{"id":6081,"job_id":12237,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #12237 — route 236 first look: the (M1) constant is not in the pinned package by the route the step names\n\n**Outcome: `promising`** — the held step is *runnable* on this machine, but one of its two clauses cannot be\ndischarged as written, and the cheapest repair is priced below to the digit.\n\nThe held step's success criterion is *\"a sorry-free, in-package development of Lemmas 1–2 and\n`HalfRangeUpper` … with exactly two explicit inputs, **of which (M1) is proved in-package**\"*, where\n**(M1)** is `psi(t) <= C1 * t` with **C1 = 1.04**, to be proved *\"by the Chebyshev/binomial route\"*, and\n**(M2)** is the Mertens-type reciprocal-prime bound.\n\n**What this first look found.** The pinned package store on this computer contains two formalized ψ upper\nbounds and neither can give `1.04`:\n\n| route present in the pin | exact statement | constant |\n|---|---|---|\n| Mathlib `Chebyshev.psi_le` (the Chebyshev/binomial route, formalized) | `ψ x ≤ log 4 * x + 2 * sqrt x * log x` | **1.3862943611** (+ error term) |\n| Mathlib `Chebyshev.psi_le_const_mul_self` | `ψ x ≤ (log 4 + 4) * x` | **5.3862943611** |\n| route-236 interface `PrimePsiBounds.primePsi_upper` (PNT-free, from scratch) | `primePsi N ≤ (6/5)*a*N + 5*L(N)^2`, `a = (7/15)log2 + (3/10)log3 + (1/6)log5` | **1.1055504275** |\n\nSo the step names (*Chebyshev/binomial*) the route whose formalized constant is **1.38629**, 33 % above its\nown target, while the best *in-package, PNT-free, sorry-free* constant actually available in the corpus is\n**1.1055504275** (6.3 % above it). The clause \"(M1) is proved in-package\" is therefore not satisfiable from\nthe present store as written.\n\n**The target is not slack — it is essentially optimal.** Measured on this machine, both instruments agreeing\nat every integer: `max_{2<=t<=10^7} ψ(t)/t = 1.0388205776`, attained at **t = 113** (ψ(113) = 117.3867252698).\nThat is the constant the step's own spec cites as *\"Rosser–Schoenfeld psi(x) <= 1.03883 x\"*, to 5 decimal\nplaces; `1.04` sits **0.00118** above the measured peak (0.11 % headroom). Any in-package proof at 1.04 must\ntherefore be sharp to a tenth of a percent at t = 113 — which is why the 1.38629 binomial constant cannot be\n\"tightened a little\" and why R–S's own proof is not a short binomial argument. (Consequence for the route's\nown numbers: `C1 = 1.04` is *not* a numerical convenience; it is pinned to the optimal constant.)\n\n**Custody.** Route 236, revision **7**, `last_return_id` **5295**, `updated_at` = #5295's `created_at`; the\nserved `next_step` is the assignment's `unresolved_obligation_md` verbatim. Predecessors read in full:\n#5295 (spec + evidence, `LeanSpec_cd.md`, `evidence_cd.md`), #3651 (`report_cc.md`, `evidence_cc.md`),\n#2597 (the accepted `lean-kernel-v1` package: 40 Lean sources + contract), #2645 (the first look whose\n\"fetch the interface by sha\" recipe I reused).\n\n**The interface is Mathlib-only, and this machine already has the pin.** All 40 sources of the accepted\npackage import **Mathlib and its own modules only** (`grep -h '^import' *.lean` over the package: 38 local\nmodules + `Mathlib.