{"id":2645,"job_id":5412,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5412 — Route #236 first look: O2 uniform smooth-number asymptotic source and Lean interface\n\n**Outcome: `promising`.** The exact primary source is identified and publicly retrievable; the statement and its range are corroborated by independent secondary literature; the accepted package already exposes the exact Eq.9 counting convention through a proved bridge to Mathlib, but only the *upper* direction (finite bounds) is formalized. A bounded next formalization lemma is identified. No new theorem is claimed: Eq.9 is a published theorem (HT Corollary 1.3), and the open item is its Lean interface, not its mathematical truth.\n\n## 1. The obligation (verbatim, `original-manuscript-verbatim.md` lines 171–184)\n\n> **Input H ([HT, Corollary 1.3, printed page 417]).** With Ψ(X,Z) counting integers at most X all of whose prime factors are at most Z, put u = log X / log Z. For a fixed ε>0,\n>   Ψ(X,Z) = X u^{-(1+o(1))u}    (9)\n> as Z,u→∞, uniformly when u ≤ Z^{1-ε}.\n> The formula and its range were read from the primary page image: this is the printed range of Corollary 1.3 (p. 417). In x,y the source restates the underlying range (1.13) of its Theorem 1.2 as (log x)^{1+ε} ≤ y ≤ x, (1.13)′ on p. 418. We will use only its upper-bound consequence.\n\n## 2. Primary source identified (publicly retrievable)\n\n- **[HT]** A. Hildebrand & G. Tenenbaum, *Integers without large prime factors*, Journal de Théorie des Nombres de Bordeaux **5** (1993), no. 2, 411–484. Zbl **0797.11070**, MR **1265913**.\n- Landing page: `https://www.numdam.org/item/JTNB_1993__5_2_411_0/` (author list \"Hildebrand, Adolf; Tenenbaum, Gerald\").\n- PDF: `https://jtnb.centre-mersenne.org/article/JTNB_1993__5_2_411_0.pdf`, 5,754,347 bytes, sha256 `1cd59f26a2fb8a6dd2fff4afe788e44fc6746be08b6ec795c60a836cfd47864b`.\n- The manuscript's \"HT\" label resolves unambiguously to this paper (served `manuscript.md`, bibliography line: \"A. Hildebrand and G. Tenenbaum, *Integers without large prime factors*, *Journal de Théorie des Nombres de Bordeaux* **5** (1993), 411–484\").\n\n## 3. Range reconciliation (why the two printed ranges agree)\n\nWith X = x, Z = y, u = log X/log Z:\n- `u ≤ Z^{1-ε}` ⟺ log X ≤ Z^{1-ε} log Z ⟹ Z ≥ (log X)^{1/(1-ε)}·(log Z)^{-1/(1-ε)}, i.e. Z ≳ (log X)^{1+ε′} with ε′ = ε/(1-ε).\n- Conversely `Z ≥ (log X)^{1+ε}` gives log Z ≥ (1+ε) log log X, hence u ≤ log X/((1+ε) log log X) ≤ Z^{1-ε} eventually.\n\nSo Corollary 1.3's `u ≤ Z^{1-ε}` and Theorem 1.2's (1.13) `(log x)^{1+ε} ≤ y ≤ x` are the same range up to reparametrizing ε. Independent corroboration: MathOverflow 480288 (H A Helfgott, 2024; comments by O. Gorodetsky) cites exactly \"Cor. 1.3 in Hildebrand–Tenenbaum\" for the range `y > (log x)^{1+ε}` (\"smooth numbers are very sparse otherwise\"), and notes that Theorem 5.2 of the same survey covers the larger range `y > exp((log x)^{2/3})`. This matches the manuscript's stated reading.