AlphaPrecisionCert
AlphaPrecisionCert packages the geometric seed equality to 4π×11, seed positivity, curvature correction positivity, and gap correction positivity into a single certificate. A physicist deriving the fine-structure constant in Recognition Science would cite it to support the 60 ppm interval claim for α⁻¹. The structure is assembled by a constructor that invokes the four supporting lemmas without further reduction.
claimThe structure certifies that the geometric seed satisfies $α_{seed} = 4π × 11$, that $0 < α_{seed}$, that the curvature correction $δ_κ = φ^{-5}$ is positive, and that the gap correction $δ_g(w,s) = s exp(-w ln φ / s)$ is positive whenever $w > 0$ and $s > 0$.
background
In the Alpha-Inverse Precision Refinement module the inverse fine-structure constant is built from an additive geometric seed and an exponential resummation. The seed is defined as $4π × 11$, representing the baseline spherical closure cost over 11-edge interaction paths. The curvature correction is $φ^{-5}$ and the gap correction is the function $s exp(-w ln φ / s)$ for weight $w$ and seed $s$ (see upstream alpha_seed in Constants.Alpha and the local definitions of curvature_correction and gap_correction).
proof idea
This declaration is a pure structure definition that packages four field assertions. No tactics or lemmas are applied inside the structure itself; the fields are populated downstream by the theorem that constructs a witness from the seed-equality, seed-positivity, curvature-positivity, and gap-positivity lemmas.
why it matters in Recognition Science
AlphaPrecisionCert supplies the certificate required to establish existence of a structure witnessing the precision bounds on α⁻¹. It fills the Q8 refinement step in the Recognition Science derivation of constants, where the alpha band (137.030, 137.039) is obtained from the geometric seed 4π×11 together with curvature and gap corrections. The construction relies on the phi-ladder from the forcing chain, though higher-order terms remain open for 1 ppm accuracy.
scope and limits
- Does not derive the exponential resummation formula for α⁻¹
- Does not compute numerical bounds on α⁻¹
- Does not include the full mass ladder or Berry creation threshold
- Does not address the D=3 spatial dimension forcing
formal statement (Lean)
60structure AlphaPrecisionCert where
61 seed_from_geometry : alpha_seed = 4 * Real.pi * 11
62 seed_positive : 0 < alpha_seed
63 curvature_positive : 0 < curvature_correction
64 gap_positive : ∀ w seed, 0 < w → 0 < seed → 0 < gap_correction w seed
65