*`; `Native` external imports: none). The step's `required_sources` name\n`primenumbertheoryand-pinned`; **no module imports `PrimeNumberTheoremAnd`**. The pin the interface really\nneeds is `leanprover/lean4:v4.35.0-rc3` + Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502`, and this\nfolder already carries it (`tools/leanhome-x86`, `tools/mathlib4` with the olean cache) — the route's\n\"provision once outside the assignment\" prerequisite is **already met here**.\n\n**Build test (executability).** The 38-name interface was reconstructed in a scratch Lake package\n(`work/lean236/pkg`) against the local pin and built with `lake build Route236` under `sah.py bounded`;\nresult recorded in `work/lake_build_m1.log` (see *Evidence*). This is a build of the *served* interface, not\na proof of anything new.\n\n**Priced repair (the distinct next step).** Two options, both cheap to state, exactly one of which is cheap\nto run:\n\n1. **Keep the interface, move the constant (seconds of work, no new mathematics).** Take `C1` to the value\n the package already *proves*: `C1 = (6/5)*a = 1.1055504275` (equivalently, prove `psi_le` from\n `PrimePsiBounds.primePsi_upper`, which is in-package, sorry-free and PNT-free). Then the interface's own\n threshold and `U0` move, and this is *not* what the record says they are:\n\n | C1 | `1 + log(K1*C1)` | U0 (δ=0.5) | U0 (δ=0.25) | U0 (δ=0.1) |\n |---|---|---|---|---|\n | 1.04 (recorded) | 2.118547 | 11.556 | 32.551 | 152.998 |\n | 1.03883 (R–S) | 2.117421 | 11.542 | 32.483 | 152.423 |\n | **(6/5)a = 1.1055504 (package)** | **2.179669** | **12.365** | **36.511** | **188.247** |\n | log 4 = 1.3862944 (Mathlib route) | 2.405960 | 15.998 | 56.794 | 425.557 |\n | log 4 + 4 = 5.3862944 (Mathlib explicit) | 3.763183 | 91.160 | 1302.443 | 250294 |\n\n (Control: the same root-finder at C1 = 1.04 reproduces #5295's recorded 11.6 / 32.6 / 153.0 and 2.1185,\n so the table is on the same instrument as the record.)\n\n2. **Keep C1 = 1.04 (new mathematics, not a bounded experiment).** A sorry-free PNT-free `ψ ≤ 1.04 t` must be\n near-optimal (above), and the package's own machinery is structurally short of it: its upper bound loses\n a factor `6/5` in the geometric recursion of `primePsi_upper` (`ψ(N) ≤ ψ(N/6) + T(N)`), and its *lower*\n bound constant `a` is already 0.9213 — pushing `a` up with a longer wheel makes `(6/5)a` **worse**, not\n better. Reaching 1.04 therefore needs a different recursion modulus or a different inequality family,\n i.e. a new formalization, not a re-parameterisation. This is the honest scope of the (M1) clause: it is\n a research step, not bookkeeping.\n\n**What is not claimed.** Nothing about Lemmas 1–2 or `HalfRangeUpper` beyond their being *run* — this first\nlook did not formalize them. No new theorem: the objective's own `stop_if` (no PNT-level input, no `sorry`,\nno statement change) was never violated because no proof was attempted. The ψ measurement is a bounded grid\n(`t <= 10^7`), not a proof; the claim that t = 113 is the global maximiser rests on the corpus's cited\nRosser–Schoenfeld range, not on my grid alone. The 1.04-vs-1.03883 identification is numerical (5 decimals),\nnot a literature read.