{"id":1830,"job_id":1945,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #1945 (pursue route 83): DGS Theorem 1.2 is refused on the whole band, the \"support on the modulus\" question does not arise, and the input (*) needs is BDH-type, not BV-type\n\n**Caveat first.** Nothing here proves (*) or closes the varE cell. The refusal of DGS rests on its statement, read at the page. The corrections to #1035 are exact arithmetic. The replacement route (section 3) is a derivation sketch at the **heuristic** rung, with two named open obligations.\n\n## The step's question, answered\n\nCan DGS Theorem 1.2 (arXiv:1704.04831v2) be restated with the friable support moved from the summed function to the modulus, for u in (1.2, 4/3)? **The question rests on a misreading, and DGS does not apply anywhere in the band.**\n\n1. **No role swap is needed.** In the served varE note, section 4, the cell remainder is `E_d = sum_{e ~ L/d, e y-smooth squarefree, (e,30d)=1} lam1(e) R_d(e, -2 e^{-1} mod d)`. The summed variable `e` is itself y-smooth, so the support is on the summed sequence, as in DGS. The modulus `d` is *additionally* y-smooth squarefree. That is a subset of moduli, which costs nothing in any sum over `q <= Q` of non-negative terms. #1035's finding 3 (\"friable restriction sits on the modulus\") misreads the roles.\n2. **DGS Thm 1.2 is refused on three separate clauses** (`job1945-bdh-reduction.md` section 1):\n   (i) **y-range.** Hypothesis (1.2) is `x^delta > y` for *sufficiently small* delta. The band `u in (1.2,2]` means `log y / log x in [1/2, 5/6)`, so the theorem says nothing at any u in the band, not only below 4/3. #1035's \"coverage exactly for u >= 4/3\" used the modulus cap and dropped this hypothesis.\n   (ii) **Normalisation.** With `f = 1_sqfree * lam1` (about 1/e), `sum_{n<=x}|f(n)|` is polylogarithmic, so the bound `Psi(x,y)/(log x)^A` is vacuous by a factor of `x^{1-o(1)}`. Measured at u = 1.5: Psi/sum|f| = 15745 (X = 1e5) and 131437 (X = 1e6). The mass-carrying weight `a(e) = e lam1(e) = prod p/(p-4)` has `a(p) > 1`, so it is **not** in class C. In addition, #1035's C4 \"`Lambda_f(p^k) = 0` for k >= 2\" is false: `Lambda_f(p^k) = (-1)^{k+1}(p-4)^{-k} log p`, which is `-(1/9) log 7` at 49. `f` is still in C.\n   (iii) **Classes.** DGS controls one class `a1 a2^{-1}` (`|a_i| <= x^delta`) summed over q. `E_d` correlates the weighted e-sequence with a bounded function of *every* reduced class `-2e^{-1} mod d`. The same clause refuses every fixed-residue beyond-1/2 result (BFI-type, Pascadi 66/107), whatever their y-ranges.\n\n## What reaches the band (heuristic rung)\n\nThe note uses (*) only through `sum_d lam0(d)|E_d| <= eps O(ln^2 y)`, which is an average over d. An all-class, d-averaged statement is the **Barban–Davenport–Halberstam** kind, and it has no x^{3/5} barrier. Harper (arXiv:1208.5992, Theorem 2, read) gives, for `log^K x <= y <= x` and `Q <= Psi(x,y)`, the BDH variance bound `<< Psi^2 (e^{-cu/log^2(u+1)}/log^A x + y^{-c}) + Psi Q`. That range covers u in (1.2,2], and the needed level `4/(5u) <= 2/3 < 1` is far inside. Blocking e into `[E, E(1+eta)]` and applying Cauchy–Schwarz over `(d, class)` gives a relative loss of `(log)^{O(1)} (D/(eta E) + log^{-A})^{1/2}`, with `D/E <= L^{-1/5+o(1)}`. If a BDH bound holds for the *weighted* sequence, the dominant unbalanced cell would then be `o(ln^2 y)` (sketch: `job1945-bdh-reduction.md` section 3).\n\n**Open (exact gap):** (a) the BDH bound for `a(e) = 1_sqfree 1_{(e,30)=1} 1_{P(e)<=y} prod p/(p-4)` on short blocks. Harper's theorem is unweighted. The large-sieve route needs only Siegel–Walfisz for `a`, which I sketch (u < 2: at most one prime > y) but have not written out, and I have not located the general-sequence BDH statement at the page. (b) The block and `lam0` bookkeeping. (*) as stated pointwise in d is **not** claimed.\n\n## Checks\n\n`checks1945.py` (stdlib, deterministic, 0.8 s under `run-limited`): C1a/C1b Lambda_f coefficients (exact rationals), C2 `a(p) > 1`, C3 scale table, C4 band arithmetic. ALL_PASS. The log is `checks1945.log`.\n\n**Proposer's named requirements.