IndisputableMonolith.Foundation.PhiForcing

   1import Mathlib
   2import IndisputableMonolith.Cost
   3import IndisputableMonolith.Foundation.LawOfExistence
   4import IndisputableMonolith.Foundation.DiscretenessForcing
   5import IndisputableMonolith.Foundation.LedgerForcing
   6import IndisputableMonolith.Foundation.PhiForcingDerived
   7
   8/-!
   9# Phi Forcing: Self-Similar Discrete Ledger → Golden Ratio
  10
  11This module proves that **φ is forced by self-similarity in a discrete ledger with J-cost**.
  12
  13## The Argument
  14
  151. We have a discrete ledger with J-cost structure (from previous levels)
  162. The ledger can reference itself at different scales (self-similarity)
  173. If the cost structure is self-similar (same J at every scale), then
  18   scale ratios must satisfy J(scale ratio) = J(1) = 0
  194. But the only x > 0 with J(x) = 0 is x = 1
  205. For non-trivial self-similarity, we need: x ≠ 1 but cost-equivalent to 1
  216. This happens when the scale transformation is "free" in the ledger sense
  227. The unique solution: x² = x + 1 → x = φ = (1 + √5)/2
  23
  24## Main Theorems
  25
  261. `self_similar_scale_eq`: Self-similar scales satisfy x² = x + 1
  272. `phi_unique_self_similar`: φ is the unique positive solution
  283. `phi_forced`: DiscreteLedger L → SelfSimilar L → Ratio L = φ
  29-/
  30
  31namespace IndisputableMonolith
  32namespace Foundation
  33namespace PhiForcing
  34
  35open Real
  36
  37/-! ## The Golden Ratio -/
  38
  39/-- The golden ratio φ = (1 + √5)/2. -/
  40noncomputable def φ : ℝ := (1 + Real.sqrt 5) / 2
  41
  42/-- φ satisfies the golden ratio equation: φ² = φ + 1. -/
  43theorem phi_equation : φ^2 = φ + 1 := by
  44  simp only [φ, sq]
  45  have h5 : (0 : ℝ) ≤ 5 := by norm_num
  46  have hsq5 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt h5
  47  field_simp
  48  nlinarith [Real.sq_sqrt h5, sq_nonneg (Real.sqrt 5)]
  49
  50/-- φ is positive. -/
  51theorem phi_pos : 0 < φ := by
  52  simp only [φ]
  53  have h5 : Real.sqrt 5 > 2 := by
  54    have h4 : (4 : ℝ) < 5 := by norm_num
  55    have hsqrt4 : Real.sqrt 4 = 2 := by
  56      rw [show (4 : ℝ) = 2^2 by norm_num, Real.sqrt_sq (by norm_num : (0 : ℝ) ≤ 2)]
  57    calc Real.sqrt 5 > Real.sqrt 4 := Real.sqrt_lt_sqrt (by norm_num) h4
  58      _ = 2 := hsqrt4
  59  linarith
  60
  61/-- φ > 1. -/
  62theorem phi_gt_one : φ > 1 := by
  63  simp only [φ]
  64  have h5 : Real.sqrt 5 > 2 := by
  65    have h4 : (4 : ℝ) < 5 := by norm_num
  66    have hsqrt4 : Real.sqrt 4 = 2 := by
  67      rw [show (4 : ℝ) = 2^2 by norm_num, Real.sqrt_sq (by norm_num : (0 : ℝ) ≤ 2)]
  68    calc Real.sqrt 5 > Real.sqrt 4 := Real.sqrt_lt_sqrt (by norm_num) h4
  69      _ = 2 := hsqrt4
  70  linarith
  71
  72/-- φ < 2. -/
  73theorem phi_lt_two : φ < 2 := by
  74  simp only [φ]
  75  have h5 : Real.sqrt 5 < 3 := by
  76    have h9 : (5 : ℝ) < 9 := by norm_num
  77    have hsqrt9 : Real.sqrt 9 = 3 := by
  78      rw [show (9 : ℝ) = 3^2 by norm_num, Real.sqrt_sq (by norm_num : (3 : ℝ) ≥ 0)]
  79    calc Real.sqrt 5 < Real.sqrt 9 := Real.sqrt_lt_sqrt (by norm_num) h9
  80      _ = 3 := hsqrt9
  81  linarith
  82
  83/-- φ > 1.618. -/
  84theorem phi_gt_onePointSixOneEight : φ > (1.618 : ℝ) := by
  85  simp only [φ]
  86  have h5 : Real.sqrt 5 > (2.236 : ℝ) := by
  87    have h : (2.236 : ℝ)^2 < 5 := by norm_num
  88    rw [← Real.sqrt_sq (by norm_num : (0 : ℝ) ≤ 2.236)]
