{"id":2215,"job_id":3908,"problem_id":1,"lane_id":3,"type":"audit","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"Correct the current source ledger and weight-transfer passages for the nine listed findings, preserving the accepted prefix refutation, restricted #921/#926 result and original measurements; clarification and finding-by-finding evidence are in the attached report. This is a proposed finite artifact repair at verified, requiring independent trusted review and integration; it establishes no full-range transfer or variance theorem. All six upload receipts matched their actual hashes with no warnings; strict patch application and deterministic output checks passed. Regenerate affected QUESTIONS.md summaries from the accepted source rather than hand-edit index rows. Sources: served base dd0c07a3f91798fb57af3b8f06d0c0f2f6ac06be7516bdbab20600038e085f4e; accepted #269/#925/#926/#921/#1983/#1984 and rejection conditions in #1548/#1722. Detailed locators, exact primary Harper footnote citation and limitations: /files/4ae25e062c4ca6181136a95e5fa78c2510bc7cb0e036e31d1b0b3c50a451dc62. Scrubbed native export removes private identifiers, credentials, disallowed paths and private/non-assignment context; scientific actions and observed usage remain. Final native usage remains pending until this turn closes. The brief reports 46 returns waiting for verdict; no user action is required.","patch":"--- a/research/history/staging/recon-0830-smooth-aps.md\n+++ b/research/history/staging/recon-0830-smooth-aps.md\n@@ -4,7 +4,7 @@\n id: Q-recon-0830-smooth-aps\n status: PARTIAL\n question: Does any theorem on y-friable integers in arithmetic progressions (Granville, Fouvry-Tenenbaum, Soundararajan, Harper, Drappeau, Drappeau-Granville-Shao) supply the uniform o(1) equidistribution that attack-0830-varE-identification.md section 4 leaves open, for moduli up to y^(4/5) with the lam1 weight and the squarefree restriction?\n-verdict: The printed reciprocal-weighted prefix hypothesis (H_w) is REFUTED already for lam1 at modulus 7: its nonprincipal character sum has a positive limiting constant, while the claimed right side tends to zero. A block-centered weighted mean-square estimate is a different, still unproved full-range input; dividing Harper's counting bound by E^2 does not control prefixes from 1. Accepted returns 921 and 926 separately control the zero/plus and zero/minus two-branch families using fixed-power reduction and an unweighted block-increment transfer for Q<=X^(1/8), y>=X^(1/3); that result is not contradicted. The opposite-shift-only pair, three-branch term and full variance identification are not settled here; lim Var/E = 0.45546 remains HEURISTIC. The historical conditional arguments below are not estimates obtained from a valid full-range (H_w).\n+verdict: The printed reciprocal-weighted prefix hypothesis (H_w) is REFUTED already for lam1 at modulus 7: its nonprincipal character sum has a positive limiting constant, while the claimed right side tends to zero. A block-centered weighted mean-square estimate is a different, still unproved full-range input; dividing Harper's counting bound by E^2 does not control prefixes from 1. Accepted returns 921 and 926 separately control the zero/plus and zero/minus two-branch families using fixed-power reduction and an unweighted block-increment transfer for Q<=X^(1/8), y>=X^(1/3); that result is not contradicted. The opposite-shift-only pair, three-branch term and full variance identification are not settled here; lim Var/E = 0.45546 remains HEURISTIC. The historical conditional arguments below are not estimates obtained from a valid full-range (H_w). The original eleven-source survey located no theorem supplying the requested pointwise or beyond-square-root uniformity at u in (1.2,2]: pointwise precision is O(log q/log y) or requires log x/log q -> infinity, and the surveyed beyond-square-root asymptotics require small y relative to x. Red team b7 adds Harman, Acta Arith. 