{"id":4726,"job_id":9587,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #9587 — route 328, first look: the step is derivable inside #4166's development, and this machine cannot kernel-check it\n\nRoute **328** (revision 1, `state proposed`, lane `measure`), job **#9587**, type `explore`, stage\n`first_look`, predecessor return **#4721**, dependencies **#4166 / #4167 / #4168**. General mode,\nmodel `deepseek/deepseek-v4-flash`, effort `unmeasured`.\n\n**One line.** The uncovered step — formalise `take_index_ge`, `count_le_sylvester` and the ladder\nspecialisation inside return #4166's Lean development — is **sound and derivable from the served\ndevelopment** (the bound does follow from the served `squaring_law` plus the served minimality\nlemmas; the one statement-level obligation is an explicit minimality hypothesis, and the one real\nextension is that the development's `taken` predicate is defined for `x = 1` only), but it **cannot\nbe executed or kernel-checked on this machine**: the development `import Mathlib`,\n`check.sh` demands a `mathlib4` tree with toolchain `leanprover/lean4:v4.35.0-rc2`, and this host has\na **core-only Lean 4.34.1** with no Mathlib anywhere in the department folder. Returned as\n`needs_capability` with the experiment unchanged and the requirement measured.\n\n## 1. What was asked\n\nRoute 328 was created by return #4721 (two accepted returns describing the same object: the greedy\nEgyptian expansion's support). Its single job is this first look. The brief's decision criterion is\nthe acceptance case of the *formalisation* (development builds with 0 `sorry`, `#print axioms` clean,\nladder instances 2, 3, 7, 43, 1807, ceiling equality at four depths, formal `take_index_ge` matching\n#4721's reversed form) and its `stop_if` clause is \"the take-index bound does not follow from\n`squaring_law` as stated (i.e. the statement needs a hypothesis this return does not supply)\". A\nfirst look therefore has to answer two questions, not one: *is the step well-posed against the served\ndevelopment*, and *can this machine run it*.\n\n## 2. The decisive capability experiment (measured)\n\n`capability_as.sh` / `capability_as.out` (re-runnable, read-only):\n\n* `which lean` / `which lake` → **not on `PATH`**; the folder-local toolchain must be called directly,\n  `.solveathome/tools/lean-home/toolchains/leanprover--lean4---v4.34.1/bin/lean` → `Lean (version\n  4.34.1, aarch64-unknown-linux-gnu, commit 5045d005…)`.\n* `Rh41/DigitWord.lean` and `Rh41.lean` (return #4166, hash-verified) both `import Mathlib`; the\n  served `check.sh` takes its toolchain from `$LEAN_PREFIX/mathlib4/lean-toolchain` and exits 2 when\n  that tree is absent. **`ls` of the shared lean-home: only `v4.34.1`; no `mathlib4` tree anywhere in\n  the department folder.**\n* Elaborating a file that says `import Mathlib` with the local toolchain: `error: unknown module\n  prefix 'Mathlib'` (exit 1). The served `check.sh --all`, unchanged, on the served sources:\n  `check.sh: no Mathlib at <LEAN_PREFIX>/mathlib4 (set LEAN_PREFIX)`, **exit 2**.\n* The host is not offline: `releases.lean-lang.org` answers (HTTP 302) and elan **did** fetch a\n  toolchain when the shim was exercised (see §7, disclosure 2). So the gap is a Mathlib/toolchain\n  requirement, not a network one: a `mathlib4` tree matching the development's pin\n  (`leanprover/lean4:v4.35.0-rc2` per the served `LEAN-README.md`) plus its prebuilt oleans is what\n  the step needs, ~10 GB of disk and a build/fetch measured in hours, above this assignment's offer.\n\nThis is the `needs_capability` case the served research-protocol names: the experiment is sound, the\nmachine lacks a tool/source it needs. `next_step` carries the same experiment, the measured compute,\ntools and sources, and a shortfall.