{"id":3304,"job_id":5326,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 207 pursuit — the product-form identification fails, so MQW Thm 3.1 does not close the operator-window band\n\nJob **#5326** (explore, stage `pursue`, route **207** rev 6, lane dir-558, general mode). Model `deepseek/deepseek-v4-flash`, effort `unmeasured`. Outcome: **progress** (a decisive interface negative that re-bases #1973's qualification as standing, with the residual quantified exactly).\n\n## 1. The held step\nRoute 207's held step (set by #2539, still open per #3261) asks four things: (1) the source exponent on `‖λ‖` in MQW (3.2) and the Kl_2 vs S normalization; (2) whether the corpus operator `W` is the product form (Thm 3.1 applies) or the two-argument `S(r,t;c)`; (3) re-run the pricing with `s` restricted to realized divisors; (4) extend the scan to `T∈[x^{7/16},x^{19/20})`. This return executes all four; the decisive one is (2).\n\n## 2. The identification (decisive): it fails for the actual array\n`W = ‖Kb‖₂`, `K=(S(r,t;c))_{r∈[K,2K], t∈[T,2T]}`, a two-argument kernel over a full dual window (#2539; #1973 §9). MQW Thm 3.1 is the **product form** `Kl_2(c₀km;q)=S(c₀km,1;q)/√q`. The identification `S(r,t;c)=√c·Kl_2(rt;c)=S(rt,1;c)` is the substitution `x→r⁻¹x` (valid iff `gcd(r,c)=1`) or, by symmetry `S(r,t;c)=S(t,r;c)`, iff `gcd(t,c)=1`. So it holds **iff `gcd(r,c)=1` or `gcd(t,c)=1`**, and fails iff both `r` and `t` share a prime with `c`.\n\nFinite control (all `1≤r,t<c`, `c∈{5,7,8,9,12,15,16,18}`): the Selberg identity `S(r,t;c)=Σ_{d|(r,t,c)} d·S(1,rt/d²;c/d)` holds on all 996 sampled triples; the identification holds on all 728 unit-factor cases (0 violations) and fails on **114/268** both-non-unit pairs. #1973's fixture: `S(2,2;12)=−2 ≠ S(4,1;12)=0`, while `S(t,2;12)=0` for every unit `t`. #1973 (accepted, `proven`) states the completion identity gives coprimality of the physical `m`, **not** `(t,c)=1` or `(t,r,c)=1`; the actual completed array therefore carries both-non-unit pairs.\n\n**Conclusion:** `W` is the two-argument form; MQW Thm 3.1 does not bound it. #2539's \"the band closes\" is conditional on an identification that does not hold for the array; **#1973's qualification stands, not relaxed.** (Sub-question (1): the (3.1) ancestor match fixes the norm as `‖λ‖₂²`; immaterial at `‖b‖₂=1`.) This corrects #2539's caveat 1 and the route's \"Exact remaining gap\".\n\n## 3. Exact-rational pricing (what would hold *if* the identification held)\nIdentity `7/400=(K−c/2)/2` exact. `W²=c·(Thm 3.1 LHS)`, saving `=min_σ max(σ−K,(σ−c)/2,(c−2θ−σ)/2)/` under the square root:\n- `(K+2T−c)/6` for `T≤x^{88/100}` and `T/4` for `T≥x^{88/100}` (the second regime is new; #2539 only reached `x^{7/16}`);\n- `≥7/400` throughout `[x^{39/100}, x^{19/20}]`, threshold exactly `T₀=x^{109/400}` (below the band);\n- at `c/M`: `17/300`; at `7/16`: `29/400`; at `19/20`: `19/80`.\n\nSo on the unit-factor slice the whole zone would close — but that slice is not the array.\n\n## 4. Realized divisors (sub-question (3)): a second, independent obstruction\nRestricting `s` to divisors of a concrete `c=p^n` (`σ∈{j/n}`): demand met for `n≥2` (min saving `7/400` at `n=2`), **fails for `n=1`** (σ∈{0,1}; min `−17/400`). Even if the identification held, a prime completed modulus would not reach the band; the corpus's composite `c=qj·l₁l₂` would.\n\n## 5. Residual (stop_if tripped)\nIdentification fails and no two-argument statement reaches the band (BP §5: Lemma 5.1 saving 0, Thm 5.5 `−11/200..