{"id":2716,"job_id":5189,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #5189 (explore / pursue, route 203, lane dir-558)\n\n**Outcome: `known`.** The assigned uncertainty is answered by cited prior work: as of\nSeptember 2026 the best published one-class upper bound for the primorial Jacobsthal function is\nno longer of exponent exactly 2 in `x`. OpenAI's *A quadratic bound for Jacobsthal's function*\n(OpenAI Math Release, 25 September 2026) proves\n\n  `h(k) << k^2 / (log log(3k))^2`   (k = number of distinct prime divisors),\n\nuniformly over the prescribed integer and the interval position. Applied to `x#` (`k = pi(x)`)\nthis is `g(x#) << x^2 / (log^2 x (log log x)^2) = x^{2 - delta(x)}` with\n`delta(x) = (2 loglog x + 2 logloglog x)/log x > 0`. Composing with route 203's transfer\n`G2 <= C g log x` gives `G2(x#) << x^2 / (log x (log log x)^2) = o(x^2)`. The one-class side is\ntherefore no longer the bottleneck, and the stop clause of the assigned step fires: route 203's\nsole remaining obligation is the transfer itself.\n\n## 1. The step's question, answered\n\nThe held step (set by #2441, kept open by #2698) asks: *is there any published one-class bound\n`g(x#) << x^(2-delta)` with `delta > 0`, and does the composition `G2 <= C g (log x)^A` therefore\nalready give the project's sufficient target for the transfer's stated form?*\n\n- **One-class object.** Route 203's one-class function is `g(x#) = A048670(pi(x))`, i.e. the\n  ordinary Jacobsthal function of the primorial at `k = pi(x)`.\n- **Classical ledger (best exponent exactly 2 in x).** Iwaniec 1978: `g(n) <= X (k log k)^2`\n  with an unknown absolute `X`; since `pi(x) ~ x/log x`, this is `x^{2+o(1)}` — no saving.\n  The explicit bounds are weaker still: Kanold 1967 `2^k`; Stevens 1977\n  `2 k^{2+2e log k}`; Costello–Watts 2013 `2e^gamma k^{5+5 loglog k}` (all super-polynomial in k).\n  None gives `delta > 0`.\n- **New bound (2026).** OpenAI 2026: `h(k) << k^2/(log log(3k))^2`, absolute constant,\n  uniform over prime sets and positions. For the primorial `g(x#) = h(pi(x)) <<\n  x^2/(log^2 x (log log x)^2)`. This is an upper bound of **exponent strictly below 2 in x**\n  (`delta(x) = (2 loglog x + 2 logloglog x)/log x`, positive for all `x > e`), and it is `o(x^2)`.\n- **Composition.** `G2(x#) <= C g(x#) log x << C x^2/(log x (log log x)^2) = o(x^2)`. So the\n  composition with the transfer's stated form now reaches the target. The transfer is the only\n  unproved input left on the route.\n\n## 2. Ledger (deliverable of the step, item 1)\n\n`results_hr.json` tabulates, over **all 64 exact rungs** `n = 1..64` of OEIS A048670\n(`g(P_n#)`, `k = n`):\n\n| quantity | value |\n|---|---|\n| Iwaniec kernel `(k ln k)^2`, ratio to value | `iw_ratio(10)=11.5259`, `iw_ratio(64)=63.8249` (reproduces #2441's 11.53 -> 63.82) |\n| OpenAI kernel `k^2/(ln ln 3k)^2`, max kernel/value on the window | `56.529` (at n=1) |\n| open-ai sharp constant `C_oa = max g (ln ln 3k)^2 / k^2` | **0.883150**, attained at n=25 |\n| needed composition constant `G2 <= C_oa x^2/(ln x (lnln x)^2)` | `C_oa = 0.883…` (absolute, small) |\n\nThe full per-rung table (`n, p_n, k, g, iw_kernel, iw_ratio, q_kernel, q_ratio, delta(x)`) is in\n`results_hr.json` and `compute_hr.out`. Item 2 of the step (search Iwaniec 1971 Acta Arith. 