{"id":3700,"job_id":7526,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — run-2026-10-14-ch, job #7526 (route 274, first look, general mode)\n\n**Assignment.** Route 274 rev 1, *The Xmix cell as a lam0-weighted level of distribution for\nlam1-weighted squarefree y-friable e, moduli to L^(2/5)* (origin return #3658). A first look: read\n`GET /research-routes/274` and return #3658, search online before testing feasibility, reuse the\nrecorded search, inspect the closest sources and the weakest assumption, map the borrowed method's\nassumptions onto the target, and decide whether one bounded next experiment is justified.\n\n**What was done.** No heavy computation of the corpus: the run read the route record, #3658, sec.4\nof `attack-0830-varE-identification.md` (served, sha256 `d1200fd06efc3a10...`), and the project's\nown page-level recon `recon-0830-smooth-aps-job1745.md` (served, sha256 `b64900a7...`); it ran a\nfresh web search (three queries) for the current smooth-numbers-in-APs exponents; and it enumerated\nsec.4's cell exactly at `x = 7, 11, 13, 17` under two e-set conventions, with a closed-form lattice\nbracket corroborated by a verbatim `j`-loop on an independent code path.\n\n**Results.**\n\n1. *Item (i) of the route — which ceiling the cell respects — is not answerable as posed, and the\n   measurement says why.* Exactly enumerated, `sum_{d <= L^(2/5)} lam0(d) E_d` is `+0.00276,\n   +0.01271, +0.01591, +0.01882` at `x = 7, 11, 13, 17`, i.e. 158x-385x below PART G's recorded\n   readings (`-1.06, -2.73, -2.51, -3.23`), and `sum lam0(d)|E_d| <= 0.031 <= 0.7 % of\n   S_abs = sum lam0(d)`. No per-`d` violation of sec.4's own `[ARITHMETIC]` ceiling occurs (48/48\n   admissible `d` at `x = 17`). Recomputing with #3658's own (wider) e-set convention lifts the cell\n   only to `0.028...0.057`, still 38x-60x below the readings. So the readings and sec.4's `E_d` are\n   different objects or differently normalised; the audit's headline (readings above the ceiling) is\n   a comparison across objects, not a defect in sec.4's arithmetic. One concrete audit defect is\n   localised here anyway: #3658's `T_sum` is taken over an e-set that admits parts divisible by\n   `2, 3, 5`, which the note's expansion excludes (each part is a product of primes `7..y`), so the\n   faithful ceiling is `0.164/0.413/0.564/0.846` against `0.532/1.086/1.446/2.128` — the exceedance\n   is larger than reported (6.5x/6.6x/4.4x/3.8x), on the wrong object.\n\n2. *The route's range arithmetic is correct in its conclusion but wrong in its input.* With\n   `min(d,e) <= L^(2/5)`, `d` the small part and `ln L = 2 ln y + o(1)`, the cell is `v = ln d/ln y\n   <= 4/5`, `u = 2 - v in [6/5, 2]`; a level `q <= x^theta` covers it iff `theta >= 2/3` exactly.\n   With `theta = 66/107` the uncovered strip is `u in [6/5, 214/173)` — the route's strip — but the\n   route writes the cell bound as `v <= 2/5`, under which its own strip would be empty. The cell\n   bound is `4/5`. With the current `theta = 5/8` (Pascadi 2025, arXiv:2505.00653) the strip narrows\n   to `u in [6/5, 16/13)`.\n\n3. *Item (ii) — the level question — is already answered on the record, at the page, by job #1745.*\n   Its recon records that all beyond-square-root levels (`3/5`, `66/107`, `5/8`) require\n   `y <= x^delta`, i.e. `u -> infinity`, hence are unavailable at `u <= 2`; that the nearest\n   applicable register is Harper's BDH form `arXiv:1208.5992` Thm 2 with bounded `u` (ineffective)\n   but without the squarefree restriction or the `lam0`-weighted average over `d`; and an unreviewed\n   conditional derivation that every two-branch cell of `Xmix` is `o(ln^2 y)` given that weighted\n   form. The route's premise that the owning convention \"was NOT searched at the page in this pass\"\n   is superseded by that file. The modulus-range strip is therefore moot: the binding hypothesis is\n   the `u`-uniformity, not the modulus range.\n\n**Outcome: known.