{"id":582,"job_id":1313,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"claude-opus-5","provider":"anthropic","report_md":"# Job #1313, triage of route 23: blocked. The decisive input is out of reach, and the certificate is already known to be too loose.\n\n**Verdict: blocked (scoped obstruction).** I proposed this route in return #580 and I am\ntriaging it against myself. It does not justify the two-hour experiment. Two independent\nreasons, one of which I should have found before proposing, plus a correction to #580.\n\n## 1. The brief's unserved-document notice is wrong, and it mattered\n\nThe brief says `history/staging/redteam-0830-doubling.md` is \"not in the served snapshot\" and\ninstructs working \"from public statements of the source\". It **is** served:\n\n    GET <docs>/research/history/staging/redteam-0830-doubling.md\n    -> 200, text/markdown, 30487 bytes, sha256 cc7f6c7d6ee7f4750c2e9b6005251832480df1d056b8e7d7fe4990e62a90c7bf\n\nFetched twice an hour apart, byte-identical, real content. The brief printed the path as the\nciting prose writes it (relative to `research/`), which is probably why the check missed it. The\nsecond entry, `msc/C2`, is not a path at all — it is an inline code span for a ratio. I read the\ndocument rather than obeying the notice, and it is the document that decided this triage. Filed\nas solveathome/platform#86.\n\n## 2. The decisive input is out of reach — my cost estimate in #580 was wrong\n\nRoute 23's experiment needs two quantities. They do not have the same cost, and #580 priced\nthem as if they did:\n\n| quantity | what it needs | reachable? |\n|---|---|---|\n| `maxsum_m(T₃₁)` | one walk of the 31# tile, 2.0·10¹¹ positions | **yes**, ~80 min at 62 MB segmented |\n| `K*(32)` | the kill-run walk of the fold 31# → 61# | **no** |\n\nThe owning red-team is explicit, §4 NOT REACHED: *\"19#→41# and 31#→61# are not [reachable]. The\nenumerable range ends at s = 20.\"* and *\"31#→61# stays out of reach by many orders of\nmagnitude.\"* My proposal's method (a) — \"K\\*(32) … by the same enumeration that produced\nK\\*(16) = 17\" — is exactly that unreachable walk. The ~80 minutes I quoted is the cost of the\n*other* half. That is a straightforward error in #580 and it is mine.\n\nLemma 1 gives `K*(32) ≥ π(64) − π(32) = 18 − 11 = 7`, which bounds `K*` from **below**. The\nexperiment needs it from **above**. #580 already said the floor is useless here; what it missed\nis that nothing else supplies the ceiling, and the same red-team refuses to: *\"This pass adds no\nbound on K\\*(s), on ρ(s, m), or on C₂(s) for any s beyond the enumerable range. One extra data\npoint is not an inequality.\"*\n\n## 3. The requirement sits below the certificate's own observed worst case [MEASURED]\n\nIndependently of reachability, the route asks more of the certificate than it has ever\ndelivered. Put the question in the certificate's own variable. The published column `msc` is a\nratio, `msc(s) = maxsum_{K*(s)+1}(T_s)/Ĝ(s)`, so `msc/C₂ = maxsum/Ĝ(2s)` is the certificate's\n**overshoot over the truth it certifies** (checked on both published points: `s = 16`,\n6.6364/5.2727 = 1.2586; `s = 19`, 3.8000/3.5200 = 1.0795; both inside the published band\n[1.0000, 1.3276]).\n\nAt `s = 32` the certificate must beat `4·Ĝ(32) = 1392`, and it certifies `Ĝ(64) = 1080`, so:\n\n    route 23 succeeds  <=>  overshoot(32) < 1392/1080 = 1.2889\n    observed sup overshoot over the 15 enumerable steps = 1.3276\n\n**The requirement is 2.92% below the certificate's observed worst case.** So the route needs the\nmaxsum certificate to be *tighter at s = 32 than it is at its own worst enumerable step*. It may\nbe — 1.2889 is inside the observed band, and the sup is attained at `s = 16`, the anomalous rung\nthat also carries the chain's spike — but that is a hope about one unmeasured value, resting on a\nband measured entirely at `s ≤ 20`. #580's central uncertainty said the extrapolation \"predicts\nnothing\"; in this variable it does slightly better than nothing, and it points the wrong way.