The table of first products of the modulo-30 wheelratio lines and Lehman's multipliers, exhaustive checks to new bounds, and the zeros ofthe zeta function in the Goldbach and prime-pair counts

The base table of our paper on prime gaps is the eight vectors VR = R + 30n of themodulo-30 wheel read row by row. Its first products form an 8 × 8 table of first products:the row is the vector RN of the product, the column the vector a of the smaller factor, andthe entry κ the position of the first product aˆb,ˆb ≥ a, on VRN . Every composite entry N ofthe wheel follows from it by one formula, nN = κ + m(a +ˆb) + 30m2 + σ(a + 30m), with therow m of the smaller factor and the stage σ; the stage is the diagonal j of Paper III shiftedby [ˆb > 31], so the blocks (m, σ) are the cells (j, k) of the series in another order. We provea chain of equivalent forms of the formula in the notation of the series: the position formulanN = n p+m b+c; the one-unknown equation n = (nN −c−mb)/(a+30m), whose integralityis exactly p | N; the square form nN = n□(p) + p σ + e, which generates every stage fromthe table of squares; and N = p(p + 2d) = s2 − d2, which gives p = −d +√d2 + N oncethe half-distance d is known. The blocks σ = m form the ratio-2 line q ≃ 2p, which coversN ≃ 1800 m2; there the search along the row m of p and the search along d cost the same,at m∗ =pN/1800, and a third path starting on this line is added. Continued to every ratioq/p ≃ v/u, the table gives the ratio lines with the multipliers kL = uv of Lehman, used inthe ratio stage of Papers II and V, with cost N1/3. With the table as a sieve, written in C, wecheck that every even 18 ≤ E ≤ 1012 is a sum of two wheel primes, 500 times the bound 2·109of Paper V, and count the twin primes to 1013, with the counts equal to the published valuesat every power of 10. We then apply the spectral method of our zeta paper to the Goldbachand prime-pair counts up to X = 108. The cumulative Goldbach count carries the zeros ofζ(s) with the amplitudes and phases of Fujii’s formula (first six zeros within 1.3% and 0.6◦,no fitted parameter), 99 of the first 100 zeros are matched, the mean distance of the first 30peaks to the zeros falls from 0.233 to 0.030 as X grows from 106to 108, and four controlsexclude chance (probability of the phase agreement alone 9 · 10−16). The twin count and thetriplets to sextuplets, streamed to 1012, carry no detectable zero, alone or summed; summingmany gaps makes the zeros appear, through the term 2 Ph S(h)(ψ(x) − x) inherited fromthe density of primes. We separate throughout what is proved, what is verified to a statedbound, and what is measured.