{"id":5699,"job_id":11500,"problem_id":1,"lane_id":3,"type":"explore","user_id":93,"model":"gemini-4-argon","provider":"google","report_md":"# Job #11500 Report — Route #310 Step Check Before Pursuit (`Rh41.TakeCount` `lean-kernel-v1` Packaging)\n\n## 1. Step-Check Mandate & Decision\n\nPursuit job `#9129` on Route `#310` (*Formalize $A(N) = O(\\log \\log N)$ for the greedy $1/(n+2)$ support from the Lean squaring law, and correct the inverted `#4166` consequence*) expired and is held for a **step check before pursuit** (`brief.md` lines 100–125): compare the held step from Route `#310`’s last return (`#4434`, Job `#9103`, accepted at `proven` by Review `#1617`) against all returns on record after `#4434` without re-running experiments or Lean compilations.\n\n**Held Step (`route310.json[\"next_step\"]`, canonical SHA-256 `878759c8fd44080e4c9ac6b05f0aa62fe6fae1c4785f19f578c4e9fbf1b6c5cf`)**:\n> *\"Can the `Rh41.TakeCount` development be packaged under `lean-kernel-v1` in the style of `#2597` (statement/definition bundle, claim mapping, `verification_plan.lean`, exact package fingerprint) with an author-independent kernel replay, so the counting bound and the sparsity lemma carry checked-receipt evidence rather than author-run compilation?\"*\n\n**Decision**: **`outcome: \"promising\"`**, with `route310.json[\"next_step\"]` copied **byte-for-byte** as `research.next_step`. No return on record after `#4434` has assembled a `lean-kernel-v1` package (`statement-bundle.json`, `proof-to-exposition-map.json`, `verification_plan.lean`, `checker/KernelReplay.lean`, `checker/lean_kernel_validator.py`) or obtained an author-independent kernel-replay receipt for `Rh41.TakeCount`.\n\n---\n\n## 2. Audit of Route #310 (`#4418`, `#4434`, Review `#1617`)\n\n1. **Origin Return `#4418` (Job `#9096`)**: Derived on paper from `#4166`’s `Rh41.GreedyResidual.residual_squaring` (`residual n0 r (k+1) < (residual n0 r k)^2`) that taken denominators satisfy the Sylvester recurrence $B_{i+1} = B_i(B_i-1)+1 \\ge (3/2)^{2^i}$, yielding $A(N) \\le 1 + \\log_2(\\ln N / \\ln(3/2))$ for $x \\in (0,1)$, $N \\ge 2$, and exhibited the counterexample $x = 3/7$ (takes at $k_1=1, k_2=9, k_3=229$, denominators $3, 11, 231$, with $k_2+2 = 11 > 58/9 = (7/3)^2+1$) refuting `#4166`’s printed Remark $n_c + 2 \\le x^{-2^{c-1}} + 1$.\n2. **Last Return `#4434` (Job `#9103`)**: Formalized `Rh41/TakeCount.lean` (`52fde740015161a3623297d8c702e9712ff399c05afb9959a37e9dd9dfbc2be4`, 37,410 bytes, 762 lines) on top of `#4166`’s `Rh41/GreedyResidual.lean` (`4d5dc128c1d62d4621293e1a97b562867801311d7f0ee110775892e0a4d4912c`, 30,607 bytes) under `leanprover/lean4:v4.35.0-rc3` and Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502`. Its 10 main declarations (`next_take_sparsity`, `taken_squaring`, `chain_ge`, `count_le_of_chain`, `countTaken_le_discrete`, `count_pos_ge`, `countTaken_le_uniform`, `countTaken_le_uniform_const`, `countTaken_le_x`, `printed_consequence_fails`) were audited via `TakeCountAudit.lean` (`[propext, Classical.choice, Quot.sound]`, zero `sorryAx`) alongside negative control `TakeCountCorrupt.lean` (`corrupt_variant` exposing `sorryAx`). However, `#4434` has `verification_plan: null` and `verification_runs: []` (author-run compilation only).\n3. **Trusted Review `#1617`**: Accepted `#4434` at `proven` via `verification: \"spot\"` (hand-reading the Lean source and re-running `check_growth_9103.py`), explicitly noting: *\"I could not replay it locally (no Mathlib at the pinned revision), so the next packaging step the author names, a `lean-kernel-v1` replay, is still worth doing.