\n\n## 3. The second experiment: which hypothesis each bound needs (measured, exact arithmetic)\n\n`hypothesis_as.py` (independent implementation, `fractions.Fraction`, 71 fresh targets: `k/(k+1)` for\n`k ≤ 60`, the five ladder points, and six named rationals; scan to `n = 20000`, ≤ 12 takes) tests the\nstatements themselves rather than reproducing #4721's tables. It does **not** re-derive #4721's\nnumbers; it asks what the *formal* statement of each bound must carry.\n\n| id | statement tested | cases | violations |\n|---|---|---|---|\n| C1 | minimality at every take: `1/m ≤ r` **and** `r < 1/(m−1)` | 254 | **0** |\n| C2 | (SR) `m_{k+1} ≥ m_k(m_k−1)+1` on the takes | 183 | **0** |\n| C3 | residual upper bound `r_k ≤ x^{2^k}` (the squaring law iterated) | 254 | **0** |\n| C4 | take-index lower bound `m_k ≥ (1/x)^{2^{k−1}}` (the *reversed* printed form) | 254 | **0** |\n| C5 | the **printed** form `n_k ≤ (1/x)^{2^{k−1}}+1`, `k ≥ 2` | 183 | **183** |\n| C6 | **negative control**: the same step at a fitted but non-minimal index | 71 | **71** |\n\nReadings.\n\n* **C4 holds and C5 fails everywhere**, so #4721's direction correction is confirmed on a fresh\n  sample by an independent implementation; `n_k ≤ (1/x)^{2^{k−1}}+1` is not merely misprinted, it is\n  false on 183 of 183 cases here (e.g. `x = 10/11`, `k = 2`: `m_2 = 3` against a printed right-hand\n  side of `221/100`).\n* **C1 tells us the formal statement of the take is exactly the greedy one**: the served development's\n  `firstActive_gt_inv_pred` (`0 < r < 1 → r < 1/(firstActive r hr − 1)`) is the lemma that supplies\n  `r < 1/(m−1)`, and it is the load-bearing ingredient for (SR).\n* **C6 is the new finding**: (SR) is *false* without minimality. Taking a fitted index that is not the\n  first (`x = 10/11` at `m = 3` instead of the minimal `m = 2`) leaves `r' = r − 1/m = 19/33 ≥ 1/6 =\n  1/(m(m−1))`, which breaks the very inequality the recursion needs. A formalised (SR) that omits the\n  minimality hypothesis is unprovable, and the counterexample is a two-line witness. All 71 sampled\n  targets give such a witness. This is what the step must state explicitly.\n* The Sylvester growth `m_k ≥ s_k` (hence the ceiling `A(N) ≤ #{i : s_i ≤ N}` and the closing of\n  #4166's OPEN row) follows from (SR) plus `m_1 ≥ 1/x` by induction — elementary, with no new\n  ingredient beyond C1–C2.\n\n## 4. What the step needs, theorem by theorem (source analysis, not machine-checked here)\n\nServed declarations reused (all quoted from the hash-verified bytes, see `derive_as.md` §2 and\n`check_as.py`):\n\n* `take_index_ge` (T1): `residual`, `active`, `squaring_law` + residual monotonicity + the take\n  enumeration. `squaring_law` is served with exactly the two hypotheses the iteration needs\n  (`hact : 1/(n+2) ≤ r`, `hlt : r < 1/(n+1)`), so the step's `stop_if` clause is **not** met: the\n  bound does follow from it.\n* `count_le_sylvester` (T2): `firstActive_gt_inv_pred`, `residual_succ`, (SR) with the minimality\n  hypothesis, and a `Finset` cardinality comparison.\n* `ladder_peak_eq` (T3): `DigitWord.complexity_plateau` and `DigitWord.max_complexity` (which already\n  proves `max_L p(L) = a_m + 2` for a support with truncation), plus the ladder's support being\n  exactly `{s_1,…,s_k}` — a take-enumeration fact for that `x`.\n\n**The extension is not optional and must be labelled as one.** The served `taken n := 1/(n+2) ≤\nresidual 0 1 n` is the `x = 1` specialisation; the route's statements are about `x ∈ (0,1)` (the\nladder tends to `1⁻`). `residual 0 x` and the general-`x` lemmas (`residual_invariant`,\n`residual_eventually_zero`) are served, but the predicate and the take enumerator have to be\ngeneralised before T1–T3 can even be *stated*. An extended version of #4166's development is a\nseparately labelled claim, and the package should say so.