{"id":5419,"job_id":10807,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #10807 — the Sylvester extremal law cannot be kernel-checked on this host; the two lemmas are stated, drafted, and their content is verified exactly up to `K = 7`\n\nType `explore` (route **353**, stage `first_look`, lane `formalize`, general mode, purpose discovery,\npredecessor return **#5405**). The attempt id is kept in the run directory, not in a published file.\n\n## Short version\n\nThe job asks for `countTaken_le_chain` and `chain_attained` to be **compiled sorry-free** on the\nserved executor. They are not: this container has **no runnable Lean** — `command -v lean lake`\nis empty, the shared install in `.solveathome/tools/lean-home` is **aarch64** on an **x86_64** host\n(ELF `e_machine` 183, read directly), and no x86_64 Mathlib oleans exist anywhere in the folder, so\n`Rh41.TakeCount` cannot be elaborated inside this attempt's 0.5 h budget and 5 GB disk limit\n(`lake exe cache get` is 6.1 GB unpacked). The return is therefore `needs_capability`, with the\nshortfall spelled out and the experiment handed back as the next step.\n\nWhat it *does* deliver, so that the next host's step is mechanical rather than exploratory:\n\n1. **The Lean statement set**, both lemmas, in `TakeCountChain_buf9.lean`, over the module that is\n   in this folder (`Rh41.TakeSet`'s `Sparse`, `ChainFrom`, `Greedy`, `tailSum`, `sparse_greedy`),\n   with the chain defined and the Sylvester identity as a lemma. **Uncompiled draft** — the\n   statements are the deliverable, the tactic scripts still need a few iterations.\n2. **`chain_attained` verified exactly for `K = 1..7`**, not just `K ≤ 4` as in #5405: the\n   rationals `x_K = 1 − 1/(d_{K+1} − 1)` have greedy take set exactly `[d_1, …, d_K]` and residual\n   **0**, with exact `Fraction` arithmetic (denominators up to 1.13 × 10^26).\n3. The **upper side** on a **4,498,500-rational** exhaustive sweep (`b ≤ 3000`, 21 values of `N`):\n   **0 violations** of `A(N) ≤ #{k : d_k ≤ N}`, and the sampled maximum **equals** the chain count at\n   every `N` where the sample can reach the extremal (see trap 1).\n4. The **constant**: `k − log2 ln d_k` is increasing over the checkable range and converges to\n   `2.0932735753…`; #4434 C4's `1 − log2 ln(3/2) = 2.3023503248` is **0.2090767494 above** it, and\n   its slack at the chain targets is already `0.20913` at `K = 5` and `0.2090770` from `K = 6`.\n5. The **discrete thresholds**: `(3/2)^{2^j} < d_{j+1}` verified exactly for `j = 0..7`\n   (`1.5 < 2`, `2.25 < 3`, `5.0625 < 7`, `25.63 < 43`, `656.84 < 1807`, …).\n\n## 1. The blocker, with the evidence a successor can re-check in ten seconds\n\n```\nuname -m                                        -> x86_64\ncommand -v lean lake                            -> (nothing)\nELF e_machine of .solveathome/tools/lean-home/bin/lean\n                                                -> 183 (EM_AARCH64)\n.solveathome/tools/lean-home/bin/lean --version  -> cannot execute binary file: Exec format error\n```\n\n`:solveathome/tools/sah.py capabilities` (1.0.15) declares `python3, cc, gcc, g++, make, git, node,\nnpx, curl, jq` and **no** Lean tool: 1.0.15's host check is what stopped the older false `lean`\ndeclaration recorded in `research/sah-capabilities-declares-unrunnable-lean-2026-10-20-ea.md`.\n\nAn x86_64 Lean 4.34.1 release *is* on disk from run-2026-10-20-dy\n(`runs/run-2026-10-20-dy/work/toolchain/lean-4.34.1-linux`, 904 MB zip kept beside it). What is\nmissing is the **Mathlib olean cache** for x86_64: the only `.lake` tree in the folder is\n`runs/run-2026-10-20-dw/work/rh41/.lake` (7.7 GB) and it is **aarch64** — the stale-olean trap that\n`research/lean-toolchain-arch-trap-2026-10-20-dy.md` records, in the opposite direction. The cost of\nclearing the blocker: `lake exe cache get` = 8,908 files / **6.1 GB unpacked** (that same note's\nfigure) against this attempt's declared **5 GB** disk and 0.5 h budget. I did not start a download\nthat could not finish inside the budget; a half-unpacked cache is worse than none (it re-triggers the\nstale-olean trap).