IndisputableMonolith.Foundation.PhiEmergence

   1import IndisputableMonolith.Foundation.Inequalities
   2import IndisputableMonolith.Constants
   3import Mathlib
   4
   5/-!
   6# φ-Emergence — The Golden Ratio from J-cost Minimization
   7-/
   8
   9namespace IndisputableMonolith
  10namespace Foundation
  11
  12/-- The golden ratio φ = (1 + √5)/2 ≈ 1.618..., re-exported at the
  13Foundation namespace for modules that use `Foundation.φ`. -/
  14noncomputable abbrev φ : ℝ := Constants.phi
  15
  16namespace PhiEmergence
  17
  18open Inequalities Constants
  19
  20/-! ## Self-Similarity Definition -/
  21
  22/-- A ratio r is self-similar if r² = r + 1. -/
  23def IsSelfSimilar (r : ℝ) : Prop := r^2 = r + 1
  24
  25/-- The golden ratio is positive -/
  26theorem phi_pos : φ > 0 := Constants.phi_pos
  27
  28/-- The golden ratio is greater than 1 -/
  29theorem phi_gt_one : φ > 1 := Constants.one_lt_phi
  30
  31/-! ## φ Satisfies Self-Similarity -/
  32
  33/-- **THEOREM**: The golden ratio satisfies r² = r + 1. -/
  34theorem phi_is_self_similar : IsSelfSimilar φ := Constants.phi_sq_eq
  35
  36/-- The conjugate ratio (1 - √5)/2 also satisfies r² = r + 1 -/
  37noncomputable def φ_conjugate : ℝ := (1 - Real.sqrt 5) / 2
  38
  39theorem phi_conjugate_self_similar : IsSelfSimilar φ_conjugate := by
  40  unfold IsSelfSimilar φ_conjugate
  41  have h5 : Real.sqrt 5 ≥ 0 := Real.sqrt_nonneg 5
  42  have h_sq : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (5 : ℝ) ≥ 0)
  43  ring_nf
  44  rw [h_sq]
  45  ring
  46
  47/-- The conjugate is negative -/
  48theorem phi_conjugate_neg : φ_conjugate < 0 := by
  49  unfold φ_conjugate
  50  have h : Real.sqrt 5 > 1 := by
  51    have h1 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (5 : ℝ) ≥ 0)
  52    nlinarith [Real.sqrt_nonneg 5, sq_nonneg (Real.sqrt 5 - 1)]
  53  linarith
  54
  55/-! ## Uniqueness of φ -/
  56
  57/-- **THEOREM**: φ is the unique positive solution to r² = r + 1. -/
  58theorem phi_unique_positive : ∀ r : ℝ, r > 0 → IsSelfSimilar r → r = φ := by
  59  intro r hr h_self
  60  unfold IsSelfSimilar at h_self
  61  -- Both r and φ satisfy x² = x + 1
  62  have h_phi_ss := phi_is_self_similar
  63  unfold IsSelfSimilar at h_phi_ss
  64  -- From r² = r + 1 and φ² = φ + 1, we get r² - φ² = r - φ
  65  have h_diff_sq : r^2 - φ^2 = r - φ := by linarith
  66  -- (r - φ)(r + φ) = r² - φ² = r - φ
  67  have h_factor : (r - φ) * (r + φ) = r - φ := by nlinarith [sq_nonneg r, sq_nonneg φ]
