The Goldbach Comet under omega-Stratification: A Pointwise Verification of the Hardy-Littlewood Conditional Singular Series at the 5e7 scale

Part XIII of the 6N project. The Goldbach comet -- the scatter plot of r(2N), the number of ways to write 2N = p1 + p2 with both prime -- shows pronounced bands and heavy oscillation. We apply the omega-stratified, factor-resolved viewpoint developed for the twin skeleton in Parts I-XII to the sum 2N, and show the comet's scatter is, pointwise, the classical Hardy-Littlewood conditional singular series.

Writing S_G(N) = 2*Pi_2 * prod_{q|N, q>2} (q-1)/(q-2), with Pi_2 = prod_{q>2}(1 - 1/(q-1)^2) the twin constant, and base(2N) = 2N/ln^2(2N), the ratio R(N) = r(2N) / (base(2N) * S_G(N)) is constant: mean 0.566, pointwise coefficient of variation 0.22% across the ~1.25e7 even numbers with N in [1.25e7, 2.5e7], and at most 0.24% within each omega_{>2}(N) band. The comet's vertical structure is exactly the values of S_G(N); dividing by it collapses the comet to a single line. The measured Pi_2 = 0.66016182 matches the literature value to eight figures, confirming S_G is used with no free normalisation.

The factor prod_{q|N, q>2} (q-1)/(q-2) is the sum-analogue of the twin-difference enrichment prod (q-1)/(q-3) found in Part I: divisibility of 2N by a small prime q raises the number of admissible residue pairs, lifting the representation count. The same factor-resolved mechanism evaluated on the two arithmetic constraints (difference and sum) accounts for both.

ATTRIBUTION. S_G is the classical Hardy-Littlewood Goldbach singular series (1923), NOT a new formula. The contribution here is methodological and empirical: the omega-stratified viewpoint that reads the comet's bands directly as prod (q-1)/(q-2), and the pointwise verification at the 5e7 scale (CV 0.22%), carrying the factor-resolved framework of Parts I-XII from the prime difference to the prime sum.

SCOPE. This concerns only the shape of the conditional count. It makes NO statement about the Goldbach conjecture, i.e. that r(2N) >= 1 for every even 2N >= 4.

RESIDUAL TREND. R drifts monotonically by decile of 2N, from 0.5674 to 0.5647 (about 0.5%). This is the known logarithmic correction: the crude weight N/ln^2 N differs from the sharper integral weight, and the difference varies with N. It is a deficiency of the chosen base, not of S_G, and is absorbed into R; a logarithmically exact weight flattens it. We report it rather than hide it, as the natural entry point for a finer post-main-term analysis.