Documentation

FUST.Astronomy.Cosmology

FUST Cosmological Scale Structure #

Scale lattice {φⁿ} and hierarchical suppression from time evolution (TimeStructure.lean).

Scale Lattice from φ-Invariance #

The scale lattice {φⁿ : n ∈ ℤ} is derived from FUST time evolution:

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theorem FUST.Cosmology.scaleLattice_from_timeEvolution (n : ℕ) :
(fun (f : ℂ → ℂ) (t : ℂ) => f (↑φ ^ n * t)) = fun (f : ℂ → ℂ) (t : ℂ) => f (↑φ ^ n * t)

Scale lattice is derived from time evolution: φⁿ = n-fold time scaling

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@[reducible, inline]
noncomputable abbrev FUST.Cosmology.scaleLattice (n : ℤ) :
ℝ

Scale lattice point at level n: notation for φ^n (n-fold time evolution)

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Instances For
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    theorem FUST.Cosmology.scaleLattice_pos (n : ℤ) :
    scaleLattice n > 0

    Scale lattice is positive

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    theorem FUST.Cosmology.scaleLattice_shift (n : ℤ) :
    scaleLattice (n + 1) = φ * scaleLattice n

    Scale lattice respects time evolution: φ^{n+1} = φ · φⁿ

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    theorem FUST.Cosmology.inverseScale_from_psi :
    φ⁻¹ = |ψ|

    Inverse scale uses |ψ| = φ⁻¹: derived from φ · |ψ| = 1

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    @[reducible, inline]
    noncomputable abbrev FUST.Cosmology.inverseScaleLattice (n : ℤ) :
    ℝ

    Inverse scale lattice point: notation for φ^{-n} (ψ-evolution)

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    Instances For
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      theorem FUST.Cosmology.inverseScaleLattice_eq (n : ℤ) :
      inverseScaleLattice n = (scaleLattice n)⁻¹

      Inverse lattice equals reciprocal

      Part 3: Hierarchical Suppression (FUST Derivation) #

      From time evolution f ↦ f(φ·), each scale level is separated by factor φ. This gives natural hierarchy suppression: higher levels are φⁿ times larger.

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      theorem FUST.Cosmology.hierarchy_suppression_factor (n m : ℤ) (h : n < m) :
      scaleLattice n < scaleLattice m

      Hierarchy suppression factor between levels

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      theorem FUST.Cosmology.inverse_hierarchy (n m : ℤ) (h : n < m) :
      inverseScaleLattice m < inverseScaleLattice n

      Inverse hierarchy: higher level means smaller inverse

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      theorem FUST.Cosmology.adjacent_level_ratio (n : ℤ) :
      scaleLattice (n + 1) / scaleLattice n = φ

      Ratio between adjacent levels is exactly φ

      Part 4: Time Evolution and Entropy #

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      theorem FUST.Cosmology.entropy_time_connection (f : ℂ → ℂ) (t : ℂ) :
      TimeStructure.entropyAtFζ (TimeStructure.timeEvolution f) t = TimeStructure.FζLagrangian (TimeStructure.timeEvolution f) t

      Entropy of time-evolved state equals Lagrangian

      Part 5: Golden Ratio Mathematical Identities #

      These are properties of φ = (1+√5)/2, not FUST derivations.

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      theorem FUST.Cosmology.golden_partition :
      φ⁻¹ + φ⁻¹ ^ 2 = 1

      φ⁻¹ + φ⁻² = 1: The fundamental partition identity

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      theorem FUST.Cosmology.golden_partition_div :
      1 / φ + 1 / φ ^ 2 = 1

      Equivalent form: 1/φ + 1/φ² = 1

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      theorem FUST.Cosmology.sqrt5_bounds :
      2 < √5 ∧ √5 < 2.5

      √5 is between 2 and 2.5

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      theorem FUST.Cosmology.phi_inv_sq_eq :
      φ⁻¹ ^ 2 = 2 - φ

      φ⁻² = 2 - φ

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      theorem FUST.Cosmology.geometric_series_sum :
      (1 - φ⁻¹)⁻¹ = φ ^ 2

      Sum of geometric series: (1 - φ⁻¹)⁻¹ = φ²

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      theorem FUST.Cosmology.phi_plus_phi_inv :
      φ + φ⁻¹ = √5

      φ + φ⁻¹ = √5

      Part 6: Scale Power Properties (Mathematical) #

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      theorem FUST.Cosmology.zpow_neg_pos (n : ℕ) :
      0 < φ ^ (-↑n)

      φ^{-n} is positive for all n

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      theorem FUST.Cosmology.zpow_neg_decreasing (n : ℕ) :
      φ ^ (-(↑n + 1)) < φ ^ (-↑n)

      φ^{-(n+1)} < φ^{-n}: strict monotonic decrease

      Part 7: Scale Structure #

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      theorem FUST.Cosmology.scaleAtLevel_from_inverse_evolution (k : ℕ) :
      φ ^ (-↑k) = |ψ| ^ k

      Scale factor derived from inverse time evolution