{"id":2651,"job_id":5413,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# O3 map: which fixed-A>4 smooth input is missing (route #237, job #5413)\n\n**Outcome: `promising`.** This first look returns the exact dependency/quantifier map route #237\nasked for. It shows the Eq.23 parameter limits are complete for every fixed A≥4 with B=chooseB(A),\nand that the single missing input is the smooth-number estimate for Ψ, which splits into a\n**finite-Rankin upper-bound route** (closes the consequence only for A>240/23≈10.435) and the\n**printed Input-H equality** (Eq.9, OPEN as O2; needed for 4<A≤240/23). No proof was executed; the\nprinted equality is left OPEN.\n\n## 1. What Eq.23–24 require\n\nSet X=m+2, Z=z₁=exp(L·L₃/(A·L₂)), u=log X/log Z, with L=log y, L₂=log L, L₃=log L₂ and\nB=chooseB(A)=1+6·survivorConstant·A⁴. Eq.23 (checked limits) and Eq.24 (the claim) are, verbatim\nfrom the retained manuscript (`manuscript.md` = served `kk-lower-bound.md`, sha256 `6c287c18…`):\n\n- Eq.23: `log X = L+O(L₂)`, `u = log X/log Z ∼ A L₂/L₃`; with ε=1/2 the range `u ≤ Z^{1-ε}` holds\n  eventually; `u log u = (A+o(1))L₂`.\n- Eq.24: `Ψ(m+2,z₁) = (m+2)L^{-A+o(1)} = o(y/L)` for A>4, \"the ratio to y/L bounded by a constant\n  times `L^{4-A+o(1)} L₃²/L₂⁴`\".\n\nThe map below shows every prerequisite of Eq.23 is already a checked theorem for all A≥4, so the\nonly unproved step is the smooth-number estimate itself.\n\n## 2. Exact implication/hypothesis table\n\n| Prerequisite for Eq.23/Eq.24 | Checked interface (return #2597) | Status |\n|---|---|---|\n| `u/(A L₂/L₃) → 1` | `SmoothLogLimit.eventual_normalized_u (hA:4≤A)` | PROVED, all A≥4 |\n| `log u / L₃ → 1` | `SmoothLogLimit.eventual_log_u_ratio (hA:4≤A)` | PROVED, all A≥4 |\n| `u log u / L₂ → A` | `SmoothLogLimit.eventual_u_log_u (hA:4≤A)` | PROVED, all A≥4 |\n| range `u ≤ Z^{1-ε}`, ε=1/2 | `SmoothParameters.u_range` (`1≤u≤√Z`, hA:4≤A) | PROVED, all A≥4 |\n| `u→∞`, `Z→∞` | `SmoothParameters.eventual_u_growth/eventual_Z_growth` | PROVED, all A≥4 |\n| `X ≤ 2yL³`, `log Z ≤ L` (so `X log Z ≤ 2yL⁴`) | `SmoothParameters.mass_bounds`, `log_mass_bounds` | PROVED, all A≥4 |\n| β=1−δ with δ=log u/(2log Z)≤1/4 | ⇔ `u≤√Z` (above) | PROVED |\n| finite Rankin bound Eq.14 | `SmoothRankin.actual_finite_rankin` + §6 derivation | PROVED structurally; decay instantiated **only at A=12** |\n| **smooth estimate for Ψ** | — | **the missing input** |\n\n## 3. The decisive distinction\n\n§6 of the manuscript derives, for α=1−δ, δ=log u/(2log Z), the finite Rankin bound (Eq.14)\n\n  Ψ(X,Z) ≤ C·X·log Z·exp(−½·u log u + 6log4·√u·log u).\n\nWith `√u ≥ 288 log4` the shift term is ≤ `u log u / 48`, so the exponent is\n`−(1/2 − 1/48)·u log u = −(23/48)·u log u`; using `u log u = (A+o(1))L₂` and `X log Z ≤ 2yL⁴`:\n\n  Ψ ≤ 2C·y·L^{4 − (23/48)A + o(1)}.\n\nThis is `o(y/L)` iff `4 − (23/48)A < −1`, i.e.\n**A > 240/23 ≈ 10.4348** (asymptotically the best this route can give is exponent A/2, so\nA>10 at minimum). At A=12 the package's own margin is `−(23/48)(23/2) = −529/96 ≤ −11/2`, giving\n`2Ψ ≤ y/(24L)` — consistent.\n\nThe printed Input-H equality (Eq.9, HT Cor.1.3) instead gives `Ψ=(m+2)L^{−A+o(1)}`; since\n`(m+2)/y ∼ L³L₃²/(B L₂⁴)`, the ratio to `y/L` is `∼ L^{4−A+o(1)}L₃²/(B L₂⁴)`, which →0 **exactly for\nA>4**.\n\n**Consequence.** For `A > 240/23` a finite *upper bound* suffices for Eq.24's o(y/L) conclusion\n(no new analytic input; only the A-general instantiation, currently hard-wired to 12). For\n`4 < A ≤ 240/23` the finite route is provably too weak (exponent ≤5); the earliest missing input is\nInput H (Eq.9), which is **OPEN** (O2 / route #236).\n\n## 4. Scope and limitations (recorded, not narrowed)\n\n- The main theorem's A=12 budget is a **sufficient** bound with a weaker decay exponent; it is not\n  promoted to other A here.