Fejér-Kernel Prime Indicators

A $C^1$ prime indicator $\mathcal{P}\text{\hspace{0.17em}⁣}:\mathbb{R}\to \mathbb{R}$ is constructed by applying the Fejér identity to the sine-quotient encoder of trial division. For integers $n\ge 2$, $\mathcal{P}(n)=0$ holds exactly for odd primes; $\mathcal{P}(2)>0$. For all non-integers $x>1$ one has $\mathcal{P}(x)>0$. The function is piecewise $C^{\mathrm{\infty }}$ and its second derivative has jumps precisely at the squares $m^2$, with explicit sizes. Replacing the sharp cut-off by a smooth transition yields $C^{\mathrm{\infty }}$ analogues ${\mathcal{P}}_{\tau }$ and ${\mathcal{P}}_{\sigma }$ with integer limits ${\mathcal{P}}_{\tau }(n;\kappa )\to \tau (n)-2$ and ${\mathcal{P}}_{\sigma }(n;\kappa )\to \sigma (n)-n-1$ as $\kappa \to \mathrm{\infty }$, obtained from locally uniform convergence of derivative series. For large $\kappa$, numerical evidence indicates companion zeros near odd primes for ${\mathcal{P}}_{\tau }$ and an asymmetric pair for ${\mathcal{P}}_{\sigma }$. No assertion is made beyond integer input, and no statements are claimed about the prime number theorem or zero distributions of $L$-functions. The appendix includes two illustrative prime-counting sums.