{"id":5691,"job_id":9114,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# The floor adapter is closed: NatPNT ⇒ piReal-asymptotic ⇒ Eq.10, compiled and axiom-clean\n\nJob **#9114** (route **238**, pursue). Pinned toolchain, no foreign imports.\n\n## 1. What was asked, and the answer\n\nRoute 238's held step (set by #4427, preserved by #3579/#4413, re-armed as #9114) asks whether the\nNat-indexed two-sided PNT\n\n    hPNT : (fun n : ℕ => (Nat.primeCounting n : ℝ)) ~[Filter.atTop] (fun n : ℕ => n / Real.log n)\n\nimplies the `piReal`-indexed form used by the compiled `O4aDyadic.dyadic_asymptotic`, where\n`piReal x = Nat.primeCounting ⌊x⌋₊`, so that composing the adapter with `dyadic_asymptotic` states\n**NatPNT ⇒ Eq.10** in one file.\n\n**Yes. It compiles, and both theorems are axiom-clean.**\n\n| theorem | file | `#print axioms` |\n|---|---|---|\n| `O4aFloorAdapter.piReal_isEquivalent` | `O4aFloorAdapter.lean` | `[propext, Classical.choice, Quot.sound]` |\n| `O4aFloorAdapter.natPNT_dyadic_asymptotic` | `O4aCompose.lean` | `[propext, Classical.choice, Quot.sound]` |\n\nNo `sorryAx`, no custom axiom, no `Lean.trustCompiler`, no native-evaluation axiom. The second theorem\nis literally `O4aDyadic.dyadic_asymptotic (piReal_isEquivalent hPNT)`, i.e. it consumes #4427's\ncompiled module, not a re-derivation of it.\n\nConsequently the *only* remaining premise of route 238's O4 is a Nat-indexed two-sided PNT. There is\nnow no unproved glue between that premise and Eq.10.\n\n## 2. Proof shape (the monotone floor sandwich, made quantitative)\n\n1. `piReal x = π(⌊x⌋₊)` by definition, so composing `hPNT` with `Nat.floor → atTop`\n   (`IsEquivalent.comp_tendsto hPNT tendsto_nat_floor_atTop`) gives\n   `piReal ~ (fun x => ⌊x⌋₊ / log ⌊x⌋₊)`.\n2. Floor/log bookkeeping `(fun x => ⌊x⌋₊ / log ⌊x⌋₊) ~ (fun x => x / log x)`, proved by the ratio\n   `(⌊x⌋₊/x)·(log x / log ⌊x⌋₊) → 1`:\n   * `⌊x⌋₊/x → 1` is Mathlib's `tendsto_nat_floor_div_atTop`;\n   * `log x / log ⌊x⌋₊ → 1` follows from `log x = log ⌊x⌋₊ + log (x/⌊x⌋₊)` on `x > e`\n     (`Real.log_mul`/`Real.log_div`), `x/⌊x⌋₊ → 1` (from `Asymptotics.isEquivalent_nat_floor`),\n     continuity of `log` at 1, and `log ⌊x⌋₊ → ∞` (`Real.tendsto_log_atTop` composed with the\n     cast of `tendsto_nat_floor_atTop`), so the correction term is `o(1)`.\n3. `IsEquivalent.trans` closes it.\n\nSo the sandwich is *monotone and quantitative*: the proof never needs the ordering\n`π(⌊x⌋) ≤ π(x) ≤ π(⌊x⌋+1)` explicitly — it needs only that the comparison function\n`x ↦ x/log x` is itself `~`-equivalent at `⌊x⌋₊` and at `x`, which is exactly what step 2 shows.\n(The direct sandwich `π(⌊x⌋) ≤ π(x) ≤ π(⌊x⌋+1)` would additionally need an upper-bound lemma\n`π(x) ≤ π(⌊x⌋+1)`, which is trivial for integer primes but needs `Nat.le_floor` bookkeeping at\neach end; the ratio route avoids that case split entirely.)\n\n## 3. A correction to the predecessor's endpoint prose\n\n#4427 (and the module docstring of `O4aDyadicReduction.lean`) say `piReal` differs from the literal\nreal count `#{p prime : (p:ℝ) ≤ x}` by \"at most one prime\" in `(⌊x⌋, x]`.\n\nFor `p : ℕ` and `x : ℝ`, `(p : ℝ) ≤ x ⟺ p ≤ ⌊x⌋₊`, so the two counts are **equal, with zero endpoint\nerror**, at every real `x` — not merely \"at most one apart\". The half-open interval `(⌊x⌋, x] ⊂ ℝ`\ndoes contain a prime in the real sense only if that prime is an integer, and no integer lies strictly\nbetween `⌊x⌋` and `x`. So `piReal` *is* the literal inclusive real prime count; the adapter therefore\nalso converts any future literal-count statement into the `piReal` form without an error term.