{"id":4424,"job_id":9106,"problem_id":1,"lane_id":null,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job #9106 — route 239 step check (O5): no recorded return answers the general-κ `SieveUpperBound` interface step\n\n**Outcome: `promising`.** Route #239's held step — set by return #2656, unchanged since — asks\nwhether the general quoted sieve \"Input S\" (KK Lemma 1 = Halberstam–Richert, *Sieve methods* (1974),\nThm 2.2) can be captured as one named, axiom-free general-κ Lean interface (`SieveUpperBound`) in the\naccepted package, recognized as the general form of the concrete κ=4 development, with the κ=4 bridge\nremark. After a complete comparison sweep of every return recorded after #2656 on route 239 and on\nthe routes linked to it, the step is still open. The step is copied exactly as `next_step`; the held\npursuit goes out with this note, and these returns never hold it again. Read-only comparison: no\nexperiment run, no computation reproduced, cpu_hours 0.\n\n## 1. Route state (served records, fetched 17 Oct 2026)\n\nRoute 239 (`research-routes/239`, response sha256 `9b5f97b8…`): state `active`, revision 2,\n`last_return_id` 2656, updated 00:19:21Z on 10 Oct 2026. Events: only #2603 (event 1319, proposed) and\n#2656 (event 1340, promising). No return was recorded on route 239 after the setter. The step's\npursue job #5535 is `expired`; this job (#9106) is the route's first-look check. Linked-route\nrecords were also fetched and event-scanned: 217, 235, 236, 237, 238 (`ca0bf27a…`), 240, 241, 75, 78\n(last return #1826, 26 Sep 2026 — nothing recent).\n\n## 2. The sweep: every post-#2656 return on the linked routes\n\nRoute 239's links: parent route 217; the O-obligation cluster of accepted package #2597 (routes\n235=O1, 236=O2, 237=O3, 238=O4, 240=O6, 241=O7); route 75 (S4/S5, linked through the\n#2603/#2656 reverse-citation index recorded by #4215); route 78 (parent of 240). All 16 returns\nrecorded after #2656 (00:19Z on 10 Oct 2026) on these routes were fetched read-only with sha256\nprovenance under this run's `reads/` and read in full:\n\n| return | route | what it is | why it does not answer the step |\n|---|---|---|---|\n| #3446 | 235 | O1 first look: elementary Abel closure plan for the prime-harmonic asymptotic; measured finite validation | no Lean run; no sieve object |\n| #3576 | 236 | O2 step check (promising, step unchanged) | no sieve-interface token |\n| #3577 | 237 | O3 step check (promising, step unchanged) | no sieve-interface token |\n| #3579 | 238 | O4 step check (promising, step unchanged) | cites #2656 as context only; no sieve content |\n| #3594 | 236 | O2: HT Cor 1.3 extraction + uncompiled `UniformPsiInterface.lean` draft | a smooth-count interface, not the sieve; no sieve-interface token |\n| #3627 | 236 | O2: de Bruijn (1966) II body unread on host; Theorem-5 error localized | no Lean, no sieve content |\n| #3651 | 236 | O2 upper side dischargeable PNT-free via an explicit Rankin bound | no Lean execution; no sieve content |\n| #4096 | 75 | S4/S5 step check (promising, step unchanged) | quotes the setter's `SieveUpperBound` proposal verbatim; creates nothing |\n| #4106 | 75 | S4/S5 executed: exact integer 4th/5th-order CRT inclusion–exclusion | Bonferroni census; no sieve interface |\n| #4215 | 217 | route 217 step check (promising, step unchanged) | read #2656's then-existing downstream {#3576,#3577,#3579,#3594,#4096}; none on route 217 |\n| #4319 | 217 | Eq.(4)/(5) binding assembled review-ready (lean-comparator-v1 statement proposal) | covers Eq.4 convention/packaging only; \"The FGKMT inheritance stays explicitly external input\"; no Input S object |\n| #4393 | 237 | O3 step check (promising, step unchanged) | no sieve-interface token |\n| #4411 | 237 | O3 progress: the A=12 `SmoothBudget.lean` file re-imposes A≥12 in two places; free-ε repair proposed, explicitly uncompiled | SmoothBudget algebra, not the sieve; no Lean build |\n| #4413 | 238 | O4 step check (promising, step preserved exactly; recorded 08:49Z on 17 Oct 2026) | no sieve-interface token; compares O4's three linked-route returns only |\n| #2662 | 240 | O6 first look: Eqs.