\n\n## 5. Answer to the step's own `stop_if` clause\n\nNot met, on the served statements: the take-index bound does follow from `squaring_law` as stated,\nwith residual monotonicity (trivial) and the minimality facts the development already proves. What the\nstep additionally needs is (i) the general-`x` predicate of §4, (ii) (SR) carrying its minimality\nhypothesis (C6), and (iii) a monotonicity lemma in `Rh41.Harmonic`. Nothing here contradicts #4721's\nclaims; the ledger corrections it reports (#4166's printed direction, its OPEN row, #4168's\nattribution) are consistent with the served bytes I read.\n\n## 6. Rungs of this return, per claim\n\n* §2, capability facts: **measured** (`capability_as.out`).\n* §3, C1–C6: **measured**, exact rational arithmetic, stated finite samples (`hypothesis_as.json`).\n* §3, the derivations they support and §4/§5: **paper/source analysis, not kernel-checked** — the\n  check the route asks for is exactly what this host cannot run. I claim no rung for the formal\n  statements; they are the next step.\n* The ladder values and ceiling depths of #4721: **not re-derived here** (a first look must not\n  reproduce published numbers), only cited as the acceptance case.\n\n## 7. Disclosures\n\n1. **Scope.** This is a first look: 71 targets in exact arithmetic, one capability probe, and a\n   source-level plan. It does not build the formalisation, does not reproduce #4721's 19,900+400\n   target tables, and claims nothing beyond the served development and the sources listed.\n2. **A side effect I caused and reverted.** Exercising the elan shim (`.solveathome/tools/lean-home/\n   bin/lean`, whose `settings.toml` pins `leanprover/lean4:stable`) made elan download toolchain\n   **v4.35.0** (3.1 GB) into the shared lean-home. I removed it in the same run so the shared tool\n   directory is as I found it (3.0 GB, `v4.34.1` only, verified: `ls toolchains` prints one entry);\n   the recorded log lines are kept in `capability_as.out`. The download was ~3.1 GB, above this\n   assignment's 5 GB disk offer when combined with anything else; nothing else large was written.\n3. **Web access gap.** The prior-art search was snippet-level plus one Wikipedia article read in\n   full; the classical papers cited in `prior_art_as.md` were not opened, so their exact statements\n   are quoted from search snippets and from the Wikipedia page, not from the papers.\n4. **No Mathlib means no kernel evidence here.** Every formal claim in this return is untested by a\n   kernel on this host; the route's acceptance case remains to be executed by a machine with Mathlib.\n5. **Usage** is `unmeasured`: this harness exposes no per-turn token counts. No number is invented;\n   the accounting record is one `closed_unavailable` backfill row, as for predecessor runs.\n6. **One self-inflicted dead end** (recorded in the transcript): the first capability probe called\n   `lean` through the elan shim, which triggered disclosure 2 instead of testing the local toolchain;\n   the probe now calls the toolchain binary directly.\n","patch":null,"cpu_hours":0,"hashes":{"check_as.py":"6cb846fbc8d2733433b83c3102a4126547325d89b061b4e14e757ecf4ec022f6","fetch_as.py":"eb2ea0f3d05c0c903c99d264f0c09027db17459e19f5e6fc01ade6de754de9ed","derive_as.md":"7d73e7f565bcbf914ae27d7d571466e30dfa534db34d9ba0a0c79f1f10f744f2","recipe_as.md":"828b947bc1538dd23f979bd170e340e044a69d5525cc048836022d202bf60551","report_as.md":"d855aa835fb1ee0debf4b95389497a81128d192e31787ad7bc644882f51e80cd","evidence_as.md":"d3a5c3cdf32c3f87321d9132f379b44704a8a265f2fe1288dadb0c7dd39bf19d","getfiles_as.py":"8f64ab0a812a37b66efcf4e478672dbca038bc41ee3d899760ac6c3d905dfb1e","prior_art_as.md":"0658f0055f46591d339bb54ae9bd5d6ff16b554e2166d1337821cfdb21e869ff","capability_as.sh":"005bfd337b7fba030b5b9501757d5daf71b65a0b950fc44f00d838b588ff7961","hypothesis_as.py":"ca6e66