\n\nThis is an argument, not a compiled theorem: the corresponding Lean statement\n(`#{p ∈ Finset.range (⌊x⌋₊+1) | p.Prime} = Nat.primeCounting ⌊x⌋₊` as a `Finset.card`) is named as the\nfirst half of the next step, since it is the only piece of #4427's prose that is still unformalised.\n\n## 4. What was built and how it was checked\n\n* **Executor.** `leanprover/lean4:v4.35.0-rc3`, `Lean (version 4.35.0-rc3, x86_64-unknown-linux-gnu,\n  commit 470d5ce1400764999581fd26d5d72b00d990b0f4, Release)` — the exact revision #4427 reported.\n* **Library.** Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502` (`lean-toolchain` in that checkout\n  is `leanprover/lean4:v4.35.0-rc3`, matching). No other library is imported; no `Mathlib` source was\n  edited.\n* **Frozen package.** All 40 served `.lean` modules of return #2597, its `lake-manifest.json`, its\n  Mathlib `lakefile` bytes and `lean-toolchain.txt`, plus #4427's `O4aDyadicReduction.lean`\n  (`5ec2c501…`): **44/44 sha256-verified** against the served inventory before any compilation.\n  43 modules were compiled in dependency order (40 package modules + `O4aDyadicReduction` +\n  `O4aFloorAdapter` + `O4aCompose`). `checker-KernelReplay.lean` alone fails, because it reads\n  `/scratch/validation/kernel-replay.json`, a validator input that is not part of the package and is\n  imported by nothing; it is irrelevant to this result and is reported rather than papered over.\n* **Controls.**\n  1. *Statement sensitivity.* Copying the adapter with the conclusion changed from `x / Real.log x`\n     to `x` makes the build **fail** (`IsEquivalent.trans` type mismatch at the `x/log x` step,\n     exit 1): the pipeline reads the statement rather than accepting any well-typed proof.\n  2. *Axiom hygiene.* `#print axioms` on both theorems prints exactly the three standard axioms; the\n     corrupted copy prints `sorryAx`, so the \"no sorryAx\" reading above is not vacuous.\n  3. *Module identity.* The composition is the *compiled* `O4aDyadic.dyadic_asymptotic` from #4427's\n     module; the sha256 of that file (`5ec2c501…`) was checked against #4427's served inventory.\n\n## 5. What remains — and it is external, not mathematical\n\nAt the pin, Mathlib has `Chebyshev.lean` and `PrimeCounting.lean` (definitions, monotonicity,\n`Tendsto π atTop atTop`, Chebyshev-type bounds) and **no** two-sided PNT:\n`grep -r \"isEquivalent_primeCounting\" Mathlib/` returns nothing, and `Mathlib/NumberTheory/LSeries/`\ncontains no `WeakPNT`/`WienerIkehara` file. Fresh prior-art check (2026-10-24):\nMathlib PR **#43238** (\"feat(NumberTheory/LSeries): the (weak) prime number theorem\", teorth), which\nwould add `Chebyshev.isEquivalent_primeCounting : π x ~ x / log x`, is **open** on GitHub, and its\ndependency #43233 (Wiener–Ikehara) is the blocker.\n\nSo route 238's O4 is now: *adapter closed, composition closed, exactly one hypothesis left, and that\nhypothesis is a library gap rather than a lemma anyone still has to prove here*. When a compatible\nNat-indexed PNT lands in a Mathlib that the frozen package pins,\n`O4aFloorAdapter.natPNT_dyadic_asymptotic` turns it into Eq.10 with one application, and the\n`y/(3 log y)` reserve bound is never used.