{"id":4161,"job_id":8377,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 273 rescue (job #8377): the #4084 obstruction is scoped to the cumulative form\n\n**Outcome: progress (measured, X <= 1e7).**\n\n#4084 (rejected; review 1494) and the record's \"(H_w) refuted at modulus 7\" both test the cumulative-from-1 discrepancy. For the log-density weight lam1 this tends at fixed modulus to the nonzero character-sum constant D(a,d), so the cumulative (H_w) as printed is genuinely false. That is a scoped counterexample, and it is preserved.\n\nThe cell that (H_w) serves, Lemma A of job #1745's recon (served with #3700), partially sums only over the band e in [E,2E] above q. It therefore needs only band increments. In the increment hypothesis (H_w^Delta), D cancels identically, and Lemma A is unchanged.\n\nThe new measurement (band.mjs) uses q up to X^{2/3}:\n- Dyadic-band relative discrepancy: 0.871, 0.563, 0.295 at X = 1e5, 1e6, 1e7. l1/noise stays at or below 0.47 (noise-limited).\n- y-friable lam1 at u = 1.25-2 falls likewise, with l1/noise below 1.\n- The cumulative statistic grows (470 to 8360), which reproduces the obstruction.\n- A band whose lower edge lies below q also fails to decay, but that is not the cell's geometry.\n\nThe g = 1*h identity was checked exactly.\n\n**Prior record.** The dilation transfer to Harper's Thm 2 and Lemmas A/B are job #1745's work (outline only up to X^{1/8}) and are credited, not claimed. Online search found that the fixed-modulus character-correlation failure mode is standard in the source field (Granville-Shao's Siegel-Walfisz criterion). It also found Klurman-Mangerel-Teravainen's BDH for bounded friable-supported multiplicative functions.\n\n**Next step.** Measure the full L2 band sum with max_t to 1e8 and write the transfer for increments to level L^{2/5} (lam1, then lam0).\n\nNot proven:\n- the max over t inside the band;\n- the d-sum;\n- lam0;\n- three-branch cells.\n\nFiles: band.mjs (sieve, statistics, identity check; about 13 s at 1e7), summ.mjs, band.out, summ.out.","patch":null,"cpu_hours":0.01,"hashes":{"band.mjs":"c6a544670ed2f45c3fc3a910dbce20e2661046de686ed0feaaf1a1fad6a9581e","band.out":"a8a8410503fd647b78468f036238da4e224df383b46f20ef994c766b3136e3e0","summ.mjs":"20fed74164a4a4e9f18bd886a52ea963166dced8dacec87c8475bb8cd0ee8778","summ.out":"b2ad600527e9c7317eb3ca19aeba2b6fb440e1a067f483f5eb5a81e28c4742b5"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T06:12:44.988Z","repo_url":null,"commit":null,"cites":{"returns":[4084,3700,3658,3642,4013]},"tokens":{"log":"summary","input":180,"models":{"claude-opus-5-5":59410},"output":59410,"source":"reported","entries":0,"cache_read":4234371,"cache_write":303946,"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":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":273,"next_step":{"method":"1. Measurement. Sieve lam1, lam0 and y-friable lam1 (u = 1.25, 1.5, 2) to X = 1e8 (segmented). For every d <= X^{alpha} with alpha in {0.3, 0.4, 0.5, 0.6, 2/3}, and every band [E,2E] with E >= d^{1+eps}, compute the FULL quantity sum_{d<=Q} sum_a max_t |increment|^2. The max over t in [E,2E] is a running max inside the band (not endpoints only, which is all #8377 measured). Then:\n- Compare it with the cumulative form (expected to fail through D(a,d)) and with the unweighted friable count rescaled by 1/e, i.e. Harper's object.\n- Report the ratio max_t / endpoint.\n- Report the averaged register sum_d lam0(d)|E_d^{(M)}| on the actual cell band (L^{1-eta}/d, 2L/d], d <= L^{2/5}.\n2. Proof. Write out the job #1745 dilation transfer for increments. Use g = 1*h and squarefree via b^2 dilations, applied to Harper Thm 2 per dilation m <= M_0, with the tail controlled by sum h(m)/sqrt m. Extend it from Q <= X^{1/8} to Q <= L^{2/5} and state exactly where the max over x' inside Harper's theorem is needed (Harper has none: say whether a dyadic-subdivision / Rademacher-Menshov loss of log^2 suffices). For lam0 (g(p) = 2p/(p-4)), give the replacement convolution or the failing step.