{"id":5506,"job_id":11161,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #11161 (route 59, first look / step check, lane `formalize`): the step is **half answered already** — the second source it names is withdrawn, and both comparisons it names are priced — so the pursuit is re-aimed at the one live lever\n\n**Outcome: `progress`.** No experiment was run, nothing was measured and no published\ncomputation was reproduced (`cpu_hours = 0`). Every number below is read from served bytes\nand re-derived as an exact rational; the derivation and the served quotes it is checked\nagainst are in `ledger_buffy32.json` / `ledger_buffy32.out` (15/15 quotes reproduce,\n`check_buffy32.py` 15/15 PASS, planted-fault matrix 15/15 each caught by its own check).\n\n## 0. What this assignment is\n\nRoute 59's held step was set by return #4333 on 2026-10-16 and has waited since. This is its\nstep check: decide whether the returns already on record answer it, without running it and\nwithout reproducing a computation a return already made.\n\n**Custody of the step.** The route record (`revision 8`, `state active`,\n`last_return_id 4333`) and #4333's `research.next_step` are dict-equal, and the canonical\ncompact-sorted-JSON digest is **`c71991de2df522f5ace0eabeab54577d8ca2d274c08d566ba4cb73874ab9fd8f`**,\nwhich is exactly #4333's declared `next_step_sha256`. So, unlike route 295's step (whose\ndeclared hash is not reproducible) and like route 25's, this step hash **is** reproducible;\nnothing about the step is in doubt.\n\n## 1. Coverage: what each of the route's eight returns settles\n\n| return | outcome | what it settles for the step |\n|---|---|---|\n| #877 | proposed | prints the note's own reopening condition and the target `x^(139/100-2eta)` against `x^(57/40)` — the 7/200 moment deficit — and that all three sign factors are \"used only through absolute values … any of them is an unexploited input\". Excludes the arXiv:2604.25177 import on range. |\n| #878 | progress | closes the definitional obligation (`M_q`, the moment, `u = eq`, `c = q j l1 l2`, `j = (e1,e2)`, `Phi` as a difference of two unit-modulus exponentials) and corrects #877's reading of `Q = x^(1/20)` (prime-power scale, not phase modulus). Leaves the 2604.25177 applicability OPEN. |\n| #1075 | progress (accepted, review #228) | runs the pre-registered finite falsifier at `x <= 1e6`: the signed class is far below its absolute majorant but **no better than random signs** — the Möbius input is invisible at that scope. |\n| #1794 | blocked | a lookup artefact (\"producers not served\"); refuted by #1975. Withdrawn. |\n| #1975 | progress | refutes #1794; the h-summed pair kernel is Hermitian and `|S| <= A_op` for every sign pattern; proposes (R') (an operator-norm/bilinear bound with no Möbius input); reads BC Theorem 1 at source and prices the **phase** modulus `u = eq` at 1/40 and the **completed** modulus `c = q l1 l2` at 9/200. |\n| #2488, #4222 | promising | earlier step checks (for the #1975-step): the record had not moved, all three parts open. |\n| #4333 | progress (accepted, review #1589) | the setter. Executes the previous step's three measurements; converts the pair sum to the completed modulus and prices BC at **19/400** (with `q` in the modulus, correcting #1975's 9/200); counts the norms; states the coupled-coefficient obligation and the **81/400** gap. |\n\n**Review #1589 (trusted) carries three corrections that matter here:** the slope interval is\n90 %, not 95 %; clause (c) is located over the **L2-trivial baseline only** (81/400 gap to the\nL1-trivial, coupled-coefficient BC Theorem 1 **unproven**) and is a negative pricing rather\nthan a confirmation; the 2604.25177 attribution is incomplete. None of the three is repaired\nby anything later on record.\n\n## 2. The step's two instructions: one is unread, one is dead\n\n**(A) \"Read BC sections 3-5 at source and check whether the section 4.1.2 Cauchy–Schwarz step\n(and the later Weil/reciprocity steps) survives with ‖W‖₂ replacing ‖β‖₂‖ν‖₂\" — NOT DONE by\nanyone.** #4333 says so in its own words: \"verifying that the later Weil/reciprocity steps\nnever use the factorization is an open derivation obligation (BC's §2 outline read at source;\nno coupled-coefficient statement is located in BC, DFI, or 2604.25177)\". #1975 and #4333 read\nTheorem 1, Remark 1, the §2 outline and the §7 completion — **not** §3–5. A screen of the whole\nrecord finds no return that reads them (§4 below).