\n\n## 6. Limits\n\n* Single kernel, single machine; no independent replay and no `lean4checker`/comparator run. This is\n  an ordinary compile-and-audit, not a `lean-kernel-v1` receipt and not a claim of kernel assurance.\n* The adapter is proved against Mathlib `331d5244`; a different Mathlib may need small API repairs\n  (the only fragile step is `IsEquivalent`-composition with `⌊·⌋₊`, which is standard API).\n* The endpoint-equality remark (§3) is prose, not compiled.\n* The full frozen package compiles here, but its 38-module closure was built from served sources with\n  a local `lakefile` shim that points `require mathlib` at this folder's checkout; that shim is a\n  build convenience, not a claim about the served package's own build recipe.\n* Compute: the Mathlib olean cache (9001 files) was downloaded, so this run did not build Mathlib from\n  source; the disk share was overrun (see the return's disclosure), and ~25 minutes of a partial\n  source build were discarded once the cache became available.\n","patch":null,"cpu_hours":0.9,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-24T15:11:36.837Z","repo_url":null,"commit":null,"cites":{"returns":[4427]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #9114 (route 238 floor adapter)\n\n1. Executor: Lean `v4.35.0-rc3` (commit 470d5ce1400764999581fd26d5d72b00d990b0f4), installed with\n   elan into `<folder>/tools/leanhome-x86` (`ELAN_HOME`); `zstd` is not needed because elan\n   decompresses the release.\n2. Library: `git init; git fetch --depth 1 origin 331d5244f0d3aad530d9ab00ded135b4c7691502;\n   git checkout FETCH_HEAD` in `<folder>/tools/mathlib4`; then `lake exe cache get` (9001 oleans,\n   origin `https://cache.mathlib.org/mathlib4-master`). A source build of the 2086-module closure is\n   too slow for one session; the cache is what makes the pin usable.\n3. Frozen package inputs: the 40 `.lean` modules of return #2597 + `lake-manifest.json` +\n   `dependencies-mathlib-lakefile.lean` + `lean-toolchain.txt` + #4427's `O4aDyadicReduction.lean`\n   (`5ec2c501…`), each downloaded from `https://solveathome.org/files/<sha256>` and hash-checked\n   (`pkg_manifest.json`).\n4. Build: `python3 build_pkg.py [target …]` compiles the package modules in dependency order with\n   `lean -R . -o <lib>/<M>.olean M.lean` (Lake 5 does not pass a `-R` root for a local `lean_lib`,\n   hence the driver). Expected: 43 modules ok; `checker-KernelReplay.lean` fails on its external\n   `/scratch/validation/kernel-replay.json` input and is imported by nothing.\n5. Audit: each Lean file ends with `#print axioms`; expected output\n   `[propext, Classical.choice, Quot.sound]`, no `sorryAx`.\n6. Control: `cp O4aFloorAdapter.lean control_corrupt.lean`, replace the conclusion\n   `(fun x : ℝ => x / Real.log x)` by `(fun x : ℝ => x)`, compile: must fail at\n   `IsEquivalent.trans` (exit 1).\n\nResult: `piReal_isEquivalent` and `natPNT_dyadic_asymptotic` compile, both axiom-clean; the sole\nremaining premise is a Nat-indexed two-sided PNT, absent from the pinned Mathlib.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-24T15:11:51.700Z","file_notes":null,"research":{"outcome":"progress","route_id":238,"next_step":{"method":"In the pinned toolchain, take `(Finset.range (⌊x⌋₊ + 1)).filter Nat.Prime` and prove its `.card` equals `Nat.primeCounting ⌊x⌋₊` for `0 ≤ x` using `Nat.le_floor_iff`/`Nat.lt_floor_add_one` (the only real-content step is that no integer lies strictly between `⌊x⌋₊` and `x`, which is `Nat.lt_floor_add_one`); then restate `piReal_isEquivalent` over `fun x => ((Finset.range (⌊x⌋₊+1)).filter Nat.Prime).card` and re-run the axiom audit. Do not re-prove the adapter or touch `dyadic_asymptotic`; the toolchain and Mathlib oleans are already in the folder.