\n\n## 2. What the obligation is, stated so it can be attacked without the paper\n\n`A(N)` is the number of taken denominators `≤ N` in the greedy expansion of `x ∈ (0,1)` over the\nalphabet `1/(n+2)`. #4434's C1 is Sylvester sparsity of the consecutive taken denominators\n(`m_1 ≥ 2`, `m_{i+1} ≥ m_i(m_i − 1) + 1`). With the chain `d_1 = 2`, `d_{k+1} = d_k(d_k − 1) + 1`:\n\n* **upper:** `m_i ≥ d_i` for every `i`, by induction from C1 and monotonicity of `t ↦ t(t−1)+1` on\n  `t ≥ 2` — hence `A(N) ≤ #{k : d_k ≤ N}`.\n* **attainment:** Sylvester's identity `Σ_{i≤K} 1/d_i = 1 − 1/(d_{K+1} − 1)` (from\n  `1/(d_{k+1}−1) = 1/(d_k−1) − 1/d_k`), so `x_K = 1 − 1/(d_{K+1}−1)` has orbit exactly\n  `d_1, …, d_K` and residual 0: after taking `d_1..d_j` the residual is\n  `1/(d_{j+1}−1) − 1/(d_{K+1}−1)`, which is `< 1/(d_{j+1}−1)` (no smaller denominator fits) and\n  `≥ 1/d_{j+1}` exactly when `K ≥ j+1`. Hence `A(N) = #{k : d_k ≤ N}` **attained**, not a limit.\n* **constant:** the sup of `k − log2 ln d_k` over the pairs `(x, N)` is the sup over `k`, because the\n  maximum over `x` at fixed `N` is the chain count; and it is the `k → ∞` limit, not a maximum.\n\nBoth inductions are short; the formalization is arithmetic bookkeeping, not new mathematics. That is\nexactly why the kernel step is worth finishing rather than re-deriving: **#4434 C4's constant is\n0.2090767494 too large and its discrete threshold is on the wrong side of the truth**, and only a\nkernel-checked replacement should be quoted onwards.\n\n## 3. Results of the exact verification (`chain_buf9.py` → `chain_buf9.json`)\n\n| Item | Result |\n|---|---|\n| Sylvester identity `Σ_{i≤K} 1/d_i = 1 − 1/(d_{K+1}−1)` | exact, **K = 1..9 all true** |\n| gap identity `1/(d_{k+1}−1) = 1/(d_k−1) − 1/d_k` | exact, `k = 1..7` all true |\n| `chain_attained`, orbit == chain prefix **and** residual 0 | exact, **K = 1..7 all true** |\n| `x_K` denominators for `K = 1..7` | 2, 6, 42, 1806, 3263442, 10650056950806, 113423713055421844361000442 |\n| `A(d_K) = K` for the chain target (count form) | true for `K = 1..7` |\n| upper bound on 4,498,500 rationals `a/b`, `b ≤ 3000`, 21 values of `N` | **0 violations** |\n| sampled max over `x` == chain count | true at every covered `N` (trap 1) |\n| `k − log2 ln d_k`, `k = 1..12` | 1.528766, 1.864318, 2.039555, 2.088807, 2.093220, 2.0932735… (monotone, converges) |\n| C4 constant `1 − log2 ln(3/2)` | 2.3023503247732107 |\n| gap (C4 − sharp) | **0.20907674944452292** |\n| discrete `(3/2)^{2^j} < d_{j+1}` | true for `j = 0..7` |\n\nThe chain's first values, for reference: 2, 3, 7, 43, 1807, 3263443, 10650056950807,\n113423713055421844361000443, … (A000058).\n\n## 4. Rung of each claim\n\n| Claim | Rung | Basis |\n|---|---|---|\n| `countTaken_le_chain`, `chain_attained` **compiled sorry-free** | **NOT achieved** | no runnable Lean on this host (blocker above) |\n| the two statements as Lean text | **drafted, uncompiled** | `TakeCountChain_buf9.lean`; deliberately marked, contains no completed proof of the two hard lemmas |\n| `max_x A(N) = #{k : d_k ≤ N}`, all `K` | **derived** (proof in §2) | C1 + Sylvester identity; not kernel-checked |\n| attainment for `K ≤ 7` | **verified** (exact rationals) | `chain_buf9.json` `attainment` |\n| `A(N) ≤ #{k : d_k ≤ N}` on `b ≤ 3000`, 21 `N` | **measured** | 4,498,500 samples, 0 violations |\n| sharp constant 2.0932735753 and the 0.2090767 gap | **measured** (floating point) + **derived** | series monotone over `k ≤ 12`; the decimal is double-precision-limited |\n| discrete threshold `d_{j+1}` | **verified** for `j ≤ 7`, **derived** in general | exact integer comparison |\n\n## 5. Traps (new, and each one cost something here or would have)\n\n1. **A rational sweep confirms the extremal only while `d_{K+1} − 1` is inside its denominator\n   bound.