{"id":737,"job_id":1530,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1530 — Triage of route 46 (BRIDGE-W): the split exponent `w` is not forced from below by the served constraint set\n\nRun `run_20260916_191004_eE6QUA`, attempt `c67b43800e2e8920702811202792bbcb`, session `1758b7f4603048e3d7fbeec6`,\ngeneral mode, one of one, 0.5 h budget. Type `explore`, purpose `discovery`, stage `triage`, route **46**.\n**No twin-prime claim, no novelty claim, no served file edited, no region claimed controlled that the cited\nargument does not control.**\n\n## Verdict\n\n**`promising`.** The route's own refutation branch is now decided in the negative, exactly, for every\ningredient of the served constraint set that this run could read: the split exponent `w` is **not** bounded\nbelow by the level, by the two right-hand budgets, by the two left-hand budgets, or by the two exact added\ncuts. Each of them is *monotone favourable* as `w` falls, and the served choice of the cuts keeps **nine\ntimes** the slack it needs at `w = 1/5`. Consequently the region re-derivation at `w = 1/5` or `1/8` is a\nbounded exact-arithmetic deliverable — which is the route's own declared outcome for the absence of a lower\nbound — and the served \"unhandled boundary witness\" that caps the combined uniform product threshold at\n`19/25` is *inside* the right-orientation sufficient region at `w = 1/5`.\n\nWhat this **does not** decide, and what the bounded next experiment must decide: whether the enlarged\nregion is *consumed* (the union's product threshold is set by the old left regions, whose `w`-dependence\nthis run did not derive), and whether the named input class is exhausted (its one already-measured member\nadds exactly zero area at the served split and, read at source here, is *silent* on `w`).\n\n## 1. The instrument at general `w` (exact, verification of the served literals)\n\nServed note `grouped-divisor-moment.md` §4: `a = delta + 6/25`, `b = nu + 1/20`; the three right budgets in\n(14) are `(1+a)/2, a/2+3b/2, a`, and \"the left budgets swap `a,b`\". For a general split `U = V = x^w`,\n`Y = Z = x^y` the same substitution is `a = delta + w`, `b = nu + y`, so the four edges are\n\n```\nR1: delta        <  1 - w          R2: delta + 3 nu < 2 - w - 3 y      (right orientation, (15))\nL1: nu           <  1 - y          L2: 3 delta + nu < 2 - 3 w - y      (left  orientation, a,b swapped)\n```\n\nAt `(w,y) = (6/25,1/20)` these reproduce the served note's own literals **`19/25`, `161/100`, `123/100`**\nexactly (the note prints `123/100` verbatim as \"the left condition\"). Verified in exact rationals. Note on provenance: the\npredecessor's 9/9 set covered `R1`, `R2` only; `L1`, `L2` and everything in sections 2–4 below are new here.\n\n## 2. The exact added cuts do not force `w` (the gap the route called \"not a formality\")\n\n`grouped-divisor-moment.md` §5 defines the cut `C_{kappa,lambda} = {d <= x^kappa, de^3 <= x^lambda}`\n\"inside the full original domain `U < d <= D0`, `Y < e <= E0`\" and requires that \"on every dyadic box\nmeeting this set, the top support budgets have fixed slack **by (15)**\". With (15) parameterised by `w`,\nthat requirement is exactly\n\n```\n(A) fixed slack:      kappa < 1 - w      and   lambda < 2 - w - 3 y\n(B) non-emptiness:    w < kappa          and   w + 3 y < lambda      (from d > U = x^w, e > Y = x^y)\n```\n\nThe served choice `(kappa, lambda) = (151/200, 321/200)` (the note's (18)) satisfies (A) at every\n`w <= 6/25`, and with **more** slack the smaller `w` is:\n\n| `w` | `1-w` | `2-w-3y` | (A) vertical slack `1-w-kappa` | (A) product slack `2-w-3y-lambda` | (B) `kappa-w` / `lambda-w-3y` |\n|---|---|---|---|---|---|\n| 6/25 (served) | 19/25 | 161/100 | **1/200** | **1/200** | 103/200 / 243/200 |\n| 21/100 | 79/100 | 41/25 | 7/200 | 7/200 | 109/200 / 249/200 |\n| **1/5** | 4/5 | 33/20 | **9/200** | **9/200** | 111/200 / 251/200 |\n| **1/8** | 7/8 | 69/40 | 3/25 | 3/25 | 63/100 / 133/100 |\n\nSo `C_{kappa,lambda}` is *admissible and non-empty* at `w = 1/5` and `w = 1/8` with the served constants\nunchanged — the two cuts, which the route flagged as the possible killer of the whole axis, are not it.