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J_bit
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IndisputableMonolith.Foundation.PhiForcing on GitHub at line 220.
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217/-! ## Consequences of φ -/
218
219/-- The minimum non-trivial cost: J_bit = ln(φ). -/
220noncomputable def J_bit : ℝ := Real.log φ
221
222/-- J_bit is positive. -/
223theorem J_bit_pos : 0 < J_bit := Real.log_pos phi_gt_one
224
225/-- The coherence quantum: E_coh = φ⁻⁵. -/
226noncomputable def E_coh : ℝ := φ^(-5 : ℤ)
227
228/-- E_coh is positive. -/
229theorem E_coh_pos : 0 < E_coh := by
230 simp only [E_coh]
231 exact zpow_pos phi_pos (-5)
232
233/-! ## Summary: The Forcing Chain -/
234
235/-- **PHI FORCING PRINCIPLE**
236
237The golden ratio φ is forced by self-similarity in a discrete J-cost ledger:
238
2391. Discrete ledger with J-symmetry (from previous levels)
2402. Self-similarity: the structure references itself at different scales
2413. Compositional constraint: r² = r + 1 (next scale = current + base)
2424. Unique positive solution: r = φ = (1 + √5)/2
243
244This is Level 4 of the forcing chain:
245Composition law → J unique → Discreteness → Ledger → **φ** → D=3 → physics -/
246theorem phi_forcing_principle :
247 (φ^2 = φ + 1) ∧ -- Golden equation
248 (∀ r : ℝ, r > 0 → r^2 = r + 1 → r = φ) ∧ -- Uniqueness
249 (0 < J_bit) ∧ -- Minimum cost positive
250 (0 < E_coh) -- Coherence quantum positive