{"id":2458,"job_id":5213,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 207 first look — the coprimality-free l2 interface exists, but it does not close the whole window band\n\nJob **#5213** (explore, stage `first_look`, route **207** rev 1, lane dir-558, general mode).\nModel `deepseek/deepseek-v4-flash`, effort `unmeasured`, general mode. Outcome: **promising** (with a\nscoped residual).\n\n## 1. What was asked\n\nRoute 207 asks whether the coprimality-free l2 bilinear/large-sieve Kloosterman interface of the newer\nliterature reaches the structured-dispersion **dual-window band** at the corpus's dictionary and delivers\nthe required **7/400** saving over the trivial operator bound. The lead came from return **#2455** and\nsharpens accepted return **#1973**: #1973 showed the recorded window arithmetic of returns **#762-#765**\nis conditional because Pascadi's `arXiv:2511.08445v2` Cor. 8.1 needs **bounded** coefficients over\n`(t,c)=1` and Thm 7.1/7.8(i) needs joint `(t,r,c)=1`, while the completion identity supplies only\n`(m,c)=1`. The open question is whether some statement supplies the same saving with the **l2** norm and\n**without** the dual coprimality.\n\nThis first look does the smallest decisive experiment on that uncovered step: a bounded source read of the\ncandidate statements plus **exact-rational pricing** of each at the corpus's own dictionary (0 CPU-h).\n\n## 2. Dictionary (the corpus's own constants, returns #762-#765 / §2)\n\n| symbol | value | source |\n|---|---|---|\n| shift range `K = A_h E` | `x^{51/100}` | #762 Finding 2 |\n| completed modulus `c` | `x^{19/20}` (`c^{1/2} = x^{19/40}`) | #762 (`c^{1/2}=x^{0.475}`) |\n| block length `M` (so `c/M`) | `x^{14/25}` | #764 (`[c/M,c]` is `14/25` wide) |\n| requirement zone | `T ∈ [c/M, c] = [x^{39/100}, x^{19/20}]` | #762/#764 |\n| required saving | exactly `7/400` over the trivial bound | §8 reopening condition |\n| recorded deficit band | `[x^{39/100}, x^{0.5043})`, `T* = x^{0.504331}` | #762 |\n| corpus identity | `7/400 = (A_h E − c^{1/2})/2` **exact** | #763 |\n\nThe object is `W = ||K b||_2`, `||b||_2 = 1`, `K = (S(r,t;c))_{r∈[K,2K],t∈[T,2T]}`, so its trivial\n(operator/Hilbert-Schmidt) bound is `sqrt(KcT)` and the demand is\n`W ≤ sqrt(KcT)·x^{−7/400}` **uniformly** in `T` (#764: the requirement is uniform; the worst octave decides).\n\n## 3. Interface checklist (pre-registered, then applied)\n\nFor each candidate: (A) coefficient norm type, (B) the exact coprimality condition on the summed\nvariables, (C) admissible index sets (dyadic intervals?), (D) range conditions and priced saving.\n\n| statement | (A) norm | (B) coprimality | (C) index sets | verdict at the band |\n|---|---|---|---|---|\n| Pascadi Cor. 8.1 | `l∞` (bounded) | `(t,c)=1` outer, `r | q` | — | fails (A) and (B) — #1973 |\n| Pascadi Thm 7.1 / 7.8(i) | `l2` | **joint** `(t,r,c)=1` | — | fails (B) — #1973 |\n| **Milićević–Qin–Wu Thm 1.1** | **`l2`** | **none on the summed variables** (only `gcd(a,q)=1`) | support `[1,M]`,`[1,N]` — dyadic fits | **passes (A),(B),(C)**; ranges hold on the whole band |\n| Milićević–Qin–Wu Thm 3.1 | `l2` (`||λ||_2`) | none (`(m,q)=1` removed) | — | **not priced here** (statement/range not obtained) |\n| **Blomer–Pascadi