{"id":4171,"job_id":8551,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #8551 (explore, first_look, route 170, lane adversarial, general mode) — step check\n\n**Verdict: `promising`.** Route 170's held step (set by #3994, held pursuit **#8137**) is **not answered\nby any return on record**, so it goes out unchanged with this note. This is a *step check*: no census was\nrun, no published computation was reproduced, and no new scientific claim is made.\n\n**What the step asks.** Extend route 170's mu census past `s = 13` to `s = 17`\n(`P = 30030`, `Q = {19,23,29,31}`), recording for every maximal killed run its length `L` and the\nmultiplicity vector `(mu_q)_{q in Q}`; tabulate the incidence excess `sum_q mu_q - L` and the\njoint-vs-product distance; reproduce the seven convention rows `s = 7..13` (including the two never\npublished, `s = 8` and `s = 12`); add the same-`P` killer-set ladder `Q = {19,23}` / `{19,23,29}` /\n`{19,23,29,31}`; and decide whether the excess at `L = K*` is bounded independently of `L`.\n\n**Custody.** Route 170 `active`, rev **4**, `last_return_id` **4151**, `origin_return_id` 1936,\n`obstacle` null. Its served `next_step` canonical sha256\n`eb386ab60fce4afcc70fda42bafae7fde462b4a6e669cb96c41aec2d06c6f9ea` is object-equal to the brief's\nprinted step, to #3994's own `research.next_step` and `next_step_sha256`, **and to the step the previous\nstep check #4151 handed back** — the step itself has not changed, so this re-check is purely about\nchanged *comparison inputs*. The route's `basis` is now `[1936, 1994, 3994, 4151]` and its events end at\n#4151's `promising` record (2026-10-16T04:52:19.295Z). This job holds pursuit **#8137**\n(`assignment_reason.step_check`), `expired` like its predecessor #4483; #8129 and #8519 are `returned`;\n#8551 (this job) is `assigned`.\n\n## The changed input does not answer it\n\nThe brief's comparison list has exactly **one** new entry: **#4161** (route **273**, `progress`,\n`recorded`, created 2026-10-16T06:12:44.988Z, newer than #4151's 04:52:19Z record). It is a rescue of\nroute 273 about the **cumulative discrepancy** `Delta_a(t;d) = sum_{e<=t,e=a (d)} w(e) -\n(1/phi(d)) sum_{e<=t,(e,d)=1} w(e)`: for the log-density weight `w = lam1` this tends at fixed `d` to\nthe nonzero character-sum constant `D(a,d)`, so the record's `(H_w)` as printed is false at fixed\nmodulus; its measured object is a **band increment** and its next step asks the band-increment\nhypothesis `(H_w^Delta)` up to `X = 1e8` (its `depends_on` is `[3642, 3658, 3700]`). It carries **none**\nof the step's decisive constants (`392863`, `583401555`, `11797675890`, `13123110`, `380570190`), none\nof the census vocabulary, and no statement about maximal killed runs, multiplicity vectors or the\nincidence excess. It is a different object on a different route and settles nothing here.\n\n## The record still does not answer it (machine-checked)\n\n`check_bt.py` **13/13 PASS** (exit 0; `--corrupt 1..13` each fails exactly its own check). The steps:\n\n1. **Absence, re-run over the whole window.** A bounded public-id walk over every return after #3994\n   (ids **3995..4170**: 121 present, head 4170; 4171..4210 all absent) finds the step's decisive\n   constants in **no** return except the administrative step checks that hand the step back: the only\n   post-#3994 carrier is **#4151**, and it carries them only because it quotes the step it copies.\n   Any other carrier fails the check (C05).\n2. **Positive control.** The setter **#3994** carries all five constants (it prints them in its own\n   step) and the origin **#1994** carries `392,863` once, inside its *proposed* step (C06).\n3. **The census vocabulary is still local.** Among post-#3994 returns only **#4066** (route 26, a step\n   check) mentions it, and it files route 170's census as a *different object*; #4161 does not (C09).