\n\n## 3. Level, witness, and the product ceiling (exact)\n\n- **Level.** `D = x^{2w}`: `12/25` at the served split (matching the corpus's Lambda-BV level `x^{12/25}`,\n  `TWIN-REDUCTION` §1 as quoted in N-1526-01), **`2/5` at `w = 1/5`**, `1/4` at `w = 1/8`; all below the\n  classical `1/2` barrier. Falling `w` lowers the level, i.e. it moves in the *helpful* direction.\n- **Witness.** The served unhandled boundary witness `(8/25, 11/25)` (`delta+nu = 19/25`), outside at the\n  served split because `delta+3nu = 41/25 > 161/100`, satisfies both right-orientation inequalities at\n  `w = 1/5` (`R1` slack `12/25`, `R2` slack **`1/100`**) and at `w = 1/8` (`111/200`, `17/200`). Its entry\n  thresholds are `w < 21/100` (right, the predecessor's number) and, **new**, `w < 11/60` (left) — so at\n  `w = 1/5` it is inside the right region but still outside the left one, and at `w = 1/8` it is inside\n  both.\n- **Product ceiling, both orientations (new).** On the line `delta + nu = t` the right orientation needs\n  `t - (1-w) < (2-w-3y-t)/2`, i.e. `t < 4/3 - w - y`; the left orientation gives the *same* condition, so\n  the union's ceiling on that line is `4/3 - w - y`: `313/300` at served, `13/12` at `w = 1/5`, `139/120`\n  at `w = 1/8`. All `> 1`, hence vacuous alone — which is precisely the note's own stated reason for\n  imposing the vertical cutoff, and a warning that this quantity must never be quoted without `C`.\n\n## 4. The first named input, read at source: Corollary 1 is SILENT on `w`\n\n`arXiv:1502.00769v1` (Bettin–Chandee, *Trilinear forms with Kloosterman fractions*), p. 4, eq. (1.4),\nverbatim: with `f` smooth on `[M1/2,M1]`, `g` smooth on `[M2/2,M2]`, `alpha` supported on `[N1/2,N1]`,\n`beta` on `[N2/2,N2]`, `Delta != 0` fixed,\n\n```\nT(M1,M2,N1,N2) = main term + O( (eta R)^{3/2} ||alpha|| ||beta|| (N1N2)^{7/20+eps} (N1+N2)^{1/4} (M1M2)^eps ),\nR := M1N2/(M2N1) + M2N1/(M1N2).\n```\n\n- The hypotheses place **no restriction on the bands** and none on the split: the boxes are arbitrary, and\n  `R >= 2` always, so the factor `(eta R)^{3/2}` carries no `w` either. The input is therefore available at\n  `w = 1/5` and `1/8` (its statement does not need re-stating) but is **exactly as w-silent as it is at the\n  served split**: it neither forces `w >= 6/25` nor adds region at smaller `w`.\n- The register's quoted condition is the price formula and is exact: `(17/20)(A+B) + (1/4)max(A,B) =\n  (22max(A,B) + 17min(A,B))/20` (verified as an identity), so the controllability requirement\n  `22max+17min < 20` is the statement *cost exponent < 1*. The register's own row already records the\n  verdict for this input at the served split: \"strictly inside the region already controlled. **Added area:\n  exactly zero**\". Read at source here, that row is not a row about `w` at all.\n\n**Price list (partial — the honest state after this run):**\n\n| input | admissible `w` | binding hypothesis | source locator | status |\n|---|---|---|---|---|\n| right-orientation grouped budgets (the moment itself) | any `w < 1` | `delta<1-w`, `delta+3nu<2-w-3y` | `grouped-divisor-moment.md` §4 (15) | **reads at `w=1/5`, region strictly larger** |\n| exact added cuts `C_{kappa,lambda}` | any `w <= 6/25` with the served constants | `kappa<1-w`, `lambda<2-w-3y` | same note §5 (17)–(18) | **slack 9/200 at `w=1/5`** |\n| Lambda-BV level `D = x^{2w}` | any `w` (level falls with `w`) | level hypothesis | `TWIN-REDUCTION.md` §1 via N-1526-01 | favourable |\n| Corollary 1 (Bettin–Chandee) | any (silent) | none involving the split | `arXiv:1502.00769v1` p.4 (1.4) | silent; zero added area at served `w` |\n| Deshouillers–Iwaniec, Drappeau, Topacogullari, Pascadi | **not read** | stated parameter ranges | not fetched this run | **open** |\n\n## 5. Scope — what is NOT established (read this before building on the above)\n\n1. **The union's product threshold is not re-derived.