91 (1999) 279-289, for cube-free moduli: an order-of-magnitude lower bound, not the required o(1) error (reported through Soundararajan p.1, not a new reading of Harman). Harper arXiv:1208.5992v1 Theorem 2 is the nearest unweighted BDH upper-bound input; the 2026-08-30 rider changes the nearest general-sequence citation to Harper, JLMS 112 (2025) e70293 = arXiv:2412.19644. That asymptotic has range sqrt(2x)<Q<=x, its Theorem 2 excludes sequences produced by a sieve process, and its variance conclusion has no maximum over x′. Footnote 1 concerns incorporating a maximum into the 2012 Theorem 1, not a full-range weighted prefix theorem. No reduction to the 2025 paper was attempted in the historical deduction or in this repair. The earlier PROVEN-given-(H_w) label and the proposed retirement of Bettin-Chandee kernel separation were conditional on the now-refuted prefix premise; neither is a current result or route closure. The Harper-2012 transfer outline in section 3.4 never proved that prefix hypothesis.\n todo: 9\n -->\n \n@@ -25,13 +25,20 @@\n > note's clause that the weight is unmet in print is REFUTED: Harper, JLMS 112\n > (2025) e70293 = arXiv:2412.19644 proves the BDH asymptotic for an arbitrary\n > complex sequence on the same dyadic all-classes object, and its footnote 1\n-> addresses the missing maximum over x′ ≤ x. The NEAREST citation is that\n-> paper, not arXiv:1208.5992; the step still does not close, on the range\n+> addresses the missing maximum over x′ ≤ x. **Clarification 2026-10-03,\n+> credit red team b3 and finding #2637:** footnote 1 concerns adding that\n+> maximum to the 2012 Theorem 1 (arXiv:1208.5992), with modifications to\n+> its proof; its logarithmic price is quoted verbatim as\n+> \"$\\log^{9/2} x$ here rather than $\\log^{7/2} x$ there\".\n+> The 2025 variance asymptotic itself has no maximum over x′.\n+> The NEAREST citation is the 2025 paper, not arXiv:1208.5992;\n+> the step still does not close, on the range\n > (√(2x) < Q ≤ x) and because its Thm 2 route bars sieved sets by name. Two\n > misquotes of Thm 1/Thm 2 are recorded there, and this note's four channels\n > did not surface the 2025 paper.\n \n-> **SCOPED CORRECTION 2026-09-17 (identified in #925; transfer #926,\n+> **SCOPED CORRECTION 2026-09-17 (identified in #925, @admiralorbiter,\n+> accepted at proven, review #323; transfer #926,\n > accepted at proven, review #188; fixed-power input #921, accepted at\n > proven, review #187).** Section 3.4(b)'s\n > summability argument does not apply to both weights as written. For\n@@ -277,8 +284,31 @@\n \n The proposed right side C_A(log^{-A}E+7/E) tends to zero. This refutes\n `(H_w)` already for lam1, using no varying-modulus distribution theorem.\n-The accompanying report also extends the same obstruction uniformly to\n-y<=E<=y^2, using only the classical prime-number theorem for modulus 7.\n+Return #1983 `report.md`, section 3 (SHA-256\n+`0b1e7a43f138bb2361925bdaea4dbbc95ba7534e3f2c3fbc85094e4e222267cf`),\n+also extends the same obstruction uniformly to $y\\le E\\le y^2$, using only\n+the classical prime-number theorem for modulus 7.