Cosmology Cosmic Aging Amplitude Sharp Cosmic Aging Sharp One Statement

The sharpened bound

Cosmology measures the accelerating expansion of the universe through the dark energy density parameter Ω_Λ. The Planck 2018 survey gave a central value of 0.6889. Recognition Science's own midpoint estimate is 0.6845, leaving a gap of about 0.0044 between the two. The ledger, a discrete record of recognition events, implies a specific correction to the dark energy equation of state, and that correction must account for this gap.

The correction is expressed as δw, the deviation of the dark energy pressure-to-density ratio from the cosmological constant value of -1. A simple linear model translates the observed gap into an implied δw: divide the gap by a time-averaged normalized cosmic Z, a framework quantity that tracks the aging effect, which lies between 0.3 and 0.5. The resulting implied δw falls between 0.008 and 0.017, with a midpoint near 0.011. This is far below the older, looser ceiling of J(φ) ≈ 0.118, which came from a thermodynamic efficiency bound.

The declaration cosmic_aging_sharp_one_statement bundles four machine-checked facts. First, the observed gap lies strictly between 0.004 and 0.005. Second, the maximum implied δw times six is still less than J(φ), so the gap requires a correction at least six times smaller than the old ceiling. Third, the sharpened falsifier threshold, set at twice the maximum implied δw, lies between 0.026 and 0.034, which is less than one third of J(φ). Fourth, a strong falsifier value of 0.005 sits below the smallest implied δw of 0.008, meaning a null result at that level would directly contradict the framework's explanation of the gap.

In Recognition Science, this sharpens the operational test for DESI Y3+, Roman, and Euclid. A null result at δw = 0.005 refutes the BIT explanation of the Planck-RS gap, while a null result at 0.05 would only rule out the extreme thermodynamic ceiling, not the framework itself. The theorem is a statement about numerical bounds and their logical consequences, not a measurement of δw.

/-- 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 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 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

This theorem does not measure δw; it only bounds the value implied by the gap under a linear-kernel assumption. This theorem does not prove that the linear translation Δ ≈ δw · ⟨Z⟩_norm is exact; that relation is a HYPOTHESIS pending full nonlinear treatment. This theorem does not claim that a null result at 0.005 would falsify all of Recognition Science, only the BIT explanation of the Planck-RS gap.

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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