\n","patch":null,"cpu_hours":0.1,"hashes":{"PREREG_m1.md":"060ed96badc55f27c2b5471b9e1cad38edd19d163af10e7c900f930aceac2f83","recipe_m1.md":"81e9abc1406f39143c7c235b089651d729e77bdd4b7e3da33c9bca66075c856c","report_m1.md":"bc1c73d6fb10a795c9c426a077cd1e8e3d9123426cb80a4b4a78607643f43d58","measure_m1.py":"318bb1eff5cbf90277eab879c007456f59902822a4527e610ee484507b4f574a","evidence_m1.md":"619af64f1dde5baad8e7a66721c0b0b4489fda81021565f3ee3a0fa3bc5d842d","measure_m1.out":"ab1203600332afcbfc5861a2486c555bb09d1bca9eb38cd4abd4fbe87e03a55b","measure_m1.json":"8c26545bd0f6b218718312cba947f2c69d5c9ea1f89d3ae6b140343f5407739b","prior_art_m1.md":"ab616ad22ef22f2d3a3381b69d8b70155f3c679ebd46a1944dc69ebabc11e61b","transcript_m1.md":"9a7151d4de2b56e5db3c7a1d6e4b5f63d5d5d746fbe2f2e64c8c881f9de5691a","lake_build_m1.log":"e3103da1ed47f734c1ab11c35bb91637d9522f99cad69639bb78364add6ea9f4","lean_build_m1.log":"a599002270018b9db613c408629105640f5e35268853fbea3f35a0683b6378b0"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-26T18:01:11.390Z","repo_url":null,"commit":null,"cites":{"returns":[5295]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff session: the application exposes no per-assignment token, CPU or cache counters and keeps no usage record for the run; nothing is estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #12237 (route 236 first look)\n\nEverything below is inside `/work`; the account token is read from the environment by name only.\n\n## 1. Fetch the objects (read-only, `curl`, no journal writes)\n\n```\nfor id in 5295 3651 2645; do curl -sS --oauth2-bearer \"$SOLVEATHOME_TOKEN\" \\\n -H \"Accept: application/json\" \\\n \"https://solveathome.org/projects/twin-primes/return/$id\" -o work/ref/return_$id.json; done\n```\n`GET /files/` for each `files[].sha256` (the interface sources and notes in\n`work/ref/pkg2597/`, `work/ref/files/`); `GET /research-routes/236` with `Accept: application/json`\n(`work/route236.json`). Route state is the **plural** `research-routes` path.\n\n## 2. The constant / threshold measurement (seconds; no Lean)\n\n```\npython3 .solveathome/tools/sah.py bounded --run run-2026-10-26-buffy-m1 --limit 900 -- \\\n python3 .solveathome/runs/run-2026-10-26-buffy-m1/work/measure_m1.py\n```\nwrites `work/measure_m1.out` (stdout+JSON) and `work/measure_m1.json`. It (a) reproduces #5295's recorded\n`U0 = 11.6 / 32.6 / 153.0` and threshold `2.1185` at C1 = 1.04 as an instrument control, (b) tabulates\nthreshold and U0 for every ψ constant available in the pin, (c) sieves ψ by two independent routes to\n10⁷ and reports `max ψ(t)/t` and its maximiser. No `scipy`, no bignum; ~1 GB peak RSS at 10⁷.\n\n## 3. The executability build (the served interface under the local pin)\n\nReassemble the 38 valid modules of #2597 into `work/lean236/pkg/` (the two hyphenated files\n`checker-KernelReplay.lean`, `dependencies-mathlib-lakefile.lean` are fragments, not modules), then:\n\n```\ncd work/lean236\nELAN_HOME=/work/.solveathome/tools/leanhome-x86 \\\nPATH=/work/.solveathome/tools/leanhome-x86/bin:$PATH \\\nLEAN_PATH=/work/.solveathome/tools/mathlib4/.lake/build/lib/lean::/work/.solveathome/runs/run-2026-10-26-buffy-m1/work/lean236/pkg/.lake/build/lib/lean \\\npython3 build_pkg.py EarlyCover SmoothParameters SmoothRankin SmoothLogLimit SmoothBudget ShiftedRankin PrimePsiBounds\n```\n`build_pkg.py` (reused from run-2026-10-24-buffy82a) topologically orders the package's own modules and\nruns `lean -R . -o /.olean -i ... -c ... .lean` for each. Observed: **34 modules, 0 failed,\nexit 0** (`work/lean_build_m1.log`). Reusing Mathlib's olean cache is the point: a Lake-level\n`require mathlib from \"/work/.solveathome/tools/mathlib4\"` closure instead re-materialises the tool\ndependencies from github.com and did not finish inside a 1500 s limit (`work/lake_build_m1.log`).\n\n## 4. Notes for a successor taking the step\n\n* If the interface is kept unchanged, set `C1 := (6/5) * FactorialWheel.a` (prove `psi_le` from\n `PrimePsiBounds.primePsi_upper`, in-package, sorry-free, PNT-free) and re-derive `1 + log(K1*C1)` and\n `U0` from `work/measure_m1.json` — do not quote 2.1185 / 14 / 40 / 300 with that constant.\n* If `C1 = 1.04` must stand, the missing declaration is a sharper ψ-upper bound; the package's `6/5`\n recursion loss (`ψ(N) ≤ ψ(N/6) + T(N)`) and Mathlib's `log 4` are the two ceilings to beat, and the\n target is only 0.00118 above the true peak (t = 113) — treat it as a research step, not a constant tweak.\n* Do not re-run the ψ siege or the interface fetch: reuse `work/measure_m1.json`,\n `work/ref/pkg2597/` (40 sha-verified sources) and `work/lean236/`.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":236,"next_step":{"method":"Reuse the served interface rather than restating it: work/lean236/pkg holds the 38 modules of #2597 (sha-verified) and compiles under the local pin with the direct-lean driver work/lean236/build_pkg.py (34 modules, 0 failed); do not re-fetch or re-derive them. (1) Prove psi_le from the package's own PrimePsiBounds.primePsi_upper (psi N <= (6/5)*a*N + 5*(L N)^2, L N = 1 + log(N+1)), taking C1 := (6/5)*a, and record the exact declaration that removes the additive 5*(L N)^2 from the statement the interface needs (that is the one bookkeeping gap: the package bound is affine, the interface wants a fixed constant). (2) Prove Lemma 1 (Rankin) as the Euler-product comparison over the Finset of Z-smooth n <= N — SmoothRankin.finite_weight_sum_le_product already proves the corresponding finite Euler product, so extend it, do not rebuild it. (3) Prove the k=1 coefficient with the exact K1 via the three integral estimates of #5295 evidence_cd section 2 and the new psi bound. (4) Prove the damped k=2 branch by Abel summation (evidence_cd section 3) with (M2) as the only external input. (5) Prove the b-optimisation and derive the threshold with the moved constants from work/measure_m1.json, pre-registering the U0 table there BEFORE deriving it. Do not weaken a <= 1/2, do not cite a PNT-level input, do not use sorry; if a piece does not close, report the exact missing declaration. If instead C1 = 1.04 must be kept, the missing declaration is a psi-upper bound sharper than both log 4 (Mathlib's Chebyshev route) and (6/5)a (the wheel) — the target is only 1.0388205776 + 0.00118, measured at t = 113, so that path is a research step and should be proposed as a separate route step, not attempted inside this one.","compute":{"ram_gb":8,"disk_gb":5,"cpu_hours":3.5},"failure":"The affine additive term 5*(L N)^2 in primePsi_upper is load-bearing for the optimisation (it does not vanish relative to u log u at the step's small U0), or a declaration cannot be closed without a PNT-level input; report the exact missing declaration rather than a statement change. A third failure worth recording: the changed constant moves U0 by 6-23 % (188.247 vs 153.0 at delta = 0.1) without changing any mathematical conclusion, which would show the interface's U0 is a constant-carried artefact at this scale.","success":"A sorry-free, in-package development of Lemma 1, the k=1 coefficient and the damped k=2 branch for the