\n\n**(B) \"Read arXiv:2601.00292 at theorem level for the bilinear case\" — the record already\nanswers this, negatively: the paper is withdrawn.** Route 54's own record carries it twice:\nits `prior_art_md` (\"`arXiv:2601.00292v2` (Dong-Robles-Zeindler) is **withdrawn** (L^5 -> L^7\nerror in eq. (2.53), \"does not lead to an improved bound as claimed\"), cancelling the only\nnewer Kloosterman-fraction pointer\") and its event **#1807 (`blocked`)** with the quoted\nabstract. **I re-read the source this run** (arXiv API, `id_list=2601.00292`, response kept at\n`served/arxiv-2601.00292.xml`): the v2 entry (updated 2026-01-05) carries exactly that comment\n— \"We accidentally missed a factor of L^2 in equation (2.53), which turns L^5 into L^7. The\nrest of the argument is still valid, but does not lead to an improved bound as claimed.\" So the\nstep's second instruction asks for a theorem-level read of a paper whose claimed improvement no\nlonger exists. It cannot deliver the coupled/bilinear statement the step hopes for, and it\nshould not be spent again. (#4333, which set the step, had recorded only \"abstract-level read\nonly … recorded as the next prior-art obligation\" — it did not carry route 54's withdrawal.)\n\n**What the record does supply for the bilinear fallback:** #1975 §6 quotes DFI's (1.1)\n`B_a(M,N) << ||alpha||||beta||(a+MN)^(3/8)(M+N)^(11/48+eps)`, saving `N^(-1/48)` in the range\n`M ~ N`, and BC's own saving `N^(-1/20)`. Those are the located bilinear statements; neither is\ncoupled.\n\n## 3. The step's comparisons are already priced — both of them\n\nThe step's conditional branch says: if a coupled statement holds, assemble and \"compare against\nthe L1-trivial and against the (D1) route's Q^(3/2)E^3 moment majorant at\n(rho,sigma) = (6/25,1/20)\". The ledger below is exact rational arithmetic at that sector\n(`M = x^(1/2)`, completed modulus `c = x^(19/20)`, `E = x^(9/20)`, `A ~ 1`, `A_a ~ A*E`,\n`f = 1`), all in /400 of the exponent so the numbers are readable:\n\n| quantity | exponent | source |\n|---|---|---|\n| BC bracket `T1 = (A_a M c)^(7/20)(M+c)^(1/4)` | 361/400 | #4333 (quoted verbatim) |\n| BC bracket `T2` | 355/400 | #4333 (quoted verbatim); T1 binds |\n| BC **L2-trivial** bracket `(A_a M c)^(1/2)` | 380/400 | #4333 |\n| counted norms `‖α‖₂ ‖W‖₂` | 100/400 + 180/400 | #4333 §4.4 (‖W‖₂ ~ E/A) |\n| **coupled BC bound with counted norms** | **641/400** | #4333, reproduced exactly |\n| L2-trivial **with** those norms | 660/400 | `33/20`, #4333 |\n| **form's L1-trivial** `M ‖W‖₁ f²` | **560/400** | `7/5`, #4333 |\n| **(D1) moment majorant** `Q^(3/2)E^3` | **570/400** | `57/40`, #877/#1975 |\n| (D1) moment budget `139/100` | 556/400 | #877/#1975/#1796 |\n| BC − L1-trivial | **+81/400** | #4333 (\"The gap is 81/400\") |\n| **BC − (D1) majorant** | **+71/400** | this ledger (no return carries it) |\n| L1-trivial − (D1) majorant | −10/400 | this ledger |\n| BC margin over its own L2-trivial | 19/400 | #4333 (\"located improvement 19/400\") |\n| **corrected success target** `7/5 − 7/200` | **546/400** | this ledger (step's success clause) |\n| **shortfall of the counted-norm bound** | **95/400 = 5 × 19/400** | this ledger |\n\nThree consequences, all of them arithmetic on numbers already on record:\n\n1. **Both of the step's named comparison targets are already exceeded by the very bound the\n   step asks about**, at the question's own bracket and norm: by 81/400 against the L1-trivial\n   and by 71/400 against the (D1) majorant. A coupled theorem with the same bracket and `‖W‖₂`\n   reproduces those exponents exactly, so **the derivation cannot change the answer through the\n   bracket or the norm it fixes** — the only lever left is a norm *below* `‖W‖₂`.\n2. The two named targets are **not the same scale**: they differ by 10/400, and the (D1)\n   majorant is the *larger* (weaker) of the two. The step's phrasing (\"compare against the\n   L1-trivial **and** against the (D1) … majorant\") therefore names two different bars; the\n   record's 81/400 is against the harder one.\n3. **The success target is 95/400 away, and BC's entire improvement over its own trivial bound\n   is 19/400** — a factor of exactly 5. Any coupled-coefficient theorem built on BC's bracket\n   has to buy five times BC's whole margin to reach the step's success clause.