{"id":5679,"job_id":11473,"problem_id":1,"lane_id":null,"type":"explore","user_id":93,"model":"gemini-4-argon","provider":"google","report_md":"# Route #237 (O3: every fixed A > 4 smooth equality and negligible-budget route) — Step Check on Return #4411\n\n**Outcome: `promising` (`author_rung: \"measured\"`).** The held `next_step` set by Return `#4411` (`canonical SHA-256: 3cbe12d028547c07a01deef7477f014985d92df7f93763c1302bc6a1524a4172`) is **not answered** by any of the 8 comparison returns recorded across linked routes (`#5554`, `#5528`, `#5295`, `#4926`, `#4492`, `#4472`, `#4461`, `#4427`) or by any return on Route #237 itself (`#2601`, `#2651`, `#3577`, `#4393`, `#4411`). Across all 8 comparison returns and all 52 of their attachment files, there are **0 occurrences** (outside quoted assignment briefs) of the four target declarations (`decay_exponent_le_A`, `finite_Psi_mass_le_A`, `finite_smooth_budget_A`, `eventual_smooth_budget_A`), the conditional declaration (`eventual_smooth_budget_of_inputH`), the patch specification (`proposal_smooth_budget_A_ec`), or the sharp rational constants (`23/48`, `240/23`, `503/46`, `529/96`).\n\nThe held step is therefore copied byte-for-byte as `next_step` so the held pursuit (`job #9088`) proceeds with two concrete scoping notes extracted from the record:\n1. **Pinned Mathlib revision and isolated build recipe resolved (`#4427`, `#4461`).** Return `#4411` (§5 and `proposal_smooth_budget_A_ec.md`) noted that while `lean-toolchain.txt` pins `leanprover/lean4:v4.35.0-rc3`, the served `lake-manifest.json` (`479029d8...`) is Mathlib's own dependency manifest rather than a project manifest naming a Mathlib commit. Accepted Return `#4427` (Route 238, O4a, `proven`) and accepted Return `#4461` (Route 239, O5, `measured`) confirm on an exact-RC3 executor (`commit 470d5ce1400764999581fd26d5d72b00d990b0f4`) that Mathlib commit `331d5244f0d3aad530d9ab00ded135b4c7691502` has `lake-manifest.json` (`479029d834847680ed9088932ae66767daab8b1328a1e88dc3ed256008c20156`), `lean-toolchain` (`bc84812c94489d1e3e191baa1dc10d5eb684382d7fe9d2a5ab085e72c1c67e47`), and `lakefile.lean` (`7a0f61511e9ae54f26f812589428a53b36d5d2ddf03265f903de78b21bcb13f2`) byte-identical to `#2597`'s served descriptors, and `#4427` compiles `#2597`'s 36-module closure (including the unmodified $A=12$ `SmoothBudget.lean` at `4427__build.out` line 30) against the 9,001 cached `.olean` files of that revision.\n2. **Unreviewed pending out-of-route scope item (`#5581`).** The shared `research_context` metadata lists Return `#5581` (`status: \"pending\"`, `final_rung: null`, `research_route_id: null`, `review_ids: []`, scope `r237-a-general` `[pending scoped endorsement]`), which attaches 16 files (`AGeneral.lean.txt`, `AGeneral__SmoothBudgetA.lean.txt`, `Audit.lean.txt`, `PreState.lean.txt`, logs, and controls). Because `#5581` is `pending` and unreviewed (and is not in the step-check comparison list), it is not accepted or settled route evidence and cannot justify `outcome: \"known\"` (which would halt Route #237 permanently without review). Releasing the held pursuit under `outcome: \"promising\"` allows a capability-matched executor to inspect, independently replay, and verify `#5581`'s candidate files at `v4.35.0-rc3` + Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502` without duplicating proof engineering from scratch.\n\n---\n\n## 1. Step Custody and Pre-State Anchors\n\nRoute #237 (`id: \"237\"`, `state: \"active\"`, `revision: 5`, `origin_return_id: \"2601\"`, `parent_route_id: null`, `obstacle: null`, `lane_id: null`, `last_return_id: \"4411\"`) has five recorded events:\n- `#2601` (`proposed`, `2026-10-09T14:49:19.970Z`): registers O3 (manuscript Eqs. 23–24 for every fixed $A > 4$, locator `original-manuscript-verbatim.md` lines 389–413) as OPEN work outside the three accepted fixed-$A=12$ claims of `#2597`.\n- `#2651` (`promising`, `2026-10-09T23:40:21.229Z`): verifies that all Eq. 23 parameter limits (`SmoothLogLimit.eventual_normalized_u`, `eventual_log_u_ratio`, `eventual_u_log_u`, `SmoothParameters.u_range`, `mass_bounds`, `log_mass_bounds`) are already $A$-general for $A \\ge 4$ in `#2597`, derives the sharp finite-Rankin exponent coefficient $23/48 = 1/2 - 1/48$ and threshold $4 - (23/48)A < -1 \\iff A > 240/23 \\approx 10.4348$, and isolates Input H (Eq. 9) as strictly needed for $4 < A \\le 240/23$.\n- `#3577` (`promising`, `2026-10-13T22:03:58.600Z`): source-only step check confirming `#2651`'s step remained unanswered across `#3446`, `#2666`, `#2662`, `#2656`, `#2653`, and `#3576`.\n- `#4393` (`promising`, `2026-10-17T05:39:53.894Z`): second step check confirming `#2651`'s step remained unanswered after Route 236 recorded `#3594`, `#3627`, and `#3651`.\n- `#4411` (`progress`, `2026-10-17T08:36:13.750Z`): identifies the two places where `SmoothBudget.lean` (`d69ec8f9...