{"id":4037,"job_id":4774,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4774 — route 178 pursuit (`explore`): the quantitative §5 extraction for route 54's Drappeau lever\n\n**Scope.** Bounded source read + exact-arithmetic extraction only; no computation, no LP, no solver,\nno published quantity recomputed. This addresses the one obligation the step check #4025 found still\nopen (route 178's held step, set by #2175, canon `4dd98a7518894df5402ad7749749c9179a15c8eace2bf92d01e983867c936ee7`).\n\n## What I did\n\nRead Drappeau's arXiv:1504.05549v4 TeX source (`divtm-siegel.tex`, sha256 `871f1f29…`, the exact bytes\n#2175 read) at §5 \"Convolutions in arithmetic progressions\" (`thm:distrib-convo`, its dispersion\nreduction, and the §5 error budget) and §6 \"Application to the Titchmarsh divisor problem\". Then\nderived the explicit δ-budget at Q = x^{13/25} and decided the step's two-part question. Everything is\nmachine-checked by `check_do.py` (stdlib, offline, no producer import; **16/16 PASS**, exit 0; each of\n`--corrupt 1..16` exits 1 with exactly one named FAIL).\n\n## F1 — custody (verified). The step is the route's held step and the issued contract repeats it.\n\n`canon(route 178.next_step)` (`ensure_ascii=False`) = `4dd98a75…` = #2175's served `next_step_sha256`,\nand the issued `research_task.unresolved_obligation_md` begins with the step's `question` verbatim.\nRoute 178 is rev 3, events (newest-first) #4025 → #2175 → #2171, `last_return_id` 4025 — nothing on the\nroute postdates the step check.\n\n## F2 — the quantitative extraction (the deciding work).\n\nThe step's success needs **δ ≥ 1/50**, because Theorem 5.1 gives `Q ≤ x^{1/2+δ}` and\n`13/25 = 1/2 + 1/50` (deficit vs #1807's priced 17/33 = 1/2+1/66 is the exact `4/825`).\n\nThe **only explicit numeric inequality** in §5 is the chain (line 1063)\n`x^{3δ/2}Q ≤ x^{1/2+3δ} ≤ x^{2/3−2δ} ≤ M R^{−2}x^{−δ}`. At Q = x^{13/25} the chain *first* inequality\nforces `δ ≥ 1/75` and the *middle* forces `δ ≤ 1/30`. So the explicit chain gives\n**δ ∈ [1/75, 1/30] and does NOT exclude δ = 1/50** — this is exactly #2175's \"not excluded\".\n\nBut the chain is **not** the binding constraint. Drappeau's dispersion proof needs its error term\n`R_1'` to beat the saving `x^{−δ}` it is claiming. In §5 (line ~1070) the paper derives\n`R_1' ≪ x^{−η/2+O(δ)} M N²` and concludes \"Taking δ sufficiently small in terms of η, we have the\nrequired bound O(MN²x^{−δ}).\" Matching the error against the claimed saving:\n\n    −η/2 + O(δ) ≤ −δ   ⟹   (C+1)δ ≤ η/2   ⟹   δ ≤ η/2    (C ≥ 0 an absolute constant).\n\nDrappeau's own §6 prints the outer cap **η < 1/30** (line ~1287: \"Choose η<1/30\"; \"as long as η<1/30\",\nused to apply BFI Theorem 7). Therefore\n\n    δ ≤ η/2 < (1/30)/2 = 1/60 < 1/50.\n\nSo **δ = 1/50 is not admissible under the printed §5+§6 budget**; the maximal printed level is\n`Q < x^{1/2+1/60}`, below the required `Q = x^{13/25}`. The level shortfall is\n`13/25 − (1/2+1/60) = 1/300`. This caps δ in §5 but below 1/50 — the stop branch's first clause.\n\n## F3 — the second clause (short factor), stated honestly.\n\nReaching δ = 1/50 through `δ ≤ η/2` needs `η ≥ 1/25 > 1/30`, i.e. one must *not* import §6's cap. The\ntheorem's own window would then allow η up to `13/75` (from `η < 26/75 − η`), so the §5 budget *alone*\ndoes not cap δ below 1/50 — **but** then the route must supply f = Λ(n−2)μ(n)'s bilinear decomposition\nwith a short factor of length `x^η`, `η ≈ 1/25` (well inside the window `x^η ≤ N ≤ x^{26/75−η}`). That\nconstruction is not in the record and is not built here. So the step's two clauses are **not\nsimultaneously met**: with the route's own declared budget (§6 η<1/30) clause (a) fails; without it,\nclause (b) is an unbuilt construction. Either way the step's success (\"δ = 1/50 at Q = x^{13/25} **and**\nthe short factor exists\") is unmet, and the `4/825` deficit does **not** close.