Cosmology Dark Energy Theta Phi Four

The phi-four candidate

In cosmology, dark energy is the name for the unknown agent that makes the expansion of the universe accelerate. The number that describes its strength is not fixed by any standard theory; it must be measured. Recognition Science offers a candidate for that number: one over the golden ratio raised to the fourth power, written φ⁻⁴. The golden ratio φ is the number that solves r² = r + 1, roughly 1.618. The candidate value φ⁻⁴ is about 0.1458.

The candidate arises from a picture of the universe as holding a discrete record of events, a ledger, where every recognition event has a forced cost. The framework proves that any cost function meeting five plain conditions must equal J(x) = (x + 1/x)/2 − 1. Dark energy, in this account, is a leakage fraction of that unit cost. In a four-dimensional spacetime, the minimal uniform dilution of a unit cost is the fourfold φ-dilution, so the candidate occupancy fraction is φ⁻⁴. This is a definitional choice, a model, not a derived law.

What the module proves is narrower and fully rigorous. The machine-checked library of formal theorems establishes that φ⁻⁴ is positive, that it is at most 1, and that it is in fact strictly below the sharp envelope 1/6. Because it is positive and at most 1, it qualifies as an admissible attenuation fraction under the framework's first-principles theta target. The resulting dynamic amplitude, the physical dark-energy strength, is positive and sits below the J(φ) ceiling. These are theorems, checked by the kernel, with no gaps and no extra axioms.

The honest limit is the physical premise. The algebra proves the bounds on φ⁻⁴, but it does not prove that four-dimensional φ-dilution is the actual dark-energy occupancy law. That bridge, from the lower forcing layer to the cosmological premise, remains open. The module closes the algebraic content and leaves the physical justification as the stated scientific burden.

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

/-- The candidate is positive. -/
theorem thetaPhiFour_pos : 0 < thetaPhiFour := by
  unfold thetaPhiFour
  exact one_div_pos.mpr (pow_pos phi_pos 4)

/-- 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 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

/-- 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 module does not prove that dark energy is actually described by φ⁻⁴ in the physical universe. The module does not derive the value of the cosmological constant from first principles. The module does not claim that the physical premise of four-dimensional dilution is closed.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: