IndisputableMonolith.RRF.Hypotheses.PhiLadder

   1import Mathlib.Analysis.SpecialFunctions.Log.Basic
   2import Mathlib.Data.Int.Basic
   3import Mathlib.Data.Real.Basic
   4
   5/-!
   6# RRF Hypotheses: φ-Ladder
   7
   8The φ-ladder hypothesis: physical scales are organized by powers of φ.
   9
  10This is an EXPLICIT HYPOTHESIS, not a definitional truth.
  11It generates prediction obligations that must be tested empirically.
  12
  13## The Hypothesis
  14
  15Privileged physical scales satisfy: X = X₀ · φⁿ for integer n.
  16
  17Examples:
  18- Particle masses: m = B · E_coh · φʳ
  19- Timescales: τ = τ₀ · φⁿ
  20- Semantic amplitudes: A = φ^level
  21
  22## Falsification Criteria
  23
  241. Find a privileged scale that doesn't land near a φ-rung
  252. Find two privileged scales whose ratio is not φ^k for any integer k
  263. Find a derivation that requires non-integer rung shifts
  27-/
  28
  29
  30namespace IndisputableMonolith
  31namespace RRF.Hypotheses
  32
  33/-! ## φ Definition and Basic Properties -/
  34
  35/-- The golden ratio φ = (1 + √5) / 2. -/
  36noncomputable def phi : ℝ := (1 + Real.sqrt 5) / 2
  37
  38/-- φ is positive. -/
  39theorem phi_pos : 0 < phi := by
  40  unfold phi
  41  have h : 0 < Real.sqrt 5 := Real.sqrt_pos.mpr (by norm_num : (0:ℝ) < 5)
  42  linarith
  43
  44/-- φ > 1. (√5 > 1, so (1 + √5)/2 > 1) -/
  45theorem phi_gt_one : 1 < phi := by
  46  unfold phi
  47  -- We need: 1 < (1 + √5) / 2  ⟺  2 < 1 + √5  ⟺  1 < √5
  48  have h5pos : (0 : ℝ) < 5 := by norm_num
  49  have hsqrt5 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (le_of_lt h5pos)
  50  have h1sq : (1 : ℝ) ^ 2 = 1 := by norm_num
  51  -- √5 > 1 because 5 > 1 and sqrt is monotone
  52  have h1 : (1 : ℝ) < Real.sqrt 5 := by
  53    have : Real.sqrt 1 = 1 := Real.sqrt_one
  54    rw [← this]
  55    exact Real.sqrt_lt_sqrt (by norm_num) (by norm_num : (1:ℝ) < 5)
  56  linarith
  57
  58/-- φ² = φ + 1 (the defining property). -/
  59theorem phi_sq : phi ^ 2 = phi + 1 := by
  60  unfold phi
  61  have h5 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 5)
  62  ring_nf
  63  rw [h5]
  64  ring
  65
  66/-! ## The φ-Scaling Action -/
  67
  68/-- Scale a value by φⁿ. -/
  69noncomputable def phiScale (n : ℤ) (x : ℝ) : ℝ := x * phi ^ n
  70
  71/-- φ-scaling is a group action. -/
  72theorem phiScale_add (m n : ℤ) (x : ℝ) :
  73    phiScale m (phiScale n x) = phiScale (m + n) x := by
  74  simp only [phiScale, mul_assoc]
  75  congr 1
  76  rw [zpow_add₀ (ne_of_gt phi_pos), mul_comm]
  77
  78theorem phiScale_zero (x : ℝ) : phiScale 0 x = x := by
  79  simp [phiScale]
  80
  81theorem phiScale_neg (n : ℤ) (x : ℝ) :
  82    phiScale (-n) (phiScale n x) = x := by
  83  rw [phiScale_add, neg_add_cancel, phiScale_zero]
  84
  85/-! ## The φ-Ladder Hypothesis Class -/
  86
  87/-- A value is on the φ-ladder if it equals X₀ · φⁿ for some base X₀ and integer n. -/
  88structure OnPhiLadder (x : ℝ) (base : ℝ) where
  89  rung : ℤ
  90  eq : x = base * phi ^ rung
  91
  92/-- The rung of a value relative to a base (computed via logarithm). -/
  93noncomputable def computeRung (x : ℝ) (base : ℝ) : ℝ :=
  94  Real.log (x / base) / Real.log phi
  95
  96/-- A value is "near" a φ-rung if its computed rung is close to an integer. -/
  97def nearPhiRung (x : ℝ) (base : ℝ) (tolerance : ℝ) : Prop :=
  98  ∃ n : ℤ, |computeRung x base - n| ≤ tolerance
  99
 100/-! ## The φ-Ladder Hypothesis (Explicit) -/
 101
 102/-- The φ-ladder hypothesis for a collection of scales.
 103
 104This is the explicit claim that "privileged scales land on integer rungs."
 105It generates prediction obligations.
 106-/
 107class PhiLadderHypothesis (α : Type*) where
 108  /-- The base scale of the ladder. -/
 109  base : ℝ
 110  /-- Extract the scale value from an element. -/
 111  scale : α → ℝ
 112  /-- All elements are on the ladder (the hypothesis). -/
 113  on_ladder : ∀ x : α, ∃ n : ℤ, scale x = base * phi ^ n
 114
 115/-- Prediction obligation: if two elements are on the ladder, their ratio is φ^k. -/
 116theorem ratio_is_phi_power [PhiLadderHypothesis α] (x y : α) (hy : PhiLadderHypothesis.scale y ≠ 0) :
 117    ∃ k : ℤ, PhiLadderHypothesis.scale x / PhiLadderHypothesis.scale y = phi ^ k := by
 118  obtain ⟨m, hm⟩ := PhiLadderHypothesis.on_ladder x
 119  obtain ⟨n, hn⟩ := PhiLadderHypothesis.on_ladder y
 120  refine ⟨m - n, ?_⟩
 121  simp only [hm, hn]
 122  have hbase : PhiLadderHypothesis.base (α := α) ≠ 0 := by
 123    intro h; rw [h, zero_mul] at hn; exact hy hn
 124  have hphi_ne : phi ≠ 0 := ne_of_gt phi_pos
 125  have hphi_n_ne : phi ^ n ≠ 0 := zpow_ne_zero _ hphi_ne
 126  field_simp [hbase, hphi_n_ne]
 127  rw [zpow_sub₀ hphi_ne]
 128  field_simp [hphi_n_ne]
 129
 130/-! ## Falsification Interface -/
 131
 132/-- A falsification witness: a scale that doesn't land on the ladder. -/
 133structure PhiLadderFalsifier where
 134  /-- The measured scale value. -/
 135  observed : ℝ
 136  /-- The expected base. -/
 137  base : ℝ
 138  /-- Tolerance for "near integer" (e.g., 0.1). -/
 139  tolerance : ℝ
 140  /-- The observed value is NOT near any rung. -/
 141  not_near : ¬ nearPhiRung observed base tolerance
 142
 143/-- If a falsifier exists, the hypothesis is falsified. -/
 144theorem falsifier_falsifies (f : PhiLadderFalsifier) :
 145    ¬ (∃ n : ℤ, f.observed = f.base * phi ^ n ∧
 146       |computeRung f.observed f.base - n| ≤ f.tolerance) := by
 147  intro ⟨n, heq, htol⟩
 148  apply f.not_near
 149  exact ⟨n, htol⟩
 150
 151end RRF.Hypotheses
 152end IndisputableMonolith
 153