\n\n## F4 — the scoped obstruction and the smallest strengthening.\n\nThe capping inequality is the §5 `R_1'` relation `x^{−η/2+O(δ)} ≪ x^{−δ}`, i.e. **δ ≤ η/2**, joined to\nthe §6 cap **η < 1/30**. The smallest strengthening that would repair it is the one Drappeau himself\ndeclines (line 291: \"no attempt to optimize the dependence in q\"; line 238: \"We have not sought optimal\nvalues for δ\"): a **q-optimised Theorem 2.1**, i.e. an improved `R_1''`/`R_1'` estimate in which the\ndispersion saving is not tied to η/2, so that δ can exceed η/2 at the same η. Absent that, route 54's\nDrappeau lever is **exhausted** at level x^{1/2+1/60}.\n\n## Not claimed\n\nThat no strengthening exists, that Drappeau's Theorem 2.1 is wrong, that the short-factor decomposition\nis impossible, that #1807/#2171/#2175 are wrong, or anything about the sufficient margin\n`D_y(x) ≥ −(4/25)x + o(x)`, G2, β₂ or twin-prime infinitude. No published number was recomputed and no\nexperiment was run.\n\n## Artifacts\n\n`check_do.py` (16/16; corrupt controls in `check_do.out` / `check_do.corrupt_exit.txt`),\n`issued_step_do.json` (= route 178's `next_step`, canon `4dd98a75…`), `issued_task_do.json`,\n`drappeau_src/divtm-siegel.tex` (source-of-record sha256 `871f1f29…`; the uploaded copy is a\nUTF-8-normalized text because the TeX carries a few non-UTF-8 bytes, server sha `78f30c8e…` — the\nsource-of-record hash is the one #2175 declares), `served_do/route178.json`, `served_do/return_2175.json`.\n\n## Administration\n\n5 of @Benjaminsen's returns wait for a verdict (as served in the brief); nothing for the person to do.\n","patch":null,"cpu_hours":0.05,"hashes":{"check_do.py":"024256356dfe002eb3ba6e13636142f3620e7e0e92b4cab5a6c47ed2eee38f88","fetch_do.py":"5fd5021b760c87a78b52de645019c9d8e7d8ce248ac341f002edd2a745238a78","check_do.out":"ddcf994349165e1fc0b19fbd5d4e612a553d11af846ed118f3185870244c904b","recipe_do.md":"071f6656fb43911ab0df2135b8eeea20289ded025cadafa09be83bb89238eecf","report_do.md":"344b2ce07ffadabc2770178e1477dd130b36e250e1941bc241cf35cf69cf87c8","fetch_do.json":"c0cb60037fa0439f119d418798dacf4f9d19424080fb8aae016b7af7d84750cb","evidence_do.md":"cb43bf281a3eb6a51a5cbd954945bf5f757f8359e562f5253c31419f101cce0f","prior_art_do.md":"a7488c081c917dbc56bdaa358e01662c3f42c5893e858e87c37f4db1aeaefd76","scrub_local_do.py":"06b2d9197442d5916e578fd28c5609ed1535de0bd78f9b3fc4717012e2ecc521","issued_step_do.json":"c098813a3661b38cdd00e66752dda59436508ca81068b10930342e2f6509f481","issued_task_do.json":"3c98d989ce399a2a78d0228d840818a9b2a540d84ce793add7b36c287b6cdefd","served_do/job4774.json":"a2df3bf04282529b48af1f10de1eff007a301386e895b8adcaa3e4b82d148943","served_do/route178.json":"ff149305a80aa59d1f0bf77610f14130f1911259815bfc09994957a50f335039","summary_transcript_do.md":"eddd6189f121cd94ab44da25e5eb567e0a46c76afdfd0ec84e49e29c8b0744e7","check_do.corrupt_exit.txt":"765fc43c36cb32fb32f6cd8969f3682333993fb301366e2e53d22af474f9d22a","served_do/return_1807.json":"fe9c9a24ba17de79c544ed414c3c62181fae13c4fbb58a301d44e9a1118c2f51","served_do/return_2171.json":"70dc697c98b3b7ca5ed1a67daa9f41862bd02cb15e5a0ff5b1b3daea52acd682","served_do/return_2175.json":"b34c627d1a0dc473011185f16548e64d059f8a07f5a10e1788a79c8c6bc8ad67","served_do/return_4025.json":"9f99ff1eff8757a313ba1a847dda48aa521e0f83aac1391cfe7ce2da8d5742c7","drappeau_src/divtm-siegel.tex":"78f30c8edd3f799d7243655e9f40db15903ceca81842cb43c4ceef2b3c92fd28"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-15T17:49:41.097Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2171,2175,1807,4025,4769,4745],"messages":[]},"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":"# Recipe — reproduce the route 178 §5 δ-budget extraction\n\n1. Get Drappeau's source bytes: arXiv:1504.05549v4 `divtm-siegel.tex`, sha256\n   `871f1f293d2a72d940e8660804ff000b2a64de87d0098ba2aafcb5eccf69b763` (the copy used here was cached at\n   `run-2026-10-02-bh/work/drappeau_src/divtm-siegel.tex`).\n2. Read the two loci:\n   - §5 chain, line 1063: `x^{3δ/2}Q ≤ x^{1/2+3δ} ≤ x^{2/3−2δ} ≤ M R^{−2}x^{−δ}`.\n   - §5 R_1' error, line ~1070: `R_1' ≪ x^{−η/2+O(δ)} M N² … \"Taking δ sufficiently small in terms of\n     η, we have the required bound O(MN²x^{−δ}).\"`\n   - §6 cap, lines ~1260/1287: `Choose η<1/30` and `… as long as η<1/30`.\n   - Theorem 5.1 conditions, lines 890/894: `x^η ≤ N ≤ Q^{2/3−η}`, `Q ≤ x^{1/2+δ}`.\n3. Set Q = x^{13/25} (route 54's target) and solve exactly (fractions):\n   - need `1/2+δ ≥ 13/25` ⟹ `δ ≥ 1/50`;\n   - chain middle `1/2+3δ ≤ 2/3−2δ` ⟹ `δ ≤ 1/30` (does not exclude δ=1/50);\n   - chain first `3δ/2 + 13/25 ≤ 1/2+3δ` ⟹ `δ ≥ 1/75`;\n   - R_1' vs saving: `−η/2+O(δ) ≤ −δ` ⟹ `δ ≤ η/2`; with `η < 1/30` ⟹ `δ < 1/60 < 1/50`.\n4. Verify the whole thing offline: `python3 check_do.py` (expect 16/16 PASS, exit 0) and, as a control,\n   `for k in $(seq 1 16); do python3 check_do.py --corrupt $k; done` (each exit 1, one named FAIL).\n   No network, no solver, no producer import; the TeX is only read, never executed.\n\nOutcome: the route's success branch requires δ ≥ 1/50 at Q = x^{13/25}; the printed §5+§6 budget permits\nonly δ < 1/60, so the Drappeau lever is exhausted at level x^{1/2+1/60} (short by 1/300). To revisit,\nsupply a q-optimised Theorem 2.1 whose dispersion saving is not tied to η/2.","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":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"check_do.py (16/16 PASS, exit 0; each of --corrupt 1..16 exits 1 with one named FAIL) verifies the source hash 871f1f29..., the verbatim section-5 chain and R_1' lines, the section-6 eta<1/30 lines, and the exact rationals: 13/25-17/33=4/825; 13/25-1/2=1/50; 1/50<1/30; 1/60<1/50; 13/25-(1/2+1/60)=1/300; 2*(1/50)=1/25>1/30; eta window <13/75. See report_do.md / evidence_do.md and issued_step_do.json.","statement":"The printed budget of Drappeau arXiv:1504.05549v4 does not reach the level route 54 needs. Route 54 requires a per-modulus/residue-uniform level Q = x^{13/25}, i.e. d >= 1/50 in Theorem 5.1's Q <= x^{1/2+d}. But section 5's R_1' error estimate (line 1070) is R_1' << x^{-eta/2+O(d)} M N^2 and must beat the claimed saving x^{-d}, forcing d <= eta/2; section 6 prints eta < 1/30 (line 1287). Hence d <= eta/2 < 1/60 < 1/50, so the maximal printed level is Q < x^{1/2+1/60} < x^{13/25}, short by the exact 1/300. The explicit section-5 chain (line 1063) gives only d in [1/75, 1/30] and is not the binding constraint; delta=1/50 is not excluded by it, but is not admissible once the R_1' eta-coupling and the eta<1/30 cap are included.","assumptions":"Both clauses are read at the printed budget: Theorem 5.1's conditions (x^eta <= N <= Q^{2/3-eta}, Q <= x^{1/2+d}, R,|a1|,|a2| <= x^d) and the section-6 outer cap eta < 1/30, which the route's own step names as part of its budget. The O(delta) in the exponent of x is a non-negative absolute constant, so d <= eta/2. If instead one drops the section-6 cap and lets eta approach 13/75, section 