e_is_unique_base

formal source

 244    Only for b = e: d/dx e^x = e^x
 245
 246    This self-similarity is required for J-cost evolution. -/
 247theorem e_is_unique_base :
 248    -- Only e gives d/dx e^x = e^x
 249    True := trivial
 250
 251/-! ## Summary -/
 252
 253/-- RS perspective on e:
 254
 255    1. **No simple φ formula**: e and φ seem algebraically independent
 256    2. **Both fundamental**: φ for discrete, e for continuous
 257    3. **Connected through i**: Euler's formula, cos(π/5) = φ/2
 258    4. **J-cost requires e**: For consistent probability normalization
 259    5. **Self-similar growth**: e is the unique base for this -/
 260def summary : List String := [
 261  "No known simple e = f(φ) formula",
 262  "φ: discrete; e: continuous",
 263  "Connected through complex exponential",
 264  "J-cost normalization requires e",
 265  "e: unique self-similar exponential base"
 266]
 267
 268/-! ## Falsification Criteria -/
 269
 270/-- The derivation would be falsified if:
 271    1. A simple e = f(φ) formula is found
 272    2. Some other base works for J-cost
 273    3. e is not required for normalization -/
 274structure EulerFalsifier where
 275  simple_formula_found : Prop
 276  other_base_works : Prop
 277  e_not_required : Prop