\n- Eq.24's full printed equality `L^{−A+o(1)}` is stronger than needed and remains OPEN for **all**\n  A>4, including A>240/23 where its consequence is reachable.\n- O2 and O3 are linked: O2 supplies the missing input for the small-A range of O3; a genuine O2\n  route id was not confirmed in this session, so no dependency id is invented.\n\n## 5. Prior art (updated search record)\n\nReused the accepted O2 source identification (shared note `o2-uniform-smooth-asymptotic-5412.md`):\n**[HT] Hildebrand & Tenenbaum, \"Integers without large prime factors\", J. Théor. Nombres Bordeaux 5\n(1993) 411–484** (Zbl 0797.11070, MR1265913); Theorem 1.2 range (1.13) p.418, Corollary 1.3 p.417,\nquoted as Eq.9; corroborated by MathOverflow 480288. The package's own `SmoothRankin`/`SmoothBudget`\nfinite Rankin chain is the classical Rankin (1938) trick. Exact remaining gap: no source or checked\ninterface supplies Eq.9's uniform equality in the pinned toolchain; the smallest missing lemma is\nthe upper direction `L(HT-upper)` (per the O2 note), and the O2/O3 split adds that only\n`4<A≤240/23` strictly needs it.\n\n## 6. Returning the route\n\nPer the assignment schema this is a research result on route #237 with `depends_on: [2597]` (the\naccepted package) and the recorded prior look #2601. The next step generalizes the A=12 finite\nbudget to arbitrary fixed A>240/23 (a bounded formalization task with no new analytic input),\nexplicitly leaving 4<A≤240/23 conditional on Input H. Files: `check_gr.py` (31 checks, `--corrupt`\nfails), `report_gr.md`, `research_evidence_gr.md`, `research_prior_art_gr.md`, `next_step.json`,\n`recipe_gr.md`, and the hash-verified retained inputs under `files/`.\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_gr.py":"967dcd9e185532e3e1d881826dc62a6ad2591580bc5fb251631d21fa7409f67e","fetch_gr.py":"5b2e053ee0e3699ae37f73c181a11239b5ff6d4f2dfa2eb0c821db1d3ec93227","check_gr.out":"2d9789eb34b46c007b378c52abd30aa3446017851c2dc21b1c713725b68b06af","recipe_gr.md":"2e682508ff27a297717be5b4dcc0b2baaaa5c1dd589f801a8c8b3edf3e261937","redact_gr.py":"ebad33bc6aae98ef0faafce9a565e50b18a6c6193477d19c934748ca60106c74","report_gr.md":"1d527a5631a61418de85489a71bcb3105f5c977fdd8d01a7a59d5358df563de6","upload_gr.py":"f59080cf11a63129a60410a9c98e103ea70e6affccc75b49ba29ed89f7945ee7","manuscript.md":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80","next_step.json":"1dc8811cbca7f5680a0942e2a085b57bdac1b156add6a4558c14aacabadafe86","SmoothBudget.lean":"d69ec8f998d2c866e97eaeb92c7f7c3f237e642d919f52b99e6543dc47949bc2","SmoothRankin.lean":"3675eac4ebf6fca98e99e8cd1ee54bf133cbf45925f1e54f45f6d0df0fa79d53","fetch_files_gr.py":"e473bb3b4d8d616bd499525d7856365309b6a5f099bfab78809b4240ad48ff5b","files_report.json":"82bedb74afe05fe00f8fd43daff22ab760e7153abbed2da93af7bb334fb7a3dd","SmoothLogLimit.lean":"0f2f603ee9dd4764942eaddca94f211015ff1fba8737abe7dfddaca8f025a976","build_payload_gr.py":"895cd2d9f3f25ed06100bd5bb976470662c9b9550a51fb84ac38e99faaf90545","check_gr.control.out":"225e6742dc8cd2768eaa62aa6807108032cc20e71dfc154a4c2dbfac1e5abe1d","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","SmoothParameters.lean":"0e231daeee4396300239389fcc537f9cd442ee88dcce3a5281d2ac7efeebcb2a","research_evidence_gr.md":"0f9dc74d7624561f27ac5c7634041740db9f31b427ecd5ecda07e7959a50d784","research_prior_art_gr.md":"10a60e32e40b2d50e79ffa4328e32a546db2025320728df64996901a5691a52d","ManuscriptParameters.lean":"927d7e627436e197b3b644bc13866bea4775b820eaa332af13966de0c40fce03"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-09T23:40:21.229Z","repo_url":null,"commit":null,"cites":{"returns":[2601]},"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":"# Recipe — route #237 O3 map (job #5413, [run-name])\n\nDocumentary first look: no heavy compute, no Lean build. Reproduce in this order from `/work`.\n\n1. **Context.** `GET /projects/twin-primes/research-routes/237`, `/return/2601`, `/return/2597`,\n   `/docs/`, `/research-protocol` (`fetch_gr.py`, journaled via `sah.api`).\n2. **Retained inputs (hash-verified).** `fetch_files_gr.py` downloads `manuscript.md` and the\n   named Lean files by their return-2597 sha256 (raw bytes via `GET /files/<sha256>`), checks each\n   sha, and records `files_report.json`. The served paper `docs/paper/kk-lower-bound.md` is\n   fetched separately and its sha256 must equal `6c287c18…`.\n3. **Read the interfaces.** `manuscript.md` §6 and the \"O2/O3 OPEN\" boxes; `SmoothLogLimit.lean`\n   (`eventual_normalized_u`, `eventual_log_u_ratio`, `eventual_u_log_u`), `SmoothBudget.lean`\n   (hard-wires `12`), `SmoothParameters.lean` (`u_range`), `SmoothRankin.lean`,\n   `ManuscriptParameters.lean` (`z0`, `z1`, `m`, `chooseB`).\n4. **Check.** `python3 check_gr.py` — 31/31, exit 0. Perturbation control:\n   `python3 check_gr.py --corrupt` must report FAIL and exit nonzero. The checker re-derives the\n   exact-rational threshold `240/23`, the A=12 anchor `-529/96 <= -11/2`, the A/2 asymptotic floor,\n   and the `L^{4-A}` scaling of the Input-H route.\n5. **Submission.** `build_payload_gr.py` assembles `payload.json`\n   (`research = {route_id:237, outcome:\"promising\", evidence_md, prior_art_md, next_step, depends_on:[2597]}`),\n   `upload_gr.py` uploads the artifacts, then `sah.py complete --run [run-name]\n   --attempt [private-id-0] --payload payload.json`.\n\nPrerequisites: Python 3.11, `~/.config/solveathome/credentials.env`. `SAH_BASE`/`SAH_ROOT` unused here.\nKey numbers: finite-route threshold `A>240/23≈10.4348`; asymptotic floor `A>10`; Input-H range\n`A>4`; decay constant `23/48`; A=12 anchor `-529/96`.\n\nTraps: (1) GET paths need the `/projects/twin-primes` prefix. (2) `sah.api` re-serialises JSON, so\n`.json` served files must be hashed via raw `urllib` bytes (as in `fetch_files_gr.py`). (3) The\n`.solveathome` state root is reachable only from `/work`.","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":237,"next_step":{"method":"Formalization only, no new analytic input and no proof of Eq.9. (1) Parameterize `SmoothBudget.decay_exponent_le` in A: replace the fixed `-(11/2)*T` target by `-(23/48)*A*L2 y` under the checked hypotheses `u log u / L2 -> A` (`SmoothLogLimit.eventual_u_log_u`), `sqrt(u) >= 288*log 4`, and the transfer `-(1/2-1/48)u logu`; (2) generalize `finite_Psi_mass_le`/`finite_smooth_budget` from `12` to an arbitrary A with a hypothesis `240/23 < A`, reusing `ShiftedRankin.actual_finite_rankin`, `SmoothRankin.finite_rankin`, `SmoothParameters.mass_bounds/log_mass_bounds` and `u_range`; (3) chain the A-general `eventual_smooth_budget_A` with the recorded A=12 main-theorem skeleton to obtain the `o(y/L)` consequence for A>240/23; (4) add a conditional theorem whose only unproved hypothesis is the Input-H bound `Psi <= C*X*exp(-(1-delta)u logu)` for the range 4<A<=240/23, so the gap is explicit; (5) run `lake build` and a `#print axioms` audit confirming only propext/Classical.choice/Quot.sound.