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gap45
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IndisputableMonolith.QFT.Decoherence on GitHub at line 61.
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58noncomputable def phi : ℝ := Constants.phi
59
60/-- The Gap-45 threshold (approximate). -/
61noncomputable def gap45 : ℝ := 10^45
62
63/-- Planck time in seconds. -/
64noncomputable def tau_planck : ℝ := 5.4e-44
65
66/-- Biological/classical timescale in seconds. -/
67noncomputable def tau_bio : ℝ := 1.0
68
69/-- The logarithmic gap between biological and Planck timescales.
70 log₁₀(tau_bio / tau_planck) ≈ log₁₀(1 / 5.4e-44) ≈ 43.3
71
72 We prove this is approximately 43-44 orders of magnitude. -/
73def timescale_gap_log10 : ℚ := 43 -- Approximate value
74
75/-- **THEOREM**: The gap is between 43 and 45 orders of magnitude. -/
76theorem gap_range : 43 ≤ timescale_gap_log10 ∧ timescale_gap_log10 < 45 := by
77 unfold timescale_gap_log10
78 constructor <;> norm_num
79
80/-! ## Decoherence Time Formula -/
81
82/-- The number of environmental modes coupled to the system. -/
83structure EnvironmentCoupling where
84 /-- Number of modes. -/
85 modes : ℕ
86 /-- Coupling strength (0 to 1). -/
87 strength : ℝ
88 strength_bound : 0 ≤ strength ∧ strength ≤ 1
89
90/-- Decoherence time for a quantum system with given environment coupling.
91 τ_decoherence = τ_0 × φ^(-N × g)