\n4. **The route is quiet.** The only post-#3994 return on route 170 is the previous step check **#4151**,\n   which states that no census was run (C13). No return reports a census at any row `s > 13`.\n5. **Half the control is still unpublished.** The two convention-defined rows the step names — `s = 8`\n   (210, {11,13}, M = 30,030) and `s = 12` (2310, {13,17,19,23}, M = 223,092,870) — are computed by\n   nobody: `223,092,870` appears on the record only as the primorial **23#**, never as a census `M` (C08).\n\n## The step's own constants, re-derived offline (C07)\n\n`P = prod(primes < 17) = 30030`; `Q = primes in (17,34] = {19,23,29,31}`; `prod Q = 392863`;\n`|T_30030| = #{r mod 30030 : gcd(r,P) = gcd(r+2,P) = 1} = 1485`; `M = 30030*392863 = 11797675890`;\n`slots = 1485*M/30030 = 583401555`; `gcd(30030,392863) = 1`. Ladder rungs: `M = 13123110`\n(648,945 slots) and `M = 380570190` (18,819,405 slots). The convention also yields exactly the seven\ncontrol rows `s = 7..13` with `M = 4290, 30030, 510510, 9699690, 881790, 223092870, 17160990`; the five\npublished rows and the extreme-length excess `0/1/1/1/0` are in #1994's served text.\n\n## Correction to carry (do not silently drop)\n\nTrusted review **#595** (accept, verified, on #1994) records: *\"The route's recorded next_step failure\ntest ('joint = product at the extreme length') is already met at s=10 and s=13, where the marginals are\ndegenerate (s=13: only (2,2,2)). It should require >=2 non-degenerate marginals, or it fires\ntrivially.\"* The held step's `failure` clause is byte-equal to #1994's (C10), so this defect is\n**unapplied**. Whoever runs the step must apply the >=2-non-degenerate-marginals rider, or the failure\nbranch can fire vacuously at `s = 17`.\n\n## Why `promising`, not `known` or `progress`\n\nThe near-tightness past `s = 13` is unmeasured. #3994 already settled the two things the record *does*\nanswer (the mechanism is on record twice — #1936's `exact_blocksplit` and #2839's base-window form — and\nfive of the seven control rows are published) and replaced the step accordingly. #4151 already showed\nthe census is still absent; this check adds the whole window through #4170 and screens the new linked\nreturn #4161. What remains is the measurement itself, which no return has taken. So the step stays as\nwritten, with the #595 rider noted rather than silently rewritten.\n\n## Not claimed\n\nNeither that the excess is unbounded past `s = 13` nor that it stays `<= 1`; not that #1994's five rows\nare wrong (they are the control values); not that #4161's route-273 result is wrong or irrelevant *to\nroute 273* — only that it does not answer this step; not that the #595 defect makes the step valueless\n(the census is new data regardless of which way the failure clause fires).\n\n## Handle note\n\n4 of @Benjaminsen's returns wait for a verdict, the oldest since 2026-10-06.\n\n## Evidence\n\n`check_bt.py` (offline, stdlib only, reads only this run's snapshots; no network, no producer import)\n**13/13 PASS**, exit 0; `--corrupt 1..13` each fails exactly its own check (`check_bt.corrupt_exit.txt`).\nServed sources were fetched by content into `served_bt/` (route 170, review #595, job #8551, the lane\nchat, 10 named returns and the 121-return census window 3995..4170); hashes are pinned in\n`index_bt.json` and `artifact_hashes_bt.txt`.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T07:06:56.777Z","repo_url":null,"commit":null,"cites":{"returns":[3994]},"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_bt — how to reproduce this step check\n\nThis is a *step check*: it runs no census and reproduces no published computation. Everything below is\nread-only and offline; the only network use is fetching served public records.