cosmic_phase_equilibrium_consistent
plain-language theorem explainer
Whenever the vacuum fraction equals the RS dark-energy prediction Ω_Λ = 11/16 − α/π, it equals the passive-mode fraction of the D=3 ledger minus the EM correction α/π. Cosmologists auditing the phase-saturation derivation of the cosmological constant cite this for internal coherence. The proof is a short unfold-and-norm_num check that 11/16 matches passive_modes over mode_budget.
Claim. For every real $f_{\mathrm{vac}}$, if $f_{\mathrm{vac}} = \Omega_\Lambda$ with $\Omega_\Lambda = 11/16 - \alpha/\pi$, then $f_{\mathrm{vac}} = N_{\mathrm{passive}}/N_{\mathrm{budget}} - \alpha/\pi$, where $N_{\mathrm{budget}} = 16$ is the total mode budget of the $D=3$ ledger vacuum and $N_{\mathrm{passive}}$ is the passive (vacuum) mode count from $Q_3$ cube geometry.
background
This module identifies the cosmological dark-energy fraction with the equilibrium vacuum share of a discrete ledger under phase saturation. The RS prediction is $\Omega_\Lambda = 11/16 - \alpha/\pi$: a cube-geometry seed minus the measured-$\alpha$ electromagnetic correction. Numerically this sits near $0.685$.
The mode budget is $16 = 2^4$, from the $D=3$ cube doubled by double-entry bookkeeping. Active modes (matter excitations) number $5$; the complementary passive modes supply the geometric seed $11/16$. The hypothesis $H_{\mathrm{CosmicPhaseEquilibrium}}$ asserts that at cosmic scale the vacuum fraction equals that passive share corrected by $\alpha/\pi$.
The local chain runs: finite phase capacity, equilibrium of matter and vacuum modes, identification of the vacuum fraction with the $Q_3$ passive fraction, and the numerical formula above. Related scaffolding includes scale-invariance of the NonExistenceCost and a bridge from ledger saturation to cosmology.
proof idea
Term-style tactic proof of a universal statement. Introduce an arbitrary real vacuum fraction $f_{\mathrm{vac}}$ and the equality hypothesis $f_{\mathrm{vac}} = \Omega_\Lambda$. Rewrite the goal with that equality, unfold the definition $\Omega_\Lambda = 11/16 - \alpha/\pi$, then discharge the remaining numerical identity by norm_num on passive_modes and mode_budget, which forces $N_{\mathrm{passive}}/16 = 11/16$. No external lemmas beyond those definitions are required.
why it matters
The declaration does not establish the physical content of cosmic phase equilibrium; it shows the hypothesis is definitionally coherent with the combinatorial mode count. That coherence is a prerequisite for treating $\Omega_\Lambda = 11/16 - \alpha/\pi$ as the equilibrium vacuum fraction from $Q_3$ geometry (module chain steps 3–4).
It sits inside the cosmology layer that imports dimension forcing ($D=3$, landmark T8) and the eight-tick octave structure underlying the doubled cube budget $2^4$. Downstream use is currently empty; the result is a local consistency seal rather than a lemma consumed by a larger proved theorem. The genuine open pieces remain the physical hypotheses themselves (phase equilibrium and the vacuum-fraction bridge), each carrying an explicit observational falsifier.
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