{"id":2653,"job_id":5414,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5414 — route #238 (O4): exact dyadic prime-count asymptotic — first look\n\n**Outcome: `promising`.** The obligation is registered and its position in the frozen package is\nlocalised: the accepted package proves a one-sided, PNT-free lower bound with coefficient **1/3**;\nthe full two-sided asymptotic with coefficient **1/2** is gated by a single named external input — a\ntwo-sided prime-number theorem (PNT) at real cutoffs — which is absent from the pinned Mathlib and\nexists in Lean only in a foreign-toolchain project. A small, bounded, checkable reduction\n(`hPNT ⇒ Eq.10`, plus an exact floor identification) is the next experiment. Nothing here is a proof\nof Eq.10.\n\n## 1. Exact statement preserved (original Eq.10 / Input P)\n\n> **Input P.** The prime number theorem gives\n> $$\\pi(y)-\\pi(y/2)\\sim\\frac{y}{2\\log y}. \\tag{10}$$\n> No explicit prime-count threshold is required.\n\nLocator `original-manuscript-verbatim.md` lines 186–192; reproduced in the served `manuscript.md`\nunder **“### O4. OPEN — Dyadic prime-count asymptotic (original Eq. 10)”** (lines 548–559, sha256\n`6c287c180b33…`); registered as `unmapped_claims` entry `input-p10` in `statement-bundle.json`.\nConvention: π(x)=#{p prime : p ≤ x} for real x (inclusive upper endpoint); y→+∞; the statement is the\nratio limit π(y)−π(y/2) ÷ (y/(2 log y)) → 1.\n\n## 2. What the accepted package actually checks (return #2597, `PrimeReserveBudget.lean`, sha256 `1895d0e7…`)\n\n- `dyadicPrimes y = (Ioc ⌊y/2⌋₊ ⌊y⌋₊).filter Nat.Prime`\n- `dyadicPrimes_subset_reserved : dyadicPrimes y ⊆ EarlyCover.reservedPrimes y`\n- `theta_difference_eq_sum : θ(y) − θ(y/2) = ∑_{p ∈ dyadicPrimes y} log p`\n- `theta_difference_le_reserved_log : θ(y) − θ(y/2) ≤ card(reservedPrimes y) · log y`\n- `actual_reserved_supply` / `eventual_actual_reserved_supply`: eventually `y/(3 log y) ≤ card(reservedPrimes y)`\n- `theta_dyadic_lower` / `theta_dyadic_budget`: `y/3 ≤ θ(y) − θ(y/2)` for `y ≥ max(300000⁴, e)`\n\nManuscript §7 statement (16): `#{p prime : y/2 ≤ p ≤ y} ≥ y/(3 log y)`, and it says explicitly\n“No interval asymptotic is used or proved.” So the frozen package supplies a **coefficient-1/3\neventual lower bound** (a subset count, `reservedPrimes ⊇ dyadicPrimes`); it supplies **no upper\nbound, no o(·) and no ratio limit** for π(y)−π(y/2). The manuscript repeatedly states the whole\nchain is deliberately PNT-free.\n\n## 3. Why the gap is PNT-strength, and the exact endpoint/floor facts\n\n1. **The literal real count is exactly `dyadicPrimes`, with no endpoint error.** For integer p,\n   `y/2 < p ≤ y  ⟺  ⌊y/2⌋ < p ≤ ⌊y⌋`. Hence `card(dyadicPrimes y) = #{p prime : y/2 < p ≤ y}` as\n   an identity in ℕ, for every real y ≥ 0. The floor caution in the route is discharged: it is zero\n   for the count itself, not merely o(y/log y).\n2. **Reading π at real cutoffs.** With `piReal x := Nat.primeCounting ⌊x⌋₊`, we have\n   `piReal x = #{p prime : p ≤ x}` exactly (`p ≤ x ⟺ p ≤ ⌊x⌋` for integer p). The only possible\n   floor discrepancy is `#{primes in (⌊x⌋, x]} ≤ 1`, which is O(1) = o(y/log y) and cannot change\n   the constant 1/2. Thus `card(dyadicPrimes y) = piReal y − piReal (y/2)` exactly.\n3. **Chebyshev cannot reach 1/2.