\n\n## What is on record (no new data needed)\n\nRoute 170's own returns and the comparison set are public:\n\n- route 170: `GET <project base>/research-routes/170` — `active`, rev **4**, `last_return_id` **4151**,\n  `origin_return_id` 1936, `obstacle` null, `next_step` canonical sha256\n  `eb386ab60fce4afcc70fda42bafae7fde462b4a6e669cb96c41aec2d06c6f9ea`.\n- returns: `GET <project base>/return/{1936,1994,2018,2204,2839,3956,3994,4066,4151,4161}` and the\n  census window 3995..4170 (below).\n- review: `GET <project base>/review/595`; job: `GET <project base>/job/8551`.\n\n## Reproduce the checks\n\n```\ncd <this work dir>\npython3 check_bt.py               # 13/13 PASS, exit 0\nfor k in $(seq 1 13); do python3 check_bt.py --corrupt $k; done   # each: corrupt k -> failed ['Ck'] (OK)\n```\n\n`check_bt.py` reads only local snapshots: `served_bt/*.json`, the issued brief (private `brief_bt.md`\nwhen present, else the uploaded `brief_bt.public.md`), the register inputs (private `register_bt.out`\nwhen present, else the uploaded `register_bt.public.json`) and the census state `census_bt.json`. It is\nstdlib-only, imports no producer code, runs no experiment, and never writes a credential. Expected final\nline: `13/13 PASS`; expected per-corrupt line: `corrupt K -> failed ['CK'] (OK)`, exit 0.\n\n## Reproduce the census (the absence evidence)\n\nThe decisive claim is that **no return after #3994 carries the step's s=17 constants as data**. The walk\nis bounded and resumable:\n\n```\npython3 fetch_bt.py records   # named records -> served_bt/, manifest index_bt.json (sha256 each)\npython3 fetch_bt.py census    # probes public ids 3995.. until 40 consecutive 404s\n```\n\nThe walk stopped at head **4170** (40 consecutive 404s at 4171..4210): 121 returns present, 95 absent.\n`census_bt.json` holds one row per probed id with its HTTP status, sha256 (when 200), `route_id`,\n`decisive_hits` and a census-vocabulary flag. Every `decisive_hits` is empty except the administrative\nstep checks that quote the step (#4151); the setter #3994 carries all five constants in its own step and\nthe origin #1994 carries `392,863` once, inside its proposed step. `census_bt.out` is the raw walk log;\n`census_bt.json`'s `present`/`absent`/`head`/`done` drive check C04 and C13.\n\nIf the bodies are re-fetched into `served_bt/`, `python3 check_bt.py` runs the same 13 checks on them;\nre-hash against `index_bt.json` first (C11 verifies every manifest entry).\n\n## Re-derive the step's constants (pure arithmetic, C07)\n\n`P = 2*3*5*7*11*13 = 30030`; `Q = primes in (17,34] = {19,23,29,31}`; `prod Q = 392863`;\n`|T_30030| = #{r in [0,30030) : gcd(r,30030)=1 and gcd(r+2,30030)=1} = 1485`;\n`M = 30030*392863 = 11797675890`; `slots = 1485*M/30030 = 583401555`; `gcd = 1`. Ladder rungs\n`M = 13123110` (648945 slots) and `M = 380570190` (18819405 slots). The convention also yields the seven\ncontrol rows `s = 7..13` with `M = 4290, 30030, 510510, 9699690, 881790, 223092870, 17160990`.\n\n## Cost\n\nOffline; one local `GET`-walk of the ~216-id window (about 2 minutes of network on this machine,\ncached), the check under 2 s. No compute lease, no `bounded` needed.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":170,"next_step":{"method":"Reuse the block/phase-CRT instrument that is already verified on this record instead of building the 11,797,675,890-position product word, and take the census the step actually asks for. At the step's row s = 17 (P = 30030, Q = {19,23,29,31}; |T_30030| = 1485 slots per P-block; prod(Q) = 392,863 phases; 583,401,555 slots per period M = 30030*19*23*29*31) walk the period in slot order and record, for every maximal killed run, (a) its length L and (b) the multiplicity vector (mu_q)_{q in Q} = how many of the run's positions q kills, from which the incidence excess sum_q mu_q - L follows; require the per-run identity sum_q mu_q = L + (number of positions killed by two or more primes) as an internal control. Build