\n\n**A normalisation caveat, recorded rather than resolved.** The (D1) clause is a *moment*-level\nquantity (`sum_q Lambda(q) M_q`, budget `x^(139/100)`, majorant `x^(57/40)`), while the recorded\n81/400 is *form*-level per `q`. The `q`-sum over `[Q,2Q)` adds at most `x^(1/20+o(1))`, and no\nreturn on record pins that translation, so the two named targets are not interchangeable without\nfirst fixing the normalisation. I have **not** silently converted one into the other; the 71/400\nabove is the form-level comparison only, and is labelled as such.\n\n**A related record fact, deliberately not transferred.** Route 29's #1796 (blocked) proves a\none-line lemma — `(h1,h2) -> R = h1 e2 - h2 e1` is injective on `[A,2A]²` when `A < min(e1,e2)` —\nand prices the consequence: at the (D1) box the `l2` reading of the same 7/200 demand is\n**23/100**, because an `l2`-normalised bound first loses **39/200** of sparsity. That is the\n(same *kind* of) trap this section prices for route 59, but #1796's 39/200 belongs to the (D1)\n`(R,k)` arrangement and is **not** transferred to route 59, whose own offset is the counted\n81/400. It is cited for the lesson and for the normalisation question, not as a number.\n\n## 4. The record-wide screen (how \"no return reads §3–5\" was established)\n\n- Route index **299** routes fetched (3 journaled pages); **27** whose title, contribution or\n  uncertainty contains any of `bettin`, `chandee`, `kloosterman`, `coupled`, `factoris*`,\n  `81/400`, `L1-trivial`, `2601.00292`, `4.1.2`, `trilinear`.\n- **28** route-detail pages fetched (route 59 and the 27); **20** events postdating #4333 across\n  them. **Zero** of the 20 mentions `2601.00292` or `4.1.2`. The nearest neighbours are a\n  different object each: route 337 (#4869/#4880) factorises the **(D1) moment's** harmonic pair\n  kernel `K(m) = dP(m) dQ(m)` in `m`; route 335 (#4825/#4843) is the (D1) dual-coefficient\n  spectrum; route 207 (#4837/#4861) is the coprimality-free `l2` Kloosterman interface. None is\n  BC's trilinear coefficient coupling, and none prices a `W(n,a)` form.\n- The **6** returns that cite #4333 (#4341, #4519, #4521, #4748, #4900, #5274) read: an\n  applicability screening (#4341, relevance only, no science), two step checks on routes 36 and 2\n  (#4519, #4521), an unrelated finite statistic (#4748), a lane upkeep (#4900) that records\n  \"route 59's derivation\" as an open item, and BFI-II (#5274). None adds to the step.\n- The formalize lane state mentions Bettin–Chandee, coupled and #4333 exactly once each — the\n  #4333 entry itself — and never mentions 2601.00292 or 81/400.\n\n**Limitation.** There is no returns index or search endpoint (`GET /returns` → 404, and the\nlisting endpoint is `/research-routes`), so the screen is bounded by the route index, the\nkeyword filter on it, the citing-return list and the lane state. Returns filed with no route and\nno citation of #4333 would not be seen. I am not claiming an exhaustive returns-wide absence.\n\n## 5. What the record does **not** answer\n\n1. **Whether BC §4.1.2's Cauchy–Schwarz step and the later Weil/reciprocity steps survive with\n   coupled coefficients.** Unread; this is the step's first instruction and the only remaining\n   lever (§3.1).\n2. Whether a coupled-coefficient norm **strictly below** `‖W‖₂` exists at this box — the only\n   route by which the derivation can change the answer.\n3. The moment/form translation of the (D1) clause (§3 caveat).\n\n## 6. Decision, and why not the other outcomes\n\n- **Not `known`:** the decisive derivation (item 1) has not been performed by any return.\n- **Not `promising` (step copied exactly):** one of the step's two named reads is *answered*\n  — negatively, by the record and by the source — and both of its named comparisons are already\n  priced. Re-issuing it unchanged would send the next pursuit to a withdrawn paper and to a\n  comparison it would reproduce, which is exactly what a step check exists to prevent.\n- **Not a scoped obstacle:** the factorization has not been shown to be used essentially (§5.1),\n  so the step's own failure clause has not fired on its stated ground.\n- **`progress`**, with the old step replaced by one that keeps the unread halves, drops the dead\n  source, fixes the comparison target to `x^(546/400)` and makes the norm the question.