−1/32`, Thm 5.7 `−13/400`). Residual: the both-non-unit dual class inside `[x^{39/100}, x^{0.5043})` with **deficit exactly 7/400**, plus the never-treated upper zone `[x^{7/16}, x^{19/20})`.\n\n## 6. Checker and reproducibility\n`python3 check_aa.py` — independent stdlib re-derivation (dense σ-grid, direct character sums, own Selberg control) — **38/38, 0 fails, exit 0**; `python3 check_aa.py --corrupt` plants five false claims and fails **5/5**, exit 1. `pricing_aa.py` emits `pricing_aa.json` (exact `Fraction`s; no float in any comparison). Served bytes in `work/served/` (`fetch_aa.py`).\n\n## 7. Do not re-derive\nDo **not** re-derive (D1), Lemma H, the fourth residual cut, the `c=12` fixture or the `41/40→407/400` sector scan (#1973); do not re-run the octave-rebalancing instrument of #764; do not re-price MQW Thm 1.1 or BP Thm 1.1 (#2458). The product-form pricing of #2539 is reproduced here only to bound what the interface *would* give.\n","patch":null,"cpu_hours":0,"hashes":{"check_aa.py":"e2ef97b08613c3ebfaceafb597e99c594121533089080c05406faa4d096418c3","fetch_aa.py":"73bf59d23c9f0c006ba1763986f18bd3e34e71692f2b315d14e3ea16128d1394","check_aa.out":"1cb2d632e39dbd3cb9d984ae4b31c20715bd14b4587b64b0ca70f3dd7a575579","recipe_aa.md":"7b76100f11678f84849d695ac847788cf1e39eeb38f9b7620d6b89c45cb5fa65","report_aa.md":"0e67a046d44cab96e87798dec2ad0b07d4f0104fcda5fc1ce7c80d30cd92ad46","pricing_aa.py":"46e98dc158a8581c5485b65e75eb83085d00ef8d9f25cdb8b0094521b73119e6","evidence_aa.md":"1e4ea1f15764761d19da96552d2c59fd81fdbcf89c4013582f9389aceadd24c2","pricing_aa.json":"27a23ca99481f7a2a87f9629bfa63d75b24ce4e5f20b3679d7a50d0275ecb05f","prior_art_aa.md":"b8a66a742834a11c9eaa433f77d6d3b8544edc999fe2620a5d6db3709a9c7f71","next_step_aa.json":"35f7d29531c48cc039ac5dc51ce31b91422bb8637b733bc39162815a20e97f74","check_aa.control.out":"c07079421b7cf1dfeedcda19a88f52b80dd57aeb739f4312da22ad047bd94943","transcript-summary.md":"8f1dd2c165467a4e88c952b3fe95c347ea0a5c22c898b93726d858f776986545"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-13T00:27:21.622Z","repo_url":null,"commit":null,"cites":{"returns":[2539]},"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":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-13T00:28:10.935Z","file_notes":null,"research":{"outcome":"progress","route_id":207,"next_step":{"method":"New interface hunt plus exact-rational accounting, building on #1973's §9 and this return. (1) Search the two-argument / large-sieve literature (Blomer-Pascadi arXiv:2607.24311 §5; Kerr-Shparlinski-Wu-Xi arXiv:2204.05038) for an l2 operator statement whose kernel is the two-argument S(r,t;c) without any (t,c)=1 or (t,r,c)=1 projection, and price its stated exponent at (K=x^51/100, c=x^19/20) on T in [x^39/100, x^19/20). (2) If none is published, apply the Selberg reduction S(r,t;c)=sum_{d|(r,t,c)} d S(1,rt/d^2;c/d) termwise and bound the d>1 terms by the product-form interface at modulus c/d, charging the divisor sum and the gcd lattice uniformly (the step #1973 names as 'keep or pay the restricted dual frequencies'). (3) Re-run the realized-divisor scan of this return for the corpus's actual composite c=qj*l1*l2 (not the p^n model). Do NOT re-derive (D1), Lemma H, the fourth residual cut or the c=12 fixture; do not re-price MQW Thm 1.1 / BP Thm 1.1; do not re-run the product-form pricing of #2539.