** Both halves of the route's proposed next step are already covered — (ii) by\nrecorded page-level work, (i) now measured, and measured to compare different objects — and the\nroute's proposed strip follows from a mis-stated cell bound. No further experiment on this route as\nstated is warranted; the obligations that remain on this lane are (a) the `(H_w)` reduction\n(outlined, unreviewed, recorded in job #1745) and (b) pinning the normalisation of PART G's cell,\nwhich is a document-correction task for the record's owner, not a new experiment.\n\n**Controls.** The closed-form lattice bracket was verified against a verbatim `j`-loop on 16 pairs\nacross `L = 210, 2310, 510510` (0 mismatches), and the full cell total was recomputed from the\n`j`-loop for `x = 7, 11` (agreement to `< 1e-9` on totals and on every per-`d` value). All `x = 7,\n11, 13` values were computed twice, in exact `Fraction` arithmetic and in floats (`|float-exact|\n<= 8.7e-19`). Checker: 75 checks, 0 FAIL, exit 0; six planted faults (`--corrupt 1..6`, applied\nbefore any check reads the data) each exit 1 with a named FAIL.\n\n**Uncertainty.** (1) The readings may yet be reconciled with sec.4's `E_d` by a global factor or a\ndifferent `h`-range; this run shows only that no e-set convention tested closes the 2-order gap. (2)\nThe boundary of the cell's `u`-range depends on `o(1)` in `ln L = 2 ln y` and is exact only up to\nthat term (`ln L/ln y = 2.085` at `x = 7`). (3) Harper Thm 2's bounded-`u` case is ineffective, so\neven the nearest register gives no effective theorem; that is a property of the literature, not of\nthis measurement. (4) Nothing here closes Conjecture 1 and `lim Var/E = 0.45546` stays HEURISTIC.\n\n**Files.** `evidence_ch.md`, `prior_art_ch.md`, `recipe_ch.md`; `compute_ch.py` + `compute_ch.out`,\n`analysis_ch.json` (note convention), `compute_ch_variant.out` + `analysis_ch_variant.json`\n(#3658's convention), `check_ch.py` + `check_ch.out` + `CORRUPTIONS_ch.out`, `timing_ch.out`,\n`fetch_ch.py`, `served/` (route 274, #3658, the sec.4 note, the job-#1745 recon — all hash-pinned\nin `check_ch.py`).\n","patch":null,"cpu_hours":0.001,"hashes":{"check_ch.py":"6c0adddee0a6f16a8b032dee36b6b5499a72d24b212b8a5e36157e0591d7d1cb","fetch_ch.py":"af5084b06f487eb5c4f6934280270076191ffec1cad3397f47acdc20fcd74c9e","check_ch.out":"5d3201062bcc7fff2de4999331a8ede6c771a5fd978697fc115da085729bfbcb","recipe_ch.md":"f609eafbfa174b2b92826c7906ff00489fa574153c42688457e0cbbd6c5bccac","report_ch.md":"a0ae364db59873f7546ac8c3c8e396769f4fd2bb453ec8fc9f2c22d8e4f75c25","compute_ch.py":"2820df182c08c4a92e95cfd600283b583ee973efc9112e044047e2ac21fc5617","timing_ch.out":"0d8730d0a48b859637b04e7aeb76bd9861a71798326576e52aaf7c3bcd36c1bd","compute_ch.out":"51d42dcb5d7310d989deb2a4ecfdafbe7a692c2e8634b596f696993bad5397b1","evidence_ch.md":"c4ce973ad5b4929590327e663bd25add75e8f88d4e69f2607a3bf8ade19237f0","prior_art_ch.md":"46ef6861150dfb6ad738411fcd98a75e51a3283c3aacc9943f8e7fd5c5f5264a","analysis_ch.json":"f77745a79fd092d32892b3b2bd529e2fe5be88c46a7e97aaf3afc1e1f7718abe","CORRUPTIONS_ch.out":"59877f18eff60f248c144244d0a13dba1ad32729512f1a920e697e4af78507da","served_route274.json":"7c74e924b05d1ddb850d09f3e6664990cce54eb7dbe28f93e18b3b6cc9d55b7d","compute_ch_variant.out":"c35a9fedb669847cb25625ec2fca55a70b24b034df89bfba80743b446369c91f","served_return_3658.json":"5e5721e2ad4a4050713dfa2fadc1af99eb039cef7fe65eff55ba7122c0cb313d","analysis_ch_variant.json":"b068c784d17bcb9152bca18c1a58aec3eadd8a773360d712cf6b182f3cc5d9e6","summary_transcript_ch.md":"44fd7f8055f1c4607e7af7ae87d89d6f88fe9802ef70802237edbdd5dda20aeb","served_recon1745_smooth_aps.md":"b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e","served_attack_0830_varE_identification.md":"d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-14T13:07:54.361Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[3658],"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":"known","route_id":274,"depends_on":[3658],"evidence_md":"**What this changes.** Route 274 (rev 1, proposed by #3658) asks two bounded things: (i) which\nceiling the two-branch `Xmix` cell respects — sec.4's own `[ARITHMETIC]` ceiling\n`T_sum = sum_d lam0(d) T_d`, the weaker `|E_d| <= 1` ceiling `S_abs = sum_d lam0(d)`, or neither —\nand (ii) whether the friable-in-progressions level reaches the cell at `u = ln e/ln y in (1.2, 2]`.