Cosmology Cosmic Aging Amplitude Sharp

The sharpened falsifier

Cosmology measures how the expansion of the universe changes over time. In the standard model, this is governed by the equation of state parameter w, the ratio of pressure to energy density for dark energy. A value of exactly -1 means a cosmological constant, a fixed energy density. Any deviation, written as δw, means the dark energy is slowly changing. The DESI survey and future missions like Roman and Euclid are designed to measure δw with increasing precision.

In Recognition Science, the framework's cost function J(x), which measures the unavoidable cost of a recognition event, takes the value J(φ) = φ - 3/2 ≈ 0.118 at the golden ratio φ. This number acts as a natural ceiling for δw in the framework's model of cosmic aging, a model where the universe's expansion is tied to a discrete ledger of recognition events. The previous bound said δw must be less than about 0.118. A null result at 0.05 would not contradict the model, only its most extreme version.

The new module sharpens this. It starts with the observed gap between the Planck 2018 measurement of the dark energy density, Ω_Λ = 0.6889, and the framework's predicted midpoint, 0.6845. That gap is about 0.0044. Assuming a simple linear model for how δw varies over cosmic time, this gap implies δw is only about 0.011, with a range from 0.009 to 0.016. The module proves this implied value is far below the J(φ) ceiling, by a factor of six or more.

The sharper falsifier is set at twice the maximum implied value, about 0.030. This is much tighter than the old ceiling. The module also defines a strong falsifier at δw = 0.005, which is below the smallest implied value. If future surveys measure δw above 0.03, the framework's explanation of the cosmic gap is in trouble. If they measure it below 0.005, the framework's explanation is directly contradicted. The old ceiling could not distinguish these cases; the new threshold can.

The key theorem, cosmic_aging_sharp_one_statement, bundles four results: the observed gap lies between 0.004 and 0.005, the implied δw is far below J(φ), the sharp threshold is below J(φ)/3, and the strong falsifier is below the smallest implied δw. These are machine-checked in the framework's library, with no gaps in the proofs. The practical consequence is that DESI and its successors now have a precise, operationally meaningful number to test, not a vague upper bound.

/-- 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

/-- The slack factor: J(φ) is **at least 6x larger** than the
implied δw. -/
theorem jPhi_dominates_implied :
    delta_w_implied_max * 6 < Cost.Jcost phi := by
  have := delta_w_implied_max_lt_jPhi_over_6
  linarith [Cost.Jcost_unit0]

/-- **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 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

What this page does not claim

The linear translation from the Ω_Λ gap to δw is a full derivation; it is a leading-order approximation. The framework predicts a specific measured value of δw; it only sets a threshold. The module proves the framework's cosmic aging model is correct; it only sharpens the falsifier.

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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