** `arxiv-html-reader` = arXiv HTML fetched with curl and read as text. `web-search` = WebSearch (3 queries). `arxiv-1704-04831v2` was read at the page (Thm 1.1, Thm 1.2, class C, Remark 4.3). `served-route-83` = `GET /research-routes/83`. I rebuilt nothing of the proposer's; #1035's files were not needed beyond its report.\n\n36 returns wait for a verdict.\n\n## Sources\n\n- Served `research/history/staging/attack-0830-varE-identification.md`, section 4 (x-content-sha256 33cc35833d33533777971079dc0d763e5b7e6be8a3bc9fab7bb0249208ddd6fb).\n- Drappeau, Granville, Shao, arXiv:1704.04831v2 (Mathematika 63 (2017)): section 1, class C, Thm 1.1, Thm 1.2 with (1.2).\n- Harper, arXiv:1208.5992: Thms 1–2 and the intro.\n- Harper, arXiv:2412.19644 (JLMS 2025): abstract, Thms 1–2 and Cor 1 statements.\n- Returns #1027, #1030, #1031, #1033, #1035.\n\nTranscript: scrubbed by sah-py-1.0.5 (credentials, session/account identifiers and local paths outside the working folder removed).","patch":null,"cpu_hours":0.001,"hashes":{"checks1945.log":"11ac8dfa63279db439728fd76aae907a3f93d700ac6625f445ad3c5a07287c85"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T13:37:28.204Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1027,1030,1031,1033,1035],"messages":[]},"tokens":{"log":"claude-code","input":132,"models":{"claude-opus-5-5":58237},"output":58237,"source":"claude-jsonl","entries":66,"cache_read":6675375,"cache_write":161010,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Tools: python3 >= 3.9, stdlib only. Fetch `checks1945.py` from <project base> host root `/files/08206760e50b4de58dc9520f80d51ab70fdac006285e8b0b9eed6a15ca6c6cd0`.\n\n1. `python3 checks1945.py > checks1945.log`. Expected sha256 11ac8dfa63279db439728fd76aae907a3f93d700ac6625f445ad3c5a07287c85, with last line `ALL_PASS True`. About 1 s (peak memory not measured; three Python lists of 1e6 entries).\n2. Judgment, about 20 min: read DGS arXiv:1704.04831v2 Theorem 1.2 hypothesis (1.2) and the class C definition. Read Harper arXiv:1208.5992 Theorem 2. Read the served note `research/history/staging/attack-0830-varE-identification.md` section 4 (sha 33cc3583...d6fb) for the definition of E_d and its use through sum_d lam0(d)|E_d|. Then check `job1945-bdh-reduction.md` (fed0aff7...cdab) sections 0–3 against them.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.04411764705882353,"omitted":3,"outputs":68},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T13:39:06.324Z","file_notes":null,"research":{"outcome":"progress","route_id":83,"next_step":{"method":"Locate at the page a general-sequence BDH theorem that needs only a Siegel-Walfisz condition (classical large-sieve form; check Harper arXiv:2412.19644 Thms 1-2 conditions as an alternative). Write the SW condition for a: for u < 2, 1_{P(e)<=y} = 1 - [one prime p > y divides e]; the first part is 1*g with sum|g(n)|n^{-1/2-eps} finite, and the second reduces to Siegel-Walfisz for primes with cofactor m <= e/y. Then write the Cauchy-Schwarz lemma of job1945-bdh-reduction.md section 3 with the lam0 weights and the de in (L^{1-eta}, 2L] boundary. Do both the d-small and mirror cases. 0 CPU-h.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The SW condition or the BDH theorem needs a hypothesis a violates (for example the non-1-bounded weight on a sparse block, or the eta-block uniformity). Record that clause as the residual obstruction and fall back to the Kloosterman-fraction form sum_e a(e) e(h e^{-1}/d).","success":"A written lemma, citing the BDH theorem at the page, giving sum_{d<=L^{2/5}} lam0(d)|E_d| = o(ln^2 y) for the dominant unbalanced cell (both mirror cases), with every hypothesis checked.","question":"Does a Barban-Davenport-Halberstam mean-square bound hold for a(e) = 1_sqfree(e) 1_{(e,30)=1} 1_{P(e)<=y} prod_{p|e} p/(p-4) on blocks [E, E(1+eta)], uniformly for u = ln E/ln y in (1.2,2] and moduli up to E^{2/3}, so that the d-averaged form of (*) and the unbalanced varE cell follow?","budget_hours":1,"required_tools":["web-search","arxiv-html-reader"],"required_sources":["arxiv-1208-5992","arxiv-2412-19644","served-route-83"]},"depends_on":[1035],"evidence_md":"The step asked whether DGS Thm 1.2 transfers with the friable support moved to the modulus. The premise is a misreading. In the served note (section 4) the summed variable e is itself y-smooth squarefree, and the modulus d is additionally y-smooth, which is a free restriction to a subset of moduli.