209 rw [deriv_alphaInv_of_gap] 210 rw [alphaInv_linear_term] 211 field_simp 212 213/-! ## Part 5: The Uniqueness Question (Open) 214 215A full forcing argument would prove that the exponential form is the 216UNIQUE form satisfying certain structural constraints. The simplest 217candidate uniqueness statement: 218 219Given a function g : ℝ → ℝ such that: 2201. g is smooth (C^∞) 2212. g(0) = α_seed and g'(0) = -1 (so leading-order behavior matches 222 α_seed - f_gap) 2233. The logarithmic derivative (log g)'(x) is CONSTANT (equal to -1/α_seed) 224 225Then g(x) = α_seed · exp(-x/α_seed). 226 227Condition (3) is the distinctive feature: it says the relative rate of 228change of g is scale-free (same at all x). This IS a forcing property 229(standard ODE uniqueness), but it is also a STRUCTURAL ASSUMPTION that 230needs physical justification in the RS context. 231 232Alternative formulas like α_seed / (1 + x/α_seed) have non-constant log 233derivative ((d/dx) log(α_seed/(1+x/α_seed)) = -1/(α_seed + x), which 234depends on x), so they don't satisfy (3). 235 236Whether RS structure FORCES the log-derivative to be constant is the 237genuine open question. 238-/ 239 240/-- **Open question as a Prop**: the exponential form is uniquely 241 determined by constant logarithmic derivative. -/
used by (1)
From the project-wide theorem graph. These declarations reference this one in their body.