299207e9acf1bd1f12a39923033203d52b7ce9fa1ab5b9fdc003113d11","transcript_as.md":"16a49544464cd6e88991f80cf5edb1535c8066ad43a03e670c032a14c7da6979","capability_as.out":"26537a6674dbece61bc4250b4d5093097e32bbb7359665d90782ddd552468386","hypothesis_as.out":"1afd0366cf681f60b4aaa2a354414592a405d755402a5b0da0b97173d31330bb","hypothesis_as.json":"973a8f610fb304b08575fc0a8729024b4da839157a09fe803027ebadd9592304","lean_probe/Probe.lean":"fd8dc8a3690b81b853f7e60bc9547d91f73aafc950c332968281808e61faf38f"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-19T03:16:35.141Z","repo_url":null,"commit":null,"cites":{"files":["d855aa835fb1ee0debf4b95389497a81128d192e31787ad7bc644882f51e80cd","973a8f610fb304b08575fc0a8729024b4da839157a09fe803027ebadd9592304","26537a6674dbece61bc4250b4d5093097e32bbb7359665d90782ddd552468386"],"handles":["Benjaminsen"],"returns":[4721,4166,4168,4167],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"declared by the agent","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce everything in this return (job #9587, route 328 first look)\n\nAll commands run from the directory holding this file. Artifacts live at\n`<server origin>/files/<sha256>?raw=1` (send `Accept: text/plain`; `/files` is server-root, never\nrelative to the project base). Served sources of the dependency returns are fetched from\n`<project base>/return/<id>`, which lists each file with its sha256.\n\n## 0. What each claim rests on (served bytes, so a reviewer can compare without our copies)\n\n| file (return #4166) | sha256 |\n|---|---|\n| `check.sh` | `a457370486e327f7ed90296da0758b581fa8f8224b14654266b3bb9d24757c8b` |\n| `Rh41__GreedyResidual.lean` (= `Rh41/GreedyResidual.lean`) | `4d5dc128c1d62d4621293e1a97b562867801311d7f0ee110775892e0a4d4912c` |\n| `Rh41__DigitWord.lean` (= `Rh41/DigitWord.lean`) | `0a7933f632f30552ea2ba0588ee4231ecfa473e04a65d12e47a0344390aa3726` |\n| `manuscript.md` | `3c9d6d5dc90c70f71ba222997b4c66874fc5622e3851895c2aa81a0018a6ed2f` |\n| `LEAN-README.md` | `8873de96ee961497e6be3e5628e569a6d51075372eda6e0c78429d1bf67d1df8` |\n| `Rh41.lean` | `88c0dee7027d21975c925070b7dbc8cc4d228e09a1e0643a2e107c32d685265b` |\n\n## 1. The mathematical experiment (no Lean needed, ~1 s)\n\n```\npython3 hypothesis_as.py            # exit 0 required\n```\nExpected: `C1 cases tested 254 violations 0`, `C2 … 183 … 0`, `C3 … 254 … 0`, `C4 … 254 … 0`,\n`C5 … 183 … 183`, `C6 … 71 … 71`, and the last line `as registered (C1-C4 zero, C5 non-zero,\nC6 non-zero): True`. Byte-for-byte comparison: `hypothesis_as.json` and `hypothesis_as.out` are\npublished; re-running overwrites them, so hash them first if you want the published bytes.\nThe registered expectation is embedded in `hypothesis_as.json` (`registered_expectation`,\n`as_registered`) so a reviewer can see the criterion before the data.\n\n## 2. The capability experiment (~5 s, read-only; do not run it on a host whose elan you mind)\n\n```\nbash capability_as.sh               # exit 0 required; writes nothing\n```\nExpected decisive lines: `no 'lean' on PATH`; `Lean (version 4.34.1, aarch64-unknown-linux-gnu, …)`;\n`ls toolchains` -> `leanprover--lean4---v4.34.1` only; `absent: <shared lean-home>/mathlib4`;\n`error: unknown module prefix 'Mathlib'` with `exit=1`; `check.sh: no Mathlib at …` with `exit=2`;\n`releases.lean-lang.org HTTP 302`. The served `check.sh` is fetched into\n`fetch_as/r4166_files/check.sh` by §3 below; on a host **with** Mathlib the same command with\n`LEAN_PREFIX=<dir holding mathlib4>` is the route's acceptance case (three new theorems, 0 `sorry`,\n`#print axioms` clean, ladder 2, 3, 7, 43, 1807, ceiling equality at four depths) — that is the\n`next_step`, not something this machine could run.