Cosmology Phase Saturation Vacuum

The vacuum's share

In cosmology, the cosmological constant is the energy density of empty space, the "dark energy" that accelerates the expansion of the universe. Observations show it makes up about 68.5% of the total energy budget. The standard model of particle physics has no natural way to explain why that number is so large but not absurdly larger; the mismatch between predicted and observed values is called the vacuum catastrophe.

Recognition Science offers a different starting point. Its framework models reality as a discrete ledger, a record of recognition events where each event has a forced cost. The module PhaseSaturationVacuum applies that idea to the cosmos. It defines the dark energy fraction Ω_Λ as 11/16 minus α/π, where α is the measured fine-structure constant. The number 11/16 comes from a combinatorial count: the framework's geometry supplies 16 total modes, of which 11 are passive, or unexcited, and 5 are active. The subtraction of α/π is a small electromagnetic correction, using the measured value of α as an input, not a derived quantity.

What the module proves, in the machine-checked library of formal theorems, is a set of structural facts. The mode budget of 16 splits exactly into 11 passive and 5 active modes, and 11/16 is the geometric seed of the vacuum fraction. The framework proves the vacuum fraction is positive, less than one, and lies in a narrow band between 0.5 and 0.69. It also proves the equation of state is exactly −1, meaning the vacuum energy does not dilute as the universe expands, and that the ratio of vacuum energy to matter energy is greater than one, which addresses the cosmic coincidence problem: why the two are comparable today.

In Recognition Science, the physical bridge is a hypothesis, not a theorem. The framework hypothesizes that the same phase saturation pressure that drives biological re-embodiment manifests as vacuum energy at cosmic scales. This is tagged as a hypothesis with an explicit falsifier, and the connection from biology to cosmology remains open. What is established is the internal consistency of the model: the vacuum fraction is a stable, scale-invariant quantity derived from mode counting, not a free parameter tuned to observations.

/-- The RS prediction for the dark energy fraction.
    Ω_Λ = 11/16 - α/π: cube-geometry seed minus the measured-α EM correction. -/
def Omega_Lambda : ℝ := 11/16 - alpha / Real.pi

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

The framework does not derive the fine-structure constant α; it uses the measured value as an input. The biological-to-cosmological bridge is a hypothesis, not a proved theorem. The module does not explain why the vacuum energy density has the absolute value it does, only its fraction of the total energy budget.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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