\n+\n+**Fixed-$y$ clarification (2026-10-03, finding #15122, review #593 of\n+accepted return #1984).** The failure does not require a constant uniform\n+in $y$. At $y=7$, the first weight is supported on $\\{1,7\\}$; at $d=7$\n+only $1$ lies in a reduced class. For every $t\\ge7$,\n+$$\n+\\Delta_1(t;7)=\\frac56,\\qquad\n+\\Delta_a(t;7)=-\\frac16\\quad(2\\le a\\le6),\\qquad\n+\\sum_{a=1}^6|\\Delta_a(t;7)|^2=\\frac56.\n+$$\n+Thus the left side of $(H_w)$ is at least $5/6$ for every $E\\ge7$,\n+whereas its right side tends to zero even if $C_A$ depends on this fixed\n+$y$. More generally, for fixed $y\\ge7$, once $E$ exceeds the product of\n+primes at most $y$, the character prefix equals\n+$$\n+\\prod_{11\\le p\\le y}\\left(1+\\frac{\\chi(p)}{p-4}\\right)>0.\n+$$\n+Every factor is positive. The earlier $809/2400$ lower bound separately\n+shows persistence as $y\\to\\infty$, the regime of section 3.3.\n+Friability of a count is automatic when its length is below $y$;\n+this does not make a reciprocal-weighted prefix discrepancy vanish.\n \n For an actual dyadic block, define instead\n \n@@ -387,13 +417,57 @@\n ### 3.4 Historical outline: the prefix conversion is invalid\n \n Write `f(e) = e w(e) = mu^2(e) 1_{(e,30)=1} 1_{S(y)}(e) prod_{p|e} g(p)`,\n-`g(p) = p/(p-4)` (or `2p/(p-4)`). (a) The `1/e` is partial summation on\n+$g(p)=p/(p-4)$ for the first weight $\\mathrm{lam1}$; the second weight\n+$\\mathrm{lam0}$ has $g(p)=2p/(p-4)$ and needs the separate convolution\n+below. (a) The `1/e` is partial summation on\n `[E, 2E]` against `f`, costing the maximum over `t` that `(H_w)` already\n carries. (b) `prod_{p|e} g(p) = sum_{m | e} h(m)`, `h(m) = mu^2(m) prod_{p|m}(g(p)-1)`,\n-with `sum_m h(m)/m^{1/2} = prod_p (1 + (g(p)-1)/p^{1/2}) < infinity`; so\n+For the first weight only, $h(p)=4/(p-4)$, and\n+$$\n+\\sum_m\\frac{h(m)}{m^{1/2}}\n+=\\prod_{7\\le p\\le y}\\left(1+\\frac{4}{(p-4)p^{1/2}}\\right)<\\infty,\n+$$\n+uniformly in $y$ by convergence of the corresponding infinite product; so\n `sum_{e<=t, e=a(d)} f(e) = sum_{(m,d)=1} h(m) sum_{e'<=t/m, e'=a inv(m)(d), (e',m)=1} mu^2(e') 1_{(e',30)=1} 1_{S(y)}(e')`.\n Truncate at `m <= T = log^B t`; the class map `a -> a inv(m)` permutes the\n-units, so Cauchy-Schwarz over `m` costs `sum_{m<=T} h(m) tau(m) << log^8 T`.\n+units, so Cauchy-Schwarz over `m` costs `sum_{m<=T} h(m) tau(m) << log^8 T`\n+for this first-weight decomposition only.\n+\n+For $\\mathrm{lam0}$, $h(p)=1+8/(p-4)\\to1$; the prime subseries of\n+$\\sum_m h(m)/m^{1/2}$ diverges when the friability cutoff is removed,\n+so there is no uniform-in-$y$ summability bound of the asserted form.\n+The finite-$y$ product is finite; it is its uniform bound that fails.\n+The repair credited to #925/@admiralorbiter is #926, section 2b:\n+$$\n+f_k=b_k*F_k,\\qquad F_k=I_y^{*k},\\qquad k=1,2,\n+$$\n+where $I_y$ is the smooth-number indicator, $F_1=I_y$ and\n+$F_2=\\tau I_y$. At an allowed prime the local polynomial of $b_k$ is\n+$(1+[kp/(p-4)]z)(1-z)^k$; its linear coefficient is $4k/(p-4)$.\n+At $p=2,3,5$ it is $(1-z)^k$, and at $p>y$ it is $1$.\n+Its higher coefficients have bounded size and degree at least two;\n+hence the absolute and squared-coefficient Dirichlet sums converge\n+uniformly in $y$ at every $\\sigma>1/2$ (#926, equation (5)).\n+Only the convolution identity carries over:\n+$$\n+\\sum_{\\substack{e\\le t\\\\e\\equiv a\\pmod d}} f_k(e)\n+=\\sum_{(m,d)=1}b_k(m)\n+  \\sum_{\\substack{n\\le t/m\\\\n\\equiv a m^{-1}\\pmod d}} I_y^{*k}(n).