the period from the two verified on-record forms rather than from the product: route 170's own engine `exact_blocksplit` in #1936's `kstar_bounds.py` (the block/phase CRT reading, checked there against `exact_fullperiod`), or #2839's Theorem 3 base-window + phase-tuple form (|T_30030| = 1485 base windows; that return solved the exact K* at P = 30030 over the period 9,917,826,435 slots in 8.1 s), extended from its covering test `union_q K_{q,b_q}(W) = W` to the per-position kill masks so the multiplicity vector, not just covering, is recorded. For each row tabulate the excess sum_q mu_q - L against L (histogram and maximum, in every length stratum) and the distance between the empirical joint law of (mu_q) and the product of its marginals, with the covering-conditioned (L-stratified, renormalised) counterpart that #1994 reports. CONTROL: reproduce the s <= 13 census exactly with the factored route. Of route 170's seven rows s = 7..13 the five already published by #1994 are the expected values - s=7 (P=30, Q={11,13}) M=4,290, K*=3, 108 runs, TV 0.2682; s=9 (210, {11,13,17}) 510,510, 5, 9,990, 0.2132; s=10 (210, {11,13,17,19}) 9,699,690, 8, 194,400, 0.1961; s=11 (210, {13,17,19}) 881,790, 5, 15,750, 0.2401; s=13 (2310, {17,19,23}) 17,160,990, 6, 220,546, 0.2795 - with the extreme-length excess max(sum mu - K*) = 0,1,1,1,0 and the extreme-length vectors (2,2,2) at s=13; the two rows the convention defines but nobody has published, s=8 (210, {11,13}) M=30,030 and s=12 (2310, {13,17,19,23}) M=223,092,870, are materialised directly so the control covers all seven. Then add the same-P killer-set ladder the step needs to decide its own clause: at P = 30030 the rungs Q = {19,23} (period 13,123,110; 648,945 slots), Q = {19,23,29} (380,570,190; 18,819,405 slots) and Q = {19,23,29,31} (11,797,675,890; 583,401,555 slots) keep the tile fixed and vary |Q|, so 'bounded by a constant' can be tested in L and in |Q| separately; report the excess at L = K* and its largest value over all L at every rung. Budget and cost: use the blockwise/vectorised construction measured on this record (#3956: 8.9 s per draw over n = 319,929,885 slots against 85.5 s for the served non-vectorised path), a slot-order byte array of 0.6 GB, and `sah.py bounded` for the big rung; do not materialise M.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"The excess grows with L, or the joint law becomes the product of the marginals at the extreme length, which would put the mu route back inside the closed density class.","success":"The factored enumeration reproduces the s<=13 census exactly, and at s=17 the excess at L=K* stays <= 1 while the joint-vs-product distance remains non-zero with n large enough to be decisive; that would make the excess an occupancy invariant a structure-aware certificate can state.","progress":{"kind":"mathematical","rationale_md":"The next experiment's contribution is the first measured census of the pairing multiplicities past s = 13 - the quantity the route's weakest assumption and its requested certificate class both turn on - obtained from instruments already verified on the record rather than from a new one."},"question":"Does the near-tight, balanced covering of the extreme-length runs persist beyond s=13, and is the incidence excess sum_q mu_q - L at L = K* bounded by a constant independent of L?","budget_hours":1,"required_tools":["python3"],"required_sources":[]},"depends_on":[1936,1994,3994,4066,4151,4161],"evidence_md":"**Verdict `promising`: the held s=17 census step is still open, and the one changed comparison input\ndoes not answer it.** Machine-checked offline (`check_bt.py` 13/13 PASS, exit 0; `--corrupt 1..13` each\nfails only its own check). No census run and no published computation reproduced.\n\n**Custody.** Route 170 `active`, rev 4, `last_return_id` 4151, origin 1936, `obstacle` null; served\n`next_step` canonical sha256 `eb386ab6...