Cosmology Dark Energy Phi Dilution Law Predicted Present Amplitude Above Strong

The falsifiability threshold

In cosmology, the present amplitude of dark energy is a number that observations constrain. The Recognition Science framework derives a specific value for this amplitude from a dilution law: a unit of dark energy transmitted through a four-dimensional carrier is attenuated by one factor of the golden ratio's reciprocal per dimension. The framework's library proves this amplitude, in its closed form, equals (phi - 3/2) / (3 * phi + 2), which lies between 0.015 and 0.018.

The declaration predictedPresentAmplitude_above_strong_falsifier establishes that this predicted amplitude is greater than a pre-defined threshold called the strong falsifier. This threshold is not fitted to the prediction; it is a sharp boundary set by the framework's own analysis of what would contradict its account. The theorem proves the predicted value sits above that boundary, meaning the framework's prediction is not trivially small or zero; it is a concrete, testable number in a specific range.

This result is a theorem in the machine-checked library, meaning it is derived from the framework's axioms with no gaps. The theorem also proves the predicted amplitude is below a sharp threshold derived from DESI observations, so the prediction occupies a narrow window: above the falsifier, below the observational bound. This is the structure of a genuine prediction: a number that could be wrong, with a clear criterion for what would count as being wrong.

What this declaration does not claim is that the predicted amplitude matches any measured value. The framework does not assert an empirical match here; it asserts a logical consequence of its own premises. The theorem does not prove that the dilution law is true, only that if the law holds, the amplitude must fall in this band. The falsifier threshold itself is a definition within the framework, not an independent observational quantity.

/-- The predicted amplitude lies strictly above the BIT strong-test falsifier `0.005`: the
dynamic signature is large enough to be a real DESI/Roman/Euclid target. -/
theorem predictedPresentAmplitude_above_strong_falsifier :
    CosmicAgingAmplitudeSharp.bit_strong_falsifier < predictedPresentAmplitude := by
  have h := predictedPresentAmplitude_band.1
  unfold CosmicAgingAmplitudeSharp.bit_strong_falsifier
  linarith

/-- Closed form: `δw(0) = (φ − 3/2)/(3φ + 2)`. -/
theorem predictedPresentAmplitude_closed :
    predictedPresentAmplitude = (phi - 3/2) / (3 * phi + 2) := by
  unfold predictedPresentAmplitude DarkEnergyAmplitudeDerivation.attenuatedAmplitude
  rw [darkEnergyThetaFromDimension_eq_phiFour]
  unfold DarkEnergyThetaPhiFour.thetaPhiFour
  rw [jcost_phi_closed, phi_pow_four_eq]
  ring

/-- Numeric band: `0.015 < δw(0) < 0.018`, robust to the `(1.61, 1.62)` φ enclosure. -/
theorem predictedPresentAmplitude_band :
    0.015 < predictedPresentAmplitude ∧ predictedPresentAmplitude < 0.018 := by
  rw [predictedPresentAmplitude_closed]
  have hlo : (1.61 : ℝ) < phi := phi_gt_onePointSixOne
  have hhi : phi < (1.62 : ℝ) := phi_lt_onePointSixTwo
  have hden : (0 : ℝ) < 3 * phi + 2 := three_phi_plus_two_pos
  constructor
  · rw [lt_div_iff₀ hden]; nlinarith [hlo]
  · rw [div_lt_iff₀ hden]; nlinarith [hhi]

/-- The predicted amplitude lies strictly below the sharpened DESI threshold, i.e. inside the
"consistent with the BIT explanation of the Ω_Λ gap" regime, not the diagnostic-large
regime. -/
theorem predictedPresentAmplitude_below_sharp_desi :
    predictedPresentAmplitude < CosmicAgingAmplitudeSharp.desi_sharp_threshold := by
  have hb := predictedPresentAmplitude_band.2
  have hd := (CosmicAgingAmplitudeSharp.desi_sharp_threshold_band).1
  linarith

What this page does not claim

This theorem does not claim the predicted amplitude matches any measured value. This theorem does not prove the dilution law is true; it proves a consequence conditional on it. The strong falsifier threshold is a framework definition, not an independent observational quantity.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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