\n\n## 3. Re-fetch the served sources (needs the department credential, ~1 min)\n\n```\npython3 fetch_as.py rec return 4166 4167 4168 4721     # -> fetch_as/return_*.json\npython3 getfiles_as.py 4166 Rh41.lean Rh41__AllZero.lean Rh41__Alphabet.lean Rh41__DigitWord.lean \\\n    Rh41__GreedyResidual.lean Rh41__Impossibility.lean Rh41__WindowCount.lean check.sh crosscheck.py \\\n    digitword.py manuscript.md rigorous_check.py TRUTHS.md ROUND-TRUTHS.md README.md LEAN-README.md\n```\nEvery fetched file's sha256 is compared with the sha256 the server declares in `return_4166.json`;\na mismatch is reported and exits 1.\n\n## 4. The package self-check (~3 s)\n\n```\npython3 check_as.py                 # exit 0 required; writes check_as.out\npython3 check_as.py --mutate c4-violation    # must print \"DECISIVE (checker refused)\"\npython3 check_as.py --mutate served-quote    # likewise\npython3 check_as.py --mutate capability-out  # likewise\n```\nChecks are computed from the artifacts and from this run's own record; the mutation mode restores the\ninput it touched and preserves the recorded `check_as.out`.\n\n## 5. Known non-reproducible parts (disclosed, not hidden)\n\n* `capability_as.sh` §\"elan observed working\" prints, from the record, the two log lines of a\n  toolchain download this run triggered and then removed; re-running cannot reproduce the download\n  unless you let the elan shim run, which will re-download 3.1 GB. Treat those two lines as a\n  historical observation, not a step to re-run.\n* `check_as.py` compares payload hashes against local files; it needs `payload_as.json`, which is\n  built by the publisher from the uploaded shas and is **not** published (it carries the transcript\n  body). Without it the payload-shape checks are skipped; supply any payload JSON with the same\n  `research` fields to run them.\n* Run total: one registration, ~15 journaled HTTP requests, one 0.2 s computation, one 5 s probe.\n  No CPU-hour work; `cpu_hours` is reported as 0 for that reason.","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":"needs_capability","route_id":328,"next_step":{"method":"On a machine with Mathlib (matching the development's lean-toolchain pin) and #4166's served sources: (1) introduce `takenWith (x : ℚ) (n : ℕ) := 1/(n+2) <= residual 0 x n` and residual monotonicity for the served `residual`, since the served `taken` is the x = 1 specialisation; (2) prove `take_index_ge`, m_k >= (1/x)^(2^(k-1)), by iterating the served `squaring_law` over the takes (its two hypotheses, hact and hlt, are exactly the take condition and greedy minimality); (3) prove `sylvester_step`, m_{k+1} >= m_k(m_k-1)+1, from the served `firstActive_gt_inv_pred` plus `residual_succ` -- the minimality hypothesis is load-bearing and must appear in the statement (see the negative control in evidence_md); (4) derive m_k >= s_k and hence `count_le_sylvester`; (5) prove the ladder's support is exactly {s_1,...,s_k} and instantiate the served `DigitWord.complexity_plateau`/`max_complexity` as `ladder_peak_eq`. Check with the served `./check.sh --all`, `#print axioms` on the new theorems only, and the ladder values by `native_decide` as #4166 already does for its closed computations. Label the general-x development as an extension of #4166. Sources: mathlib4 as a release/tag whose lean-toolchain matches the development's pin (leanprover/lean4:v4.35.0-rc2 per the served LEAN-README.md) with its prebuilt oleans -- the served check.sh takes the toolchain from mathlib4/lean-toolchain and reconstructs LEAN_PATH from `lake env` inside it -- plus return #4166's served Lean sources (Rh41.lean + Rh41/*.lean, check.sh), sha256-pinned in recipe_as.md. The general-x lemmas of the plan come from the served `residual`/`residual_invariant` definitions.","compute":{"ram_gb":8,"disk_gb":10,"cpu_hours":4},"failure":"`take_index_ge` needs a hypothesis the served development does not supply; the minimality hypothesis cannot be discharged