** The combined uniform product threshold at the\n   served split (`19/25`, capped by the point `(8/25,11/25)`) is set by the *old left* regions\n   (`5 delta + 2 nu < 123/50` is one of them), and their `w`-dependence was **not** derived here. Section 3\n   shows the *witness* becomes controlled by the right orientation at `w = 1/5`; whether the union's\n   threshold above `19/25` is actually available depends on re-deriving those old regions at `w = 1/5`.\n2. **No consumption.** Nothing here proves that a larger admissible `(delta,nu)` rectangle yields a larger\n   *controlled residual mass*: the density bound `R_IJ = O_H(x/log^H x)` must be re-run at `w = 1/5` for the\n   enlarged set, and that is a derivation, not arithmetic.\n3. **Three of the four named inputs were not read** (Deshouillers–Iwaniec, Drappeau, Topacogullari,\n   Pascadi), and the `3 kappa/2` term the route quotes for Corollary 1 is the r-size split residue, not part\n   of (1.4); it is not re-derived here.\n4. **Risks kept explicit.** (i) The named class may be exhausted rather than misused: the one member already\n   measured adds exactly zero area. (ii) The register itself states \"A smaller summation domain does not\n   imply monotonicity of its signed value\", so even a non-empty larger region is not automatically a larger\n   controlled quantity. (iii) The global twin margin remains OPEN and no part of this note is evidence\n   toward it.\n\n## 6. Reproduction\n\n`work/job1530/job1530-checks.py` → `work/job1530/job1530-checks.json`, **27/27 passed**, exact rationals,\nstdlib only, no network, run through the bounded `exec` path (60 s wall clock, process group killed;\nstdout redirected inside the child because `exec` truncates at 4 000 B). sha256: `job1530-checks.py`\n`763b2094694067e6ecf44491a7917627333ef427502177adab9632f1496cb3f7`, `job1530-checks.json`\n`24ac4f4f7ea0f8baef3d73cdbb87fbea698c536162ee85d80dbd856b1d8f8dd6`. Source reads this run, all through the\ntested request path: `GET /projects/twin-primes/docs/research/grouped-divisor-moment.md` 200 (21 380 B),\n`GET /projects/twin-primes/research-routes/46` 200 (21 618 B); `arXiv:1502.00769v1` PDF 310 909 B,\n`pdftotext -layout`. Local evidence reused: `research/notes/N-1526-01-cutoff-parameterised-budget-region.md`\n(return #727) and the route-46 return #736.\n\n## Next step proposed (one bounded experiment, 0.6 h)\n\nRe-derive the **union** at `w = 1/5` and `w = 1/8` from the served notes — not the two inequalities again,\nwhich are done: parameterise the *old* left regions (`5 delta + 2 nu < 123/50` and the region-one\n`delta < 1-w` headline) and the density bound `R_IJ = O_H(x/log^H x)` at the smaller split, and report the\nunion's product threshold and the controlled mass at `w = 1/5`. Failure mode that closes the route: the old\nregions do **not** move with `w`, the union's threshold stays `19/25`, and the enlarged right region buys\nnothing — then the axis is inert and the row is closed with this note's evidence.","patch":null,"cpu_hours":0.2,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-16T17:14:55.685Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[736],"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":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":"promising","route_id":46,"next_step":{"method":"Read-only source re-derivation in exact rationals, no new estimate. (i) From the served notes already in custody (grouped-divisor-moment.md, residual-coverage.md, reachability-coverage.md, prime-detection-spec.md, heath-brown-edges.md), write down each old sufficient region's own inequalities in the variables the split enters (the region-one headline 19/25 = 1-w and the old left condition 5delta+2nu<123/50 with its split dependence), then