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 y\), and gives a positive lower bound for\ prime, p-2 P ₂, ^- (p-2) >p^-o (1) a certified theorem-level exponent\=0. 2765. , if the counted shift is composite, then-2=qr, ^0. 2765-o (1) <q r<p^0. 7235+o (1). 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 \ (=/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.