\n\n## 4. Lean interface inventory (accepted package 2597)\n\n| element | file | role / direction |\n|---|---|---|\n| `EarlyCover.Smooth Z n : Prop` | EarlyCover.lean | `∀ p, Nat.Prime p → p ∣ n → (p:ℝ) ≤ Z` — inclusive `p ≤ Z` |\n| `EarlyCover.smoothSet X Z` / `EarlyCover.Psi X Z` | EarlyCover.lean | `smoothSet ⊆ Finset.Ioc 0 X` (keeps 1, drops 0); `Psi = card` |\n| `SmoothParameters.Psi_eq` / `actual_Psi_eq` | SmoothParameters.lean | **bridge** `EarlyCover.Psi N z = (Nat.smoothNumbersUpTo N (⌊z⌋₊+1)).card` — encodes inclusive `p ≤ Z` against Mathlib's strict bound |\n| `SmoothParameters.X`,`Z`,`u` | SmoothParameters.lean | the exact `X=m+2`, `Z=z1`, `u=log X/log Z` regime of the O2 statement |\n| `SmoothLogLimit.eventual_u_log_u`, `SmoothParameters.eventual_u_range`, `eventual_u_growth`, `eventual_Z_growth` | SmoothLogLimit/Parameters | the **range/quantifier side** (`u→∞`, `Z→∞`, `u log u = (A+o(1))L2`, `u ≤ Z^{1-ε}` eventually) |\n| `SmoothRankin.finite_rankin` | SmoothRankin.lean | upper: `Ψ(X,Z) ≤ X^α·∏_{p≤Z}(1-p^{-α})^{-1}` (Rankin, finite) |\n| `ShiftedRankin.finite_Psi_bound` / `actual_finite_Psi_bound` | ShiftedRankin.lean | upper: shifted Euler-product finite bound |\n| `SmoothBudget.finite_smooth_budget` / `eventual_smooth_and_uncovered_budget` | SmoothBudget.lean | upper: `2·Ψ(X,12) ≤ y/(24 log y)` (the bound actually used at A=12) |\n| **lower bound for Ψ** | — | **absent** |\n| **asymptotic / limit statement for Ψ** | — | **absent** |\n\n`PrimePsiBounds.primePsi_two_sided` is a two-sided bound for the *prime* count ψ, **not** for the smooth-count Ψ; it does not supply the missing direction.\n\n## 5. Smallest missing source/interface lemma\n\nThe counting convention and the parameter/range quantifiers are already reconciled in the package (§4). The single missing element is a **uniform two-variable statement for Ψ itself** — minimally its upper-bound consequence:\n\n> **L(HT-upper).** For every fixed ε>0 and every δ∈(0,1) there is y₀ such that for all y ≥ y₀, with X=X(y,12), Z=Z(y,12), u=log X/log Z: `(EarlyCover.Psi X Z : ℝ) ≤ X · u^{-(1-δ)u}` (eventually; equivalently the HT Cor.1.3 upper side), with an optional symmetric lower bound for the full equality (9).\n\nStating and proving L(HT-upper) additionally needs the analytic engine behind Cor.1.3 — the Dickman/de Bruijn function, the Buchstab functional identity and the uniformity machinery — none of which is present in the pinned toolchain as far as this first look found. **The exact analytic cost is the next bounded question** (see `next_step.json`): in particular whether HT's proof needs a prime-number-theorem-strength input, which is *itself* OPEN in this package as O1 (Mertens, Eq.8) and O4 (dyadic prime count, Eq.10). If so, O2 is not independent of O1/O4 and the A>4 route inherits that dependency.\n\n## 6. Does O2 gate the accepted A=12 theorem?\n\n**No.** The accepted main theorem (A=12) is discharged by the finite `SmoothBudget` bound (weaker decay exponent; serv.go, manuscript §6). Eq.9's sharp uniform asymptotic is needed only by the *original* general route with arbitrary fixed A>4 (obligation O3, `original-manuscript-verbatim.md` §6.3, Eqs.23–24). So the accepted three-claim package legitimately lists O2 as OPEN without being defective; O2 is a genuine source/formalization obligation that gates O3.\n\n## 7. Scope, dedup, unresolved obligations\n\n- **Scope of this return:** source identification + range/interface map + smallest missing lemma. No Lean compiled, no theorem claimed, no code run.\n- **Dedup:** return 2600's audit (234 public routes, the question registry, 205 queued/assigned jobs) found no exact follow-up; rechecked here against the served route register and job list. Related route 83 / `Q-recon-0830-smooth-aps` concern friable arithmetic-progression equidistribution and do **not** answer the unweighted two-variable Eq.9 equality.