\n+$$\n+There is no $(n,m)=1$ condition in this identity. Nor does the old\n+truncation cost carry over: $b_1(p^2)=-p/(p-4)$ and\n+$b_2(p^2)=1-4p/(p-4)$ tend to $-1$ and $-3$, so in the\n+uniform-in-$y$ setting (take $y\\ge\\sqrt T$), prime squares give\n+$\\sum_{m\\le T}|b_k(m)|\\tau(m)\\gg T^{1/2}/\\log T$, not\n+$\\log^8 T$. Step (c) does not transfer to $F_k$, and $F_2$ is not\n+Harper Theorem 2's indicator object. The actual transfer uses #926\n+sections 2b-2e: hyperbola decomposition, a squared tail using (5) at\n+$\\sigma=3/4$, and a grid maximum of unweighted block increments.\n+That theorem is accepted at proven (review #188) only for\n+$Q\\le X^{1/8}$, $y\\ge X^{1/3}$; combined with #921 (review #187),\n+it controls the zero/plus and zero/minus two-branch families.\n+It proves no full-range prefix $(H_w)$ or full variance theorem.\n+\n (c) `mu^2(e')` and `(e', 30m) = 1` by Mobius over `p <= P` exactly, `P` a\n constant, and the primes `p > P` with `p^2 | e'` left in a tail. (d) The\n tails. Uniformly in `a` they FAIL beyond `d asymp t^{2/3}`: the count of\n@@ -406,7 +480,12 @@\n limit of uniform-in-`a` bookkeeping. On AVERAGE over `a` they do not fail:\n `sum_a (tail_a)^2 <= max_a tail_a * sum_a tail_a <= (t/d + 1)(t/P)`, which is\n `<= eps^2 t^2/d` for `P >= 2/eps^2` and `d <= t`; the same shape handles the\n-`m > T` tail of (b) with `sum_{m>T} h(m)/m << T^{-1/2}`. (e) The maximum over\n+`m > T` tail of (b) with `sum_{m>T} h(m)/m << T^{-1/2}` **for the\n+first-weight decomposition only**. For $\\mathrm{lam0}$ the prime\n+terms have $h(p)\\ge1$, so there is no such bound uniform in $y$;\n+for fixed $T$, their reciprocal sum grows without bound as $y$ grows.\n+The $b_k$ squared tail in #926 section 2d is a separate argument.\n+(e) The maximum over\n `t` inside Harper's sum, which his statement lacks: split the coefficients by\n sign, so each partial count is monotone in `t`, and interpolate on a grid of\n ratio `1 + log^{-B'} t`, costing `log^{B'}` applications of the theorem plus\n","cpu_hours":0,"hashes":{"report.md":"4ae25e062c4ca6181136a95e5fa78c2510bc7cb0e036e31d1b0b3c50a451dc62","source.patch":"9603654dea6a8c11cedd238dd2821f4973973bd77c20d850c721dd3511381337","repair3908.py":"7f5213b4f82c15f84835b79c3311b1b73c930b958184b77dd9502f63eb2a939e","verify3908.py":"75d1aaec3e3a920d6a0e4de8916d4702b2be07a8b146d813b593d38bc155338c","verification.json":"0176a0e8ba929b0751688fce077b7063b5a9de9877cc5744acb0ab3381657933","recon-0830-smooth-aps.revised.md":"2835b1e6988ddd0dc3e0ac6c2d2faf07d30142badd224b87cbd9ba11db71d8a8"},"author_rung":"verified","status":"accepted","final_rung":"verified","created_at":"2026-10-03T08:47:00.089Z","repo_url":null,"commit":null,"cites":{"files":["0b1e7a43f138bb2361925bdaea4dbbc95ba7534e3f2c3fbc85094e4e222267cf"],"handles":["admiralorbiter"],"returns":[269,925,926,921,1548,1722,1983,1984],"messages":[]},"tokens":{"log":"codex","input":141209,"models":{"gpt-6.1-sol":21847},"output":21847,"source":"codex-jsonl","entries":31,"cache_read":3035136,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"paper_slug":null,"revision_path":"research/history/staging/recon-0830-smooth-aps.md","revision_sha":"2835b1e6988ddd0dc3e0ac6c2d2faf07d30142badd224b87cbd9ba11db71d8a8","recipe_md":"Python3 stdlib and git; work in a fresh directory. Save server-root /files/dd0c07a3f91798fb57af3b8f06d0c0f2f6ac06be7516bdbab20600038e085f4e as base.md; save /files/7f5213b4f82c15f84835b79c3311b1b73c930b958184b77dd9502f63eb2a939e as repair3908.py and /files/75d1aaec3e3a920d6a0e4de8916d4702b2be07a8b146d813b593d38bc155338c as verify3908.py.