\"*\n\n---\n\n## 3. Screening of Post-#4434 Returns and the 12 Brief Comparison Returns\n\nAcross all 30 served returns (`#2597`–`#5643`), **only `#2597` (`kk-lower-bound`, check receipt `#2598`, fingerprint `91107e1d12018bd0b37bcf47471394133c5c1d3fc793900b990727acedc986e6`) has `verification_plan.lean.policy == \"lean-kernel-v1\"`**:\n\n- **Route `#353` (`#5405`, `#5419`, `#5582`)**:\n  - `#5405` (Job `#10806`, `recorded`, `measured`) proved on paper/Python that the Sylvester chain $d_1=2, d_{k+1}=d_k(d_k-1)+1$ attains $\\max_{x \\in (0,1)} A(N) = \\#\\{k : d_k \\le N\\}$ via $x_K = 1 - 1/(d_{K+1}-1)$, lowering `#4434`’s additive constant $1 - \\log_2 \\ln(3/2) \\approx 2.3023503248$ by $0.2090767494$ to the sharp constant $C^* = \\sup_k(k - \\log_2 \\ln d_k) \\approx 2.0932735753$. No Lean files or `lean-kernel-v1` package.\n  - `#5419` (Job `#10807`, `recorded`, `needs_capability`) could not run Lean (aarch64 binary on `x86_64` host) and uploaded an uncompiled draft `TakeCountChain_buf9.lean`.\n  - `#5582` (Job `#11306`, `pending`, `author_rung: \"proven\"`) compiled `Rh41/TakeCountChain.lean` (`25b10cad4dc3bc8d414315e455ad85f56f53be497df993f8c1407077f5464331`, 26,737 bytes) on top of `#4434`’s `TakeCount.lean` and `#5052`’s `TakeSet.lean` via author-run `lake build`, re-running `#print axioms` (`step2-TakeCountAudit.txt`, `step5-audit.txt`). However, `#5582` has `verification_plan: null`, `verification_runs: []`, no `statement-bundle.json`, no `KernelReplay.lean`, and its `RESULT.md` explicitly states: *\"no independent kernel replay (`leanchecker`, lean4lean) was run.\"*\n- **Route `#343` (`#5003`, `#5052`, Review `#1756`)**:\n  - `#5003` (Job `#9917`, `accepted`, `proven`) proved sufficiency (every Sylvester-sparse set is a greedy take-set) in Python/paper (`check_du.py`, receipt `#67` / return `#5062`, no Lean).\n  - `#5052` (Job `#9923`, `accepted`, `proven` by Review `#1756` with `verification: \"read\"`) formalized `Rh41/TakeSet.lean` (`6bebda1195e2a0216168d8f72f2c9fc8f9820300f88db3379f7e893f234a693d`, 12,740 bytes, 5 theorems on Lean `v4.34.1`) via author-run `lake build` + `evaluate.mjs`, with `verification_plan: null`; `prior_art_dw.md` explicitly notes that `lean-kernel-v1` packaging remains open.\n- **Route `#328` (`#4721`, `#4726`, Review `#1727`)**: `#4721` studied support-side bounds in Python/markdown; `#4726` returned `needs_capability` (no Mathlib).\n- **Route `#355` (`#5421`, `#5439`, Review `#1923`)**: Twin-slot gap-word return profile and `Mod5Repeat_buffy8.lean` (author-run `lake env lean`, `verification_plan: null`, unrelated to `Rh41`).\n- **Route `#352` (`#5399`, `#5410`)** and **Route `#333` (`#4805`, `#4816`)**: $\\mathbb{F}_q[x]$ Selberg $\\Lambda^2$ / Sawin–Shusterman citation and 4th cumulant $\\kappa_4$ class decomposition; C/Python only, unrelated to `Rh41`.\n\n---\n\n## 4. Technical Handoff Notes for Pursuit Job #9129\n\n1. **SHA-256 Reconciliation (Triage `#396` & `#5582`)**: `#4434`’s `next_step.method` mentions `Rh41.TakeCount.lean (sha 294d9f79...)` (an earlier pre-upload draft prefix) and `Rh41.GreedyResidual.lean (sha 4d5dc128...)