Cosmology Dark Energy Phi Dilution Law Dimension Forced Theta Amplitude Le Ceili

The amplitude ceiling

The declaration dimensionForcedTheta_amplitude_le_ceiling is a theorem in the framework's machine-checked library of formal theorems. It states that a certain dark-energy amplitude, derived from first principles, is less than or equal to a specific cost ceiling. The amplitude in question is the predicted present-day strength of dark energy, and the ceiling is the cost function J(φ) = (φ + 1/φ)/2 - 1, evaluated at the golden ratio φ ≈ 1.618. The theorem is not an empirical measurement; it is a logical consequence of the framework's axioms and definitions.

To understand what the theorem establishes, one must see how the amplitude is built. The framework models dark energy as a ledger, a discrete record of recognition events, transmitted through a carrier of four dimensions. The dilution law states that a unit of this occupancy is attenuated by one factor of φ⁻¹ per dimension, yielding an occupancy fraction of φ⁻⁴. From this, the framework derives the amplitude's closed form: (φ - 3/2) / (3φ + 2), which numerically falls between 0.015 and 0.018. The ceiling theorem then proves that this amplitude never exceeds J(φ), which equals φ - 3/2 ≈ 0.118. The amplitude is not merely below the ceiling; it is far below it, by more than a factor of six.

The theorem's significance lies in what it rules out. It shows that the dark-energy amplitude, as derived from the framework's first principles, cannot grow without bound. It is constrained by a universal cost function that the framework proves is the unique function satisfying five plain conditions. This is a structural result: the amplitude is not a free parameter fitted to data, but a forced consequence of the framework's logic. The proof is conditional on the named dimension-uniform φ-dilution law, which is an admissibility rule the framework chooses to impose, not a fact about the physical universe.

What the theorem does not claim is equally important. It does not claim that the amplitude equals any measured value of dark energy. The framework provides a band for the amplitude, 0.015 to 0.018, but this is a prediction to be tested against observations, not a measurement with a receipt. The theorem also does not claim that the ceiling J(φ) is a physical constant of nature; it is a mathematical bound derived from the cost function. The theorem does not assert that the dilution law is true; it is a premise, and the theorem's validity is conditional on it. Finally, the theorem does not claim that the amplitude is the actual dark-energy density in our universe; it is a model-derived quantity whose empirical status remains open.

/-- The dimension-forced theta gives an amplitude below the `J(φ)` ceiling. -/
theorem dimensionForcedTheta_amplitude_le_ceiling :
    DarkEnergyAmplitudeDerivation.attenuatedAmplitude darkEnergyThetaFromDimension ≤
      Cost.Jcost phi :=
  DarkEnergyThetaStatus.thetaDerived_amplitude_le_ceiling dimensionForcedThetaFirstPrinciples

/-- Dimension-uniform φ attenuation through a carrier dimension `n`: one factor of `φ⁻¹`
per dimension. -/
def phiDilution (n : ℕ) : ℝ := 1 / phi ^ n

/-- 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]

/-- `δw(0)` is more than six times below the ceiling, confirming `J(φ)` is a ceiling and not
the value (occupancy `θ = φ⁻⁴ < 1/6`). -/
theorem predictedPresentAmplitude_lt_ceiling_div_six :
    predictedPresentAmplitude < Cost.Jcost phi / 6 := by
  have hb := predictedPresentAmplitude_band.2
  have hlo : (1.61 : ℝ) < phi := phi_gt_onePointSixOne
  rw [jcost_phi_closed]
  have hnum : (0.11 : ℝ) < phi - 3/2 := by linarith
  linarith

What this page does not claim

The theorem does not claim the amplitude equals any measured value of dark energy. The theorem does not claim the dilution law is true; it is a conditional premise. The theorem does not claim the ceiling J(φ) is a physical constant of nature.

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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