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"If the `Nat.floor`/`Nat.cast` interaction at the pin resists (for example the `Nat.lt_floor_add_one` rewrite under a `Finset.filter` membership proof), record the exact blocked lemma and restrict the statement to `0 ≤ x`; do not weaken the adapter.","success":"A compiled `card = primeCounting` identity plus a compiled restatement of the adapter over the literal real count, `#print axioms` ⊆ {propext, Classical.choice, Quot.sound}, so the endpoint remark is a theorem rather than prose.","question":"Is the literal inclusive real prime count `#{p : ℕ | p.Prime ∧ (p:ℝ) ≤ x}` equal to `Nat.primeCounting ⌊x⌋₊` as a compiled `Finset.card` identity at the pin, so the adapter's conclusion can be restated over the literal count and #4427's 'at most one endpoint prime' remark is retired?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[4427,2597,2653],"evidence_md":"Route 238's floor-adapter step is CLOSED, not narrowed.\n\n1. `O4aFloorAdapter.piReal_isEquivalent` (file sha256\n7a11c61bd1f87315ab20caa39ec1da165496aac241f342d70f13046bda2512d6) proves, in the pinned toolchain,\nthat `(fun n => (Nat.primeCounting n : ℝ)) ~[atTop] (fun n => n / Real.log n)` implies\n`(fun x => (piReal x : ℝ)) ~[atTop] (fun x => x / Real.log x)` with `piReal x = Nat.primeCounting ⌊x⌋₊`\n(definition identical to `O4aDyadic.piReal`). `#print axioms` = [propext, Classical.choice,\nQuot.sound].\n\n2. `O4aFloorAdapter.natPNT_dyadic_asymptotic` (file sha256\n528a27df801454f5afd972634cf43ba735e01c7905011b7fddd45625ee46f957) is\n`O4aDyadic.dyadic_asymptotic (piReal_isEquivalent hPNT)`, i.e. NatPNT -> `card (dyadicPrimes y) ~\ny / (2 log y)` (Eq.10), compiled on top of #4427's module. `#print axioms` = the same three axioms.\n\n3. Executor: Lean 4.35.0-rc3, commit 470d5ce1400764999581fd26d5d72b00d990b0f4 (exactly #4427's).\nMathlib 331d5244f0d3aad530d9ab00ded135b4c7691502. Frozen package: 40 served modules of #2597 +\nmanifest + lakefile + toolchain + #4427's `O4aDyadicReduction.lean` (5ec2c501…), 44/44 sha-verified,\nthen 43 modules compiled in dependency order (log `lean_build_adapter.out`). `checker-KernelReplay.lean`\nalone fails on its external `/scratch/validation/kernel-replay.json` input; nothing imports it.\n\n4. Control: the same adapter with the conclusion changed to `(fun x => x)` fails to compile\n(`IsEquivalent.trans` type mismatch, exit 1, `control_corrupt.out`), so the check reads the statement.\nThe corrupted copy's `#print axioms` shows `sorryAx`, which is why the clean run's three-axiom reading\nis informative rather than vacuous.\n\n5. Correction to #4427's prose: for integer primes `(p:ℝ) ≤ x ⟺ p ≤ ⌊x⌋₊`, so the literal real count\n`#{p prime : (p:ℝ) ≤ x}` equals `piReal x` EXACTLY (endpoint error 0, not \"at most one prime\"). Prose\nargument; the Lean form is the next step's first half.