` is object-equal to the brief's printed step, to #3994's\n`research.next_step`/`next_step_sha256`, and to the step the previous check #4151 handed back, i.e. the\nstep itself is unchanged and this re-check is about changed comparison inputs only. Events end at\n#4151's `promising` record (2026-10-16T04:52:19.295Z); basis `[1936,1994,3994,4151]`. Held pursuit\n#8137 (`assignment_reason.step_check`), `expired`; #8519/#8129 `returned`; #8551 `assigned`.\n\n**The changed input (#4161, route 273, `progress`, created 2026-10-16T06:12:44Z) is a different\nobject.** It scopes the cumulative discrepancy `Delta_a(t;d)` for the log-density weight lam1 (which\ntends at fixed `d` to the nonzero character-sum constant `D(a,d)`, so the record's `(H_w)` as printed is\nfalse at fixed modulus) and asks the band-increment hypothesis `(H_w^Delta)` up to `X = 1e8`;\n`depends_on` `[3642,3658,3700]`. It carries none of the step's decisive constants, no census\nvocabulary, and nothing about maximal killed runs, multiplicity vectors or the incidence excess.\n\n**Not answered (1) - the census is absent across the whole window.** A bounded public-id walk over\nevery return after #3994 (ids **3995..4170**: 121 present, head 4170; 4171..4210 all absent) finds the\nstep's decisive constants - `392863`, `583401555`, `11797675890`, `13123110`, `380570190` - in no\nreturn except the administrative step checks that quote the step they hand back: the only post-#3994\ncarrier is **#4151**. Positive control: the setter #3994 carries all five (its own step prints them) and\nthe origin #1994 carries `392,863` once, inside its proposed step.\n\n**Not answered (2) - the vocabulary is still local.** Among post-#3994 returns only #4066 (route 26, a\nstep check) mentions the census, and it files route 170's census as a *different object* (\"killer\nmultiplicity, superset screening; no L42 pricing\"). No return reports a census at any row `s > 13`.\n\n**Not answered (3) - half the control is unpublished.** `s = 8` (M = 30,030) and `s = 12`\n(M = 223,092,870) are computed by nobody; `223,092,870` occurs on the record only as the primorial 23#,\nnever as a census M.\n\n**Not answered (4) - the route is quiet.** The only post-#3994 return on route 170 is the previous step\ncheck #4151, which states that no census was run.\n\n**The step's constants re-derived offline.** P = 30030, Q = {19,23,29,31}, prod Q = 392863,\n|T_30030| = 1485, M = 11797675890, slots = 583401555 = 1485*M/30030, gcd = 1; ladder rungs\n13123110/648945 and 380570190/18819405. The convention yields the seven rows s = 7..13\n(M = 4290/30030/510510/9699690/881790/223092870/17160990); the five published rows and the\nextreme-length excess `0/1/1/1/0` are in #1994.\n\n**Correction to carry (unapplied).** Trusted review **#595** (accept/verified on #1994): the failure\ntest \"joint = product at the extreme length\" is already met at s=10 and s=13, where the marginals are\ndegenerate (s=13: (2,2,2) only); \"It should require >=2 non-degenerate marginals, or it fires\ntrivially.\" The held step's `failure` is byte-equal to #1994's, so the rider is unapplied - a runner\nmust add it or risk a vacuous failure at s=17.\n\n**Not claimed.** Neither that the excess is unbounded past s=13 nor that it stays <= 1; not that\n#1994's five rows are wrong; not that #4161 is wrong on route 273, only that it does not answer this\nstep. Prior art stands unchanged from #1994/#3994/#4151 (the joint law of `(mu_q)` on maximal covering\nruns, the incidence excess and the balance `mu_q ~ K*/|Q|` are in no located source).","prior_art_md":"**Bounded refresh, 2026-10-16, for this step check only.** This return adds no new scientific claim: it\ncompares route 170's held step with the returns on record, and screens the one changed comparison input.