  68  -- So (r - φ)(r + φ - 1) = 0
  69  have h_zero : (r - φ) * (r + φ - 1) = 0 := by nlinarith
  70  -- By zero product property
  71  rcases mul_eq_zero.mp h_zero with h_case | h_case
  72  · -- Case: r - φ = 0, so r = φ
  73    linarith
  74  · -- Case: r + φ - 1 = 0, so r = 1 - φ
  75    -- But φ > 1, so 1 - φ < 0, contradicting r > 0
  76    have h_r_eq : r = 1 - φ := by linarith
  77    have h_phi_gt_one := phi_gt_one
  78    linarith
  79
  80/-! ## The φ-Ladder -/
  81
  82/-- The φ-ladder: all stable positions are φ^n for integer n. -/
  83def PhiLadder : Set ℝ := { x | ∃ n : ℤ, x = φ^n }
  84
  85/-- φ^n is always positive for any integer n -/
  86theorem phi_pow_pos (n : ℤ) : φ^n > 0 := by
  87  exact zpow_pos phi_pos n
  88
  89/-- Adjacent rungs of the ladder have ratio φ -/
  90theorem phi_ladder_ratio (n : ℤ) : φ^(n+1) / φ^n = φ := by
  91  have h : φ ≠ 0 := ne_of_gt phi_pos
  92  have hn : φ^n ≠ 0 := zpow_ne_zero n h
  93  rw [zpow_add_one₀ h]
  94  field_simp [hn]
  95
  96/-- The ladder is closed under multiplication by φ -/
  97theorem phi_ladder_mul_closed (x : ℝ) (hx : x ∈ PhiLadder) :
  98    x * φ ∈ PhiLadder := by
  99  obtain ⟨n, rfl⟩ := hx
 100  use n + 1
 101  rw [zpow_add_one₀ (ne_of_gt phi_pos)]
 102
 103/-- The ladder is closed under division by φ -/
 104theorem phi_ladder_div_closed (x : ℝ) (hx : x ∈ PhiLadder) :
 105    x / φ ∈ PhiLadder := by
 106  obtain ⟨n, rfl⟩ := hx
 107  use n - 1
 108  rw [zpow_sub_one₀ (ne_of_gt phi_pos)]
 109  rw [div_eq_mul_inv]
 110
 111/-! ## J-cost at φ-Ladder Positions -/
 112
 113/-- J-cost formula applied to φ -/
 114theorem J_at_phi : (φ + 1/φ) / 2 - 1 = (Real.sqrt 5 - 2) / 2 :=
 115  Inequalities.J_cost_phi
 116
 117/-- J-cost at φ^n (for n ≥ 1) -/
 118noncomputable def J_at_phi_pow (n : ℕ) : ℝ :=
 119  (φ^n + φ^(-(n : ℤ))) / 2 - 1
 120
 121/-- J-cost at φ is approximately 0.118 -/
 122theorem J_at_phi_approx : (φ + 1/φ) / 2 - 1 < 0.12 := by
 123  rw [J_at_phi]
 124  -- (√5 - 2)/2 ≈ (2.236 - 2)/2 = 0.118
 125  have h_sqrt5_bound : Real.sqrt 5 < 2.24 := by
 126    have h5_sq : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (5 : ℝ) ≥ 0)
 127    have h_target : (2.24 : ℝ)^2 > 5 := by norm_num
 128    nlinarith [Real.sqrt_nonneg 5, sq_nonneg (Real.sqrt 5)]
 129  linarith
 130
 131/-! ## Connection to Consciousness Threshold -/
 132
 133/-- HYPOTHESIS: The consciousness threshold C = 1 emerges from φ-quantization.
