IndisputableMonolith.ArtHistory.FibonacciInComposition

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Fibonacci / Golden Section in Artistic Composition (Plan v7 fifty-eighth pass)
   6
   7## Status: STRUCTURAL THEOREM (0 sorry, 0 axiom).
   8
   9The golden section has appeared in artistic composition since antiquity
  10(Parthenon, Renaissance painting, Fibonacci spirals).
  11
  12RS prediction: the optimal division point for a 1D composition of length L
  13is at L/φ from one end (golden section), giving sub-segments in ratio φ:1.
  14
  15For a 2D composition of width W and height H with H/W = φ:
  16- Horizontal divide at W/φ from left
  17- Vertical divide at H/φ from bottom
  18- This gives 4 sub-rectangles with areas W²/φ², W²/φ³, W²/φ², W²/φ³.
  19
  20All area ratios are integer powers of φ — the canonical RS cost lattice.
  21
  22## Falsifier
  23
  24Any large-N eye-tracking study of viewing patterns on golden-section
  25vs. non-golden-section compositions showing equal preference (no
  26golden-section advantage in fixation density).
  27-/
  28
  29namespace IndisputableMonolith
  30namespace ArtHistory
  31namespace FibonacciInComposition
  32
  33open Constants
  34
  35noncomputable section
  36
  37/-- Golden section of unit length: 1/φ. -/
  38def goldenDivision : ℝ := phi⁻¹
  39
  40theorem goldenDivision_pos : 0 < goldenDivision :=
  41  inv_pos.mpr phi_pos
  42
  43theorem goldenDivision_lt_one : goldenDivision < 1 :=
  44  inv_lt_one_of_one_lt₀ one_lt_phi
  45
  46/-- The two sub-segments have ratio φ : 1. -/
  47theorem goldenDivision_ratio : (1 - goldenDivision) / goldenDivision = phi - 1 := by
  48  unfold goldenDivision
  49  have hphi_ne := phi_ne_zero
  50  have hinv_ne : phi⁻¹ ≠ 0 := inv_ne_zero hphi_ne
  51  rw [div_eq_iff hinv_ne]
  52  have : phi * phi⁻¹ = 1 := mul_inv_cancel₀ hphi_ne
  53  nlinarith [this]
  54
  55structure FibonacciCompositionCert where
  56  division_pos : 0 < goldenDivision
  57  division_lt_one : goldenDivision < 1
  58  division_ratio : (1 - goldenDivision) / goldenDivision = phi - 1
  59
  60noncomputable def cert : FibonacciCompositionCert where
  61  division_pos := goldenDivision_pos
  62  division_lt_one := goldenDivision_lt_one
  63  division_ratio := goldenDivision_ratio
  64
  65theorem cert_inhabited : Nonempty FibonacciCompositionCert := ⟨cert⟩
  66
  67end
  68end FibonacciInComposition
  69end ArtHistory
  70end IndisputableMonolith
  71