H_CosmicPhaseEquilibrium
plain-language theorem explainer
Named hypothesis that the cosmic vacuum fraction equals the passive-mode share of the D=3 ledger: 11/16 minus the electromagnetic correction α/π. Cosmologists working the RS dark-energy derivation cite it as the physical identification step. It is a Prop definition, not a proved theorem; structural consistency is a one-line unfold of the combinatorial mode counts.
Claim. Cosmic phase equilibrium is the statement that for every real $f_{\mathrm{vac}}$, if $f_{\mathrm{vac}}=\Omega_\Lambda$ then $f_{\mathrm{vac}}=N_{\mathrm{passive}}/N_{\mathrm{budget}}-\alpha/\pi$, where $\Omega_\Lambda=11/16-\alpha/\pi$ is the RS dark-energy fraction, $N_{\mathrm{passive}}=11$ (vacuum modes), $N_{\mathrm{budget}}=16$ (total D=3 ledger modes), and $\alpha$ is the measured fine-structure constant.
background
The module derives the cosmological dark-energy fraction from phase saturation of the discrete recognition ledger. Dark energy is identified with the equilibrium share of ledger modes that remain unexcited (vacuum voxels); matter excitations are the active patterns. The predicted fraction is
$$\Omega_\Lambda=11/16-\alpha/\pi\approx 0.6852.$$
Combinatorial inputs are fixed upstream: total mode budget $16=2^4$ (D=3 cube doubled by double-entry bookkeeping), passive modes $11=8$ vertex ground states $+3$ unexcited face-pair contributions, and active modes $5$. The geometric seed $11/16$ is therefore pure mode counting. The single measured input is $\alpha$ (CODATA), entering only as the EM correction $\alpha/\pi$. Dimension forcing (T8) supplies $D=3$, which sets the cube geometry behind the $16$-mode budget.
proof idea
No proof: the declaration is a def whose body is a Prop. It packages the equality
$$f_{\mathrm{vac}}=\Omega_\Lambda\implies f_{\mathrm{vac}}=\frac{\texttt{passive_modes}}{\texttt{mode_budget}}-\frac{\alpha}{\pi}$$
as a named hypothesis. Downstream, structural consistency is discharged by introducing $f_{\mathrm{vac}}$, rewriting with the hypothesis equality, unfolding $\Omega_\Lambda:=11/16-\alpha/\pi$, and closing with norm_num on the concrete naturals $11$ and $16$.
why it matters
This is the physical identification step in the phase-saturation origin of the cosmological constant. Module chain: finite ledger capacity $\to$ cosmic equilibrium of matter vs vacuum modes $\to$ vacuum fraction fixed by $Q_3$ geometry $\to$ $\Omega_\Lambda=11/16-\alpha/\pi$. Downstream, cosmic_phase_equilibrium_consistent shows the Prop is algebraically coherent with the mode counts, and no_dark_energy_evolution records the companion claim $w(z)=-1$ (no dynamical dark energy). Framework landmarks: T8 forces $D=3$ (cube geometry), eight-tick structure underlies the discrete ledger, and $\alpha$ is the same fine-structure input used elsewhere in RS. Open: the separate bridge hypothesis that ledger saturation pressure really is cosmic vacuum energy; this Prop only names the equilibrium fraction, it does not derive the bridge.
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