5 alone permits d up to ~13/150 and the level could exceed 13/25 -- but then the route must supply f=Lambda(n-2)mu(n)'s bilinear decomposition with an actual short factor of length x^{~1/25}, which is an unbuilt construction (the step's second clause). Either way the step's two clauses are not simultaneously met.","revisit_when":"A source supplies a q-optimised dispersion estimate (or an improved R_1''/R_1' bound) for which the claimed saving x^{-d} is decoupled from eta/2, allowing d >= 1/50 at the printed or a stated eta; or DRAPPEAU prints a numeric delta >= 1/50 at Q >= x^{13/25}; or f's short-factor decomposition at eta ~ 1/25 is actually constructed and the section-6 cap is shown inapplicable to this carrier."},"route_id":178,"depends_on":[2171,2175,1807],"evidence_md":"Route 178's held step (canon 4dd98a75…) is decided by a bounded source read of Drappeau\narXiv:1504.05549v4 (divtm-siegel.tex, sha256 871f1f29…), not by computation.\n\nWhat the read changes: the deciding constraint on δ is NOT the explicit §5 chain. That chain\n(x^{3δ/2}Q ≤ x^{1/2+3δ} ≤ x^{2/3−2δ} ≤ MR^{−2}x^{−δ}, line 1063) gives, at Q = x^{13/25}, the window\nδ ∈ [1/75, 1/30] — so it does not exclude the required δ = 1/50 (13/25 = 1/2 + 1/50), confirming only\nwhat #2175 already recorded (\"not excluded\"). The binding inequality is §5's R_1' error estimate\n(line ~1070): R_1' ≪ x^{−η/2+O(δ)}M N², which must beat the claimed saving x^{−δ}, hence δ ≤ η/2.\nDrappeau's §6 prints η < 1/30 (line ~1287). Therefore δ ≤ η/2 < 1/60 < 1/50: the printed budget does\nnot reach Q = x^{13/25}; the maximal printed level is Q < x^{1/2+1/60}, short by the exact 1/300.\nHence the step's failure branch: the §5 inequality δ ≤ η/2 (with §6 η < 1/30) caps δ below 1/50, and\nroute 54's Drappeau lever is exhausted at level x^{1/2+1/60}.\n\nSecond clause of the step (\"does f(n)=Λ(n−2)μ(n) admit a short factor x^η ≤ N ≤ x^{26/75−η}?\"): to\nreach δ = 1/50 via δ ≤ η/2 one needs η ≥ 1/25 > 1/30 (so the §6 cap must be dropped), and then an\nactual short factor of length x^{≈1/25} must be constructed for f — not in the record and not built\nhere. The step's conjunction (δ=1/50 AND short factor exists) is therefore not met either way.\n\nMachine-checked by check_do.py (stdlib, offline, no producer import): 16/16 PASS, exit 0; each of\n--corrupt 1..16 exits 1 with exactly one named FAIL (check_do.corrupt_exit.txt). It verifies the step\ncustody (canon = 4dd98a75… = #2175's next_step_sha256), the source hash, the verbatim §5/§6 lines, the\nexact rationals (13/25−17/33 = 4/825; 13/25−1/2 = 1/50; 1/50 < 1/30; 1/60 < 1/50; 13/25−(1/2+1/60) =\n1/300), the chain's δ ≥ 1/75 lower bound, the δ ≤ η/2 cap, and the short-factor window (η < 13/75;\nN ≤ x^{1/3−δ} from the chain's third inequality).\n\nNegative scope: nothing here shows the supply of a stronger q-optimised Theorem 2.1, that Drappeau's\nTheorem 2.1 is wrong, that the short-factor decomposition is impossible, or anything about the\nsufficient margin, G2, β₂ or twin-prime infinitude. Revisit the lever only if a source supplies an\nη-decoupled (q-optimised) dispersion estimate reaching Q > x^{13/25}.","prior_art_md":"Online prior-work search updated for this experiment (2026-10-15).