passive_mode_decomposition
plain-language theorem explainer
Passive vacuum modes on the Q₃ ledger split as eight vertex ground states plus three unexcited face-pair modes, giving the integer identity 11 = 8 + 3. Cosmologists citing the geometric seed 11/16 for Ω_Λ need this count. The proof is a one-line native_decide on the three constant definitions.
Claim. The number of passive (vacuum) modes equals the sum of the vertex ground-state count and the unexcited face-pair mode count: $11 = 8 + 3$.
background
The module identifies the dark-energy fraction $\Omega_\Lambda$ with the equilibrium passive-mode fraction of a discrete ledger at cosmic scale, aiming at $\Omega_\Lambda = 11/16 - \alpha/\pi$. Spatial dimension is forced to $D = 3$ (T8/T9), so the recognition geometry is the 3-cube $Q_3$ with $2^D = 8$ vertices.
Passive modes are the vacuum contribution to the mode budget. By definition they total 11: eight vertex ground states (one per vertex of $Q_3$) plus three unexcited face-pair modes. Active modes make up the complementary part of the budget; the geometric seed $11/16$ is the passive share of that budget.
Upstream, $D := 3$ is taken from dimension forcing and the alpha-derivation constants. The three naturals passive_modes, vertex_ground_states, and unexcited_face_modes are closed combinatorial definitions in this file.
proof idea
Term-mode proof by native_decide. The three sides are definitionally the naturals 11, 8, and 3, so the equality $11 = 8 + 3$ is discharged by kernel evaluation with no lemmas or rewriting.
why it matters
This identity is the combinatorial content behind the passive-mode count that feeds the geometric seed $11/16$ in the $\Omega_\Lambda$ chain. Module status lists passive_modes and the mode budget as proved combinatorial steps; the decomposition makes the split 8 + 3 explicit rather than leaving 11 as an opaque constant.
It sits under the phase-saturation story: ledger phase capacity, equilibrium between matter excitations and vacuum modes, and the identification of the vacuum fraction with the $Q_3$ passive fraction. Downstream hypotheses (CosmicPhaseEquilibrium, vacuum_fraction_bridge) still connect that fraction to cosmology; this theorem only locks the integer arithmetic. Framework landmarks in play are T8 ($D = 3$) and the eight-vertex cube $2^3$.
No used_by edges are recorded yet; sibling results bound $\Omega_\Lambda$ and compare it to the seed $11/16 - \alpha/\pi$.
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