alpha_lt_half

In the phase-saturation cosmology module, the dark-energy fraction is identified with the equilibrium vacuum share of the discrete ledger:

$$\Omega_\Lambda = \frac{11}{16} - \frac{\alpha}{\pi}.$$

The seed $11/16$ comes from Q₃ mode counting (passive versus active modes). The single measured input is the fine-structure constant, taken here as the CODATA 2022 anchor rather than an RS-derived value. Because $\alpha$ is a few parts in $10^{-3}$, the correction $\alpha/\pi$ is small, but positivity and upper bounds on $\Omega_\Lambda$ still need a concrete comparison of $\alpha$ against simple rationals such as $1/2$.

Sibling facts already record $\alpha > 0$. This lemma supplies the matching upper bound used by the ratio comparisons against $\pi$ and against the geometric seed.

Term-mode numeric proof. Unfold the local alpha definition to the external CODATA anchor $7.2973525643 \times 10^{-3}$, then discharge $\alpha < 1/2$ by norm_num. No analytic lemmas about $\pi$ or $\varphi$ are required; the inequality is pure decimal comparison.

This bound is the arithmetic gate for the positivity chain in phase-saturation vacuum energy. Downstream, alpha_over_pi_lt_seed uses it with $\pi > 1$ to get $\alpha/\pi < 11/16$, hence $\Omega_\Lambda > 0$; the tighter form alpha_over_pi_lt_tight sharpens to $\alpha/\pi < 1/6$ via $\pi > 3$. The twin derivation module reuses the same fact for omega_lambda_positive. In the GR-limit parameter bundle a parallel (but $\varphi$-native) $\alpha < 1/2$ feeds the perturbative product bound $|\alpha, c_{\mathrm{Lag}}| < 0.1$. Within the module status list this sits on the proved side of the $\Omega_\Lambda$ bounds, upstream of the still-open CosmicPhaseEquilibrium and vacuum_fraction_bridge hypotheses that would connect ledger saturation to cosmology.