phi_irrational_information

formal source

 247/-- **THEOREM IC-001.19**: φ (the golden ratio) is irrational.
 248    This means φ-based information cannot be exactly represented with rational arithmetic.
 249    In RS: the fundamental ledger constant φ encodes "transcendent" information. -/
 250theorem phi_irrational_information : Irrational phi :=
 251  phi_irrational
 252
 253/-- **THEOREM IC-001.20**: φ satisfies x² = x + 1 (the ledger self-similarity equation).
 254    This means φ is the unique positive real encoding the information that
 255    "the next level contains the previous two" — the Fibonacci property. -/
 256theorem phi_self_similar : phi ^ 2 = phi + 1 :=
 257  phi_sq_eq
 258
 259/-- **THEOREM IC-001.21**: The J-cost of φ is positive (φ ≠ 1).
 260    The golden ratio represents non-trivial information in the ledger. -/
 261theorem phi_has_positive_info_cost : Jcost phi > 0 := by
 262  rw [Jcost_eq_sq phi_pos.ne']
 263  apply div_pos
 264  · exact pow_pos (by linarith [one_lt_phi]) 2
 265  · linarith [phi_pos]
 266
 267/-! ## Summary Certificate -/
 268
 269/-- IC-001 Status Certificate: Information IS the Ledger — DERIVED -/
 270def ic001_certificate : String :=
 271  "═══════════════════════════════════════════════════════════\n" ++
 272  "  IC-001: INFORMATION IS THE LEDGER — STATUS: DERIVED\n" ++
 273  "═══════════════════════════════════════════════════════════\n" ++
 274  "✓ info_cost_nonneg:           J(x) ≥ 0 for all x > 0\n" ++
 275  "✓ info_cost_zero_iff_unit:    J(x) = 0 ↔ x = 1\n" ++
 276  "✓ info_cost_pos_of_ne_one:    J(x) > 0 for x ≠ 1\n" ++
 277  "✓ info_cost_symmetric:        J(x) = J(1/x)\n" ++
 278  "✓ balanced_has_minimum_cost:  J(1) ≤ J(x)\n" ++
 279  "✓ balanced_is_unique_minimum: J(e) = 0 → e.ratio = 1\n" ++
 280  "✓ nothingness_infinite_cost:  ∀ M, ∃ x, J(x) > M\n" ++