Alternating series of n over the nth prime

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evidence/: Review records for the four reconstruction pages of Tao's conditional convergence proof; no executable evidence is held.

lemma_3_1_reconstruction: Reconstructs the two-sided Bonferroni-type inequality that sandwiches the sign (-1)^N between truncations of the binomial expansion of (1-2)^N.

lemma_3_2_reconstruction: Reconstructs the random sifted model of the primes, its product formula for tuple probabilities, and the mean and variance bounds for the number of survivors, importing the pair singular-series average.

relation_2_1_reconstruction: Reconstructs the unconditional equivalence, credited to Said, between the convergence of the alternating series of n over the nth prime and of the series of the parity of the prime counting function over n log n.

theorem_1_4_reconstruction: Reconstructs the conditional proof that the parity of the prime counting function is equidistributed enough for the series of (-1)^n n over the nth prime to converge, assuming the quantitative Hardy-Littlewood prime tuples conjecture, through the van der Corput step, Bonferroni truncation, and the bias recursion in the random sifted model.

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This folder holds the author-recorded reconstruction of the one substantial result on Problem 15: Tao's proof that $\sum_{n\ge1}(-1)^nn/p_n$ converges assuming a quantitative Hardy--Littlewood prime tuples conjecture, from the library source Tao (2023). The Theorem 1.4 page carries the main argument with the conjecture stated as its imported hypothesis; the Lemma 3.1 and Lemma 3.2 pages carry the same-paper inputs (the Bonferroni-type parity bounds, and the random sifted model with its mean and variance); the relation (2.1) page carries the unconditional equivalence with the convergence of $\sum_{n\ge2}(-1)^{\pi(n)}/(n\log n)$. Each page names the source pages and labels it reads, the external theorems it imports at the strength used, and the places where it fills or reads past the source's wording.

Where things stand

Reviewed. Each reconstruction page was independently reviewed as it stood on 2026-09-28T05:03:27Z by a focused review filed under evidence/verify/, with a distinct grade of the four reports. As the grade records them, the verdicts are: Lemma 3.1, fidelity faithful with the commentary correction C1 and argument sound; Lemma 3.2, fidelity faithful with corrections C2 and C3 and argument defective as stated at display (3.8) for general $k$ but sound after those corrections; relation (2.1), fidelity faithful with the labeling sentence C4 and argument sound; Theorem 1.4, fidelity faithful with corrections C5--C7 and argument sound conditional on Conjecture 1.3 exactly as the page states. No report was graded void. The seven corrections C1--C7 were applied, so the current text differs from the reviewed text at the places the grade names: one commentary sentence under "The two-step differences" on the Lemma 3.1 page; the passage deriving display (3.8) and the statement's scope with its two proof phrases on the Lemma 3.2 page; one labeling sentence in the Standing paragraph of the relation (2.1) page; and the tiling display of Step 2, the indexing remark of Step 8 with its compilation note, and the two consumer interfaces in Steps 6 and 8 of the Theorem 1.4 page. No tier is assigned and the problem's status is unchanged. The reconstruction establishes only the implication from Conjecture 1.3 of the source (Kuperberg's uniform prime tuples conjecture with the range of tuple sizes widened to $(\log\log x)^5$) to the convergence of both series. The problem's dated assessment is unchanged: no unconditional result is known, the absolute series diverges, and the 2026 Lean acceptance concerned a misformalized statement. After the review, line wrapping was normalized on the reconstruction pages; no formula or sentence changed.

Imported inputs. Besides the hypothesis, the argument imports Kuperberg's uniform singular-series bound (Theorem 1.2 of Kuperberg (2023)), the pair singular-series average $2\sum_{h_1<h_2\le H}\mathfrak S(\{h_1,h_2\})\le H^2$ for large $H$ (Montgomery, Croft), Mertens' theorems, Bertrand's postulate and the prime number theorem. None of these proofs was reread; the pages state each at the version consumed.

Mechanism. The van der Corput $A$-process turns the parity bias of $\pi$ over a long interval into the average bias of $(-1)^{\pi(n+\lambda\log x)-\pi(n)}$ over short intervals, and a Bonferroni truncation at $r\approx(\log\log x)^{4.5}$ terms together with the uniform prime tuples conjecture replaces the primes in those intervals by the random sifted model of Banks, Ford and Tao, one uniformly random residue class per prime up to $z\asymp x^{1/e^\gamma}$. In the model, sifting by one more prime $q$ multiplies the parity bias $\mathbf E(-1)^{\mathbf S}$ by $1-2\,\mathbf E\mathbf S/q<1$ up to an error controlled by the variance of the survivor count, so a discrete Gronwall iteration over the primes in $(\lambda\log x,z]$ drives the bias down to $O(\lambda^{-1/2})$, which is just enough for the series to converge.