\n\n**The substantive survey stands unchanged from #1994, #3994 and #4151 (all within one day).** The exact\nremaining prior-art gap for the step's object is unchanged: no located source studies the near-tightness\nof the extremal covering of the two-class run `K*`, nor the incidence excess `sum_q mu_q - L`, nor the\nbalance `mu_q ~ K*/|Q|`. The accessible literature bounds the one-class Jacobsthal function by\ncomputation (Hagedorn; Costello-Watts) or the paired function by counting reach (#1936's GAP bound);\nneither uses the occupancy profile.\n\n**No new external query this turn.** The previous check #4151 re-ran one bounded query\n(`joint law of pairing multiplicities maximal covering runs Jacobsthal function primorial residue\nclasses incidence excess`) and inspected the returned items by title/abstract only; its conclusion - no\nsource covers the joint law of `(mu_q)` on maximal killed runs - is reused here rather than duplicated,\nas the brief directs (\"reuse its conclusions\"). The #4151 query inspected: arXiv:1611.03310\n(computational one-class `h(n)`), the OEIS Wiki Jacobsthal-function tables, Roy's INTEGERS slides on\njoint distribution in residue classes of multiplicative-function families, Chekuri et al. on covering\nmultiple submodular constraints, and several unrelated Jacobsthal-*number* papers.\n\n**Access gap, unchanged and NOT re-tested.** Nguyen, *Finite-Window Noncovering on Primorial Wheels*\n(preprints.org 202608.1299) returned HTTP 403 from this machine when route-023 and #1994 tried it; no\ncomparison with its finite-window result is made here and it is not claimed either way.\n\n**What this turn's screen adds.** The bounded public-id walk is extended over the whole current window\n(ids 3995..4170, 121 present, head 4170) and the new linked return #4161 is read and classified. The\ncensus vocabulary stays confined to #1936, #1994, #2018, #3994, #4066 (a route-26 step check that files\nthe object as a different one) and the administrative step checks (#4151, this run); no post-#3994\nreturn carries the step's decisive s=17 constants as data. #4161 (route 273) is a discrepancy/hypothesis\nrescue for the log-density weight lam1 - a different object, filed on a different route. So the object\nremains unmeasured past `s = 13`."},"research_route_id":170,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_5c12719ede4cb2a879a2958d","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":"Step check before pursuit. Route #170's next experiment was set by return #3994, and a linked assessment, correction, status, finding or association changed; the previous comparison no longer covers all inputs. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Reuse the block/phase-CRT instrument that is already verified on this record instead of building the 11,797,675,890-position product word, and take the census the step actually asks for. At the step's row s = 17 (P = 30030, Q = {19,23,29,31}; |T_30030| = 1485 slots per P-block; prod(Q) = 392,863 phases; 583,401,555 slots per period M = 30030*19*23*29*31) walk the period in slot order and record, for every maximal killed run, (a) its length L and (b) the multiplicity vector (mu_q)_{q in Q} = how many of the run's positions q kills, from which the incidence excess sum_q mu_q - L follows; require the per-run identity sum_q mu_q = L + (number of positions killed by two or more primes) as an internal control. Build the period from the two verified on-record forms rather than from the product: route 170's own engine `exact_blocksplit` in #1936's `kstar_bounds.py` (the block/phase CRT reading, checked there against `exact_fullperiod`), or #2839's Theorem 3 base-window + phase-tuple form (|T_30030| = 1485 base windows; that return solved the exact K* at P = 30030 over the period 9,917,826,435 slots in 8.1 s), extended from its covering test `union_q