A Conditional Proof of the Cîmpeanu Conjecture 1 under the Riemann Hypothesis

Optimized Explicit Bounds for Prime Gap Exponents $α_n = ln g_n / ln p_n$ under the Riemann Hypothesis (RH). Under RH, this paper derives optimal explicit upper bounds for prime gap exponents $α_n$ on finite intervals [2657, B]. Key results: - Global bound: $α_n ≤ 0.514637 ∀ p_n ≥ 2657$ (Schoenfeld's constant $8π$ optimal vs. Carneiro et al. 1/2.63)- Practical bounds: $α_n ≤ 0.50343$ on $[2657, 10^12]$; $α_n ≤ 0.51207$ on $[2657, 10^18]$ - Finite exception set: ${p_n : α_n > 0.5}$ = ${3,7,13,23,31,113}$ exactly (exhaustive verification $pn < 2657$)- Proves Cîmpeanu Conjecture 1 (CC1) under RH Contains:- Complete PDF paper with theorems, proofs, tables- Reproducible Python verification code (sieve + α_n computation) - High-precision H(B) computations for $B = 10^k$, $k=3..30$ Improves: Previous version (https://doi.org/10.5281/zenodo.18491499) + explicit practical bounds beyond asymptotic 0.5146. Keywords: prime gaps, Riemann Hypothesis, Schoenfeld bound, explicit estimates, number theory