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theorem proved term proof

log_alphaInv_eq

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formal statement (Lean)

 102theorem log_alphaInv_eq :
 103    Real.log alphaInv = Real.log alpha_seed - f_gap / alpha_seed := by

proof body

Term-mode proof.

 104  have h := log_alphaInv_seed_ratio
 105  rw [Real.log_div (ne_of_gt alphaInv_positive) (ne_of_gt alpha_seed_positive)] at h
 106  linarith
 107
 108/-! ## Part 3: The Differential Equation
 109
 110The exponential form α⁻¹ = α_seed · exp(-f_gap/α_seed) satisfies the ODE
 111(treating α⁻¹ as a function of f_gap with α_seed fixed):
 112
 113    d(α⁻¹)/d(f_gap) = -α⁻¹/α_seed
 114
 115This is the defining characteristic of the exponential family: the
 116logarithmic derivative is constant.
 117
 118This ODE is analogous to the renormalization-group equation for a running
 119coupling, with α_seed playing the role of a "scale" setting the logarithmic
 120derivative.
 121-/
 122
 123/-- The alphaInv function parameterized by f_gap value. -/

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