coincidence_ratio_structural
plain-language theorem explainer
The dark-energy to matter density ratio exceeds unity under the Recognition Science vacuum fraction. Cosmologists treating the coincidence problem inside the phase-saturation derivation would cite this bound. The argument is short linear arithmetic: vacuum fraction above one half plus flat closure forces the matter complement below one half, hence the ratio above one.
Claim. With $\Omega_\Lambda = 11/16 - \alpha/\pi$ the RS dark-energy fraction and $\Omega_m$ its flat-universe complement (so $\Omega_\Lambda + \Omega_m = 1$), one has $\Omega_\Lambda / \Omega_m > 1$.
background
This module identifies the cosmological constant fraction with the equilibrium vacuum share of a discrete ledger at phase saturation. The seed $11/16$ comes from passive-mode counting on the $Q_3$ cube geometry; subtracting the electromagnetic correction $\alpha/\pi$ yields $\Omega_\Lambda = 11/16 - \alpha/\pi \approx 0.685$.
The matter fraction is the complement $\Omega_m = 5/16 + \alpha/\pi$ (equivalently active modes over the mode budget plus the same EM term). Flat closure $\Omega_\Lambda + \Omega_m = 1$ is an algebraic identity once both definitions are unfolded.
Upstream bounds already give $\Omega_\Lambda > 1/2$ (from $\alpha/\pi < 1/6$) and $\Omega_\Lambda < 1$ (subunitarity, since the seed $11/16 < 1$). Those two facts plus closure pin the coincidence ratio.
proof idea
Start from the proved lower bound $\Omega_\Lambda > 0.5$. Flat closure together with $\Omega_\Lambda < 1$ yields $\Omega_m > 0$ by linear arithmetic. The same closure with the half-bound yields $\Omega_m < 0.5$. Rewrite the target as $\Omega_\Lambda > \Omega_m$ via the positive-denominator form of division, then finish by nonlinear arithmetic on the two half-bounds.
why it matters
The classical coincidence problem asks why dark energy and matter densities are comparable today. Inside Recognition Science the vacuum fraction is fixed by ledger phase saturation at the geometric seed $11/16$ corrected by $\alpha/\pi$, so the ratio is $O(1)$ by construction and in fact strictly greater than one. The result feeds the module certificate that packages positivity, subunitarity, seed bounds, mode partition, and closure for the phase-saturation vacuum story. It does not discharge the still-open hypotheses connecting ledger saturation to cosmology (cosmic phase equilibrium and the vacuum-fraction bridge); those remain explicit hypothesis interfaces with falsifiers.
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