A Short-Interval Refinement of Brocard's Conjecture at Products of Consecutive Primes

Let $p < q$ be consecutive primes with p ≥ 3. Brocard’s conjecture asserts that at least four primes lie between p² and q². We propose a more structured conjectural distribution of these primes by introducing the intermediate points $p(q - 1)$ and $pq$. Our principal conjecture asserts that there is at least one prime in each of (p², p(q - 1)), $(p(q - 1), pq)$, and at least two primes in (pq, q²). Thus the four-prime lower bound of Brocard’s conjecture is decomposed as a $1 + 1 + 2$ lower bound across three subintervals. The middle interval has the particularly short length p, while pq p². We formulate the conjecture in terms of the prime-counting function, compare it with Bertrand’s, Legendre’s, Oppermann’s, Andrica’s, and Brocard’s conjectures, and give elementary implications between them. An exhaustive computation is reported for all relevant consecutive-prime pairs up to q² 1.05 × 10⁹ (extending an initial check at q² ≤ 10⁶); no counterexample was found in either range. This paper is intended as a conjectural note: no proof of the proposed statements is claimed.