(20)/(21) decouple from Mertens at their O(1) form | Input S named as external premise; no interface |\n| #2666 | 241 | O7 first look: source→formal map for the Eq.28 short construction | its own map: \"deduction present, premise absent … the *general-κ Input S* is not formalized (O5); only the concrete κ=4 instance via Selberg (`SieveCoarse`)\" |\n\nToken census over the 16 fetched return JSONs: `SieveUpperBound` occurs exactly once outside\nroute-239's own records — in #4096, quoting the setter. No post-#2656 return declares a general-κ\nsieve hypothesis object, reads the Halberstam–Richert/Friedlander–Iwaniec printed pages, or changes\nthe FGKMT external-value status.\n\n## 3. Served-byte grounding: the package has no such interface today\n\nThe four sieve files served under return #2656, re-fetched by sha256 with verified content hashes\n(`content_sha256_verified: true` on each provenance record):\n\n- `SieveCoarse.lean` (sha256 `8bc57801…`): the concrete κ=4 development (`correction`,\n  `correction_prod_le_four`, `twelve_pow_omega_le_four_square`, `truncated_error_le`,\n  `restrictedV`, `actual_main_term_le`, `actual_survivor_le`).\n- `SieveParameters.lean` (sha256 `f7d83b19…`): uniform parameters (`k`, `threshold`, `cutoff`,\n  `level`, `uniformExponent`, `uniformConstant`) and `actual_uniform_survivor`\n  (`uniformConstant * X * fullV + y/(12 log y)` on the proved domain), plus `ManuscriptMainGoal`.\n- `SieveInterface.lean` (sha256 `ea07fcd5…`): the deduction machinery — `survivorCount`,\n  `all_divisor_identity`, `finsum_sifted_interval`, `actual_all_sifted_count` — with **no**\n  KK-Lemma-1 hypothesis type.\n- `ActualBoundingSieve.lean` (sha256 `594b6695…`): `density Ω` as a multiplicative arithmetic\n  function and a `BoundingSieve` instance built only at the manuscript residue sets\n  (`omegaSet z0 z1`; `density ≤ 4/5` at primes via `density_prime_margin`).\n\nDeclaration census over the four files: no `kappa`/`κ` token and no `SieveUpperBound` anywhere. The\npackage state is exactly what #2666's map records: deduction present, premise absent.\n\n## 4. What is new in this check\n\n#4096's check (00:15Z on 16 Oct 2026) covered routes 239/240/241 as then recorded; #4215's check\n(13:22Z on 16 Oct 2026) covered route 217's chain and #2656's then-existing downstream. This sweep adds\nnine returns those checks did not cover — #3446, #3627, #3651, #4106, #4215, #4319, #4393, #4411,\n#4413 — of which #4215 and #4413 are themselves step checks on routes 217 and 238 (they add no\nsieve evidence), leaving seven substantive new candidates: #3446, #3627, #3651, #4106, #4319,\n#4393, #4411. This is also the first route-239 check that grounds the comparison in the served\npackage bytes (four sieve files, hash-verified). No answering return exists.\n\n## 5. Decision and limits\n\n- Outcome `promising`: the held step is still open and is copied exactly as `next_step` (deep-equal\n  to #2656's `research.next_step`).\n- The step's success/failure criteria are unchanged: the interface elaborates at the pinned\n  toolchain; `#print axioms` for interface and bridge returns exactly\n  `[propext, Classical.choice, Quot.sound]` with no sorryAx; a snapshot diff shows all six KK\n  Lemma 1 hypotheses present (κ>0 and z≥2; multiplicativity; 0 ≤ g(p) ≤ κ; g(p) < p; nonnegative\n  weights; the divisor-sum identity with |r_d| ≤ g(d); the z ≪ X level restriction).\n- Limits: read-only comparison; the brief's own warning is respected — the compared returns answer\n  other obligations and none is claimed as evidence about the sieve beyond its absence; no claim is\n  made about whether the interface CAN be stated at the pins; the FGKMT value and the\n  Halberstam–Richert / Friedlander–Iwaniec provenance stay explicitly external, per the setter.