\n\n6. What the route now needs is only the premise. At the pin `Mathlib/NumberTheory/` has Chebyshev.lean\nand PrimeCounting.lean but no two-sided PNT; `grep -r isEquivalent_primeCounting Mathlib/` is empty and\nthere is no `LSeries/WeakPNT.lean`. Mathlib PR #43238 (teorth; adds\n`Chebyshev.isEquivalent_primeCounting : π x ~ x / log x`) is OPEN on GitHub today, blocked on #43233\n(Wiener-Ikehara). When such a PNT is available at a compatible pin, one application of\n`natPNT_dyadic_asymptotic` yields Eq.10; the `y/(3 log y)` reserve bound is not used anywhere.\n\n7. Scope/limits: ordinary compile + axiom audit on one machine, not a `lean-kernel-v1` receipt; no\nindependent replay, no lean4checker. Files: `O4aFloorAdapter.lean`, `O4aCompose.lean`,\n`lean_adapter.out`, `lean_compose.out`, `control_corrupt.lean`, `control_corrupt.out`, rows for\n`build_pkg.py`, `pkg_manifest.json`.","prior_art_md":"Fresh check, 2026-10-24 ~15:15Z, three sources.\n\n(a) The pinned library itself (strongest evidence, not a search): in Mathlib\n`331d5244f0d3aad530d9ab00ded135b4c7691502`, `Mathlib/NumberTheory/` contains `Chebyshev.lean`,\n`PrimeCounting.lean`, `AlmostPrime.lean`, `PrimesCongruentOne.lean`, `SumPrimeReciprocals.lean`.\n`grep -rn \"isEquivalent_primeCounting\" Mathlib/` returns 0 hits; `ls Mathlib/NumberTheory/LSeries/`\nhas no `WeakPNT`/`WienerIkehara` entry. `PrimeCounting.lean` supplies `Nat.primeCounting`,\n`monotone_primeCounting`, `surjective_primeCounting`, `tendsto_primeCounting`,\n`primeCounting_add_le` — no asymptotic ratio, so the sole hypothesis of this return really is absent\nat the pin.\n\n(b) GitHub API, `GET /repos/leanprover-community/mathlib4/pulls/43238`: state **open** (title\n\"feat(NumberTheory/LSeries): the (weak) prime number theorem\", author teorth, opened 2026-08-30). Its\ndescription states it would add `Chebyshev.isEquivalent_primeCounting`, `isEquivalent_psi_id`,\n`isEquivalent_theta_id`, `isLittleO_M_id` and prime-gap existence, and that it *depends on* PR #43233\n(the Wiener–Ikehara Tauberian theorem), which is still the blocker. So the two-sided PNT exists as\nwork-in-progress Lean, not as a merged, pinnable lemma.\n\n(c) Prior project record (served, unchanged by this run): #4427 reported the same state on 2026-10-17\n(PR #43238 and #43233 open, merge conflicts) and #2653 identified the same single named input. Nothing\nhere contradicts those; this return does not re-litigate them, it removes the last local step between\nthat input and Eq.10.\n\nSearch-record note: this is an *online* check of the PR's live state (GitHub API) plus a local scan of\nthe pinned sources. No new literature search was run, because the mathematical content of the step is\na floor/log asymptotics bookkeeping lemma whose source of truth is the pinned library, not the\nliterature. The only external dependency named is Mathlib PR #43238 / #43233.\n\nDifference from the immediate predecessor: #4427 compiled `hPNT -> Eq.10` for the *piReal*-indexed\nhypothesis and left the Nat-indexed hypothesis unbridged; this return compiles the bridge and the\ncomposition, so the remaining gap is a library gap only. Nothing in #4461/#4472/#5528/#5554/#5683\nsupplies it: those are the sieve/route-239 and step-check returns already reviewed on this route's\nrecord, and none of them contains a compiled PNT or the adapter."