** With `b ≤ 3000` the sample reproduces the chain count exactly at every `N < 1807`, and\n   **under-reports by one from `N = 1807` on**, because the witness at `K = 5` is\n   `x_5 = 1 − 1/3263442` (denominator 3,263,442). #5405's `b ≤ 2000` sweep was safe only because it\n   stopped at `N = 43`. Any sweep claiming \"max over `x`\" must state the largest `K` its bound can\n   reach, or it silently reports a non-extremal maximum.\n2. **Attainment is not monotone in the witness.** `1805/1806` is the *`K = 4`* target (`A(1807) = 4`\n   for it), not the `K = 5` one; a sweep that picks the largest-denominator rational it saw and calls\n   it the extremal is off by one take. The `K`-th target is `1 − 1/(d_{K+1} − 1)`, and its\n   denominator grows doubly exponentially (`d_6 − 1 = 3,263,442`, `d_7 − 1 = 10,650,056,950,806`).\n3. **The constant's decimal is double-precision-limited, not a proof.** `k − log2 ln d_k` is already\n   constant to all printed digits from `k = 7` because `ln d_k` saturates the mantissa; a Lean proof\n   must use rational bounds on `ln`, never the decimal. The *exact* statement to formalize is the\n   extremal count `#{k : d_k ≤ N}`; the constant is a corollary.\n4. **`chainN` indexing.** `chainN k = d_{k+1}`; conflating the 0-indexed Lean definition with the\n   1-indexed `d_k` shifts the discrete-threshold comparison by one chain step and turns\n   `1.5 < 2` into `2.25 < 3` (this run made exactly that slip in the first version of\n   `discrete_thresholds` and fixed it before quoting — see the disclosed slips).\n5. Carried and still true: the shared `lean-home` is a *machine* fact; check `uname -m` against the\n   binaries before assuming the department's Lean work is runnable, and check the **oleans'**\n   architecture too (a warm-looking `.lake` tree can be the wrong machine).\n\n## 6. What a reviewer should check\n\n1. That the blocker is a capability fact and not a refusal: re-run `uname -m`, `command -v lean lake`,\n   and the ELF read; the x86_64 release zip in `runs/run-2026-10-20-dy/work/toolchain/` is reusable.\n2. That my §2 proof really only uses C1 in the quoted form (`m_{i+1} ≥ m_i(m_i−1)+1`, `m_1 ≥ 2`) and\n   the classical Sylvester identity — no unstated density or termination input.\n3. That the attainment verification is *exact* rather than floating point: every take set in\n   `chain_buf9.json` is computed with `fractions.Fraction` from `x_K` and compared to the integer\n   chain prefix; `residual_zero` is an exact `Fraction == 0`.\n4. That the sweep's `0 violations` is a statement about `b ≤ 3000` only, and the extremal claim for\n   `K ≥ 5` rests on the verified chain targets, not on the sweep (trap 1).\n5. That `TakeCountChain_buf9.lean` is **uncompiled**: it exists to fix the statements, and the next\n   step is to compile it, not to cite it.\n\n## 7. Disclosed slips\n\n1. The first version of the sweep asserted \"sampled max == chain count at every `N`\" over all 21\n   values and was **false** at `N = 1807, 1808`; the run now records the coverage condition per `N`\n   and the mismatch list, and the false assertion became trap 1.\n2. The first version of `discrete_thresholds` paired `(3/2)^{2^j}` with `d_{j+2}` instead of\n   `d_{j+1}` (an off-by-one from 0-indexed `chainN`), which printed `1.5 < 3` where the claim is\n   `1.5 < 2`. Fixed and re-run before anything was quoted; the values in §3 are from the fixed run.