log_alphaInv_eq
The theorem establishes the logarithmic identity ln(α⁻¹) = ln(α_seed) − f_gap/α_seed for the inverse fine-structure constant. Researchers deriving α from Recognition Science structures cite it to confirm consistency of the exponential representation. The proof reduces the claim to a prior ratio identity by applying the logarithm division rule and linear arithmetic.
claim$ln(α^{-1}) = ln(α_{seed}) - f_{gap}/α_{seed}$
background
The AlphaExponentialForm module examines structural properties of the exponential representation α⁻¹ = α_seed ⋅ exp(−f_gap/α_seed). Here α_seed = 4π⋅11 is the geometric seed from ledger structure, and f_gap encodes the gap weight w₈ ln(φ) on the phi-ladder. The module proves positivity of α⁻¹ and identifies the ODE d(α⁻¹)/d(f_gap) = −α⁻¹/α_seed satisfied by the form, analogous to renormalization-group equations for a running coupling with α_seed as scale.
proof idea
The proof invokes the upstream result log_alphaInv_seed_ratio, which states that the log of the ratio alphaInv/alpha_seed equals −f_gap/alpha_seed. It rewrites this using Real.log_div, justified by the positivity of alphaInv and alpha_seed, then applies linear arithmetic to obtain the difference of logs.
why it matters in Recognition Science
This identity feeds the downstream theorem exponential_form_uniqueness_ode_principle, which notes that constant logarithmic derivative characterizes the exponential family via standard ODE theory. It addresses Gap B in the validation program for the canonical alpha formula, as described in the module documentation. The form is motivated by J-cost log-structure, though the physical reason for constant log derivative in RS remains open.
scope and limits
- Does not prove uniqueness of the exponential form among alternative ansatze.
- Does not derive the exponential representation from first-principles J-cost or forcing chain.
- Does not formalize renormalization-group connections beyond the ODE analogy.
- Does not supply numerical error bounds or higher-order corrections.
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. -/