Cosmology Cosmic Aging Amplitude Sharp Desi Sharp Tighter Than Carnot

The sharpened falsifier

In cosmology, the dark energy equation of state parameter w describes how the pressure of dark energy relates to its density. A value of w = -1 corresponds to a cosmological constant, while deviations are written as w(z) = -1 + δw · Z(z)/Z_today, where Z(z) is a cosmic aging function and δw measures the deviation. The Recognition Science framework's ledger, a discrete record of recognition events, implies a maximum possible deviation, the Carnot saturation ceiling, equal to J(φ) = φ - 3/2 ≈ 0.118.

The framework's model of cosmic aging predicts a specific midpoint for the dark energy density parameter, Ω_Λ = 0.6845. The Planck 2018 central value is Ω_Λ = 0.6889. The gap between these, Δ ≈ 0.0044, is the observed discrepancy that the framework's BIT (background interaction theory) must explain. Under the linear-kernel assumption, this gap translates to an implied δw of roughly δw_implied ≈ Δ / ⟨Z⟩_norm, where ⟨Z⟩_norm is the time-averaged normalized cosmic Z over the late-time dark-energy-dominated era, lying between 0.3 and 0.5.

The theorem desi_sharp_tighter_than_carnot establishes that the sharpened DESI falsifier threshold, defined as desi_sharp_threshold = 2 · δw_implied_max, is strictly less than one third of the Carnot ceiling: desi_sharp_threshold < J(φ)/3. Numerically, the sharp threshold lies in the band 0.026 < desi_sharp_threshold < 0.034, while the Carnot ceiling is above 0.11. This means the new threshold is about eight times tighter than the old one, which was simply δw < J(φ) ≈ 0.118.

The practical consequence is operational. With the old loose threshold, a null result at δw = 0.05 would not falsify the BIT explanation; it would only rule out the extreme Carnot saturation. With the sharp threshold, a null result at δw = 0.005, the strong falsifier, directly contradicts the BIT explanation of the Planck-RS Ω_Λ gap, because 0.005 is below the smallest implied δw of 0.008. This sharp threshold is the number that DESI Y3+, Roman, and Euclid analyses should use to test the framework.

The theorem is a statement about the relationship between two numbers. It proves that the sharp threshold is smaller than the Carnot ceiling divided by three. It does not prove that the implied δw is the actual value, nor that the linear-kernel translation is exact; the full nonlinear treatment remains a hypothesis. It also does not claim that the sharp threshold has been measured; it is a prediction for future experiments to test.

/-- **The sharpened threshold is at least a factor 3 tighter than
the Carnot ceiling.** -/
theorem desi_sharp_tighter_than_carnot :
    desi_sharp_threshold < Cost.Jcost phi / 3 := by
  -- desi_sharp_threshold < 0.034 from band
  have h := desi_sharp_threshold_band.2
  -- J(φ) > 0.11
  have h_jphi : 0.11 < Cost.Jcost phi := by
    rw [jcost_phi_closed]
    have hphi : 1.61 < phi := phi_gt_onePointSixOne
    linarith
  have : (0.034 : ℝ) < 0.11 / 3 := by norm_num
  linarith

/-- The sharpened threshold is in `(0.026, 0.034)`. -/
theorem desi_sharp_threshold_band :
    0.026 < desi_sharp_threshold ∧ desi_sharp_threshold < 0.034 := by
  unfold desi_sharp_threshold
  have h1 : 0.013 < delta_w_implied_max := delta_w_implied_max_gt
  have h2 : delta_w_implied_max < 0.017 := delta_w_implied_max_lt
  refine ⟨?_, ?_⟩ <;> linarith

/-- The strong falsifier is below the smallest implied `δw_min`. -/
theorem strong_falsifier_below_implied_min :
    bit_strong_falsifier < delta_w_implied_min := by
  unfold bit_strong_falsifier
  exact lt_trans (by norm_num : (0.005 : ℝ) < 0.008) delta_w_implied_min_gt

/-- Numerical: the observed gap satisfies `0.004 < Δ_obs < 0.005`. -/
theorem gap_observed_band : 0.004 < gap_observed ∧ gap_observed < 0.005 := by
  unfold gap_observed planck_omega_lambda_central rs_omega_lambda_mid
  refine ⟨?_, ?_⟩ <;> norm_num

What this page does not claim

The theorem does not prove that the implied δw is the actual value of the deviation. The linear-kernel translation from the Ω_Λ gap to δw is a hypothesis, not a theorem. The sharp threshold is a prediction, not a measurement. The theorem does not claim that the Carnot ceiling itself is the correct upper bound for all possible models.

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

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