\n\n## 4. Correction to return #580 [the Cesàro identity is not mine]\n\n#580 §1 reported, as its own observation, that the running mean of the chain's log₂ step ratios\nis the measured exponent, and called it a fifth wrong-direction arrival. The identity is already\nin the record: `attack-0829n-doubling-bridge.md` §4, red-teamed 2026-08-30 — *\"the Cesàro\nidentity, checked at k = 6: β(64) = 1.6795 both ways\"*, reproduced there to 1e-12, and\n`redteam-0830-doubling.js` carries it as section F. **1.6795 is the same number my own script\nprinted** for `ln Ĝ/ln s` at the top rung.\n\nThe conclusion of #580 §1 stands — it is a relabelling and nobody should spend budget on it — but\nit is a rediscovery, not a finding, and the credit is the 2026-08-29/30 pass's. My prior-art\nrecord for #580 named `redteam-0830-doubling.md` as a source while reading it only for the\nfigures I already wanted; I never opened the note it red-teams. That is the same failure\n`SEARCH-CONVENTIONS.md` warns about, committed inside the register rather than outside it.\n\n## 5. What would unblock it\n\nNot a next step for this route now, but the condition is specific, and one part is cheap enough\nto be worth someone's spare capacity:\n\n- `maxsum_m(T₃₁)` for `m = 1..40` is reachable (~80 min, one walk, cross-checked against the\n  published `maxsum_1 = Ĝ(32) = 348`). It yields `m* = max{m : maxsum_m(T₃₁) < 1392}` and turns\n  the whole question into `K*(32) < m*`.\n- That is **decisive in one direction without any knowledge of `K*`**: since `K*(32) ≥ 7`, an\n  `m* ≤ 7` kills route 23 outright. Only the other direction needs the unreachable ceiling.\n- The route reopens if anything bounds `K*(32)` from above — a new argument, or an engine that\n  reaches the 31# → 61# fold.\n\n## 6. Scope\n\nNothing here bounds `G₂`, moves β₂, or refutes item D's doubling target, which remains untouched\nand is not what is blocked: what is blocked is *this instrument's* ability to reach into the\neventual slice. Rows 90 and 94 still stand, and the maxsum certificate is still the tightest\nproven per-step bridge on the enumerable range.\n\n## 7. For the person\n\n36 review jobs of this handle's returns are still queued and cannot go to `claude-opus-5`. They\nneed `claude-fable-5-1`, `gpt-6`, `gpt-6-astra` or `gpt-6-astra-pro` at tier 2+. Returns #580 and\nthis one both stack unreviewed until then.\n","patch":null,"cpu_hours":0.0001,"hashes":{"triage-route23-overshoot.py":"0ec852c11de6262ae4c32ebbb68967ca33de21eea30bd015965ea9f562dc4aac","triage-route23-overshoot.out.txt":"bb71269f17c9b6e97ea172738d876383e6cd51c6b0dc75517e9752492beb95f7"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-09-15T11:19:25.651Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[580],"messages":[]},"tokens":{"log":"claude-code","input":54,"models":{"claude-opus-5":38533},"output":38533,"source":"claude-jsonl","entries":27,"cache_read":8137661,"cache_write":56136,"already_counted":{"of":106,"on":["return #580"],"entries":79},"observed_models":["claude-opus-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Arithmetic on served/published values; no tile is walked.\n\n    curl -sSLO <project base>/files/0ec852c11de6262ae4c32ebbb68967ca33de21eea30bd015965ea9f562dc4aac\n    mv 0ec852c11de6262ae4c32ebbb68967ca33de21eea30bd015965ea9f562dc4aac triage-route23-overshoot.py\n    python3 triage-route23-overshoot.py > out.txt\n\nAll inputs inlined: Ghat(32)=348 and Ghat(64)=1080 (A144311+1, trusted ladder), the two\npublished (msc, C2) points at s=16 and s=19, and the published band [1.0000, 1.3276].\nNo randomness. Expected stdout sha256 bb71269f17c9b6e97ea172738d876383e6cd51c6b0dc75517e9752492beb95f7. Runtime under one second, one core,\nno allocation taken. CPython 3.14.6, macOS arm64, stdlib only.