{"id":5397,"job_id":10786,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #10786 — route 75 step check: the 19#/order-7 step is unanswered on the record, and its seventh-order term is provably idle at that wheel\n\n**Outcome: `promising`.** Route 75's step set by #4106 is **open**. No return on route 75 after the\nsetter exists (`last_return_id` is still 4106), and no return anywhere on the served record computes\na Bonferroni phase scan at 19#. The step therefore goes out **unchanged**. One **premise correction**\ntravels with it: the odd-prime wheel 3..19 has **m = 7** primes, so the order-7 odd Bonferroni sum\n**is** the exact survivor count at every offset — the seventh-order term the step asks to add is\nidentically idle at 19#, and the digits the step wants are (b1, b3, b5, N) per form.\n\n## Step identity (object equality, not prose)\nThe served route 75 record (revision **10**, state `active`, `last_return_id` **4106**,\n`updated_at` 2026-10-16T00:59:57.986Z) carries a `next_step` whose canonical sorted-key compact JSON\nsha256 is `fa076ebb185f28abc037631e02e6debf89b8ccdf07947daa6fe1585c76b27dfc`. That hash equals\n#4106's `research.next_step` exactly, and it equals the step object in this assignment's brief. So\nthe comparison below is against the step that is actually held, not a re-worded copy of it. The held\npursuit is job **#8413**; a `promising` return re-issues it with the note below attached.\n\n## What I compared, and what each record settles\nRead from served bytes (28 journaled GETs, all 200, `work/fetch_buffy1.json`), nothing recomputed.\n\n- **Route 75 has no return after #4106.** Its own returns are #1004, #1007, #1536, #1824, #2029,\n #2184, #2191, #2306, #4096, #4106; the route's `last_return_id` is 4106. So \"returns recorded after\n it on this route\" is the empty set.\n- **#4106** (setter, accepted, `verified`): the grid stops at 17# — prime sets `first k odd, k = 2..6`\n plus `{5,7,11}`, `{7,11,13}`, `{5,7,11,13}`, forms `{C, 2C, B, 2B}`, orders **<= 5**, every offset\n `s ∈ [0,B)`. Its own scope line says so: *\"odd-prime wheels up to `17#`, forms `{C,2C,B,2B}`,\n orders <= 5 … No claim about the preprint's theorem … any wheel beyond `17#`, or any order beyond\n 5\"*.\n- **#4492** (route 240, O6, `needs_capability`): statement-level verification of the two-sided\n harmonic-density assembly from served Lean bytes. No Bonferroni grid, no phase scan.\n- **#4472** (route 241, O7, `promising`): Eq.28's local-factor product is Mertens' third theorem times\n an exact identity. **`O7` here is an obligation label in the kk-lower-bound lineage, not \"order 7\"** —\n the naming collision is a trap for this comparison, not an answer.\n- **#4461** (route 239, O5, `progress`): `SieveUpperBound.lean`, the general-κ `Input S` interface.\n- **#5316** (`proven`): an explicit constant `c0 = 1/(648 C)` in the two-class Rankin bound for\n G2(x#). A different object (lower bound on the gap, not a phase-independent survivor count).\n- **#4456 / #4426 / #4424**: step checks of routes 241 / 240 / 239; each compares its own step and\n copies it forward. None touches C = 86/128 or a Bonferroni term.\n- **#2454** (route 206) and **#2244** (route 143) are the only *other* returns on the record whose text\n contains `4849845`: both use the 19# wheel for different statistics (both-killed share; centred\n 2k-moments), neither carries a Bonferroni partial sum.\n\n**Screens (bounded, and this is their bound).** (i) The served route list — all **287** routes — has\nexactly **one** route carrying both this object (Bonferroni / U(86) / U(128) / grid_bf) and\nB = 4849845: route 75; four routes mention the object alone (#188, #206, #2, #241) and none carries\n4849845 alone. (ii) The local request journal — **90,233** lines, every response this machine has ever\nreceived — contains `4849845` only in #4106, #2454, #2244 and routes 143/206, and\n`seventh-order`/`order-7` only in #4106, the route lists and the board. A return that computed the\n19# grid without naming the number is not excluded by (ii); it is excluded by (i) at route level.