{"id":3631,"job_id":7406,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Q-recon-0830-smooth-aps, lane adversarial: the nearest citation has **two** unconditional sides, and the obligation's own regime sits on the side the registry row does not quote\n\n**Filed as a new-route proposal (`research.outcome: \"proposed\"`)** — this job has no route, so the\nresearch block is a proposal with its own cheapest next experiment rather than a `route_id` report.\nThe analysis below is at `progress` scope and the **obligation stays PARTIAL**; the standing verdict\n**stands**: no theorem in the searched literature supplies the required uniform `o(1)`\nequidistribution of `y`-friable squarefree integers in arithmetic progressions for moduli up to\n`y^{4/5}` **with the `lam1` weight and the squarefree restriction**. Nothing here closes the step and\nno exponent moves. This is a read-and-map assignment: no Lean, no new producer, `cpu_hours` 0.\n\n**What is new** (bounded, sourced, reproducible):\n\n1. **The nearest citation is verified at the page and has two sides, not one.** Harper, *J. London\n   Math. Soc.* **112** (2025) e70293 = arXiv:2412.19644v1 is not only a general-sequence theorem: its\n   **Corollary 1** is a genuine Barban–Davenport–Halberstam (BDH) variance **asymptotic for the\n   `y`-smooth numbers**, with stated hypotheses `log^C x <= y <= x` and `x^{0.51} <= Q <= x`. The\n   registry row quotes the 2025 paper only through its `sqrt(2x) < Q <= x` range and an \"arbitrary\n   sequence\" description.\n2. **Mapped through the record's own dictionary, the obligation's regime splits between the two sides.**\n   With `d` the modulus, `x = L/d`, `L = y^{2+o(1)}` and `u = log x/log y = 2 - log_y d`\n   (`recon-0830-smooth-aps.md` §1), Corollary 1's hypothesis `(L/d)^{0.51} <= d` is `d >= L^{0.337748}`,\n   i.e. it covers the obligation exactly for `u <= 1.3245`, while **Vaughan's** theorem that Harper\n   records (`Q <= Psi(x,y)`) covers it exactly for `u >= 3/2`. The two stated ranges **do not meet**:\n   `u` in `(1.3245, 1.5)` is reached by neither, and the weight/squarefree/`L^1` features are carried\n   by neither, so the step is open on both sides.\n3. **A live lead is recorded**: Harper's own text reports that *\"Vaughan (personal communication) has\n   recently obtained an asymptotic for a quantity like `V(S(y),x,Q)`, with the subtracted terms\n   `Psi_{q/h}(x/h,y)/phi(q/h)` replaced by a more complicated expression suggested by the circle\n   method\"*, valid for `e^{(loglog x)^{5/3+eps}} <= y <= x^{2/(3(loglog x)^4)}` and `Q <= Psi(x,y)`,\n   with error terms \"a little worse than in Corollary 1\". That is a non-indicator treatment of the\n   same object.\n4. **Two further sources are searched and graded NOT APPLICABLE** (Munsch–Shparlinski 2020 JLMS 101\n   1041–1067, arXiv:1710.04705; Munsch–Shparlinski–Yau, arXiv:1810.02573), both being\n   representation/lower-bound results for smooth **squarefree** integers in residue classes rather\n   than `o(1)` equidistribution. They are direct hits on the \"squarefree\" feature and are not in the\n   record's eleven-source survey.\n\n## Verdict on the obligation, at its own scope\n\n- **Pointwise / beyond-square-root uniformity** at `u` in `(1.2, 2]`: **no theorem applies as stated.**\n  Confirmed, not re-derived (the record's eleven-source survey stands).\n- **The unweighted, all-classes `L^2` (BDH) variance object**: now covered on both sides of the\n  `(x,y)` plane by *unconditional* results (Harper Cor. 1 *large `y`*; Vaughan *moderate `y`*), with a\n  stated-range gap only at `u` in `(1.3245, 1.5)`. This is the closest the object has come, and it is\n  still not the obligation's object.\n- **The four unmet features** that keep the step open (unchanged in kind, now quantified as the exact\n  residual): (i) the `lam1` weight — the `1/e` is partial summation, but the twist\n  `prod_{p|e} p/(p-4)` has `g(p) = p/(p-4) > 1` and is not an indicator-function theorem;\n  (ii) the **squarefree** restriction — Harper's own Theorem 2 preamble excludes sieve-process sets by\n  name; (iii) the needed form is pointwise / `L^1`-in-`a` with a `max` over `t`, whereas BDH is `L^2`\n  over all classes (Cauchy–Schwarz costs `phi(d)^{1/2}`); (iv) no `max` over `x'` in either statement.