alpha_seed_eq
The equality establishes the geometric seed for the additive inverse fine-structure constant as exactly 4π times 11. Researchers refining α^{-1} precision bounds in Recognition Science cite this baseline when assembling certified intervals. The proof is a one-line wrapper that unfolds the definition and applies the ring tactic to confirm the arithmetic identity.
claimThe geometric seed satisfies $α_{seed} = 4π × 11$.
background
The Alpha-Inverse Precision Refinement module (Q8) defines the additive formula for the inverse fine-structure constant as α^{-1}_add = 4π × 11 ≈ 138.23. This seed represents the baseline spherical closure cost over 11-edge interaction paths from the ledger structure. The module also states the exponential resummation α^{-1} = α_seed × exp(−w_8 ln φ / α_seed), with the proved interval (137.030, 137.039) against the CODATA 2022 value 137.035999177(21).
proof idea
The proof is a one-line wrapper. It unfolds the definition of alpha_seed (set to 44 * Real.pi in the module) and applies the ring tactic to verify the identity 44π = 4π × 11.
why it matters in Recognition Science
This theorem supplies the seed_from_geometry field in the AlphaPrecisionCert record constructed by alpha_precision_cert_exists. It anchors the additive seed step in the Q8 refinement of α^{-1} within Recognition Science, where constants derive from the phi-ladder and forcing chain (T0-T8). The result enables the current 60 ppm interval before curvature corrections δ_κ are applied toward the 1 ppm target.
scope and limits
- Does not incorporate curvature or gap corrections.
- Does not compute the exponential resummation for α^{-1}.
- Does not establish numerical bounds on the final α^{-1} interval.
- Does not reference phi, w_8, or higher-order terms.
Lean usage
example : alpha_seed = 4 * Real.pi * 11 := alpha_seed_eqformal statement (Lean)
31theorem alpha_seed_eq : alpha_seed = 4 * Real.pi * 11 := by
proof body
One-line wrapper that applies unfold.
32 unfold alpha_seed; ring
33