alpha_over_pi_lt_tight
plain-language theorem explainer
The measured fine-structure constant over π is strictly less than 1/6. Anyone bounding the RS dark-energy fraction Ω_Λ = 11/16 − α/π cites this to force Ω_Λ above one half. The proof is a three-step calc: π > 3 and 0 < α < 1/2 give α/π < α/3 < (1/2)/3 = 1/6.
Claim. The measured fine-structure constant $\alpha$ satisfies $\alpha/\pi < 1/6$.
background
In the phase-saturation account of vacuum energy, the dark-energy fraction is the equilibrium vacuum share of the discrete ledger: $\Omega_\Lambda = 11/16 - \alpha/\pi$. Here $\alpha$ is the CODATA fine-structure constant (the single measured input in this module), and $11/16$ is the geometric seed from $Q_3$ mode counting.
The bound $\alpha/\pi < 1/6$ is elementary numerical control on the electromagnetic correction. Upstream lemmas already give $\alpha > 0$ and $\alpha < 1/2$; Mathlib supplies $\pi > 3$. No Recognition-forcing content is used beyond those facts.
proof idea
Short calc chain. From Mathlib $\pi > 3$ and positivity of $\alpha$, divide to get $\alpha/\pi < \alpha/3$. From the upstream bound $\alpha < 1/2$, divide by the positive constant 3 to get $\alpha/3 < (1/2)/3$. Norm-num closes $(1/2)/3 = 1/6$.
why it matters
Direct input to the theorem $\Omega_\Lambda > 1/2$: unfold $\Omega_\Lambda = 11/16 - \alpha/\pi$ and feed this inequality into linear arithmetic. That lower bound sits in the proved numerical envelope around the RS prediction $\Omega_\Lambda \approx 0.685$ (seed $11/16$ minus the $\alpha/\pi$ correction).
The module still leaves open the bridge from ledger phase saturation (capacity scale $\varphi^{45}$) to cosmic vacuum energy. This lemma is pure arithmetic support for the proved side of the $\Omega_\Lambda$ bounds, not a resolution of those hypotheses.
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