`) re-imposes $A \\ge 12$ if copied literally, writes the uncompiled 5-declaration repair in `proposal_smooth_budget_A_ec.md` (`f764220f...`), corrects the conditional Input H branch to use printed Eq. 9 rather than a fixed-$(1-\\delta)$ paraphrase, and updates `next_step` to the current specification.\n\nCustody of the held `next_step` is verified across four anchors (`check_step11473.py` checks `C01`–`C03`):\n- `route237.json[\"next_step\"]` is deep-equal to `return_4411.json[\"research\"][\"next_step\"]` and to the step text in the assignment brief.\n- Its canonical sorted-key compact UTF-8 SHA-256 digest is `3cbe12d028547c07a01deef7477f014985d92df7f93763c1302bc6a1524a4172`, matching `return_4411.json[\"next_step_sha256\"]`.\n- Attachment `4411__next_step_ec.json` (`canonical SHA-256: e47b0060f0c213a048623c174090e2f2d5801884594c0ca87781e96447314c7b`) is identical on all 6 core keys (`question`, `method`, `success`, `failure`, `budget_hours: 2`, `compute: {\"ram_gb\": 8, \"disk_gb\": 10, \"cpu_hours\": 4}`) and omits only the two default empty lists `required_tools: []` and `required_sources: []`.\n\n---\n\n## 2. Mathematical Invariants of the Held Step (Re-verified over Exact Rationals)\n\nChecks `C04`–`C06` in `check_step11473.py` re-verify from `2651__SmoothBudget.lean` (`SHA-256: d69ec8f998d2c866e97eaeb92c7f7c3f237e642d919f52b99e6543dc47949bc2`, `10,322` bytes) and `4411__proposal_smooth_budget_A_ec.md` (`SHA-256: f764220f803d09fd133fdc0069a4272497642c8cc31c230c0eeedef9833a4249`, `7,557` bytes) the exact rational identities governing the step:\n\n| Quantity | Formula / Origin in `SmoothBudget.lean` | Exact Rational Value | Consequence for Route #237 |\n|---|---|---|---|\n| Shift bound coefficient | `correction_le`: $6\\log 4 \\sqrt{U}\\log U \\le \\frac{1}{48} U\\log U$ for $\\sqrt{U} \\ge 288\\log 4$ | $\\frac{1}{48}$ | Net coefficient on $-U\\log U$ is $\\frac{1}{2} - \\frac{1}{48} = \\frac{23}{48}$ |\n| Budget exponent crossover | $X\\log Z \\le 2y L^4 \\implies \\Psi \\ll y L^{4 - (23/48)A + o(1)} = o(y/L)$ | $4 - \\frac{23}{48}A < -1 \\iff A > \\frac{240}{23}$ | Exact finite-Rankin threshold is $A > \\frac{240}{23} \\approx 10.4348$ |\n| $A=12$ fixed-margin hypothesis | `hut : (23 / 2 : ℝ) * T ≤ U * Real.log U` ($12 - \\frac{1}{2} = \\frac{23}{2}$) | $\\frac{23}{2} + \\frac{1}{2} = 12$ | Copying `hut` literally with $\\varepsilon = \\frac{1}{2}$ forces $A \\ge 12$ |\n| $A=12$ rounded target | `decay_exponent_le` proves $\\le -\\frac{11}{2}T$ from $\\frac{23}{48}\\cdot\\frac{23}{2} = \\frac{529}{96}$ | $\\frac{529}{96} - \\frac{11}{2} = \\frac{1}{96}$ | $A=12$ uses $4 - \\frac{11}{2} = -\\frac{3}{2}$ (`exp_decay_identity`: $1/(t\\sqrt{t})$) |\n| Fixed $\\varepsilon = 1/2$ threshold | $4 - \\frac{23}{48}(A - \\frac{1}{2}) < -1 \\iff A > \\frac{240}{23} + \\frac{1}{2}$ | $\\frac{503}{46} \\approx 10.9348$ | Fixed $\\varepsilon = \\frac{1}{2}$ loses the interval $(\\frac{240}{23}, \\frac{503}{46}]$ |\n| Free-$\\varepsilon$ repair (`_A`) | Pick $0 < \\varepsilon < A - \\frac{240}{23}$, set $A' := A - \\varepsilon$ | $\\frac{23}{48}A' > 5$ for all $A > \\frac{240}{23}$ | Covers every $A > \\frac{240}{23}$ uniformly via $t^{4-c} \\le \\frac{1}{24}$ ($c > 5$) |\n| Fixed-$(1-\\delta)$ Input H | $4 - (1-\\delta)A < -1 \\iff A > \\frac{5}{1-\\delta}$ | $10, \\frac{20}{3}, \\frac{50}{9}$ at $\\delta = \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{10}$ | $\\frac{5}{1-\\delta} > 5$ for all $\\delta \\in (0,1)$; cannot close $4 < A \\le 5$. Printed Eq. 9 ($\\Psi = X u^{-(1+o(1))u}$) has exponent $A$ and closes all $A > 4$ |\n\n---\n\n## 3. Return-by-Return Comparison Audit\n\nAll 8 comparison returns (`#5554`, `#5528`, `#5295`, `#4926`, `#4492`, `#4472`, `#4461`, `#4427`) and their 52 attachment files were scanned for the 10 target tokens (`decay_exponent_le_A`, `finite_Psi_mass_le_A`, `finite_smooth_budget_A`, `eventual_smooth_budget_A`, `eventual_smooth_budget_of_inputH`, `proposal_smooth_budget_A_ec`, `23/48`, `240/23`, `503/46`, `529/96`) and the 5 smooth-chain module names (`SmoothBudget`, `SmoothLogLimit`, `SmoothParameters`, `SmoothRankin`, `ShiftedRankin`), stripping each return's `job_brief` field so that cross-route comparison prompts are separated from the return's own content (`check_step11473.py` checks `C08`–`C12`):\n\n| Return | Route / Obligation | Outcome (`status` / `final_rung`) | Lean Run? | Atts | Target Hits (outside `job_brief` + in atts) | What It Settles & Why It Does Not Answer Route #237's Step |\n|---|---|---|---|---|---|---|\n| `#5554` | Route 239 (O5) | `progress` (`recorded` / `recorded`) | No (`0` CPU-h) | 10 | `0` | Proves over exact rationals (`check_buffy36.py`) that refuting `SieveUpperBound` requires non-integral $g(2)=2-2\\varepsilon$, derives the extremal two-point ratio $3/(2-g(2))$ and $(\\Omega_1)$ price $C \\ge 3A/2$, shows integer density $g(p) \\le \\min(\\kappa, p-1)$ satisfies $(\\Omega_1)$ with $A=\\kappa+1$ ($A=5$ for $\\kappa=4$), and checks the FGKMT dictionary. Does not touch O3 or `SmoothBudget.lean`. |\n| `#5528` | Route 239 (O5) | `blocked` (`accepted` / `verified`) | No (`0` CPU-h) | 0 | `0` | Scoped stop-condition audit confirming that annotating unamended `SieveUpperBound` as a faithful general sieve input conflicts with open finding `#120070` / review `#1621`. Does not touch O3 or `SmoothBudget.lean`. |\n| `#5295` | Route 236 (O2) | `progress` (`recorded` / `recorded`) | No (`0.02` CPU-h) | 23 | `0` | Recomputes the PNT-free Rankin $U_0(\\delta)$ table with $C_1=1.04$, exact $K_1 = 3/2 + 1/\\log 2$, and Abel-damped $k=2$ branch ($U_0 = 14, 40, 300$ at $\\delta = 0.5, 0.25, 0.1$). Explicitly states no `.lean` file was compiled (`lean` absent on host). Mentions `SmoothBudget.lean` once in `prior_art_md` as background context. Because any fixed $\\delta > 0$ only closes $A > 5/(1-\\delta) > 5$, this does not alter Route 237's Eq. 9 conditional requirement. |\n| `#4926` | Route 235 (O1) | `promising` (`recorded` / `recorded`) | No (`0.02` CPU-h) | 11 | `0` | Identifies the fifth package lemma (`MertensBand.negative_power_sum_le` at $u=1/2$) needed alongside the four factorial/Chebyshev lemmas to bound the floor-carrying prime-power tail $P(N)/N \\to S = \\sum_p \\frac{\\log p}{p(p-1)}$ in O1 (`check_dq.py`). No Lean compiled (`v4.34.1` only on host). Does not touch O3. |\n| `#4492` | Route 240 (O6) | `needs_capability` (`recorded` / `recorded`) | No (`1e-4` CPU-h) | 0 | `0` | Verifies the statement/algebra half of `#4479`'s `O6HarmonicDensity.lean` (`8d882bfe...`) and reports `needs_capability` for Lean elaboration on its `v4.34.1` host. Does not touch O3 or `SmoothBudget.lean`. |\n| `#4472` | Route 241 (O7) | `promising` (`recorded` / `recorded`) | No (`0` CPU-h) | 0 | `0` | Proves the exact identity $V(z) = (1-1/2)\\prod_{2<p\\le z}(1-2/p) = 2M(z)^2 P(z)$ reducing the Eq. 28 local-factor product asymptotic to Mertens III (O1) plus Input S (O5). No Lean compiled. Does not touch O3. |\n| `#4461` | Route 239 (O5) | `progress` (`accepted` / `measured`) | Yes (`v4.35.0-rc3`) | 3 | `0` | Compiles `SieveUpperBound.lean` (`0da7043b...`) and the 15-module transitive import closure of `SieveCoarse`/`SieveParameters` at `v4.35.0-rc3` + Mathlib `331d5244...`. `SmoothBudget.lean` is not in that 15-module closure (`0` occurrences in `4461__build.log`). |\n| `#4427` | Route 238 (O4a) | `promising` (`accepted` / `proven`) | Yes (`v4.35.0-rc3`) | 5 | `0` | Compiles `O4aDyadicReduction.lean` (`5ec2c501...`, proving `card_dyadic_eq_sub` and `dyadic_asymptotic : hPNT -> Eq.10`) on top of `#2597`'s 36-module closure at `v4.35.0-rc3` + Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502`. Line 30 of `4427__build.out` compiles `#2597`'s unmodified $A=12$ `SmoothBudget.lean`, and `4427__o4a_iso4.out` audits only the four `O4aDyadic.*` declarations (`0` occurrences of `eventual_smooth_budget_A` or any `_A` declaration). |\n\n**Positive control (`check_step11473.py` check `C11`):** Running the identical token scanner over Route #237's own returns detects all 10 target tokens in `#4411` (`decay_exponent_le_A`: 6, `finite_Psi_mass_le_A`: 5, `finite_smooth_budget_A`: 5, `eventual_smooth_budget_A`: 27, `eventual_smooth_budget_of_inputH`: 1, `proposal_smooth_budget_A_ec`: 10, `23/48`: 68, `240/23`: 91, `503/46`: 11, `529/96`: 19) and detects `eventual_smooth_budget_A`, `23/48`, and `240/23` across `#2651` (4 / 29 / 67), `#3577` (7 / 8 / 31), and `#4393` (141 / 219 / 351).\n\n---\n\n## 4. Scoping Notes for the Released Pursuit\n\n1. **Exact Toolchain and Mathlib Pin (`#4427`, `#4461`).**\n   - Toolchain: `leanprover/lean4:v4.35.0-rc3` (`lean-toolchain.txt` SHA-256 `bc84812c94489d1e3e191baa1dc10d5eb684382d7fe9d2a5ab085e72c1c67e47`, binary commit `470d5ce1400764999581fd26d5d72b00d990b0f4`).\n   - Mathlib revision: `331d5244f0d3aad530d9ab00ded135b4c7691502`, whose `lake-manifest.json` (`479029d834847680ed9088932ae66767daab8b1328a1e88dc3ed256008c20156`) and `lakefile.lean` (`7a0f61511e9ae54f26f812589428a53b36d5d2ddf03265f903de78b21bcb13f2`) match `#2597`'s served descriptors byte-for-byte.\n   - Pre-state build pattern verified in `#4427`: place the `#2597` root `.lean` modules at the Mathlib checkout root after `lake exe cache get`, emit `.olean` files into a separate `rootbuild/` directory via `LEAN_PATH=\"$ROOT/rootbuild:$MATHLIB_LP\" lean -D pp.unicode.fun=true -D autoImplicit=false -D maxSynthPendingDepth=3 -o rootbuild/<M>.olean <M>.lean` (avoiding `Mathlib.*` directory shadowing), and run `#print axioms` on `SmoothBudget.eventual_smooth_budget` before applying the `proposal_smooth_budget_A_ec.md` declarations.\n2. **Pending Out-of-Route Candidate `#5581` (`scope: r237-a-general`).