Cosmology Dark Energy Phi Dilution Derivation Occ Eq Pow

The dilution law

In cosmology, dark energy is often described by its energy density, but in Recognition Science the relevant quantity is an occupancy: a discrete record of how much of a physical state survives as it is carried across independent dimensions. The framework's library of machine-checked theorems contains a declaration, occ_eq_pow, that establishes the exact form of this dilution. It proves that if an occupancy occ n for n independent dimensions obeys two plain premises, then it must equal (1/φ)^n, where φ is the golden ratio, approximately 1.618.

The first premise is multiplicative composition: the occupancy carried through m + n independent dimensions is the product of the occupancies through m and through n separately. This mirrors how independent channels add in log-cost, so their survival ratios multiply. The second premise is single-dimension self-similar attenuation: one dimension attenuates by a ratio ρ that satisfies ρ = 1/(1+ρ). The unique positive solution to that equation is ρ = 1/φ, which follows from the same forcing theorem that fixes the golden ratio as the self-similar scaling in the framework. From these two premises alone, the theorem derives by induction that occ n = (1/φ)^n for every natural number n.

The exponent is not a free parameter. The framework forces the spacetime dimension to be 4, composed of one temporal octave and three spatial dimensions, and this forced dimension becomes the exponent in the dilution law. The dark-energy occupancy at that dimension is therefore θ = φ⁻⁴, approximately 0.146. The declaration occ_eq_pow is part of a certificate showing that this value is derived from the two premises, not asserted as an admissibility choice. The certificate also exhibits a canonical inhabitant, occ n = (1/φ)^n, proving the premises are consistent and the derivation is not vacuous.

What the declaration does not claim is just as important. It does not assert that the two premises are physically true; it proves a conditional statement. If a dilution law satisfies composition and self-similar attenuation, then it has this form. The theorem does not identify dark energy with any specific physical field or mechanism, and it does not predict a measurable value for the cosmological constant. The framework's derivation of the spacetime dimension 4 is a separate theorem, and the bridge from recognition events to physical spacetime is itself an open target. The declaration establishes a mathematical consequence, not an empirical observation.

/-- **The dilution law, derived.** `n` independent dimensions dilute by `φ⁻ⁿ`. -/
theorem occ_eq_pow : ∀ n : ℕ, L.occ n = (1 / Constants.phi) ^ n := by
  intro n
  induction n with
  | zero => rw [pow_zero]; exact L.occ_zero
  | succ k ih =>
      have hc := L.composes k 1
      rw [hc, ih, L.occ_one_eq_inv_phi, pow_succ]

/-- **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]

/-- **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 canonical dilution law `occ n = φ⁻ⁿ`, exhibiting that the two premises are satisfiable
(so the derivation is not vacuous). -/
def canonicalDilution : DimensionUniformDilution where
  occ := fun n => (1 / Constants.phi) ^ n
  occ_pos := fun n => by
    have hφ : (0 : ℝ) < Constants.phi := Constants.phi_pos
    have hpos : (0 : ℝ) < 1 / Constants.phi := by positivity
    exact pow_pos hpos n
  composes := fun m n => by rw [pow_add]
  one_dim_self_similar := by
    show (1 / Constants.phi) ^ 1 = 1 / (1 + (1 / Constants.phi) ^ 1)
    simp only [pow_one]
    exact inv_phi_self_similar

What this page does not claim

The declaration does not assert that the two premises are physically true. It does not identify dark energy with any specific physical field or mechanism. It does not predict a measurable value for the cosmological constant.

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.

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