Cosmology Dark Energy Theta Phi Four Six Lt Phi Four

The fourfold dilution bound

The golden ratio φ, roughly 1.618, appears throughout mathematics and physics. One of its less famous powers is φ⁴, about 6.854. The inequality 6 < φ⁴ is a simple numerical fact, but in the Recognition Science framework it is a load-bearing bound. It says that the fourth power of the golden ratio exceeds six, which is exactly what one needs to show that the reciprocal 1/φ⁴ is smaller than 1/6.

The number 1/φ⁴, about 0.1459, is proposed as a candidate for the dark-energy occupancy fraction. The idea is that dark energy acts like a leakage or occupancy correction, a fraction of a unit ceiling set by the framework's cost function. In a four-dimensional spacetime setting, the minimal dimension-uniform attenuation of a unit cost is a fourfold dilution, hence the fourth power. The bound 6 < φ⁴ guarantees that this candidate fraction is strictly below the sharp 1/6 envelope, meaning it is a positive, sub-saturation value that fits the first-principles target.

The framework's machine-checked library of formal theorems proves the inequality using a previously established lower bound, 1.61 < φ. From that bound, the proof derives 6 < φ⁴, and from that, the strict inequality 1/φ⁴ < 1/6. The chain is closed: the candidate is positive, at most one, and strictly below the envelope, so it qualifies as an admissible attenuation fraction. The library also shows that the resulting dynamic amplitude is positive and below the J(φ) ceiling.

What the declaration does not claim is the physical premise. The algebraic content is closed, but the scientific burden remains open: justifying that four-dimensional φ-dilution is the actual dark-energy occupancy law. The framework proves the candidate and its bounds as a theorem; it does not prove that this candidate is the true description of dark energy. That physical identification is a separate step, explicitly flagged as the remaining work.

/-- `φ⁴` is larger than `6`, using the established lower bound `1.61 < φ`. -/
theorem six_lt_phi_four : (6 : ℝ) < phi ^ 4 := by
  have h : (1.61 : ℝ) < phi := phi_gt_onePointSixOne
  have hpos : (0 : ℝ) < phi := phi_pos
  nlinarith [sq_nonneg (phi - 1.61), sq_nonneg (phi ^ 2 - (1.61 : ℝ) ^ 2)]

/-- The candidate dark-energy occupancy fraction: four-dimensional φ-dilution. -/
def thetaPhiFour : ℝ := 1 / phi ^ 4

/-- The candidate is strictly below the sharp `1/6` envelope. -/
theorem thetaPhiFour_lt_one_sixth : thetaPhiFour < (1 : ℝ) / 6 := by
  unfold thetaPhiFour
  have h6 : (6 : ℝ) < phi ^ 4 := six_lt_phi_four
  have hphi4 : (0 : ℝ) < phi ^ 4 := pow_pos phi_pos 4
  have h6pos : (0 : ℝ) < (6 : ℝ) := by norm_num
  rw [div_lt_div_iff₀ hphi4 h6pos]
  nlinarith

/-- The candidate's dynamic amplitude is positive. -/
theorem thetaPhiFour_amplitude_pos :
    0 < attenuatedAmplitude thetaPhiFour :=
  thetaDerived_amplitude_pos thetaPhiFourFirstPrinciples

/-- The candidate's dynamic amplitude is below the `J(φ)` ceiling. -/
theorem thetaPhiFour_amplitude_le_ceiling :
    attenuatedAmplitude thetaPhiFour ≤ Cost.Jcost phi :=
  thetaDerived_amplitude_le_ceiling thetaPhiFourFirstPrinciples

What this page does not claim

The physical premise that four-dimensional φ-dilution is the dark-energy occupancy law is not proved. The candidate 1/φ⁴ is not claimed to match any measured dark-energy value. The inequality 6 < φ⁴ is not claimed to hold for any number other than the golden ratio.

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

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