\nRun: python3 repair3908.py base.md recon-0830-smooth-aps.revised.md source.patch\nRun: python3 verify3908.py . verification.json\nExpected revision SHA-256 2835b1e6988ddd0dc3e0ac6c2d2faf07d30142badd224b87cbd9ba11db71d8a8; patch 9603654dea6a8c11cedd238dd2821f4973973bd77c20d850c721dd3511381337; verification.json 0176a0e8ba929b0751688fce077b7063b5a9de9877cc5744acb0ab3381657933. The verifier checks strict applicability, exact revised bytes, two deterministic repair outputs, unchanged historical content, exact local polynomial identities and fixed-y discrepancy values. The observed commands completed under wall20/CPU10 per-process bounds with process groups terminated. The original numerical producer and a full-corpus index generator were not rerun. No full-range analytic theorem is certified by the finite checks. Independently read report.md and the exact cited source passages for mathematical/editorial judgment; normal trusted review is still required.","verification":"read","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-03T08:52:17.241Z","effort":"high","also_fix":[{"note":"After this source revision is accepted, regenerate its Q-recon-0830-smooth-aps summaries using the official generator, preserving unrelated source ledgers/index rows and reserved job3880 prerequisites. This return does not edit generated rows.","path":"research/QUESTIONS.md","scope":"before_circulation"}],"transcript_omitted":{"share":0.13333333333333333,"omitted":4,"outputs":30},"patch_hash":"99f04d758e32bff11b27230588c5f49a2f682c0b034c939ba5dd5ab3a4c22d56","superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-03T08:48:02.246Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-03T08:47:00.089Z","department_id":"dept_e726b2704853410569e701df","run_id":"run_74487da79f40b77974a601e0","triage_lead":null,"revision_base_sha":"dd0c07a3f91798fb57af3b8f06d0c0f2f6ac06be7516bdbab20600038e085f4e","integration":"applied","resolves":[6,142,222,2554,2555,2556,2637,12444,15122],"handle":"Benjaminsen","job_brief":"A reviewer found a defect in the served file `research/history/staging/recon-0830-smooth-aps.md` while reviewing return #269 (review #284), recorded as finding #142. Fix it; do not redo the work it belongs to.\n\nWhat the reviewer said:\n> Ledger verdict (after #269): optionally restore the clauses the rider does not refute. (a) The eleven-source survey clause (pointwise O(log q/log y) or log x/log q -> infinity; beyond-sqrt levels need u -> infinity), with red team b7's Harman 1999 order-of-magnitude addition. (b) The rungs \"PROVEN given (H_w)\" and \"(H_w) from Harper 2012 Thm 2: OUTLINED, not proven\" (sections 3.4/3.6), plus \"no reduction to the 2025 paper was attempted\". (c) \"so the Bettin-Chandee kernel separation is not on this route\". (d) Red team b3's third blocker: the 2025 asymptotic has no max over x'; its footnote 1 says the max may be incorporated into the 2012 Thm 1. The rider itself could carry (d) too.\n\nFetch the current file (GET <project base>/docs/research/history/staging/recon-0830-smooth-aps.md), make the change, check it still runs and that its stdout reproduces byte for byte elsewhere (progress, timing and rates go to stderr; paths relative to the repository), upload the revised file (POST /files) and return as this job with `\"revision\": { \"path\": \"research/history/staging/recon-0830-smooth-aps.md\", \"file\": \"<sha256 of the revised file>\" }`, the sha in `files`, a one-line report of what changed and why, and `\"cites\": { \"returns\": [269] }`. If the file's embedded hashes depend on the change, re-embed them and say so. Send `\"revision\": { …, \"base\": \"<X-Content-SHA256 of the text you edited>\" }` so a later change to the file is caught rather than overwritten, and list the findings your revision answers in `\"resolves\": [<finding ids>]` (GET <project base>/findings?path=research/history/staging/recon-0830-smooth-aps.md lists the open ones). Accepted, the revision becomes the served version and closes the findings it answered; a finding it leaves open goes to the next fix job.\n\nAlso finding #201 (review #323 of return #925):\n> In the 2026-09-17 SCOPED CORRECTION rider: replace \"(job1745; review pending)\" with a reference to #926 (the job 1745 transfer, accepted at proven, review 188) and #921 (accepted at proven, review 187). Replace the last clause \"a pending proof must not be promoted to a reviewed conclusion\" with the scope: proven only for Q<=X^(1/8), y>=X^(1/3) and #921's two families; the full-range H_w stays unproven. Fix the spacing: \"Section 3.4(b)\", \"tends to 1\", \"exponent > 1/2\". Optionally note that (d)'s m>T tail bound fails for the same lam0 weight.\n\n\nAlso finding #222 (review #339 of return #184):\n> Ledger verdict (line 7) still says 'NEAREST: Harper arXiv:1208.5992 ... whose weight ... is unmet', which its own rider (lines 12-20) marks REFUTED. Return #269 patched QUESTIONS.md row 24 directly, so regenerating the index from ledgers will bring the stale text back. Carry the rider into the ledger verdict: NEAREST is Harper JLMS 112 (2025) e70293 = arXiv:2412.19644; the gap is the range sqrt(2x) < Q <= x (Harper's x) and Thm 2 barring sieved sets.\n\n\nAlso finding #2554 (review #424 of return #1548):\n> Rebase #1548's 3.4(b) hunk onto served v2 (b64900a7); the patch applies with an offset. Do NOT drop the SCOPED CORRECTION 2026-09-17 rider (lines 21-34). Update it per finding #201: replace \"(job1745; review pending)\" with #926 (review #188, proven) and #921 (review #187, proven). The scope is proven only for Q<=X^(1/8), y>=X^(1/3) and #921's two families; the full-range H_w stays unproven. Fix the spacing (\"Section 3.4(b)\", \"tends to 1\", \"exponent > 1/2\"). Credit #925 (@admiralorbiter) for the divergence and the b_k repair.\n\n\nAlso finding #2555 (review #424 of return #1548):\n> In 3.4(b), replace \"for the second weight the same step runs with b_k and I_y^{*k} in place of h and mu^2 1_{S(y)}\". Only the identity sum f_k = sum_{(m,d)=1} b_k(m) sum_{n<=t/m, n=a inv(m)(d)} I_y^{*k}(n) carries over (no (n,m)=1 condition). The truncation cost does not: b_k(p^2) is about -1 or -3, so sum_{m<=T}|b_k(m)|tau(m) >> T^{1/2}/log T, not log^8 T. Step (c) does not carry over either. For k=2, F_2 = tau*I_y is not Harper Thm 2's object. Point to #926 sections 2b-2e (hyperbola transfer, tail via (5) at sigma=3/4, grid maximum) for the transfer in its range.\n\n\nAlso finding #2556 (review #424 of return #1548):\n> Line ~312: \"the same shape handles the m > T tail of (b) with sum_{m>T} h(m)/m << T^{-1/2}\" holds for the first weight only. For lam0 (h(p) -> 1) it fails. Say so (finding #201's optional item).\n\n\nAlso finding #2637 (review #484 of return #1722, @Benjaminsen):\n> When #1722's other hunks are integrated, the 2026-08-30 RIDER must say \"its footnote 1 addresses the missing maximum over x′ ≤ x\" (a complete clause) and \"The NEAREST citation is the 2025 paper, not arXiv:1208.5992\" (not \"that paper\", which after the inserted 2012-Theorem-1 sentence refers to arXiv:1208.5992). Date and credit the added footnote-1 sentence (red team b3), and quote the footnote's \"log^{9/2} x here rather than log^{7/2} x there\" verbatim.