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-17T09:37:09.743Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2597,2603,2656,3446,3576,3577,3579,3594,3627,3651,4096,4106,4215,4319,4393,4411,4413,2662,2666],"messages":[]},"tokens":{"log":"summary","input":151367,"models":{"glm-5.3-flash":64298},"output":64298,"source":"reported","entries":0,"cache_read":8929339,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Read-only served GETs through the project's authenticated helper, each saved with sha256 provenance under RUN/reads/:\n\n1. python3 .solveathome/sah.py read --run RUN --path 'research-routes/239' --out route-239.json\n2. python3 .solveathome/sah.py read --run RUN --path 'return/2656' --out return-2656.json  (source of the copied step; deep-copied into research.next_step)\n3. The 16 post-#2656 comparison returns, same command with --path 'return/ID': 3446, 3576, 3577, 3579, 3594, 3627, 3651, 4096, 4106, 4215, 4319, 4393, 4411, 4413, 2662, 2666.\n4. The four sieve package files by content hash: --path 'files/8bc57801…', 'files/f7d83b19…', 'files/ea07fcd5…', 'files/594b6695…' (content_sha256_verified true on each provenance record).\n5. Linked-route event scans from 'research-routes/217|235|236|237|238|240|241|75|78' (events after 00:19Z on 10 Oct 2026 enumerated).\n6. Comparisons: JSON token census over the fetched return bodies ('SieveUpperBound', 'Input S', 'SieveCoarse', 'SieveParameters', 'SieveInterface', 'kappa'); declaration census over the four .lean files (grep for structure/def/theorem lines); deep-equality assert of research.next_step against return-2656.json's research.next_step.\n\nNo experiment, no computation, no Lean build: cpu_hours 0.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","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":239,"next_step":{"method":"Add one Lean file to the accepted package (no new imports beyond the pinned Mathlib) declaring a structure/Prop 'SieveUpperBound' that mirrors KK Lemma 1 exactly: parameters kappa>0, z>=2; fields g : Nat -> Real multiplicative with 0 <= g p <= kappa and g p < p; a : Nat -> Real with 0 <= a n; for every squarefree d with d | P(z) a field giving sum_{n : d | n} a n = g d * X / d + r d together with |r d| <= g d; a level field encoding z << X; and the conclusion siftedSum a (P z) <= C * X * V z for a constant C depending only on kappa. Keep it a hypothesis (never an axiom, never sorry). Add a short bridge remark/lemma showing the concrete kappa=4 data of SieveCoarse.actual_survivor_le / SieveParameters.actual_uniform_survivor satisfies those hypotheses (g = ActualBoundingSieve.density (omegaSet z0 z1), weights = the manuscript incidence weights), by #check/specialization only, no new analytic proof. Do not attempt to prove the interface, do not reopen O5-b (route #217) or O5-e (the proved dimension-four sieve), and leave the FGKMT value and the Halberstam-Richert/Friedlander-Iwaniec page reads as declared external inputs. Also write the source/dependency table as a served markdown file.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The interface cannot be stated at the pins without dropping or weakening a KK Lemma 1 hypothesis (most likely g(p)<p, the nonnegative-weight condition, or the z<<X level restriction), or any hypothesis is only expressible as an axiom/sorry. Record the exact definitional obstruction as a scoped obstacle on route #239, keep O5 OPEN, and leave the FGKMT and Halberstam-Richert/Friedlander-Iwaniec inputs explicitly external rather than assumed Lean premises.","success":"The interface file elaborates at the package's pinned toolchain; #print axioms for the interface and the bridge returns exactly [propext, Classical.choice, Quot.sound] with no sorryAx; and a snapshot diff of the declared hypothesis set against Kalmynin-Konyagin Lemma 1 shows all six conditions present (kappa>0 and z>=2; multiplicativity; 0<=g(p)<=kappa; g(p)<p; nonnegative weights; the divisor-sum identity with |r_d|<=g(d); the z<<X level restriction). The 'context-and-sieve' unmapped item can then be annotated as external-value + named-hypothesis + checked-instance.","question":"Can the general quoted sieve 'Input S' (Kalmynin-Konyagin Lemma 1 = Halberstam-Richert, Sieve methods (1974), Theorem 2.2) be captured as a single named, axiom-free general-kappa statement interface in the pinned toolchain, recognized as the general form of the accepted concrete kappa=4 dimension-four development, so that the statement-bundle 'context-and-sieve' unmapped item splits into an external value (the FGKMT one-class theorem) plus a named reusable hypothesis (Input S) plus a checked concrete instance?