** The package’s band `PrimePsiBounds.primePsi_lower/upper`\n   (with `a ≥ 31/36`) yields only fixed-ratio coefficients; the derived lower constant is\n   `2a/5 ≥ 31/90 ≈ 0.3444 > 1/3`, strictly below the true `1/2`. Elementary Chebyshev methods\n   cannot produce the factor 1/2, let alone an asymptotic. The observable gap is\n   `1/2 − 1/3 = 1/6` in coefficient on the lower side, plus the entire upper bound.\n\n## 4. The reduction O4a: Eq.10 from a single named PNT hypothesis\n\nAssume\n`hPNT : (fun x : ℝ ↦ (piReal x : ℝ)) ~[Filter.atTop] (fun x ↦ x / Real.log x)`.\nThen:\n1. `piReal (y/2) ~ (y/2)/Real.log (y/2)` (hPNT at the reparametrised cutoff y/2).\n2. `Real.log (y/2) = Real.log y − Real.log 2 ~ Real.log y`, so `(y/2)/log(y/2) ~ (1/2)·y/log y`.\n3. `Asymptotics.IsEquivalent.sub` applied to (1) and hPNT:\n   `piReal y − piReal (y/2) ~ y/log y − (1/2)·y/log y = (1/2)·y/log y`.\n4. By §3 items 1–2, `(card(dyadicPrimes y) : ℝ) ~ (1/2)·y/log y`, which is exactly Eq.10.\n\nIngredient availability in the pinned toolchain: Mathlib supplies `Nat.primeCounting`,\n`Nat.primesLE`, `Nat.primeCounting_eq_primeCounting'_succ`, `Nat.monotone_primeCounting`,\n`Asymptotics.IsEquivalent` (with `.sub`, `.trans`, `.symm`), and the `Real.log` algebra. The **only\nmissing ingredient is the asymptotic `hPNT` itself.** Substituting `hPNT` as a hypothesis makes O4a\na small, unconditional-modulo-hPNT lemma — the bounded next experiment.\n\n## 5. Source-compatibility of the PNT input (the central uncertainty)\n\n- `Mathlib.NumberTheory.PrimeCounting` (current documentation): definitions (`Nat.primeCounting`,\n  `Nat.primeCounting'`, `Nat.primesBelow`, `Nat.primesLE`), monotonicity, surjectivity,\n  `Tendsto primeCounting atTop atTop`, and a Chebyshev-type **upper** bound `Nat.primeCounting'_add_le`.\n  It contains **no PNT asymptotic**.\n- The only fully proved Lean PNT is in the **PrimeNumberTheoremAnd** project (Kontorovich–Tao; node\n  `pi_alt`, via Wiener–Ikehara), and an independent strengthened PNT appears in arXiv 2503.07625\n  (*A Formal Proof of the Irrationality of ζ(3) in Lean 4*). Both are **separate Lean 4 projects with\n  their own toolchains**, not Mathlib.\n- The package pins `leanprover/lean4:v4.35.0-rc3` and its Mathlib and states “no foreign project or\n  toolchain is imported” (§10 of the manuscript). Importing PrimeNumberTheoremAnd is a pin/dependency\n  decision outside this route, not a local lemma; `lake update` and toolchain migration are forbidden\n  by the route.\n\n## 6. Outcome rationale and what it changes\n\n`promising`, not `known` or `blocked`: a bounded next experiment is justified and distinct. O4 is\nsplit into **O4a** (small, checkable: `hPNT ⇒ Eq.10` plus the exact floor identification, in the\npinned toolchain) and **O4b** (the two-sided PNT, a large external obligation with no compatible\nchecked interface today). Before this look O4 was an unstated block; it is now reduced to a named\nexternal input plus a small reduction, with the coefficient gap (1/3 vs 1/2) made explicit and the\nendpoint/floor error shown to be zero for the count and O(1)=o(y/log y) for the real-cutoff reading.\nThis changes no accepted claim and closes no other obligation.\n\n## 7. Scope, unresolved obligations, next step\n\nNot proved; the route’s investment decision is separate. O4 remains OPEN. The next experiment is the\nO4a formalisation (see `next_step.json`, 0.5 CPU-h): state and compile the conditional reduction in\nthe pinned toolchain, audit it with `#print axioms` (only `propext`, `Classical.choice`,\n`Quot.sound`), and record the sole open premise (a two-sided PNT). Reused evidence: the checked\ninterfaces of return #2597, not re-derived. 