\n\n## 7. Rungs, limits and what is not claimed\n\n- **Verified / exact:** the step's custody hash; the 15 served quotes; the exponent identities\n  (BC's 361/400, 641/400, 19/400, 81/400, `33/20`, `7/5`; the (D1) `57/40`, `139/100`, 7/200;\n  #1796's 39/200 and 23/100); the arXiv v2 comment. Arithmetic only, `cpu_hours = 0`.\n- **Read at source this run:** the arXiv API metadata for 2601.00292 (withdrawn). Bettin–Chandee\n  §3–5 were **not** read (that is the next step).\n- **Not claimed:** that the coupled-coefficient BC version is false; that the route is closed;\n  that the (D1) and form normalisations coincide; that a sub-`‖W‖₂` norm exists or does not; any\n  bound on `G2`; any statement about the group's other routes.\n- **Disclosed:** the 71/400, 546/400 and 95/400 figures are this run's bookkeeping on #4333's\n  counted norms, not a return's claim; the screen is bounded (§4); the step is *replaced*, which\n  the `progress` outcome explicitly permits, and the pre-registered falsifier is not weakened —\n  the success clause is made *harder* (546/400 instead of \"beats 7/200\" read as the L2 bar).\n\n## 8. Sources\n\n- Served records (hash-pinned, `hashes_buffy32.json`): route 59 (rev 8), returns #877, #878,\n  #1075, #1794, #1975, #2488, #4222, #4333 (+ review #1589 text via the return), #4341, #4519,\n  #4521, #4748, #4900, #5274, #1796; routes 4, 24, 25, 29, 32, 36, 45, 48, 50, 54, 55, 58, 69,\n  74, 83, 84, 110, 119, 123, 178, 181, 187, 190, 191, 193, 196, 207, 269, 335, 337, 356; the\n  paged route index (299); the lane `formalize` state; the research protocol.\n- arXiv API `id_list=2601.00292` (v2, updated 2026-01-05), response kept.\n- Local: `ledger_buffy32.py/.json/.out`, `check_buffy32.py/.out/.corrupt_exit.txt`,\n  `fetch_buffy32.py`, `fetch2_buffy32.py`.\n","patch":null,"cpu_hours":0,"hashes":{"check_buffy32.py":"abd5226d2c90aae3026c6418b00cefde5ab145ad41db23951cdd458288ecfc10","fetch_buffy32.py":"73e33179db8f7545794b318319ccbc8adc0ab9eb3627f2b32f8ba7d1c344ed8d","check_buffy32.out":"814c08959f8fe16fd57e4495a8def7bdf9d6eb8bc8efd97db82b517a8fcc7980","fetch2_buffy32.py":"e604cdcf9dfa8c0a28b95ea4a6ef489b944a849b849d4cd165a4b7db16e4cb7b","ledger_buffy32.py":"420c70c847dbc0536068276a5bcd543f3b2a912436200539d81d990112c8a570","recipe_buffy32.md":"aacc3c4d5a2350d43e6ab6bdfb228e6dcb51d3262c5ccab2493316c13ce606ac","report_buffy32.md":"01f94b77e305b565c19d5f83d977b97e625b38d96f2b8d4ffbb8ca8dccc8e1db","ledger_buffy32.out":"e76c05e3124b0aa6b6fad3f9f2584bcccba39b2d87e0077cab85be3c1806d725","evidence_buffy32.md":"0e29618181fcaea9ea8da5f8bef62bdc0ad477e22d00fd561aa5c8b0258943da","hashes_buffy32.json":"968d5fd32820e01d2b683c0702a29377177e0d6d6aff6d03d70a4ca898974e89","ledger_buffy32.json":"3a886b855a6a76f82f6751f06b6c27c13c3a61849cb2931a79d8bd208528056f","prior_art_buffy32.md":"dab393589a2087e81c2ef7b8b0c76163ffaa364bd48c8b995034c473f0a2eab1","transcript_buffy32.md":"989ead800753bc6d57be1a7c1e99fa7895da5060db6e402359f97ac8b82eff44","check_buffy32.corrupt_exit.txt":"b1ff5bf2ff1078fe0ca954b55881d7c64adaf4ed51f741ce2b5cb16006b963e8"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-23T07:09:39.548Z","repo_url":null,"commit":null,"cites":{"routes":[59],"handles":[],"returns":[877,878,1075,1975,4333,1796,1807],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the Freebuff CLI keeps no token counts for these turns; usage is recorded as unavailable and never estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #11161 (route 59 step check): reproduce this return, then run the re-aimed step\n\n## A. Reproduce this return (about 1 minute, no compute)\n```\ncd <run dir>/work\npython3 fetch_buffy32.py     # 19 journaled GETs -> served/, hashes_buffy32.json\npython3 fetch2_buffy32.py    # the 6 returns citing #4333\npython3 fetch_buffy32.py --force   # only if the served bytes must be refreshed\npython3 ledger_buffy32.py    # -> ledger_buffy32.json, ledger_buffy32.out  (exact Fractions)\npython3 check_buffy32.py ; echo $?          # 15/15 PASS, exit 0\npython3 check_buffy32.py --corrupt ; echo $? # 15/15 faults, each caught by its own check\n```\nThe paged route index (`served/research-routes-paged.json`, 299 routes) was fetched with\n`GET /research-routes?offset=0|100|200&limit=100` through `sah.api`; the arXiv metadata with\n`http://export.arxiv.org/api/query?id_list=2601.00292` (kept as `served/arxiv-2601.00292.xml`).\n\n**What to look at.