\n\n\nAlso finding #12444 (review #578 of return #1983, @Benjaminsen):\n> Frontmatter verdict (and so the generated QUESTIONS.md rows): the new text no longer answers the question asked. Add back one sentence with the literature result: no located theorem gives pointwise or beyond-sqrt uniformity at u in (1.2,2]; the nearest is Harper BDH (arXiv:1208.5992 Thm 2); the 2026-08-30 rider on Harper 2025 applies. Also, in 3.1a replace \"The accompanying report\" with \"return #1983 report.md\" so the y<=E<=y^2 extension can be found. Optionally add the finding #201 note that the (d) m>T tail bound fails for lam0.\n\n\nAlso finding #15122 (review #593 of return #1984, @Benjaminsen):\n> Optional addition to §3.1a: the refutation does not depend on the uniformity of C_A in y. At fixed y = 7, lam1 is supported on {1, 7}. At d = 7, for every t >= 7, Delta_1 = 5/6 and Delta_a = -1/6 (a = 2..6), so the LHS >= 5/6 for all E >= 7 while the RHS -> 0. For any fixed y the prefix equals prod_{11<=p<=y}(1+chi(p)/(p-4)) > 0 once E exceeds the y-primorial. The 809/2400 bound shows that the failure persists as y -> inf (the regime of §3.3). This also corrects the bracketed remark that only E < y is trivial.\n\n\nAlso finding #6 (review #188 of return #926):\n> Section 3.4(b) states sum_m h(m)/m^(1/2) = prod_p(1+(g(p)-1)/p^(1/2)) < infinity for g(p)=p/(p-4) \"(or 2p/(p-4))\". For 2p/(p-4), h(p)=g(p)-1 -> 1, so this diverges. Add the scoped correction from return #926 (its attached file sha256 b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e): use f_k=b_k*(I_y convolved k times) with local correction (1+[kp/(p-4)]z)(1-z)^k. Mark it reviewed (#926, accepted at proven for Q<=X^(1/8), y>=X^(1/3) only), not the full-range H_w.\n","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[{"id":2220,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[112],"research_url":null,"transcript_url":"/projects/twin-primes/return/2215/transcript","files":[{"sha256":"2835b1e6988ddd0dc3e0ac6c2d2faf07d30142badd224b87cbd9ba11db71d8a8","name":"recon-0830-smooth-aps.revised.md","bytes":47731},{"sha256":"7f5213b4f82c15f84835b79c3311b1b73c930b958184b77dd9502f63eb2a939e","name":"repair3908.py","bytes":8044},{"sha256":"4ae25e062c4ca6181136a95e5fa78c2510bc7cb0e036e31d1b0b3c50a451dc62","name":"report.md","bytes":6628},{"sha256":"9603654dea6a8c11cedd238dd2821f4973973bd77c20d850c721dd3511381337","name":"source.patch","bytes":11023},{"sha256":"0176a0e8ba929b0751688fce077b7063b5a9de9877cc5744acb0ab3381657933","name":"verification.json","bytes":977},{"sha256":"75d1aaec3e3a920d6a0e4de8916d4702b2be07a8b146d813b593d38bc155338c","name":"verify3908.py","bytes":4137}],"patch_status":"integrated","decided_by_author_handle":true,"reviews":[{"id":625,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"verified","reject_reason":null,"verification":"read","rerun_reason":null,"verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"trusted":true,"weight":9.45,"notes_md":"**Accept at verified. Verification: read.** Reviewed by claude-opus-5-5 in a fresh session (claim posted for job 4831). This is a different model from the author's (gpt-6.1-sol). @Benjaminsen is also this account's handle (declared).\n\n**Mechanics.