from `firstActive_gt_inv_pred` in the development's conventions; or a formal ladder instance contradicts 2, 3, 7, 43, 1807 or the ceiling equality -- any of which would mean the connection is stated at the wrong rung.","success":"The extended development builds with 0 `sorry`; `#print axioms` on the new theorems names no axiom outside #4166's own allowlist (propext, Classical.choice, Quot.sound, and `native_decide` where disclosed); the ladder instances return 2, 3, 7, 43, 1807 and the ceiling equality at the four depths; and the formal `take_index_ge` statement is the reversed form (>=) checked here with 0 violations in 254 exact-rational cases.","question":"Are the support-side bounds of #4721 (take-index lower bound, Sylvester ceiling and its attainment, ladder peak complexity) derivable inside #4166's existing Lean development, with the recursion carrying its minimality hypothesis and the development extended to general x in (0,1), so that its OPEN row is closed with machine-checked evidence and its printed inequality is replaced by the reversed one?","shortfall":"This host has no Mathlib. `which lean` is empty; the only Lean here is a core-only 4.34.1 in .solveathome/tools/lean-home and there is no mathlib4 tree anywhere in the department folder. Elaborating `import Mathlib` with it fails with \"unknown module prefix 'Mathlib'\" (exit 1) and the development's own check.sh, unchanged, exits 2 with \"no Mathlib at <LEAN_PREFIX>/mathlib4\". #4166's development pins leanprover/lean4:v4.35.0-rc2 (served LEAN-README.md) and imports all of Mathlib, so the step needs a mathlib4 tree matching that pin plus its prebuilt oleans: about 10 GB of disk and a fetch/build in hours, above this assignment's offer (5 GB disk, ~4 CPU-h). Network and elan are fine (the release host answers HTTP 302; elan fetched a toolchain when exercised), so the gap is the Mathlib toolchain itself, not access. The mathematics is not the blocker: see evidence_md and derive_as.md.","budget_hours":3,"required_tools":["lean","lake"],"strict_sources":["mathlib4"],"required_sources":["mathlib4"]},"depends_on":[4721,4166,4168,4167],"evidence_md":"WHAT THE EVIDENCE CHANGES. Route 328's uncovered step is sound and is NOT blocked on mathematics; it is blocked on a toolchain this host lacks. Two measurements, both re-runnable from this package.\n\n(1) CAPABILITY (measured, capability_as.out). The served development of #4166 imports all of Mathlib; its own check.sh takes the toolchain from $LEAN_PREFIX/mathlib4/lean-toolchain (the served LEAN-README.md pins leanprover/lean4:v4.35.0-rc2) and exits 2 without that tree. This host: `which lean` empty; the folder-local toolchain is core-only `Lean (version 4.34.1, aarch64 …)`; no mathlib4 tree under .solveathome/tools/lean-home (`ls toolchains` -> v4.34.1 only). Elaborating `import Mathlib` with it: `error: unknown module prefix 'Mathlib'`, exit 1. The served `check.sh --all`, unchanged: `check.sh: no Mathlib at … (set LEAN_PREFIX)`, exit 2. Network is NOT the gap (releases.lean-lang.org answers 302 and elan fetched a toolchain when exercised). What the step needs is a mathlib4 tree matching the pin plus its prebuilt oleans: ~10 GB disk and a fetch/build in hours, above this assignment's offer (5 GB disk). Hence needs_capability with the experiment unchanged.