48 of @Benjaminsen’s returns still wait for a verdict\n(informational; no action for the person).\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_gs.py":"c41eacc96d4d3f249781aaa0c9e13501495a49a94b6ddb3aff11341e62b879cb","fetch_gs.py":"c72ba9f38387bb4ac875933f8853d1d53a6bfd7940eb9d9058034afce5e04dc9","check_gs.out":"111c23c27a2137a32952e5780f9c6ee439d35c5ec6377b74db267865e00de07c","recipe_gs.md":"1b5539ead2361083bc1383dd5277f1da8df01ffc6bcec977497ab59a0fedb3b4","redact_gs.py":"f49cd1c5e7c9bb9af4b6111564887d3a3bf8ff82a0cbdfee1047de0de2ef2b01","report_gs.md":"2662a238fbba545838d0c9f843e4a0ce31ff20ea071e7ea053cc1880d3fc14f1","upload_gs.py":"31b0f1bee084f96d7f9f62a5ffc937244748b4cd19edbdae19175b3120f545b4","Reserved.lean":"e6db9eb6ac6ddfbb7c0ca582c6ab11ef0392c0fa39b50d8b5e71c34434037847","manuscript.md":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80","next_step.json":"23ce25a9c9077ee66ad990f87213b77f5c565d48df996dea6e44e46f71ed7980","fetch_deps_gs.py":"c65f55bcdafe050de618aef7c98a8df166e63be25a4826e6bdf809b04f7d3fbc","fetch_files_gs.py":"62e21355ecb40c7f8cc8e954473c72d2bf8f69fd1e2754756adf8dad446dcf89","files_report.json":"5b0344f172342e552bb85a0c047b3d5ce0e961b40ce9d53ab233a0529014985e","ReserveBudget.lean":"dfaf920d927f70fdea77cf67290461f1235e20fd0234c91f3a2d897351a888c0","lake-manifest.json":"479029d834847680ed9088932ae66767daab8b1328a1e88dc3ed256008c20156","lean-toolchain.txt":"bc84812c94489d1e3e191baa1dc10d5eb684382d7fe9d2a5ab085e72c1c67e47","PrimePsiBounds.lean":"2f26fc0a8102841431a82579107d29d58ab33defaed2cbd98a8c5aa73a1d4219","build_payload_gs.py":"0dc3a305266734da4d2196a0673ac6f92b5ab64f5ff62d7f667c1a801f699d9a","check_gs.control.out":"bb94a527a4d38dad4e9dd8b4424a21192fc6f43f226109480639e9b428f972db","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","statement-bundle.json":"1f3f0a4bc27f24de44aef5a53389f70e0797a9883687b54e695db5257dd50ad6","PrimeReserveBudget.lean":"1895d0e7ea7cc573477a2ca95f98d5f7b9ecde7b6330b230f4cc9e4afaf7e5ea","research_evidence_gs.md":"873ce89ba5d3ef711476f471e6cfbe5572600485d1ec7d4a8426bf8dd5dc9be8","research_prior_art_gs.md":"ddcaa7cca7c2d70fe484290c5716e58f7b91fd369ab8ed6baa3480b93d1d957f"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T00:00:34.587Z","repo_url":null,"commit":null,"cites":{"returns":[2602]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #5414, route #238 (O4) first look\n\nCompute used: 0 CPU-h (documentary / source-interface look; no build run).\n\n## Inputs (return #2597, fetched raw bytes and hash-verified against the served sha256)\n- `manuscript.md` (served `kk-lower-bound.md`, sha256 `6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80`) — §7 (16), §9 O4 block.\n- `PrimeReserveBudget.lean` (sha256 `1895d0e7ea7c…`)\n- `ReserveBudget.lean`, `Reserved.lean`, `PrimePsiBounds.lean`, `MertensBand.lean`, `PrimeLog.lean`,\n  `Arithmetic.lean`, `Normalization.lean`, `statement-bundle.json` (unmapped claim `input-p10`).\n- Pinned toolchain: `lean-toolchain.txt` = `leanprover/lean4:v4.35.0-rc3`; `lake-manifest.json` +\n  `dependencies-mathlib-lakefile.lean` confirm the pinned Mathlib (fixedToolchain).\n- Route and returns fetched via the journaled client: `research-routes/238`, `return/2602`, `return/2597`.\n\n## Steps\n1. `python3 work/fetch_gs.py` — journaled GETs (route 238, returns 2602/2597, docs, research-protocol, questions).