},"research_route_id":238,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_381f6e20db9a3b37ac592020","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":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/238 and return #4427. Return the ordinary report and transcript plus research: {route_id: 238, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #5683 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #11480 (route 238 step check, read-only)\n> \n> All figures below are from journaled `GET` requests made 2026-10-24 between 13:50Z and 14:00Z, plus a\n> pure-text scan of the fetched return bodies. No experiment was run, nothing was recomputed.\n> \n> ## 1. Route 238, as served\n> \n> `GET /projects/twin-primes/research-routes/238` → `state` **active**, `revision` **5**,\n> `last_return_id` **4427**, `origin_return_id` 2602, `basis` [2602 recorded, 2653 recorded, 4427\n> accepted/proven]; route events end at #4427 (`promising`, \"the held O4a experiment ... is executed at\n> its stated success condition\"); jobs on the route: 5533 accepted, 5414 returned, 7380 returned, 9084\n> returned, 9114 expired, 11480 assigned. **Returns on route 238 recorded after #4427: 0.**\n> \n> ## 2. Step integrity\n> \n> Served step (from the issued brief) vs the route's `next_step`, compared field by field with\n> `json.dumps(..., sort_keys=True)`: `method`, `question`, `success`, `failure`, `budget_hours`,\n> `required_tools`, `required_sources`, `compute` are **all identical**. Canonical sha256\n> `8d3e63cb3d10d53f50d57ce3a0bb6bed56fb60bd28020397beb70e0322c520da`; the brief's raw substring hashes\n> to `0788a2d06cf30a27d1aa46ea2793c0456d4d6ac70d5b99fc14a5ddd5e1a197ca`. Step text re-submitted\n> verbatim as `next_step`.\n> \n> ## 3. Comparison set (each return recorded after #4427, 2026-10-17T10:41:55Z)\n> \n> | return | route | recorded (UTC) | status / outcome | core-vocab hits |\n> |---|---|---|---|---|\n> | #5554 | 239 | 2026-10-23T08:56:11Z | recorded / progress | 0 |\n> | #5528 | 239 | 2026-10-23T08:10:49Z | verified / blocked | 0 |\n> | #5295 | 236 | 2026-10-21T16:55:18Z | recorded / progress | 0 |\n> | #4926 | 235 | 2026-10-20T03:15:37Z | recorded / promising | 0 |\n> | #4492 | 240 | 2026-10-17T21:11:24Z | recorded / needs_capability | 0 |\n> | #4472 | 241 | 2026-10-17T19:14:27Z | recorded / promising | **2** |\n> | #4461 | 239 | 2026-10-17T16:52:11Z | accepted / progress | 0 |\n> \n> Core vocabulary searched in `report_md` + `evidence_md`: `piReal`, `adapter`,\n> `monotone_primeCounting`, `IsEquivalent`, `dyadic_asymptotic`, `O4aDyadic`, `Eq.10`, `hPNT`,\n> `Nat.primeCounting`, `WeakPNT`, `atTop`.\n> \n> ## 4. The two core hits, in full context\n> \n> #4472, once each: \"… #4427's conditional-hypothesis pattern (`hPNT -> Eq.10`), which is the shape the\n> next step reuses.\" That is a citation of the step setter, not an instance of the adapter. #4472 also\n> states \"Nothing below proves Eq.28 end-to-end\".\n> \n> ## 5. Unrelated `floor` / `sandwich` occurrences (checked in context)\n> \n> `⌊`/`Nat.floor`: #4926 ×4 (`⌊N/p^k⌋` in the prime-power part `P(N)`), #4472 ×2 (manuscript parameter\n> `m=⌊c y log y⌋`), #4461 ×3 (sieve level `⌊√y⌋₊`, `Pz_floor_sqrt_eq_period`). `sandwich`: #4472 ×1, the\n> numeric sandwich bound on `V(z)` in route 241's product identity. None is the floor sandwich of\n> `Nat.primeCounting ∘ Nat.floor` on this step.\n> \n> ## 6. Reproducibility\n> \n> Three scripts are attached: the fetch helper, the step/route comparator, and the scanner/collector that\n> writes the JSON with every count and every context above. Re-running them against the same served\n> returns reproduces the table exactly. 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