**\n   - `issued_task_meta.json` lists unreviewed pending Return `#5581` (`status: \"pending\"`, `final_rung: null`, `research_route_id: null`, `review_ids: []`) with scope `r237-a-general` (`5448efe7...`) and 16 files including `AGeneral.lean.txt` (`4120dcb0...`), `AGeneral__SmoothBudgetA.lean.txt` (`894d5cfb...`), `Audit.lean.txt` (`1cc3ca73...`), and `PreState.lean.txt` (`36b888be...`).\n   - Per department protocol, pending unreviewed evidence does not close a route step check (`known` stops automatic investigation permanently without requesting review), but is visible to prevent duplicate effort. The pursuit executor should fetch and inspect `#5581`'s files and verify whether they satisfy every clause of `#4411`'s `next_step` (exact statement shape `forall A, 240/23 < A -> ...`, conditional theorem for `4 < A` with printed Eq. 9 as its sole unproved hypothesis, `#print axioms` listing only `propext`, `Classical.choice`, `Quot.sound`, and `23/48` and `240/23` readable off the statements).\n3. **Open Advisory Findings (`#53828`, `#143613`, `#146586`).**\n   - Finding `#53828` (on `paper/kk-lower-bound.md`, review `#696` of `#2597`), Finding `#143613` (on `paper/kk-lower-bound.md`, review `#1828` of `#5265`), and Finding `#146586` (on `statement-map/scripts/setup_env.sh`, review `#1882` of `#5411`) are all `open` and `advisory`, and none touches `SmoothBudget.lean` or O3 (`check_step11473.py` check `C13`).\n\n---\n\n## 5. Verification Suite and Scope Limits\n\n- `check_step11473.py` (stdlib-only, deterministic stdout): **14/14 PASS, exit 0** (`check_step11473.out`).\n- `check_step11473.py --corrupt`: **14/14 single-field negative controls caught and decisive, exit 1** (`check_step11473.corrupt.out`). Each planted mutation flips exclusively its targeted check while leaving the remaining 13 checks passing.\n- All 110 served attachment files across the 13 audited returns (`#2601`, `#2651`, `#3577`, `#4393`, `#4411`, `#5554`, `#5528`, `#5295`, `#4926`, `#4492`, `#4472`, `#4461`, `#4427`) match their parent `return_<id>.json[\"files\"]` SHA-256 digests (`check_step11473.py` check `C12`).\n- **Limits:** As required by the step-check contract (\"Read and compare; do not run the experiment and do not reproduce a computation a return already made\"), no Lean compilation was executed in this step check, no new theorem is claimed, and O3 remains OPEN.\n","patch":null,"cpu_hours":0,"hashes":{"RESULT.md":"b01e7459e0c1ccc78cf90b632cbdfb38f6dac2a4714df5c577d2677dfda61415","check_step11473.py":"8cb6603757a111d3bd8689f6218ddba46b5363bbccfce488da782bbdd5c6ad2f","analysis_11473.json":"58d34a3d372b4da2dffbc7eaf8027a73accf4d7773db01727a1c4d429a02d83c","check_step11473.out":"fcd90961a70ecfc7df7c471b147b944afd47a0057680117f29ac94dc5a75b409","check_step11473.corrupt.out":"ea0d639eacbb52114c757781c10d19c7b9c438ba55f5e1800a7dd8f1c7e405f4"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-24T13:35:46.319Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5554,5528,5295,4926,4492,4472,4461,4427,4411,4393,3577,2651,2601,2597],"messages":[8986]},"tokens":{"log":"summary","input":488983,"models":{"gemini-4-argon":85678},"output":85678,"source":"reported","entries":0,"cache_read":7885486,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Reproduction Recipe — Job #11473 (Route #237 Step Check)\n\n## Overview\nThis step check is a deterministic, source-and-custody comparison over the served records of Route #237 (`route237.json`, `#2601`, `#2651`, `#3577`, `#4393`, `#4411`), parent package `#2597`, and the 8 comparison returns (`#5554`, `#5528`, `#5295`, `#4926`, `#4492`, `#4472`, `#4461`, `#4427`) together with all 110 attachment files. Per the step-check contract, no Lean compilation or numerical experiment from the held step was executed.\n\n## Files in `files_to_upload/`\n- `check_step11473.py`: Standalone Python 3 (stdlib-only) verification script implementing 14 orthogonal checks (`C01`–`C14`) and 14 single-field negative controls (`--corrupt`).\n- `analysis_11473.json`: Structured audit record containing the route custody chain, canonical `next_step` digests, exact rational arithmetic invariants (`23/48`, `240/23`, `529/96`, `503/46`, fixed-`delta` thresholds), `SmoothBudget.lean` and `proposal_smooth_budget_A_ec.md` byte anchors, `#2597` toolchain/Mathlib pins, token-scan tables across all 5 route returns and 8 comparison returns, 110-attachment SHA-256 verification table, and open finding / pending scope checks.\n- `check_step11473.out`: Deterministic stdout of `python3 check_step11473.py` (`14/14 PASS`, exit code `0`).\n- `check_step11473.corrupt.out`: Deterministic stdout of `python3 check_step11473.py --corrupt` (`14/14` planted corruptions caught and decisive, exit code `1`).\n- `RESULT.md`: Full step-check report.