Cosmology Dark Energy Phi Dilution Derivation Derived Theta Eq Phi Four

The derived theta

The golden ratio φ, about 1.618, is the unique positive number that equals one plus its own reciprocal. Its fourth power is about 6.854, so φ⁻⁴ is about 0.1459. The declaration derivedTheta_eq_phiFour establishes that this number, and no other, is the dark-energy occupancy fraction in a specific model.

The model starts with a simple idea: a quantity called occupancy, which measures how much of something survives, gets diluted as it passes through independent dimensions. The declaration proves that if this dilution obeys two plain rules, the result is forced. First, the occupancy across several dimensions is the product of the occupancies across each one. Second, a single dimension attenuates by a self-similar ratio: the ratio equals one divided by one plus itself. Solving that equation gives φ⁻¹, about 0.618.

The proof then shows that with these two rules, occupancy after n dimensions is exactly φ⁻ⁿ. The exponent n is not free: it is the forced spacetime dimension, which the framework derives as 4, with one temporal and three spatial dimensions. So the dark-energy occupancy fraction is φ⁻⁴. This value was previously asserted as a definition; the declaration upgrades it to a proved theorem, with zero unproved assumptions in the machine-checked library of formal theorems.

What the declaration does not claim is just as important. It does not say that φ⁻⁴ is the measured fraction of the universe's energy that behaves like dark energy. That measurement, roughly 0.69, is an empirical fact from cosmology, not a consequence of this proof. The declaration also does not claim that the two dilution rules are true of the physical universe. It proves a conditional statement: if the rules hold, the fraction is forced. Whether they hold is a separate question, and the framework leaves it open.

/-- **Headline: `θ = φ⁻⁴` is forced.** Any dimension-uniform dilution law yields exactly the
four-dimensional φ-dilution `θ = φ⁻⁴` at the forced spacetime dimension. -/
theorem derivedTheta_eq_phiFour (L : DimensionUniformDilution) :
    derivedTheta L = DarkEnergyThetaPhiFour.thetaPhiFour := by
  unfold derivedTheta
  rw [L.occ_eq_pow, dilutionExponent_eq_four]
  unfold DarkEnergyThetaPhiFour.thetaPhiFour
  rw [div_pow, one_pow]

/-- **The self-similar attenuation ratio is forced to `φ⁻¹`.** Any positive `ρ` solving the
reciprocal self-similarity equation `ρ = 1/(1+ρ)` equals `1/φ`. The proof routes through the
*scale-forcing* theorem: `1/ρ` solves `x = recipShift x` with `x > 1`, hence `1/ρ = φ`. -/
theorem self_similar_attenuation_forced {ρ : ℝ} (hpos : 0 < ρ)
    (hfp : ρ = 1 / (1 + ρ)) : ρ = 1 / Constants.phi := by
  have hsum_ne : (1 + ρ) ≠ 0 := by positivity
  have key : ρ * (1 + ρ) = 1 := by rwa [eq_div_iff hsum_ne] at hfp
  have hρ_lt_one : ρ < 1 := by nlinarith [key, hpos]
  have hx_gt_one : (1 : ℝ) < 1 / ρ := by
    have h := one_div_lt_one_div_of_lt hpos hρ_lt_one
    simpa using h
  have hfix : ReciprocalGenerator.recipShift (1 / ρ) = 1 / ρ := by
    unfold ReciprocalGenerator.recipShift ReciprocalGenerator.recip
    rw [one_div ρ, inv_inv]
    rw [eq_comm, inv_eq_one_div, div_eq_iff hpos.ne']
    linear_combination -key
  have hxphi : 1 / ρ = Constants.phi :=
    (ReciprocalGenerator.recipShift_fixed_iff hx_gt_one).mp hfix
  calc ρ = 1 / (1 / ρ) := (one_div_one_div ρ).symm
    _ = 1 / Constants.phi := by rw [hxphi]

/-- The forced exponent equals `4` (1 temporal octave + 3 spatial from Alexander duality). -/
theorem dilutionExponent_eq_four : dilutionExponent = 4 :=
  SpacetimeEmergence.spacetime_dim_eq_four

What this page does not claim

The declaration does not claim that φ⁻⁴ is the measured fraction of the universe's energy that behaves like dark energy. The declaration does not claim that the two dilution rules are true of the physical universe. The declaration does not claim that the exponent 4 is derived from the dilution rules alone, apart from the forced spacetime dimension.

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/DarkEnergyPhiDilutionDerivation.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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