured input is α (CODATA), entering only through the IR correction α/π.
In this file, Omega_Lambda_RS is exactly that real number. The sibling lemma alpha_pos records α > 0 by unfolding the external CODATA anchor and norm_num. The same upper-bound claim appears in the unification registry as a calculated prediction.
proof idea
Unfold Omega_Lambda_RS to 11/16 − α/π. From alpha_pos and Real.pi_pos obtain α/π > 0 by div_pos. Then linarith closes 11/16 − (positive) < 11/16. No further arithmetic or numeric evaluation is required.
why it matters
This is the strict upper half of theorem C-010.4 (bounds 0 < Ω_Λ < 11/16) and is re-exported verbatim as C-010.2. Downstream, Omega_Lambda_bounds pairs it with positivity; Omega_Lambda_lt_upper_bound is a one-line alias. The unification registry reuses the same inequality inside omega_lambda_bounds and as a field of registry_predictions_cert_exists, so the certificate that RS predictions are inhabited depends on it.
Framework-wise it locks the geometric seed from T8 (D = 3, eight-tick) against the α/π correction: the vacuum fraction cannot reach the bare ledger value once electromagnetism is present. It does not itself derive 11/16 or compute the tighter observational window (0.683, 0.686); those live in sibling results.
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