Thm 1.1** | **`l2`** | `(m,n,c)=1`, **dropped only for `I=J={1,…,N}`** | intervals `|I|,|J| ≤ N` | `l2` ✓, but the coprimality-free clause needs **full** intervals, not the corpus's dyadic windows |\n| Blomer–Pascadi Thm 1.6 | `l2` large sieve | removes `(n,q)=1` | `N ≍ sqrt(q)` | different (spectral) object — not the operator window |\n\nSources read at the primary text (2026-10-07): `arXiv:2511.07550` HTML, `arXiv:2607.24311` HTML/abs,\n`arXiv:2511.08445` (via #1973's served `source-interface.patch`).\n\nSo #1973's qualification **is** sharpened: an `l2`, dual-coprimality-free bilinear Kloosterman estimate\nexists (MQW Thm 1.1, verbatim: \"for any integer `c` coprime with `q`\", no restriction on the summed\nvariables; and, on the `(m,c)=1` reading, BP Thm 1.1). It is not an intrinsic limit of the `l2` operator\nestimate.\n\n## 4. Pricing result (exact rationals; `compute_cn.py`, checked by `check_cn.py` 27/27)\n\nWith `M,N` the two lengths of the bilinear form, `q = c`, and the corpus demand\n`sqrt(KcT)·x^{−7/400}` (trivial exponent `(K+c+T)/2`):\n\n| interface | least `T` meeting the demand | note |\n|---|---|---|\n| MQW Thm 1.1, `(M,N)=(K,T)` | `x^{1463/3000} = x^{0.487667}` | worse than trivial below `x^{161/375}` |\n| MQW Thm 1.1, `(M,N)=(T,K)` | `x^{109/240} = x^{0.454167}` | ranges hold throughout the band |\n| BP Thm 1.1, `N = K` | `x^{7/16} = x^{0.4375}` | bound `x^{149/160}`, saving `43/800`; **conditional** on (B) |\n| BP Thm 1.1, `N = T` (forced for `T>K`) | `x^{19/24}` | but BP is non-trivial only for `N < c^{7/12} = x^{0.554167}`, so moot |\n| BP Thm 1.1 critical length `N = c^{1/2}` | saving `c^{−1/32} = 19/640 = x^{0.029687}` | exceeds `7/400` at the square-root length |\n\n**Consequences.**\n\n1. The band does **not** close. Residual: `[x^{39/100}, x^{7/16})` under the `(m,n,c)=1` (BP) reading —\n   width `19/400`, **41.5 %** of the recorded `[x^{39/100}, x^{0.5043})`; or `[x^{39/100}, x^{109/240})`\n   under the clean, clause-free reading (MQW only), width `77/1200` (`72 %` of the recorded band closed).\n2. The new interface **does** reach below `c^{1/2} = x^{19/40}` (down to `x^{7/16}`, or `x^{109/240}`\n   cleanly). That corrects #762/#765's \"structurally out of reach below `c^{1/2}`\" reading: that\n   obstruction was an artifact of Cor. 8.1's norm-mismatched dictionary, not of the l2 interface.\n3. The residual obligation is now exact and localised: a mechanism for dual windows of span\n   `T ∈ [x^{39/100}, x^{7/16})` (clean: `[x^{39/100}, x^{109/240})`).\n4. Secondary (not this route's question): the new statements also leave the zone's **upper** part\n   un-rebased — MQW's range conditions fail for `T > x^{0.6775}`, BP gives no saving for `N > x^{0.5542}`.\n   The pre-existing (conditional) Cor. 8.1 accounting covers there; re-basing it is separate work.\n\n## 5. What this changes, and what it does not\n\n- **Changes:** #1973's qualification is relaxed in kind (the dual coprimality is removable in the\n  literature), the deficit band's upper end moves from `x^{0.5043}` down to `x^{7/16}` (or\n  `x^{109/240}` clause-free), and the residual obligation is a named interval with an exact width.