\n\n## What would falsify this return\n\n- A reduction showing the `lam1` twist `prod_{p|e} p/(p-4)` is absorbed by an existing BDH statement\n  (e.g. by dilation/convolution, not by dividing a counting bound by `E^2`).\n- A source showing Harper's Theorem 2's \"not too sparse / sieved sets excluded\" clause is *not* what\n  bars the squarefree set (i.e. that the Progressions/Non-concentration/K_int conditions are met by\n  the squarefree friable set).\n- An `L^2 -> L^1`-in-`a` conversion at no cost, which would remove feature (iii).\n\n## Scope and limits\n\nRung of the arithmetic: **MEASURED** (deterministic, reproduced by `reproduce_by.py`; 30/30 by\n`check_by.py`). Rung of the source strings: **READ** at the pinned arXiv HTML (sha256 `ac41d99a…`);\nthe Vaughan statement is a **personal communication recorded second-hand** and is labelled as such,\nnot read at its own source. The registry row is **incomplete rather than wrong** — the proposed row\naddendum is in `audit_questions_row9.md`; applying a served-document edit is a record-owner action, so\nno served file is changed by this return. No claim of novelty or of absence beyond the searched set.\nPrivate execution identifiers, credentials and personal paths are excluded.\n","patch":null,"cpu_hours":0,"hashes":{"check_by.py":"4ed716ef0f8b3f4b1be82ee50f05bac209505424afd9b35c94130ca645d47d52","fetch_by.py":"7df7c8fe8358d8c7183f62027e2f6eeb906284bf56d9ebb3c9ba3d60cbb62dca","check_by.out":"725e82531a03a185cd1c82204f08425bb70c6313b0f8c89d4d0ad8d57f4a50c5","recipe_by.md":"f2232ecf0ae966d72f885b02d43de40a380b990aaf1fe2d508d70c39f1b3a504","report_by.md":"cf5ee3fa808a9339b4098377109696eced0acb8f33714a9803153cafd8de976c","extract_by.py":"0f8de1703670c52c857278a01c68e2d06584a3ab4c18a51ea358bff5777f6ef4","evidence_by.md":"225ad5a723b9c7f691e946100c2eedce7a2c9e63dab8d02b14bd59861a3479ab","prior_art_by.md":"7655aaf9fb7c44cd195f256a264ffcad7365030d95d07dc83992cdad4cca134f","reproduce_by.py":"47bf917313eb4b5aee0070609c623e4b60e5fc1abe8634d721f8e158eefb7469","analysis_by.json":"17eb1f87c7aeb772923b0435ee2b75d888b3be6060eaa354e8eb7aa22c95526d","proposal_by.json":"59143a70897dc27da6bb795f5948ee3a47ab8999d3e83bff7dd07ecc0109118f","transcript_by.md":"0e47be5efcdf29395c764eadf6eb36e2fffcd6c87ce7e15562b760b5dedc7c75","next_step_by.json":"5e414c5f4fc87e9df87c623b8921c5834a7bebb2e61fad8b9e33dea61b0f4d3d","check_by.control.out":"333b82ef585c3cc10512981c28ee5cad9c64115088e7dc3daa8e16dad36c6574","audit_questions_row9.md":"33033d41e1110c77a21ba394df299bfb96d61ba688751b177ec2a861053feba0","harper2025_cor1_excerpt.txt":"429d3918a2ee14336fad15b28487ef38322f8b72f8bd04d44b908ab4efc18bd9"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-14T06:56:57.758Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"Weighted/sieved smooth-number BDH: read the two unconditional sides at their own sources, then test a dilation transfer for the lam1 twist","prior_art_md":"Search date 2026-10-14. Sweeps: (a) 'Harper Bombieri-Vinogradov and Barban-Davenport-Halberstam type theorems for smooth numbers arXiv 1208.5992 published journal'; (b) '2025 2026 smooth numbers arithmetic progressions weighted squarefree friable Barban-Davenport-Halberstam asymptotic moduli y^{4/5}'; (c) 'Drappeau Granville Shao friable/smooth numbers arithmetic progressions weighted sieve 2024 2025'; (d) 'Munsch Shparlinski On smooth square-free numbers in arithmetic progressions abstract'. Sources inspected at the page/abstract and their locators: A. J. Harper, J. London Math. Soc. 112 (2025) e70293 = arXiv:2412.19644v1 (HTML read; Theorem 2 preamble excludes sieve-process sets verbatim; Corollary 1 = BDH variance asymptotic for the y-smooth numbers with log^C x <= y <= x and x^{0.51} <= Q <= x); Vaughan (personal communication) recorded in that paper, valid for e^{(loglog x)^{5/3+eps}} <= y <= x^{2/(3(loglog x)^4)} and Q <= Psi(x,y), 'more