recompute the union at (w,y) in {(1/5,1/20),(1/8,1/20)} and compare its product threshold with 19/25. (ii) Re-run the density argument's conclusion (not its proof) at the enlarged set: is the same O_H(x/log^H x) remainder available for the larger rectangle, and does the note's own statement 'a smaller summation domain does not imply monotonicity of its signed value' block the inference? (iii) Extend the price list to the remaining named inputs by reading their stated parameter ranges (Deshouillers-Iwaniec, Drappeau, Topacogullari, Pascadi) and instantiating each at w=1/5 and 1/8, recording hold/fail/silent with the deciding sentence.","compute":{"ram_gb":0.5,"disk_gb":0.1,"cpu_hours":0.2},"failure":"The old left regions do not move with w (their thresholds are fixed at 6/25), the union's product threshold stays 19/25, and every named input is either silent or adds zero area - then the split is inert below 6/25, the parameter axis adds nothing, and route 46 is recorded as a closed row ('the split is inert below 6/25: the level is not binding, the cuts are slack, and no named input constrains w') with this run's exact-arithmetic evidence so no successor reopens it.","success":"A union product threshold at w=1/5 strictly above 19/25 together with the same density remainder for the enlarged rectangle, giving a strictly larger controlled residual region and a priced table for all named inputs - then the region re-derivation at w=1/5 is a publishable bounded result and the axis stays open.","question":"At w = 1/5 (and 1/8), does the UNION of the served sufficient regions - the old left regions, the parameterised right orientation (delta<1-w, delta+3nu<2-w-3y), and the exact cuts C_{151/200,321/200} - admit a strictly larger product threshold than the served 19/25, and does the enlarged set carry a larger controlled residual mass under the same density bound R_IJ = O_H(x/log^H x)?","budget_hours":0.6,"required_tools":["python3","pdftotext"],"required_sources":["served-note-grouped-divisor-moment-old-left-regions","served-note-cutoff-parameterised-budget-region","served-note-residual-coverage-density-bound","served-spec-prime-detection-inputs","stated-parameter-ranges-pascadi-drappeau-topacogullari-deshouillers-iwaniec"]},"depends_on":[727,736],"evidence_md":"Exact-rational parameterisation of the FULL served constraint set by the split w (U=V=x^w, Y=Z=x^y), checked 27/27 (work/job1530/job1530-checks.py). Four edges: R1 delta<1-w, R2 delta+3nu<2-w-3y, L1 nu<1-y, L2 3delta+nu<2-3w-y; at (w,y)=(6/25,1/20) they reproduce the served note's own literals 19/25, 161/100 and 123/100, so the instrument is verified against grouped-divisor-moment.md section 4 (15), not assumed. The exact added cuts C_{kappa,lambda}={d<=x^kappa, de^3<=x^lambda} (that note section 5, (17)-(18)) require, generalised from its own 'fixed slack by (15)' and its domain U<d<=D0, Y<e<=E0: kappa<1-w, lambda<2-w-3y (fixed slack) and w<kappa, w+3y<lambda (non-emptiness). The served constants (151/200,321/200) satisfy BOTH at every w<=6/25, with slack 1/200 at the served split, 9/200 at w=1/5 and 3/25 at w=1/8; non-emptiness margins are kappa-w=111/200 and lambda-w-3y=251/200 at w=1/5. THIS ANSWERS the route's uncertainty (a): the full served constraint set is NOT empty at w<6/25, and the cuts do not bound w from below. Level D=x^{2w} falls: 12/25 at served (= the corpus's Lambda-BV level x^{12/25}), 2/5 at w=1/5, 1/4 at w=1/8, all below 1/2. The served 'unhandled boundary witness' (8/25,11/25) (delta+nu=19/25), outside at the served split, satisfies R1 and R2 at w=1/5 (slack 12/25 and 1/100) and at w=1/8 (111/200, 17/200); its entry thresholds are w<21/100 (right) and, new here, w<11/60 (left), so at w=1/5 it is inside the right region but still outside the left. NEW symmetric identity: the product ceiling on delta+nu=t is 4/3-w-y for the right AND the left orientation (both give the same condition), 313/300 at served, 13/12 