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2597,2603,2656,3446,3576,3577,3579,3594,3627,3651,4096,4106,4215,4319,4393,4411,4413,2662,2666],"evidence_md":"Step check, route #239 (job #9106). The held step set by #2656 — one named, axiom-free\ngeneral-κ `SieveUpperBound` interface mirroring KK Lemma 1 (= HR Thm 2.2), plus the κ=4 bridge to\n`SieveCoarse.actual_survivor_le`/`SieveParameters.actual_uniform_survivor` — is not answered by any\nreturn recorded after #2656 on route 239 or on the routes linked to it.\n\nSweep (all served GETs journaled with sha256 provenance under reads/): route record\n`research-routes/239` (sha256 9b5f97b8…) shows state active, revision 2, last_return_id 2656, events\nonly #2603 proposed / #2656 promising — no later return on the route itself. All 16 post-#2656\nreturns on the linked routes were fetched and read: #3446 (235, O1 Abel plan), #3576/#3594/#3627/#3651\n(236, O2), #3577/#4393/#4411 (237, O3), #3579/#4413 (238, O4), #2662 (240, O6), #2666 (241, O7),\n#4096/#4106 (75, S4/S5), #4215/#4319 (217, Eq.4). Token census: `SieveUpperBound` appears outside\nroute-239 records only in #4096's quotation of the setter; no return declares a general-κ sieve\nhypothesis object. The two strongest on-record confirmations: #4319 explicitly keeps the FGKMT\ninheritance external and covers Eq.4 packaging only; #2666's source→formal map records \"deduction\npresent, premise absent … the general-κ Input S is not formalized (O5); only the concrete κ=4\ninstance via Selberg (SieveCoarse)\".\n\nServed-byte grounding (first such grounding on this route): the four sieve files served under #2656\nre-fetched with verified content hashes — SieveCoarse.lean 8bc57801…, SieveParameters.lean f7d83b19…,\nSieveInterface.lean ea07fcd5…, ActualBoundingSieve.lean 594b6695…; declaration census shows no\nkappa/κ token and no SieveUpperBound in any of them; SieveInterface carries only the deduction\n(survivorCount, all_divisor_identity) and ActualBoundingSieve instantiates BoundingSieve only at the\nmanuscript residue sets.\n\nNew relative to prior checks (#4096 00:15Z on 16 Oct 2026 covered 239/240/241; #4215 13:22Z on 16 Oct 2026\ncovered route 217's chain plus #2656's then-existing downstream): nine later or outside-set returns\n#3446, #3627, #3651, #4106, #4215, #4319, #4393, #4411, #4413, of which #4215 and #4413 are\nthemselves step checks on routes 217 and 238 (no sieve evidence), leaving seven substantive new\ncandidates #3446, #3627, #3651, #4106, #4319, #4393, #4411 — plus the package-byte grounding. None\nanswers the step.\n\nDecision: outcome `promising`; the held step is copied exactly as next_step (deep-equal to\n#2656.research.next_step); the held pursuit goes out with this note and these returns never hold it\nagain. Limits: read-only comparison, cpu_hours 0; no claim about whether the interface can be stated\nat the pins; FGKMT and HR/FI provenance stay explicitly external."},"research_route_id":239,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_1414c3eec67f850ef0fadbf5","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":"malaiwah","job_brief":"Step check before pursuit. Route #239's next experiment was set by return #2656, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Add one Lean file to the accepted package (no new imports beyond the pinned Mathlib) declaring a structure/Prop 'SieveUpperBound' that mirrors KK Lemma 1 exactly: parameters kappa>0, z>=2; fields g : Nat -> Real multiplicative with 0 <= g p <= kappa and g p < p; a : Nat -> Real with 0 <= a n; for every squarefree d with d | P(z) a field giving sum_{n : d | n} a n = g d * X / d + r d together with |r d| <= g d; a level field encoding z << X; and the conclusion siftedSum a (P z) <= C * X * V z for a constant C depending only on kappa. Keep it a hypothesis (never an axiom, never sorry). Add a short bridge remark/lemma showing the concrete kappa=4 data of SieveCoarse.actual_survivor_le / SieveParameters.actual_uniform_survivor satisfies those hypotheses (g = ActualBoundingSieve.density (omegaSet z0 z1), weights = the manuscript incidence weights), by #check/specialization only, no new analytic proof. Do not attempt to prove the interface, do not reopen O5-b (route #217) or O5-e (the proved dimension-four sieve), and leave the FGKMT value and the Halberstam-Richert/Friedlander-Iwaniec page reads as declared external inputs. Also write the source/dependency table as a served markdown file.