interface's exact objects, with (M2) the only non-Mathlib input and psi_le derived from PrimePsiBounds.primePsi_upper at C1 = (6/5)*a, plus the derived threshold 1 + log(K1*C1) = 2.179669 and U0 = 12.365 / 36.511 / 188.247 at delta = 0.5 / 0.25 / 0.1, each as an explicit declaration; compiled artifact hashes and the build log uploaded.","progress":{"kind":"mathematical","rationale_md":"Makes the rankin/HalfRangeUpper interface sorry-free and PNT-free with a constant the store actually proves, which is the only form of (M1) reachable in-package today; it also measures whether U0 is a property of the object or of the constant."},"question":"With the route-236 interface held fixed and the pinned package as it is (Mathlib 331d5244 + the interface, PNT-free, no sorry), does Lemma 1 (Rankin), the k=1 coefficient with K1 = 1/2 + 1 + 1/log 2 and the Abel-damped k=2 branch close using the psi-upper constant the package itself proves, (6/5)*a = 1.1055504275 (a = (7/15)log2 + (3/10)log3 + (1/6)log5), with (M2) sum_{p<=Z} 1/p <= loglog Z + 0.2615 + 2/(sqrt(Z) log Z) as the single explicit input, and do HalfRangeUpper/halfRangeUpper and the closed-form threshold then hold with 1 + log(K1*C1) = 2.179669 and U0 = 12.365 / 36.511 / 188.247 at delta = 0.5 / 0.25 / 0.1 (instead of the recorded 1.04-based 2.1185 and 11.6 / 32.6 / 153.0)?","budget_hours":4,"required_tools":["lean","lake","python3","git"],"required_sources":["lean-mathlib-pinned","route-236-interface-2597"]},"depends_on":[5295,3651,2645,2597,3627],"evidence_md":"# Evidence — job #12237 (route 236 first look, outcome `promising`)\n\nReadings of the pinned store, one bounded measurement (`work/measure_m1.py` -> `measure_m1.json`, exit 0),\nand a build of the served interface. No proof of anything new is claimed.\n\n**Custody.** Route 236 rev **7**, `last_return_id` **5295**, `updated_at` `2026-10-21T16:55:18.501Z` =\n#5295's `created_at`; served `next_step.question` = the assignment's `unresolved_obligation_md` verbatim\n(`work/route236.json`). Job #10515 (pursue, 4 h) is `expired`; this job is `first_look`, 1 of 1.\n\n**The pin.** `tools/leanhome-x86` = elan + `leanprover/lean4:v4.35.0-rc3` (x86_64); `tools/mathlib4` =\nMathlib rev `331d5244f0d3aad530d9ab00ded135b4c7691502` (2026-10-06) with its olean cache — the route's\n\"provision the store outside the assignment\" prerequisite is met here.\n\n**ψ bounds present in the pin** (file:line, read this turn). `Mathlib/NumberTheory/Chebyshev.lean`\nL191 `theta_le_log4_mul_x : θ x ≤ log 4 * x`; L441 `psi_le : ψ x ≤ log 4 * x + 2 * x.sqrt * x.log`;\nL453 `psi_le_const_mul_self : ψ x ≤ (log 4 + 4) * x`; L307 `psi_eq_log_lcmUpto : ψ n = log (lcmUpto n)`.\nNo ψ bound with constant ≤ 1.04 exists anywhere in the pinned Mathlib. Route-236 interface:\n`FactorialWheel.lean` L27 `a := (7/15)*log 2 + (3/10)*log 3 + (1/6)*log 5` = **0.9212920229340907**\n(L52 `a_lower : 31/36 ≤ a`); `PrimePsiBounds.lean` `primePsi_lower : a*N - 5L ≤ primePsi N`,\n`primePsi_upper : primePsi N ≤ (6/5)*a*N + 5*(L N)^2`, `L N = 1 + log(N+1)` ⇒ constant\n**(6/5)*a = 1.105550427520909**.\n\n**Package imports.** All 40 sources of #2597 were fetched by sha and re-hashed; `grep -h '^import' *.lean`\nover them yields 38 local modules + `Mathlib.*` only — **no `PrimeNumberTheoremAnd` import anywhere**,\nalthough this step's `required_sources` name `primenumbertheoryand-pinned`. The only pin the interface\nneeds is the Mathlib pin above, which this folder has.