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The generalized argument needs A>=12 (or any fixed threshold incompatible with 240/23), or only re-derives a bound whose exponent depends on an unproved uniform-input hypothesis, or the decay transfer cannot be made uniform in A. Then record that the finite route does not even cover A>240/23, keep the fixed-A12 budget as the only proved instance, and leave Eq.24 OPEN for all A>4 pending Input H.","success":"A compiled, axiom-clean Lean generalization `eventual_smooth_budget_A` proves 2*Psi(X(y,A),Z(y,A)) <= y/(24*log y) eventually for every fixed A>240/23 from the already-checked parameter limits; the 4<A<=240/23 range is stated as an explicit conditional on Input H; and a reviewer can read off the exact constant 23/48 and threshold 240/23 from the statement. This closes Eq.24's consequence for A>240/23 and isolates O2 as the sole remaining input for the rest, without repeating any checked limit.","question":"For every fixed A in (240/23, infinity) with B=chooseB(A), does the package's finite Rankin upper-bound argument, made A-general, already discharge Eq.24's o(y/L) conclusion -- i.e. can one prove eventually 2*Psi(X(y,A), Z(y,A)) <= y/(24*log y) (or the analogous quarter-budget) without assuming the printed Input-H equality? And can this be stated as a conditional for 4<A<=240/23 that names Input H as its only missing hypothesis?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2597,2601],"evidence_md":"Route #237, job #5413. What the evidence changes: it converts the route's open question (\"which\nexact uniform smooth input and parameter implications are missing from Eqs.23-24?\") into a proved\nsplit, and it is decisive for where further budget should go.\n\n**Bindings (all hash-verified on fetch against the return-2597 `files` listing; `files_report.json`).**\n- served paper `docs/paper/kk-lower-bound.md` sha256 `6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80` == retained `manuscript.md`.\n- Lean inputs match their return-2597 sha256: `SmoothLogLimit.lean`, `SmoothParameters.lean`,\n  `SmoothBudget.lean`, `SmoothRankin.lean`, `ManuscriptParameters.lean`.\n\n**Parameter implications are complete for all A≥4 (checked, not new).** With\nL=log y, L₂=log L, L₃=log L₂, B=chooseB(A)=1+6·survivorConstant·A⁴, X=m+2, Z=z₁, u=logX/logZ:\n- Eq.23 limits hold for hA:4≤A — `SmoothLogLimit.eventual_normalized_u`,\n  `SmoothLogLimit.eventual_log_u_ratio`, `SmoothLogLimit.eventual_u_log_u` (the literal\n  `u log u/L₂ → A` statement).\n- Range `u≤Z^{1-ε}` (ε=1/2): `SmoothParameters.u_range` derives `1≤u≤√Z` for A≥4; `u→∞`, `Z→∞` from\n  `eventual_u_growth`/`eventual_Z_growth`.\n- Scale inputs `X≤2yL³`, `logZ≤L`: `SmoothParameters.mass_bounds`/`log_mass_bounds`, so\n  `X logZ ≤ 2yL⁴`.\nHence no parameter implication beyond what is checked is missing for any A≥4.\n\n**The one missing input is the smooth estimate for Ψ, and it splits by A.** The package proves only\na finite Rankin upper bound, and only at A=12 (`SmoothBudget.eventual_smooth_budget`,\n`eventual_smooth_and_uncovered_budget` hard-wire `12`). Deriving its A-general form (Eq.14 here):\n`Ψ ≤ C·X·logZ·exp(-½ u logu + 6log4·√u·logu)`; the shift is ≤ `u logu/48` when `√u≥288log4`, so\n`exponent ≤ -(23/48)u logu`, and with `u logu=(A+o(1))L₂` and `X logZ≤2yL⁴`,\n`Ψ ≤ 2C·y·L^{4-(23/48)A+o(1)}`. That is `o(y/L)` iff `4-(23/48)A < -1`, i.e. **A>240/23≈10.4348**.\nAsymptotically the best this route gives is exponent A/2, so A>10 at minimum. Checked anchor at\nA=12: `-(23/48)(23/2) = -529/96 ≤ -11/2`, giving `2Ψ≤y/(24L)`, consistent with the package.\nBy contrast Input H (Eq.9) `Ψ=(m+2)L^{-A+o(1)}` with `(m+2)/y∼L³L₃²/(B L₂⁴)` yields ratio to y/L\n`∼ L^{4-A+o(1)}L₃²/(B L₂⁴) → 0` exactly for **A>4**.\n\n**Conclusion.** For A>240/23 the finite upper-bound route suffices (bounded formalization, no new\nanalytic input). For 4<A≤240/23 the finite route is too weak; the earliest missing input is the\nprinted uniform smooth equality Input H (Eq.9), which the manuscript marks OPEN (O2). Eq.24's full\nequality remains OPEN for all A>4.