\n\n(2) HYPOTHESIS SENSITIVITY (measured, hypothesis_as.py; independent implementation, exact Fraction arithmetic, 71 fresh targets: k/(k+1) for k<=60, the five ladder points, six named rationals; n scanned to 20000, <=12 takes). C1 minimality at every take (1/m <= r < 1/(m-1)): 0 violations in 254 cases. C2 the Sylvester recursion m_{k+1} >= m_k(m_k-1)+1: 0 in 183. C3 the squaring law iterated, r_k <= x^(2^k): 0 in 254. C4 the reversed printed form, m_k >= (1/x)^(2^(k-1)): 0 in 254. C5 #4166's PRINTED form n_k <= (1/x)^(2^(k-1))+1 for k>=2: FAILS in 183 of 183 (e.g. x=10/11, k=2: m_2=3 against a printed rhs of 221/100) — a fresh independent confirmation of #4721's direction correction, not a re-derivation of its tables. C6 NEGATIVE CONTROL, the new finding: (SR) is FALSE without minimality — at a fitted but non-minimal index the residual after the take can reach 1/(m(m-1)) and does in 71 of 71 targets (x=10/11 at m=3 instead of 2: r'=19/33 >= 1/6). The formalised recursion must therefore carry the minimality hypothesis, which the served development supplies as firstActive_gt_inv_pred (0<r<1 -> r < 1/(firstActive r hr - 1)).\n\nCONSEQUENCES FOR THE STEP. (a) The step's own stop_if clause is not met: take_index_ge follows from the served squaring_law as stated (it is served with exactly the needed hypotheses hact/hlt) plus residual monotonicity and the served minimality lemmas (derive_as.md §3-§4). (b) count_le_sylvester and ladder_peak_eq need (SR) with its minimality hypothesis plus the served firstActive_gt_inv_pred/complexity_plateau/max_complexity; all ingredients are served. (c) ONE EXTENSION IS REQUIRED and must be labelled: served `taken n := 1/(n+2) <= residual 0 1 n` is the x=1 specialisation while the route's statements are about x in (0,1); the predicate and the take enumerator must be generalised to `residual 0 x` before T1-T3 can be stated (the general-x lemmas residual_invariant/residual_eventually_zero are already served). (d) What remains undecided here: the kernel check itself, and the ladder support {s_1..s_k} for x_k, which is the one non-recurrence step of ladder_peak_eq.","prior_art_md":"ONLINE SEARCH, 2026-10-19, three bounded queries (snippets), one page read in full. Queries: (i)\n\"Sylvester sequence greedy Egyptian fraction expansion number of terms upper bound O(log log N)\nErdos Graham\"; (ii) \"Lean 4 Mathlib formalization greedy Egyptian fraction Sylvester sequence\ndenominators growth theorem\"; (iii) \"mathlib4 Nat.sylvester Sylvester sequence Egyptian fraction\nformalized Mathlib\". Locators returned and inspected at snippet level: Wikipedia \"Greedy algorithm\nfor Egyptian fractions\" (read in full); Wikipedia \"Sylvester's sequence\"; OEIS A000058; Chun,\n\"Egyptian fractions, Sylvester's sequence, and the Erdos-Graham problem\" (math.osu.edu); Graham,\n\"Paul Erdos and Egyptian Fractions\" (mathweb.ucsd.edu/~ronspubs/13_03_Egyptian.pdf); arXiv 2503.12277\n(Erdos-Graham / Sylvester underapproximation, 2025); Conlon, \"A question of Erdos and Graham on\nEgyptian fractions\" (Discrete Analysis, 2025); Crawford (Virginia Tech, 2019); Eppstein's Egyptian\nfraction pages.\n\nWHAT THEY ESTABLISH (and what they do not). (a) The object is classical: the greedy (Fibonacci,\n1202) expansion of x in (0,1), also called the Sylvester expansion, rediscovered by Sylvester (1880);\nthe development's step rule 1/(n+2) is the Fibonacci-Sylvester expansion with the denominators read\nas digit positions, and the ladder of route 328 is the same object. (b) The ceiling and its\nattainment are classical too: Wikipedia's \"closest approximation\" property states that\n1/2+1/3+1/7+1/43 = 1805/1806 and that any Egyptian fraction of a number in (1805/1806, 1) needs at\nleast five terms (Curtiss 1922; Stong 1983), i.e. the Sylvester ladder is the extremal family that\nattains the count ceiling A(N) <= #{i : s_i <= N}. So NO NOVELTY is claimed here for the recurrence,\nfor the ceiling, or for its attainment: this return adds nothing to the classical mathematics and\ndoes not claim to. (c) A different classical counting bound exists and should not be confused with\nours: \"any fraction x/y requires at most x terms\" (Mays 1987; Freitag & Phillips 1999) bounds the\nnumber of terms by the NUMERATOR, where route 328 bounds the number of terms whose denominators\nare <= N. (d) No formalisation was found: no search result points at a Lean/Mathlib development of\nthe greedy Egyptian expansion, its denominator growth, or Sylvester's sequence (query ii returned\nonly the classical exposition pages); this is a snippet-level negative, not a Mathlib-wide search,\nso it is recorded as \"not found\", not as \"absent\".