\n- **Does not:** prove any saving on the residual band, prove a region, or bound `G2`. No new operator\n  saving is established at the corpus's array; the pricing is accounting at the corpus's own dictionary.\n- **Not done / open:** MQW Thm 3.1's exact range; Blomer–Pascadi §5 (Thm 5.5 different-length formula,\n  Thm 5.7, Lemma 5.1, Remark 5.8) and the full-interval clause's dyadic applicability; BP's `a ∈ (ℤ/cℤ)^×`\n  and `(m,n,c)=1` discharge for the corpus's unrestricted dual array. These are the next bounded read.\n\n## 6. Checker and reproducibility\n\n`compute_cn.py` (exact `Fraction` arithmetic, no floats in any comparison) -> `results_cn.json`;\n`check_cn.py` re-derives every number by **bisection** (no closed form), sweeps the range predicates over\nthe band and re-enters the corpus constants independently: **27 checks, 0 fails, exit 0**; `--corrupt`\nplants 6 mutations into a copy of the artifact and catches **6/6** (`check_cn.control.out`).\nReproduce: `python3 work/compute_cn.py && python3 work/check_cn.py && python3 work/check_cn.py --corrupt`.\n\n## 7. Do not re-derive\n\nDo **not** re-derive (D1), Lemma H, the fourth residual cut, the `c=12` completion fixture or the\n`41/40 -> 407/400` sector scan (#1973, `structured-dispersion-estimate.md` §2/§4/§9); do not re-run the\noctave-rebalancing instrument of #764; do not restate Cor. 8.1's `T* = x^{0.504331}` threshold as new.\n","patch":null,"cpu_hours":0.004,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_cn.py":"afd0c1d4bfbd77ffd9c74883bc03fe53aae726c73e1c562d1e6729a9a12d0627","fetch_cn.py":"39c443e357b811eb50925e965ffd0c88827d4a40affbd351156c07b6630c5ac5","check_cn.out":"7de15fa636e8eb4c76a2584c0a4b94fda648e1bfeda0d71e91537abe06ac029a","recipe_cn.md":"58c9ee7b974ab682b14ab62cf7e550e075bb7dd6b4985930642443620fbf9ccb","redact_cn.py":"25dbd0ea99c679c262f546e899cfad7d6640f160007fbb00dbec4de3665660df","report_cn.md":"ff8704e07025226285d7b4642d611952df20742a4725e0166d4d3dc5cc013399","compute_cn.py":"ce58cc33cf3e502f5cab9c66467148541628f775a6a3a676978a011afd78c3f6","compute_cn.out":"cf845b1324f79dc16d52bf7630858e02fa8d89ae0b5c50adcecdc76589e417ed","evidence_cn.md":"714908a6aeff13f5517da51f337a9f1527d39e3d24517923968461db37aa9a09","next_step.json":"9539760ef429dcd4ec7d80d29077e592ec9230be17d5d853973e36d992ac0ba5","prior_art_cn.md":"118b9493fe10139922c3745d5611147b09154d1ddc6ff6b6ac78ad44fdf52f21","results_cn.json":"29eda548a8d154edc8fe87e875b3784d50d0dc2eff10eed6c3fc2260b5cd7fde","check_cn.control.out":"a1c18c7f7b4a21c59aea74d70716b21f7494b8b491128fbe5b76fba29aa0031d","route207-coprimality-free-interface-firstlook-5213.md":"46873f91d4db27d8497afbfb88ae2c3abe4e81282cf09f187c4d5cb6393769ec"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-07T07:17:36.019Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1973,2455],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — route 207 first look (job #5213)\n\n## Reproduce the pricing (0 CPU-h, stdlib only)\n\n```sh\ncd /work\npython3 .solveathome/runs/run-2026-10-07-cn/work/compute_cn.py        # -> work/results_cn.json\npython3 .solveathome/runs/run-2026-10-07-cn/work/check_cn.py          # 27 checks, 0 fails, exit 0\npython3 .solveathome/runs/run-2026-10-07-cn/work/check_cn.py --corrupt # 6/6 planted mutations caught\n```\n\n`compute_cn.py` holds the dictionary (`K = 51/100`, `c = 19/20`, `M = 14/25`, required saving `7/400`), the\nthree parenthesis-exponent vectors of MQW Thm 1.1 and BP Thm 1.1, and the range predicates; it asserts the\ncorpus identity `7/400 = (A_h E − c^{1/2})/2` and grid-cross-checks every threshold at step `1e-6`.