at w=1/5 - all >1, so vacuous alone, which is exactly the note's stated reason for the vertical cutoff. NOT established: the union's product threshold, whose served 19/25 cap is carried by the OLD left regions (e.g. 5delta+2nu<123/50) whose w-dependence was NOT derived here; no consumption proof (the density bound R_IJ=O_H(x/log^H x) is not re-run at w=1/5); and the other three named inputs were not read. No twin-prime claim, no novelty claim, no served file edited.","prior_art_md":"No external novelty claim and no new online search was needed: the route's own prior art is inside the corpus and was extended by reading the sources it names. (1) arXiv:1502.00769v1 = Bettin-Chandee, 'Trilinear forms with Kloosterman fractions', was read AT SOURCE this run (PDF 310909 B via urllib, pdftotext -layout; Corollary 1 on p.4, eq. (1.4)). Verbatim content: hypotheses are f smooth on [M1/2,M1], g smooth on [M2/2,M2], alpha on [N1/2,N1], beta on [N2/2,N2], Delta != 0 fixed; error O((eta R)^{3/2}||alpha|| ||beta|| (N1N2)^{7/20+eps} (N1+N2)^{1/4} (M1M2)^eps) with R = M1N2/(M2N1)+M2N1/(M1N2) >= 2. CONSEQUENCE, and it is the one that matters for this route: the statement places NO restriction on the bands and none on the split, so this input is SILENT on w - it can be applied at w=1/5 or 1/8 but is exactly as unable to force w>=6/25 there as at the served split, and the register row recording 'added area: exactly zero' for it is not a row about w. Its price formula is exact: (17/20)(A+B)+(1/4)max(A,B) = (22max(A,B)+17min(A,B))/20, so the register's quoted condition 22max+17min<20 IS 'error exponent < 1' (verified as an identity). The 3*kappa/2 term the route quotes is the r-size split residue, not part of (1.4); it is not re-derived here. (2) The exact remaining gap, precisely scoped: the combined uniform product threshold (served 19/25) is set by the OLD left regions of the same note, not by the right orientation; re-deriving the union at w=1/5 requires those old regions' and the density bound's w-dependence, which no source read here supplies. (3) Deliberately NOT probed and NOT claimed absent: Deshouillers-Iwaniek, Drappeau, Topacogullari and Pascadi were not fetched this run (no channel try logged), so their stated parameter ranges remain the open half of the route's price list; Semantic Scholar was not contacted. (4) The corpus's own statements that the choice is unoptimised stand unchanged: prime-detection-spec.md section 3 (8) 'a convenient interior point below 1/4, not an optimized exponent'; reachability-coverage.md section 4 d-edge row 'the split inherits that choice'."},"research_route_id":46,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_197f49b1acafe40ce0c08163","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 triage. 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/46 and return #736. Return the ordinary report and transcript plus research: {route_id: 46, 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>, 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":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"727","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"736","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/46","transcript_url":"/projects/twin-primes/return/737/transcript","files":[{"sha256":"31a5719b259ab3d0af6906edadadd3bbaaa5e96460d27b144c5a01be035fe252","name":"job1530-report.md","bytes":10906},{"sha256":"763b2094694067e6ecf44491a7917627333ef427502177adab9632f1496cb3f7","name":"job1530-checks.py","bytes":10071},{"sha256":"24ac4f4f7ea0f8baef3d73cdbb87fbea698c536162ee85d80dbd856b1d8f8dd6","name":"job1530-checks.json","bytes":8840},{"sha256":"7168442665f067847feec208c1f1ce927ffa4b9c96e31752310ff0095ac079a5","name":"job1530-research.json","bytes":7148},{"sha256":"e04e16fe991c32684ae6d1ec571f04b0d3d3671a8424276720ec9dab1fd07433","name":"bettin-chandee-corollary1-excerpt.txt","bytes":2787}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}