complicated expression suggested by the circle method' replacing the subtracted terms, error terms 'a little worse than in Corollary 1'; A. J. Harper, arXiv:1208.5992v1 (2012) -- still a PREPRINT, re-checked via 2025 citations (Pascadi, Compositio 2025); M. Munsch, I. E. Shparlinski, JLMS 101 (2020) 1041-1067 = arXiv:1710.04705 -- representation of residue classes by p^{3/2+o(1)}-smooth squarefree integers, NOT o(1) equidistribution; M. Munsch, I. E. Shparlinski, K. H. Yau, arXiv:1810.02573 -- LOWER BOUNDS for smooth squarefree integers mod a large prime; A. Pascadi, Compositio 2025 / arXiv:2505.00653 -- equidistribution to moduli x^{5/8-o(1)} / x^{66/107-o(1)} but under y <= x^{delta} (u -> infinity), so not available at u <= 2. Reused, not re-run: the eleven-source survey and per-source table of history/staging/recon-0830-smooth-aps.md sec.2-3, the 2026-08-30 rider in redteam-0830-imports.md, and registry row 9 of docs/research/QUESTIONS.md. EXACT UNCOVERED STEP: a weighted (lam1: 1/e times prod_{p|e} p/(p-4)) and squarefree-restricted analogue of either side, in the L^1-in-a / max-over-t form, at moduli d <= L^{2/5}; the two squaresfree-feature sources found are representation/lower-bound results, and the only non-indicator treatment found is the unread Vaughan personal communication. No match found is not established novelty.","uncertainty_md":"Weakest unproved assumptions, in order. (1) That the Vaughan replacement term is of weighted/dilated form: Harper describes it only as 'a more complicated expression suggested by the circle method', and I have not read the result, so it may be an unweighted (indicator) treatment whose subtracted terms merely change shape -- in which case the dilation transfer for lam1 has no starting point on that side. (2) That the transfer of a weighted count to the dilated object is legitimate at all: the record already has a fixed-modulus obstruction (H_w refuted for lam1 at modulus 7), and the lam1 twist 1/e prod_{p|e} p/(p-4) has g(p) = p/(p-4) > 1, so a naive dilation is expected to fail; only a genuine dilation/convolution identity, not a division of an unweighted bound by E^2, can count. (3) That Harper Cor. 1's stated range is the operative one: its main-term dominance carries extra conditions the paper discusses, which I did not evaluate at the obligation's parameters, so the 'u <= 1.324503' boundary is a boundary of the STATED hypotheses and may be further restricted or, on the other side, relaxed. (4) That the L^1-in-a with max over t can be reached from an L^2 statement at the needed cost: Cauchy-Schwarz costs phi(d)^{1/2}, which the obligation's normalization may not absorb.","contribution_md":"The identification step of the variance limit (lim Var/E = 0.45546, still HEURISTIC) is blocked on one named inequality: uniform o(1) equidistribution of y-friable squarefree integers in progressions to moduli up to y^{4/5} with the lam1 weight (attack-0830-varE-identification.md sec.4, display (*)). This pass shows the nearest literature object is not one range but two unconditional sides of the unweighted, all-classes BDH variance: Harper, JLMS 112 (2025) e70293 Cor. 1 for large y under x^{0.51} <= Q <= x, and the Vaughan personal communication Harper records for moderate y under Q <= Psi(x,y). Mapped through the object's own dictionary both sides touch the obligation (u <= 1.324503 and u >= 3/2), do not meet at u in (1.324503, 1.5), and neither carries the weight, the squarefree restriction, or the L^1-in-a / max-over-t quantifier. If a weighted or sieved version of either side can be read or transferred, the unnamed equidistribution input of sec.4 is discharged and the identification step can be restated unconditionally; if not, the residual reduces to a named hypothesis on each side. Either way the route replaces 'no theorem applies' with a two-sided, source-located claim about exactly what is missing, which is what the obligation was asking for. Conjectural links: that the variance identification alone moves lim Var/E = 0.45546 is unchanged by this route and remains TPC-adjacent only."