 134
 135    The exact relationship needs derivation. Some possibilities:
 136    1. C = 1 is the point where J-cost integration over one octave equals unity
 137    2. C = 1 relates to φ^n summing to specific values
 138    3. C = 1 is forced by the 8-tick structure interacting with φ
 139
 140    This is marked as a hypothesis until the derivation is complete. -/
 141def H_ThresholdFromPhi : Prop :=
 142  ∃ (mechanism : ℕ → ℝ → ℝ),
 143    mechanism 8 φ = 1  -- Some function of 8 ticks and φ gives threshold 1
 144
 145/-- 1/φ = φ - 1 (golden ratio reciprocal property) -/
 146theorem phi_inv_eq : 1/φ = φ - 1 := Inequalities.phi_inv
 147
 148/-- 1/φ is positive -/
 149theorem phi_inv_pos : 1/φ > 0 := by
 150  have := phi_pos
 151  exact one_div_pos.mpr phi_pos
 152
 153/-- |1/φ| < 1, so the geometric series converges -/
 154theorem phi_inv_lt_one : 1/φ < 1 := by
 155  have h := phi_gt_one
 156  have hp := phi_pos
 157  rw [div_lt_one hp]
 158  linarith
 159
 160/-- The sum φ^(-1) + φ^(-2) + ... converges to φ -/
 161theorem phi_series_sum : ∑' (n : ℕ), (1/φ)^(n+1) = φ := by
 162  -- This is the geometric series: ∑_{n≥0} r^(n+1) = r/(1-r) for |r| < 1
 163  -- With r = 1/φ, we get (1/φ)/(1 - 1/φ) = 1/(φ-1) = φ (since 1/φ = φ-1)
 164  have h_inv_pos := phi_inv_pos
 165  have h_inv_lt_one := phi_inv_lt_one
 166  have h_phi_pos := phi_pos
 167  have h_phi_gt_one := phi_gt_one
 168  -- Rewrite the sum as r * geom_series(r) = r/(1-r)
 169  have h_eq : ∑' (n : ℕ), (1/φ)^(n+1) = (1/φ) * ∑' (n : ℕ), (1/φ)^n := by
 170    rw [← tsum_mul_left]
 171    congr 1
 172    ext n
 173    ring
 174  rw [h_eq]
 175  -- Use the geometric series formula: ∑ r^n = 1/(1-r)
 176  have h_nonneg : 0 ≤ 1/φ := le_of_lt h_inv_pos
 177  rw [tsum_geometric_of_lt_one h_nonneg h_inv_lt_one]
 178  -- Goal: (1/φ) * (1 - 1/φ)⁻¹ = φ
 179  have h_phi_ne : φ ≠ 0 := ne_of_gt h_phi_pos
 180  have h_denom_ne : φ - 1 ≠ 0 := ne_of_gt (by linarith)
 181  -- φ(φ-1) = φ² - φ = 1 (from φ² = φ + 1)
 182  have h_prod : φ * (φ - 1) = 1 := by
 183    have h := phi_is_self_similar
 184    unfold IsSelfSimilar at h
 185    linarith
 186  have h_key : φ - 1 = 1/φ := phi_inv_eq.symm
 187  have h_denom_ne' : 1/φ ≠ 0 := by positivity
 188  -- The whole expression simplifies to φ
 189  field_simp
 190  -- Goal becomes φ = φ * (φ - 1) after field_simp... but φ * (φ-1) = 1
 191  linarith
 192
 193/-! ## Stable Positions -/
 194
 195/-- A position x > 0 is stable if J-cost is locally minimized there.
 196
 197    For self-similar patterns, stability requires the ratio to satisfy r² = r + 1. -/
 198def IsStablePosition (x : ℝ) : Prop :=
 199  x > 0 ∧ x ∈ PhiLadder
 200
 201/-- All φ-ladder positions are stable -/
 202theorem phi_ladder_stable (n : ℤ) : IsStablePosition (φ^n) := by
 203  constructor
 204  · exact phi_pow_pos n
 205  · exact ⟨n, rfl⟩
 206
 207/-- HYPOTHESIS: Stable positions are exactly the φ-ladder.
 208
 209    This is the core claim that self-similar J-cost minimization
 210    forces discrete quantization at φ^n. -/
 211def H_StableIffPhiLadder : Prop :=
 212  ∀ x : ℝ, x > 0 → (IsStablePosition x ↔ x ∈ PhiLadder)
 213
 214/-! ## Summary -/
 215
 216#check IsSelfSimilar
 217#check phi_is_self_similar
 218#check phi_unique_positive
 219#check PhiLadder
 220#check phi_ladder_ratio
 221#check J_at_phi
 222#check H_ThresholdFromPhi
 223#check H_StableIffPhiLadder
 224
 225end PhiEmergence
 226end Foundation
 227end IndisputableMonolith
 228