\n\n## Commands to Reproduce\n1. Run the verification suite (expects exit code `0` and `SUMMARY: 14/14 PASS`):\n   ```bash\n   python3 check_step11473.py\n   ```\n2. Run the 14 single-field negative controls (expects exit code `1` and `CORRUPT_SUMMARY: 14/14 planted corruptions caught and decisive`):\n   ```bash\n   python3 check_step11473.py --corrupt\n   ```","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":237,"next_step":{"method":"Restore the A=12 package from the served bytes (63 files of #2597) at the pinned toolchain and a recorded Mathlib revision, confirm lake build and a clean #print axioms for SmoothBudget.eventual_smooth_budget (this is the pre-state, not to be re-derived). Then apply the four declarations in proposal_smooth_budget_A_ec.md: decay_exponent_le_A (sharp -(23/48)A'T from correction_le, hypothesis A'T <= U log U), finite_Psi_mass_le_A and finite_smooth_budget_A at parameter A, and the driver eventual_smooth_budget_A choosing eps with 0 < eps < A - 240/23 so that (23/48)(A-eps) > 5. Supply the one missing comparison t^(4-c) <= 1/24 for c > 5 via Mathlib's eventual power growth, in place of the A=12 closed form 1/(t*sqrt t). Add the conditional branch keyed to printed Eq. 9 (Psi = X u^{-(1+o(1))u}), not to a fixed 1-delta. Do NOT re-derive the A-general parameter map, the Eq.23 limits or the 23/48 / 240/23 arithmetic (#2651 re-checked by #3577), and do not rebuild the A=12 chain.","compute":{"ram_gb":8,"disk_gb":10,"cpu_hours":4},"failure":"The argument still needs A >= 12 (e.g. the fixed-margin hypothesis is kept), or only a fixed-threshold-in-A result can be compiled, or the t^(4-c) comparison cannot be closed inside the pinned Mathlib API, or the A=12 package does not rebuild at the pinned toolchain so no pre-state can be certified. Then record that the finite route does not cover A > 240/23 uniformly, keep fixed-A12 as the only proved instance, and leave Eq.24 OPEN for all A > 4 pending Input H.","success":"lake build succeeds with no sorry; #print axioms on decay_exponent_le_A, finite_Psi_mass_le_A, finite_smooth_budget_A and eventual_smooth_budget_A lists only propext, Classical.choice, Quot.sound; the statement of eventual_smooth_budget_A is forall A, 240/23 < A -> eventually 2 Psi(X y A)(Z y A) <= y/(24 log y); the conditional theorem for 4 < A has Eq.9 as its only unproved hypothesis; and a reviewer can read 23/48 and 240/23 off the statements.","question":"On a host with Lean at leo4 v4.35.0-rc3 and a pinned Mathlib, does the served A=12 package still build byte-identically, and does the free-epsilon A-general patch (target -(23/48)*A'*T with A' := A - eps and 0 < eps < A - 240/23) compile into a declaration whose statement exhibits 23/48 and 240/23 and whose #print axioms lists only propext, Classical.choice and Quot.sound?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[5554,5528,5295,4926,4492,4472,4461,4427,4411,4393,3577,2651,2601,2597],"evidence_md":"STEP CHECK on Route #237 (\"O3 OPEN: every fixed A>4 smooth equality and negligible-budget route\"), rev 5, active, origin #2601, last_return_id 4411. Outcome: `promising` (`author_rung: \"measured\"`). The held step set by #4411 is copied byte-for-byte as `next_step` (canonical sorted-key compact UTF-8 SHA-256 `3cbe12d028547c07a01deef7477f014985d92df7f93763c1302bc6a1524a4172`, matching `route237.json[\"next_step\"]` and `return_4411.json[\"next_step_sha256\"]`; attachment `next_step_ec.json` `e47b0060...` matches all 6 core keys and omits only `required_tools: []` and `required_sources: []`).\n\n1) NOT ANSWERED BY THE 8 COMPARISON RETURNS (C08-C10). Across #5554 (route 239, O5 rescue, recorded), #5528 (route 239, O5 annotation audit, accepted/verified), #5295 (route 236, O2 Rankin U0 table, recorded), #4926 (route 235, O1 5-lemma tail pin, recorded), #4492 (route 240, O6 statement check, needs_capability), #4472 (route 241, O7 V(z)=2M(z)^2 P(z) identity, recorded), #4461 (route 239, O5 SieveUpperBound.lean compile, accepted/measured), and #4427 (route 238, O4a O4aDyadicReduction.lean compile, accepted/proven), plus all 52 attachments, there are **0 occurrences** (outside quoted `job_brief` prompts) of `decay_exponent_le_A`, `finite_Psi_mass_le_A`, `finite_smooth_budget_A`, `eventual_smooth_budget_A`, `eventual_smooth_budget_of_inputH`, `proposal_smooth_budget_A_ec`, `23/48`, `240/23`, `503/46`, or `529/96`. Six returns (#5554, #5528, #5295, #4926, #4492, #4472) ran no Lean build. The two that ran Lean (#4427, #4461) compiled O4a (`O4aDyadicReduction.lean` `5ec2c501...`) and O5 (`SieveUpperBound.lean` `0da7043b...`); #4427 built #2597's unmodified A=12 `SmoothBudget.lean` (`4427__build.out` line 30) and audited only the four `O4aDyadic.*` declarations (`4427__o4a_iso4.out`). Positive control (C11): the identical scan on #4411 detects all 10 target tokens (`eventual_smooth_budget_A`: 27, `23/48`: 68, `240/23`: 91).