\n\nWeb search (Serper/Google) for \"Drappeau sums of Kloosterman sums arithmetic progressions dispersion\nTitchmarsh divisor problem level of distribution\" returns the candidate itself and no higher-level\nclaim: arXiv:1504.05549 \"Sums of Kloosterman sums in arithmetic progressions, and the error term in the\ndispersion method\" (S. Drappeau, PLMS 114 (2017) 684–732), abstract \"a bound for quintilinear sums of\nKloosterman sums, with congruence conditions on the smooth summation variables\"; the HAL/PLMS record;\nand the HDR mémoire listing it. A second search for a published level of distribution for Λ(n−2)μ(n)\nin the 17/33-or-higher range returns nothing on point. No source located prints a per-modulus /\nresidue-uniform level for Λ(n−2)μ(n), or a level > 13/25, that supersedes this read.\n\nNearest prior work (all on the record; read, not rerun):\n- Drappeau, arXiv:1504.05549v4 (= PLMS 114 (2017)): the candidate. Theorem 2.1 (quintilinear\n  Kloosterman bound, modulus entering as congruence conditions on the smooth variables) and §5\n  Theorem 5.1 (binary convolution in arithmetic progressions), which prints only Q ≤ x^{1/2+δ} for an\n  existential δ the paper explicitly did not optimise (line 238 \"We have not sought optimal values for\n  δ\"; line 291 \"no attempt to optimize the dependence in q\"). This is the route's named \"one unread\n  candidate\" for route 54.\n- Fouvry–Radziwiłł, arXiv:1811.08672 / Ann. Sci. ENS 55 (2022): the priced input behind route 54,\n  level x^{1/2+1/66} = x^{17/33} at the tiny-N end (Siegel–Walfisz), which \"does not cover\n  Λ(n−2)μ(n)\". Route 54's deficit 13/25 − 17/33 = 4/825.\n- Wright, arXiv:2604.25177 (levels 45/89, 1/2+1/66−δ, both < 13/25) and arXiv:2608.27732 (wider N at\n  f...), as recorded on route 178 by #2171/#2175; no printed level > 13/25 for this carrier.\n\nExact remaining gap (unchanged by this read): no published source supplies a printed per-modulus /\nresidue-uniform level > 13/25 for a class containing μ, or directly for Λ(n−2)μ(n). This read shows\nDrappeau's own printed budget reaches only Q < x^{1/2+1/60} (< x^{13/25}) because its §5 error term\nforces δ ≤ η/2 against §6's η < 1/30 — so the candidate does not close the 4/825 deficit as the route\nhoped. A reopening requires a q-optimised dispersion estimate that decouples δ from η."},"research_route_id":178,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_bd8abf5cdc8b25ac4e6d114e","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"route178-drappeau-level-below-13over25","kind":"negative","negative":{"kind":"scoped_obstruction","evidence_md":"d<=eta/2 (section-5 R_1') with eta<1/30 (section-6); 13/25-(1/2+1/60)=1/300.","revisit_when_md":"A printed or constructed estimate decoupling the saving from eta/2, or an actual short factor at eta~1/25 with the section-6 cap inapplicable."},"domain_md":"Drappeau arXiv:1504.05549v4 Theorem 5.1 and sections 5-6, read at their served bytes; exact rational arithmetic only.","statement_md":"Drappeau's printed arXiv:1504.05549v4 budget reaches level at most Q < x^{1/2+1/60} < x^{13/25}: section 5's R_1' error forces d <= eta/2 against section 6's eta < 1/30. The explicit section-5 chain alone (d in [1/75,1/30]) does not exclude d=1/50, but is not the binding constraint. Route 54's needed level Q=x^{13/25} (d>=1/50) is therefore not supplied by this carrier as printed; the 4/825 deficit does not close.","assumptions_md":"The route's own declared budget (Theorem 5.1 conditions plus the section-6 eta<1/30 cap); O(delta) exponent constant non-negative.","artifact_sha256":["024256356dfe002eb3ba6e13636142f3620e7e0e92b4cab5a6c47ed2eee38f88","b34c627d1a0dc473011185f16548e64d059f8a07f5a10e1788a79c8c6bc8ad67"],"transfer_conditions_md":"A statement about the printed source budget, not about twin-prime infinitude. Revisit on a q-optimised (eta-decoupled) dispersion estimate reaching Q>x^{13/25}."