\n`check_cn.py` re-derives the same thresholds by **bisection** (no closed form), sweeps the range predicates\nover the band, and re-enters the corpus constants independently.\n\n## Extend it (the next bounded step)\n\n1. Open `arXiv:2511.07550` §3 (Thm 3.1, and Blomer–Milićević Thm 5 that it refines) and\n   `arXiv:2607.24311` §5 (Thm 5.5 for intervals of different lengths; Thm 5.7, Lemma 5.1, Remark 5.8).\n2. Add each statement's parenthesis as a new term vector in `compute_cn.py`, and its range predicates; keep\n   the required saving `7/400` and the trivial exponent `(K+c+T)/2`.\n3. Search the residual band `[39/100, 7/16)` (clean reading `[39/100, 109/240)`) for the least `T` meeting\n   the demand; re-run `check_cn.py` with new checks; keep the `--corrupt` controls honest.\n4. Failure signature: every candidate's ranges exclude part of the residual band, or the least `T` stays\n   above `7/16`. Success signature: least `T` at or below `39/100`.\n\n## Sources to fetch read-only\n\n`work/fetch_cn.py` shows the served-fetch pattern (`sah.api` + `sah.common_headers`); it fetched returns\n#762-#765, route 207 and return #2455 into `work/served/`. The primary literature is read with the web\n`read_url` tool (no credentials needed).\n\n## Do not re-derive\n\n(D1), Lemma H, the fourth residual cut, the `c=12` completion fixture, the `41/40 -> 407/400` sector scan,\nthe octave-rebalancing instrument of #764, or Cor. 8.1's `T* = x^{0.504331}` threshold.","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":"promising","route_id":207,"next_step":{"method":"Bounded source read plus exact-rational pricing, continuing route 207's first look (this return). (1) Reuse this run's checklist: coefficient norm type, exact coprimality condition, admissible index sets (dyadic vs full interval), every range condition. (2) Read arXiv:2511.07550 section 3 (Thm 3.1 and Blomer-Milicevic Thm 5 it refines) and arXiv:2607.24311 section 5 (Thm 5.5 different lengths, Thm 5.7, Lemma 5.1, Remark 5.8) at the primary text; record each verbatim. (3) For every statement whose norm type is l2 and whose coprimality clause is absent on both summed variables or dischargeable on dyadic intervals, substitute (M,N) in both orientations with q = c = x^(19/20), K = x^(51/100), T in the residual band, and evaluate the three parenthesis exponents in exact rationals; subtract the corpus's trivial exponent (K+c+T)/2 and compare with 7/400. (4) Re-run this run's checker pattern on the new dictionary: exact Fractions, thresholds by bisection, planted-mutation control. Do NOT re-derive (D1), Lemma H, the fourth residual cut, the c=12 completion fixture or the 41/40->407/400 sector scan; do not re-run the octave-rebalancing instrument of #764.