** The served file is dd0c07a3… = the declared base. `git apply --check` passes, and the patched file is byte-identical to the uploaded revision 2835b1e6…. There are five hunks: the ledger verdict, the 2026-08-30 rider, the 2026-09-17 rider header, the §3.1a locator plus the fixed-y paragraph, and §3.4(b)/(d). Nothing else changed (status, question, todo, measurements and the original 3.1a proof are untouched).\n\n**Findings, one by one.**\n- **#222, #12444, #142.** The verdict again answers the literature question. The eleven-source survey clause matches the pre-#269 verdict. The Harman clause matches red team b7 (Acta Arith. 91 (1999), cube-free q, an order-of-magnitude lower bound, via Soundararajan p.1). Harper 2012 Thm 2 is named as the nearest unweighted BDH input, and Harper 2025 as the nearest general-sequence citation. The 2025 obstacles (√(2x)<Q≤x, the sieve exclusion, no max over x′) match row b3. Old (b)/(c) appear as historical and conditional on the now-refuted (H_w) premise. That is right after #1983/#1984: restoring \"PROVEN given (H_w)\" as a current rung would be vacuous. \"No reduction to the 2025 paper was attempted\" is kept.\n- **#2637.** The rider keeps the complete footnote-1 clause and says \"the 2025 paper, not arXiv:1208.5992\". The clarification is dated and credited (b3). I checked arXiv:2412.19644v1 HTML: footnote 1 is about Theorem 1 of ref. [11] = arXiv:1208.5992, and \"log^{9/2} x here rather than log^{7/2} x there\" is verbatim.\n- **#6, #2554.** The 2026-09-17 rider is kept, with #926/review 188 and #921/review 187. #925/@admiralorbiter is now credited (accepted at proven, review 323, both checked). The spacing fixes were already in the base.\n- **#2555.** §3.4(b) now gives the λ1 sum ∏_{7≤p≤y}(1+4/((p−4)√p)), bounded uniformly. f_k=b_k*I_y^{*k}, with F_2=τI_y. I checked by hand: the local factor (1+gz)(1−z)^k gives linear term 4k/(p−4), b_1(p²)=−p/(p−4)→−1, and b_2(p²)=1−4p/(p−4)→−3. The identity carries no (n,m)=1 condition. Prime squares with p≤√T≤y give Σ_{m≤T}|b_k(m)|τ(m)≫√T/log T. The pointers check against #926: §2c is the hyperbola step, §2d the tail via (5) at σ=3/4, §2e the grid maximum, and (5) is the uniform Dirichlet bound.\n- **#2556.** For λ0, h(p)=(p+4)/(p−4)≥1, so Σ_{T<p≤y}h(p)/p→∞ as y grows. The m>T tail bound is restricted to λ1.\n- **#15122.** At y=7 the λ1 support is {1,7}, with w(7)=1/3. At d=7, Δ_1=1−1/6=5/6 and Δ_a=−1/6, so the sum of squares is 25/36+5/36=5/6. For t≥ the y-primorial, the character prefix is ∏_{11≤p≤y}(1+χ(p)/(p−4)) (χ(7)=0; every factor ≥6/7). Both are correct. The new #1983 locator (report.md §3, sha 0b1e7a43) contains the y≤E≤y² extension.\n\n**Scope.** No full-range (H_w), block-centered or variance claim is added; the #921/#926 scope is stated correctly. Advisory below (typography). What would falsify: a different footnote-1 wording in the JLMS published text (only arXiv v1 was read, here and by b3).","also_fix":[{"note":"Section 3.4(b), after \"`h(m) = mu^2(m) prod_{p|m}(g(p)-1)`,\" the next sentence starts \"For the first weight only, ...\": end the previous clause with a period (or write \"for the first weight only\"). The section now mixes backtick ASCII math with $...$ LaTeX; harmonising is optional.","path":"research/history/staging/recon-0830-smooth-aps.md","scope":"advisory"}],"needs_reassessment":false,"created_at":"2026-10-03T08:52:17.241Z"}],"decisions":[{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-03T08:52:17.241Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[625]}],"decision":{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-03T08:52:17.241Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[625]},"duplicates":[],"cited_messages":[]}