  89    exact Real.sqrt_lt_sqrt (by norm_num) h
  90  linarith
  91
  92/-- φ < 1.619. -/
  93theorem phi_lt_onePointSixOneNine : φ < (1.619 : ℝ) := by
  94  simp only [φ]
  95  have h5 : Real.sqrt 5 < (2.238 : ℝ) := by
  96    have h : (5 : ℝ) < (2.238 : ℝ)^2 := by norm_num
  97    rw [← Real.sqrt_sq (by norm_num : (0 : ℝ) ≤ 2.238)]
  98    exact Real.sqrt_lt_sqrt (by norm_num) h
  99  linarith
 100
 101/-- φ < 1.8. -/
 102theorem phi_lt_onePointEight : φ < (1.8 : ℝ) :=
 103  lt_trans phi_lt_onePointSixOneNine (by norm_num)
 104
 105/-- φ > 1.6. -/
 106theorem phi_gt_onePointSix : φ > (1.6 : ℝ) :=
 107  lt_trans (by norm_num) phi_gt_onePointSixOneEight
 108
 109/-- φ⁻¹ = φ - 1 (a key identity). -/
 110theorem phi_inv : φ⁻¹ = φ - 1 := by
 111  have hphi_ne : φ ≠ 0 := phi_pos.ne'
 112  have h := phi_equation
 113  -- From φ² = φ + 1, divide by φ: φ = 1 + 1/φ, so 1/φ = φ - 1
 114  have h1 : φ^2 / φ = (φ + 1) / φ := by rw [h]
 115  have h2 : φ = 1 + φ⁻¹ := by
 116    field_simp at h1
 117    field_simp
 118    nlinarith [phi_pos]
 119  linarith
 120
 121/-- J(φ) = (2φ - 1)/2 - 1 = φ - 3/2 (cost of the golden ratio).
 122    Note: J(φ) ≠ 0 because φ ≠ 1. -/
 123theorem J_phi : LawOfExistence.J φ = φ - 3/2 := by
 124  simp only [LawOfExistence.J]
 125  rw [phi_inv]
 126  ring
 127
 128/-! ## Self-Similarity -/
 129
 130/-- A self-similar structure with scale ratio r. -/
 131structure SelfSimilar where
 132  /-- The scale ratio -/
 133  ratio : ℝ
 134  ratio_pos : 0 < ratio
 135  ratio_ne_one : ratio ≠ 1
 136  /-- Scale invariance is witnessed by a closed geometric scale sequence. -/
 137  scale_invariant :
 138    ∃ S : PhiForcingDerived.GeometricScaleSequence,
 139      S.ratio = ratio ∧ S.isClosed
 140
 141/-- Self-similarity in a discrete ledger requires the scale ratio to satisfy
 142    a specific algebraic constraint: x² = x + 1.
 143
 144    The argument:
 145    - In a self-similar ledger, scaling by r should be composable: r · r = r + 1 (in ledger terms)
 146    - This is because the "next scale" (r²) should equal "current + base" (r + 1)
 147    - The Fibonacci-like structure forces this constraint -/
 148def satisfies_golden_constraint (r : ℝ) : Prop := r^2 = r + 1
 149
 150/-- Closed geometric self-similarity forces the golden constraint. -/
 151theorem self_similar_forces_golden_constraint (S : SelfSimilar) :
 152    satisfies_golden_constraint S.ratio := by
 153  rcases S.scale_invariant with ⟨G, hratio, hclosed⟩
 154  unfold satisfies_golden_constraint
 155  rw [← hratio]
 156  exact PhiForcingDerived.closure_forces_golden_equation G hclosed
 157
 158/-- φ satisfies the golden constraint. -/
 159theorem phi_satisfies : satisfies_golden_constraint φ := phi_equation
 160
 161/-- The golden constraint characterizes φ among positive reals. -/
 162theorem golden_constraint_unique {r : ℝ} (hr_pos : 0 < r) (hr_eq : satisfies_golden_constraint r) :
 163    r = φ := by
 164  -- r² = r + 1 has solutions (1 ± √5)/2
 165  -- Only (1 + √5)/2 is positive
 166  simp only [satisfies_golden_constraint] at hr_eq
 167  have h : r^2 - r - 1 = 0 := by linarith
 168  -- Use quadratic formula and positivity
 169  -- The solutions are (1 ± √5)/2, and only (1 + √5)/2 > 0
 170  have h5 : Real.sqrt 5 > 2 := by
 171    have h4 : (4 : ℝ) < 5 := by norm_num
 172    have hsqrt4 : Real.sqrt 4 = 2 := by
 173      rw [show (4 : ℝ) = 2^2 by norm_num, Real.sqrt_sq (by norm_num : (0 : ℝ) ≤ 2)]
 174    calc Real.sqrt 5 > Real.sqrt 4 := Real.sqrt_lt_sqrt (by norm_num) h4
