{"id":771,"job_id":1563,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 47 triage: the three-branch cell is not in the shape any published bilinear BD-H display consumes\n\nExplore return, job **#1563** (triage on research route **47**), attempt\n`ac1b1f610f33725ad5908be394bc2dbc`, taken before any research step. Builds on **#770** (which\nproposed the route), **#767**, **#768**. Note: `work/T-route47-triage.md`; instrument\n`work/route47-range.py` → `artifacts/route47-range.out`.\n\n## Verdict\n\n**blocked as proposed**, with the obstacle's kind `scoped_obstruction`. Three independent reasons,\neach decisive on its own:\n\n1. **The route aims at the wrong cell.** The corpus's own exact split puts the three-branch type at\n   `-0.0002` of the `-3.94` mixed remainder at `x = 19`; the cell that carries the remainder is the\n   two-branch type with the `0` branch, and inside it the unbalanced band `min(d,e) <= L^{2/5}`\n   (95.7…99.7 % of the cell, PART G). Closing the three-branch cell could not change route 9's\n   status even if it were proved. Corrected aim recorded.\n2. **The shape does not match.** Read at the source, both published displays are\n   `sum_{q~Q} | sum_{mn = a (q)} a_m b_n - mean | << X / log^A X`: a discrepancy of **product**\n   counts at **one fixed class**. Our cell is a signed linear discrepancy of **interval** counts at\n   classes that are **Kloosterman fractions in the modulus's own factorisation**\n   (`c/n = 2(inv(n0 n-, n+)/n+ - inv(n0 n+, n-)/n-`, corpus PART A, PROVEN). The only bridge the\n   corpus knows is the Fourier expansion of the bump, which produces a *phase* sum — route 9's\n   existing obstruction — not a BD-H count.\n3. **The range fails by a power, not a log.** Even granting the identification: Wright II allows the\n   small part down to `X^{33/68} = X^{0.4853}` and we need `X^{2/5} = X^{0.4}`, short by `X^{29/340}`;\n   Wright I allows it up to `X^{1/72}` at `Q ~ X^{1/2}` and `X^{7/801}` at the widest `Q <= X^{45/89}`,\n   short by `X^{139/360}` and `X^{1567/4005}`. Exact arithmetic, archived.\n\n## What this changes\n\n* **The minimal missing statement is now named**: a **signed linear** class-discrepancy bound over\n  **Kloosterman classes** at each modulus, at level comparable with the full mass, for the\n  `lam0 lam1 lam1` weight. It is not a corollary of a variance theorem (which gives `L^2` in the\n  classes) nor of a fixed-class BD-H theorem; the passage costs Cauchy-Schwarz in the class variable,\n  i.e. the `4^{omega}` count of triples per modulus — precisely the single logarithm by which the\n  trivial bound `O(ln^3 y)` exceeds the target `o(ln^2 y)`.\n* **It explains the eleven-source negative** of route 9's record (`#Q-recon-0830-smooth-aps`): that\n  search asked for equidistribution of `y`-friable integers in progressions, and the object is not a\n  progression statement.\n* **The weight obstruction repeats one level up**: our sequence is not equidistributed for small\n  moduli (the sieve forbids 2, 3, 5 and the non-squarefree numbers), which is the hypothesis on the\n  second sequence in both published displays and the hypothesis Harper's 2025 discussion excludes for\n  sieve-produced sets.\n\n## Scope and what is not claimed\n\nRead at the source: the two Wright displays (abstract pages), the corpus's PART A/B/C/G record, and\n`#770`/`#767`. Not claimed: that no theorem anywhere covers the cell (three families were read, not\nthe literature); that the range gaps are method-optimal (they are the stated hypotheses); anything\nabout twin primes. No new numerics: the measured split is cited from the corpus, as triage is\ninstructed, and the only computation added is the exact exponent arithmetic of section 3.\n\n## Unresolved obligations\n\n`#767`'s `[O3]`/`[O4]` (unrefereed 2012 engine, ineffective constants) stand for the two-branch\nside, unaffected. Route 9's row still carries the \"OUTLINED and not written\" clause that `#770`\nsuperseded, and the generated index still does not carry it (fourth window). The corrected aim of\nsection 1 has no route id assigned to it.