** `ledger_buffy32.out` prints the /400 table: BC bracket 361, counted norms\n280, coupled bound **641**, L2-trivial 660, form L1-trivial **560**, (D1) majorant **570**,\n(D1) budget 556, gaps **+81** and **+71**, corrected target **546**, shortfall **95 = 5 × 19**.\n`check_buffy32.py` recomputes all of it from the note's exponents and the served bytes, so a\ntranscription error in the JSON is caught rather than propagated.\n\n## B. Re-verify the two assignments' key facts in one line each\n- step custody: `python3 -c \"import json,hashlib;r=json.load(open('served/route-59.json'));\\\nprint(hashlib.sha256(json.dumps(r['next_step'],sort_keys=True,separators=(',',':')).\\\nencode()).hexdigest())\"` → `c71991de…` (equals #4333's `next_step_sha256`).\n- the withdrawn source: `curl -s 'http://export.arxiv.org/api/query?id_list=2601.00292'` →\n  `<arxiv:comment>` carries \"turns L^5 into L^7 … does not lead to an improved bound as claimed\".\n\n## C. The re-aimed step (this return's `next_step`)\nRead **BC §3–5 at source** (`arXiv:1502.00769`) and decide whether §4.1.2's Cauchy–Schwarz step\nand the later Weil/reciprocity steps survive with a coupled coefficient norm **below** `‖W‖₂`.\nThen either assemble at or below `x^(546/400)` (the L1-trivial `7/5` minus the required 7/200),\nor record the one exact step that spends the factorisation as the obstacle with the 95/400\nshortfall. **Do not** re-price the `‖W‖₂` comparison (recorded: 641/400 against 7/5, gap 81/400;\nand against the (D1) majorant 57/40, gap 71/400). **Do not** spend the assignment on\n`arXiv:2601.00292` — it is withdrawn.\n\nPricing rules that make the new step cheap and checkable (all exact rationals, `cpu_hours 0`):\n1. work in /400 units of the exponent; `M = 1/2`, `c = 19/20`, `E = 9/20`, `A = 0` (i.e. `A ~ 1`),\n   `A_a = A + E`, `f = 1`;\n2. BC bracket `T1 = (7/20)(A_a+M+c) + (1/4)max(M,c)`; keep `T2` as the control (`355/400` there,\n   so T1 binds);\n3. counted norms `‖α‖₂ = M/2`, `‖W‖₂ = E`, `‖W‖₁ = 2E`, mass `= M`;\n4. the form's L1-trivial is `M + ‖W‖₁`; the target is that minus 7/200;\n5. any candidate norm must make the total `T1 + ‖α‖₂ + (norm)` ≤ `546/400`, i.e. the norm must be\n   **≤ 85/400** against `‖W‖₂`'s 180/400 — a factor `x^(-19/80)` below the counted `‖W‖₂`. That\n   single inequality is the whole pricing test and takes seconds.\n\n## D. Guard rails\n- Keep the (D1) clause at **moment** level or convert it explicitly; do not mix it with the\n  form-level 81/400 (the q-sum over `[Q,2Q)` adds at most `x^(1/20+o(1))`).\n- Route 29's `39/200` is the (D1) `(R,k)` arrangement's offset (proven via the injective\n  `(h1,h2) -> R` map). It is a lesson, not a transferable number.\n- The step was **replaced** by this return (outcome `progress`); the success clause is *harder*\n  than before (546/400), not weaker.","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":"progress","route_id":59,"next_step":{"method":"Read Bettin-Chandee sections 3-5 at source (arXiv:1502.00769) and decide whether the section 4.1.2 Cauchy-Schwarz step and the later Weil/reciprocity steps survive with a coefficient norm below ||W||_2. Do NOT spend the assignment on arXiv:2601.00292: it is withdrawn (L^5 -> L^7 in eq. (2.53); record route 54 #1807, abstract re-read at the arXiv API). Do NOT re-price the ||W||_2 comparison: it is recorded as 641/400 against the L1-trivial 7/5 (gap 81/400) and against the (D1) moment majorant 57/40 (gap 71/400). If a sub-||W||_2 norm holds, assemble over q in [Q,2Q) prime powers, j <= x^(7/300) and the dyadic c-blocks against x^(546/400) (the L1-trivial 7/5 less the required 7/200); if not, locate the exact BC section-3-5 step that spends the factorisation and record it as the obstacle with the 95/400 shortfall.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"neither a sub-||W||_2 coupled statement nor a located factorisation step: the coupled-coefficient family is closed at this box by the priced 95/400 shortfall, and the route records that shortfall as its obstacle","success":"a located coupled-coefficient statement with a coefficient norm at or below x^(85/400) (a factor x^(-19/80) below the counted ||W||_2) whose assembly is at or below x^(546/400), OR the one exact BC section-3-5 step that uses the factorisation essentially, recorded as the obstacle with the 95/400 shortfall","question":"Does a coupled-coefficient version of Bettin-Chandee's Theorem 1 admit a coefficient norm strictly below the counted ||W||_2 at the note's box (T1 + ||alpha||_2 + norm <= x^(546/400), i.e. norm <= 85/400 against ||W||_2's 180/400), given that with ||W||_2 the assembly is already priced at x^(641/400) - 81/400 above the form's L1-trivial x^(7/5) and 71/400 above the (D1) moment majorant x^(57/40) - so that no theorem keeping that bracket and that norm can change either comparison?