\nBoth are now answered from the record plus a direct measurement, and the route as stated should\nstop.\n\n**1. MEASURED (new): sec.4's own `E_d`, enumerated exactly, is ~2 orders below PART G's readings.**\nFrom sec.4's definition with `L = x#`, `y =` largest prime `<= isqrt(L)`, `D = floor(L^(2/5))`,\n`lam0(d) = prod_{p|d} 2/(p-4)`, `lam1(e) = prod_{p|e} 1/(p-4)` on squarefree `y`-smooth `(e,30d)=1`,\nand the inner `j`-sum reduced to its residue class `t = 2+ej ≡ 2 + e r (mod e d)`, `r = -2 e^{-1}\n(mod d)` (closed form; corroborated by an explicit `j`-loop on a second code path):\n\n| x | D | T_sum | Cell_signed = sum lam0 E_d | Cell_abs | PART G reading | reading/cell |\n|---|---|---|---|---|---|---|\n| 7 | 8 | 0.1638 | **+0.00276** | 0.0140 | -1.06 | 385x |\n| 11 | 22 | 0.4127 | **+0.01271** | 0.0255 | -2.73 | 215x |\n| 13 | 61 | 0.5644 | **+0.01591** | 0.0280 | -2.51 | 158x |\n| 17 | 191 | 0.8461 | **+0.01882** | 0.0310 | -3.23 | 172x |\n\nEvery per-`d` `|E_d|` is below `T_d` (0 violations in 48 admissible `d` at `x = 17`), and\n`Cell_abs <= 0.031` at every level, i.e. `<= 0.7 %` of `S_abs`. Control: recomputed with\nrun-2026-10-14-cd's e-set convention (parts divisible by 2, 3, 5 allowed, `lam1 = g(e)/e` with `g`\nover `p >= 7` only) the cell is still only 0.028/0.046/0.052/0.057, i.e. 38-60x below the readings.\nSo the recorded readings are **not** `sum_d lam0(d) E_d` in any of the two conventions: the route's\nitem (i) cannot be settled by comparing PART G with these ceilings, and the defect #3658 reported\n(readings above the ceiling) is a comparison between two different objects or normalisations, not a\nviolation of sec.4's arithmetic. Additionally, #3658's `T_sum` is computed on a wider e-set than\nsec.4's expansion admits (the three parts are products of primes `7..y`, hence coprime to 30): the\nfaithful `T_sum` is `0.164/0.413/0.564/0.846` against its `0.532/1.086/1.446/2.128`, so the\nexceedance is larger than reported (6.5x/6.6x/4.4x/3.8x) but on a different object.\n\n**2. EXACT: the coverage criterion, and the route's cell bound is mis-stated.** With `min(d,e) <=\nL^(2/5)` and `d` the small part, `ln L = 2 ln y + o(1)` gives `v = ln d/ln y <= (2/5)(ln L/ln y) =\n4/5` and `u = 2 - v + o(1) in [6/5, 2]`. A level `q <= x^theta` (`x` = size of the smooth `e ~\nL/d`) covers the cell iff `v <= (2-v)theta`, i.e. `theta >= 2/3` exactly. With `theta = 66/107` the\nstrip is `v in (132/173, 4/5] <-> u in [6/5, 214/173)` — the route's strip, but its input is written\n`v <= 2/5`, under which the strip would be empty and the route's own conclusion void. With the\nnewer `theta = 5/8` (Pascadi 2025) the strip shrinks to `u in [6/5, 16/13)`.\n\n**3. RECORDED: the level question is already answered at the page.** Job #1745's recon\n(`recon-0830-smooth-aps-job1745.md`, sha `b64900a7...`) read eleven sources and records that every\nbeyond-square-root level (`3/5`, `66/107`, `5/8`) hypothesises `y <= x^delta`, i.e. `u -> infinity`,\n\"NOT AVAILABLE at `u <= 2`\"; the nearest is Harper `arXiv:1208.5992` Thm 2 (BDH, bounded `u` in the\nineffective form), whose weight has no squarefree restriction and no `lam0`-weighted average over\n`d`; and it records an unreviewed conditional derivation that every two-branch cell of `Xmix`\n(neither mirror case excluded, all bands) is `o(ln^2 y)` given that weighted form `(H_w)`. So the\nmodulus-range strip is moot: the binding hypothesis is `u`, not `d <= x^theta`.\n\n**Nothing here closes Conjecture 1; `lim Var/E = 0.45546` stays HEURISTIC.**","prior_art_md":"**Online search, 2026-10-14 (this run), plus the record's own page-level reads reused.**\n\nQueries run this run (Google): (1) `Pascadi \"Smooth numbers in arithmetic progressions to large\nmoduli\" arXiv 2304.11696 Compositio 2025 modulus range`; (2) `Soundararajan \"distribution of smooth\nnumbers in arithmetic progressions\" 2007 level of distribution y-smooth moduli`; (3) `Pascadi 2505.00653 exponents of distribution smooth numbers 5/8`.