A Prime-Composite Difference Function and a Conjecture on Zeros in Prime Gaps

We define an infinite product over primes derived from the classical Wallis product, and its logarithmic derivative P(x), the prime pressure. We prove that P(x) and its composite counterpart Q(x) satisfy P(x) + Q(x) + 2x/(1-x²) = 1/x - π cot(πx), a decomposition of Euler's partial fraction expansion of the cotangent into prime and composite contributions. This identity shows that the prime pressure is precisely the prime component of the classical cotangent partial fractions. We then study the difference function D(x) = P(x) - Q(x) and state the following conjecture, supported by numerical verification of all consecutive prime gaps with 3 ≤ p ≤ 9,973 (gap sizes 2 through 36, totalling 1,228 gaps), with no counterexample found: for every pair of consecutive primes (p, q) with p ≥ 3 and gap g = q - p, the open interval (p, q) contains exactly g - 2 non-integer zeros of D(x). A partial proof is given: the lower bound g - 2 is established rigorously via the Intermediate Value Theorem, using a monotonicity argument for the local composite pole contributions. 2020 Mathematics Subject Classification: 11A41, 11B83, 26A99.