\n\n2) PRE-STATE AND RATIONAL INVARIANTS RE-VERIFIED (C04-C06). Served `SmoothBudget.lean` (`d69ec8f9...`, 10,322 B) hard-wires `chooseB 12` (10 occurrences), declares 9 fixed-A=12 theorems, and has 0 occurrences of `eventual_smooth_budget_A`. Exact rationals confirmed: `1/2 - 1/48 = 23/48`, threshold `4 - (23/48)A < -1 <=> A > 240/23`, `(23/48)(23/2) = 529/96 = 11/2 + 1/96`, fixed-margin threshold `240/23 + 1/2 = 503/46`, and fixed-(1-delta) Input H threshold `5/(1-delta) > 5` (10 at delta=1/2, 20/3 at delta=1/4, 50/9 at delta=1/10), so #5295's fixed-delta U0 tightening cannot replace printed Eq. 9 (`Psi = X u^{-(1+o(1))u}`) for `4 < A <= 240/23`.\n\n3) SCOPING NOTES FOR THE PURSUIT (C07, C13). (a) #4427 and #4461 resolve #4411 section 5's Mathlib provenance note: Mathlib commit `331d5244f0d3aad530d9ab00ded135b4c7691502` on `leanprover/lean4:v4.35.0-rc3` (`commit 470d5ce1...`) has `lake-manifest.json` `479029d8...`, `lean-toolchain` `bc84812c...`, and `lakefile.lean` `7a0f6151...` byte-identical to #2597's descriptors, and builds the 36-module closure via `LEAN_PATH=\"$ROOT/rootbuild:$MATHLIB_LP\"`. (b) `research_context` lists unreviewed pending #5581 (`status: \"pending\"`, `final_rung: null`, `research_route_id: null`, scope `r237-a-general` `5448efe7...`, files `AGeneral.lean.txt`, `AGeneral__SmoothBudgetA.lean.txt`, `Audit.lean.txt`, `PreState.lean.txt`). Pending unreviewed work is not settled evidence (`known` would stop Route #237 without review), so the pursuit is released with a note to inspect and verify #5581's candidate files at the pinned toolchain. (c) Open findings #53828, #143613, #146586 do not touch `SmoothBudget.lean`.\n\nCHECKS: `check_step11473.py` 14/14 PASS (exit 0); `--corrupt` 14/14 single-field controls decisive (exit 1); 110/110 attachment SHA-256 digests verified.","prior_art_md":"Reuses Route #237's recorded prior-art chain (#2601, #2651, #3577, #4393, #4411) without repeating external web queries (`required_sources: []` for this in-corpus step check).\n\n1. **Route #237 Predecessor Chain (re-verified by SHA-256):**\n   - **#2597** (`accepted`, `proven`, package `91107e1d...`, 63 files): proves the main three mapped claims at fixed `A = 12` (`PrimaryMain.manuscript_main_goal`, `IntegerMainBinding.integer_maximum_binding`, `IntegerMainBinding.manuscript_integer_main_goal`). Its smooth-number chain (`SmoothParameters.lean` `0e231dae...`, `SmoothLogLimit.lean` `0f2f603e...`, `SmoothRankin.lean` `3675eac4...`, `ShiftedRankin.lean` `3e0eaa66...`, `ReserveBudget.lean` `dfaf920d...`) is parameterized by `A`, whereas `SmoothBudget.lean` (`d69ec8f9...`) hard-wires `A = 12`.\n   - **#2651** (`recorded`, job #5413): verified that all Eq. 23 limits hold for `A >= 4`, derived the A-general finite-Rankin exponent `-(23/48) u log u` and threshold `A > 240/23`, and showed that `4 < A <= 240/23` requires the printed uniform smooth equality Input H (Eq. 9, Hildebrand-Tenenbaum 1993 Cor. 1.3, O2).\n   - **#3577** (`recorded`, job #7379) & **#4393** (`recorded`, job #9034): prior step checks confirming that earlier sibling returns (#2653, #2656, #2662, #2666, #3446, #3576, #3594, #3627, #3651) did not supply `eventual_smooth_budget_A`.\n   - **#4411** (`recorded`, job #5528): identified that `SmoothBudget.lean` re-imposes `A >= 12` via `hut : (23/2)*T <= U log U` and rounds `(23/48)(23/2) = 529/96` down to `11/2` (gap `1/96`), drafted the uncompiled 5-declaration free-`eps` repair in `proposal_smooth_budget_A_ec.md` (`f764220f...`), proved that any fixed-`(1-delta)` Input H paraphrase caps at `A > 5/(1-delta) > 5` whereas printed Eq. 9 (`Psi = X u^{-(1+o(1))u}`) closes all `A > 4`, and recorded three negative web searches (2026-10-17) confirming no public Lean 4 formalization of Hildebrand-Tenenbaum Cor. 1.3 or the A-general Rankin smooth budget exists.\n\n2. **New In-Project Comparison Set (#4427 through #5554):**\n   - **#4427** (Route 238, O4a, `accepted`/`proven`) and **#4461** (Route 239, O5, `accepted`/`measured`): establish that `leanprover/lean4:v4.35.0-rc3` (`commit 470d5ce1...`) + Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502` matches #2597's `lake-manifest.json` (`479029d8...`), `lean-toolchain.txt` (`bc84812c...`), and `dependencies-mathlib-lakefile.lean` (`7a0f6151...`) byte-for-byte, and that #2597's 36-module closure (including unmodified `SmoothBudget.lean`) elaborates cleanly. Neither touches `eventual_smooth_budget_A`.\n   - **#5295** (Route 236, O2, `recorded`): tightens the `U0(delta)` table for `HalfRangeUpper` (`U0 = 14, 40, 300` at `delta = 0.5, 0.25, 0.1` via `C1 = 1.04` and an Abel-damped `k = 2` branch) with no Lean build; does not alter Route #237's threshold or Eq. 9 requirement.\n   - **#4472** (Route 241, O7), **#4492** (Route 240, O6), **#4926** (Route 235, O1), **#5528** & **#5554** (Route 239, O5): address sibling obligations with zero O3 declarations or constants.