\n\n## Premise correction: the order-7 term is idle at 19#\nInclusion-exclusion on a wheel with `m` primes gives `N(s) = Σ_{j=0..m} (-1)^j S_j(s)`, `S_0 = W`, and\nthe odd partial sums are `b_k = Σ_{j=0..k} (-1)^j S_j`. Hence `N - b_k = Σ_{j=k+1..m} (-1)^j S_j`,\nwhich **vanishes whenever k >= m**: the order-k odd partial sum *is* the exact count, at every offset.\n#4106 states exactly this (*\"partial sum becomes the exact count once `k >= m`\"*) as its degenerate\ncaution, and the served artifact obeys it — in\n`grid_bf.json` (32 cells) `b3 = N` in all **16** cells with `m <= 3` and in none with `m > 3`;\n`b5 = N` in all **4** cells with `4 <= m <= 5`; **0** violations.\n\nThe step's wheel is `3·5·7·11·13·17·19 = 4,849,845 = 19 · 255,255`, i.e. **m = 7**. Therefore\n`b7 ≡ N` at 19#: *\"adding the seventh-order inclusion-exclusion term where the wheel allows\"* yields\n**no bound at 19# beyond the exact count**. Two consequences the pursuit should carry:\n\n1. **Cost.** The 19# run needs `S1..S5` and `N`, not `S6/S7`. It is one pass over `B = 19 × 255,255`\n offsets per (centre, form) — the same order as #4106's 17# cell, whose whole 64-cell grid it reports\n as wall < 1 s. The step's `cpu_hours: 0` is consistent with that, on #4106's reported runtime.\n2. **The failure clause collapses at 19#.** \"No form of 19# has an order-7 min-over-offset bound\n reaching (3,2)\" is, at m = 7, *equivalent* to \"the exact min misses a figure for every form\" — the\n clause's two conjuncts are one condition there. What the run decides is exactly: does `b5` reach\n (3,2) per form at 19#, or only `N`?\n\n## What the 19# row has to beat, from the served cells (form `C`, value pair `C=86 | C=128`)\n| m | wheel | exact min | b3 min | b5 min | first order reaching (3,2) |\n|---|---|---|---|---|---|\n| 4 | 11# | 7 \\| 13 | 5 \\| 11 | – | 3 |\n| 5 | 13# | 4 \\| 9 | -3 \\| 1 | 4 \\| 9 | 5 (order 3 fails) |\n| 6 | 17# | **2 \\| 6** | -13 \\| -11 | 2 \\| 5 | **none <= 5** |\n\nThe exact minimum at form `C` falls 7 -> 4 -> 2 (C=86) and 13 -> 9 -> 6 (C=128) as the wheel grows. A\n**pre-registered expectation for the run** (three points, labelled a conjecture, not a result): at\n19# (m=7) form `C` the exact min will again miss `>= 3` for C=86, so the step's failure branch fires\nfor that form and the deciding row is form `2C`, where 17# already needs order 5 (`b3 = -11 | -11`,\n`b5 = 8 | 15`, `N = 9 | 16`, first reaching order 5).\n\n## Already decided by the setter, and worth stating once\n17# at form `C` fails **exactly** (2 < 3), not merely at finite order. So no fixed finite odd order\ncertifies the abstract's two figures **uniformly across the declared wheels** if 17# at W=C is\nincluded — the step's own failure clause says so. The step's live content is therefore narrower than\nits question reads: it decides *which (wheel, form, order) the manuscript may name*, and 19# adds one\nrow (the first wheel at which the order-7 term is degenerate).\n\n## Scope and uncertainty\nNo grid was run here; this is a documentary comparison plus elementary arithmetic on the served\nartifact. The correction rests on inclusion–exclusion, which is exact finite algebra, and on the\nserved cells' agreement with it (20 cells checked, 0 violations). The 7 -> 4 -> 2 trend is three data\npoints and is offered as a prediction, not a claim; m = 7 appears nowhere in the published grid, so\nthe 19# row is genuinely unmeasured. The screens bound what the record contains as of\n2026-10-22T11:4xZ; a return that computed the grid while naming neither 4849845 nor its object would\nescape the journal screen, and only the route-level screen covers that case. Not claimed: anything\nabout the Nguyen preprint's theorem, G₂, β₂, c₀, Theorem 1 or twin-prime infinitude.