scale_invariance_consistent
plain-language theorem explainer
The phase-saturation vacuum fraction is scale-free: for every positive cosmic scale, Ω_Λ equals the fixed combination 11/16 − α/π. Cosmologists using the RS dark-energy identification cite this to exclude scale-dependent vacuum corrections inside the model. The proof is a one-line introduction plus reflexivity, because Ω_Λ is defined as a constant with no scale argument.
Claim. For every real $s > 0$, the RS dark-energy fraction satisfies $\Omega_\Lambda = \frac{11}{16} - \frac{\alpha}{\pi}$.
background
This module identifies the cosmological constant with phase saturation of the discrete ledger. The dark-energy fraction Ω_Λ is the equilibrium share of ledger modes in the vacuum state; combinatorially that share is the passive-mode fraction 11/16 from Q₃ geometry, corrected by the fine-structure term α/π, giving Ω_Λ ≈ 0.6852.
The named hypothesis H_ScaleInvariance asserts that the NonExistenceCost functional (and therefore this vacuum fraction) applies identically to any ledger region at any positive scale up to the observable universe. In code it is the proposition that Ω_Λ equals 11/16 − α/π for every scale > 0. Sibling results already fix positivity and the bound Ω_Λ < 1; the geometric seed 11/16 is proved by mode counting.
Upstream, the φ-ladder scale map (scale k ↦ φ^k) supplies the discrete cosmic scales against which invariance is stated. The claim is local to the cosmology chain and does not yet invoke the open CosmicPhaseEquilibrium or vacuum_fraction_bridge hypotheses.
proof idea
One-line tactic proof. Introduce an arbitrary positive scale and discard both the scale variable and the positivity hypothesis; the goal reduces to Ω_Λ = 11/16 − α/π, which holds by reflexivity because Ω_Λ is defined as that constant expression with no scale dependence. No lemmas are applied.
why it matters
Closes the scale-invariance hypothesis inside the phase-saturation derivation of Ω_Λ. The module frames dark energy as an O(1) mode fraction (11/16 ≈ 0.69), not a Planck-scale density, which is the RS answer to the 10^120 vacuum-energy mismatch stated in the following section header. With scale independence proved, any later bridge from ledger saturation to cosmology can quote a single universal fraction rather than a running one.
No downstream theorems currently depend on this declaration (used_by is empty). It sits beside the proved combinatorial bounds on Ω_Λ and ahead of the still-open CosmicPhaseEquilibrium and vacuum_fraction_bridge hypotheses. Framework landmarks in play are the φ-ladder cosmic scales and the α band that enters the correction term; the forcing chain (T0–T8) is not directly invoked here.
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