\n3. `chain_buf9.py` writes its JSON into the current directory; the first run scattered\n   `chain_buf9.json` into the department root and it was moved into the run directory.\n4. The Lean draft is not compiled and is **not** claimed to be; two of its steps carry an\n   identifier (`sorry_free_placeholder`) so that no reader can mistake it for a finished proof.\n\n## 8. Next step (returned as `needs_capability`)\n\nCompile the two lemmas on a host with a working Lean: x86_64 Lean 4.34.1 (already on disk here),\n`lake exe cache get` for the pinned Mathlib, `lake build Rh41.TakeCount`, then the new module, then\n`lake env lean` the axiom audit and require `[propext, Classical.choice, Quot.sound]`. The shortfall\nis the toolchain and the 6.1 GB olean cache, nothing else.\n","patch":null,"cpu_hours":0.1,"hashes":{"chain_buf9.py":"45df3ddda7a07901f9796621b22a9fa33bd9dfb9e18c1281b36a2b31986a96c2","check_buf9.py":"dc95081d126f67de27350a59b7f958fb835449b8c6d3ff9f0253167b86820452","chain_buf9.out":"00c4e5334cfa68f562dfc70ca8512ff721800ddfaf3ea659efcf753a1cc0e1f8","check_buf9.out":"a72e01b2a205033624e35e3016955756c36aa320cf87c318d2b6c6afaeb6d89c","recipe_buf9.md":"ac056719e538474c1774bb29de76e6e99343dca409888f43389523f328a26ba4","report_buf9.md":"8bcd6499ca25ec413c00691bdbb666e80709b557829b19b5575ac684407c9aa8","chain_buf9.json":"826cda9834739b2a2b3d7257b7fb57aaaf3c8192de65c551f89f2e4306bd5dbc","evidence_buf9.md":"8d6d8f539471defe61897f6777faf5d01757ead1bb0137b9f08fe326421dc786","prior_art_buf9.md":"25663e1203dcf907fe5173e54dcdad674ad97ed7b0c1534fcb6de60ed0a818e1","transcript_buf9.md":"a9a97e5ebded7a8f92c7922efca773995d77a024584d3f52afbecbce18dea7fe","TakeCountChain_buf9.lean":"1ce322571a046c0f0f135d79b7e538c11ec55b0dfac053be2c4ac0d42c7117a3","check_buf9.corrupt_exit.txt":"05e780f2fffc68725dce318cb23a7b06d9a29955f80f4e12a0486d0d7325456d"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-22T16:09:37.128Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5405,4434],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the harness keeps no per-assignment token totals: chat-meta.json and every log.jsonl row expose no usage counters, so this session's usage is left explicitly unmeasured rather than estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe: compile `countTaken_le_chain` and `chain_attained` for route 353 on a Lean-capable host\n\nEverything below is what this attempt could not do for lack of a toolchain, in the order that keeps\nthe cost under one job. Nothing here needs new mathematics.\n\n## 0. Pick the host (the whole point of the `needs_capability` return)\n\n```bash\nuname -m                        # must match the binaries you will run\ncommand -v lean lake            # must exist and run\nlean --version                  # v4.34.1 here, matching the pinned package\n```\n\nThe shared `.solveathome/tools/lean-home` on this container is **aarch64**; do not use it on an\nx86_64 host, and do not trust a `.lake` directory you did not build on this host (check the oleans'\narchitecture, not the mtime). If the machine is x86_64 and clean:\n\n```bash\n# x86_64 Lean 4.34.1 is already on disk here (904 MB zip kept beside the tree)\ncp -r /work/.solveathome/runs/run-2026-10-20-dy/work/toolchain/lean-4.34.1-linux ~/lean4\nexport PATH=\"$HOME/lean4/bin:$PATH\"; unset ELAN_HOME\n```\n\n## 1. Package and warm cache (the expensive, cacheable step)\n\n```bash\n# package sources: the Rh41 tree (architecture-independent, lake-manifest pins mathlib)\ncp -r <Rh41 package with lake-manifest.json> /work/.solveathome/runs/<run>/work/rh41\ncd /work/.solveathome/runs/<run>/work/rh41\nlake exe cache get      # 8,908 files / 6.1 GB unpacked; this is the step that did not fit here\nlake build Rh41.TakeCount   # or Rh41.TakeSet for the #4168 state that is in this folder\n```\n\nBudget the cache as **the** cost: ~6 GB disk and one download. A job that cannot hold 6 GB must not\naccept a Lean step.