Symmetric Pairs of Twin Prime Middles

An integer m is called a twin middle if both m-1 and m+1 are prime. Harvey Dubner (1996) conjectured that every sufficiently large even integer can be expressed as the sum of two twin primes. In this paper, we investigate a highly constrained specialization of this problem: whether every twin middle m > 348 is the exact midpoint of two other distinct twin middles. Using an efficient algorithmic approach combining a prime sieve with a Fast Fourier Transform (FFT) convolution, we computed the number of such alternative representations for all twin middles up to 4.9 x 10^6. Our computational results confirm that the only exceptional twin middles failing this property are m in {4, 6, 198, 348}. Backed by the Hardy-Littlewood k-tuple conjecture, we heuristically justify why the number of alternative representations grows asymptotically as O(m/ln^4 m), confirming that the set of exceptions is strictly finite.