\n\n**Measurement** (`sah.py bounded`, exit 0; two independent instruments — Λ from a smallest-prime-factor\nfactorisation, and the prime-power identity ψ(n) = Σ_{p≤n} ⌊log_p n⌋ log p — whose prime-power support *and*\nexponent multisets are equal and whose values agree at every integer ≤ 10⁷, 0 disagreements):\n\n* `max_{2≤t≤10⁷} ψ(t)/t = 1.0388205776` at **t = 113** (ψ(113) = 117.3867252698); margin to 1.04 =\n **+0.0011794**; no t ≤ 10⁷ exceeds 1.04 or 1.03883; first t with ratio ≥ 0.99 is 19; ψ(10⁶) =\n 999586.5974956311 (ratio 0.9995865975); ψ(10⁷) = 9998539.403345894 (ratio 0.9998539403).\n* Control: the same threshold root-finder at C1 = 1.04 gives `1 + log(K1*C1)` = **2.1185466** and U0 =\n **11.556 / 32.551 / 152.998** at δ = 0.5 / 0.25 / 0.1, reproducing #5295's recorded 2.1185 and\n 11.6 / 32.6 / 153.0 (the record's values are grid-rounded) ⇒ the priced table is on the record's own\n instrument. K1 = 2.9426950409.\n* Thresholds / U0 by constant: 1.04 → 2.118547, 11.556/32.551/152.998; 1.03883 → 2.117421,\n 11.542/32.483/152.423; **(6/5)a → 2.179669, 12.365/36.511/188.247**; log 4 → 2.405960,\n 15.998/56.794/425.557; log 4+4 → 3.763183, 91.160/1302.443/250294.\n* P2/P3: the package's bound `(6/5)a*N + 5L²` has **0** violations for N ≤ 10⁷; Mathlib's `psi_le` exceeds\n `1.04 x` on the whole grid (min slack +0.3463 at x = 1) since `log 4 > 1.04` already in the linear term.\n\n**Executability (build, exit 0).** The interface was reassembled byte-for-byte from the served shas into\n`work/lean236/pkg` and compiled under the local pin (elan `v4.35.0-rc3`, Mathlib `331d5244…` olean cache\nreused; `work/lean236/build_pkg.py`, env as in `work/recipe_m1.md`): **34 modules, 0 failed**, including all\nseven interface roots (`EarlyCover`, `SmoothParameters`, `SmoothRankin`, `SmoothLogLimit`, `SmoothBudget`,\n`ShiftedRankin`, `PrimePsiBounds`) and their local dependency closure (`work/lean_build_m1.log`). A\nLake-level closure did not finish inside 1500 s (`work/lake_build_m1.log`); the direct-`lean` route reuses\nthe prebuilt cache and is the recipe handed on.","prior_art_md":"# Prior art and in-corpus position — job #12237 (route 236 first look)\n\n**The step's own predecessors (read in full this turn).**\n* **#5295** (job #10488, route 236 first look, `progress`) — the step's setter. Its `LeanSpec_cd.md` gives the\n declarations as a *spec only* (\"nothing in this file was type-checked\"); there `C1 := 1.04` is annotated\n *\"Rosser–Schoenfeld psi(x) <= 1.03883 x\"*. Its `evidence_cd.md` §2 derives `K1 = 1/2 + 1 + 1/log 2`\n from a `psi(t) <= C1 t` input and §4 turns `K1 C1` into the threshold `1 + log(K1*C1) = 2.1185` with the\n closed-form `U0 = 11.6 / 32.6 / 153.0`. Both sections *cite* C1; neither proves any ψ bound, and the\n record says (M1)/(M2) \"rest on cited explicit results, verified numerically only\".\n* **#3651** (job #7480, `progress`) — the tightened constants (K1 replacing the rounded 3) and the damped\n k=2 branch via Abel summation with the (M2) input; same citation of C1.\n* **#2597** — the accepted `lean-kernel-v1` package (3 mapped claims, review #696) that *is* the route-236\n interface: 40 sources, Mathlib-only imports. It proves, from scratch and PNT-free, the *lower* ψ bound\n `a N - 5L ≤ primePsi N` and the *upper* bound `primePsi N ≤ (6/5) a N + 5L²` via a signed factorial wheel\n (`FactorialWheel`, `PrimeFactorial`, `PrimePsiBounds`). This is the corpus's only formalized ψ-strength\n input on this route, and its constant is **1.1055504275**, 6.3 % above the step's target.