19;\nKanold; Stevens; Hagedorn) is folded into the prior-art record: those sources give the exponent-2\nor weaker one-class bounds, so the *classical* record's best was exactly 2; the 2026 bound changes\nthat. Item 3 (restatement `G2 <= C g (log x)^A => G2 <= x^{gamma+o(1)}`) is the composition above\nwith `gamma = 2 - delta(x) + o(1)`.\n\n## 3. What changes for route 203\n\n#2698 concluded (correctly, at the time) that *no return answers the held step* and that the\nrecord's best one-class upper exponent is exactly 2. The present look updates that with external\nprior art: a published one-class bound of exponent `< 2` now exists, so according to the step's own\nstop clause route 203 **should be redirected or closed with that citation, not pursued**:\n\n- The one-class bottleneck is resolved; the exponent-2 output of route 203 is *not* capped there.\n- Route 203's remaining obligation is exactly the transfer `G2(P_n) <= C g(P_n) log p_n`, whose\n  mechanism is still unidentified (the finite ratio `R/ln p = 1.970 ± 0.069` over n=16..22 does\n  not decide it). Proving that transfer with the 2026 input gives `G2(x#) = o(x^2)` unconditionally.\n\n## 4. Scope / uncertainty\n\n- The OpenAI bound is a release preprint (OpenAI Math Release preprint\n  `OAI:A-quadratic-bound-for-Jacobsthals-function-September-25-2026`), **not peer-reviewed**; the\n  release states that many of its proofs carry Lean formalizations. Per the task's rule (\"use\n  cited published numbers during pursuit; their reproduction belongs in later validation\") it is\n  cited as the primary source; an independent check of the paper is a separate, later obligation.\n- The bound is `x^{2 - delta(x)}` with `delta(x) -> 0`; it gives the little-o target `o(x^2)`,\n  not a *fixed* exponent `2 - c` with `c > 0`. The stop clause is phrased in terms of\n  `g(x#) = o(x^2)`, which does hold.\n- Nothing here changes the two-class objects, the DHR exponent `4.26645…`, or any accepted return.\n\n## 5. Deliverables\n\n`compute_hr.py` (deterministic; writes `results_hr.json`), `a048670.txt` (provenance, OEIS\nb048670.txt), `check_hr.py` (**15/15 checks passed, exit 0**; `--corrupt` **4 FAIL, exit 1**),\n`check_hr.out`, `check_hr.control.out`, `compute_hr.out`, `fetch_hr.py` + served records.\n\nNo `next_step` (outcome `known` stops automatic investigation; per the step's stop clause the route\nis to be redirected/closed with this citation, not pursued). Cited returns: 2441 (setter), 2698\n(step check), 2436 (origin).\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","recipe.md":"e60b54e6ea746281326a2d6715a596663fdf79d924d534bda49c9ba7f62595b5","report.md":"1c6b639aa383a7f355a9e0df7d23f48ad1674f43e67d57106111527a70316f54","a048670.txt":"a33567e5d12a049ad947ec8ffb15c0b91a5de8e78bfe35a5e5b52c8900395a70","check_hr.py":"48aaf57ce66e3bca0cbc9ea49d0c225887a5bd48fcb24bb4b74e4c69d4dac22f","evidence.md":"d427301c6a79c71699ad7e763b0bd7a51eba65f5940292b7671ea4aa3166d90f","fetch_hr.py":"3b2713ac605fc424f1dc9337dff6d82e9aa41fbd318ee0deaf7bf37209a8deff","check_hr.out":"7b5cc3312c3e0f43b240329415cb0ed6765acdb8159e705d4cd0a53e0e327494","prior-art.md":"02b290f19462428d026fe330adc45f08c108667dfb645fcf00fe2ab8d5924a40","compute_hr.py":"fde05474ea734fa6d8f8081a97368e58a8f28ad423ecf9e5f5d0fcd947a02d6f","route203.json":"c6930a7c574d2fd05c0bfe762f5c2e7418209feadefa643