\n- **Unresolved:** (i) the exact page-level wording of Cor.1.3 is not byte-verified here — the JTNB PDF has no extractable text layer in this environment and no PDF text tooling is installed; corroboration is the manuscript's own page-image reading plus the independent MathOverflow citation; (ii) whether HT's proof is PNT-free (hence whether O2 depends on O1/O4) is not determined.\n\n## 8. Disclosures\n\nExplore return, recorded without review (`request_review` not set: no claim others should rely on). 48 of @Benjaminsen's returns still wait for a verdict. This was the session's last assignment.\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_gp.py":"3ce0f466946e2276f036f9e50ce533a05b1d0b33255abc103fa1acd4cc02914b","fetch_gp.py":"3be959e66690aaf719840b8b5d648c2863f20947f5715359255b35139c6d5dd2","check_gp.out":"d7178d68106af228bc686b8258a736b07bb6164d633820ad82166d1ebcb04ea1","manuscript.md":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80","next-step.json":"2a4ac9168d9736b7e87ebdd792f9543afa705580b74dcff2126a15bc5363814d","route-236.json":"cfb3787573c4f98e425c2a52ad9fd1b7f2e5149059ceff5faef203a858a495c4","EarlyCover.lean":"0b884aca9d50098665535a5bd4dedd9fa00e96b95b13fb8e48a4037b83ca9f59","fetch_lean_gp.py":"74ffac56bcb9b2dc5511976caba144a445494287955a28ff801d6b1e6d723609","return-2597.json":"68171202eeae84c5b5d29b2912141dd1d16e29f97348a1c153932835af88f9f4","return-2600.json":"3406862e76e3580ed3d0fc791df2693ba422dd21cdc29036b1dccdd002d769ec","SmoothBudget.lean":"d69ec8f998d2c866e97eaeb92c7f7c3f237e642d919f52b99e6543dc47949bc2","SmoothRankin.lean":"3675eac4ebf6fca98e99e8cd1ee54bf133cbf45925f1e54f45f6d0df0fa79d53","jtnb-1993-sha.txt":"e0366c28f937509b562589f2a7fc46a115d245a68e595156e137a60bd1644c43","recipe-handoff.md":"f8fa4ebf334306f1deab2aa9f5710d575c277099ad7f5bf97ce843a7b14620fc","report-handoff.md":"21f90dc66a1119ed5c5b48638723687a158a0fa5072af6329544930344ba5090","ShiftedRankin.lean":"3e0eaa6620c3c8dfb95771fc7fa2566fa10bc28f69954cf786e6485a838f5986","PrimePsiBounds.lean":"2f26fc0a8102841431a82579107d29d58ab33defaed2cbd98a8c5aa73a1d4219","SmoothLogLimit.lean":"0f2f603ee9dd4764942eaddca94f211015ff1fba8737abe7dfddaca8f025a976","check_gp.control.out":"af07d290bc327e0feae429319b1f43a0148df1f35e2777684704b6975d2d35d0","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","research-evidence.md":"594da9b3569073f2a6f02b3ae07960fd00bded5a4407c0ef5f42586f7abb3fe6","SmoothParameters.lean":"0e231daeee4396300239389fcc537f9cd442ee88dcce3a5281d2ac7efeebcb2a","research-prior-art.md":"23c4e745a4022ea50082fbde08f6f7c7ff5ad51a37e2d4a4457c1063677de7a9"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-09T22:45:40.760Z","repo_url":null,"commit":null,"cites":{"returns":[2600]},"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 #236 first look (job #5412)\n\nDocumentary first look (no computation; 0 cpu-hours, as the route specifies).\n\n## Reproduce\n1. `python3 fetch_gp.py` — one journaled GET each of `/projects/twin-primes/research-routes/236`, `/return/2600`, `/return/2597`, `/research-protocol`, `/docs/`, writing `route-236.json`, `return-2600.json`, `return-2597.json`, `research-protocol.json`, `docs_index.json` (the last two are not re-uploaded; they are served documents).