{"id":2137,"job_id":4702,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Reassessment of return #1722\n\n**Source audit; heuristic.** Review #484 refutes a damaged document revision, not the accepted Euler-factor repair or the entire smooth-AP approach. The current owner already incorporates the stronger accepted prefix-normalization refutation from #1983/#1984. No distinct supported rescue experiment emerged; stop without a research proposal.\n\n#1722's rejected revision `9bcd35cc…` changes the rider to “says the missing maximum” and puts a 2012 citation immediately before “that paper”, causing the referent failure identified by review #484. The review preserves the lam0 divergence, the convolution coefficients, the truncation objection and #926's restricted transfer. Its rejection must not be converted into a mathematical closure. The fetched original transcript has 304 parseable records with no external-search marker detected; the original literature search is recorded in the owning note §§2,4–5 and red-team b3. Recorded return #2110 already reassessed the related #1548 revision and modulus-7/block-centering distinction; this report preserves that prior result. This reassessment refreshed the primary-source comparison on 2026-10-02 before any experiment.\n\nPreserve the algebra: for lam0, `h(p)=2p/(p−4)−1=1+8/(p−4)>1`, so the prime subseries of `sum h(m)/sqrt(m)` diverges as the prime cutoff grows. Changing the base to `F_k=I_y^{*k}` gives `f_k=b_k*F_k` and local correction `(1+cz)(1−z)^k`, `c=kp/(p−4)`. The linear coefficient is `4k/(p−4)`; the square coefficients are `−p/(p−4)` and `1−4p/(p−4)` for k=1,2. Absolute correction series converge for exponent >1/2, but, when y>=sqrt(T), the prime-square terms force `sum_{m<=T}|b_k(m)|tau(m) >> sqrt(T)/log(T)`. The old `log^8(T)` cost does not carry over, and the convolution class identity has no `(n,m)=1` condition. Accepted #926/review #188 supplies separate hyperbola, squared-tail and maximal transfers only for `Q<=X^(1/8), y>=X^(1/3)` and its stated two-branch families.\n\nThe current source §3.1a supplies the stronger obstruction: at fixed modulus 7 and E=y, the lam1 character prefix tends to a constant at least `809/2400`. Cauchy on six reduced classes gives discrepancy liminf at least `(809/2400)^2/6>0`, whereas the printed right side tends to zero. This preserves #1983/#1984's accepted refutation. Subtracting `Delta(E)` from `Delta(t)` removes the constant prefix bias; it does not itself prove the required full-range centered-block estimate.\n\nThe inspected primary methods do not provide that missing estimate. [Harper 2012 v1, Theorem 2, p.3](https://arxiv.org/pdf/1208.5992v1) is an unweighted smooth-counting mean-square bound. [Harper 2024 v1, Theorems 1–2, §1.1 and footnote 1, §5.2](https://arxiv.org/html/2412.19644v1), published in JLMS 112 (2025), e70293, treats an all-class dyadic variance for `sqrt(2x)