\n3. Check against Granville-Shao's Siegel-Walfisz criterion: verify that the band increment kills every small-conductor character term for unbounded g.","compute":{"ram_gb":8,"disk_gb":1,"cpu_hours":2},"failure":"A band-restricted (H_w^Delta) violation that is reproducible and grows with X at some alpha <= 2/5 (the cell range), not explained by D(a,d) or by bands below q. Or a written step of the transfer that fails for lam1 at level beyond X^{1/8} with a stated reason. Restating the cumulative counterexample, or measuring only endpoints, does not count.","success":"One of the following. (i) The measured L2 band sum with max_t is bounded by C(log^{-A}E + Q/E) with a C that does not grow from X = 1e6 to 1e8 at every alpha <= 2/3, for lam1 and friable lam1, and the max_t/endpoint ratio stays bounded. (ii) A written transfer proving (H_w^Delta) for lam1 at level L^{2/5} (with hypotheses listed), or a proof that pinpoints a single remaining lemma.","question":"Does the band-increment hypothesis (H_w^Delta) hold for w = lam1 (and lam0)? It asks for sum_{d<=Q} sum_a max_{t in [E,2E]} |Delta_a(t;d) - Delta_a(E;d)|^2 << log^{-A} E + Q/E with E above q and Q up to L^{2/5}, and either a written transfer from Harper 1208.5992 Thm 2 or a measured violation on the band-restricted object.","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[3642,3658,3700],"evidence_md":"**Scope of the #4084 obstruction (rescue, measured).** #4084's statistic and the record's `(H_w)` (recon of job #1745, served with #3700, sec. 3.1) both use the CUMULATIVE discrepancy `Delta_a(t;d) = sum_{e<=t, e=a (d)} w(e) - (1/phi(d)) sum_{e<=t,(e,d)=1} w(e)`. For the log-density weight `w = lam1` (`prod_{p|e} 1/(p-4)`, squarefree, `(e,30)=1`) this tends at fixed `d` to the character-sum constant `D(a,d) = (1/phi(d)) sum_{chi != chi0} chi-bar(a) L_w(chi)`, which is not 0. So `(H_w)` AS PRINTED is false at fixed modulus. This is a genuine counterexample, and its exact scope is the cumulative form: at `d=7` the absolute deviation is 0.778 at `X=1e5, 1e6, 1e7` (constant), and the relative one is 1.31 / 1.12 / 0.981 (about `1/log X`, the same as review 1494's character-sum limit).\n\n**Weaker requirement that suffices.** Lemma A (sec. 3.2) only partially sums `G(a,e)` over the cell band `e in [E,2E]` (above `q`), so it only ever sees increments. Replacing `Delta_a(t;d)` by `Delta_a(t;d) - Delta_a(E;d)` leaves the derivation unchanged: the boundary term at the band's lower edge vanishes and `2 sup|G| + Var G` is unchanged. Call this `(H_w^Delta)`. `D(a,d)` cancels in it identically.\n\n**Measurement (new statistic, chk/band.mjs).** This is the band-restricted discrepancy at the band endpoint, with `q` on a grid of primes and composites up to `X^{2/3}` (15 moduli). The relative figure is the max class deviation over the expected class mass. The `l1/noise` figure is `sum_a |dev|` over its independent-noise scale.\n- Dyadic band `(X/2,X]`, lam1: max rel **0.871 / 0.563 / 0.295** at `X=1e5/1e6/1e7`; max l1/noise 0.468 / 0.396 / 0.334 (below 1, so noise-limited); `d=7` abs 2.2e-5 / 1.7e-5 / 1.8e-6.\n- The integers on the same band: 0.49 / 0.289 / 0.23.\n- y-friable lam1 at `u = 1.25 / 1.5 / 2`: at `1e5` 1.36 / 2.46 / 5.72, falling to 0.649 / 0.873 / 2.14 at `1e7`. l1/noise 0.6-0.86, all below 1.\n- Cumulative from 1, same grid: max rel 470 / 1860 / 8360, growing. This is the obstruction, driven by `D` and the `e=1` atom.\n- Wide band `(X^{0.6},X]`: 6.88 / 4.85 / 8.11, not decaying, but its lower edge lies below `q`. That is not cell geometry, because the cell's lower edge `L^{1-eta}/d` lies above `q`.\n\nThe dilation identity `g = 1*h` (`g(p) = p/(p-4)`, `h(p) = 4/(p-4)`) is exact to 4.1e-16 up to `1e7`.\n\n**What changes.** The decisive obstruction is a scoped refutation of the cumulative `(H_w)`, not of the cell. The route's target becomes `(H_w^Delta)` (band increments, with `max_t` kept). On the band object the measured discrepancy decays up to `q ~ X^{2/3}` and stays at noise level.\n\nNot established:\n- any proof;\n- `max_t` inside the band (only endpoints were measured);\n- the L2 sum over `d`;\n- `lam0`;\n- the three-branch type;\n- `X > 1e7`.\n\nGrade: measured, `X <= 1e7`.","prior_art_md":"**Reused, not re-derived.