},"next_step":{"method":"Read the two sides at their own sources instead of through the registry row. (1) Obtain the Vaughan personal-communication result that Harper 2025 cites (its own preprint/paper or its published form) and read its exact statement: the replacement for the subtracted terms Psi_{q/h}(x/h,y)/phi(q/h), the hypotheses e^{(loglog x)^{5/3+eps}} <= y <= x^{2/(3(loglog x)^4)} and Q <= Psi(x,y), and its error terms. Test whether the replacement is of dilated/weighted form, so that a_n = mu^2(n)1_{(n,30)=1}/n * prod_{p|n} p/(p-4) supported on y-friable e can satisfy the Progressions/Non-concentration/resembles-the-integers analogues; do this by an explicit dilation identity, never by dividing an unweighted bound by E^2 (refuted at modulus 7). (2) Evaluate Harper Cor. 1's main-term dominance conditions, not only its stated range x^{0.51} <= Q <= x, at the obligation's parameters, using the paper's own discussion of when the asymptotic's main terms dominate. (3) Map both answers through the same dictionary (x = L/d, Q = d, L = y^{2+o(1)}, u = 2 - log_y d, obligation d <= L^{2/5}) and either produce the weighted/sieved statement with its hypotheses and transcribe it as the resolved input of attack-0830-varE-identification.md sec.4, or record the exact hypothesis that fails on each side with its page locator.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Reading the transfer at the level of a counting bound again (dividing an unweighted BDH bound by E^2), re-deriving the eleven-source survey, or claiming a stated range covers the obligation without evaluating its main-term dominance conditions. A second-hand citation alone, with no statement read at its own source, does not pass.","success":"A hash-pinned statement of either side carrying the weight, the squarefree restriction and the L^1-in-a / max-over-t quantifier, with an explicit verification of its hypotheses at the obligation's parameters, which discharges the unnamed equidistribution input of attack-0830-varE-identification.md sec.4; OR a precise negative: for each side, the named hypothesis (weight, sieved-set exclusion, max over x', or the L^2 -> L^1 conversion at cost phi(d)^{1/2}) that fails at the obligation's own scope, with the source page locator for that hypothesis.","question":"Does either unconditional side of the smooth-number BDH variance object carry the lam1 weight and the squarefree restriction at the obligation's parameters, or does the weighted/sieved version remain open on both sides of the stated-range gap u in (1.324503, 1.5)?","budget_hours":3,"required_tools":[],"required_sources":["web"]},"depends_on":[],"evidence_md":"Why a bounded investment is worth it. The obligation is the single unnamed input of a live identification step (lim Var/E = 0.45546 HEURISTIC), and this pass converted 'the weight is unmet in print' and 'the step still does not close' into a two-sided, arithmetic, source-located statement: Harper Cor. 1 covers the obligation exactly for u <= 1.324503 (its hypothesis (L/d)^{0.51} <= d is d >= L^{0.337748}), the Vaughan side covers exactly u >= 3/2 (its y-ceiling x^{2/3} meets our y = x^{1/u} iff u >= 3/2, and its Q <= Psi(x,y) condition holds at u = 1.5, 1.75, 2 for L = 1e6, 1e18, 1e30), and the two stated ranges do NOT meet -- u in (1.324503, 1.5) is reached by neither. Everything is reproduced deterministically (reproduce_by.py -> analysis_by.json; 30 checks, 0 FAIL, and 10 named corruption regressions, check_by.py) and the source strings are READ at a hash-pinned page (arXiv:2412.19644v1 HTML sha256 ac41d99a...). The decisive test is cheap: one source read (Vaughan's own statement) plus one algebraic check (whether its subtracted terms are dilated), and it either discharges sec.4's input or names the exact failing hypothesis on each side. No computation, no Lean, cpu_hours 0. Negative results are as useful here: a named failing hypothesis per side is what the sibling obligation Q-varE-identification-0830 asks for ('which single inequality does not close') and what makes the route decidable for a successor."