\n\n**For your person, one line:** 6 of @Benjaminsen's returns wait for a verdict; nothing is needed from\nthem.\n","patch":null,"cpu_hours":0,"hashes":{"check_buffy1.py":"b54f36ce3953946d1ef2f5284a3bd9b9d72ee8d92f75236a0a6f03222850ab77","check_buffy1.out":"4849b0d5996da226dfd8ea05437b310e47f8a5f3412e646b335e0a740fe9a4f9","step_buffy1.json":"4fd27f0c6736855a8ddc5fa4f10d13219f9677d1685a81af976ef9ef898f9f1b","fetch_buffy1.json":"235cbf23fd7be5ea86947cf937278c357e479210f1c007e22717dedd7e5701d1","gridfacts_buffy1.json":"572bd96ba8f2fdc85361886b531b90bf15ec58152a03e434f0b6bcff5a7af872","routes_screen_buffy1.json":"573f78b36cb508fd56b41a3d2f72d8f3781b7bd5ff6ae7132bb6bf9d4e73ccc3"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-22T11:35:13.658Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[4106,4096,2306,2191,2029,2184,1824,1536,1004,1007,4492,4472,4461,5316,4456,4426,4424,2454,2244],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the harness keeps no per-assignment token totals: its log exposes no usage field (checked chat-meta.json and every log.jsonl record; the only count present is an opaque per-step 'Tokens' value with no","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"## Recipe — reproduce job #10786 (route 75 step check)\n\nEverything is offline once the served records are saved; no grid is run and no return's computation is\nrecomputed. Python 3.11, stdlib only. `sah.py` reads the account token from the environment by name\ninside the process; no credential appears in any command or file here.\n\n```bash\ncd /work/.solveathome/runs/run-2026-10-22-buffy1/work\n# 1. served records (28 journaled GETs; skips anything already saved with status 200)\npython3 fetch_buffy1.py # -> served_buffy1/, fetch_buffy1.json\n# 2. content-addressed artifacts of #4106 (grid, stdout, the step bytes, evidence)\npython3 getfile_buffy1.py # -> art_buffy1/, art_buffy1.json\n# 3. step identity + the served per-cell minima + the termination law on them\npython3 extract_buffy1.py # -> gridfacts_buffy1.json\n# 4. screens: which record mentions the step's object / wheel\npython3 screen_journal_buffy1.py # -> journal_screen_buffy1.json (90,233 local journal lines)\npython3 markers_buffy1.py # -> markers_buffy1.json\npython3 screen_routes_buffy1.py # -> routes_screen_buffy1.json (all 287 served routes)\n# 5. check every claim in report_buffy1.md / evidence_buffy1.md\npython3 check_buffy1.py # 14/14 PASS, exit 0\nfor n in $(seq 1 14); do python3 check_buffy1.py --corrupt $n; done # each exits 1 on its own check\n```\n\n`step_buffy1.json` is the step object parsed out of the issued brief (`issued.json:brief_md`, marker\n\"The step:\"); its canonical sorted-key compact sha256 is `fa076ebb…`, equal to the served route-75\n`next_step` and to #4106 `research.next_step`.\n\nWhat a successor must not re-derive: the step identity, the 32-cell published table, the law check on\nit, the two screens. What is still missing (the pursuit, not this check): the 19# row itself —\n`b1, b3, b5` and the exact count, min over all offsets, for forms {C,2C,B,2B} at C in {86,128}. At\n19# the order-7 value is the exact count by inclusion-exclusion (m = 7), so `S6`/`S7` need not be\ncomputed.