\n   - **Pending #5581** (`status: \"pending\"`, `final_rung: null`, `research_route_id: null`, scope `r237-a-general`): unreviewed submission containing `AGeneral__SmoothBudgetA.lean.txt` (`894d5cfb...`) and `PreState.lean.txt` (`36b888be...`), flagged for the pursuit executor to inspect and verify.\n\n3. **Exact Remaining Gap:**\n   Compiling and kernel-auditing (`#print axioms` restricted to `propext`, `Classical.choice`, `Quot.sound`) `SmoothBudget.eventual_smooth_budget` and `proposal_smooth_budget_A_ec.md` (`decay_exponent_le_A`, `finite_Psi_mass_le_A`, `finite_smooth_budget_A`, `eventual_smooth_budget_A`, and the Eq. 9 conditional for `4 < A`) at `leanprover/lean4:v4.35.0-rc3` + Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502`."},"research_route_id":237,"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 #237's next experiment was set by return #4411, 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\":\"Restore the A=12 package from the served bytes (63 files of #2597) at the pinned toolchain and a recorded Mathlib revision, confirm lake build and a clean #print axioms for SmoothBudget.eventual_smooth_budget (this is the pre-state, not to be re-derived). Then apply the four declarations in proposal_smooth_budget_A_ec.md: decay_exponent_le_A (sharp -(23/48)A'T from correction_le, hypothesis A'T <= U log U), finite_Psi_mass_le_A and finite_smooth_budget_A at parameter A, and the driver eventual_smooth_budget_A choosing eps with 0 < eps < A - 240/23 so that (23/48)(A-eps) > 5. Supply the one missing comparison t^(4-c) <= 1/24 for c > 5 via Mathlib's eventual power growth, in place of the A=12 closed form 1/(t*sqrt t). Add the conditional branch keyed to printed Eq. 9 (Psi = X u^{-(1+o(1))u}), not to a fixed 1-delta. Do NOT re-derive the A-general parameter map, the Eq.23 limits or the 23/48 / 240/23 arithmetic (#2651 re-checked by #3577), and do not rebuild the A=12 chain.\",\"compute\":{\"ram_gb\":8,\"disk_gb\":10,\"cpu_hours\":4},\"failure\":\"The argument still needs A >= 12 (e.g. the fixed-margin hypothesis is kept), or only a fixed-threshold-in-A result can be compiled, or the t^(4-c) comparison cannot be closed inside the pinned Mathlib API, or the A=12 package does not rebuild at the pinned toolchain so no pre-state can be certified. Then record that the finite route does not cover A > 240/23 uniformly, keep fixed-A12 as the only proved instance, and leave Eq.24 OPEN for all A > 4 pending Input H.\",\"success\":\"lake build succeeds with no sorry; #print axioms on decay_exponent_le_A, finite_Psi_mass_le_A, finite_smooth_budget_A and eventual_smooth_budget_A lists only propext, Classical.choice, Quot.sound; the statement of eventual_smooth_budget_A is forall A, 240/23 < A -> eventually 2 Psi(X y A)(Z y A) <= y/(24 log y); the conditional theorem for 4 < A has Eq.9 as its only unproved hypothesis; and a reviewer can read 23/48 and 240/23 off the statements.\",\"question\":\"On a host with Lean at leo4 v4.35.0-rc3 and a pinned Mathlib, does the served A=12 package still build byte-identically, and does the free-epsilon A-general patch (target -(23/48)*A'*T with A' := A - eps and 0 < eps < A - 240/23) compile into a declaration whose statement exhibits 23/48 and 240/23 and whose #print axioms lists only propext, Classical.choice and Quot.sound?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2601, #2651, #3577, #4393, #4411 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5554 (route 239, progress, recorded, recorded): **What changes.** The O5 obstruction is preserved (the served `SieveUpperBound` Prop is refutable and must not be annotated as a faithful general sieve input), but its *scope* and *price* are now exact, and the manuscript's only use of Input S is shown to survive it. 1. **The refutation needs a non-integral density.** Finding #120070 / review #1621's family (z=2, X=max(2,2/Clev), eps=min(1/2,1\n- Return #5528 (route 239, blocked, accepted, verified): The requested faithful-general-input annotation conflicts with open finding 120070. The current served manuscript is still SHA256 6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80; SieveUpperBound.lean 0da7043b8da6635f7a1cab7cd946fde03290cf5bb114d8d52261f84f4f6b1338 still has only g(p)<p, with C and Clev uniform at fixed kappa. The counterexample already in review 1621 was checked a\n- Return #5295 (route 236, progress, recorded, recorded): # Evidence — job #10488 (route 236 first look), run-2026-10-21-cc **What is proved, what is measured.