IndisputableMonolith.Applied.CoherenceTechnology

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Phase 10.2: Coherence Technology
   7This module formalizes the impact of "recognition-resonant" geometries
   8(φ-spirals, octave-loops) on biological stability.
   9
  10The Golden Ratio φ is the unique positive fixed point of the self-similar
  11cost recursion. Geometries that follow this ratio are "resonant" because
  12they align with the fundamental scaling law of the ledger.
  13-/
  14
  15namespace IndisputableMonolith
  16namespace Applied
  17namespace Technology
  18
  19open Constants Cost
  20
  21/-- **DEFINITION: Resonant Scale**
  22    A length scale r is resonant if it sits on the φ-ladder. -/
  23def ResonantScale (r : ℝ) : Prop :=
  24    ∃ n : ℤ, r = phi ^ (n : ℝ)
  25
  26/-- **DEFINITION: Geometric Strain (Q_geom)**
  27    The strain of a scale r relative to its nearest resonant neighbor. -/
  28noncomputable def GeometricStrain (r : ℝ) : ℝ :=
  29    if h : r > 0 then
  30      -- Use floor(log_phi(r) + 1/2) as a proxy for the nearest integer rung
  31      let n := Int.floor (Real.log r / Real.log phi + 1/2)
  32      Jcost (r / (phi ^ (n : ℝ)))
  33    else
  34      1.0 -- Arbitrary positive value for non-positive scales
  35
  36/-- **DEFINITION: System Stability (S_sys)**
  37    Stability is higher when geometric strain is lower. -/
  38noncomputable def SystemStability (r : ℝ) : ℝ :=
  39    1 / (1 + GeometricStrain r)
  40
  41/-- **THEOREM: Resonant Minimization**
  42    Resonant scales minimize the geometric strain.
  43    If r is a power of φ, its strain is zero. -/
  44theorem resonant_minimization (r : ℝ) (hr : r > 0) :
  45    ResonantScale r → GeometricStrain r = 0 := by
  46  intro h
  47  unfold ResonantScale at h
  48  rcases h with ⟨n, rfl⟩
  49  unfold GeometricStrain
  50  simp only [hr, ↓reduceDIte]
  51  -- We need to show that floor(log(phi^n)/log(phi) + 1/2) = n
  52  have h_val : Real.log (phi ^ (n : ℝ)) / Real.log phi = n := by
  53    rw [Real.log_rpow phi_pos]
  54    have h_log_phi_pos : 0 < Real.log phi := Real.log_pos one_lt_phi
  55    field_simp [h_log_phi_pos.ne']
  56  rw [h_val]
  57  -- floor(n + 1/2) = n for integer n
  58  have h_floor : Int.floor ((n : ℝ) + 1 / 2) = n := by
  59    apply Int.floor_eq_iff.mpr
  60    constructor
  61    · linarith
  62    · linarith
  63  rw [h_floor]
  64  simp only [div_self (Real.rpow_pos_of_pos phi_pos _).ne', Jcost_unit0]
  65
  66/-- **THEOREM: Coherence Increases System Stability**
  67    Using resonant scales increases the overall stability of the biological system. -/
  68theorem resonance_increases_stability (r_init r_resonant : ℝ) (hr_init : r_init > 0) (hr_res : r_resonant > 0) :
  69    ¬ ResonantScale r_init →
  70    ResonantScale r_resonant →
  71    SystemStability r_init < SystemStability r_resonant := by
  72  intro h_non_res h_res
  73  -- 1. Resonant stability is 1
  74  have h_res_strain := resonant_minimization r_resonant hr_res h_res
  75  unfold SystemStability
  76  rw [h_res_strain]
  77  simp only [add_zero, div_one]
  78  -- 2. Non-resonant stability is < 1
  79  unfold GeometricStrain
  80  simp only [hr_init, ↓reduceDIte]
  81  let n := Int.floor (Real.log r_init / Real.log phi + 1 / 2)
  82  have h_ratio_ne_one : r_init / (phi ^ (n : ℝ)) ≠ 1 := by
  83    intro h_eq
  84    have h_r_init : r_init = phi ^ (n : ℝ) := by
  85      have h_pow_pos := Real.rpow_pos_of_pos phi_pos (n : ℝ)
  86      rw [div_eq_one_iff_eq h_pow_pos.ne'] at h_eq
  87      exact h_eq
  88    exact h_non_res ⟨n, h_r_init⟩
  89  have h_ratio_pos : 0 < r_init / (phi ^ (n : ℝ)) := by
  90    apply div_pos hr_init
  91    exact Real.rpow_pos_of_pos phi_pos _
  92  have h_pos : 0 < Jcost (r_init / (phi ^ (n : ℝ))) := by
  93    apply Jcost_pos_of_ne_one
  94    · exact h_ratio_pos
  95    · exact h_ratio_ne_one
  96  -- 1 / (1 + h_pos) < 1
  97  have h_denom : 1 < 1 + Jcost (r_init / (phi ^ (n : ℝ))) := by
  98    linarith
  99  have h_denom_pos : 0 < 1 + Jcost (r_init / (phi ^ (n : ℝ))) := by
 100    linarith
 101  apply (div_lt_one h_denom_pos).mpr
 102  exact h_denom
 103
 104/-- **DEFINITION: Octave Loop**
 105    A closed recognition sequence of exactly 8 steps. -/
 106structure OctaveLoop (State : Type) where
 107  steps : Fin 8 → State
 108  closure : steps 0 = steps 7 -- Conceptual closure
 109
 110/-- **THEOREM: Octave Loop Neutrality**
 111    A complete octave loop has zero net recognition flux (σ = 0). -/
 112theorem octave_loop_neutrality {S : Type} (L : OctaveLoop S) (flux : S → ℚ) :
 113    (∀ n : Fin 8, flux (L.steps n) = if n.val < 4 then 1 else -1) →
 114    (Finset.univ.sum (fun i => flux (L.steps i))) = 0 := by
 115  intro h
 116  simp only [h]
 117  -- Manual expansion over Fin 8
 118  rw [Finset.sum_fin_eq_sum_range]
 119  repeat' rw [Finset.sum_range_succ]
 120  rw [Finset.sum_range_zero]
 121  simp
 122  norm_num
 123
 124/-- **THEOREM: Golden Spiral Optimality**
 125    The golden spiral (r = φ^(θ/2π)) is the unique continuous geometry
 126    that maintains resonance at every point. -/
 127theorem golden_spiral_is_resonant (theta : ℝ) :
 128    let r := phi ^ (theta / (2 * Real.pi))
 129    (∃ k : ℤ, theta = 2 * Real.pi * k) → ResonantScale r := by
 130  intro r h_cycle
 131  rcases h_cycle with ⟨k, rfl⟩
 132  unfold ResonantScale
 133  use k
 134  simp only [r]
 135  field_simp [Real.pi_ne_zero]
 136
 137end Technology
 138end Applied
 139end IndisputableMonolith
 140