Cosmology Phase Saturation Vacuum Vacuum Energy Is Mode Fraction

The vacuum's ledger share

The dark energy fraction, written Ω_Λ, is the portion of the universe's energy budget that drives the observed accelerated expansion. In the Recognition Science framework, this fraction is not a free parameter but a piece of bookkeeping. The framework models the vacuum as a discrete record of possible states, a ledger with a fixed capacity of 16 modes. Of these, 11 are classified as passive, meaning they remain unexcited, and 5 as active. The theorem vacuum_energy_is_mode_fraction proves that the dark energy fraction equals the passive share, 11/16, minus a small correction α/π, where α is the measured fine-structure constant. In plain numbers, this gives Ω_Λ ≈ 0.6852.

The 11 and 16 are not arbitrary. The mode budget of 16 derives from the framework's three-dimensional geometry: it equals 2 raised to the power of the physical dimension plus one. The passive modes split further into 8 vertex ground states and 3 unexcited face modes, a combinatorial decomposition that is itself proved. The theorem also shows the fraction stays within a narrow band: greater than 0.5 and less than 0.69, with the vacuum energy density positive and less than one, avoiding the so-called vacuum catastrophe where predicted energy vastly exceeds observation.

In Recognition Science, this is the labeled turn: the framework identifies the dark energy fraction with the equilibrium fraction of the ledger residing in its light memory state. The same phase saturation pressure that drives re-embodiment at biological scales is said to manifest as vacuum energy at cosmic scales. That bridging claim, from biology to cosmology, is a hypothesis with an explicit falsifier, not a proved theorem. The machine-checked library of formal theorems proves the arithmetic: the mode counts, the partition, the bounds, and the identity linking Ω_Λ to the passive fraction.

What the declaration does not claim is just as important. It does not derive the fine-structure constant α; that value enters as a measured input, a boundary datum from external anchors. It does not prove that the cosmic equilibrium actually holds; that is the hypothesis CosmicPhaseEquilibrium. The theorem establishes a structural identity and its consistency, not the physical mechanism that would realize it. The equation of state is proved to be exactly -1, meaning the vacuum energy does not evolve with redshift, a result consistent with standard cosmology's cosmological constant.

/-- The "10^120 discrepancy" dissolves because the vacuum energy is a
    mode fraction (dimensionless, O(1)), not an energy density requiring
    renormalization against M_Planck^4. -/
theorem vacuum_energy_is_mode_fraction :
    Omega_Lambda = (passive_modes : ℝ) / (mode_budget : ℝ) - alpha / Real.pi := by
  unfold Omega_Lambda
  norm_num [passive_modes, mode_budget]

/-- Mode budget derives from D=3: 2^(D+1) = 2^4 = 16. -/
theorem mode_budget_from_D3 : mode_budget = 2 ^ (Foundation.DimensionForcing.D_physical + 1) := by
  rfl

/-- Passive mode decomposition. -/
theorem passive_mode_decomposition :
    passive_modes = vertex_ground_states + unexcited_face_modes := by
  native_decide

/-- **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⟩

theorem w_is_minus_one : equation_of_state = -1 := rfl

What this page does not claim

This answer does not claim that the fine-structure constant is derived within the framework. This answer does not claim that the cosmic equilibrium hypothesis is proved. This answer does not claim that the vacuum energy fraction is a prediction independent of measured inputs.

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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