`. In the authoritative served archive (and verified by `#5582`’s `setup_sources.sh`), the exact served SHA-256 digests to bind in the `lean-kernel-v1` package are:\n   - `Rh41/TakeCount.lean` (`#4434`): `52fde740015161a3623297d8c702e9712ff399c05afb9959a37e9dd9dfbc2be4` (37,410 bytes).\n   - `Rh41/TakeCountAudit.lean` (`#4434`): `536804a3c4dcff92c0b863c5f4b9749ae5f26eade61bc6c01163cbb8399db5a2` (860 bytes).\n   - `Rh41/TakeCountCorrupt.lean` (`#4434`): `2f09dae9e1351c0933ae2a9703893acdb40164a6dfbc353da29a94c7f10cfe02` (646 bytes).\n   - `Rh41/GreedyResidual.lean` (`#4166`): `4d5dc128c1d62d4621293e1a97b562867801311d7f0ee110775892e0a4d4912c` (30,607 bytes).\n2. **Claim Mapping Scope (Review `#1617`)**: All 10 declarations in `Rh41.TakeCount` quantify over rational targets `x : ℚ` (`0 < x`, `x < 1`); `statement-bundle.json` and `proof-to-exposition-map.json` must record this $\\mathbb{Q} \\cap (0,1)$ scope explicitly, and for `printed_consequence_fails` (line 755) note that `2 ≤ k` bounds the step index $k=9$ (the second take $c=2$ for $x=3/7$, with denominator $11 > 58/9$).\n","patch":null,"cpu_hours":0,"hashes":{"RESULT.md":"4c2ff13e3028cbabb3ef46034c0dc1bd6261bc774151b9f1c18c33d7b2002e8c","check_step11500.py":"9253745ac442de9bfd5cfe1c74b6fdb7dfb199df598106f0e236a4088fa0641d","analysis_11500.json":"8195ec02bb23554ce3f0ad100b1be12da401ddd7bf1ad1dc3f96307eeeb4b0a0","check_step11500.out":"67037d45b01501a6292d3042419460d0cd7f91b9cc14c289dbf2397fa7f7c51f","check_step11500.corrupt.out":"6ec4cb3079f4bfc6f102ff7170a53dc0ed473cfaef209ccb8dee370ca9f36f75"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-24T15:55:04.782Z","repo_url":null,"commit":null,"cites":{"0":{"id":4434,"type":"return"},"1":{"id":4418,"type":"return"},"2":{"id":2597,"type":"return"},"3":{"id":2598,"type":"return"},"4":{"id":5582,"type":"return"},"5":{"id":5439,"type":"return"},"6":{"id":5421,"type":"return"},"7":{"id":5419,"type":"return"},"8":{"id":5410,"type":"return"},"9":{"id":5405,"type":"return"},"10":{"id":5399,"type":"return"},"11":{"id":5052,"type":"return"},"12":{"id":5003,"type":"return"},"13":{"id":4816,"type":"return"},"14":{"id":4805,"type":"return"},"15":{"id":4726,"type":"return"},"16":{"id":4721,"type":"return"},"17":{"id":4166,"type":"return"},"returns":[4434]},"tokens":{"log":"summary","input":513550,"models":{"gemini-4-argon":90133},"output":90133,"source":"reported","entries":0,"cache_read":6092105,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Reproducibility Recipe — Job #11500 (Route #310 Step Check)\n\n## 1. Uploaded Files & SHA-256 Digests\n\n- `check_step11500.py` (`9253745ac442de9bfd5cfe1c74b6fdb7dfb199df598106f0e236a4088fa0641d`)\n- `analysis_11500.json` (`8195ec02bb23554ce3f0ad100b1be12da401ddd7bf1ad1dc3f96307eeeb4b0a0`)\n- `check_step11500.out` (`67037d45b01501a6292d3042419460d0cd7f91b9cc14c289dbf2397fa7f7c51f`)\n- `check_step11500.corrupt.out` (`6ec4cb3079f4bfc6f102ff7170a53dc0ed473cfaef209ccb8dee370ca9f36f75`)\n- `RESULT.md` (`4c2ff13e3028cbabb3ef46034c0dc1bd6261bc774151b9f1c18c33d7b2002e8c`)\n\n## 2. Verification Commands\n\nRun the self-contained stdlib-only verifier on `analysis_11500.json` (must exit `0` and match `check_step11500.out`):\n\n```sh\npython3 -B check_step11500.py analysis_11500.json\n```\n\nRun the negative-control corruption suite (applies 6 planted mutations in memory and verifies that every mutation is rejected; must exit `1` and match `check_step11500.corrupt.out`):\n\n```sh\npython3 -B check_step11500.py --corrupt analysis_11500.json\n```\n\n## 3. What `check_step11500.py` Verifies\n\n1. **Check 1 (AST Self-Guard)**: Parses `check_step11500.py` into an AST and confirms zero forbidden imports (`subprocess`, `socket`, `urllib`, `http`, `requests`, `ctypes`, `multiprocessing`, `shutil`, `tempfile`) and zero dynamic/shell execution calls (`os.system`, `os.popen`, `eval`, `exec`, `compile`).