Cosmology Cosmic Aging Amplitude Sharp Delta W Implied Mid Band

The implied shift

In cosmology, dark energy is often modeled as a fluid whose pressure changes slowly over time. The standard model treats it as a constant, but many alternatives allow a small deviation, written as δw, from that constant value. A positive δw means the fluid's pressure changes slightly with cosmic time, and a negative one means the opposite. The question is how large that deviation could be while still matching what telescopes actually see.

The Planck satellite's 2018 measurement of the dark energy density gives a central value of Ω_Λ = 0.6889. The Recognition Science framework's own midpoint for that density is 0.6845. The difference, about 0.0044, is the gap the framework's cosmic aging model must explain. That gap is real in the framework's accounting, and it is small: less than half a percent of the density itself.

The framework's ledger, a discrete record of recognition events, implies a specific relationship between that density gap and the dark energy deviation. If the deviation follows a simple linear pattern over the late universe, the implied δw is the gap divided by a time-averaged factor. The declaration delta_w_implied_mid_band pins that middle value down: it proves the implied δw lies strictly between 0.010 and 0.012, using the framework's midpoint for the time average. The full range, using the lower and upper bounds of that average, runs from just above 0.008 to just below 0.017.

That number matters because it is far smaller than the framework's older ceiling. The older bound, derived from a thermodynamic efficiency limit, allowed δw up to about 0.118. The new implied value is roughly eight times smaller. A future experiment that measures δw and finds it around 0.05 would not contradict the framework's core model, only its extreme version. But a measurement below 0.005 would directly falsify the framework's explanation of the Planck gap. The sharp threshold for that test is set at twice the implied maximum, between 0.026 and 0.034, a much tighter target than the old ceiling.

What the declaration does not do is claim that the linear relationship is the full story. The translation from density gap to δw is a leading-order approximation; a complete treatment requires solving the full modified expansion equation. The framework states this openly. The declaration also does not claim that the Planck value itself is wrong, or that the framework's midpoint is the true density. It records the gap as observed and derives what the framework's model would need to explain it.

/-- Numerical: `δw_implied_mid` lies in `(0.010, 0.012)`. -/
theorem delta_w_implied_mid_band :
    0.010 < delta_w_implied_mid ∧ delta_w_implied_mid < 0.012 := by
  unfold delta_w_implied_mid gap_observed planck_omega_lambda_central
    rs_omega_lambda_mid z_norm_late_mid
  refine ⟨?_, ?_⟩ <;> norm_num

/-- Numerical: `δw_implied_min > 0.008` (smallest plausible implied δw). -/
theorem delta_w_implied_min_gt : 0.008 < delta_w_implied_min := by
  unfold delta_w_implied_min gap_observed planck_omega_lambda_central
    rs_omega_lambda_mid z_norm_late_upper
  norm_num

/-- Numerical: `δw_implied_max < 0.017` (largest plausible implied δw). -/
theorem delta_w_implied_max_lt : delta_w_implied_max < 0.017 := by
  unfold delta_w_implied_max gap_observed planck_omega_lambda_central
    rs_omega_lambda_mid z_norm_late_lower
  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 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 relationship between the density gap and δw is the complete physical model. The Planck 2018 central value is incorrect or the framework's midpoint is the true density. The framework derives the value of δw from first principles without any input from the observed density 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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