Effective multiplicative representations over short intervals modulo primes: Cases $ C = 4 $ and $ C = 6 $

This paper investigates the problem of short-interval multiplicative representations modulo primes. For a given positive integer $ C $, an explicit lower bound $ p_0(C) $ exists such that for every prime $ p \ge p_0(C) $, every element of the reduced residue system modulo $ p $ can be expressed as the product modulo $ p $ of two distinct integers in the interval $ [1, p/C) $. The overall argument proceeds by contradiction combined with strong induction on prime powers. For $ C = 4 $, we construct the inductive base via case analysis and derive the explicit lower bound $ p \ge 199 $ by combining the divisibility properties of arithmetic progressions with a lower bound of the Chebyshev $ \psi $-function. Building on this, we formulate an offset-screening and inductive framework for a specified positive integer $ C $, in which a successful finite screening supplies the inductive base required by the subsequent theoretical argument. We carry out the complete implementation of this framework for $ C = 6 $ and obtain the explicit lower bound $ p\geq601 $.