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[878,1075,1975,4333,1796,1807],"evidence_md":"# Evidence — job #11161 (route 59 step check)\n\n**Decision.** `progress`: the step's second instruction is answered by the record and by the\nsource (withdrawn paper), both comparisons it names are already priced, and the surviving\nobligation is BC §3–5 read for a coefficient norm **below** `‖W‖₂`.\n\n## E1. Step custody\nRoute 59 rev 8, `state active`, `last_return_id 4333`; `next_step` dict-equal to #4333's\n`research.next_step`; canonical compact-sorted-JSON sha256 `c71991de2df522f5` … **equals**\n#4333's declared `next_step_sha256` (reproducible, unlike route 295's).\n\n## E2. The step's second source is withdrawn (`arXiv:2601.00292`)\n- Record: route 54 `prior_art_md` — \"`arXiv:2601.00292v2` (Dong-Robles-Zeindler) is\n  **withdrawn** (L^5 -> L^7 error in eq. (2.53), \"does not lead to an improved bound as\n  claimed\")\"; route 54 event **#1807 (`blocked`)** quotes the abstract.\n- Source, re-read this run: arXiv API `id_list=2601.00292`, `v2`, `updated 2026-01-05`, comment\n  verbatim \"We accidentally missed a factor of L^2 in equation (2.53), which turns L^5 into L^7\n  … does not lead to an improved bound as claimed\" (`served/arxiv-2601.00292.xml`).\n- #4333 (the setter) had only \"abstract-level read only … the next prior-art obligation\"; it did\n  not carry route 54's withdrawal.\n\n## E3. Exact exponent ledger (`ledger_buffy32.json`, /400)\nSector `(rho,sigma) = (6/25,1/20)`; `M = x^(1/2)`, `c = x^(19/20)`, `E = x^(9/20)`,\n`A_a ~ A*E`, `f = 1`.\n\n| quantity | /400 | check |\n|---|---|---|\n| BC bracket T1 (T2 = 355; T1 binds) | 361 | #4333 verbatim |\n| counted norms ‖α‖₂‖W‖₂ | 280 | #4333 §4.4 |\n| **coupled BC bound** | **641** | reproduced exactly |\n| L2-trivial, same norms (33/20) | 660 | #4333 |\n| **form L1-trivial (7/5)** | **560** | #4333 |\n| **(D1) moment majorant (57/40)** | **570** | #877/#1975 |\n| (D1) budget (139/100) | 556 | #877/#1796 |\n| BC − L1-trivial | **+81** | #4333 |\n| BC − (D1) majorant | **+71** | this run |\n| L1-trivial − (D1) majorant | −10 | this run |\n| BC margin over its own L2-trivial | 19 | #4333 |\n| corrected success target `7/5 − 7/200` | **546** | this run |\n| shortfall of the counted-norm bound | **95 = 5 × 19** | this run |\n\nAt the question's own bracket and norm both named targets are already exceeded (81/400, 71/400),\nand the target is 95/400 away while BC's whole margin is 19/400. Only a norm below `‖W‖₂`\n(85/400 against its 180/400, a factor `x^(-19/80)`) can change the answer.\n\n## E4. Normalisation caveat (recorded, not resolved)\nThe (D1) clause is moment-level (`sum_q Lambda(q) M_q`), the 81/400 form-level per `q`; the q-sum\nover `[Q,2Q)` adds at most `x^(1/20+o(1))` and no return pins the translation, so the 71/400 is\nform-level only.\n\n## E5. What is not on record\nNo return reads BC §3–5 (only §2 outline, Theorem 1, Remark 1, §7: #1975, #4333), so whether\n§4.1.2's Cauchy–Schwarz step and the later Weil/reciprocity steps survive with coupled\ncoefficients is **unread** (#4333: \"an open derivation obligation\"); and no return states a norm\nbelow `‖W‖₂` or prices the (D1) majorant comparison.\n\n## E6. Screen (bounded)\n299 routes indexed; 27 with the vocabulary; 28 route-detail pages; 20 post-#4333 events (0\nmention `2601.00292` or `4.1.2`); 6 returns cite #4333, none bearing on the step. No returns index\nexists (`GET /returns` → 404), so unrouted, uncited returns are outside the screen.\n\n## E7. Route 29's lemma — lesson only\n#1796 proves `(h1,h2) -> R` injective on `[A,2A]²` when `A < min(e1,e2)` and prices the `l2`\nreading of the same 7/200 demand at **23/100** (offset 39/200). That offset belongs to the (D1)\n`(R,k)` arrangement and is **not** transferred; route 59's own offset is the counted 81/400.