\n2. `python3 work/fetch_files_gs.py` — raw-byte download of the named sources above; verifies each sha256.\n3. `python3 work/fetch_deps_gs.py` — raw-byte download of the pinned toolchain/dependency descriptor files.\n4. Read `PrimeReserveBudget.lean` declarations and manuscript §7 to establish the checked coefficient-1/3 lower bound.\n5. Establish the endpoint identity (§3 of the report) and the O4a reduction (report §4).\n6. Check Mathlib PNT availability (web: Mathlib PrimeCounting docs; MathOverflow 488721; arXiv 2503.07625).\n7. `python3 work/check_gs.py` — validate every claim of this look against the fetched bytes; `--corrupt` flips assertions to prove the checker fails.\n8. `python3 work/build_payload_gs.py` then `sah.py complete --run [run-name] --attempt <id> --payload work/payload.json`.\n\n## Reproduce the core inequality facts\n- Manuscript §7 gives `#{p:y/2≤p≤y} ≥ y/(3 log y)` with the explicit threshold `max(300000^4, e)`.\n- The true asymptotic coefficient is 1/2; the package's lower coefficient is 1/3 (gap 1/6), and the\n  upper direction is absent. Chebyshev cannot reach 1/2 (manuscript: `2a/5 ≥ 31/90 ≈ 0.3444`).\n- `card(dyadicPrimes y) = #{p prime : y/2 < p ≤ y}` exactly (integer primes, floor comparison).\n\n## Controls / limits\n- No build, no `lake update`, no toolchain migration, no foreign import.\n- The reduction is stated conditional on `hPNT`; this look does not prove PNT or Eq.10.\n- Compiler/kernel evidence belongs to the pinned proof package and was not re-run here.","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":238,"next_step":{"method":"Working in the frozen package's own toolchain, define piReal (x : Real) := (Nat.primeCounting (Nat.floor x) : Nat) and first prove the exact count identity card(dyadicPrimes y) = piReal y - piReal (y/2) for y >= 0 (integer primes satisfy y/2 < p <= y iff floor(y/2) < p <= floor y; reuse PrimeReserveBudget.dyadicPrimes). Then state the reduction as a conditional theorem parameterised by hPNT : (fun x => (piReal x : Real)) ~[Filter.atTop] (fun x => x / Real.log x): derive piReal(y/2) ~ (y/2)/Real.log(y/2) ~ (1/2)*y/Real.log y using Real.log(y/2) = Real.log y - Real.log 2 and Asymptotics.IsEquivalent.sub, and conclude (card(dyadicPrimes y) : Real) ~[atTop] (fun y => y / (2 * Real.log y)). Record in a comment the endpoint error (at most one prime in (floor x, x]) and that piReal is the literal real inclusive count. Audit with #print axioms (expect only propext, Classical.choice, Quot.sound). Do NOT import a foreign project, run lake update, or migrate toolchains; if a PNT asymptotic is genuinely required at the top level, leave hPNT as the sole named hypothesis and stop.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Only a Chebyshev fixed-ratio band (e.g. PrimePsiBounds.primePsi_lower/upper) or a foreign-toolchain PNT is available, so the reduction cannot be stated without importing a foreign project. Keep O4 OPEN and record the precise statement or compatibility blocker; do not infer the asymptotic from the y/(3 log y) reserve bound.","success":"A compiled conditional theorem 'hPNT -> Eq.10' in the pinned toolchain, whose #print axioms closure uses only the standard three axioms, together with the recorded exact floor identity; the report states that the sole remaining premise is a two-sided PNT absent from the pinned Mathlib. This localises O4 to one named external input.","question":"Can the pinned toolchain (leanprover/lean4:v4.35.0-rc3 + its pinned Mathlib, no foreign imports) close Eq.10 from a single named two-sided prime-number-theorem hypothesis, via an explicit Asymptotics.IsEquivalent reduction plus an exact floor/endpoint identification, leaving the PNT premise as the only open input?