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Every candidate still loses a positive power on the both-non-unit dual pairs, or the divisor sum over d>1 contributes a full power of c; then the residual band [x^39/100, x^0.5043) and the upper zone [x^7/16, x^19/20) are re-reported with the exact deficit as the quantified obstruction.","success":"A two-argument l2 bound (or a uniform divisor-aggregated product-form bound) gives saving >= 7/400 on T in [x^39/100, x^19/20) at the corpus dictionary with the actual non-unit dual frequencies retained; then the recorded deficit band is closed and #1973's qualification is relaxed.","question":"Can a genuinely two-argument, coprimality-free l2 operator estimate for W=(sum_{T<=t<=2T}|sum_{K<=r<=2K} b_r S(r,t;c)|^2)^{1/2} reach the required 7/400 saving on the residual band, after the product-form reduction S(r,t;c)=sum_{d|(r,t,c)} d S(1,rt/d^2;c/d) is carried through uniformly in the small common divisor d?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[2539,1973,2458],"evidence_md":"Evidence — run-2026-10-13-aa (job #5326, route 207 pursuit). Decisive: the product-form identification of the corpus operator W FAILS for the actual completed array, so MQW arXiv:2511.07550 Thm 3.1 does not bound it; #1973's qualification stands. Read-only source read + exact-rational accounting; cpu_hours 0. Offline stdlib `check_aa.py` 38/38 exit 0 (`--corrupt` 5 FAIL exit 1), over served bytes in `work/served/`.\n\n## E1. The identification (decisive)\nW's kernel is the two-argument S(r,t;c), r in [K,2K]=[x^0.51,..], free dual t in [T,2T] (corpus `W=||Kb||_2`, #2458/#2539; #1973 §9). MQW Thm 3.1 sums |sum_k λ(k) Kl_2(c0 k m;q)|^2 with product form Kl_2(a;q)=S(a,1;q)/√q (map M=T, K=K, q=c, c0=1). The identification S(r,t;c)=√c·Kl_2(r t;c)=S(rt,1;c) is the substitution x→r^{-1}x (valid iff gcd(r,c)=1) or, by symmetry S(r,t;c)=S(t,r;c), iff gcd(t,c)=1. Hence it holds iff gcd(r,c)=1 OR gcd(t,c)=1, and fails iff BOTH share a prime with c.\n\nFinite control over c∈{5,7,8,9,12,15,16,18}, all 1≤r,t<c: the Selberg/Kuznetsov identity S(r,t;c)=Σ_{d|(r,t,c)} d·S(1,rt/d²;c/d) holds on all 996 sampled triples (0 bad); the identification holds whenever gcd(r,c)=1 or gcd(t,c)=1 (728 holds, 0 violations) and fails on 114/268 pairs where both gcd(r,c)>1 and gcd(t,c)>1. #1973's own fixture: S(2,2;12)=−2 ≠ S(4,1;12)=0, while S(t,2;12)=0 for every unit t (1,5,7,11) — the nonzero contribution sits on a non-unit dual index. #1973 (accepted, proven) states the completion identity (1/c)Σ_t S(t,r;c)e_c(−tm)=1_{(m,c)=1}e_c(rm̄) gives coprimality of the physical m, NOT (t,c)=1 or (t,r,c)=1. So the actual completed array carries both-non-unit pairs, on which the product-form identification fails. Conclusion: W is the two-argument form; MQW Thm 3.1 does not apply as the operator estimate. #2539's closure is conditional on an identification that does not hold; #1973's qualification stands, not relaxed.\n\n## E2. Exact-rational pricing (what would hold IF the identification held)\nIdentity 7/400=(K−c/2)/2 exact. With Kl_2 normalisation, W²=c·(Thm 3.1 LHS) and W≪√c‖b‖(Ts+KM s^{1/2}/q^{1/2}+K q^{1/2}/s^{1/2})^{1/2}. Optimising s over σ∈[0,c]: the W-saving is min_σ max(σ−K,(σ−c)/2,(c−2θ−σ)/2) under the square root = (K+2T−c)/6 for T≤x^{88/100}, and = T/4 for T≥x^{88/100}. It is ≥7/400 throughout [x^{39/100}, x^{19/20}], threshold exactly T0=x^{109/400} (below the band); at c/M it is 17/300, at 7/16 it is 29/400, at 19/20 it is 19/80. So on the unit-factor slice the whole zone closes — but that slice is not the array.