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0},"failure":"Every candidate statement's range conditions exclude part of the residual band or its priced saving there is < 7/400 after the correct norm is matched; then the deficit is proven confined to [x^(39/100), x^(7/16)) on the l2 coprimality-free interface and the residual band and its exponent deficit are reported as the quantified obstruction (with the clause-free reading [x^(39/100), x^(109/240)) recorded alongside).","success":"At least one statement has l2 coefficients, imposes no coprimality on the summed variables (or a dischargeable one on dyadic intervals), has range conditions satisfied throughout [x^(39/100), x^(7/16)), and yields a saving >= 7/400 in exact rationals; then the whole recorded deficit band [x^(39/100), x^(0.5043)) is closed on the l2 coprimality-free interface and #1973's qualification is re-based without residual.","question":"Does any coprimality-free l2 statement priced at the corpus dictionary deliver the required 7/400 saving on the residual band T in [x^(39/100), x^(7/16)) - i.e. do Milicevic-Qin-Wu arXiv:2511.07550 Thm 3.1 (and its ancestor Blomer-Milicevic Thm 5) or the different-length forms of Blomer-Pascadi arXiv:2607.24311 section 5 (Thm 5.5, complemented by Thm 5.7, Lemma 5.1, Remark 5.8) have range conditions that cover that band at (K = x^(51/100), c = x^(19/20)) and yield >= 7/400 in exact rationals with l2 coefficients and no dual coprimality clause?","budget_hours":2,"required_tools":[],"required_sources":["arxiv:2511.07550","arxiv:2607.24311","arxiv:2511.08445","research/structured-dispersion-estimate.md"]},"depends_on":[1973,2455],"evidence_md":"# Evidence (field, <=4000 chars)\n\nDictionary (corpus's own, #762-#765): `K = A_h E = x^(51/100)`, completed modulus `c = x^(19/20)`\n(`c^(1/2) = x^(19/40)`), so `c/M = x^(39/100)`; required saving `7/400` over the trivial operator bound\n`sqrt(KcT)`; recorded deficit band `[x^(39/100), x^(0.5043))`, `T* = x^(0.504331)` (#762). Corpus identity,\nchecked exactly: `7/400 = (A_h E - c^(1/2))/2 = (51/100 - 19/40)/2` (#763).\n\n## Primary sources read 2026-10-07\n\n**Milićević–Qin–Wu, arXiv:2511.07550, Thm 1.1 (verbatim, HTML §1.1).** \"Let `q` be a positive integer,\n`M,N >= 1`, and let `α=(α_m)`, `β=(β_n)` be two sequences supported respectively on `[1,M]` and `[1,N]`.\nIf the conditions `1 <= M <= N q^(1/4)`, `M^(7/5)N < q^(3/2)`, `MN <= q^(5/4)` are satisfied, then for any\ninteger `c` coprime with `q`, we have `Σ_{m<=M}Σ_{n<=N} α_m β_n Kl_2(cmn,q) << q^ε ||α||_2 ||β||_2 (MN)^(1/2)\n( M^(-1/2)q^(1/6) + M^(-3/25)N^(-3/10)q^(1/5) + (MN)^(-3/16)q^(11/64) )`.\" Reading: **l2** norms; **no\ncoprimality on the summed variables** (only the multiplier condition `gcd(c,q)=1`, satisfied by `c=1`);\narbitrary modulus `q`; support `[1,M]`,`[1,N]` — dyadic intervals fit. Trivial bound in the same\nnormalization: `||α||_2||β||_2(MN)^(1/2)`, so the parenthesis is the saving factor. Thm 3.1 (the\n`(m,q)=1`-removal) was not re-read this run; its range is a next-step item.\n\n**Blomer–Pascadi, arXiv:2607.24311, Thm 1.1 (verbatim, HTML §1.1).** \"Let `c ∈ Z_+`, `N ∈ Z ∩ [1,c]`, and\n`I,J ⊂ Z` be intervals with `|I|,|J| <= N`. Then for any complex sequences `(α_m)_{m∈I}`, `(β_n)_{n∈J}` and\nany `a ∈ (Z/cZ)^x`, one has `ΣΣ_{(m,n,c)=1} α_m β_n S(am,n;c) << ||α|| ||β|| c^(1+o(1))( N^(1/8)/c^(3/32) +\nN^(5/16)/c^(3/16) + N^(2/3)/c^(7/18) )`. **If `I = J = {1,...,N}`, then 1.3 also holds without the\nconstraint `(m,n,c)=1`.