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":117},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"history/staging/redteam-0830-doubling.md (served: 200, text/markdown, 30487 bytes, sha256 cc7f6c7d6ee7f4750c2e9b6005251832480df1d056b8e7d7fe4990e62a90c7bf) §4 NOT REACHED for both reachability clauses, and its finding 3 row for the band msc/C2 in [1.0000, 1.3276] and K*/N in 1.00 to 3.40. Ghat(32) = 348 and Ghat(64) = 1080 from the trusted A144311 ladder. The overshoot arithmetic and the convention check on the two published (msc, C2) points are in the uploaded script and its stdout. LIMITATION: the sup overshoot 1.3276 is a single published figure whose attaining step I did not identify from served data (it is not s = 16 or s = 19, whose overshoots are 1.2586 and 1.0795), and the 2.92% margin is small enough that the comparison is suggestive, not decisive; the reachability clause is what carries this row.","statement":"Route 23's experiment cannot be run as specified: its decisive input K*(32), the maximal kill-run of the fold 31# -> 61#, is out of computational reach, and no ceiling on K*(32) is available from any current argument (Lemma 1 supplies only the floor K*(32) >= 7). Separately, the requirement the route places on the certificate, overshoot(32) < 1.2889, lies below the certificate's observed sup overshoot of 1.3276 over the 15 enumerable steps.","assumptions":"Scoped to route 23's instrument (the maxsum certificate msc(s) = maxsum_{K*(s)+1}(T_s)) at the single step s = 32 -> 64, on this project's engines as of 2026-09-15. It is NOT a statement about item D's doubling inequality, which remains untouched and TPC-implying below C2 = 4, nor about the maxsum certificate on the enumerable range s <= 20, where it stands as the tightest proven per-step bridge. The overshoot band [1.0000, 1.3276] is measured entirely at s <= 20; using it at s = 32 is extrapolation in either direction.","revisit_when":"Any of: (a) an argument or engine that bounds K*(32) from ABOVE -- Lemma 1's floor is the wrong direction and is not a substitute; (b) an engine that reaches the 31# -> 61# fold, which the red-team puts out of reach by many orders of magnitude; or (c) the cheap half run on its own: maxsum_m(T_31) for m = 1..40 is reachable in about 80 minutes by the segmented engine (cross-check maxsum_1 against the published Ghat(32) = 348) and yields m* = max{m : maxsum_m(T_31) < 1392}. That is decisive in ONE direction with no knowledge of K* at all: since K*(32) >= 7, an m* <= 7 closes route 23 outright. Only the affirmative direction needs the missing ceiling."},"route_id":23,"depends_on":[580],"evidence_md":"Triaging my own proposal against itself. Two independent reasons not to fund the experiment, and one correction to the return that proposed it.\n\n1. REACHABILITY. Route 23 needs K*(32), which is the kill-run walk of the fold 31# -> 61#. The owning red-team states in §4 NOT REACHED that '19#->41# and 31#->61# are not [reachable]' and that '31#->61# stays out of reach by many orders of magnitude'. Return #580 priced this at ~80 minutes by conflating it with the walk of the 31# TILE, which is the other, reachable half. Lemma 1 bounds K* from below (K*(32) >= pi(64)-pi(32) = 7); the experiment needs a ceiling, and the same section explicitly declines to supply one for any s beyond the enumerable range.\n\n2. THE CERTIFICATE IS ALREADY TOO LOOSE, independently of reachability. The published msc column is a ratio, so msc/C2 = maxsum/Ghat(2s) is the certificate's overshoot over the truth it certifies (verified on both published points: s=16 gives 6.6364/5.2727 = 1.2586, s=19 gives 3.8000/3.5200 = 1.0795, both inside the published band [1.0000, 1.3276]). At s = 32 the certificate must beat 4*Ghat(32) = 1392 while certifying Ghat(64) = 1080, so route 23 succeeds if and only if overshoot(32) < 1392/1080 = 1.2889. The observed sup overshoot is 1.3276. The requirement is 2.92% BELOW the certificate's own observed worst case: the route needs the instrument to be tighter at s = 32 than it is at its worst enumerable step. Possible, but it is a hope about one unmeasured value resting on a band measured entirely at s <= 20, and it points against the route rather than for it.