Cosmology Cosmic Aging Amplitude Sharp Delta W Implied Max Lt J Phi Over 6

The sharpened bound

Cosmologists describe dark energy by a parameter equation of state w, the ratio of its pressure to its energy density. A constant w = -1 means a cosmological constant, while a deviation δw measures how much the dark energy changes over time. The Recognition Science framework models this deviation through a quantity it calls the cosmic-aging amplitude, and the declaration delta_w_implied_max_lt_jPhi_over_6 proves a sharp numerical bound on it.

The framework starts from a ledger, a discrete record of recognition events, and derives a unique cost function J(x) = (x + 1/x)/2 - 1. At the golden ratio φ, this cost takes the value J(φ) = φ - 3/2 ≈ 0.118, which the framework treats as a fundamental ceiling on certain physical deviations. The new theorem shows that the maximum implied δw, derived from the observed gap between the Planck 2018 value of the dark energy density (Ω_Λ = 0.6889) and the framework's midpoint (0.6845), is less than J(φ)/6, or about 0.017. This is a factor of about eight below the ceiling itself.

The bound matters because it sharpens the test. The older, looser threshold said δw < J(φ) ≈ 0.118, which a null result at 0.05 would not falsify. The new bound says δw < 0.017, so a measurement finding δw above that would directly contradict the framework's explanation of the Planck gap. The module also defines a sharp falsifier at δw = 0.005, below even the smallest implied value, providing a clean target for experiments like DESI Y3+, Roman, and Euclid.

What the theorem does not claim is equally important. It does not prove that the linear-kernel translation from the Ω_Λ gap to δw is exact; that translation is a leading-order approximation, and a full nonlinear treatment remains a target. It does not claim the framework's midpoint value 0.6845 is measured; it is a model output. And it does not claim that a null result at 0.005 falsifies the framework itself, only the specific BIT explanation of the Planck gap. The theorem is a machine-checked statement about the size of a derived quantity, not a claim about the universe's actual dark energy.

/-- **The implied `δw` from the Planck-RS Ω_Λ gap is well below
the BIT Carnot ceiling.** Concretely: `δw_implied_max < J(φ) / 6`.

This shows `J(φ)` is an upper bound, not a saturation prediction. -/
theorem delta_w_implied_max_lt_jPhi_over_6 :
    delta_w_implied_max < Cost.Jcost phi / 6 := by
  -- delta_w_implied_max < 0.017 (from above)
  have h_max : delta_w_implied_max < 0.017 := delta_w_implied_max_lt
  -- J(φ) = φ - 3/2 > 0.11 since φ > 1.61
  have h_jphi : 0.11 < Cost.Jcost phi := by
    rw [jcost_phi_closed]
    have hphi : 1.61 < phi := phi_gt_onePointSixOne
    linarith
  -- 0.017 < 0.11 / 6 = 0.01833...
  have : (0.017 : ℝ) < 0.11 / 6 := by norm_num
  linarith

/-- 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 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-kernel translation from the Ω_Λ gap to δw is exact. The framework's midpoint value 0.6845 is a measured quantity. A null result at δw = 0.005 falsifies the entire Recognition Science framework.

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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