We present a self-contained combinatorial and sieve-theoretic framework forestimating and bounding the number of primes of the form $x^2+1$ in a naturalfamily of intervals. The centrepiece is the emph{Coprime Identity}: for anyinteger $m$ with $n < m le n^2$, $m$ is prime if and only if $m$ is coprimeto the primorial $P_n = prod_{p le n} p$. Applying this identity to thepolynomial $f(x) = x^2+1$ and encoding the root structure of $f(x)bmod p$via quadratic residue theory yields the emph{Quadratic Totient Estimate}[ mathcal{N}(n) ;=; frac{n}{2} prod_{substack{ple n pequiv 1pmod{4}}} !frac{p-2}{p},]which is strictly increasing and diverges to infinity. We then introduce theemph{Quadratic Jacobsthal function} $gQ(n)$, the maximum gap betweenconsecutive survivors of the quadratic sieve for $f$, and give a sieve-theoreticargument --- based on the computation of the sieve dimension $kappa = 1$ andthe linear sieve lower bound --- that $gQ(n) < n^2 - n$ for all sufficientlylarge $n$. Large-scale numerical verification confirms the formula's accuracy:against exact prime counts from OEIS~A002496, the relative error of$mathcal{N}(n)$ remains below $8%$ for $n$ up to $10^{13}$, with systematicimprovement toward zero as a tail-factor correction is applied. We discuss theconnection to the Hardy--Littlewood Conjecture~F, the Bateman--Horn singularseries, and Iwaniec's 1978 $P_2$ theorem, and we carefully delineate theremaining obstacles --- the parity barrier and the rigorous control of theerror term --- that separate this heuristic from a complete proof of Landau'sFourth Problem.