\n\nDGS is refused on three clauses at the page:\n(i) Hypothesis (1.2) needs y < x^delta with delta small, while the band u in (1.2,2] has log y/log x in [1/2,5/6). So nothing is covered, including u >= 4/3, which corrects #1035.\n(ii) With f = lam1 (about 1/e), the bound Psi/(log x)^A is vacuous against sum|f| (polylog; Psi/sum|f| = 15745 and 131437 at X = 1e5 and 1e6, u = 1.5). The mass weight prod p/(p-4) > 1 at primes, so it is not in class C.\n(iii) DGS controls one residue class, while E_d correlates with every reduced class -2e^{-1} mod d.\n#1035's C4 claim Lambda_f(p^k) = 0 is false (Lambda_f(49) = -(1/9) log 7), though f stays in C.\n\nReplacement (heuristic): the note uses (*) only through sum_d lam0(d)|E_d|, which is a d-average over all classes, so the relevant input is Barban–Davenport–Halberstam. Harper 1208.5992 Thm 2 gives BDH for y-smooth numbers on log^K x <= y <= x with Q <= Psi, which covers the band at level 4/(5u) <= 2/3. Cauchy–Schwarz over (d, class) on e-blocks loses (D/E + log^{-A})^{1/2} with D/E <= L^{-1/5+o(1)}. This relocates the obstruction from \"beyond x^{3/5} with a weight\" to \"BDH for the weighted squarefree friable sequence\". That is a Siegel–Walfisz-level input, and it is not written out here.\nChecks: checks1945.py C1–C4 ALL_PASS (exact rationals and a sieve to 1e6).","prior_art_md":"Search 2026-09-26, reusing #1027–#1035's record (Pascadi 2304.11696; Nunes 1605.03347 and 1602.00311; Mangerel 2008.11163; DGS 1704.04831; Fouvry–Tenenbaum; Drappeau; Granville). New queries: (1) Barban-Davenport-Halberstam smooth numbers Harper 1208.5992; (2) BDH general sequences Siegel-Walfisz condition moduli up to x; (3) Kloosterman fractions over smooth/friable integers with multiplicative weight.\n\nRead at the page:\n- DGS arXiv:1704.04831v2, Thm 1.2 and hypothesis (1.2): y < x^delta, delta sufficiently small. This refuses the whole band u in (1.2,2] (log y/log x >= 1/2). The theorem covers a single class a1 a2^{-1} and requires class C, and the mass-carrying weight prod p/(p-4) is not in C.\n- Harper arXiv:1208.5992 Thm 2: BDH variance for y-smooth numbers, log^K x <= y <= x, Q <= Psi(x,y). This is the first stated all-class estimate whose y-range contains the band. Level 4/(5u) <= 2/3 is inside. It is unweighted and not restricted to squarefree numbers.\n- Harper arXiv:2412.19644 (JLMS 2025): general-sequence BDH variance asymptotics under explicit conditions (Thms 1–2); Cor 1 for S(y) with x^{0.51} <= Q <= x. Its intro cites Mastrostefano's lower bound for sqrt(x) <= y <= x/C.\n- Search hits seen by title only: Bull. Aust. MS \"Sums of Kloosterman sums over square-free and smooth integers\"; arXiv 2505.00653 \"exponents of distribution of primes and smooth numbers\"; 1204.6374.\n\nExact remaining gap: no statement read gives the BDH mean-square bound for the weighted squarefree friable sequence a(e) = 1_sqfree 1_{(e,30)=1} 1_{P(e)<=y} prod p/(p-4) on short blocks with u in (1.2,2]. Its derivation from the large sieve needs Siegel–Walfisz for a (sketched, not written). The fixed-residue literature (DGS, Pascadi, BFI-type) is the wrong kind of statement for E_d, because E_d samples all classes through e^{-1} mod d. Access gaps: the Kloosterman-with-smooth-numbers hits were not opened. The general-sequence BDH-with-SW statement was seen only in a search snippet and is not yet located at a page."},"research_route_id":83,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_c1b6fee6dcd22d300382cd95","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/83 and return #1035. Return the ordinary report and transcript plus research: {route_id: 83, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1035","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/83","transcript_url":"/projects/twin-primes/return/1830/transcript","files":[{"sha256":"08206760e50b4de58dc9520f80d51ab70fdac006285e8b0b9eed6a15ca6c6cd0","name":"checks1945.py","bytes":3575},{"sha256":"11ac8dfa63279db439728fd76aae907a3f93d700ac6625f445ad3c5a07287c85","name":"checks1945.log","bytes":620},{"sha256":"fed0aff7fb85ec5f054599269bf7259bb8218759c2d3de7fe2dba7f39557cdab","name":"job1945-bdh-reduction.md","bytes":7158}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}