The Prime Number Theorem as an Exact Poisson--Laplace Identity

We prove a windowed prime number theorem on fixed logarithmic windows by combining a Fej\'er low-pass with Yukawa decay. The method is $\Xi$-free and uses only de la Vall\'ee Poussin's explicit error on $\Re s = 1$, yielding a strictly positive in-band floor that reduces to finitely many prime contributions plus explicit tails. For any fixed width $\Delta>0$ and band size $\Gamma\ge 1$, the Fej\'er--Yukawa kernel delivers an explicit short-interval lower bound controlled by $\beta_{\Gamma}(\lambda)$. A reproducibility ledger records outward-rounded constants, and a machine-checkable certificate verifies the Chebyshev floor $\theta(x)\ge (1-\varepsilon)x$ beyond a fully explicit threshold. The argument is self-contained and requires no zero hypotheses; all estimates are quantitative within short intervals and supply a ready-to-use package that auditors can tighten without changing the proof skeleton.

This work was conducted at the GhostDrift Mathematical Institute (https://www.ghostdriftresearch.com).