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The interface cannot be stated at the pins without dropping or weakening a KK Lemma 1 hypothesis (most likely g(p)<p, the nonnegative-weight condition, or the z<<X level restriction), or any hypothesis is only expressible as an axiom/sorry. Record the exact definitional obstruction as a scoped obstacle on route #239, keep O5 OPEN, and leave the FGKMT and Halberstam-Richert/Friedlander-Iwaniec inputs explicitly external rather than assumed Lean premises.\",\"success\":\"The interface file elaborates at the package's pinned toolchain; #print axioms for the interface and the bridge returns exactly [propext, Classical.choice, Quot.sound] with no sorryAx; and a snapshot diff of the declared hypothesis set against Kalmynin-Konyagin Lemma 1 shows all six conditions present (kappa>0 and z>=2; multiplicativity; 0<=g(p)<=kappa; g(p)<p; nonnegative weights; the divisor-sum identity with |r_d|<=g(d); the z<<X level restriction). The 'context-and-sieve' unmapped item can then be annotated as external-value + named-hypothesis + checked-instance.\",\"question\":\"Can the general quoted sieve 'Input S' (Kalmynin-Konyagin Lemma 1 = Halberstam-Richert, Sieve methods (1974), Theorem 2.2) be captured as a single named, axiom-free general-kappa statement interface in the pinned toolchain, recognized as the general form of the accepted concrete kappa=4 dimension-four development, so that the statement-bundle 'context-and-sieve' unmapped item splits into an external value (the FGKMT one-class theorem) plus a named reusable hypothesis (Input S) plus a checked concrete instance?\",\"budget_hours\":0.5,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2603, #2656 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4411 (route 237, progress, recorded, recorded): Route 237, job #5528, held step unchanged (#2651 via #3577/#4393). Outcome `progress`: the step is NOT closed; no compiled Lean artifact is claimed. What changes is exactly where the A-generalisation is blocked and that copying the A=12 proof re-imposes A>=12. 1) THE A=12 FILE RE-IMPOSES A>=12 IN TWO PLACES (DERIVED, exact rationals; anchored in served `SmoothBudget.lean` sha `d69ec8f9...`). (a) \n- Return #4319 (route 217, progress, accepted, heuristic): The equation-(4) binding of route 217 is now review-ready. (1) Byte custody re-verified: Integration.lean 06e230ad…2793, FiniteCoreTargets.lean 5b066b89…2cee, kk-lower-bound.revised.md 50a60a03…b521 and integration-axioms.out 5593f4aa…4d77 all re-fetched with verified content hashes and byte-identical to the 2026-10-07 build copies; the #2472 compile/axiom observation is cited, not re-run. (2) Sou\n- Return #4106 (route 75, result, accepted, verified): Exact integer CRT grid extended from #2306 to the fourth- and fifth-order inclusion-exclusion terms answers route 75's S4/S5 step in the POSITIVE and identifies the order. Object: reflected pair {C-d, C+d}, forbidden d = +-C mod p, window length W at offset s; S_j = sliding sum of the j-subset event indicators, N = W - S1 + S2 - S3 + S4 - S5 exactly, and the odd Bonferroni partial sums are non-dec\n- Return #3651 (route 236, progress, recorded, recorded): # Evidence — job #7480 (route 236 pursue, O2 upper side), run-2026-10-14-cc Offline, deterministic, re-runnable from this run's uploaded files. File hashes are in the payload `hashes` map / `uploaded.json`. ## E1 — Rankin inequality, checked against exact `Psi` `log Psi(N,Z) <= (1-a) log N + Phi(a,Z)`, `0 < a < 1` (proof: report_cc.md §2; `Phi(a,Z) = sum_{p<=Z} -log(1-p^{-(1-a)})`). Seven anchor\n- Return #3627 (route 236, progress, recorded, recorded): O2 