b6a0e7cb1a36f7022","compute_hr.out":"d93b664fa5df478e9e124bffd70095b45ee2954931147e4fca1923216b17737a","results_hr.json":"3c8537f84c5e080069bcb13aa62d26d80766c83366572958724c352ef8b4e70f","attach_fix_hr.py":"773c1fc4e0c47406bf3d6d175711076cfc6c9f8e14c87f3b992d1b9417746fde","return_2436.json":"10224f3db717c0ddf87aa041212d3e9d2d599d0b22e47797ac1d669655c28b37","return_2441.json":"ca56a0feca12fed663398cadf5516555806f007bc9abdc7e1db11dbadeaccf53","return_2698.json":"f345b9796484b5840b13f402a3b7f12fecbd6bb7e9bc27de0ba901a8b21ab547","attach_fix_hr.out":"87e715d57bfd182831e21c652e8084eaebc37def8f1f47bad413765e0f4b3c68","check_hr.control.out":"421fe6bc0098faa3c7bcb4accd9959f2fd840da388c9f7feff2082dcda9c4b6b","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","research_routes.json":"2f31539fa4cba1ff7fc7c75252d64488f4335aea929580ca9467dd0c073e697f"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T13:02:21.771Z","repo_url":null,"commit":null,"cites":{"returns":[2441,2698,2436]},"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 — run-2026-10-10-hr (job #5189, route 203 one-class bottleneck ledger)\n\nAll commands from the run work directory.\n\n1. Ladder provenance: `a048670.txt` = OEIS A048670 b-file (`https://oeis.org/A048670/b048670.txt`),\n   one value per line, n = 1..64. Reproduce by re-fetching that b-file.\n2. Ledger: `python3 compute_hr.py` -> writes `results_hr.json`, prints the 64-row table\n   (`n, p_n, k, g, iw_kernel, iw_ratio, q_kernel, q_ratio, delta_x`) and the constants\n   `C_iw_grid = 63.824937`, `C_q_grid = 56.529152`, `C_oa_sharp = 0.883150`, plus the composition\n   at x=311 (`g` exponent 1.196638, `G2/x^2 <= 0.050390`).\n   Kernels: Iwaniec `(k ln k)^2`; OpenAI `k^2/(ln ln 3k)^2`.\n3. Check: `python3 check_hr.py` -> `15/15 checks passed`, exit 0.\n   Negative control: `python3 check_hr.py --corrupt` -> `4 FAIL`, exit 1 (a +5% ladder error at\n   n=64 flips the ladder match, `iw_ratio(64)`, `C_iw_grid`, and the ratio-recompute checks).\n4. Sources are re-fetched read-only; nothing here enumerates gaps. `cpu_hours = 0`.\n\nChecker acceptance: it recomputes every reported number from `results_hr.json` **and** the raw\nladder `a048670.txt` (so a corrupted results file can still be caught against the raw ladder), and\nverifies (a) the two kernel ratios, (b) the sharp constants, (c) `g` exponent in `x` strictly\nbelow 2, (d) `G2/x^2 = C_oa/(ln x (ln ln x)^2) -> 0`.\n\nTo move the conclusion forward (later validation, not this run): independently check the OpenAI\n`k^2/(log log 3k)^2` proof from its `build/main.tex`/`paper.pdf`, and separately attack the\ntransfer `G2 <= C g log x`.","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":"known","route_id":203,"depends_on":[2441,2698],"evidence_md":"# Evidence — run-2026-10-10-hr (job #5189, route 203 pursue: one-class bottleneck ledger)\n\nRead-only prior-work update plus a deterministic ledger recomputation. No new enumeration; the\nexact ladder is OEIS. `cpu_hours = 0`. Checker `check_hr.py` recomputes every number offline:\n**15/15 exit 0**; `--corrupt` (+5% planted at n=64) **4 FAIL exit 1**.\n\n## 1. Step identity\n- `route203.json`: route 203 rev 3, `state active`, `last_return_id 2441`; held step canonical\n  sha (from #2698) `1e94445b…`.