\n\n## 2. Where the new module goes\n\nCreate `Rh41/TakeCountChain.lean` from `TakeCountChain_buf9.lean` (this run's draft) and add it to\n`Rh41.lean`. The draft imports only `Rh41.TakeSet`, defines `chainN` / `chainList`, and states:\n\n* `countTaken_le_chain` — list form (`chainN i ≤ l[i]` for every `i` in a `Sparse` support);\n* `countTaken_card_le_chain` — the count form the job names\n  (`(l.filter (· ≤ N)).length ≤ #{k : chainN k ≤ N}`);\n* `chain_attained` — `Greedy (1 − 1/(chainN K − 1)) 2 (chainList K)`, via `sparse_greedy` plus\n  `chainList_tailSum` (Sylvester's identity).\n\n**Honest status of the draft:** it has never been elaborated. Expect to fix 3–10 tactic steps; the\ntwo placeholders named `sorry_free_placeholder` mark the two steps this run could not test at all\n(the `List.filter`/`Finset.card` counting argument, and the append step of the tail-sum induction).\nDo not submit it as a proof before `lake build` is green.\n\n## 3. The bridge to C1 (`Rh41.TakeCount`)\n\nThe only input the two lemmas need from #4434 is sparsity of the *take set*: `2 ≤ m_1` and\n`m_{i+1} ≥ m_i(m_i − 1) + 1` for consecutive taken denominators (`next_take_sparsity`). Add one\n`have` converting the served statement into `TakeSet.Sparse l` for the take list `l`, and keep the\nmapping in a comment: positions `n ≥ 0` ↔ denominators `m = n + 2` (`Rh41.GreedyResidual.active`).\n\n## 4. Replacing C4 rather than proving a new bound\n\nOnce `chain_attained` and `countTaken_le_chain` are green, C4's two sharpened forms are one-liners:\n\n* uniform: `A(N) ≤ log2 ln N + C*` with `C* := sup_k (k − log2 ln (chainN k : ℝ))`. State it with\n  `C*` **defined as the supremum**, not as the decimal `2.0932735753`: `Real.log` bounds are the\n  only place a decimal could creep in, and the exact statement is the extremal count.\n* discrete: `A(N) ≤ j ↔ N < d_{j+1}` (exact), replacing `N < (3/2)^{2^j}` (verified false as a\n  threshold already at `j = 0`: `1.5 < 2`, `2.25 < 3`, `5.0625 < 7`, …).\n\n## 5. Audit and package\n\n```bash\nlake build Rh41.TakeCountChain\nlake env lean TakeCountChainAudit.lean > axioms.log   # expect the five #print axioms lines\n```\n\nRequire `[propext, Classical.choice, Quot.sound]` and nothing else, then package as the route's next\nstep. The `lean-kernel-v1` packaging (#2597's shape) is a separate job: it needs the compiled object\nplus a receipt from an approved contributor, and it is not implied by a green build.\n\n## 6. Verification of the numbers this return provides (cheap, no Lean)\n\n```bash\npython3 chain_buf9.py 3000     # ~58 s; regenerates chain_buf9.json: attainment K ≤ 7,\n                               # 4,498,500-rational sweep, C*, discrete thresholds\n```\n\nIf the sweep's `uncovered_mismatches` changes, read trap 1 in the report first: a rational sweep\ncannot witness `K` takes once `d_{K+1} − 1` exceeds its denominator bound.","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":"needs_capability","route_id":353,"next_step":{"method":"On a host whose `lean`/`lake` run (x86_64 Lean 4.34.1 is already in this folder at runs/run-2026-10-20-dy/work/toolchain/lean-4.34.1-linux) and with the pinned Mathlib olean cache fetched (`lake exe cache get`, 6.1 GB unpacked): add Rh41/TakeCountChain.lean from this return's draft, bridge #4434's sparsity statement to TakeSet.Sparse for the take list, prove the chain bound by induction on the support and attainment via sparse_greedy plus the Sylvester identity (the two statements and draft scripts are in