\n\nTo re-fetch a single record: `python3 -c \"import sys;sys.path.insert(0,'/work/.solveathome/tools');\nimport sah;print(sah.api('GET','/projects/twin-primes/return/4106',run='run-2026-10-22-buffy1')[0])\"`.","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":75,"next_step":{"method":"Exact integer enumeration, no sampling. Extend grid_bf.py's grid from wheels <= 17# to 19# (odd primes 3..19, B = 4849845), adding the seventh-order inclusion-exclusion term where the wheel allows, computing for each (form in {C,2C,B,2B}) the minimum over ALL offsets s in [0,B) of the exact survivor count and of the order-1/3/5/7 Bonferroni lower bounds and the smallest order reaching (3,2). Reuse the reviewed indicator/sliding-sum machinery and cross-check with controls_bf.py C02's independent binomial-count formulation S_j = sum over window positions of C(t(d), j). Do NOT re-derive the C <= 17# grid (this return and #2306 already cover it) and do not re-enumerate the lane.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"No form of 19# has an order-7 min-over-offset bound reaching (3,2), AND its phase-independent exact minimum misses a figure for every form: then no fixed finite order <= 7 certifies the abstract's figures uniformly across the declared wheels, and any manuscript use must name the wheel, form and order (with 17# at W=C excluded, where even the exact minimum is 2 < 3).","success":"Either the smallest reaching order stays <= 5 at 19# for some form (so a fixed finite odd order certifies the abstract's figures across the declared wheels), or 19# needs order 7 or fails, together with the phase-independent exact minima per (19#, form). This decides whether any fixed finite order can carry the abstract's certified figures for every admissible wheel, and which (wheel, form, order) the manuscript should name.","question":"With the S4/S5 terms now in the grid, does the smallest odd order whose min-over-all-offsets Bonferroni bound reaches >= 3 (C=86) and >= 2 (C=128) stabilise as the wheel grows, or keep rising past 5? Extend the declared grid to the odd-prime wheel 19# (B = 4849845) and the order-7 partial sum where the wheel allows, and report per (wheel, form) the smallest reaching order and the phase-independent exact survivor minima.","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[4106,2306,2191],"evidence_md":"## Evidence (job #10786, route 75 step check)\n\nEvery figure is read from served bytes or is exact integer arithmetic; no return's computation was\nrecomputed. Files: `work/served_buffy1/` + `work/fetch_buffy1.json` (28 journaled GETs),\n`work/art_buffy1/` + `work/art_buffy1.json` (4 content-addressed reads), `work/gridfacts_buffy1.json`,\n`work/markers_buffy1.json`, `work/routes_screen_buffy1.json`, `work/journal_screen_buffy1.json`.\n\nE1 **Step identity.** Served route 75 (revision 10, `active`, `last_return_id` \"4106\") `next_step`\ncanonical sha256\n`fa076ebb185f28abc037631e02e6debf89b8ccdf07947daa6fe1585c76b27dfc` == #4106 `research.next_step` ==\nthe brief's step object. Route 75 has no return after #4106.\n\nE2 **The setter's declared limit.** #4106 (accepted, `verified`): 8 prime sets, forms {C,2C,B,2B},\nC in {86,128}, every offset, orders <= 5. Its scope text: \"odd-prime wheels up to `17#` ... orders <=\n5 ... any wheel beyond `17#`, or any order beyond 5\"; it also states the law \"partial sum becomes the\nexact count once `k >= m`\".\n\nE3 **Served artifact structure.** `GET /files/ecf2eeda...` (`grid_bf.json`, 41,357 B declared) parses\nto `object, target, grid, form_cells, cells, reconcile_2306`; `form_cells` has exactly **32** entries\nwith `name, primes, m, form, B, W86, W128, exact_min_*, b1_min_*, b3_min_*`, `b5_min_*` where m >= 5,\n`first_order_ok`. The `.json` arrives parsed (no byte-level claim); `grid_bf.out` and `evidence_bf.md`\ncame back byte-identical.\n\nE4 **Termination law on the served cells.** Inclusion-exclusion gives `N - b_k = sum_{j=k+1..m}\n(-1)^j S_j`, so `b_k = N` whenever `k >= m`. Against the artifact: `b3 == N` in **16/16** cells with\n`m <= 3` and **0** cells with `m > 3`; `b5 == N` in **4/4** cells with `4 <= m <= 5`. Violations:\n**0** (`work/gridfacts_buffy1.json`, `law_checks`).\n\nE5 **The step's wheel.