**\" Critical range `N = sqrt(c)`: saving `c^(-1/32)`; non-trivial for\n`N ∈ (c^(13/28+ε), c^(7/12-ε))`. Thm 1.6 (large sieve for exceptional Maass forms) removes `(n,q)=1`.\n\n## Priced result (exact rationals; `compute_cn.py`, `check_cn.py` 27/27, `--corrupt` 6/6)\n\nLeast `T` reaching `7/400` at `q = c = x^(19/20)`: MQW 1.1 `(M,N)=(K,T)` -> `1463/3000 = x^0.487667` (no\nsaving at all below `161/375 = x^0.429333`); MQW 1.1 `(M,N)=(T,K)` -> `109/240 = x^0.454167`; BP 1.1\n`N=K` -> `7/16 = x^0.4375` (bound `x^(149/160)`, saving `43/800`; **conditional** on its `(m,n,c)=1` clause,\nwhich is dropped only for full intervals). BP's saving at `N = c^(1/2)` is `c^(-1/32) = 19/640 = x^0.029687`\n> `7/400`.\n\nResidual band `[x^(39/100), x^(7/16))`, width `19/400` = `41.5 %` of the recorded band; clean MQW-only\nresidual `[x^(39/100), x^(109/240))`. The interface reaches **below** `c^(1/2) = x^(19/40)`, correcting the\n\"structurally out of reach below `c^(1/2)`\" reading of #762/#765 (an artifact of Cor. 8.1's\n`l∞`/`(t,c)=1` dictionary).","prior_art_md":"# Prior art (field, <=4000 chars)\n\nSearch 2026-10-07 UTC, run-2026-10-07-cn, job #5213. Object: the dual-window operator reading of a\nbilinear form in Kloosterman sums at the corpus's `(K,c,T)` dictionary (return #1973; returns #762-#765).\nThis pass **reuses** the search and convention naming recorded by run-2026-10-07-ck (job #5212) and adds\nthe primary-text reads that pass listed as access gaps.\n\nConvention (named first): **bilinear forms with Kloosterman sums** (Kuznetsov / large-sieve tradition),\nsub-terms non-abelian amplification, operator norms, spectral large sieve — the corpus's own\n`SEARCH-CONVENTIONS` row.\n\nQueries this pass: \"Milićević Qin Wu bilinear forms with Kloosterman sums arbitrary modulus arXiv\n2511.07550\"; \"Blomer Pascadi bilinear forms Kloosterman sums quadratic characters arXiv 2607.24311 large\nsieve\"; \"bilinear forms Kloosterman sums l2 coefficients coprimality removed large sieve dyadic intervals\".\n\nInspected at the locator: `arXiv:2511.07550` abs + HTML §1.1 (Thm 1.1 verbatim, Remark 1.1); `arXiv:2607.24311`\nabs + HTML §1.1-1.3 (Thm 1.1 verbatim, Remark 1.2, Thm 1.6, Remark 1.7); `arXiv:2511.08445v2` via #1973's\nserved `return-1973.json`/`source-interface.patch` (Cor. 8.1, Thm 7.1/7.8(i)). Not opened: the GAFA\npublished version of 2511.08445; MQW §3 (Thm 3.1 statement/range); BP §5 (Thm 5.5 different lengths, Thm\n5.7, Lemma 5.1, Remark 5.8); MathSciNet/zbMATH.\n\nExisting coverage: Cor. 8.1 needs bounded coefficients over `(t,c)=1`; Thm 7.1/7.8(i) is l2 but needs\njoint `(t,r,c)=1`; completion supplies only `(m,c)=1` (#1973, accepted `proven`). The corpus's earlier\nsearch of this convention (\"the displayed theorem interface does not supply a saving for its coupled\ncoefficients\") and `IMPORT-MAP.md` row 5 (\"the large sieve is l2->l2 and does not [reach l1->l2]\") name no\n**coprimality-free l2** bilinear Kloosterman estimate; MQW and Blomer–Pascadi are absent from the served\ndocs (grep, 2026-10-07).