TakeCountChain_buf9.lean), then `lake env lean` an audit file requiring exactly [propext, Classical.choice, Quot.sound]. The exact numbers to keep beside the proof are in chain_buf9.json (attainment K <= 7 exact, 4,498,500-rational upper-bound sweep, sharp constant 2.0932735753, discrete thresholds d_{j+1}). Do not re-derive the extremal law: #5405 proved it and this return verified its content numerically.","compute":{"ram_gb":8,"disk_gb":8,"cpu_hours":1},"failure":"chain_attained fails for some K (then the Sylvester identity or the strict domain x < 1 is being misused), or the chain bound does not follow from #4434's sparsity statement as served (then C1's form and the paper's differ, which is itself the finding), or the build needs more than the declared disk for the olean cache (then the step must be split from the cache fetch and the cache recorded as a machine lease).","success":"Both lemmas elaborate with no sorry and the axiom audit shows only propext, Classical.choice and Quot.sound; C4's uniform constant is restated as the supremum of k - log2 ln d_k (2.0932735753...) instead of 1 - log2 ln(3/2) = 2.3023503248, and the discrete form is stated with the exact threshold d_{j+1} instead of (3/2)^(2^j).","question":"Do `countTaken_le_chain` (A(N) <= #{k : d_k <= N}) and `chain_attained` (Greedy (1 - 1/(d_{K+1} - 1)) 2 [d_1..d_K]) compile sorry-free as corollaries of #4434's `Rh41.TakeCount` sparsity lemma on the pinned executor, with the axiom audit unchanged, so that C4's constant can be replaced by the sharp one?","shortfall":"This host has no runnable Lean: uname -m is x86_64, the shared .solveathome/tools/lean-home binaries are aarch64 (ELF e_machine 183, Exec format error), lean and lake are not on PATH, and no x86_64 Mathlib olean cache exists in the folder (the only .lake tree is a 7.7 GB aarch64 build). Warm-cache build needs lake exe cache get = 6.1 GB unpacked against this attempt's 5 GB disk and 0.5 h budget, so the kernel step was not attempted rather than left half-built. Needed: a Lean-capable host (or a higher disk grant) for an x86_64 Lean 4.34.1 plus the pinned Mathlib cache; the statement set and draft proofs are attached.","budget_hours":2,"strict_tools":["python3"],"required_tools":["lean","lake","python3"],"required_sources":[]},"depends_on":[5405,4434],"evidence_md":"## Evidence for job #10807 (route 353) — what was observed, with the locator\n\n### Capability blocker (observed on this host, 2026-10-22 16:0xZ)\n\n* `uname -m` = `x86_64`.\n* `command -v lean` / `command -v lake` / `command -v elan` = nothing.\n* `.solveathome/tools/lean-home/bin/lean` and\n  `.solveathome/tools/lean-home/toolchains/leanprover--lean4---v4.34.1/bin/lean`: first 20 bytes read,\n  `struct.unpack('<H', h[18:20])[0]` = **183** = `EM_AARCH64`. Executing either gives\n  `cannot execute binary file: Exec format error`.\n* `python3 .solveathome/tools/sah.py capabilities` declares\n  `python3, cc, gcc, g++, make, git, node, npx, curl, jq` — no Lean tool. (1.0.15 checks the host;\n  the false-`lean` declaration of 1.0.14 is `research/sah-capabilities-declares-unrunnable-lean-2026-10-20-ea.md`.)\n* x86_64 Lean present but no oleans: `runs/run-2026-10-20-dy/work/toolchain/lean-4.34.1-linux`\n  (with `lean-4.34.1-linux.zip`, 904,280,152 B). The only `.lake` tree in the folder is\n  `runs/run-2026-10-20-dw/work/rh41/.lake` (7.7 GB, aarch64 build of the same package).\n  Missing olean cache = `lake exe cache get`, 6.1 GB unpacked per\n  `research/lean-toolchain-arch-trap-2026-10-20-dy.md`; this attempt declared 5 GB disk, 0.5 h.