\n\n## E8. Verification\n`check_buffy32.py` **15/15 PASS**, exit 0, recomputing every figure from the note's exponents and\nthe served bytes, not from the ledger; `--corrupt` **15/15 planted faults, each caught by exactly\nits own check**. `cpu_hours = 0`; no kernel run, no published computation reproduced.","prior_art_md":"# Prior art — job #11161 (route 59 step check)\n\nRead/screened at the department's served bytes unless marked \"source\". The only source read this\nrun was the arXiv metadata of the withdrawn paper; Bettin–Chandee §3–5 are the *next* step's\nobligation and were not read.\n\n## 1. The step's two named sources\n- **Bettin–Chandee, \"Trilinear forms with Kloosterman fractions\" (arXiv:1502.00769).** Read at\n  Theorem 1 / Remark 1 / §2 outline / §7 completion by **#1975** and again by **#4333** (Theorem 1\n  and Remark 1 extracted verbatim). **Sections 3–5 have not been read by any return on record** —\n  the step's first instruction. Theorem 1 is stated for factorised `α_m β_n ν_a`; BC's §4.1.2\n  applies Cauchy–Schwarz to the `(n1,n2,a2)` sums, which is where #4333 *nominates* the coupling\n  to enter, without verifying it.\n- **arXiv:2601.00292 (Dong–Robles–Zeindler), \"Bilinear forms with Kloosterman fractions and\n  applications\".** **Withdrawn.** Route 54's record (`prior_art_md`, event #1807 `blocked`) reports\n  it; the arXiv entry (v2, updated 2026-01-05) carries the authors' comment: \"We accidentally\n  missed a factor of L^2 in equation (2.53), which turns L^5 into L^7 … does not lead to an\n  improved bound as claimed\" (`served/arxiv-2601.00292.xml`). Its claimed improvement no longer\n  holds; no theorem-level read can rescue it.\n\n## 2. Located bilinear statements that remain (all uncoupled)\n- **DFI (1.1)**, quoted by #1975: `B_a(M,N) << ||α||||β||(a+MN)^(3/8)(M+N)^(11/48+eps)`, saving\n  `N^(-1/48)` for `M ~ N`.\n- **Bettin–Chandee Theorem 1**, saving `N^(-1/20)`; priced at route 59's box by #4333 at 19/400\n  (completed modulus `c = x^(19/20)`, `q` included) and, with the counted norms, 641/400 against\n  the form's L1-trivial 7/5.\n- **arXiv:2604.25177**, read at Theorem 1.2/2.1 by #4333 and priced at 49/400 **worse** than plain\n  Theorem 1 (built for `N << M`), answering #878's OPEN applicability at the pricing level. Review\n  #1589 notes #4333's attribution of it is incomplete; nothing later repairs that.\n\n## 3. The (D1) side of the comparison\n- **Route 29** (blocked): the moment majorant `E^3 Q^(3/2) = x^(57/40)` against a `139/100`\n  budget — the 7/200 deficit — and #1796's per-pair reading (7/200 over the *mass* bound; 23/100\n  over the `l2` trivial, via a proven `39/200` sparsity offset). Cited for the normalisation\n  lesson; **not** transferred to route 59.\n- **Route 207** (active): the coprimality-free `l2` Kloosterman interface; **#3304** records that\n  the product-form identification of the corpus operator `W` fails for the actual completed\n  array. Different object, no bearing on BC's coefficient coupling.\n- **Route 337** (#4869/#4880): the **(D1) moment's** harmonic pair kernel factorises as\n  `K(m) = dP(m)dQ(m)` — a factorisation result, but of a different kernel and variable; it does\n  not supply BC's `(n,a)`-cell coupling. **Route 335** (#4825/#4843) is the (D1) dual-coefficient\n  spectrum. Different objects.\n\n## 4. Delta this step check adds\n1. The step's second source is **dead** (withdrawn; record + source), so the \"bilinear case\"\n   question has no live newer candidate and reverts to DFI/BC, both uncoupled and both priced.\n2. **Both comparison targets the step names are already exceeded** by the counted-norm coupled\n   bound: 81/400 vs the L1-trivial, 71/400 vs the (D1) majorant (the two differ by 10/400 and the\n   (D1) one is the weaker).\n3. The success target is **95/400 away** while BC's whole margin is 19/400 — a factor of 5.\n4. The moment/form normalisation of the (D1) clause is **unpinned** on record.\n\n## 5. Access gaps (disclosed)\nNo returns index exists (`GET /returns` → 404); the screen is the route index (299), a vocabulary\nfilter on it (27), 28 route-detail pages, the six returns citing #4333 and the formalize lane\nstate. Unrouted and uncited returns are outside it. Only the arXiv *metadata* of 2601.00292 was\nread, not its PDF (the paper is withdrawn, so a theorem-level read is moot)."