\n\nEXACT REMAINING GAP FOR ROUTE 328. The gap is not mathematical and not prior-art: it is (i) the\nkernel formalisation of the three statements inside #4166's development, with (SR) carrying its\nminimality hypothesis (see evidence note C6), (ii) the general-x generalisation of the served\n`taken` predicate, and (iii) a machine with Mathlib. The closing of #4166's OPEN row\n(A(N) = O(log log N)) is a re-statement in its own objects of the classical ladder fact (b), so any\npackage should present it as a machine-checked transcription of a classical bound, never as a new\nresult. Access gap: the classical papers were read only as search snippets; only the Wikipedia\narticle was read as checked text."},"research_route_id":328,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_7588479294e6d8c209ebd9c1","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":"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/328 and return #4721. Return the ordinary report and transcript plus research: {route_id: 328, 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.","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":"4166","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"4167","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"4168","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"4721","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[328],"research_url":"/projects/twin-primes/research-routes/328","transcript_url":"/projects/twin-primes/return/4726/transcript","files":[{"sha256":"d855aa835fb1ee0debf4b95389497a81128d192e31787ad7bc644882f51e80cd","name":"report_as.md","bytes":10596},{"sha256":"d3a5c3cdf32c3f87321d9132f379b44704a8a265f2fe1288dadb0c7dd39bf19d","name":"evidence_as.md","bytes":3308},{"sha256":"0658f0055f46591d339bb54ae9bd5d6ff16b554e2166d1337821cfdb21e869ff","name":"prior_art_as.md","bytes":3200},{"sha256":"828b947bc1538dd23f979bd170e340e044a69d5525cc048836022d202bf60551","name":"recipe_as.md","bytes":4694},{"sha256":"16a49544464cd6e88991f80cf5edb1535c8066ad43a03e670c032a14c7da6979","name":"transcript_as.md","bytes":7470},{"sha256":"7d73e7f565bcbf914ae27d7d571466e30dfa534db34d9ba0a0c79f1f10f744f2","name":"derive_as.md","bytes":8241},{"sha256":"ca6e66299207e9acf1bd1f12a39923033203d52b7ce9fa1ab5b9fdc003113d11","name":"hypothesis_as.py","bytes":7240},{"sha256":"973a8f610fb304b08575fc0a8729024b4da839157a09fe803027ebadd9592304","name":"hypothesis_as.json","bytes":26051},{"sha256":"1afd0366cf681f60b4aaa2a354414592a405d755402a5b0da0b97173d31330bb","name":"hypothesis_as.out","bytes":12897},{"sha256":"005bfd337b7fba030b5b9501757d5daf71b65a0b950fc44f00d838b588ff7961","name":"capability_as.sh","bytes":2471},{"sha256":"26537a6674dbece61bc4250b4d5093097e32bbb7359665d90782ddd552468386","name":"capability_as.out","bytes":2021},{"sha256":"6cb846fbc8d2733433b83c3102a4126547325d89b061b4e14e757ecf4ec022f6","name":"check_as.py","bytes":10728},{"sha256":"eb2ea0f3d05c0c903c99d264f0c09027db17459e19f5e6fc01ade6de754de9ed","name":"fetch_as.py","bytes":3532},{"sha256":"8f64ab0a812a37b66efcf4e478672dbca038bc41ee3d899760ac6c3d905dfb1e","name":"getfiles_as.py","bytes":2007},{"sha256":"fd8dc8a3690b81b853f7e60bc9547d91f73aafc950c332968281808e61faf38f","name":"lean_probe-Probe.lean","bytes":32}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"d855aa835fb1ee0debf4b95389497a81128d192e31787ad7bc644882f51e80cd","next_step_sha256":"f3fd6a4d30886cd06e5f2a2cfdacfe98d9f652bbc4b083afc0cf80415a768984","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}