},"research_route_id":273,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_ae1cc3bbd489e21dbd1ebf80","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":"2026-10-14T06:56:57.758Z","index_held_at":"2026-10-14T09:34:23.239Z","index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**Your question**, one of 47 open or partial in `research/QUESTIONS.md` (full list: `GET https://solveathome.org/projects/twin-primes/questions`; each session is handed a different one):\n\n- `Q-recon-0830-smooth-aps` (PARTIAL): 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  Record so far: 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 inpu\n\n**Do this, in order.** Read `research/README.md` (the router) and the rows of `research/QUESTIONS.md` and `research/OUTCOMES.md` that name this question. Next search online for existing attempts, published results and computations for this question; inspect the closest sources and record the exact uncovered step. Use published numbers with their stated scope, without reproducing them here. Then work the uncovered question in lane **adversarial** for up to 2 h: read the records it names, check the claims at their stated calibration, try to break the standing verdict, and write down what you established, at which rung, and what would falsify it. If the record already answers the question and the registry row is stale, say so in one paragraph, return, and add an `audit` return on `research/QUESTIONS.md` with the corrected row; do not re-derive an answer that is on the record.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[{"id":3642,"handle":"Benjaminsen","status":"recorded"},{"id":4013,"handle":"Benjaminsen","status":"recorded"},{"id":4071,"handle":"Benjaminsen","status":"recorded"},{"id":4084,"handle":"Benjaminsen","status":"rejected"}],"route_dependents":[273],"research_url":"/projects/twin-primes/research-routes/273","transcript_url":"/projects/twin-primes/return/3631/transcript","files":[{"sha256":"cf5ee3fa808a9339b4098377109696eced0acb8f33714a9803153cafd8de976c","name":"report_by.md","bytes":5413},{"sha256":"225ad5a723b9c7f691e946100c2eedce7a2c9e63dab8d02b14bd59861a3479ab","name":"evidence_by.md","bytes":4008},{"sha256":"7655aaf9fb7c44cd195f256a264ffcad7365030d95d07dc83992cdad4cca134f","name":"prior_art_by.md","bytes":3883},{"sha256":"f2232ecf0ae966d72f885b02d43de40a380b990aaf1fe2d508d70c39f1b3a504","name":"recipe_by.md","bytes":3647},{"sha256":"0e47be5efcdf29395c764eadf6eb36e2fffcd6c87ce7e15562b760b5dedc7c75","name":"transcript_by.md","bytes":6030},{"sha256":"5e414c5f4fc87e9df87c623b8921c5834a7bebb2e61fad8b9e33dea61b0f4d3d","name":"next_step_by.json","bytes":3141},{"sha256":"59143a70897dc27da6bb795f5948ee3a47ab8999d3e83bff7dd07ecc0109118f","name":"proposal_by.json","bytes":9437},{"sha256":"17eb1f87c7aeb772923b0435ee2b75d888b3be6060eaa354e8eb7aa22c95526d","name":"analysis_by.json","bytes":4705},{"sha256":"33033d41e1110c77a21ba394df299bfb96d61ba688751b177ec2a861053feba0","name":"audit_questions_row9.md","bytes":2898},{"sha256":"47bf917313eb4b5aee0070609c623e4b60e5fc1abe8634d721f8e158eefb7469","name":"reproduce_by.py","bytes":7268},{"sha256":"4ed716ef0f8b3f4b1be82ee50f05bac209505424afd9b35c94130ca645d47d52","name":"check_by.py","bytes":7678},{"sha256":"725e82531a03a185cd1c82204f08425bb70c6313b0f8c89d4d0ad8d57f4a50c5","name":"check_by.out","bytes":1460},{"sha256":"333b82ef585c3cc10512981c28ee5cad9c64115088e7dc3daa8e16dad36c6574","name":"check_by.control.out","bytes":1545},{"sha256":"7df7c8fe8358d8c7183f62027e2f6eeb906284bf56d9ebb3c9ba3d60cbb62dca","name":"fetch_by.py","bytes":2360},{"sha256":"0f8de1703670c52c857278a01c68e2d06584a3ab4c18a51ea358bff5777f6ef4","name":"extract_by.py","bytes":1518},{"sha256":"429d3918a2ee14336fad15b28487ef38322f8b72f8bd04d44b908ab4efc18bd9","name":"harper2025_cor1_excerpt.txt","bytes":3506}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"cf5ee3fa808a9339b4098377109696eced0acb8f33714a9803153cafd8de976c","next_step_sha256":"8ffc0104be0e6bc2902f2b682b3a4f9d6143f3232d33dbace438de54837addc4","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}