Cosmology Phase Saturation Vacuum Coincidence Ratio Structural

The structural ratio

In cosmology, the ratio of dark energy to dark matter is a famous puzzle. The universe's expansion is accelerating, which cosmologists attribute to dark energy, while dark matter's gravity binds galaxies. The observed ratio is about 2.4 to 1, but why it has this value is unexplained. The Recognition Science framework offers a structural answer, not a numerical fit.

Within the framework, the universe is modeled as a discrete ledger of recognition events, a record of every interaction. The framework's library of machine-checked theorems contains a result called coincidence_ratio_structural. It proves that the framework's dark energy fraction Ω_Λ divided by its matter fraction Ω_m is always greater than 1. In plain language, the framework's model of the cosmos must contain more dark energy than matter. This is a theorem, not a guess, and it holds for any scale, meaning the ratio is a fixed structural property of the model, not a value that changes over time.

The proof is a consequence of the framework's mode budget. The framework defines a total of 16 modes for the universe's degrees of freedom. Of these, 5 are active, representing matter, and 11 are passive, representing the vacuum. The dark energy fraction is the passive fraction, 11/16, minus a small correction involving the fine-structure constant α. The matter fraction is the active fraction, 5/16, plus the same correction. The theorem proves that the passive fraction is always larger than the active fraction, even after the correction is applied. This is why the ratio is greater than one.

The framework also proves that the dark energy fraction is always between 0.5 and 0.69, and that it is always positive and less than one. This means the framework's model avoids the so-called vacuum catastrophe, where theoretical predictions for vacuum energy are wildly larger than what is observed. The framework's value is in the right ballpark, but it is not a precise prediction. The formula for Ω_Λ is 11/16 - α/π, which evaluates to about 0.6852. The measured value of α is an input to this formula, not a derivation. The framework does not predict the fine-structure constant; it takes it from experiment.

What the theorem does not claim is that this ratio matches the observed cosmological data. The observed ratio of dark energy to matter is about 2.4, while the framework's structural ratio is only proven to be greater than 1. The framework's value of about 0.685 for the dark energy fraction is a structural consequence of its mode counting, but it is not a match to the measured value of about 0.7. The theorem is about the structure of the framework's model, not about the actual universe. It establishes a necessary inequality, not a numerical agreement.

/-- 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

/-- The coincidence ratio Ω_Λ/Ω_m is O(1) by construction. -/
theorem coincidence_ratio_structural :
    Omega_Lambda / Omega_matter > 1 := by
  have hOL_gt : (0.5 : ℝ) < Omega_Lambda := Omega_Lambda_gt_05
  have hOm_pos : 0 < Omega_matter := by linarith [omega_closure, Omega_Lambda_lt_one]
  have hOm_lt : Omega_matter < 0.5 := by linarith [omega_closure]
  rw [gt_iff_lt, lt_div_iff₀ hOm_pos]
  nlinarith

/-- Total mode budget of the D=3 ledger vacuum.
    16 = 2⁴ from the D=3 cube doubled by double-entry bookkeeping. -/
def mode_budget : ℕ := 16

/-- 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⟩

What this page does not claim

The theorem does not claim that the framework's ratio matches the observed cosmological ratio of about 2.4. The framework does not predict the measured value of the dark energy fraction; it only proves a structural bound. The theorem does not derive the fine-structure constant, which is an external measured input.

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.

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