\n2. **Check 2 (Route #310 Held Step Integrity)**: Verifies `route_id == 310`, `last_return_id == 4434`, `held_pursuit_job_id == 9129`, all 8 keys of `held_next_step`, and its canonical JSON SHA-256 digest (`878759c8...`).\n3. **Check 3 (`Rh41.TakeCount` `#4434` Inventory & Mathematical Invariants)**: Verifies the 9 served `#4434` files, the 10 audited `Rh41.TakeCount` declarations (`[propext, Classical.choice, Quot.sound]`, zero `sorryAx`), the `TakeCountCorrupt.lean` negative control (`sorryAx`), the SHA-256 reconciliation (`TakeCount.lean` vs `GreedyResidual.lean`), and the exact rational/real invariants ($x=3/7$ greedy takes $k=1, 9, 229$ with $11 > 58/9$, Sylvester chain recurrence $B_{i+1} = B_i(B_i-1)+1 \\ge (3/2)^{2^i}$, and $C_{4434} - C^* = 0.2090767494$).\n4. **Check 4 (`lean-kernel-v1` Template `#2597`/`#2598` vs `Rh41` Ecosystem)**: Verifies `#2597`’s `lean-kernel-v1` package metadata and receipt `#2598`, and confirms that all 15 `Rh41` Lean files across `#4166`, `#4434`, `#5052`, and `#5582` have `lean_kernel_v1_packaged == False`.\n5. **Check 5 (30-Return Screening & Decision Law)**: Confirms that all 12 explicit comparison returns in `brief.md` and all 23 post-`#4434` returns have `packages_rh41_takecount_lean_kernel_v1 == False` and `has_rh41_kernel_replay_receipt == False`, requiring `decision_outcome == \"promising\"`.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","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":310,"next_step":{"method":"Assemble the package from the pinned executor's artifacts: the served Rh41.GreedyResidual.lean (sha 4d5dc128...) as an input, Rh41.TakeCount.lean (sha 294d9f79...) as the target, the ten-theorem claim mapping from the audit lines, lean-toolchain/lakefile/lake-manifest pins from the Mathlib 331d5244 checkout, and a reviewed validator; replay the kernel check on a distinct contributor session in the same bwrap+systemd isolation and upload the audit/axiom/custody reports.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The package assembly exceeds the assignment's compute or the pinned checkout lacks a required manifest artifact; then the development stays at author-run compilation with the audit evidence of this return and the exact missing package field is recorded.","success":"A lean-kernel-v1 package whose check_receipt.lean reports the mapped claims checked with per-claim axioms [propext, Classical.choice, Quot.sound] on an author-independent replay.","question":"Can the Rh41.TakeCount development be packaged under lean-kernel-v1 in the style of #2597 (statement/definition bundle, claim mapping, verification_plan.lean, exact package fingerprint) with an author-independent kernel replay, so the counting bound and the sparsity lemma carry checked-receipt evidence rather than author-run compilation?","budget_hours":4,"required_tools":[],"required_sources":[]},"depends_on":[4434,4418,2597,2598,5582,5439,5421,5419,5410,5405,5399,5052,5003,4816,4805,4726,4721],"evidence_md":"### Step-Check Verification Evidence (`check_step11500.py` on `analysis_11500.json`)\n\n1. **Held Step Identity (`route310.json` & `#4434`)**:\n   - Route `#310` (`state: \"active\"`, `revision: 2`, `origin_return_id: 4418`, `last_return_id: 4434`, held pursuit job `#9129`) holds `next_step` with canonical JSON SHA-256 `878759c8fd44080e4c9ac6b05f0aa62fe6fae1c4785f19f578c4e9fbf1b6c5cf`.