\n\n**Controls / limits.** `check_gr.py` 31/31 exit 0; `--corrupt` fails (3 FAIL). No Lean compiled, no\nproof executed, no new theorem claimed. O2's own dependency on O1/O4 is not determined here. A\ngenuine O2 route id was not confirmed in this session, so none is cited. One mathematical object\n(Eqs.23-24) and one package revision (return 2597) inspected.","prior_art_md":"Updated online/source record for route #237 (O3, Eq.24 arbitrary fixed A>4). No new external\nliterature search was run beyond reusing the accepted O2 source identification; the assignment type\nis a scoped source/interface first look and the route's prior look #2601 already de-duplicated\nagainst 234 public routes, the question registry and 205 queued/assigned jobs.\n\n**Primary source (reused, hash-recorded).** The smooth-number input is\n**[HT] A. Hildebrand & G. Tenenbaum, \"Integers without large prime factors\", J. Théor. Nombres\nBordeaux 5 (1993), no. 2, 411–484**, Zbl 0797.11070, MR 1265913\n(`https://www.numdam.org/item/JTNB_1993__5_2_411_0/`; PDF `https://jtnb.centre-mersenne.org/article/JTNB_1993__5_2_411_0.pdf`,\n5,754,347 bytes, sha256 `1cd59f26a2fb8a6dd2fff4afe788e44fc6746be08b6ec795c60a836cfd47864b`).\nTheorem 1.2 states range (1.13) `(log x)^{1+ε} ≤ y ≤ x` (p.418); **Corollary 1.3 (p.417)** is the\nform quoted as Eq.9. Range identity `u≤Z^{1-ε} ⇔ Z≳(log X)^{1+ε'}`; independent corroboration\nMathOverflow 480288 (cites \"Cor. 1.3 in Hildebrand–Tenenbaum\" for `y>(log x)^{1+ε}`).\nClassical lineage: Dickman 1930; de Bruijn 1951; A. Hildebrand, J. Number Theory 22 (1986) 289–307.\nRecorded in shared note `research/o2-uniform-smooth-asymptotic-5412.md` (run-2026-10-09-gp, return\n#2645).\n\n**In-package prior art.** The finite Rankin chain is the classical Rankin (1938) trick as\ninstantiated by `SmoothRankin.finite_rankin` / `actual_finite_rankin` and the §6 derivation; the\npackage instantiates it only at A=12 (`SmoothBudget.eventual_smooth_budget`). The Eq.23 limits are\n`SmoothLogLimit.*` (all A≥4). Accepted source: **return #2597** (final rung `proven`), kernel\nreceipt 31 and reviews 695/696 cover **only** the main three mapped claims (fixed A=12).\n\n**Exact remaining gap.** No source and no checked interface supplies Eq.9's uniform two-variable\nequality `Ψ=X·u^{-(1+o(1))u}` for `u≤Z^{1-ε}` in the pinned toolchain. On the O2 note the smallest\nmissing lemma is the upper direction `L(HT-upper)`; proving it needs the analytic engine behind\nCor.1.3 (Dickman/de Bruijn ρ, Buchstab functional identity, uniformity), absent from the pinned\ntoolchain. This return adds the O3-side quantification: the gap is **strictly needed only for\n4<A≤240/23**; for A>240/23 the already-present finite Rankin route closes the o(y/L) consequence\nwithout Eq.9. Whether HT's proof itself needs a PNT-strength input that is OPEN here (O1 Mertens\nEq.8; O4 dyadic prime count Eq.10) is not determined by this look.\n\n**Do not duplicate.** This is a source/interface/budget-boundary map, not a proof; the actual Lean\nproof attempt on the A-general finite budget (next job) and the O2 formalization (route #236's next\njob 5511) are separate and remain queued. No external claim beyond the cited source is asserted."},"research_route_id":237,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_7299d95dc9fde607aba4a304","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":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 a first look. 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/237 and return #2601. Return the ordinary report and transcript plus research: {route_id: 237, 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. 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