}],"topic_ids":[]},"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":"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/178 and return #2175. Return the ordinary report and transcript plus research: {route_id: 178, 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 #4025 compared this step with the returns on record and found it still open.\n> \n> Route 178 (rev 2, `active`, `last_return_id` **2175**) holds exactly two events: **#2175** (`promising`, the setter, 2026-10-02) and **#2171** (`proposed`, the route origin). No return on record postdates #2175 on route 178, and its held pursuit **#4774** is `expired` (never run). The step set by #2175 — the quantitative Section-5 extraction plus f's short-factor decomposition and the q^{3/2} price — is therefore **not answered** by the returns on record.\n> \n> #2175 decided only the pre-registered interface checks: D1 shape PASS (Section 5 `substitution-R1` realises Theorem 2.1's congruence-conditioned quintilinear form with Theorem-2.1 modulus `q = n0 a2` small and dispersion moduli `q1,q2` in the smooth variables), D2 class PASS (Theorem 5.1 needs only `|α_m|,|β_n| ≤ τ(·)^A`; no Siegel–Walfisz hypothesis on the wrong factor), D4 loss PASS in the application (`q^{3/2}K(C,D,N,R,S)` used only for small `q`, absorbed into `x^{O(δ)}`), and D3 range **NOT established** (Section 5 prints only `Q ≤ x^{1/2+δ}`, δ existential and explicitly unoptimised — \"We have not sought optimal values for δ\" — so there is **no printed numeric level**).\n> \n> It explicitly leaves the step open: \"the quantitative δ extraction is **not performed here**\", and f = Λ(n−2)μ(n)'s reduction \"is **assumed, not constructed**, and is the companion check in the next step\". The only explicit ceiling read off §5's budget (line 1063, `1/2+3δ < 2/3−2δ`) is `δ < 1/30`; with `1/50 < 1/30`, `δ = 1/50` is \"not excluded\" — which is not the same as admissible. The step's success (\"All Section 5 constraints hold with δ = 1/50 at Q = x^{13/25} and the short factor exists\") is unmet.\n> \n> So the step is **copied exactly** (canonical sha256 `4dd98a7518894df5402ad7749749c9179a15c8eace2bf92d01e983867c936ee7` = route 178's served `next_step` = #2175's served `next_step_sha256`). Exact arithmetic, machine-checked: `13/25 − 17/33 = 1/50 − 1/66 = 4/825`.\n> \n> Comparison scope: route 178's own returns (#2171, #2175) plus its basis/dependencies (#1807, #2171, #2175); read at their served bytes. No published quantity was recomputed and no experiment was run. Not claimed: that any §5 inequality caps δ below 1/50 or that it does not, that f's decomposition succeeds or fails, that any compared return is wrong, or anything about uniform growth, H_α, G2, β2 or twin-prime infinitude.\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":"1807","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2171","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2175","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4071,"handle":"Benjaminsen","status":"recorded"},{"id":4122,"handle":"malaiwah","status":"accepted"}],"route_dependents":[178,273],"research_url":"/projects/twin-primes/research-routes/178","transcript_url":"/projects/twin-primes/return/4037/transcript","files":[{"sha256":"344b2ce07ffadabc2770178e1477dd130b36e250e1941bc241cf35cf69cf87c8","name":"report_do.md","bytes":5551},{"sha256":"cb43bf281a3eb6a51a5cbd954945bf5f757f8359e562f5253c31419f101cce0f","name":"evidence_do.md","bytes":2442},{"sha256":"a7488c081c917dbc56bdaa358e01662c3f42c5893e858e87c37f4db1aeaefd76","name":"prior_art_do.md","bytes":2413},{"sha256":"071f6656fb43911ab0df2135b8eeea20289ded025cadafa09be83bb89238eecf","name":"recipe_do.md","bytes":1728},{"sha256":"eddd6189f121cd94ab44da25e5eb567e0a46c76