Primes as Sums of Consecutive Composites Classification, Random Sampling, and a Counterexample Search

Objective: This paper introduces a constructive framework for representing primes as sums of consecutive composites. We define two representation types (Type 1: pure sum; Type 2: sum plus a window element) and propose Liu's Conjecture that every prime p > 13 admits at least one of them. Methods: Through exhaustive verification for 13 < p ≤ 10^8, we obtain a complete classification of all primes in this range. A random sampling experiment for 10^12 ≤ p ≤ 10^15 tests the prevalence of Type 2 representations with window length k = 2. Additionally, a targeted counterexample search is conducted in the interval 10^8 < p ≤ 10^9 covering over 45 million primes, using a layered scanning strategy. Results: Exhaustive verification for 13 < p ≤ 10^8 yields a complete classification of all 5,761,449 primes > 13. Random sampling shows that over 95% of large primes admit a k = 2 Type 2 representation. The counterexample search finds no prime without a Type 2 representation, with all successful representations requiring only k ≤ 5. From the exhaustive data, we observe that exclusively Type 2 primes satisfy p ≡ 1 or 11 (mod 12), coinciding exactly with the quadratic reciprocity criterion for the prime 3. Limitations: All verifications are restricted to finite numerical ranges: exhaustive verification for 13 < p ≤ 10^8, random sampling for 10^12 ≤ p ≤ 10^15, and counterexample search for 10^8 < p ≤ 10^9; a theoretical proof for all primes remains open. Conclusions: Liu's Conjecture receives strong empirical support across extensive numerical ranges. We further observe that Liu's Conjecture and the Goldbach-type problems exhibit a structural complementarity: Goldbach's conjecture asserts that composite numbers (at least the even ones) can be additively generated from primes (as sums of two primes), while Liu's Conjecture asserts that every prime p > 13 can be additively generated from composite numbers (as sums of consecutive composite windows). Together, they form a potential "additive closure" of the positive integers: if both conjectures are true, then every prime p > 13 can be built from composites via the window construction, and every even integer greater than 2 can be built from primes via binary Goldbach, forming a cyclic structure — composites → primes → composites. This perspective unifies the arithmetic of primes and composites as a mutually generative system, opening new directions for understanding the additive structure of integers.