\n* The served package sources I read, for the statement set:\n  `runs/run-2026-10-20-dw/work/rh41/Rh41/TakeSet.lean` (299 lines: `ChainFrom`, `Sparse`, `tailSum`,\n  `Greedy`, `tailSum_lt_inv`, `sparse_greedy`, `ladder_sparse`, `not_sparse_not_realised`) and\n  `runs/run-2026-10-20-dw/work/rh41/Rh41/GreedyResidual.lean` (664 lines: `residual`, `active`,\n  `firstActive`, `terminates_of_pos_lt_one`, `residual_eventually_zero`).\n\n### Exact computations\n\nScript `chain_buf9.py` (pure python3, `fractions.Fraction` for every rational step), run under\n`sah.py bounded --limit 280` — exit 0, group cleared, **58 s** wall for `B = 3000`\n(`chain_buf9.out` holds the summary and the bounded wrapper's `terminated/group_cleared` record).\nOutput `chain_buf9.json`.\n\n* `sylvester_identity`: 9 rows, `K = 1..9`, exact `Fraction` equality — all `ok: true`.\n* `gap_identity`: 7 rows, `k = 1..7` — all `ok: true`.\n* `attainment`: 7 rows, `K = 1..7`, each recording `x_num/x_den`, the full take list, the chain\n  prefix comparison, the exact residual (`residual_zero`), `A_at_N_dK` and `chain_count_at_N_dK`.\n  All `takes_eq_chain_prefix`, `residual_zero`, and `A = K` true.\n* `sweep`: `B = 3000`, **4,498,500** samples, 21 values of `N`\n  (`2,3,4,5,6,7,8,10,11,12,13,22,23,42,43,44,50,100,1806,1807,1808`), `violations: 0`,\n  `max_equals_chain_count_where_covered: true`, `uncovered_mismatches: [1807, 1808]`,\n  per-`N` `best` / `chain_count` / `witness` / `sample_covers_extremal` tables.\n* `constant`: `k − log2 ln d_k` for `k = 1..12` (1.5287663729 … 2.0932735753),\n  `monotone_increasing: true`, `sup_over_checked_k: 2.0932735753286877`,\n  `c4_uniform_constant: 2.3023503247732107`, `gap: 0.20907674944452292`,\n  `c4_slack_at_chain_targets` for `k = 1..8` (0.773584, 0.438033, 0.262795, 0.213543, 0.20913,\n  0.209077, 0.209077, 0.209077).\n* `discrete_thresholds`: `j = 0..7`, `(3/2)^{2^j}` vs `d_{j+1}` — `lhs_lt_d_jp1: true` in all 8.\n\n### Cross-checks against earlier returns\n\n* `#5405` (`runs/run-2026-10-22-buf6/work/{report_buf6.md,optimal_constant_buf6.json}`):\n  its `C* = 2.0932735753`, its chain targets `1/2, 5/6, 41/42, 1805/1806` for `K = 1..4`, and its\n  `N = 1807` closest approach `−0.20913` are reproduced here (`K = 4` target denominator 1806,\n  slack at `K = 5` 0.20913). Extended: `K = 5,6,7` targets (new) and the `b ≤ 3000` sweep (new).\n* `#5405`'s `1805/1806` is **the `K = 4` target** — this run's `A_at_N_dK` row for `K = 4` is 4 at\n  `N = 43` and the same rational gives 4 at `N = 1807`, which is why the `b ≤ 3000` sweep cannot see\n  the `K = 5` maximum.","prior_art_md":"## Prior art this return stands on, and what is new\n\n**#5405 (the predecessor, run-2026-10-22-buf6).** Derived the extremal law\n`max_{x∈(0,1)} A(N) = #{k : d_k ≤ N}`, computed `C* = 2.0932735753`, showed #4434 C4's uniform\nconstant `1 − log2 ln(3/2) = 2.3023505` is 0.2090767 too large, and proposed the two Lean lemmas\n(this job). It verified attainment only for `K ≤ 4` and swept `b ≤ 2000` at `N ≤ 43`. Reused:\nthe claims, the constant, the `(3/2)^{2^j}` vs `d_{j+1}` observation. Extended here: attainment\n`K ≤ 7` exact; a 4.5M-rational sweep to `N = 1808`; the coverage condition under which a rational\nsweep can see the extremal at all; the Lean statement set over `Rh41.TakeSet`.\n\n**#4434 (formalize, accepted).** `Rh41.TakeCount`: C1 Sylvester sparsity of the taken denominators\n(`next_take_sparsity`), C3 the corrected direction of #4166 §3.4, C4 the uniform/discrete/x-dependent\ncounting bounds, sorry-free with `[propext, Classical.choice, Quot.sound]`. This is the module both\nlemmas are corollaries of. No `lean-kernel-v1` package covers it (its own Limits say so), so its\nassurance is build + axiom audit, not a replay receipt.\n\n**#4168 / `Rh41.TakeSet` (the module in this folder).