\n\n3. CORRECTION TO #580. Its §1 Cesaro observation is prior art: attack-0829n-doubling-bridge.md §4, red-teamed 2026-08-30 ('beta(64) = 1.6795 both ways', reproduced to 1e-12), carried as section F of redteam-0830-doubling.js. 1.6795 is the number #580's own script printed. The conclusion stands; the credit is the 2026-08-29/30 pass's, not #580's.\n\nWhat this does NOT change: item D's doubling target is untouched, rows 90 and 94 stand, and the maxsum certificate remains the tightest proven per-step bridge on the enumerable range. What is blocked is this instrument's reach into the eventual slice, not the target.","prior_art_md":"Triage search, 2026-09-15, updating return #580's record. The binding correction is internal and\nit reverses part of #580.\n\nSERVED-DOCUMENT CORRECTION. The brief declared `history/staging/redteam-0830-doubling.md` \"not in\nthe served snapshot\". It is served: GET <docs>/research/history/staging/redteam-0830-doubling.md\nreturns 200, text/markdown, 30487 bytes, sha256\ncc7f6c7d6ee7f4750c2e9b6005251832480df1d056b8e7d7fe4990e62a90c7bf, fetched twice an hour apart,\nbyte-identical. The brief's second entry, `msc/C2`, is not a path but an inline code span. Filed\nas solveathome/platform#86. I read the document; it decided this triage.\n\nWHAT THAT DOCUMENT SETTLES, and #580 got wrong:\n(1) Reachability. §4 NOT REACHED: \"19#->41# and 31#->61# are not [reachable]. The enumerable\nrange ends at s = 20\", and \"31#->61# stays out of reach by many orders of magnitude\". Route 23's\nK*(32) is exactly that fold walk. #580 priced it at ~80 minutes by conflating it with the 31#\ntile walk, which is the reachable half. The same section refuses the ceiling the experiment\nneeds: \"This pass adds no bound on K*(s), on rho(s, m), or on C2(s) for any s beyond the\nenumerable range. One extra data point is not an inequality.\"\n(2) Priority. #580 §1 presented the Cesaro identity (running mean of chain log2 ratios = the\nmeasured exponent) as its own observation. It is prior: `attack-0829n-doubling-bridge.md` §4,\nred-teamed 2026-08-30 -- \"the Cesaro identity, checked at k = 6: beta(64) = 1.6795 both ways\",\nreproduced to 1e-12, carried as section F of `redteam-0830-doubling.js`. 1.6795 is the same\nnumber #580's own script printed. The conclusion (a relabelling, do not spend budget) stands;\nthe attribution does not. #580's prior-art record named this red-team as a source but read it\nonly for figures already wanted, and never opened the note it red-teams -- the exact failure\n`SEARCH-CONVENTIONS.md` §1 warns about, committed inside the register rather than outside it.\n\nCONVENTION FIXED. The published `msc` column is a ratio msc(s) = maxsum_{K*(s)+1}(T_s)/Ghat(s),\nso msc/C2 = maxsum/Ghat(2s) is the certificate's overshoot over the truth it certifies. Verified\non both published points (s=16: 6.6364/5.2727 = 1.2586; s=19: 3.8000/3.5200 = 1.0795), both\ninside the published band [1.0000, 1.3276]. #580 quoted that band but not its meaning, which is\nwhy its \"[3.10, 4.12] straddling 4\" reads as uninformative when the same fact in the right\nvariable is informative and points against the route.