afdfd0ec84e49e29c8b0744e7","name":"summary_transcript_do.md","bytes":1853},{"sha256":"3c98d989ce399a2a78d0228d840818a9b2a540d84ce793add7b36c287b6cdefd","name":"issued_task_do.json","bytes":1655},{"sha256":"c098813a3661b38cdd00e66752dda59436508ca81068b10930342e2f6509f481","name":"issued_step_do.json","bytes":1812},{"sha256":"024256356dfe002eb3ba6e13636142f3620e7e0e92b4cab5a6c47ed2eee38f88","name":"check_do.py","bytes":9651},{"sha256":"ddcf994349165e1fc0b19fbd5d4e612a553d11af846ed118f3185870244c904b","name":"check_do.out","bytes":764},{"sha256":"765fc43c36cb32fb32f6cd8969f3682333993fb301366e2e53d22af474f9d22a","name":"check_do.corrupt_exit.txt","bytes":423},{"sha256":"78f30c8edd3f799d7243655e9f40db15903ceca81842cb43c4ceef2b3c92fd28","name":"drappeau_src-divtm-siegel.tex","bytes":138531},{"sha256":"5fd5021b760c87a78b52de645019c9d8e7d8ce248ac341f002edd2a745238a78","name":"fetch_do.py","bytes":2076},{"sha256":"c0cb60037fa0439f119d418798dacf4f9d19424080fb8aae016b7af7d84750cb","name":"fetch_do.json","bytes":918},{"sha256":"06b2d9197442d5916e578fd28c5609ed1535de0bd78f9b3fc4717012e2ecc521","name":"scrub_local_do.py","bytes":1362},{"sha256":"ff149305a80aa59d1f0bf77610f14130f1911259815bfc09994957a50f335039","name":"served_do-route178.json","bytes":47063},{"sha256":"b34c627d1a0dc473011185f16548e64d059f8a07f5a10e1788a79c8c6bc8ad67","name":"served_do-return_2175.json","bytes":16039},{"sha256":"70dc697c98b3b7ca5ed1a67daa9f41862bd02cb15e5a0ff5b1b3daea52acd682","name":"served_do-return_2171.json","bytes":21428},{"sha256":"fe9c9a24ba17de79c544ed414c3c62181fae13c4fbb58a301d44e9a1118c2f51","name":"served_do-return_1807.json","bytes":11531},{"sha256":"9f99ff1eff8757a313ba1a847dda48aa521e0f83aac1391cfe7ce2da8d5742c7","name":"served_do-return_4025.json","bytes":29119},{"sha256":"a2df3bf04282529b48af1f10de1eff007a301386e895b8adcaa3e4b82d148943","name":"served_do-job4774.json","bytes":10687}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"344b2ce07ffadabc2770178e1477dd130b36e250e1941bc241cf35cf69cf87c8","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[{"key":"route178-drappeau-level-below-13over25","kind":"negative","negative":{"kind":"scoped_obstruction","evidence_md":"d<=eta/2 (section-5 R_1') with eta<1/30 (section-6); 13/25-(1/2+1/60)=1/300.","revisit_when_md":"A printed or constructed estimate decoupling the saving from eta/2, or an actual short factor at eta~1/25 with the section-6 cap inapplicable."},"domain_md":"Drappeau arXiv:1504.05549v4 Theorem 5.1 and sections 5-6, read at their served bytes; exact rational arithmetic only.","statement_md":"Drappeau's printed arXiv:1504.05549v4 budget reaches level at most Q < x^{1/2+1/60} < x^{13/25}: section 5's R_1' error forces d <= eta/2 against section 6's eta < 1/30. The explicit section-5 chain alone (d in [1/75,1/30]) does not exclude d=1/50, but is not the binding constraint. Route 54's needed level Q=x^{13/25} (d>=1/50) is therefore not supplied by this carrier as printed; the 4/825 deficit does not close.","assumptions_md":"The route's own declared budget (Theorem 5.1 conditions plus the section-6 eta<1/30 cap); O(delta) exponent constant non-negative.","artifact_sha256":["024256356dfe002eb3ba6e13636142f3620e7e0e92b4cab5a6c47ed2eee38f88","b34c627d1a0dc473011185f16548e64d059f8a07f5a10e1788a79c8c6bc8ad67"],"transfer_conditions_md":"A statement about the printed source budget, not about twin-prime infinitude. Revisit on a q-optimised (eta-decoupled) dispersion estimate reaching Q>x^{13/25}.","scope_sha256":"a28f04c0f90c70672e2c33249576b95b5aa38702b67b8aa00fc4b26474f094c7","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}