omega_closure
plain-language theorem explainer
Dark energy and matter density parameters sum exactly to one under the RS phase-saturation definitions. Cosmologists citing the RS vacuum fraction Ω_Λ = 11/16 − α/π need this algebraic closure before the coincidence-ratio and certificate theorems. The proof unfolds the two density defs and cancels by ring arithmetic on the 5+11 mode split of the D=3 cube budget.
Claim. With $\Omega_\Lambda := 11/16 - \alpha/\pi$ and $\Omega_m := 5/16 + \alpha/\pi$ (active modes over total mode budget, plus the EM correction), one has $\Omega_\Lambda + \Omega_m = 1$.
background
This module treats dark energy as the equilibrium vacuum fraction of a discrete ledger with finite phase capacity. The geometric seed is the passive-mode count on the D=3 cube: total mode budget 16 (= 2^4 from the cube doubled by double-entry), of which 11 are passive (vacuum) and 5 are active (matter: 3 face-pair/generation plus 2 charge/parity).
The RS dark-energy fraction is that seed minus an electromagnetic correction: $\Omega_\Lambda = 11/16 - \alpha/\pi$. Matter is defined as the complementary active fraction plus the same correction: $\Omega_m = 5/16 + \alpha/\pi$. The $\alpha/\pi$ terms are equal and opposite by construction, so flat-universe closure is an identity once the mode counts add to the budget.
Upstream combinatorial defs fix active_modes = 5 and mode_budget = 16; the geometric seed 11/16 is already proved in-module. The broader chain still has open hypotheses linking ledger saturation to cosmology (CosmicPhaseEquilibrium, vacuum_fraction_bridge).
proof idea
One-line algebraic identity. Unfold $\Omega_\Lambda$, $\Omega_m$, active_modes, and mode_budget to expose $11/16 - \alpha/\pi$ and $5/16 + \alpha/\pi$, then ring cancels the $\alpha/\pi$ terms and sums $11/16 + 5/16 = 1$. No external lemmas beyond the four local defs.
why it matters
Closure is the bookkeeping hinge of the phase-saturation vacuum story: without $\Omega_\Lambda + \Omega_m = 1$, the RS prediction $\Omega_\Lambda \approx 0.685$ cannot be read as a flat-universe density parameter. Downstream, coincidence_ratio_structural uses it (with $\Omega_\Lambda > 1/2$ and $\Omega_\Lambda < 1$) to prove $\Omega_\Lambda/\Omega_m > 1$ by construction, dissolving the coincidence problem as an O(1) ratio forced by the 11:5 mode split. The certificate phase_saturation_vacuum_cert packages closure as a required field alongside positivity and seed bounds.
Framework landmarks: D=3 cube geometry (T8) supplies the 16-mode budget; the eight-tick/octave structure sits behind the doubled bookkeeping. The result is fully proved combinatorially; it does not discharge the still-open bridge from ledger saturation to observed cosmology.
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