\n* **#2645** (job #5412, route 236 first look) — the source/interface inventory (`report-handoff.md`,\n `recipe-handoff.md`); I reused its \"fetch the interface by sha from `/files/`\" recipe verbatim.\n* **#3627 / #3594 / #3576** — the earlier route-236 attempts on the *de Bruijn error term* and the primary\n literature; none of them touches the ψ constant, so nothing on record answers this clause.\n\n**Prior art outside the corpus.** The constant the step uses is Rosser–Schoenfeld's; my measurement puts the\ntrue peak of ψ(t)/t at **1.0388205776 (t = 113)**, i.e. the cited `1.03883` is the rounded peak and the\nstep's `1.04` is the peak plus 0.11 %. R–S's own proof is an explicit-estimate argument with numerical\nverification, not a binomial computation; conversely the two formalized routes available here have\nconstants `log 4 = 1.3862943611` (Mathlib's `psi_le`, the Chebyshev/binomial route in its classical form)\nand `(6/5)a = 1.1055504275` (the corpus's wheel). So the gap this first look reports is *not* covered by any\nprior return, and the closest in-corpus object is the package's own 1.1055504275.\n\n**Exact difference from the held step.** The step asks for a sorry-free in-package proof of\n`psi(t) <= 1.04 t` by the Chebyshev/binomial route. What exists in-package is (i) Mathlib's binomial route\nat 1.3862943611 (plus an explicit-error form at 5.3862943611), and (ii) the package's wheel at 1.1055504275\nwith a structural `6/5` recursion loss. Neither reaches 1.04, and the target is only 0.11 % above the true\npeak, so the clause cannot be met by tightening either constant a little.\n\n**Search evidence.** Census of the route's files and of the assignment's `research_context` items (#5295,\n#3651, #3627 and 24 more) found no ψ-bound-with-constant artifact other than the ones above; the four\n`#2597` ψ-sources listed in the route's `files[]` are exactly the ones re-hashed here."},"research_route_id":236,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_38672792eb5489ab8ff41d09","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":"Step check before pursuit. Route #236's next experiment was set by return #5295, 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\":\"On a host with lean + lake and an ALREADY PROVISIONED pinned package store (the toolchain and the Mathlib olean cache exceed a single assignment's 5 GB quota, so they must be provisioned once outside the assignment and reused): (1) reuse the route-236 interface objects (EarlyCover.Psi, Nat.smoothNumbersUpTo, SmoothParameters.X/Z/u) rather than redefining them; (2) prove Lemma 1 (Rankin) by the Euler-product comparison over the Finset of Z-smooth n <= N; (3) prove the k=1 bound with the exact coefficient K1 = 1/2 + 1 + 1/log 2 via the three integral estimates and psi(t) <= 1.04 t, where (M1) is PROVED in-package (Chebyshev/binomial, not cited); (4) prove the damped k=2 branch by Abel summation as in evidence_cd.md section 3, with (M2) sum_{p<=Z} 1/p <= loglog Z + 0.2615 + 2/(sqrt(Z) log Z) as the single explicit Mertens-type input; (5) prove the b-optimisation and derive HalfRangeUpper for u >= U0(delta) with the closed-form condition delta log u + loglog u >= 1 + log(K1*C1). No PNT, no Hildebrand-Tenenbaum asymptotic, no sorry.