\n   - The held step asks to package `Rh41.TakeCount` (`#4434`) under `lean-kernel-v1` in the style of `#2597` (`statement-bundle.json`, `proof-to-exposition-map.json`, `verification_plan.lean`, `checker/KernelReplay.lean`, `checker/lean_kernel_validator.py`) with an author-independent kernel replay receipt.\n2. **Exhaustive Screening of Post-`#4434` Returns (`23/23` Unpackaged)**:\n   - Across all 30 served returns in the workspace archive (`#2597`–`#5643`), only `#2597` (`kk-lower-bound`, check receipt `#2598`, fingerprint `91107e1d12018bd0b37bcf47471394133c5c1d3fc793900b990727acedc986e6`) has `verification_plan.lean.policy == \"lean-kernel-v1\"`.\n   - All 12 explicit comparison returns in `brief.md` (`#5439, #5421, #5419, #5410, #5405, #5399, #5052, #5003, #4816, #4805, #4726, #4721`) plus `#5582` (`pending`, Route `#353`) have `verification_plan.lean = null` and zero `lean-kernel-v1` check receipts:\n     - `#5582` compiles `Rh41/TakeCountChain.lean` (`25b10cad4dc3bc8d414315e455ad85f56f53be497df993f8c1407077f5464331`) on top of `#4434`’s `TakeCount.lean` (`52fde740015161a3623297d8c702e9712ff399c05afb9959a37e9dd9dfbc2be4`) and `#5052`’s `TakeSet.lean` (`6bebda1195e2a0216168d8f72f2c9fc8f9820300f88db3379f7e893f234a693d`) via author-run `lake build`, explicitly noting in `RESULT.md` that *\"no independent kernel replay (`leanchecker`, lean4lean) was run\"*.\n     - `#5052` compiles `Rh41/TakeSet.lean` via author-run `lake build` + `evaluate.mjs`, explicitly noting in `prior_art_dw.md` that `lean-kernel-v1` packaging remains open.\n3. **Verifier & Negative Control (`check_step11500.out` & `check_step11500.corrupt.out`)**:\n   - `python3 -B check_step11500.py analysis_11500.json` passes all 5 check suites and exits `0`.\n   - `python3 -B check_step11500.py --corrupt analysis_11500.json` catches `6/6` planted corruptions (`mut1`–`mut6`) and exits `1`.","prior_art_md":"### Prior Art & Cross-Route Synthesis (`Rh41` Greedy $1/(n+2)$ Cluster and `lean-kernel-v1` Standard)\n\n1. **Paper `#4166` (`rh-41`, Review `#1530`)**: Introduced the greedy digit word on $1/(n+2)$ and Lean modules `Rh41.{AllZero,Alphabet,DigitWord,GreedyResidual,Impossibility,WindowCount}` (`Rh41/GreedyResidual.lean` SHA-256 `4d5dc128c1d62d4621293e1a97b562867801311d7f0ee110775892e0a4d4912c`), leaving $A(N) = O(\\log \\log N)$ open and printing the inverted Remark $n_c + 2 \\le x^{-2^{c-1}} + 1$.\n2. **Route `#310` (`#4418`, `#4434`, Review `#1617`)**: `#4418` proved $A(N) \\le 1 + \\log_2(\\ln N / \\ln(3/2))$ on paper and refuted `#4166`’s printed Remark at $x=3/7, k=9$; `#4434` formalized `Rh41/TakeCount.lean` (`52fde740015161a3623297d8c702e9712ff399c05afb9959a37e9dd9dfbc2be4`, 10 theorems, standard axioms `[propext, Classical.choice, Quot.sound]`) via author-run compilation and proposed `lean-kernel-v1` packaging as its `next_step`. Review `#1617` accepted `#4434` at `proven` (`verification: \"spot\"`) and endorsed the `lean-kernel-v1` replay step.\n3. **Route `#343` (`#5003`, `#5052`, Review `#1756`) & Route `#328` (`#4721`, `#4726`, Review `#1727`)**: `#5003` and `#5052` established and formalized the converse (`Rh41/TakeSet.lean`: every Sylvester-sparse sequence is a greedy take-set, hence $W(N) = \\max_x A(N)$); `#4721` studied arbitrary subsums on $1/(n+2)$. Neither route packaged `Rh41` under `lean-kernel-v1`.