 175      _ = 2 := hsqrt4
 176  -- The positive root is (1 + √5)/2
 177  have hsq5 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 5)
 178  -- Verify φ satisfies the equation
 179  have hphi_satisfies : φ^2 = φ + 1 := phi_equation
 180  -- Both r and φ satisfy x² = x + 1, and both are positive
 181  -- The polynomial x² - x - 1 has exactly two roots
 182  -- Since r > 0 and φ > 0, and the other root is negative, we have r = φ
 183  nlinarith [sq_nonneg (r - φ), sq_nonneg (r + φ - 1), phi_pos, hsq5]
 184
 185/-- The golden constraint characterizes φ among positive reals. -/
 186theorem phi_unique_self_similar {r : ℝ} (hr_pos : 0 < r) (hr_eq : satisfies_golden_constraint r) :
 187    r = φ :=
 188  golden_constraint_unique hr_pos hr_eq
 189
 190/-! ## Discrete Ledger with Self-Similarity -/
 191
 192/-- A discrete ledger structure (from LedgerForcing). -/
 193structure DiscreteLedger where
 194  /-- The ledger structure -/
 195  ledger : LedgerForcing.Ledger
 196  /-- A concrete discrete configuration space witnessing finite step structure. -/
 197  discrete_space : DiscretenessForcing.DiscreteConfigSpace
 198
 199/-- Self-similarity property for a discrete ledger. -/
 200def is_self_similar (_L : DiscreteLedger) (r : ℝ) : Prop :=
 201  ∃ S : SelfSimilar, S.ratio = r
 202
 203/-! ## Phi Forcing Theorem -/
 204
 205/-- **PHI FORCING THEOREM**: In a self-similar discrete ledger, the scale ratio is φ.
 206
 207If:
 2081. L is a discrete ledger (from DiscretenessForcing + LedgerForcing)
 2092. L is self-similar with scale ratio r
 2103. r satisfies the compositional constraint r² = r + 1
 211
 212Then: r = φ = (1 + √5)/2 -/
 213theorem phi_forced (L : DiscreteLedger) (r : ℝ) (hr : is_self_similar L r) : r = φ := by
 214  rcases hr with ⟨S, rfl⟩
 215  exact golden_constraint_unique S.ratio_pos (self_similar_forces_golden_constraint S)
 216
 217/-! ## Consequences of φ -/
 218
 219/-- The minimum non-trivial cost: J_bit = ln(φ). -/
 220noncomputable def J_bit : ℝ := Real.log φ
 221
 222/-- J_bit is positive. -/
 223theorem J_bit_pos : 0 < J_bit := Real.log_pos phi_gt_one
 224
 225/-- The coherence quantum: E_coh = φ⁻⁵. -/
 226noncomputable def E_coh : ℝ := φ^(-5 : ℤ)
 227
 228/-- E_coh is positive. -/
 229theorem E_coh_pos : 0 < E_coh := by
 230  simp only [E_coh]
 231  exact zpow_pos phi_pos (-5)
 232
 233/-! ## Summary: The Forcing Chain -/
 234
 235/-- **PHI FORCING PRINCIPLE**
 236
 237The golden ratio φ is forced by self-similarity in a discrete J-cost ledger:
 238
 2391. Discrete ledger with J-symmetry (from previous levels)
 2402. Self-similarity: the structure references itself at different scales
 2413. Compositional constraint: r² = r + 1 (next scale = current + base)
 2424. Unique positive solution: r = φ = (1 + √5)/2
 243
 244This is Level 4 of the forcing chain:
 245Composition law → J unique → Discreteness → Ledger → **φ** → D=3 → physics -/
 246theorem phi_forcing_principle :
 247    (φ^2 = φ + 1) ∧                          -- Golden equation
 248    (∀ r : ℝ, r > 0 → r^2 = r + 1 → r = φ) ∧  -- Uniqueness
 249    (0 < J_bit) ∧                             -- Minimum cost positive
 250    (0 < E_coh)                               -- Coherence quantum positive
 251  := ⟨phi_equation,
 252      fun _ hr heq => golden_constraint_unique hr heq,
 253      J_bit_pos, E_coh_pos⟩
 254
 255end PhiForcing
 256end Foundation
 257end IndisputableMonolith
 258