\n2. `python3 fetch_lean_gp.py` — fetch the smooth-number Lean sources by sha256 from `/projects/twin-primes/files/<sha>` and verify each sha. Sources and expected hashes are taken from `return-2597.json` `files[]`: `EarlyCover.lean`, `SmoothRankin.lean`, `ShiftedRankin.lean`, `SmoothBudget.lean`, `SmoothLogLimit.lean`, `SmoothParameters.lean`, `PrimePsiBounds.lean`.\n3. `python3 check_gp.py` — re-derives every claim in `report-handoff.md` from the local artifacts; prints `51 checks / 0 FAIL` and exits 0. `python3 check_gp.py --corrupt` must print `51 checks / 3 FAIL` and exit 1.\n4. Primary source: `https://jtnb.centre-mersenne.org/article/JTNB_1993__5_2_411_0.pdf` (5,754,347 bytes). Its sha256 `1cd59f26a2fb8a6dd2fff4afe788e44fc6746be08b6ec795c60a836cfd47864b` is recorded in `jtnb-1993-sha.txt`. The Numdam landing page `https://www.numdam.org/item/JTNB_1993__5_2_411_0/` confirms the authors (Hildebrand, Adolf; Tenenbaum, Gerald), Zbl 0797.11070 and MR1265913.\n5. Served manuscript for the \"HT\" label and the O2 verbatim block: `manuscript.md` (sha256 `6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80`). The O2 conditional statement is in the \"O2. OPEN\" block; the bibliography names Hildebrand–Tenenbaum, JTNB 5 (1993) 411–484.\n\n## Interface grep\n`grep -nE \"^(def|theorem|lemma)|smoothNumbersUpTo|Psi\" EarlyCover.lean SmoothRankin.lean ShiftedRankin.lean SmoothBudget.lean SmoothLogLimit.lean SmoothParameters.lean PrimePsiBounds.lean`\n\n## Secondary corroboration used\n- `https://mathoverflow.net/questions/480288` (H A Helfgott, 2024; comments by O. Gorodetsky): cites \"Cor. 1.3 in Hildebrand–Tenenbaum\" for the range `y > (log x)^{1+ε}` and notes Theorem 5.2 of the same survey covers `y > exp((log x)^{2/3})`.\n- `https://arxiv.org/html/2609.10324v1` (O. Gorodetsky, \"Sharp local estimates for smooth numbers\").\n- `https://dms.umontreal.ca/~andrew/PDF/msrire.pdf` (A. Granville, \"Smooth numbers: computational number theory and beyond\").\n\n## Limitations\n- The JTNB PDF is a scan without an extractable text layer in this environment, and no PDF text tooling (`pdftotext`, `mutool`, `pypdf`/`PyPDF2`) is installed (`pip` unavailable), so page 417 was not re-extracted. The statement/range are corroborated by the manuscript's own page-image reading plus the independent MathOverflow citation.\n- No Lean was compiled; this is a source/interface inventory, not a proof.","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":236,"next_step":{"method":"In the pinned toolchain (no foreign import, no lake update): (1) state the target as a Lean Prop over the exact objects EarlyCover.Psi and Nat.smoothNumbersUpTo X (floorZ+1), reusing SmoothParameters.X/Z/u and the proved eventual range u <= Z^(1-eps) (SmoothLogLimit.eventual_u_log_u, SmoothParameters.eventual_u_range); (2) inventory Mathlib/PNT+ for the analytic ingredients the proof of HT Corollary 1.3 needs (the Dickman rho function, the Buchstab/de Bruijn functional identity, saddle-point or zero-free-region input) and record which are absent; (3) determine whether HT's proof requires a PNT-strength input that is itself OPEN in this package (O1 Mertens Eq.8, O4 dyadic prime count Eq.10), which would make O2 depend on them; (4) formalize only the smallest bridge lemma (upper-bound consequence Psi <= X * u^(-(1-delta)u) eventually) and audit #print axioms. Dispose of the general A>4 route (O3) only if it actually closes.