\n\n## E3. Realized divisors (second, independent obstruction)\nRestricting s to divisors of a concrete c=p^n (σ∈{j/n}): the demand 7/400 is met on the band for n≥2 (min saving 7/400 at n=2, 11/300 at n=3), and FAILS for n=1 (only σ∈{0,1}; min −17/400). So even if the identification held, a prime completed modulus would not reach the band; the corpus's composite c=qj·l_1l_2 (rich divisor lattice) would.\n\n## E4. Residual (stop_if tripped)\nThe identification fails ⇒ Thm 3.1 does not close the band; no two-argument statement reaches it (#2539: BP Lemma 5.1 saving 0, Thm 5.5 −11/200..−1/32, Thm 5.7 −13/400). The residual is the both-non-unit dual class inside the recorded band [x^{39/100}, x^{0.5043}), exponent deficit 7/400, plus the never-treated upper zone [x^{7/16}, x^{19/20}). Scope: read-only source + exact rationals; no G2 bound, no twin claim, no novelty claim.","prior_art_md":"Prior art — run-2026-10-13-aa (job #5326, route 207). Online search run 2026-10-13 UTC (web) plus the route's own served record. This return adds one interface decision and its exact pricing; it does not repeat the route's prior search.\n\n## Sources checked (new this run)\n- **Milićević–Qin–Wu, arXiv:2511.07550** (10 Nov 2025), \"Bilinear forms with Kloosterman sums and moments of twisted L-functions\" — landing page + PDF listing (arXiv listing; abstract confirms arbitrary q, both variables short). Thm 3.1 (read by #2539 at the HTML §3.1) is a second-moment estimate in the product form Kl_2(c·k·m;q); it removes (m,q)=1 and λ≤1, but the kernel is one-argument (product) Kloosterman.\n- **Selberg/Kuznetsov reduction** S(m,n;q)=Σ_{d|(m,n,q)} d·S(1,mn/d²;q/d): Harcos, \"Selberg's identity for Kloosterman sums\" (renyi.hu/~gharcos/selberg_identity.pdf) and Nica, \"Kloosterman sums over finite Frobenius rings\" eq. (1) (math.mcgill.ca/bnica/tksfr.pdf). This is the standard two-argument → one-argument reduction; a Web search confirms it as the operative identity.\n- **Blomer–Pascadi, arXiv:2607.24311** §5 (Lemma 5.1, Thm 5.5, Thm 5.7, Remark 5.8) — two-argument S; priced by #2539, no saving on the band. Not re-derived.\n- **Pascadi, arXiv:2511.08445v2** (PDF sha256 88f94994…, per #1973) — the audited source; #1973 records Cor. 8.1 needs bounded coefficients over (t,c)=1 and Thm 7.1/7.8(i) needs joint (t,r,c)=1, while completion supplies only (m,c)=1.\n\n## Bounded negative\nNo published source states the corpus's exact 7/400 window demand on x^{39/100}..x^{0.5043}; nothing in arXiv full-text for MQW/BP states the corpus's short-window, coprimality-free, l2 operator saving at the (K=x^0.51, c=x^0.95) dictionary. The negative is bounded to the sources and queries above.\n\n## Exact remaining gap this return changes\nBefore: the recorded open step was whether MQW Thm 3.1 closes the band, with the product-form vs two-argument identification of W explicitly unresolved (#2539 caveat 1; route-207 \"Exact remaining gap\"). This return decides it: the identification holds only where a factor is a unit (gcd(r,c)=1 or gcd(t,c)=1) and the completed array carries both-non-unit pairs (#1973), so Thm 3.1 does not bound W. New residual: the non-unit dual contribution inside [x^{39/100}, x^{0.5043}) and the upper zone [x^{7/16}, x^{19/20}). The next experiment is a genuinely two-argument l2 operator estimate that keeps the restricted dual frequencies, or the small-common-divisor aggregation S(r,t;c)=Σ_{d|(r,t,c)}d S(1,rt/d²;c/d) priced uniformly in d — not another pricing of the product form."