geometric_seed_eq
plain-language theorem explainer
The passive (vacuum) mode count over the total D=3 ledger mode budget equals exactly 11/16. Anyone deriving Ω_Λ from phase saturation, or tracing the geometric factor 11 in the α seed, cites this identity. The proof is a one-line numerical evaluation of the two combinatorial definitions.
Claim. The ratio of passive vacuum modes to the total mode budget of the $D=3$ ledger equals $11/16$: if the budget is $16$ and the passive count is $11$, then $11/16$ is the geometric seed fraction.
background
This module identifies the cosmological dark-energy fraction with the equilibrium vacuum share of a discrete ledger under phase saturation: $\Omega_\Lambda = 11/16 - \alpha/\pi$. The geometric piece $11/16$ is pure mode counting on the $D=3$ cube.
Spatial dimension is forced to $D=3$ (T8/T9). The total mode budget is defined as $16 = 2^{D+1}$: the $2^D$ vertices of the cube, doubled by double-entry bookkeeping. Passive modes (vacuum) are $11 = 8$ vertex ground states plus $3$ unexcited face-pair contributions; the complementary active modes are the five matter/recognition channels.
Upstream, the same integer $11$ appears as the passive-channel factor in the $\alpha$ geometric seed $4\pi\cdot 11$. Here the claim is only the normalized fraction against the sixteen-slot budget.
proof idea
One-line tactic proof. Unfold passive_modes ($:= 11$) and mode_budget ($:= 16$), cast to $\mathbb{R}$, and discharge $11/16 = 11/16$ by norm_num. No lemmas beyond the two combinatorial definitions are required.
why it matters
Closes the combinatorial step in the module chain: equilibrium vacuum fraction equals the passive mode fraction from $Q_3$ geometry, fixed at $11/16$. Downstream, $\Omega_\Lambda$ bounds and the full RS formula $\Omega_\Lambda = 11/16 - \alpha/\pi$ rest on this seed. The same integer $11$ feeds AlphaDerivation.geometric_seed_eq ($\mathrm{geometric_seed} = 4\pi\cdot 11$) and channelBudget_eq in the $\alpha$-genesis loop certificate, so the cosmology and fine-structure pipelines share one mode count. Framework landmarks: T8 forces $D=3$, hence budget $2^{D+1}=16$; the eight-tick/octave structure sits behind the vertex count. Open hypotheses still bridge ledger saturation to cosmology (CosmicPhaseEquilibrium, vacuum_fraction_bridge); this theorem only locks the pure fraction.
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