euler_phi_connection

formal source

 212
 213    **Proved**: The real part of e^(iπ/5) equals φ/2, using
 214    the classical identity cos(π/5) = (1 + √5)/4 = φ/2. -/
 215theorem euler_phi_connection :
 216    -- cos(π/5) = φ/2 (the real part of e^(iπ/5))
 217    Real.cos (Real.pi / 5) = phi / 2 := by
 218  rw [Real.cos_pi_div_five]
 219  -- phi / 2 = (1 + sqrt 5) / 2 / 2 = (1 + sqrt 5) / 4
 220  unfold phi
 221  ring
 222
 223/-! ## RS Interpretation -/
 224
 225/-- RS interpretation of e:
 226
 227    1. **J-cost decay**: Probabilities involve e^(-J)
 228    2. **Continuous time**: e^(iωt) for oscillations
 229    3. **Growth rate**: Maximum sustainable rate is e
 230    4. **8-tick phases**: exp(2πik/8) uses e
 231
 232    e is the natural base for ledger dynamics. -/
 233def rsInterpretation : List String := [
 234  "Probabilities: exp(-J) for cost-weighted",
 235  "Time evolution: exp(iωt) for 8-tick phases",
 236  "Growth limit: e maximizes (1+1/n)^n",
 237  "Normalization: Required for consistency"
 238]
 239
 240/-- Why e and not some other base?
 241
 242    Because d/dx b^x = b^x × ln(b)
 243
 244    Only for b = e: d/dx e^x = e^x
 245