Part XIV of the 6N project. Part XIII collapsed the Goldbach comet (prime sums 2N = p1 + p2) onto the Hardy-Littlewood conditional singular series. Here we treat the dual object: prime differences of arbitrary even gap d (Polignac). For each even gap d, let pid (X) = #n 2 (p-1) / (p-2), Pi₂ = prod>₂ (1 - 1/ (p-1) ²), and base (X) = X/ln² (X), we find C (d) = pid (X) / (base (X) * S (d) ) collapses to a single constant across ALL gaps. Over the 500 even gaps d <= 1000 at X = 10⁸, C (d) has mean 1. 131 and coefficient of variation 0. 068%, with no trend in d (quartile means agree to 0. 015%) and clean stratification by omegaₒdd (d) (band CV at most 0. 11%). The measured Pi₂ = 0. 66016186 matches the literature to eight figures, so S carries no free normalisation. The difference side thus mirrors the sum side: dividing by the singular series collapses the comet, and here the collapse constant is moreover shared across all gaps. Gaps with wildly different absolute counts -- pi₂ = 440, 312, pi₆ = 879, 908, pi₂10 = 1, 409, 148 at X = 10⁸ -- all reduce to C ~ 1. 131 once S (d) is removed. (For d=6=2*3 the odd factor 3 contributes (3-1) / (3-2) =2, the familiar doubling of gap-6 pairs. ) SUM-DIFFERENCE SYMMETRY. Parts I-XII resolved the prime difference at fixed gap (twins d=2; cousins and sexy d=4, 6). Part XIII collapsed the prime sum. The present paper completes the pair: across all even gaps, the difference-side comet collapses onto S (d) with a gap-independent constant. The sum-side enrichment prod (q-1) / (q-2) in N and the difference-side prod (q-1) / (q-2) in d are the same singular-series mechanism read on the two linear constraints p1+p2=2N and p1-p2=d. ATTRIBUTION. S (d) is the classical Hardy-Littlewood prime-pair singular series, NOT a new formula. The contribution is the omega-stratified reading of the comet's bands as S (d), the verification of pointwise collapse to CV 0. 068% over 500 gaps at X = 10⁸, and the empirical cross-gap constancy of C (d) -- a falsifiable statement (the quartile test) that the data confirm. SCOPE. This concerns only the conditional count's shape and constant; it makes NO statement about Polignac's conjecture (that each even gap is attained by infinitely many prime pairs). The constant C ~ 1. 131 carries the logarithmic correction of the crude weight X/ln² X (it shifts with X: 1. 147 at X = 2e7, 1. 131 at X = 1e8), but -- unlike the Goldbach decile drift -- this does not break the cross-gap constancy, since X is fixed while d varies.
Ruqing Chen (Thu,) studied this question.