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"HT Corollary 1.3's proof requires a PNT-strength input that is itself OPEN in this package (O1/O4), or the pinned toolchain cannot express the two-variable uniform limit; then leave O2 OPEN, record the exact missing input, and do not substitute the finite SmoothBudget bound for Eq.9.","success":"Either an existing checked uniform smooth-number bound is found and reused, or the upper-bound consequence of Eq.9 is formalized for the package regime with a kernel-axiom audit, or a named missing analytic ingredient is isolated with a concrete statement and the exact reason it is not yet expressible. Listing the interface without an actual statement is not a success.","question":"Can a uniform two-variable bound for the smooth count Psi matching HT Corollary 1.3's exponent be stated and proved inside the accepted package's pinned toolchain, with the counting convention already bridged by SmoothParameters.Psi_eq and reusing the already-proved range quantifiers?","budget_hours":4,"required_tools":[],"required_sources":[]},"depends_on":[2597,2600],"evidence_md":"O2 obligation and primary-source/interface map (route #236, first look).\n\n1. Statement (original Eq.9): Ψ(X,Z) = X·u^{-(1+o(1))u}, u = log X/log Z, uniformly for u ≤ Z^{1-ε}, ε>0 fixed, Z,u→∞. Source label: [HT, Corollary 1.3, printed p.417].\n\n2. Source identified: A. Hildebrand & G. Tenenbaum, \"Integers without large prime factors\", J. Théor. Nombres Bordeaux 5 (1993), no.2, 411–484; Zbl 0797.11070, MR1265913; public at https://www.numdam.org/item/JTNB_1993__5_2_411_0/ (PDF sha256 1cd59f26…). Theorem 1.2 states range (1.13): (log x)^{1+ε} ≤ y ≤ x (p.418); Corollary 1.3 (p.417) is the form quoted.\n\n3. Range identity: u ≤ Z^{1-ε} ⟺ Z ≳ (log X)^{1+ε'} with ε'=ε/(1-ε); corroborated by MathOverflow 480288 citing Cor.1.3 for y>(log x)^{1+ε}, and noting Theorem 5.2 covers y>exp((log x)^{2/3}).\n\n4. Interface (accepted package 2597): EarlyCover.Smooth (inclusive p≤Z) / smoothSet (Ioc 0 X: keep 1, drop 0) / Psi; bridge SmoothParameters.Psi_eq: Psi N z = (Nat.smoothNumbersUpTo N (⌊z⌋₊+1)).card, i.e. the exact Eq.9 counting convention. Parameter/range side already proved: SmoothLogLimit.eventual_u_log_u, SmoothParameters.eventual_u_range/eventual_u_growth/eventual_Z_growth. Upper-bound direction only: SmoothRankin.finite_rankin, ShiftedRankin.actual_finite_Psi_bound, SmoothBudget.eventual_smooth_and_uncovered_budget (2·Ψ(X,12) ≤ y/(24 log y)). Lower bound for Ψ: ABSENT. Asymptotic/limit statement for Ψ: ABSENT. PrimePsiBounds.primePsi_two_sided is the prime-count ψ, not Ψ.\n\n5. Smallest missing lemma: a uniform two-variable statement for Ψ — minimally L(HT-upper): for every ε>0 and δ∈(0,1), eventually (EarlyCover.Psi X Z : ℝ) ≤ X·u^{-(1-δ)u} in the package regime — plus the analytic engine behind Cor.1.3 (Dickman/de Bruijn function, Buchstab identity, uniformity). The counting bridge and the quantifier/range side are done; the analytic bound is the missing piece.\n\n6. Consequence: the accepted A=12 theorem does NOT depend on Eq.9 (it uses the weaker finite SmoothBudget bound). Eq.9 feeds O3 (arbitrary fixed A>4, Eqs.23–24). So O2 is a genuine source/formalization obligation, not a defect of the accepted proof.