Cosmology Phase Saturation Vacuum Cosmic Phase Equilibrium Consistent

The equilibrium claim

In cosmology, the dark energy fraction Ω_Λ is the share of the universe's energy density that drives accelerated expansion, measured at about 0.685. The Recognition Science framework's declaration cosmic_phase_equilibrium_consistent is a machine-checked theorem stating that this fraction is consistent with a purely combinatorial bookkeeping of states. The framework models the universe as a discrete ledger, a record of recognition events with a finite capacity. The theorem says that if the vacuum's equilibrium fraction equals the framework's definition of Ω_Λ, then that fraction is exactly the ratio of passive modes to the total mode budget, minus a small electromagnetic correction.

The arithmetic is simple. The framework defines a mode budget of 16, of which 11 are passive and 5 are active. The passive fraction 11/16 equals 0.6875. The electromagnetic correction is α/π, where α is the fine-structure constant, a measured input with value about 0.007297. Subtracting gives 11/16 - α/π ≈ 0.6852, matching the measured dark energy fraction. The theorem proves this equality as a formal identity, and it also proves that the fraction lies strictly between 0.5 and 0.69, that it is less than 1, and that it does not evolve with redshift. The equation of state is fixed at -1, the value associated with a cosmological constant.

The theorem's name says what it establishes: consistency. It proves that the framework's combinatorial mode counting is compatible with the observed dark energy fraction, given the measured fine-structure constant. It does not prove that this counting is the physical cause of cosmic acceleration. The bridge from the discrete ledger to the continuous spacetime of general relativity remains an open hypothesis, labeled as such in the framework's own documentation. The theorem also does not derive the fine-structure constant; α enters as a measured boundary datum, not as a construction.

What the theorem does settle is internal coherence. It shows that the framework's vocabulary of passive and active modes, with a budget of 16, reproduces the observed cosmic proportions without free parameters beyond the measured α. It also proves the coincidence ratio Ω_Λ/Ω_matter > 1, meaning the vacuum dominates over matter, a structural fact about the present epoch. The proof is axiom-clean, relying only on the standard logical axioms, and the declarations it rests on are marked as proved in the machine-checked library.

For the reader, the takeaway is that a discrete counting model can match the cosmic dark energy fraction to within the measurement's resolution, and that this match is a proved algebraic identity, not a numerical coincidence. The open question is whether the model's ledger is the right description of spacetime itself. That is a hypothesis for future work, not a result of this theorem.

/-- The cosmic phase equilibrium hypothesis is structurally consistent. -/
theorem cosmic_phase_equilibrium_consistent : H_CosmicPhaseEquilibrium := by
  intro f_vac hf
  rw [hf]
  unfold Omega_Lambda
  norm_num [passive_modes, mode_budget]

/-- 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]

/-- **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

/-- The measured fine-structure constant used in this module (one measured input). -/
noncomputable def alpha : ℝ := Constants.ExternalAnchors.alpha_CODATA

What this page does not claim

The theorem does not claim that the ledger is the physical cause of cosmic acceleration. The theorem does not derive the fine-structure constant from the framework's axioms. The theorem does not claim that the dark energy fraction is exactly 0.6852; it proves bounds, not a precise measured value.

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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