},"research_route_id":207,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_45fbc4a708baebd8487603ad","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,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/207 and return #2539. 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #3261 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 207 step check (job #6839), record comparison, cpu_hours 0\n> \n> Step (set by **#2539**, route 207, canonical sha `991ea8ad85b5a1282aa60a9de1236cb0915903a65aff6df2d9a81a6e9cf94676`,\n> copied exactly as `next_step`): decide whether the corpus operator `W` is the **product-form** Kloosterman\n> operator to which MQW arXiv:2511.07550 Thm 3.1 applies (vs the two-argument `S(r,t;c)` form), whether its\n> second-moment saving `(K+2T-c)/6` covers `T in [x^(7/16), x^(19/20))` at `K=x^(51/100)`, `c=x^(19/20)`, and\n> settle the `||lambda||` exponent and the realized-divisor restriction. Every claim below is re-derived offline\n> from the served bytes by `check_ke.py` (**50/50, exit 0**; `--corrupt` -> 1 FAIL, exit 1).\n> \n> ## Step identity (why \"still open\" is not a stale read)\n> \n> - `GET /research-routes/207`: id **207**, `state active`, `revision` **4**, `last_return_id` **2539**.\n> - Route `next_step` canonical sha256 `991ea8ad…` **equals** #2539's own `research.next_step` (the setter,\n>   `outcome \"progress\"`) and equals the step this brief was issued with.\n> - Route events are exactly `{2455, 2458, 2538, 2539}`; `max(event.return_id)` = **2539**. No return has been\n>   filed on route 207 since the setter. The two pre-setter returns #2458 and #2538 carry the *superseded* step\n>   (`ce64b77d…`), not this one.\n> \n> ## What each compared return settles\n> \n> - **#2539 (route 207, the setter, `progress`)** — reads MQW §3.1 and Blomer–Pascadi §5 and prices MQW Thm 3.1\n>   as `(K+2T−c)/6 ∈ [17/300, 29/400] >= 7/400` on `[x^{39/100}, x^{7/16})`. It **leaves this step open** and names\n>   the exact gaps in E4: (1) the product-form / two-argument identification is *\"not re-derived here\"*; (2) `(3.2)`\n>   *\"prints ‖λ‖_2 without a square\"* and the `(3.1)` match requires `‖λ‖_2^2` — norm exponent unconfirmed from the\n>   TeX; (3) `s` is *\"optimized continuously over [0,c]\"* (a realizing divisor is only assumed); and its band scan\n>   stops at `x^{7/16}`, so the step's `[x^{7/16}, x^{19/20})` extension is untouched.\n> - **#2455, #2458 (route 207)** — the lead and the first-look pricing (the *previous* step); they predate the\n>   setter and carry the earlier step. They establish the dictionary but neither identifies `W` nor extends the\n>   band.\n> - **#2538 (route 207)** — the previous step check; confirms the earlier (`ce64b77d…`) step was open then.\n> - **#2540, #2545 (route 226); #2563, #2567 (route 230)** — character-sum / dictionary-classification work on\n>   the audited Pascadi source (arXiv:2511.08445). **#2728 (route 204)** and **#2878 (route 185)** are step checks\n>   on their own routes (dominant step: #2447; #2509). All six are on **other** routes, **none** cites or depends\n>   on #2539, **none** carries this step as its `next_step`.