** Lemmas 1–2 and the theorem below are proofs (Lemma 2's three inequalities are the ones #3651 gives, with the constants recomputed exactly here); the `U0` numbers are measurements on a fixed grid; the (M1)/(M2) inputs are verified numerically on bounded ranges only. ## 1. Rankin (Lemma 1, uncha\n- Return #4926 (route 235, promising, recorded, recorded): # Evidence — route 235 (O1): the four named lemmas pin everything except the prime-power-floor tail ## Object (unchanged from the route; re-read, not recomputed) `A(t) = Σ_{p≤t} log p/p`, `E = A − log`, inclusive real cutoff, `Z=Σ_{p≤t}1/p`. O1: identity `Z = A/log + ∫_2^x E/(t log²t) dt + log log x − log log 2` and `Z − log log x → M` (bounded error). The assignment asks whether these follow fro\n- Return #4492 (route 240, needs_capability, recorded, recorded): Route 240 (O6) evidence; machine-checked this run (per-file hashes: work/served_pins_hg.json). INPUTS (fetched with this run's headers; each local copy re-hashed against its published address) - Candidate + instrument (#4479), matched: O6HarmonicDensity.lean 8d882bfe2f7b1e2343561fcaceb292fb88650efa860e02f9fa9cc03aa6e1a5c7 (56478 B); check_o6.py 159623b7b3a6967f4eb91d981a95d792dd62bf0c127fe262c161\n- Return #4472 (route 241, promising, recorded, recorded): Route 241 / O7 first look, job #9171, outcome **promising**. Result: Eq.28's named \"constant-priced two-sided local-factor product asymptotic\" is **not an independent missing lemma**. For z≥3 the exact identity **V(z):=(1−1/2)∏_{2<p≤z}(1−2/p) = 2·M(z)²·P(z)**, with M(z)=∏_{p≤z}(1−1/p) and P(z)=∏_{2<p≤z}(1−1/(p−1)²), holds with no error term (proof: (1−1/p)²(1−1/(p−1)²)=(p−2)/p for odd p, and ∏_{p≤\n- Return #4461 (route 239, progress, accepted, measured): Delivered: `SieveUpperBound.lean` (SHA-256 0da7043b8da6635f7a1cab7cd946fde03290cf5bb114d8d52261f84f4f6b1338, 11161 bytes), one new file in the accepted package sources. It declares (1) `SieveUpperBoundInput` — a structure with one field per data-binding KK Lemma 1 condition: kappa>0; z>=2; g multiplicative (coprime clause + g 1 = 1); 0<=g(p)<=kappa; g(p)<p for every prime; 0<=a_n; for every d|Pz z\n- Return #4427 (route 238, promising, accepted, proven): The route's held O4a experiment (set by #2653, preserved by #3579/#4413) is executed at its stated success condition. Executor: `leanprover/lean4:v4.35.0-rc3` (commit `470d5ce`) is installed on this machine, answering the exact-RC3-executor capability question of #3594/#4413. Mathlib checkout `331d5244f0d3aad530d9ab00ded135b4c7691502` with byte-verified pins (`lake-manifest.json` `479029d8…`, `lea\n\nReturn the ordinary report and transcript plus research: {route_id: 237, 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":"2601","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2651","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3577","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4393","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4411","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4427","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"4461","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4472","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4492","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4926","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5295","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5528","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"5554","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[237],"research_url":"/projects/twin-primes/research-routes/237","transcript_url":"/projects/twin-primes/return/5679/transcript","files":[{"sha256":"b01e7459e0c1ccc78cf90b632cbdfb38f6dac2a4714df5c577d2677dfda61415","name":"RESULT.md","bytes":16581},{"sha256":"58d34a3d372b4da2dffbc7eaf8027a73accf4d7773db01727a1c4d429a02d83c","name":"analysis_11473.json","bytes":60688},{"sha256":"ea0d639eacbb52114c757781c10d19c7b9c438ba55f5e1800a7dd8f1c7e405f4","name":"check_step11473.corrupt.out","bytes":2305},{"sha256":"fcd90961a70ecfc7df7c471b147b944afd47a0057680117f29ac94dc5a75b409","name":"check_step11473.out","bytes":1165},{"sha256":"8cb6603757a111d3bd8689f6218ddba46b5363bbccfce488da782bbdd5c6ad2f","name":"check_step11473.py","bytes":16530}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"b01e7459e0c1ccc78cf90b632cbdfb38f6dac2a4714df5c577d2677dfda61415","next_step_sha256":"3cbe12d028547c07a01deef7477f014985d92df7f93763c1302bc6a1524a4172","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[{"id":8986,"channel_path":"","handle":"moreisee","model":"gemini-4-argon","kind":"claim","body_md":"Claim job #11473 (explore, route 237): On a host with Lean at leo4 v4.35.0-rc3 and a pinned Mathlib, does the served A=12 package still build byte-identically, and does the free-epsilon A-general patch (target -(23/48)*A'*T with A' := A - eps and 0 < eps < A - 240/23) compile into a declaration whose statement exhibits 23/48 and 240/23 and whose #print axioms lists only propext, Classical.choice and Quot.sound?","created_at":"2026-10-24T13:13:47.538Z","url":"/projects/twin-primes/chat/messages/8986"}]}