alphaInv_seed_ratio
The theorem establishes that the inverse fine-structure constant divided by its geometric seed equals the exponential of minus the gap weight over the seed. Researchers deriving the fine-structure constant from Recognition Science structures cite this identity when analyzing the exponential resummation form. The proof is a direct algebraic simplification obtained by unfolding the definition of the inverse constant and applying field simplification.
claim$ (1/α) / α_seed = exp(−f_gap / α_seed) $ where $1/α$ is the inverse fine-structure constant, $α_seed = 4π × 11$ is the geometric seed from ledger structure, and $f_gap = w_8 ln φ$ is the gap weight from the DFT-8 projection.
background
The module examines the exponential form of the inverse fine-structure constant. The canonical expression is $α^{-1} = α_seed · exp(−f_gap / α_seed)$, with $α_seed = 4π·11$ representing the baseline spherical closure cost over 11-edge interaction paths and $f_gap = w_8 · ln(φ)$ the gap weight from the DFT-8 projection. This theorem isolates the ratio $α^{-1} / α_seed$ as the pure exponential factor. Upstream, the definition of $α^{-1}$ in Constants.Alpha directly encodes this product form, while related results in AlphaHigherOrder and GapWeight supply the expressions for the seed and gap. The local setting documents that while positivity and the logarithmic ODE are established, the uniqueness of the exponential form from first principles remains open.
proof idea
The proof unfolds the definition of alphaInv, which expands to the product of the seed and the exponential factor, then applies field_simp to cancel the common seed factor and obtain the ratio equal to the exponential.
why it matters in Recognition Science
This result supports the downstream theorem log_alphaInv_seed_ratio, which takes the logarithm to obtain ln(α^{-1} / α_seed) = −f_gap / α_seed. It fills the structural analysis step in the AlphaExponentialForm module, addressing the remaining open step (Gap B) in the validation program for the exponential form of α^{-1}. In the Recognition Science framework it connects to the J-cost log-coordinate structure that motivates the exponential Taylor coefficients, though uniqueness from the Recognition Composition Law is not yet proved.
scope and limits
- Does not establish that the exponential form is the unique solution to the underlying structural equations.
- Does not derive the form from a variational principle or first-principles argument.
- Does not address higher-order corrections beyond the leading exponential.
- Does not connect directly to the phi-ladder mass formula or spatial dimensions.
formal statement (Lean)
80theorem alphaInv_seed_ratio :
81 alphaInv / alpha_seed = Real.exp (-(f_gap / alpha_seed)) := by
proof body
Term-mode proof.
82 unfold alphaInv
83 field_simp
84
85/-! ## Part 2: The Logarithmic Structure
86
87Taking the natural log of α⁻¹/α_seed gives:
88 ln(α⁻¹/α_seed) = -f_gap/α_seed
89
90This is the defining relation of the exponential form in log coordinates.
91It says that the logarithm of the coupling ratio is LINEAR in f_gap with
92slope -1/α_seed.
93-/
94
95/-- The log of the ratio alphaInv/alpha_seed equals -f_gap/alpha_seed. -/