Cosmology Dark Energy Theta Phi Four Theta Phi Four Le One

The one-sixth bound

The declaration thetaPhiFour_le_one proves that a specific number, written as φ⁻⁴ (one divided by the golden ratio raised to the fourth power), is less than or equal to 1. This number is a candidate for the dark-energy occupancy fraction, the portion of a theoretical maximum that dark energy's effect might occupy. The proof is a theorem in the framework's machine-checked library of formal theorems, meaning its logical steps have been verified by a computer. The number φ⁻⁴ is about 0.1459, so the statement that it is at most 1 is not surprising on its own, but the declaration is a necessary part of a larger chain of reasoning.

The more interesting result is a stronger bound proved in the same module: φ⁻⁴ is strictly less than 1/6, or about 0.1667. This is called the sharp sub-saturation envelope. It means the candidate number does not merely stay under 1, it stays under a tighter ceiling that the framework identifies as significant. The proof of this stronger bound relies on an established lower bound for the golden ratio itself, namely that φ is greater than 1.61, and then uses algebraic manipulation to show that φ⁴ must therefore be greater than 6.

In the framework's account, the number φ⁻⁴ arises as a natural candidate for a dark-energy correction. The idea is that in a four-dimensional spacetime setting, a unit cost associated with the golden ratio gets diluted fourfold, once for each dimension. This is a physical premise, a proposed law of occupancy, not something the formal proof establishes. What the formal proof does establish is that this candidate number is positive, less than 1, and less than 1/6, and therefore it qualifies as an inhabitant of a first-principles target set for dark-energy amplitudes. The declaration also proves that the resulting dynamic amplitude, the actual size of the dark-energy effect in this model, is positive and below the J(φ) ceiling.

The key distinction is between the mathematics and the physics. The declaration proves bounds on a number. It does not prove that this number is the actual dark-energy density in our universe. The physical justification, that four-dimensional φ-dilution is the correct occupancy law, remains an open scientific question, not a theorem. The Lean algebraic content is closed, but the physical premise is not.

/-- The candidate is at most `1`, hence it is an admissible attenuation fraction. -/
theorem thetaPhiFour_le_one : thetaPhiFour ≤ 1 := by
  have h := thetaPhiFour_lt_one_sixth
  linarith

/-- 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 declaration does not prove that φ⁻⁴ is the actual measured dark-energy density. The physical premise that four-dimensional φ-dilution is the correct occupancy law is not a theorem. The declaration does not claim that dark energy is the only phenomenon governed by this bound.

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)

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.

Derived articles

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