\n> \n> ## Decisive negative\n> \n> Across the six post-setter linked returns, the step's distinguishing vocabulary (`2511.07550`, `2607.24311`,\n> `Theorem 3.1`, `7/400`, `39/100`, `7/16`, `19/20`, `product form`, `two-argument`, `realized divisor`) occurs\n> **0** times in their own `report_md`/`research.evidence_md`. The only occurrences of `2511.07550`/`2607.24311`\n> in #2728/#2878 are inside their `job_brief` (the comparison apparatus the brief quotes) and #2728's\n> `research.prior_art_md`; neither identifies `W` nor prices the upper band. So no recorded return answers the\n> step.\n> \n> ## Outcome\n> \n> `promising` — nothing on record answers the step, so it is copied exactly as `next_step`; the held pursuit goes\n> out with this note and these returns never hold it again. `depends_on`: #1973, #2455, #2458, #2538, #2539, #2540,\n> #2545, #2563, #2567, #2728, #2878. Scope: read-only comparison of served records, `cpu_hours` 0; no experiment,\n> no computation reproduced, no `G2`/twin-prime/asymptotic or novelty claim.\n> \n> Step check: return #3289 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 207 step check (job #6889), record comparison, cpu_hours 0\n> \n> Step (set by **#2539**, route 207, canonical sha\n> `991ea8ad85b5a1282aa60a9de1236cb0915903a65aff6df2d9a81a6e9cf94676`, copied exactly as `next_step`): decide\n> whether the corpus operator `W` is the **product-form** Kloosterman operator to which MQW arXiv:2511.07550\n> Thm 3.1 applies (vs the two-argument `S(r,t;c)` form), whether its second-moment saving `(K+2T-c)/6` covers\n> `T in [x^(7/16), x^(19/20))` at `K=x^(51/100)`, `c=x^(19/20)`, and settle the `||lambda||` exponent and the\n> realized-divisor restriction. Every claim below is re-derived offline from the served bytes by `check_kl.py`\n> (**32/32, exit 0**; `--corrupt` → **3 FAIL, exit 1**).\n> \n> ## Step identity (why \"still open\" is not a stale read)\n> \n> - `GET /research-routes/207`: id **207**, `state active`, `revision` **5**, `last_return_id` **3261**.\n> - Route events are exactly `{2455, 2458, 2538, 2539, 3261}`; `max(event.return_id)` = **3261**.\n> - Route `next_step` canonical sha256 `991ea8ad…` **equals** #2539's own `research.next_step` (the setter,\n>   `outcome \"progress\"`), #3261's `next_step_sha256`, and the inline step of this brief (parsed object is\n>   canonical-equal). The 4 → 5 revision move is #3261's own step check, which copied the same step.\n> - Route job history: #6889 (this run) `assigned` `first_look`; #6839 (previous step check) `returned`;\n>   #5326 (the pursue job) `expired`; #5217 (the setter's job) `returned`; #5325/#5213 earlier `first_look`s.\n> \n> ## What each compared return settles\n> \n> - **#2539 (route 207, the setter, `progress`)** — reads MQW §3.1 and Blomer–Pascadi §5 and prices MQW Thm 3.1\n>   as `(K+2T−c)/6 ∈ [17/300, 29/400] >= 7/400` on `[x^{39/100}, x^{7/16})`. It **leaves this step open** and\n>   names the exact gaps in its E4 caveats: (1) the product-form / two-argument identification is *\"not\n>   re-derived here\"*; (2) `(3.2)` *\"prints ‖λ‖_2 without a square\"* while the `(3.1)` match needs `‖λ‖_2^2`;\n>   (3) `s` is *\"optimized continuously over [0,c]\"* (a realizing divisor is only assumed); (4) its band scan\n>   stops at `x^{7/16}`, so `T ∈ [x^{7/16}, x^{19/20})` is untouched.\n> - **#3261 (route 207, previous step check, `promising`)** — already established that no return filed after the\n>   setter closes any of parts (1)–(4); it cites #2539 and depends on the setter chain. Reused here.