pi_transcendence

formal source

 273
 274    In RS terms: π emerges from the INFINITE limit of 8-tick geometry.
 275    The discreteness (algebraic) gives way to continuity (transcendental). -/
 276theorem pi_transcendence :
 277    -- π is irrational (Lindemann 1882 proved it is actually transcendental,
 278    -- but irrationality was shown earlier by Lambert in 1761)
 279    -- Mathlib proves irrationality via the Niven polynomial argument.
 280    Irrational Real.pi := irrational_pi
 281
 282/-! ## RS Perspective -/
 283
 284/-- RS perspective on π:
 285
 286    1. **8-tick discreteness**: Finite approximation
 287    2. **Continuum limit**: π emerges as n → ∞
 288    3. **φ connections**: cos(π/5) = φ/2, etc.
 289    4. **Transcendence**: From discrete → continuous
 290
 291    π is the "bridge" between discrete (ledger) and continuous (geometry). -/
 292def rsPerspective : List String := [
 293  "8-tick gives discrete approximation",
 294  "π emerges in continuum limit",
 295  "Connected to φ via pentagon",
 296  "Transcendence from discreteness limit"
 297]
 298
 299/-! ## Falsification Criteria -/
 300
 301/-- The derivation would be falsified if:
 302    1. π has no 8-tick connection
 303    2. φ-π relationships don't hold
 304    3. 8-tick doesn't converge to circle -/
 305structure PiFalsifier where
 306  no_8tick_connection : Prop