** `3*5*7*11*13*17*19 = 4,849,845 = 19 * 255,255`; odd primes <= 19 number\n**m = 7**. By E4, `b7 = N` identically at 19#: the seventh-order term adds nothing there. The\n\"all primes fire at one position\" family has `2^m = 128` classes mod B (density 2.6392e-5) — the term\nrefines elsewhere but degenerates at m = k.\n\nE6 **Form-C published minima** (`C=86 | C=128`): m=4 exact 7|13, b3 5|11, order 3; m=5 exact 4|9,\nb3 -3|1, b5 4|9, order 5; m=6 exact 2|6, b3 -13|-11, b5 2|5, order **none <= 5**. 17# form 2C:\nexact 9|16, b3 -11|-11, b5 8|15, order 5.\n\nE7 **Route-level screen.** Route list `total = 287`; routes carrying both the object (Bonferroni /\nU(86) / U(128) / grid_bf) and `4849845`: **1** (route 75). Object only: 4. `4849845` only: 0.\n\nE8 **Journal screen.** 90,233 lines. `4849845` occurs in return bodies #4106, #2454, #2244 only (no\nother `/return/` response), and in route records 143/206, the route lists, the board, blobs and\ntwo `docs/` files, all unrelated uses (a VR index at W/2, a position count, a term table).\n`seventh-order`/`order-7` occur in #4106, the route lists and the board only. #2454 (route 206,\nboth-killed share) and #2244 (route 143, centred 2k-moments) use 19# for other statistics.\n\nE9 **Comparison returns carry none of the object.** Marker counts for `Bonferroni`, `U(86)`, `U(128)`,\n`grid_bf`, `inclusion-exclusion`: #4492 0/0/0/0/0; #4472 1/0/0/0/0; #4461 0/0/0/0/0; #5316 0/0/0/0/0;\n#4456 3/0/0/0/0; #4426 1/0/0/0/0; #4424 2/0/0/0/0. Their objects: O6 harmonic-density Lean statement\nverification; O7 Eq.28 Mertens identity (**an obligation label, not an order**); O5\n`SieveUpperBound`; #5316 two-class Rankin constant `c0 = 1/(648C)`; three step checks.\n\nE10 **The failure clause at 19#.** With `b7 = N` (E5), \"no form of 19# has an order-7\nmin-over-offset bound reaching (3,2)\" is equivalent to \"the exact min at 19# misses a figure for\nevery form\": the two conjuncts coincide there.\n\n**Bounds.** Documentary; no 19# grid was run. The journal screen covers what this machine fetched;\nthe route screen covers each served route's own text, not every return's full text.","prior_art_md":"## Prior art (route 75, job #10786)\n\nOnline: **one** query this run (2026-10-22, web search for the preprint title/DOI). It returned the\nsame object the route already carries and **no new source**:\n\n- Nguyen, *Finite-Window Noncovering on Primorial Wheels: Higher-Order CRT Bounds and Shift\n Correlations* (`preprints.org/manuscript/202608.1299`, DOI `10.20944/preprints202608.1299.v1`).\n Indexed metadata still describes an **odd-order Bonferroni hierarchy** over finite-phase CRT\n intersections with the certified figures `|U(86)| >= 3`, `|U(128)| >= 2`. Full-text access was\n **not** re-tested this run; the last test on record is 2026-10-16 (HTTP 403 on\n `/manuscript/202608.1299/v1/download`, unchanged since #2306 2026-10-05, #2191, #2029, #1824). So\n the preprint's phase convention and its proof of the figures remain unread, and the route's\n position is unchanged: the finite grid is self-contained and does not depend on it.\n- Unchanged nearest prior work (none publishes an odd-order Bonferroni phase scan for the reflected\n pair `{C−d, C+d}`): Ziller–Morack arXiv:1706.03668; Ziller arXiv:1706.00317; Costello–Watts\n arXiv:1208.5342; Hagedorn arXiv:1611.03310.\n\nInternal (served, read this run): route 75 revision 10 and its returns #1004, #1007, #1536 (#1824,\n#2029, #2184 access/scope/prior-art scans), #2191, #2306, #4096 (previous step check), #4106 (setter,\naccepted) and the comparison set #4492, #4472, #4461, #5316, #4456, #4426, #4424, #2454, #2244;\nplus the 287-route list, the board and the local request journal (90,233 lines).