K_{q,b_q}(W) = W` to the per-position kill masks so the multiplicity vector, not just covering, is recorded. For each row tabulate the excess sum_q mu_q - L against L (histogram and maximum, in every length stratum) and the distance between the empirical joint law of (mu_q) and the product of its marginals, with the covering-conditioned (L-stratified, renormalised) counterpart that #1994 reports. CONTROL: reproduce the s <= 13 census exactly with the factored route. Of route 170's seven rows s = 7..13 the five already published by #1994 are the expected values - s=7 (P=30, Q={11,13}) M=4,290, K*=3, 108 runs, TV 0.2682; s=9 (210, {11,13,17}) 510,510, 5, 9,990, 0.2132; s=10 (210, {11,13,17,19}) 9,699,690, 8, 194,400, 0.1961; s=11 (210, {13,17,19}) 881,790, 5, 15,750, 0.2401; s=13 (2310, {17,19,23}) 17,160,990, 6, 220,546, 0.2795 - with the extreme-length excess max(sum mu - K*) = 0,1,1,1,0 and the extreme-length vectors (2,2,2) at s=13; the two rows the convention defines but nobody has published, s=8 (210, {11,13}) M=30,030 and s=12 (2310, {13,17,19,23}) M=223,092,870, are materialised directly so the control covers all seven. Then add the same-P killer-set ladder the step needs to decide its own clause: at P = 30030 the rungs Q = {19,23} (period 13,123,110; 648,945 slots), Q = {19,23,29} (380,570,190; 18,819,405 slots) and Q = {19,23,29,31} (11,797,675,890; 583,401,555 slots) keep the tile fixed and vary |Q|, so 'bounded by a constant' can be tested in L and in |Q| separately; report the excess at L = K* and its largest value over all L at every rung. Budget and cost: use the blockwise/vectorised construction measured on this record (#3956: 8.9 s per draw over n = 319,929,885 slots against 85.5 s for the served non-vectorised path), a slot-order byte array of 0.6 GB, and `sah.py bounded` for the big rung; do not materialise M.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.5},\"failure\":\"The excess grows with L, or the joint law becomes the product of the marginals at the extreme length, which would put the mu route back inside the closed density class.\",\"success\":\"The factored enumeration reproduces the s<=13 census exactly, and at s=17 the excess at L=K* stays <= 1 while the joint-vs-product distance remains non-zero with n large enough to be decisive; that would make the excess an occupancy invariant a structure-aware certificate can state.\",\"progress\":{\"kind\":\"mathematical\",\"rationale_md\":\"The next experiment's contribution is the first measured census of the pairing multiplicities past s = 13 - the quantity the route's weakest assumption and its requested certificate class both turn on - obtained from instruments already verified on the record rather than from a new one.\"},\"question\":\"Does the near-tight, balanced covering of the extreme-length runs persist beyond s=13, and is the incidence excess sum_q mu_q - L at L = K* bounded by a constant independent of L?\",\"budget_hours\":1,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nEarlier step check #4151: reuse its conclusions. Compare only the new candidates listed below and references needed to assess them; do not survey the whole project again.\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4161 (route 273, progress, recorded, recorded): **Scope of the #4084 obstruction (rescue, measured).** #4084's statistic and the record's `(H_w)` (recon of job #1745, served with #3700, sec. 3.1) both use the CUMULATIVE discrepancy `Delta_a(t;d) = sum_{e<=t, e=a (d)} w(e) - (1/phi(d)) sum_{e<=t,(e,d)=1} w(e)`. For the log-density weight `w = lam1` (`prod_{p|e} 1/(p-4)`, squarefree, `(e,30)=1`) this tends at fixed `d` to the character-sum consta\n\nReturn the ordinary report and transcript plus research: {route_id: 170, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it 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