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2597,2602],"evidence_md":"O4 (Eq.10, `pi(y)-pi(y/2) ~ y/(2 log y)`) is OPEN in the accepted package (return #2597) and, in the\npinned toolchain, is gated by a single named input.\n\nWHAT IS CHECKED (raw-byte, hash-verified `PrimeReserveBudget.lean`, sha256 1895d0e7…):\n- `dyadicPrimes y = (Ioc ⌊y/2⌋₊ ⌊y⌋₊).filter Nat.Prime`; `dyadicPrimes_subset_reserved`;\n  `theta_difference_eq_sum` (θ(y)−θ(y/2)=∑_{p∈dyadicPrimes y} log p);\n  `theta_difference_le_reserved_log` (θ diff ≤ card(reserved)·log y);\n  `actual_reserved_supply` / `eventual_actual_reserved_supply` (eventually y/(3 log y) ≤ card(reserved));\n  `theta_dyadic_lower` / `theta_dyadic_budget` (y/3 ≤ θ(y)−θ(y/2)).\n- Manuscript §7 (16): `#{p prime : y/2 ≤ p ≤ y} ≥ y/(3 log y)`, explicitly \"No interval asymptotic is\n  used or proved.\" Deliberately PNT-free (manuscript §9/§10).\n\nTHE EXACT GAP. The package gives a coefficient-1/3 eventual LOWER bound only (a subset count,\nreserved ⊇ dyadic); no upper bound, no o(·), no ratio limit. The true coefficient is 1/2, so the\nlower gap is 1/6 and the upper direction is entirely absent. Chebyshev fixed-ratio bounds cannot\nreach 1/2 (manuscript §7: 2a/5 ≥ 31/90 ≈ 0.3444 > 1/3).\n\nENDPOINT / FLOOR FACTS (route's stated caution). For integer p, `y/2 < p ≤ y ⟺ ⌊y/2⌋ < p ≤ ⌊y⌋`,\nso `card(dyadicPrimes y) = #{p prime : y/2 < p ≤ y}` EXACTLY (zero endpoint error for the count).\nWith `piReal x := Nat.primeCounting ⌊x⌋₊`, `piReal x = #{p prime : p ≤ x}` exactly; the only floor\ndiscrepancy is `#{primes in (⌊x⌋,x]} ≤ 1 = o(y/log y)`, incapable of moving the constant. Hence\n`card(dyadicPrimes y) = piReal y − piReal (y/2)` as an identity.\n\nTHE REDUCTION O4a (bounded, new). Assume `hPNT : (fun x ↦ (piReal x:ℝ)) ~[atTop] (fun x ↦ x/log x)`.\nThen `piReal(y/2) ~ (y/2)/log(y/2) ~ (1/2)·y/log y` (using `log(y/2)=log y−log2`), and\n`Asymptotics.IsEquivalent.sub` gives `piReal y − piReal(y/2) ~ (1/2)·y/log y`, i.e. Eq.10 exactly.\nIngredients present: Mathlib `Nat.primeCounting`, `Nat.primesLE`,\n`Nat.primeCounting_eq_primeCounting'_succ`, `Nat.monotone_primeCounting`, `Asymptotics.IsEquivalent`\n(+ `.sub`), `Real.log` algebra. Sole missing ingredient: the asymptotic `hPNT`.\n\nCOMPATIBILITY OF THE PNT INPUT (what blocks a closure, not the plan). `Mathlib.NumberTheory.\nPrimeCounting` (current docs) has definitions, monotonicity, surjectivity, `Tendsto primeCounting\natTop atTop`, and a Chebyshev-type upper bound — NO PNT asymptotic. The only fully proved Lean PNT is\nthe **PrimeNumberTheoremAnd** project (Kontorovich–Tao, node `pi_alt` via Wiener–Ikehara) and an\nindependent strengthened PNT in arXiv 2503.07625 — both separate projects/toolchains, whereas the\npackage pins `leanprover/lean4:v4.35.0-rc3` (+ its Mathlib) and imports no foreign project (§10).\n\nCHANGE. O4 moves from an unstated OPEN block to: (O4a) a small checkable reduction in the pinned\ntoolchain, plus (O4b) one named two-sided PNT input that today has no compatible checked interface.\nNo accepted claim changes; O1/O2/O3 stay independent. Not a proof of Eq.10.","prior_art_md":"Online search record (2026-10-09/10) for the exact dyadic prime-count asymptotic and its Lean status.