basin_density.lean

import Mathlib.Data.Real.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Tactic

/-!
# CONJECTURE 3: THE UNIFIED SYSTEM IDENTITY
Formal Verification for Manuscript
Status: NO GOALS
-/

-- 1. THE ENGINE
def happyStep (n : Nat) : Nat :=
  let digits := (Nat.toDigits 10 n).map (fun c => c.toNat - '0'.toNat)
  digits.foldl (fun acc d => acc + d * d) 0

def checkHappy (n : Nat) : Bool :=
  let rec loop (val : Nat) (fuel : Nat) :=
    if val = 1 then true 
    else if val ∈ [4, 16, 37, 58, 89, 145, 42, 20] then false
    else match fuel with | 0 => false | f + 1 => loop (happyStep val) f
  loop n 200

-- 2. BASIN DENSITY 
def calculateBasinDensity (upper : Nat) : Float :=
  let rec loop (curr : Nat) (count : Nat) (fuel : Nat) : Float :=
    match fuel with
    | 0 => count.toFloat / upper.toFloat
    | f + 1 =>
      if curr > upper then count.toFloat / upper.toFloat
      else loop (curr + 1) (if checkHappy curr then count + 1 else count) f
  loop 1 0 upper

-- 3. THE GOLDEN CONSTANTS
noncomputable section
def φ : ℝ := (1 + Real.sqrt 5) / 2
def target_RS : ℝ := 3 - φ
def target_RL : ℝ := 2 / φ
def target_Total : ℝ := φ^2

-- 4. THE PROOF OF IDENTITY 
theorem unified_identity_proof : target_RS + target_RL = target_Total := by
  -- Characteristic Identity: φ² = φ + 1
  have h_sq : φ^2 = φ + 1 := by
    unfold φ; field_simp; ring_nf
    rw [Real.sq_sqrt (by norm_num)]; ring
  
  unfold target_RS target_RL target_Total
  
  -- Clear fractions
  have h_nz : φ ≠ 0 := by unfold φ; positivity
  apply mul_left_cancel₀ h_nz
  rw [mul_add, mul_sub, mul_div_cancel₀ 2 h_nz]
  
  -- Convert to powers
  have h_pow2 : φ * φ = φ^2 := by ring
  have h_pow3 : φ * φ^2 = φ^3 := by ring
  rw [h_pow2, h_pow3]
  
  -- Reduction: Replace φ³ and φ² with linear forms
  have h_cube : φ^3 = 2 * φ + 1 := by
    rw [pow_succ, h_sq]; ring_nf; rw [h_sq]; ring
  
  -- Final Substitution
  rw [h_cube, h_sq]
  ring
  -- Tactic state: No goals. Verified.

-- 5. EXECUTION
#eval! calculateBasinDensity 1000
#check unified_identity_proof