Cosmology Dark Energy Theta Phi Four Theta Phi Four Subsaturation

The subsaturation bound

The golden ratio φ, about 1.618, is the number that solves r² = r + 1. Its fourth power, φ⁴, is about 6.854. The reciprocal, 1/φ⁴, is therefore about 0.1459. In the Recognition Science framework, this small number is proposed as a candidate for the dark-energy occupancy fraction, the share of a theoretical maximum that dark energy's effect actually reaches.

Classically, a fraction must be between zero and one. The framework's machine-checked library of formal theorems proves that 1/φ⁴ is positive, that it is at most one, and that it is in fact strictly less than one-sixth. The last bound is the sharp sub-saturation envelope: it places the candidate clearly below the 1/6 mark, not merely below one. The proof uses the established fact that φ is greater than 1.61, which forces φ⁴ above 6.

In Recognition Science, the framework models recognition cost through a function J(x) = (x + 1/x)/2 − 1. At the golden ratio, this cost J(φ) is a ceiling. The candidate 1/φ⁴ is shown to be a valid input to the framework's first-principles theta target, and its resulting dynamic amplitude is positive and does not exceed the J(φ) ceiling. The declaration thetaPhiFour_subsaturation itself simply restates the strict one-sixth bound as a named theorem.

The framework does not claim that 1/φ⁴ is the measured dark-energy fraction. The physical premise, that four-dimensional φ-dilution is the correct occupancy law, remains an open scientific question. The machine-checked result is algebraic: given the candidate, its bounds follow. Whether nature uses this number is a separate matter, explicitly left for further justification.

/-- 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

/-- The four-dimensional φ-dilution candidate inhabits the first-principles theta target. -/
def thetaPhiFourFirstPrinciples : ThetaFromFirstPrinciples where
  theta := thetaPhiFour
  theta_pos := thetaPhiFour_pos
  theta_le_one := thetaPhiFour_le_one

What this page does not claim

The physical premise that four-dimensional φ-dilution is the dark-energy occupancy law is not proved here. The candidate 1/φ⁴ is not claimed to match any measured dark-energy density. The framework does not claim that the golden ratio itself is observed in cosmological data.

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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