Alpay Folded Prime Enumerator via gcd and Floors: Exact Enumeration, Record-Lift, and Non-Synonymy/Minimality Certificates

Abstract

A single closed expression $f_{\text{Alpay},U}(x)$ is presented which, for every integer $x\ge 0$, returns the $(x+1)$-th prime $p_{x+1}$. The construction uses only integer arithmetic, greatest common divisors, and floor functions. A prime indicator $I(j)$ is encoded through a short $\gcd$-sum; a cumulative counter $S(i)=\sum_{j\le i}I(j)$ equals $\pi(i)$; and a folded step $A(i,x)$ counts precisely up to the next prime index without piecewise branching. A corollary shows that for any fixed integer $L\ge 2$, the integer $P^\star=f_{\text{Alpay},U}(L)$ is prime and $P^\star>L$. Two explicit schedules $U(x)$ are given: a square schedule $U_{\text{sq}}(x)=(x+1)^2$ and a near-linear schedule $U_{\text{Alpay-lin}}(x)=\Theta(x\log x)$ justified by explicit bounds on $p_n$. We include non-synonymy certificates relative to Willans-type enumerators (schedule and operator-signature separation) and prove an asymptotic minimality bound: any forward-count enumerator requires $U(x)=\Omega(x\log x)$ while $U_{\text{Alpay-lin}}$ achieves $O(x\log x)$. We also provide explicit operation counts (''form complexity'') of the folded expression.