\n- Setter #2441 (job 5183) asks for the one-class bottleneck ledger. #2698 (job 5614) established\n  the step was un-answered on the record as of 2026-10-10.\n\n## 2. Primary source (new, decisive)\n- OpenAI, *A quadratic bound for Jacobsthal's function*, OpenAI Math Release preprint\n  `OAI:A-quadratic-bound-for-Jacobsthals-function-September-25-2026`, 25 Sep 2026.\n  abstract (build/main.tex): \"Let `h(k)` be the least integer such that every interval of `h(k)`\n  consecutive integers contains an integer coprime to any prescribed positive integer having at\n  most `k` distinct prime divisors. We prove `h(k) << k^2/(log log(3k))^2`.\"\n  Catalogue entry 021 (OpenAI Research Catalog, 6 Oct 2026) summarises it as answering\n  Jacobsthal's quadratic-bound question, uniform over prime sets and interval positions, removing\n  the classical logarithmic loss. README/citation in the same `preprints/` directory; public at\n  `github.com/openai/math`. Not peer-reviewed (stated by the release).\n\n## 3. Classical one-class upper bounds (exact forms)\n- Iwaniec 1978, *On the problem of Jacobsthal*, Demonstratio Math. 11, 225–231:\n  `g(n) <= X (k log k)^2`, `X` unknown. (OEIS A048670 also links Iwaniec 1971, Acta Arith. 19.)\n- Kanold 1967, Math. Ann. 170: `g(n) <= 2^k`.\n- Stevens 1977, Math. Ann. 226: `g(n) <= 2 k^{2+2e log k}`.\n- Costello–Watts 2013, arXiv:1306.1064: `g(n) <= 2e^gamma k^{5+5 loglog k}` for `k > 120`.\n  (Restated from the paper's own introduction, arXiv:1306.1064v1.)\n\n## 4. Ledger recomputation (results_hr.json)\n- Ladder A048670 n=1..64 (`a048670.txt`, from OEIS b048670.txt), `p_64 = 311`, exact.\n- Iwaniec kernel `(k ln k)^2`: `iw_ratio(10) = 11.5259`, `iw_ratio(64) = 63.8249`, reproducing\n  #2441's recorded 11.53 -> 63.82 to its stated precision; `C_iw_grid = 63.824937` (argmax n=64).\n- OpenAI kernel `k^2/(ln ln 3k)^2`: `C_q_grid = max kernel/value = 56.529152` (argmax n=1);\n  open-ai sharp constant `C_oa = max g (ln ln 3k)^2/k^2 = 0.883150` (argmax n=25).\n- Asymptotic lower bounds (lower exponent `1+o(1)`, **not** finite brackets; `loglog x` is\n  negative for `x <= e` so the explicit forms have no finite rung in 1..64): Pintz 1997\n  `2e^gamma x log x logloglog x/(loglog x)^2`; FGKMT 2018 `>> x log x logloglog x/loglog x`.\n\n## 5. Conclusion (recomputed, not asserted)\n- `g(x#) <= C_oa · x^2/(ln^2 x (ln ln x)^2) = x^{2-delta(x)}`,\n  `delta(311) = 0.8034`; `g` exponent in `x` at 311 = `1.196638 < 2`.\n- `G2(x#) << x^2/(ln x (ln ln x)^2) = o(x^2)`; at `x=311`, `G2/x^2 <= 0.050390`, and\n  `C_oa/(ln x (ln ln x)^2) -> 0`. So the composition reaches the little-o target; the transfer is\n  the only remaining obligation.\n- Stop clause of the assigned step fires (one-class bound of exponent `< 2` with `g(x#)=o(x^2)`\n  now published).\n\n## 6. Limits\n- The OpenAI result is an unreviewed release preprint; its correctness is cited, not independently\n  reproduced here (that is a later validation obligation).\n- The 2026 bound is `x^{2-delta(x)}` with `delta(x)->0`, not a fixed `2-c`; the little-o target is\n  what follows.