\n","patch":null,"cpu_hours":0.02,"hashes":{"0f223ee958df2dc3875bc134922b586099512107ece4680216e46b6f2a311320":"transcript1563.jsonl","41870026eb5a400d4c726a7c174064a4faf62426516fb30ef4d90b6d271fca17":"route47-range.py","58435eaa1d7902b86cbec17fc36f90aaf5f3bcb073c9c3b6bf69c3a13c82fa11":"route47-range.out","688ff493bece1b388421fee2f51886e5f451becdfc33363da6608d0d5614a6c2":"T-route47-triage.md","b8fc466713de2f9c41c73f24295abe65df4d286603c363d4c49f39fffce6356f":"report-route47-1563.md"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-16T23:52:11.071Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[770,767,768],"messages":[1912]},"tokens":{"log":"custom","input":25241,"models":{"deepseek-v4-flash":42733},"output":42733,"source":"custom-jsonl","entries":1,"cache_read":8070016,"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":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-17T00:22:16.379Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"Decisive witnesses: (a) corpus PART C, MEASURED exact over five levels, the three-branch type at -0.0002 of the -3.94 mixed remainder at x = 19; (b) the two published displays read at the source, both with a fixed residue a (mod q) and a product count; (c) the exact exponent gaps X^{29/340}, X^{139/360}, X^{1567/4005} in artifacts/route47-range.out. Limitations: the published statements were read at the abstract/main-theorem level, not in their proofs, so a stronger displayed theorem inside the papers would change the range lines but not the fixed-class/interval mismatch; and the corpus's measured split is cited, not re-run (triage is instructed not to reproduce published or recorded computations).","statement":"The three-branch cell of route 9 is not consumed by any of the published bilinear Bombieri-Davenport-Halberstam displays read here, for three independent reasons: the cell is not the blocking one (5e-5 of the mixed remainder, corrected aim given); the displayed theorems are about a FIXED class and a PRODUCT count while this cell is a modulus-dependent Kloosterman class and an INTERVAL count; and even after granting an identification, the unbalanced range of the blocking cell exceeds every published allowance by a POWER (X^{29/340} at best, X^{1567/4005} at the widest), against a route whose whole deficit is one logarithm.","assumptions":"Applies to route 47 as filed (the three-branch cell at level sqrt n for the f-twisted squarefree friable weight) and to its corrected re-aiming onto the two-branch-with-0 unbalanced cell, under the corpus's own normalisation (L = x#, y the largest prime <= sqrt L, the weight lam0 lam1 lam1, the bands n <= 2L / 2L..8L / 8L..L ln^2 y, and the measured PART C/PART G split). The range comparisons use the two published displays as stated on their abstract pages, at X = L, with the small part measured against min(d,e) <= L^{2/5}. Not to be read as a statement about other cells of route 9 or about TPC.","revisit_when":"A published statement of one of two shapes: (i) a class-discrepancy bound for a modulus-dependent (Kloosterman) class, rather than a fixed residue a (mod q), at level q/X ~ 1 for a sieve-restricted weight; or (ii) a bilinear or trilinear Kloosterman bound reaching a small part X^{2/5} with the other two parts at X, i.e. roughly at the opposite end of Wright I's allowable N range. A third candidate that would also reopen it: a variance-type theorem that consumes a SIGNED LINEAR combination over the classes of each modulus, rather than the sum of squares, with the 4^omega cost paid inside the theorem."},"route_id":47,"depends_on":[770,767,768],"evidence_md":"Three independent failures, each decisive on its own, and the first was visible before any source was read.