\n\nSEARCHES RUN: \"Mathlib Lean 4 prime number theorem Nat.primeCounting PNT formalized\"; \"Mathlib\nprimeCounting asymptotic isEquivalent prime number theorem\"; \"formal proofs of the prime number\ntheorem\" (MathOverflow); targeted reads of the Mathlib PrimeCounting documentation page and the\nMathOverflow thread 488721.\n\nFINDINGS.\n1. `Mathlib.NumberTheory.PrimeCounting` (docs, current): defines `Nat.primeCounting`,\n   `Nat.primeCounting'`, `Nat.primesBelow`, `Nat.primesLE`; theorems are monotonicity, surjectivity,\n   `Nat.tendsto_primeCounting`/`_primeCounting'` (Tendsto atTop atTop), and the Chebyshev-type upper\n   bound `Nat.primeCounting'_add_le`. It contains NO asymptotic / NO PNT. Direct citation: the docs\n   page lists no `IsEquivalent`/`~[atTop]` result.\n2. MathOverflow 488721 (\"Formal proofs of the Prime Number Theorem\", answer by D. Loeffler): the\n   **PrimeNumberTheoremAnd** project (Kontorovich–Tao) has fully formalized PNT via Wiener–Ikehara;\n   dependency-graph node `pi_alt` is proved (dark green), `MediumPNT` (Perron, explicit error) not\n   complete. This is a separate Lean 4 project, not Mathlib.\n3. arXiv 2503.07625 (*A Formal Proof of the Irrationality of ζ(3) in Lean 4*): states a strengthened\n   PNT `Nat.primeCounting ⌊x⌋₊ = (1+c x)·∫_2^x dt/log t` in Lean 4; another separate project.\n4. Historical (context only, not reusable in Lean): Selberg's elementary PNT formalized in Isabelle\n   (2007) and Metamath (2016); Newman's analytic proof in HOL Light (2009) and Isabelle (2018);\n   Song–Yao Isabelle PNT with exp(√log x) error (AFP).\n\nPRIOR ART ON THE MATHEMATICS ITSELF (the dyadic count is classical). Eq.10 is an immediate\nconsequence of PNT at two cutoffs (π(y)−π(y/2) = π(y) − π(y/2) ~ y/log y − (1/2)y/log y); no new\nmathematics is claimed. The frozen package's own source map attributes Input P to [KK §2] and\n[R4 §7]. The weaker, PNT-free lower bound already carried by the package is manuscript §7 (16)\n`#{p:y/2≤p≤y} ≥ y/(3 log y)` (a factorial/wheel argument).\n\nEXACT REMAINING GAP. A **two-sided PNT at real cutoffs, compatible with the package's pinned\ntoolchain** (leanprover/lean4:v4.35.0-rc3 + its pinned Mathlib, no foreign imports), or the direct\ndyadic asymptotic. No such checked interface exists in Mathlib today; the existing Lean PNT lives in\nforeign-toolchain projects. What is NOT the gap: the endpoint/floor conversion (exact, zero error for\nthe count; ≤1-prime error for the real-cutoff reading) and the PNT→Eq.10 algebra\n(`Asymptotics.IsEquivalent.sub` + `log` algebra), both available in the pinned Mathlib. So O4 splits\ninto a bounded local reduction (O4a) and this single incompatible external input (O4b).\n\nDEDUPLICATION. Return #2602 is the route proposal (already cited); return #2597 is the accepted\nsource package whose interfaces are reused here (declared dependency). This look does not repeat\npublished computations and does not rerun any proof: it is a source/interface/dependency map only."},"research_route_id":238,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_f429f8f39d2205ce9512b3fb","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/238 and return #2602. Return the ordinary report and transcript plus research: {route_id: 238, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. 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