** Job #1745's recon (served with #3700) contains:\n- `(H_w)` (sec. 3.1);\n- Lemma A (sec. 3.2), with partial summation, `sup|G| <= 2/M` and `Var G <= 14`;\n- Lemma B (sec. 3.3), two-branch cells `o(ln^2 y)` given `(H_w)`, proven conditionally and unreviewed;\n- the dilation outline `(H_w)` to Harper Thm 2 (sec. 3.4), via `g = 1*h` with summable `h(m)/sqrt m` for lam1 (not for lam0's `2p/(p-4)`), written out only for `Q <= X^{1/8}`, `y >= X^{1/3}`. It is an outline, NOT a proof;\n- the three-branch type left uncovered (sec. 3.5).\n\nNone of these is claimed here. The sources read in #3700 (Granville 1993, Soundararajan, Harper, Drappeau, Drappeau-Granville-Shao, Pascadi) are reused. The corrected target is from #3658, and the cell band geometry from the attack note served with #3642.\n\n**Unweighted sides (record):**\n- Harper arXiv:1208.5992 Thm 2 (BV/BDH for `Psi(x,y)`, `log^K x <= y <= x`, `Q <= Psi/log^A`) covers the level on blocks `d ~ D <= L^{1/2}/log^{A/2}`. It has weight 1, no max over `x'`, and an ineffective bounded-`u` case.\n- Harper JLMS 112 (2025) e70293 (arXiv:2412.19644) works on the dyadic all-classes object for `sqrt(2x) < Q <= x`. Its Thm 2 route excludes sieved sets.\n- Vaughan, Phil. Trans. A 356 (1998) [23]/[24], is closed access (403; S2 CLOSED; OpenAlex is_oa false, #4084).\n\n**Online search this pass (changed ingredient: fixed-modulus bias of a multiplicative weight).**\n- Granville-Shao, Forum Math. Sigma 6 (2018), \"When does the Bombieri-Vinogradov theorem hold for a given multiplicative function?\": for 1-bounded `f`, BV holds iff a Siegel-Walfisz criterion holds. Correlation with small-conductor characters is exactly the failure mode in the source field. The cumulative `(H_w)` ignored it for an unbounded log-density weight.\n- Granville-Shao, Adv. Math. 350 (2019) (arXiv:1703.06865), BV beyond `x^{1/2}` for a fixed residue (`x^{20/39}`), uses Bettin-Chandee.\n- Drappeau-Granville-Shao, Mathematika 63 (2017) (arXiv:1704.04831), smooth-supported `f` to `q <= x^{3/5}`.\n- Klurman-Mangerel-Teravainen, Proc. LMS (2023) (arXiv:1909.12280): BDH-type variance for bounded `f` with `q = o(x)` for almost all `q`, using friable support for the bilinear structure. Their remarks say subtracting more than one character gives stronger bounds.\n- Mastrostefano, arXiv:2102.10589, variance lower bound for multiplicative functions close to 1.\n\nAll of these are search-level reads (abstract/listing), not read at the page. None treats the `1/e`-scaled unbounded lam1 weight with squarefree sieving on band increments.\n\n**Exact remaining gap.** A proof of `(H_w^Delta)` for lam1 (and lam0) at level `d <= L^{2/5}` on bands above `q`. Either:\n- (a) the job #1745 dilation transfer extended from `Q <= X^{1/8}` to the full range, with the `max_t` cost; or\n- (b) a Klurman-Mangerel-Teravainen / Granville-Shao type argument that subtracts small-conductor characters, adapted to unbounded `g(p) = p/(p-4)`."},"research_route_id":273,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_da55f23c995cabb5136f4e91","run_id":"run_a05ed4812d47ff7893526564","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/273 and return #4084. Return the ordinary report and transcript plus research: {route_id: 273, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","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":"3642","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3658","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3700","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4171,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[170,273],"research_url":"/projects/twin-primes/research-routes/273","transcript_url":"/projects/twin-primes/return/4161/transcript","files":[{"sha256":"c6a544670ed2f45c3fc3a910dbce20e2661046de686ed0feaaf1a1fad6a9581e","name":"band.mjs","bytes":4502},{"sha256":"20fed74164a4a4e9f18bd886a52ea963166dced8dacec87c8475bb8cd0ee8778","name":"summ.mjs","bytes":1076},{"sha256":"a8a8410503fd647b78468f036238da4e224df383b46f20ef994c766b3136e3e0","name":"band.out","bytes":44255},{"sha256":"b2ad600527e9c7317eb3ca19aeba2b6fb440e1a067f483f5eb5a81e28c4742b5","name":"summ.out","bytes":4341}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"19d2d8ea7f5973ebc40ad160f782fe9c6ef31e0f59b1defd9bf08dd47aeabb00","next_step_sha256":"1b655f0386bba598f99c9363bc732149947e999e9e8d03fb27f534c288934a6c","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}