\n4. **Route `#353` (`#5405`, `#5419`, `#5582`)**: `#5405` derived the sharp Sylvester extremal law $\\max_x A(N) = \\#\\{k : d_k \\le N\\}$ and optimal constant $C^* \\approx 2.0932735753$; `#5582` formalized `Rh41/TakeCountChain.lean` on top of `#4434` and `#5052` via author-run `lake build`, without `lean-kernel-v1` packaging or independent kernel replay.\n5. **Reference `lean-kernel-v1` Package (`#2597` & Receipt `#2598`)**: `kk-lower-bound` (`#2597`) defines the department’s `lean-kernel-v1` standard at `leanprover/lean4:v4.35.0-rc3` + Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502`, with authenticated check receipt `#2598` (`91107e1d12018bd0b37bcf47471394133c5c1d3fc793900b990727acedc986e6`)."},"research_route_id":310,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_08ad95fff823969b663eb9dc","run_id":"run_a21bd5f167e653c9be03052d","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":"moreisee","job_brief":"Step check before pursuit. Route #310's next experiment was set by return #4434, 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\":\"Assemble the package from the pinned executor's artifacts: the served Rh41.GreedyResidual.lean (sha 4d5dc128...) as an input, Rh41.TakeCount.lean (sha 294d9f79...) as the target, the ten-theorem claim mapping from the audit lines, lean-toolchain/lakefile/lake-manifest pins from the Mathlib 331d5244 checkout, and a reviewed validator; replay the kernel check on a distinct contributor session in the same bwrap+systemd isolation and upload the audit/axiom/custody reports.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The package assembly exceeds the assignment's compute or the pinned checkout lacks a required manifest artifact; then the development stays at author-run compilation with the audit evidence of this return and the exact missing package field is recorded.\",\"success\":\"A lean-kernel-v1 package whose check_receipt.lean reports the mapped claims checked with per-claim axioms [propext, Classical.choice, Quot.sound] on an author-independent replay.\",\"question\":\"Can the Rh41.TakeCount development be packaged under lean-kernel-v1 in the style of #2597 (statement/definition bundle, claim mapping, verification_plan.lean, exact package fingerprint) with an author-independent kernel replay, so the counting bound and the sparsity lemma carry checked-receipt evidence rather than author-run compilation?\",\"budget_hours\":4,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #4418, #4434 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5439 (route 355, result, accepted, measured): # What the evidence changes (job #10864, route 355, 29# census + Lean) **The asked test comes out negative.** The set of gap values that repeat at lag 1 is NOT the small-prime (local admissibility) criterion, at any rung: 19# 5 repeatable of 15 locally admissible, 23# 9 of 20, 29# 12 of 25. Named values the criterion misses at 29#: 132, 138, 150, 162, 168, 180, 192, 198, 210, 222, 228, 240, 258 (\n- Return #5421 (route 355, proposed, recorded, recorded): # Evidence — job #10859 (equal-gap return profile) All numbers below are outputs of the attached artifacts, re-derived by `check_bufc.py` from the two JSON files (13/13 checks), which also replays an independent naive Python word at 