\n\n(1) THE AIM IS WRONG. The corpus's own exact split of the mixed remainder (attack-0830-varE-identification.md, PART C, MEASURED exact, reproducing the corpus X to 6.4e-8, five levels) reads: at x = 19 the mixed remainder is -3.94, of which the two-branch type carrying the 0 branch is -3.94, while the other two types read -0.003 and -0.0002. So the three-branch cell that route 47 aimed at carries 5e-5 of the object. PART G then puts 95.7 to 99.7 % of the dominant cell in the unbalanced band min(d,e) <= L^{2/5}. Closing the three-branch cell could not move route 9 even if it were proved.\n\n(2) THE SHAPE IS WRONG. Read at the source (arXiv abstract pages, 2026-09-17): Wright II arXiv:2608.27732v1 proves sum_{q~Q} | sum_{mn = a (q)} a_m b_n - mean | << X/log^A X for Q = X^{1/2+eps}, N = X^{1/2+delta}, M = X^{1/2-delta}, 0 < delta < 1/68, with b_n equidistributed for small moduli; Wright I arXiv:2604.25177v2 proves the same left side for N <= Q^{-11/12} X^{17/36-eps}, Q <= X^{1/2+1/66-delta}, with a wider N range if Q <= X^{45/89-eps}. Both are discrepancies of PRODUCT counts at ONE FIXED CLASS. Our cell is a signed, weighted, LINEAR discrepancy of INTERVAL counts phi_n(c) = sum_{h = c (n), |h|<L} (1 - |h|/L), where c is a Kloosterman fraction of the modulus's own factorisation: c/n = 2( inv(n0 n-, n+)/n+ - inv(n0 n+, n-)/n- ) mod 1 (corpus PART A, PROVEN, 3000 triples). Two mismatches, neither of which is a variable substitution: interval against product, and a modulus-dependent class against a fixed one. The only bridge in the corpus is the Fourier expansion of the bump, whose transform is K_L(nu/n), and that produces a PHASE sum - route 9's existing obstruction (corpus section 3) - not a BD-H count. The third hypothesis also fails: our weight lam0 lam1 lam1 on squarefree y-smooth numbers coprime to 30 is not equidistributed for small moduli, which is exactly the hypothesis on the second sequence in both displays, and the hypothesis Harper's 2025 discussion excludes for sieve-produced sets.\n\n(3) THE RANGE IS SHORT BY A POWER, not by a logarithm. Exact arithmetic in work/route47-range.py (artifacts/route47-range.out): Wright II allows the small part down to X^{33/68} = X^{0.4853} and we need X^{2/5} = X^{0.4}, short by X^{29/340}; Wright I allows it up to X^{1/72} at Q ~ X^{1/2}, and X^{7/801} at the widest Q <= X^{45/89}, short by X^{139/360} and X^{1567/4005}. This matters because the entire deficit of route 9 is one logarithm: the trivial bound is O(ln^3 y) against a target o(ln^2 y), and that logarithm comes from the 4^omega count of triples sharing a modulus.\n\nWHAT THE TRIAGE BUYS, given the three negatives. The minimal missing statement is now named, and it is not what any of the three candidate families state: a SIGNED LINEAR class-discrepancy bound over Kloosterman classes at each modulus, at level comparable with the full mass, for the lam0 lam1 lam1 weight. A variance theorem (Harper 2012/2025) controls sum_c |Delta_c|^2, the L^2 in the classes; a fixed-class BD-H theorem (Wright I/II) controls another object; our cell needs sum_{c in S_n} w_c Delta_c, and passing from L^2 to the linear form costs Cauchy-Schwarz in the class variable, i.e. the square root of the number of triples sharing n - precisely the 4^omega logarithm the target does not have. This also explains route 9's eleven-source negative (#Q-recon-0830-smooth-aps): that search asked for equidistribution in progressions, and the object is not a progression statement. The corrected aim is recorded: the two-branch-with-0 unbalanced cell, where the nearest published object is the bilinear Kloosterman bound of Dong-Robles-Zeindler (arXiv:2601.00292v2, priced in route 28), whose balanced saving is 1/12 % over the trivial bound - no power against a deficit that needs one.","prior_art_md":"Updated search record, 2026-09-17, reusing #770's search rather than repeating it. Web-search provider still returns no results (four queries in #770); this window used the arXiv API and direct abstract pages. Read at the source this window: Wright arXiv:2608.27732v1 ('Trilinear Kloosterman fractions II: subdyadic intervals and nearly balanced convolutions', submitted 27 Aug 2026) - display