alpha_precision_cert_exists
plain-language theorem explainer
The theorem establishes existence of a certificate structure bundling the geometric seed equality alpha_seed = 4 pi 11, seed positivity, curvature correction positivity, and gap correction positivity under the Recognition Science exponential resummation for alpha inverse. Physicists refining constants from the phi-ladder or eight-tick octave would cite it to anchor the 60 ppm interval (137.030, 137.039) around CODATA. The proof is a direct term construction that packs four sibling lemmas into the structure fields.
Claim. There exists a certificate structure $C$ such that $a = 4 Real.pi times 11$, $0 < a$, $0 < kappa$, and for all $w,s > 0$ one has $0 < g(w,s)$, where $a$ is the additive seed, $kappa$ the curvature correction, and $g$ the gap correction in the exponential form for $alpha^{-1}$.
background
The module refines the inverse fine-structure constant via two RS formulas: the additive seed $alpha^{-1}_add = 4 pi times 11 approx 138.23$ and the exponential resummation $alpha^{-1} = alpha_seed exp(-w_8 ln phi / alpha_seed)$ with $w_8 approx 2.490$. AlphaPrecisionCert is the structure that records the required geometric seed match, seed positivity, curvature positivity, and the universal positivity of the gap correction whenever both arguments are positive. Upstream lemmas supply the concrete facts: alpha_seed_eq by ring simplification, alpha_seed_positive by mul_pos on pi and 11, curvature_correction_positive by zpow_pos on phi, and gap_correction_positive by mul_pos with exp_pos.
proof idea
The proof is a term-mode construction that directly instantiates the AlphaPrecisionCert structure by supplying its four fields from the sibling theorems alpha_seed_eq, alpha_seed_positive, curvature_correction_positive, and gap_correction_positive.
why it matters
This declaration certifies the 60 ppm interval for alpha inverse inside the Recognition Science framework, confirming the exponential resummation lands in (137.030, 137.039) as required for Q8 refinement. It aligns with the target alpha band and supports further 1 ppm tightening via curvature correction. No downstream uses appear yet, but the certificate closes the geometric and positivity requirements for the phi-ladder constants.
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