Cosmology Phase Saturation Vacuum Scale Invariance Consistent

Scale invariance, stated plainly

Scale invariance is the property that a description of nature does not change when every length, time, and energy is multiplied by a common factor. In cosmology, the dark energy fraction Ω_Λ is the share of the universe's energy budget that behaves like a constant background, the part usually attributed to empty space itself. The theorem scale_invariance_consistent states that the framework's proposed value for this fraction, 11/16 − α/π, is exactly the same number no matter what positive scale factor one chooses. The proof is a direct calculation: the expression contains no length, time, or energy unit at all, so rescaling leaves it untouched.

The value itself comes from a counting argument. The framework models the universe's energy states as a discrete ledger, a record of which modes are excited. The model defines 16 total modes, of which 11 are passive, meaning unexcited, and 5 are active. The dark energy fraction is then the passive share, 11/16, with a small correction from the measured fine-structure constant α, the number that sets the strength of electromagnetic interactions. The theorem mode_budget_partition proves that the active and passive counts sum to 16, and geometric_seed_eq proves the plain ratio 11/16. Because α enters as a measured input, the correction term is not derived from first principles.

What the theorem does not claim is just as important. It does not prove that dark energy is actually scale invariant in the physical universe; that would require a hypothesis connecting the mode counting to real cosmic expansion, which the framework labels CosmicPhaseEquilibrium and marks as a HYPOTHESIS with an explicit falsifier. The theorem only proves that if one adopts the proposed formula, then that formula is consistent with scale invariance. It also does not establish that the mode budget of 16 is forced by the framework's deeper principles; the definitions of active and passive modes are MODEL choices, not derived results. The scale invariance result is a coherence check on a proposed construction, not evidence that the construction is true.

The practical payoff is a sharp bound. The theorem Omega_Lambda_band_unconditional proves that the proposed dark energy fraction lies strictly between 0.5 and 0.69, and no_vacuum_catastrophe proves it is positive and less than 1. These are unconditional results: they hold for the defined quantity without any additional assumptions. A reader can therefore see that the framework's candidate value is not absurdly large or negative, and that it sits in a plausible range compared with cosmological observations. The scale invariance theorem adds the further guarantee that this candidate value is stable under the one symmetry a constant energy density must respect.

theorem scale_invariance_consistent : H_ScaleInvariance := by
  intro _ _; rfl

/-- Mode budget is the sum of active and passive modes. -/
theorem mode_budget_partition : active_modes + passive_modes = mode_budget := by
  native_decide

/-- The geometric seed 11/16 is the passive mode fraction. -/
theorem geometric_seed_eq : (passive_modes : ℝ) / (mode_budget : ℝ) = 11 / 16 := by
  norm_num [passive_modes, mode_budget]

/-- **THEOREM**: Ω_Λ ∈ (0.5, 0.69) — unconditional precision band. -/
theorem Omega_Lambda_band_unconditional :
    0.5 < Omega_Lambda ∧ Omega_Lambda < 0.69 :=
  ⟨Omega_Lambda_gt_05, Omega_Lambda_lt_069⟩

What this page does not claim

The theorem does not prove that dark energy is scale invariant in the physical universe. The theorem does not derive the value of the fine-structure constant α. The theorem does not force the mode budget of 16 from deeper principles.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/PhaseSaturationVacuum.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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