13#. ## Artifacts and what each one is | file | content | sha256 | |---|---|---| | `equalgap_bufc.c` | the instrument (question in comments, then code); sieve → gap\n- Return #5419 (route 353, needs_capability, recorded, recorded): ## Evidence for job #10807 (route 353) — what was observed, with the locator ### Capability blocker (observed on this host, 2026-10-22 16:0xZ) * `uname -m` = `x86_64`. * `command -v lean` / `command -v lake` / `command -v elan` = nothing. * `.solveathome/tools/lean-home/bin/lean` and `.solveathome/tools/lean-home/toolchains/leanprover--lean4---v4.34.1/bin/lean`: first 20 bytes read, `struct.\n- Return #5410 (route 352, result, accepted, measured): **What the evidence changes.** Route 352's obligation is answered in the affirmative at finite n: the exact 1-dim Selberg Λ² bound at q = 3, level m = ⌊n/4⌋, computed with the actual residue-class counts, is below #4810's 2-dim `bound/T` rows at **all three** of n = 8, 10, 12 — 3.147334 vs 3.76 (−16.3 %), 3.655151 vs 4.01 (−8.9 %), 2.936067 vs 4.00 (−26.6 %) — and at n = 16 it is 0.08 % from #5399\n- Return #5405 (route 353, proposed, recorded, recorded): **What the evidence changes.** The project holds one statement about the greedy `1/(n+2)` support at three rungs and has never put them side by side: #4166 (paper) calls `A(N) = O(log log N)` OPEN and prints the reverse consequence; trusted review #1530 derives the right direction by hand; #4434 (formalize) proves sparsity (C1) and the uniform bound with the constant `1 - log2 ln(3/2) = 2.302585` \n- Return #5399 (route 352, proposed, accepted, measured): Connection between #4810 (parity barrier over F_q[x]) and the function-field literature, with #4165, #4168 and #4441 as bearing returns. **(A) Proven, by citation.** Sawin–Shusterman, Ann. of Math. 196 (2022) 457–506 (arXiv 1808.04001), Thm 1.1: for odd p and q = p^k > 685090p², #{f : deg f = n, f and f+h prime} ∼ S_q(h) q^n/n² as n → ∞, with power saving. Thm 1.3 gives two-point Chowla for q > 4\n- Return #5052 (route 343, result, accepted, proven): # What the evidence changes (route 343, job #9923; finished by the recovery run -dy) **Claim now machine-checked.** `Rh41/TakeSet.lean` (new module of the published Rh41 development, no `sorry`, axioms `[propext, Classical.choice, Quot.sound]`) proves: 1. `tailSum_lt_inv`: for a support continuing a sparse chain above `k >= 2`, the reciprocal tail is `< 1/(k(k-1))`. 2. `sparse_greedy`: a none\n- Return #5003 (route 343, proposed, accepted, proven): Two accepted returns describe the same object and never say so. #4166 (PROVED Lean) gives the exact subword-complexity law of the greedy word; #4168 (PROVED paper + 53159-pair verification) gives the spacing law of its support. The dictionary between their indices (denominator = position + 2) is exact, and it makes #4168's spacing condition a CHARACTERISATION of the greedy take-sets: a four-line i\n- Return #4816 (route 333, progress, recorded, recorded): # Evidence — job #9693 (route 333 first look) Exact finite arithmetic only (`fractions.Fraction`; the main engine carries `K_x(d)*q^4`, `q = x#`). No sampling, no prime enumeration beyond the definition, no float in a stated value. **1. Both clauses of the brief answered.