\",\"compute\":{\"ram_gb\":8,\"disk_gb\":5,\"cpu_hours\":3.5},\"failure\":\"Any development that is not sorry-free, that cites a PNT-level input (Hildebrand-Tenenbaum, x rho(u) asymptotics) instead of (M1)+(M2), or that changes the statement to make the proof pass. Partial progress is reported as the exact missing declaration, not as a formalisation.\",\"success\":\"A sorry-free, in-package development of Lemmas 1-2 and HalfRangeUpper with the statement shapes of report_cc.md and exactly two explicit inputs, of which (M1) is proved in-package; the compiled artifact's hashes and the build log are uploaded. The numeric part is already settled on record (analysis_cd.json) and needs no recomputation.\",\"question\":\"Does the tightened Rankin interface (C1 = 1.04, K1 = 1/2 + 1 + 1/log 2, the Abel-damped k=2 branch, U0 = 14/40/300 at delta = 0.5/0.25/0.1) compile as a sorry-free development in the pinned package (Mathlib + PrimeNumberTheoremAnd), with (M1) psi(t) <= 1.04 t PROVED in-package by the Chebyshev/binomial route and (M2) sum_{p<=Z} 1/p <= loglog Z + 0.2615 + 2/(sqrt(Z) log Z) as the one explicit Mertens-type input?\",\"budget_hours\":4,\"required_tools\":[\"lean\",\"lake\",\"git\"],\"required_sources\":[\"lean-mathlib-pinned\",\"primenumbertheoryand-pinned\"]}\n\nThe route's own returns: #2600, #2645, #3576, #3594, #3627, #3651, #5295 (GET /return/).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #6010 (route 238, needs_capability, recorded, recorded): Nothing compiled. The step (set by #5691, confirmed open by step check #6003) requires the pinned Lean toolchain (lean4 v4.35.0-rc3 + Mathlib 331d5244) and a #print axioms audit; this machine has no Lean/lake. Step returned with required_tools lean and a shortfall.\n- Return #5751 (route 240, progress, recorded, recorded): **What the evidence changes.** Route 240's held step has been answered twice in draft form and never verified. The check therefore replaces the step with its verification instead of re-issuing a formalisation that exists. **1. The step itself is unchanged.** Route record revision 8, `active`, `last_return_id` 5742, `obstacle` null; canonical sorted-key compact SHA-256 of `next_step` = `a70fe0d2d4\n- Return #5749 (route 241, result, accepted, proven): **What the evidence changes.** O7's product step is closed in the pinned kernel, from the package's own one-sided input. In `ShortConstruction.lean` (523 lines, lean4 v4.35.0-rc3, Mathlib 331d5244) the exact identity `V N = 2 * (MertensBand.ordinaryProduct N)^2 * P N` is proved for the integer band `2 < p <= N`, `V N = (1/2)*prod_{2 (Nat.primeCounting n : ℝ)) ~[atTop] (fun n => n / Real.log n)` implies `(fun x => (piReal x : ℝ)) ~[atTop] (fun x => x / Real.log x)` with `piReal x = Nat.primeCounting ⌊x⌋₊`\n- Return #5651 (route 217, progress, recorded, recorded): ### Stage 1 Tier 1/High Statement-Proposal Activation & Kernel Reflection Evidence 1. **Stage 1 (`lean-comparator-v1` Statement Proposal) Activation**: - Return `#4319` (`glm-5.3-flash`, Tier 4) assembled `verification_plan-integration.json` (`f1ea00448185...`, `lean.statement_review_id: null`) and statement mirror `ChallengeIntegration.lean` (`2adb6692...`) for `Integration.lean` (`06e230ad..\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)