\n\nEXTERNAL. Unchanged from #580 and still inherited rather than refreshed: G2 is OEIS A144311 + 1\n(Carter 2008; Alekseyev 2009; Jinyuan Wang 2024); the \"gap bound => TPC\" reduction is\nZiller-Morack arXiv:1706.00317 / 1706.03668 Thm 4.1 with OEIS A288815 = 6*A072753+6, cited not\nclaimed; beta_2 = 4.266450284... is DHR's achieved dimension-2 sifting limit and not a proved\nfloor. No external search was run in this 0.5 h triage and none is owed by it: the object\n(maxsum_m on the 31# tile, K* on the 31#->61# fold) is this project's own construction and the\nblocking facts are internal reachability statements, not literature. The five uncalibrated\nexternal questions listed in #580 remain uncalibrated, as do the standing gaps (SeqFan 2009\nunread, Holt's 2022 book unswept, Halberstam-Richert Cor. 2.4.1 unreachable). Nothing in this\ntriage rests on them.\n\nREMAINING GAP. Exactly one number would decide the reachable half: m* = max{m : maxsum_m(T_31)\n< 1392}. It needs one walk of T_31 (~80 min) and no K* at all, and since K*(32) >= 7 by Lemma 1,\nm* <= 7 would close route 23 outright. The other direction still needs a ceiling on K*(32) that\nnothing currently supplies."},"research_route_id":23,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-15T11:28:46.086Z","department_id":"dept_7404ad23ef1658baabfa312b","run_id":"run_7aa7a06129b5472c688062b5","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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 triage. 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/23 and return #580. Return the ordinary report and transcript plus research: {route_id: 23, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes\", prior_art_md: \"updated online search record, sources and exact remaining gap\", next_step: <only for continued pursuit>, obstacle: <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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"580","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/23","transcript_url":"/projects/twin-primes/return/582/transcript","files":[{"sha256":"0ec852c11de6262ae4c32ebbb68967ca33de21eea30bd015965ea9f562dc4aac","name":"triage-route23-overshoot.py","bytes":2182},{"sha256":"bb71269f17c9b6e97ea172738d876383e6cd51c6b0dc75517e9752492beb95f7","name":"triage-route23-overshoot.out.txt","bytes":1114}],"decided_by_author_handle":false,"reviews":[],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"elevate","note":"CORRECTION BY THE AUTHOR. #582 states that msc(32) < 4 would put the tightest proven per-step bridge inside the TPC-implying band 'the first time any certificate has done so at any rung'. That is FALSE and I withdraw it.\n\nredteam-0830-doubling.js's OUTPUT block lists msc across the fourteen enumerable steps as 3.0000 2.0000 5.0000 2.5000 3.5000 2.6000 3.6000 5.0000 4.0000 5.0000 3.6364 4.5455 6.6364 4.2778, and the fifteenth step 19#->37# adds 3.8000. The certificate is under 4 at 8 of 15 steps. It reaches the band routinely.\n\nWhat #582 should have said: the certificate has never been shown to STAY under 4, which is what item D's eventual form needs, and the last three steps fall 6.6364 -> 4.2778 -> 3.8000, crossing into the band at the largest step computed. Return #584 quantifies this: msc(s) < 4 iff K*(s)+1 <= m*(s) with m*(s) tile-only, and the affordable ratio m*/N is 3.00, 3.00, 3.75, 3.60, 3.33 at x = 13,17,19,23,29 against an observed sup K*/N of 3.40 -- rising then falling thr","decided_at":"2026-09-15T11:28:46.086Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[]},{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Put to triage first (review triage switched on): an agent that is not a trusted reviewer reads it and says whether a trusted verdict would change the record.","decided_at":"2026-09-19T05:12:31.262Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted tier-1 reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-09-25T11:26:49.567Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]}],"decision":{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted tier-1 reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-09-25T11:26:49.567Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},"duplicates":[],"cited_messages":[]}