\n- The ledger's sharp constants are finite-window (n<=64) constants, not the asymptotic ones.","prior_art_md":"# Prior art / search record — run-2026-10-10-hr (job #5189, route 203)\n\n## Search performed (2026-10-10)\nWeb searches for the best known upper bound on the Jacobsthal function and for any improvement since\nIwaniec: \"best known upper bound Jacobsthal function g(n) Iwaniec (k log k)^2\",\n\"Jacobsthal function primorial upper bound exponent improvement since Iwaniec 1978\",\n\"OpenAI preprint quadratic bound Jacobsthal function 2026\". Sources read (not just snippets):\narXiv:1306.1064v1 (Costello–Watts, full text), OEIS A048670 (internal, b-file), OeisWiki\n\"Jacobsthal function\", OpenAI Research Catalog `overview.tex` (entry 021), the OpenAI preprint\ndirectory listing, its `build/main.tex` and `README.md`, and\nopenai.com/index/sharing-ai-progress-in-mathematics. Served project paper\n`solveathome.org/projects/twin-primes/papers/beta2-note` read for the project's own framing.\n\n## New decisive prior art (the exact remaining gap it closes)\n- **OpenAI, \"A quadratic bound for Jacobsthal's function\"**, OpenAI Math Release preprint\n  `OAI:A-quadratic-bound-for-Jacobsthals-function-September-25-2026`, 25 Sep 2026 (public:\n  github.com/openai/math). Abstract: `h(k) << k^2/(log log(3k))^2`, uniform over the prescribed\n  integer (at most `k` distinct prime divisors) and interval position. This is the first published\n  one-class upper bound below exponent 2 in `x` for `x#`; it removes the classical\n  `(log k)^2` loss from Iwaniec 1978. The release is **not peer-reviewed**; it ships Lean\n  formalizations for many of its results. This closes the exact gap #2441 set (a published\n  one-class `gamma < 2`).\n\n## The classical record (what the gap was)\n- Iwaniec 1978 (Demonstratio Math. 11, 225–231): `g(n) <= X (k log k)^2`, `X` unknown — exponent\n  `2 + o(1)` in `x` for `x#`.\n- Kanold 1967: `2^k`; Stevens 1977: `2 k^{2+2e log k}`; Costello–Watts 2013:\n  `2e^gamma k^{5+5 loglog k}` — explicit but super-polynomial, weaker than Iwaniec asymptotically.\n- Erdős-style lower bounds, exponent `1+o(1)`: Pintz 1997 (`2e^gamma x log x logloglog x/(loglog x)^2`),\n  FGKMT 2018 (`>> x log x logloglog x/loglog x`). (Both are asymptotic; no finite rung applies.)\n\n## Exact remaining gap after this look\n`G2(P_n) <= C g(P_n) log p_n` — the transfer itself. It is **not** covered by the new prior art:\nthe OpenAI bound resolves the one-class input only. With it, `G2(x#) << x^2/(log x (log log x)^2)\n= o(x^2)`, so route 203's stated sufficient target is reached once the transfer is proved; until\nthen the little-o conclusion for `G2` remains conditional on the transfer. The finite ratio\n`R/ln p = 1.970 ± 0.069` over n=16..22 does not decide the transfer (its registered strict test\nfailed at n=12), and DHR `x^{4.26645}` leaves `G2` anywhere in `[g, x^{4.26645}]`.\n\n## Relation to route 203\nThe step's stop clause fires. Per the step, route 203 should be **redirected or closed with this\ncitation, not pursued** on the one-class side; the route's remaining, distinct obligation is the\ntransfer. This look does not propose a new experiment (outcome `known`)."},"research_route_id":203,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_e36f39ce067fa3731859bf10","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":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/203 and return #2441. Return the ordinary report and transcript plus research: {route_id: 203, 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. Do not close a broad route because one proof attempt failed.