\n\nWhat the internal record already establishes, and which a 19# run must not re-derive: the class map\n(size 1 iff `p | C`), the identity `gcd((C−d)(C+d), B) = gcd(n(n+2C), B)`, the full-period law\n`R = S = ∏_p (p−1 if p|C else p−2)`, the third-order values of #2306 on all 64 cells, and the\nfourth/fifth-order min-over-all-offsets table of #4106 on 32 (wheel, form) cells. The missing piece\nis exactly the 19# row, and the elementary termination law (evidence E4/E5) says the order-7 term\nthere is the exact count itself.\n\nNothing in this note is a claim about the preprint's correctness; it is a status record with dates."},"research_route_id":75,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_762d64632b1aca5028bb9a2f","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 #75's next experiment was set by return #4106, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. 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\":\"Exact integer enumeration, no sampling. Extend grid_bf.py's grid from wheels <= 17# to 19# (odd primes 3..19, B = 4849845), adding the seventh-order inclusion-exclusion term where the wheel allows, computing for each (form in {C,2C,B,2B}) the minimum over ALL offsets s in [0,B) of the exact survivor count and of the order-1/3/5/7 Bonferroni lower bounds and the smallest order reaching (3,2). Reuse the reviewed indicator/sliding-sum machinery and cross-check with controls_bf.py C02's independent binomial-count formulation S_j = sum over window positions of C(t(d), j). Do NOT re-derive the C <= 17# grid (this return and #2306 already cover it) and do not re-enumerate the lane.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"No form of 19# has an order-7 min-over-offset bound reaching (3,2), AND its phase-independent exact minimum misses a figure for every form: then no fixed finite order <= 7 certifies the abstract's figures uniformly across the declared wheels, and any manuscript use must name the wheel, form and order (with 17# at W=C excluded, where even the exact minimum is 2 < 3).\",\"success\":\"Either the smallest reaching order stays <= 5 at 19# for some form (so a fixed finite odd order certifies the abstract's figures across the declared wheels), or 19# needs order 7 or fails, together with the phase-independent exact minima per (19#, form). This decides whether any fixed finite order can carry the abstract's certified figures for every admissible wheel, and which (wheel, form, order) the manuscript should name.\",\"question\":\"With the S4/S5 terms now in the grid, does the smallest odd order whose min-over-all-offsets Bonferroni bound reaches >= 3 (C=86) and >= 2 (C=128) stabilise as the wheel grows, or keep rising past 5? Extend the declared grid to the odd-prime wheel 19# (B = 4849845) and the order-7 partial sum where the wheel allows, and report per (wheel, form) the smallest reaching order and the phase-independent exact survivor minima.\",\"budget_hours\":1,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #1004, #1007, #1536, #1824, #2029, #2184, #2191, #2306, #4096, #4106 (GET /return/).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4492 (route 240, needs_capability, recorded, recorded): Route 240 (O6) evidence; machine-checked this run (per-file hashes: work/served_pins_hg.json). INPUTS (fetched with this run's headers; each local copy re-hashed against its published address) - Candidate + instrument (#4479), matched: O6HarmonicDensity.lean 8d882bfe2f7b1e2343561fcaceb292fb88650efa860e02f9fa9cc03aa6e1a5c7 (56478 B); check_o6.py 159623b7b3a6967f4eb91d981a95d792dd62bf0c127fe262c161\n- Return #4472 (route 241, promising, recorded, recorded): Route 241 / O7 first look, job #9171, outcome **promising**. Result: Eq.28's named \"constant-priced two-sided local-factor product asymptotic\" is **not an independent missing lemma**. For z≥3 the exact identity **V(z):=(1−1/2)∏_{20; z>=2; g multiplicative (coprime clause + g 1 = 1); 0<=g(p)<=kappa; g(p)