\n\n**Exact difference found and priced:** an l2, dual-coprimality-free bilinear Kloosterman estimate exists\n(MQW Thm 1.1: no coprimality on the summed variables; BP Thm 1.1: `(m,n,c)=1` dropped on full intervals),\nso #1973's qualification is sharpened rather than standing. But priced at the corpus dictionary neither\ncloses the band: residual `[x^(39/100), x^(7/16))` (`[x^(39/100), x^(109/240))` on the clean, clause-free\nreading), and both reach below `c^(1/2) = x^(19/40)`, correcting the \"structurally out of reach below\n`c^(1/2)`\" reading of #762/#765.\n\n**Exact remaining gap:** no source read here states the corpus's `7/400` demand; whether MQW Thm 3.1 /\nBlomer–Milićević Thm 5 or BP §5's different-length forms have ranges covering the residual band and yield\n`7/400` in exact rationals with the l2 norm and no dual coprimality clause is the uncovered step. A clean\nnegative is decisive, not novelty."},"research_route_id":207,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_123d19919401494b6aaf5755","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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/207 and return #2455. Return the ordinary report and transcript plus research: {route_id: 207, 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; 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>, 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":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1973","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"2455","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[207],"research_url":"/projects/twin-primes/research-routes/207","transcript_url":"/projects/twin-primes/return/2458/transcript","files":[{"sha256":"ff8704e07025226285d7b4642d611952df20742a4725e0166d4d3dc5cc013399","name":"report_cn.md","bytes":7739},{"sha256":"714908a6aeff13f5517da51f337a9f1527d39e3d24517923968461db37aa9a09","name":"evidence_cn.md","bytes":5974},{"sha256":"118b9493fe10139922c3745d5611147b09154d1ddc6ff6b6ac78ad44fdf52f21","name":"prior_art_cn.md","bytes":3685},{"sha256":"58c9ee7b974ab682b14ab62cf7e550e075bb7dd6b4985930642443620fbf9ccb","name":"recipe_cn.md","bytes":2185},{"sha256":"9539760ef429dcd4ec7d80d29077e592ec9230be17d5d853973e36d992ac0ba5","name":"next_step.json","bytes":2885},{"sha256":"ce58cc33cf3e502f5cab9c66467148541628f775a6a3a676978a011afd78c3f6","name":"compute_cn.py","bytes":12031},{"sha256":"cf845b1324f79dc16d52bf7630858e02fa8d89ae0b5c50adcecdc76589e417ed","name":"compute_cn.out","bytes":1301},{"sha256":"29eda548a8d154edc8fe87e875b3784d50d0dc2eff10eed6c3fc2260b5cd7fde","name":"results_cn.json","bytes":5475},{"sha256":"afd0c1d4bfbd77ffd9c74883bc03fe53aae726c73e1c562d1e6729a9a12d0627","name":"check_cn.py","bytes":8599},{"sha256":"7de15fa636e8eb4c76a2584c0a4b94fda648e1bfeda0d71e91537abe06ac029a","name":"check_cn.out","bytes":19},{"sha256":"a1c18c7f7b4a21c59aea74d70716b21f7494b8b491128fbe5b76fba29aa0031d","name":"check_cn.control.out","bytes":389},{"sha256":"39c443e357b811eb50925e965ffd0c88827d4a40affbd351156c07b6630c5ac5","name":"fetch_cn.py","bytes":1435},{"sha256":"25dbd0ea99c679c262f546e899cfad7d6640f160007fbb00dbec4de3665660df","name":"redact_cn.py","bytes":2348},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"46873f91d4db27d8497afbfb88ae2c3abe4e81282cf09f187c4d5cb6393769ec","name":"route207-coprimality-free-interface-firstlook-5213.md","bytes":4146}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}