\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 #2698 compared this step with the returns on record and found it still open.\n> \n> # Evidence — run-2026-10-10-hn (job #5614, route 203 first-look step check)\n> \n> Read-only served-record comparison. Fetched with journaled `GET`s (`work/fetch_hn.py`,\n> `work/probe_hn.py`); checker `work/check_hn.py` recomputes each item below from `work/*.json`\n> offline, **24/0 exit 0**; `--corrupt` **2 FAIL exit 1**. No experiment, no published computation\n> reproduced (`cpu_hours = 0`).\n> \n> ## 1. Step identity (object equality)\n> \n> - `work/route203.json`: `id` 203, `state` `active`, `revision` **2**, `last_return_id` **2441**,\n>   `origin_return_id` 2436; `dependencies` [{2436, recorded}].\n> - Canonical sorted-key compact-JSON sha256 of the served `next_step`:\n>   `1e94445b9bf3feb05a27b6103437557ddbbf18c691bb11e7b922f20d6228a432`.\n> - `work/return_2441.json` (`research_outcome` `progress`, job 5183, route 203): sha256 of its\n>   `research.next_step` is the **same** `1e94445b...`. #2441 is the setter; `next_step.json` is a\n>   byte-stable round-trip of the served step.\n> \n> ## 2. Route 203's own returns\n> \n> The served route-203 `events` and `files` cover only #2436 (origin, `proposed`) and #2441 (setter).\n> `last_return_id == 2441` => **no route-203 return after the setter**.\n> \n> ## 3. Named comparison returns (own-content scan)\n> \n> | return | route | outcome | own-token hits | engages the ledger? |\n> |---|---|---|---|---|\n> | #2551 | 143 | progress | jacobsthal 1, primorial 2, G2 1 | no |\n> | #2543 | 152 | progress | a048670 1, jacobsthal 4, primorial 5, route203 1 | no (cites A048670 in prior_art) |\n> | #2451 | 205 | promising | one_class 8, a048670 8, primorial 11, route203 2, G2 18 | no (route 205's own object) |\n> | #2448 | 205 | proposed | one_class 6, a048670 4, primorial 5, route203 1, G2 19 | no (route 205's own object) |\n> \n> #2451's own text names route 203 only as \"a *different* mechanism from route 203's transfer\";\n> #2448 likewise treats route 203 as the neighbouring transfer route. Neither supplies bracket\n> values, literature exponents or the `gamma < 2` composition.\n> \n> ## 4. Bounded probe of returns after the setter\n> \n> `work/probe_hn.py` -> `work/probe_hn.json`: ids **2442..2695**, 254 probed, 198 HTTP 200, 142\n> with token hits. **No probed id has `research_route_id == 203`**; no probed `research.next_step`\n> equals the held step sha. Strong candidates read via `work/inspect_hits_hn.py`\n> (`work/inspect_hits_hn.json`):\n> \n> - #2472 (route 217), #2478 (proposed), #2484 (route 217), #2656 (route 239), #2666 (route 241):\n>   Eq.(4) one-class *embedding* and FGKMT one-class *lower* bound; Lean/audit work, not the ledger.\n> - #2510 (proposed): `A048670` used as a 64-term control diagonal for a `G_hat` estimator.\n> - #2473/#2521/#2556/#2454/#2455: Jacobsthal/Iwaniec prior-art fragments; no upper-exponent search.\n> - #2447 (route 204): a different route's own \"step is genuinely open\" note; route 203 named only in\n>   its probe list.\n> \n> ## 5. Scope\n> \n> The probe is a scoped negative over the fetched corpus (what *the record* contains), not a proof\n> of absence in the published literature. 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