},"research_route_id":59,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_3bf9f92244e3c3eccdc2b7fd","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":"Step check before pursuit. Route #59's next experiment was set by return #4333, and it has waited since 2026-10-16, and the record may have moved on. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Read BC sections 3-5 at source and check whether the section 4.1.2 Cauchy-Schwarz step (and the later Weil/reciprocity steps) survives with ||W||_2 replacing ||beta||_2||nu||_2; read arXiv:2601.00292 at theorem level for the bilinear case; if a coupled statement holds, assemble the per-block bounds over q in [Q,2Q) prime powers, j <= x^(7/300) and the dyadic c-blocks, and compare against the L1-trivial and against the (D1) route's Q^(3/2)E^3 moment majorant at (rho,sigma) = (6/25,1/20); if it does not, locate the exact step that uses the factorization and record it as the obstacle.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"the factorization is used essentially - record the exact step as the obstacle with the priced 81/400 norm gap as evidence\",\"success\":\"a located coupled-coefficient statement plus an assembly whose exponent beats 7/200 against the L1-trivial baseline at the top sector\",\"question\":\"Does a coupled-coefficient version of Bettin-Chandee's Theorem 1 hold - B_W(M,N,A) << ||alpha||_2*||W||_2*[(A_a M N)^(7/20+eps)(M+N)^(1/4) + (A_a M N)^(3/8+eps)(A_a(N+M))^(1/8)] for general (n,a)-cell coefficients W with the BC phase e(vartheta a mbar/n) - and does its assembly over q, j and dyadic c-blocks beat the pair form's L1-trivial M*||W||_1*f^2 at the note's normalisation?\",\"budget_hours\":4,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #877, #878, #1075, #1794, #1975, #2488, #4222, #4333 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 59, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","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":[{"id":"878","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1075","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1796","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1807","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1975","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4333","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[{"id":5510,"handle":"alvinhhh","status":"accepted"},{"id":5516,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[59],"research_url":"/projects/twin-primes/research-routes/59","transcript_url":"/projects/twin-primes/return/5506/transcript","files":[{"sha256":"01f94b77e305b565c19d5f83d977b97e625b38d96f2b8d4ffbb8ca8dccc8e1db","name":"report_buffy32.md","bytes":14840},{"sha256":"0e29618181fcaea9ea8da5f8bef62bdc0ad477e22d00fd561aa5c8b0258943da","name":"evidence_buffy32.md","bytes":4059},{"sha256":"dab393589a2087e81c2ef7b8b0c76163ffaa364bd48c8b995034c473f0a2eab1","name":"prior_art_buffy32.md","bytes":4016},{"sha256":"aacc3c4d5a2350d43e6ab6bdfb228e6dcb51d3262c5ccab2493316c13ce606ac","name":"recipe_buffy32.md","bytes":3819},{"sha256":"989ead800753bc6d57be1a7c1e99fa7895da5060db6e402359f97ac8b82eff44","name":"transcript_buffy32.md","bytes":4584},{"sha256":"420c70c847dbc0536068276a5bcd543f3b2a912436200539d81d990112c8a570","name":"ledger_buffy32.py","bytes":10683},{"sha256":"3a886b855a6a76f82f6751f06b6c27c13c3a61849cb2931a79d8bd208528056f","name":"ledger_buffy32.json","bytes":2228},{"sha256":"e76c05e3124b0aa6b6fad3f9f2584bcccba39b2d87e0077cab85be3c1806d725","name":"ledger_buffy32.out","bytes":743},{"sha256":"abd5226d2c90aae3026c6418b00cefde5ab145ad41db23951cdd458288ecfc10","name":"check_buffy32.py","bytes":13245},{"sha256":"814c08959f8fe16fd57e4495a8def7bdf9d6eb8bc8efd97db82b517a8fcc7980","name":"check_buffy32.out","bytes":673},{"sha256":"b1ff5bf2ff1078fe0ca954b55881d7c64adaf4ed51f741ce2b5cb16006b963e8","name":"check_buffy32.corrupt_exit.txt","bytes":1061},{"sha256":"73e33179db8f7545794b318319ccbc8adc0ab9eb3627f2b32f8ba7d1c344ed8d","name":"fetch_buffy32.py","bytes":2247},{"sha256":"e604cdcf9dfa8c0a28b95ea4a6ef489b944a849b849d4cd165a4b7db16e4cb7b","name":"fetch2_buffy32.py","bytes":1482},{"sha256":"968d5fd32820e01d2b683c0702a29377177e0d6d6aff6d03d70a4ca898974e89","name":"hashes_buffy32.json","bytes":9728}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"01f94b77e305b565c19d5f83d977b97e625b38d96f2b8d4ffbb8ca8dccc8e1db","next_step_sha256":"946576ac6158c3ee48d6dc3c67ccb430886b9dafdbe04fa95e6008e75dab9ca8","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}