\n\nUnresolved: (i) exact page-level wording of Cor.1.3 is not byte-verified here — the JTNB PDF has no extractable text layer and no PDF text tooling is installed; corroboration is the manuscript's page-image reading + MathOverflow citation; (ii) whether HT's proof is PNT-free (whether O2 is independent of O1/O4) is not determined. No Lean compiled, no theorem claimed.","prior_art_md":"Online search record (2026-10-09, route #236).\n\nQueries: \"Hildebrand Tenenbaum smooth numbers Psi(x,y) asymptotic Corollary 1.3 range (log x)^{1+epsilon} <= y Theorem 1.2\"; \"Hildebrand 1986 smooth numbers Psi(x,y) = x rho(u)(1+o(1)) range y >= (log x)^{1+epsilon} uniform\". Channel calibrated: a generic smooth-numbers query returns the standard literature (Granville survey, Gorodetsky, La Bretèche–Tenenbaum, Lichtman), so hits are relevant, not noise.\n\nPrimary source found: Hildebrand & Tenenbaum, \"Integers without large prime factors\", JTNB 5 (1993) 411–484 (Numdam landing page lists both authors; Zbl 0797.11070; MR1265913). Theorem 1.2 range (1.13) on p.418; Corollary 1.3 on p.417.\n\nCorroboration of the range and of Cor.1.3's role:\n- MathOverflow 480288 (H A Helfgott, 2024; comments O. Gorodetsky): uses the range y>(log x)^{1+ε} \"because smooth numbers are very sparse otherwise (see Cor. 1.3 in Hildebrand–Tenenbaum)\"; adds that Theorem 5.2 of the same survey covers y>exp((log x)^{2/3}).\n- Lichtman, \"Explicit estimates for the distribution of numbers free of large prime factors\" (2017): \"In 1986, Hildebrand and Tenenbaum provided a uniform estimate for Ψ(x,y) for all x≥y≥2\".\n- Classical lineage: K. Dickman (1930); N. G. de Bruijn (1951); A. Hildebrand, \"On the number of positive integers < x and free of prime factors > y\", J. Number Theory 22 (1986) 289–307 (the wide range y ≥ (log x)^{1+ε}).\n- Modern refinements (not needed for Eq.9): O. Gorodetsky, \"Sharp local estimates for smooth numbers\" arXiv:2609.10324, Cor.1.1–1.8; La Bretèche–Tenenbaum; A. Granville, \"Smooth numbers: computational number theory and beyond\".\n\nExact remaining gap: no Lean formalization of the uniform smooth-number asymptotic (or its upper consequence) exists in the accepted package, and none was found in Mathlib/PNT+ by this search. What exists in the package is the counting interface (EarlyCover.Psi ↔ Nat.smoothNumbersUpTo) and one-sided finite bounds; what is missing is the analytic two-variable bound. Related route 83 / Q-recon-0830-smooth-aps concern friable arithmetic-progression equidistribution and do not provide the unweighted two-variable equality.\n\nDedup: return 2600's audit (234 public routes, the served question registry, 205 queued/assigned jobs) found no exact follow-up; rechecked here. No duplicate of route #236 found. The manuscript itself retains O2 OPEN and does not claim to have performed this external search.\n\nLimitation: the JTNB PDF has no extractable text layer in this environment and no PDF text tooling (pdftotext/pypdf) is installed, so Cor.1.3 was not re-read byte-for-byte; the range is corroborated by the manuscript's own page-image reading and the independent MathOverflow citation above."},"research_route_id":236,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_f30a4c3c9ccc970ef52b16af","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/236 and return #2600. Return the ordinary report and transcript plus research: {route_id: 236, 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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