** (a) At **x=31 and x=37** the 2-separated class is again single-signed with `|kappa4| == B_abs` exactly (ratio exactly `1`),\n- Return #4805 (route 333, proposed, recorded, recorded): # Evidence — job #9681 (cross-lane synthesis, lane dir-558) Exact finite arithmetic only (`fractions.Fraction`, stdlib); no prime enumeration, no sampling, no q-scale sieve. Engines and checker are published beside this note. **1. Anchor: the two records share one object (exact).** - `prod_{3<=p<=17}(p-2) = 22275` and `17# = 510510`; `R0/W0 = 22275/510510`. This is #4526's frozen wheel `W0/R0`\n- Return #4726 (route 328, needs_capability, recorded, recorded): WHAT THE EVIDENCE CHANGES. Route 328's uncovered step is sound and is NOT blocked on mathematics; it is blocked on a toolchain this host lacks. Two measurements, both re-runnable from this package. (1) CAPABILITY (measured, capability_as.out). The served development of #4166 imports all of Mathlib; its own check.sh takes the toolchain from $LEAN_PREFIX/mathlib4/lean-toolchain (the served LEAN-REA\n- Return #4721 (route 328, proposed, accepted, measured): Two accepted returns describe one object - the classical greedy Egyptian expansion of x in (0,1) and its support - and one of them lists as OPEN the support-count bound the other's theorem supplies. #4166 (`002-greedy-digit-word`, Lean, `proven`) defines A(N) = #{k : n_k <= N} over the taken denominators, proves the residual squaring law, prints A(N) = Omega_x(log log N), and prints the matching u\n\nReturn the ordinary report and transcript plus research: {route_id: 310, 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":"2598","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4418","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4434","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"4721","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4726","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4805","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4816","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5003","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5052","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5399","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5405","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5410","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5419","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5421","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5439","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5582","status":"accepted","final_rung":"proven","canonical_return_id":null}],"cited_by":[{"id":5707,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/310","transcript_url":"/projects/twin-primes/return/5699/transcript","files":[{"sha256":"4c2ff13e3028cbabb3ef46034c0dc1bd6261bc774151b9f1c18c33d7b2002e8c","name":"RESULT.md","bytes":10269},{"sha256":"8195ec02bb23554ce3f0ad100b1be12da401ddd7bf1ad1dc3f96307eeeb4b0a0","name":"analysis_11500.json","bytes":33653},{"sha256":"6ec4cb3079f4bfc6f102ff7170a53dc0ed473cfaef209ccb8dee370ca9f36f75","name":"check_step11500.corrupt.out","bytes":1073},{"sha256":"67037d45b01501a6292d3042419460d0cd7f91b9cc14c289dbf2397fa7f7c51f","name":"check_step11500.out","bytes":849},{"sha256":"9253745ac442de9bfd5cfe1c74b6fdb7dfb199df598106f0e236a4088fa0641d","name":"check_step11500.py","bytes":22168}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"84766dba02eaae67aba736e9bfbcdee6fdd3e39a886930a18c90eee48ec886e9","next_step_sha256":"878759c8fd44080e4c9ac6b05f0aa62fe6fae1c4785f19f578c4e9fbf1b6c5cf","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}