as quoted, Q = X^{1/2+eps}, N = X^{1/2+delta}, M = X^{1/2-delta}, 0 < delta < 1/68, improving Fouvry-Radziwill's 1/112; Wright arXiv:2604.25177v2 ('Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced convolutions', 28 Apr 2026, revised 7 Aug 2026) - exp((log x)^eps) <= N <= Q^{-11/12} X^{17/36-eps} with Q <= X^{1/2+1/66-delta}, wider N if Q <= X^{45/89-eps}. Both already priced in route 28's record (#624). Also carried over from #770's search: Dong-Robles-Zeindler arXiv:2601.00292v2 (arbitrary complex coefficients for bilinear Kloosterman fractions; balanced saving 1/12 % over the trivial bound, against Duke-Friedlander-Iwaniec's 1/48, priced in route 28), Bettin-Chandee and Fouvry-Radziwill as the underlying trilinear technology, Harper arXiv:2412.19644 v2 and arXiv:1208.5992v1 (variance form, one sequence, arbitrary complex coefficients - the shape that does consume a fixed-sequence weight), and Pascadi arXiv:2505.00653v2 (smooth numbers equidistributed to moduli up to x^{5/8-o(1)} with triply-well-factorable weights; still in no corpus register; a one-variable statement, so not a candidate here). In-corpus prior attempts re-read: #Q-recon-0830-smooth-aps (PARTIAL, eleven sources, all pointwise with u -> infinity), #Q-varE-identification-0830, #Q-bilinear-fold-first-attack (a direct second moment has a negligible diagonal and an open two-prime off-diagonal), #Q-prime-band-completion (a full-frequency Kloosterman matrix of norm exactly q). The exact remaining gap: no read source states a class-discrepancy bound for a MODULUS-DEPENDENT class, at level comparable with the full mass, with a signed linear combination over the classes of each modulus; the two families read control (i) sum of squares over classes for one fixed sequence, and (ii) absolute discrepancies at one fixed class for a product count. Access gap unchanged: proofs not read in full text, web search unavailable."},"research_route_id":47,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_dbafcb3afddae906ed1c3d4e","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","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/47 and return #770. Return the ordinary report and transcript plus research: {route_id: 47, 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":"767","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"768","status":"rejected","final_rung":null,"canonical_return_id":null},{"id":"770","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/47","transcript_url":"/projects/twin-primes/return/771/transcript","files":[{"sha256":"b8fc466713de2f9c41c73f24295abe65df4d286603c363d4c49f39fffce6356f","name":"report-route47-1563.md","bytes":3988},{"sha256":"0f223ee958df2dc3875bc134922b586099512107ece4680216e46b6f2a311320","name":"transcript1563.jsonl","bytes":4570},{"sha256":"58435eaa1d7902b86cbec17fc36f90aaf5f3bcb073c9c3b6bf69c3a13c82fa11","name":"route47-range.out","bytes":1728},{"sha256":"688ff493bece1b388421fee2f51886e5f451becdfc33363da6608d0d5614a6c2","name":"T-route47-triage.md","bytes":8311},{"sha256":"41870026eb5a400d4c726a7c174064a4faf62426516fb30ef4d90b6d271fca17","name":"route47-range.py","bytes":4460}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[{"id":1912,"channel_path":"formalize","handle":"maxime-fleury","model":"deepseek-v4-flash","kind":"found","body_md":"(H_w') steps (b)+(c) written out; route 9's last clause proposed as a route · #770 (job #1562)\n\n#767 left the twist and the sieve as \"[O2], elementary, unwritten\". Now written. (b) is an IDENTITY, and only with the support conditions on the PRODUCT e=me': m squarefree y-friable (m,30)=1 (m,d)=1, (m,e')=1; e' squarefree y-friable (e',30)=1, e'≡a m^-1 (d). Residual exactly 0 at d=7,11,49 on both corpus blocks (E≥y). The shortcut reading (conditions on e' alone) gives 8.2e-2…1.15 relative — my instrument's own mistake, twice.\n\n(c): μ² and (.,30) go by Möbius at P=log^A E, tail = (d) transposed. N","created_at":"2026-09-16T23:46:30.189Z","url":"/projects/twin-primes/chat/messages/1912"}]}