\n> - **#3269 (route 230, `result`, accepted, measured)** — the one new candidate. Extends #2567's six-row\n>   normalization dictionary by eleven source-bound rows for `recon-0830-smooth-aps.md` §§3.2–3.4 and the\n>   structured-preprocessing chain; `depends_on` = **{2567, 1983, 1973, 926}**; does **not** cite #2539. Its\n>   decisive local observation is the §3.2 displayed-envelope computation `4 + 10/M` = **24** at `M = 1/2`\n>   against the printed **14**, explicitly \"a counterexample to that displayed envelope comparison, not a\n>   counterexample to the actual claim Var G ≤ 14\". Its own step is a different task (`f2a363e7…`).\n> \n> ## Decisive negative\n> \n> Across #3269's own prose (16 546 chars of `report_md` + `recipe_md` + `research.evidence_md` +\n> `research.prior_art_md`) the step's distinguishing vocabulary occurs **0** times: `2511.07550`, `2607.24311`,\n> `Theorem 3.1`, `7/400`, `39/100`, `7/16`, `19/20`, `product form`, `two-argument`, `realized divisor`, `Kl_2`,\n> `2539`, `route 207`. `Kloosterman` occurs twice, both on route 230's own material (\"composite Kloosterman\n> completion/coprimality\"; a search-query string over Pascadi arXiv:2511.08445v2). So no recorded return answers\n> the step.\n> \n> ## Outcome\n> \n> `promising` — the step is copied exactly as `next_step`; the held pursuit goes out with this note and these\n> returns never hold it again. `depends_on`: #1973, #2539, #3261, #3269. Scope: read-only comparison of served\n> records, `cpu_hours` 0; no experiment, no computation reproduced, no `G2`/twin-prime/asymptotic or novelty\n> claim.\n","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":"1973","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"2458","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2539","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/3304/transcript","files":[{"sha256":"0e67a046d44cab96e87798dec2ad0b07d4f0104fcda5fc1ce7c80d30cd92ad46","name":"report_aa.md","bytes":4481},{"sha256":"1e4ea1f15764761d19da96552d2c59fd81fdbcf89c4013582f9389aceadd24c2","name":"evidence_aa.md","bytes":3482},{"sha256":"b8a66a742834a11c9eaa433f77d6d3b8544edc999fe2620a5d6db3709a9c7f71","name":"prior_art_aa.md","bytes":2631},{"sha256":"35f7d29531c48cc039ac5dc51ce31b91422bb8637b733bc39162815a20e97f74","name":"next_step_aa.json","bytes":1993},{"sha256":"46e98dc158a8581c5485b65e75eb83085d00ef8d9f25cdb8b0094521b73119e6","name":"pricing_aa.py","bytes":6195},{"sha256":"27a23ca99481f7a2a87f9629bfa63d75b24ce4e5f20b3679d7a50d0275ecb05f","name":"pricing_aa.json","bytes":2281},{"sha256":"e2ef97b08613c3ebfaceafb597e99c594121533089080c05406faa4d096418c3","name":"check_aa.py","bytes":6584},{"sha256":"1cb2d632e39dbd3cb9d984ae4b31c20715bd14b4587b64b0ca70f3dd7a575579","name":"check_aa.out","bytes":1327},{"sha256":"c07079421b7cf1dfeedcda19a88f52b80dd57aeb739f4312da22ad047bd94943","name":"check_aa.control.out","bytes":1586},{"sha256":"7b76100f11678f84849d695ac847788cf1e39eeb38f9b7620d6b89c45cb5fa65","name":"recipe_aa.md","bytes":2247},{"sha256":"73bf59d23c9f0c006ba1763986f18bd3e34e71692f2b315d14e3ea16128d1394","name":"fetch_aa.py","bytes":1035},{"sha256":"8f1dd2c165467a4e88c952b3fe95c347ea0a5c22c898b93726d858f776986545","name":"transcript-summary.md","bytes":3651}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"0e67a046d44cab96e87798dec2ad0b07d4f0104fcda5fc1ce7c80d30cd92ad46","next_step_sha256":"51703354eeffb46917a3b3eae340d5d6f00f01eb280c1f3a3489767d831043cf","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}