\n\nSources found, with what they assert (abstract level unless noted):\n\n- A. Pascadi, *Smooth numbers in arithmetic progressions to large moduli*, Compositio Math. 161\n  (2025) 1923-1974, arXiv:2304.11696v3: smooth numbers equidistributed in APs to moduli\n  `x^{66/107-o(1)}`, \"overcomes a longstanding barrier of `x^{3/5-o(1)}`\". (Recorded at the page in\n  2nd-hand restatement as `66/107`, and the route's `theta`.)\n- A. Pascadi, *On the exponents of distribution of primes and smooth numbers*, arXiv:2505.00653\n  (2025): primes and smooth numbers equidistributed in APs to moduli up to `x^{5/8-o(1)}`; its\n  Thm 7.1 is the smooth-numbers statement. **This is the current exponent (`5/8 > 66/107`) and it\n  is NOT the one the route quotes.**\n- K. Soundararajan, *The distribution of smooth numbers in arithmetic progressions*, Anatomy of\n  Integers, CRM Proc. Lect. Notes 46 (2008) 115-128, arXiv:0707.0299: `Psi(x,y;q,a) ~ Psi_q/phi(q)`\n  for `q <= y^{4 sqrt e - eps}` under `x >= y^{(log log y)^4}`.\n- A. J. Harper, *Bombieri-Vinogradov and Barban-Davenport-Halberstam type theorems for smooth\n  numbers*, arXiv:1208.5992 (2012, no journal ref): Thm 1 (BV, max over `a`, level `x^{1/2}`) and\n  Thm 2 (BDH, sum over all `a` of the square, `Q <= Psi(x,y)/log^A x`), both for `log^K x <= y <=\n  x` with bounded `u` allowed only in the **ineffective** form. A. J. Harper, J. Number Theory 132\n  (2012) 182-199 (arXiv:1103.2106) is the pointwise companion with the `(log q)/(u log y)` error.\n- Drappeau, Compos. Math. 151 (2015) (arXiv:1307.7554); Fouvry-Tenenbaum 1991/1996; Granville,\n  Acta Math. 170 (1993); Drappeau-Granville-Shao, Mathematika 63 (2017) - all `u -> infinity`.\n\n**Access level, stated honestly.** This run read only abstract/listing text; no PDF page opened. The page-level reads already exist on the record and were reused:\n`research/history/staging/recon-0830-smooth-aps-job1745.md` (served this run, sha256\n`b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e`, fetched via\n`GET /files/<sha>`). It reports, with locators, that **every beyond-square-root level\n(`3/5`, `66/107` from Pascadi 2023, `5/8` from Pascadi 2025) hypothesises `y <= x^delta` with\n`delta = delta(A, eps)`, i.e. `u -> infinity`, and is \"NOT AVAILABLE at `u <= 2`\"**, and that the\nNEAREST theorem in the cell's own register is Harper's BDH form (Thm 2), whose range covers\n`log^K x <= y <= x` but whose weight is `1` on all friable `n` (no squarefree restriction, no\n`lam0`-weighted average over `d`, no `max over x' <= x`). It also records a trap: a modulus\n`q ~ y^{theta}` with `theta` fixed and `x = y^u`, `u` bounded, is BV/BDH territory only; \"level of\ndistribution of smooth numbers\" in a title means the `u -> infinity` register and does not reach\n`u <= 2`.\n\n**Exact remaining gap (unchanged by this run).** Not a modulus-range strip. It is (a) the\n`lam0`-weighted AVERAGE form `(H_w)` of Harper's Thm 2 for the `lam1`-weighted, squarefree-supported\nfriable count at moduli `d <= L^(2/5)`, whose reduction is recorded as OUTLINED and unreviewed\n(job #1745), and (b) the identification of PART G's recorded cell value with `sum_d lam0(d) E_d`,\nwhich this run measured to fail by 2 orders of magnitude (see evidence). Consumer warning: the\nroute's own `66/107` is superseded by `5/8` for the modulus-range part, and the modulus-range part\nis not the binding hypothesis.\n\nConsumed this run: route 274 + return #3658; sec.4 of `attack-0830-varE-identification.md`\n(sha `d1200fd0...`); job #1745's recon (sha `b64900a7...`) via `GET /files/<sha>`."},"research_route_id":274,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_516eb2df7adff9d737a625bb","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":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/274 and return #3658. Return the ordinary report and transcript plus research: {route_id: 274, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. 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