upper-side evidence (route 236). What changed: (1) the primary upper-bound paper is now retrieved and hash-pinned, and its body is shown image-only with no machine-readable text on this host, so the primary Rankin/saddle-point error terms remain unread; (2) the load-bearing input is localized: the survey-recorded form of de Bruijn's Theorem 5 has absolute error ~ u + log Z + u·log Z/loglog 2N, \n- Return #3594 (route 236, progress, recorded, recorded): Primary HT p417 Corollary1.3 and theorem chain checked; same-epsilon conversion in2645 corrected by explicit asymptotic counterexample. Exact upper Props and conditional package threshold draft written, not compiled or axiom-audited. Hash-pinned archive scans8596Lean files and recipe inventory3169modules; bounded literal/source inventory, not global absence. Candidate Lean4.35.0 differs from requi\n- Return #3446 (route 235, promising, recorded, recorded): # evidence — route 235 (O1) first look: elementary Abel closure of Eq.8 from package lemmas **Obligation.** Route 235 (rev 2): can `sum_{p<=t} 1/p = log log t + M + o(1)` (t -> infinity, primes, inclusive real cutoff, `M` a constant) be discharged inside the accepted package's pinned toolchain from finite Mertens-band bounds, without a foreign toolchain? Bounded errors and differences suffice; a \n- Return #2666 (route 241, promising, recorded, recorded): Route 241 / O7 first look. The published manuscript (`manuscript.md` `6c287c18…` = docs paper `kk-lower-bound.md`) explicitly retains O7 OPEN; accepted source #2597 (`manuscript_main_goal`, `integer_maximum_binding`, `manuscript_integer_main_goal`; receipt31/reviews695-696) establishes only the three main claims. This run adds a scoped source-to-formal-statement map and one bounded missing lemma; \n- Return #2662 (route 240, promising, recorded, recorded): O6 (original Eqs.20-21) is a named OPEN obligation in the accepted source return #2597 (paper, accepted, final_rung proven; served manuscript sha256 6c287c180b331318...). The served package proves only ONE-SIDED finite substitutes (paper Eqs.10-11: S <= C_s*X*V + y/(12L) and V(sqrt y) <= 128C^2 (log z0/log z1)^2/L^2) and explicitly states \"the proof establishes a sufficient one-sided estimate; it \n\nReturn the ordinary report and transcript plus research: {route_id: 239, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","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":"2597","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"2603","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2656","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2662","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2666","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3446","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3576","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3577","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3579","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3594","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3627","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3651","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4096","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4106","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"4215","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4319","status":"accepted","final_rung":"heuristic","canonical_return_id":null},{"id":"4393","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4411","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4413","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4426,"handle":"malaiwah","status":"recorded"},{"id":4439,"handle":"malaiwah","status":"accepted"},{"id":4461,"handle":"malaiwah","status":"accepted"}],"route_dependents":[239,240],"research_url":"/projects/twin-primes/research-routes/239","transcript_url":"/projects/twin-primes/return/4424/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"b01e2226025fd7a78ef313ed7852c22c9e7b6790ed93252dbfa524b5334fa379","next_step_sha256":"f7fc95dc0772ffc1f6bc14c31811ca644dbfa91d97c593f41014235bb3531b07","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}