alpha_positive
The theorem establishes positivity of the fine-structure constant alpha derived from the geometric seed. Unification researchers cite it to confirm consistency of the alpha band before applying bounds or inflation certificates. The proof is a term-mode one-liner that unfolds the three definitions and applies the positivity tactic.
claimThe fine-structure constant satisfies $0 < alpha$.
background
The ConstantsPredictionsProved module supplies calculated proofs for registry items including C-001 for the fine-structure constant. Alpha is defined as 1 over alphaInv where alphaInv equals alpha_seed times the exponential of minus f_gap over alpha_seed, and alpha_seed equals 4 pi times 11 from the geometric seed in Constants.Alpha. The module states that 0 < alpha < 0.1 is proved for this item.
proof idea
The term proof unfolds the definitions of alpha, alphaInv, and alpha_seed then applies the positivity tactic to conclude the result from the positive components of the expression.
why it matters in Recognition Science
This result is used in alpha_bounds to establish 0 < alpha < 0.1, in constants_predictions_cert_exists, and inside the InflationCert structure for inflation_cert in Gravity.Inflation. It fills the C-001 entry in the COMPLETE_PROBLEM_REGISTRY and supports the alpha inverse band near 137 in the Recognition Science framework.
scope and limits
- Does not derive the exact numerical value of alpha.
- Does not prove the upper bound alpha less than 0.1.
- Assumes positivity of seed and exponential terms.
- Does not address higher-order corrections.
formal statement (Lean)
34theorem alpha_positive : alpha > 0 := by
proof body
Term-mode proof.
35 unfold alpha alphaInv alpha_seed
36 positivity
37
38/-- **CALCULATED**: α < 0.1 (approximately 1/137 < 0.1). -/