A Rough Chen-type Theorem for Shifted Primes

We prove a Chen-type theorem for primes \(p\) for which the shift \(p-2\) is notonly a prime or semiprime, but is also explicitly rough. The theorem is stated in asmoothed dyadic form at heights \(p\asymp y\), and gives a positive lower bound for\[p\ \mathrm{prime},\qquad p-2\in P_{\le 2},\qquadP^{-}(p-2)>p^{\alpha-o(1)}\]with a certified theorem-level exponent\[\alpha=0.2765.\]Equivalently, if the counted shift is composite, then\[p-2=qr,\qquadp^{0.2765-o(1)}<q\le r<p^{0.7235+o(1)}.\]This rough/smooth sandwich is the main theorem-level feature of the paper. The proof combines a Rosser--Iwaniec lower linear sieve at theBombieri--Friedlander--Iwaniec well-factorable level \(4/7\) with an explicit subtractionof the exact-three-prime sector. The exact-three sector is controlled by a localizedMotohashi--Bombieri--Vinogradov convolution estimate, leading to a one-dimensionalcoefficient inequality in the roughness exponent \(\alpha=\beta/u\). A Sage/RIFcertificate verifies the displayed unconditional exponent. The improvement from the older \(0.2735\) row is supplied by Pascadi's\(3/5\)-level theorem for well-factorable upper linear-sieve weights, applied to theordered ternary branch in the exact-three subtraction after replacing the classical upperweights by Iwaniec's well-factorable upper linear-sieve variant. We also record theolder \(0.2735\) BFI--Motohashi proof path as a simpler corollary. A retained-cell\(5/8\)-level framework is discussed separately as a future conditional refinement and isnot used in the theorem-level results.