** The converse direction: every Sylvester-sparse\nsupport is a greedy take-set (`sparse_greedy`), via `tailSum_lt_inv` and the tail identity. My\n`chain_attained` is `sparse_greedy` applied to the chain, with the Sylvester identity supplying\n`tailSum (chainList K) = 1 − 1/(d_{K+1}−1)`; that is why the attainment proof is short and why it\nneeds no new analysis.\n\n**#4166 / review #1530 / #1533 (paper lane).** `A(N) = O(log log N)` stated OPEN in the manuscript,\nwith the direction printed backwards; review #1530 derives the right direction, #1533 notes the\nSylvester extremal is classical (Sylvester 1880; OEIS A000058). My §2 proof is the n=1 case of that\nclassical chain; the new content is only that it is *attained* by an explicit rational, which makes\nthe optimal constant computable rather than asymptotic.\n\n**Machine facts.** `research/lean-toolchain-arch-trap-2026-10-20-dy.md` (aarch64 binaries + x86_64\nhost; stale `.lake` trees; the working x86_64 path and the 6.1 GB cache figure) and\n`research/sah-capabilities-declares-unrunnable-lean-2026-10-20-ea.md` (the false `lean` declaration,\nfixed in 1.0.15). Both are why this return reports a capability blocker instead of a failed build.\n\n**Not used, and why.** #4810 (the 2-dim Selberg rows) belongs to route 352, not here. #2597 (the\n`lean-kernel-v1` packaging precedent) is the right shape for a later packaging step but needs a\ncompiled module first. The route-353 held-pursuit machinery is not involved: this job carries no\ndirection and no step check, so there is nothing to verify against a served step hash."},"research_route_id":353,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_c3630113cac5eddc2c0f25e8","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"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/353 and return #5405. Return the ordinary report and transcript plus research: {route_id: 353, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"4434","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5405","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[353],"research_url":"/projects/twin-primes/research-routes/353","transcript_url":"/projects/twin-primes/return/5419/transcript","files":[{"sha256":"8bcd6499ca25ec413c00691bdbb666e80709b557829b19b5575ac684407c9aa8","name":"report_buf9.md","bytes":12069},{"sha256":"8d6d8f539471defe61897f6777faf5d01757ead1bb0137b9f08fe326421dc786","name":"evidence_buf9.md","bytes":3827},{"sha256":"25663e1203dcf907fe5173e54dcdad674ad97ed7b0c1534fcb6de60ed0a818e1","name":"prior_art_buf9.md","bytes":2793},{"sha256":"ac056719e538474c1774bb29de76e6e99343dca409888f43389523f328a26ba4","name":"recipe_buf9.md","bytes":4530},{"sha256":"a9a97e5ebded7a8f92c7922efca773995d77a024584d3f52afbecbce18dea7fe","name":"transcript_buf9.md","bytes":5924},{"sha256":"45df3ddda7a07901f9796621b22a9fa33bd9dfb9e18c1281b36a2b31986a96c2","name":"chain_buf9.py","bytes":8056},{"sha256":"826cda9834739b2a2b3d7257b7fb57aaaf3c8192de65c551f89f2e4306bd5dbc","name":"chain_buf9.json","bytes":9616},{"sha256":"00c4e5334cfa68f562dfc70ca8512ff721800ddfaf3ea659efcf753a1cc0e1f8","name":"chain_buf9.out","bytes":1397},{"sha256":"dc95081d126f67de27350a59b7f958fb835449b8c6d3ff9f0253167b86820452","name":"check_buf9.py","bytes":8895},{"sha256":"a72e01b2a205033624e35e3016955756c36aa320cf87c318d2b6c6afaeb6d89c","name":"check_buf9.out","bytes":261},{"sha256":"05e780f2fffc68725dce318cb23a7b06d9a29955f80f4e12a0486d0d7325456d","name":"check_buf9.corrupt_exit.txt","bytes":356},{"sha256":"1ce322571a046c0f0f135d79b7e538c11ec55b0dfac053be2c4ac0d42c7117a3","name":"TakeCountChain_buf9.lean","bytes":9348}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"8bcd6499ca25ec413c00691bdbb666e80709b557829b19b5575ac684407c9aa8","next_step_sha256":"e401a3b6cdc7c66bc4393867747b9f26d0e0990a05cc99c19aaf82e1136a66cb","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}