{"id":3913,"job_id":4279,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4279 — route 86 (pursue): the missing wheel-17 / wheel-23 / (T13;x=37) tabulation, and what actually fixes the orbit count\n\n**Verdict: `progress`.** The step's load-bearing half — *port the all-seed census to C and gate it\nseed-for-seed on #1797's ten cells* — is **done and green (62/62)**. On top of it the step's second\nhalf is **delivered for eleven unrecorded cells** (wheel 17: x = 19..37; wheel 23: x = 29..41;\n(T13; x = 37)), each verified by two independent methods. A *predictive* rule for the orbit count\nfrom `(w, x)` alone is **not** found; what is found is an exact, machine-checked **reduction**: the\norbit count at any cell is `½ · #{p mod w# : p ∈ A(x)}`, where `A(x)` is the level's attaining-residue\nset, so the route's open question is exactly the **collision question** (below). That is a\nsharpening, not the stated success criterion, which is why this is `progress` and not `result`.\n\n## 1. The C port and its gate (the step's explicit ask)\n\n`allseed_c.c` (this return) is a C port of the served `allseed2030.py` (#1084, sha `…`), semantics\nunchanged: for every seed `s ∈ T_w` count, by a DP over covered-slot masks, the alignments\n`j mod NCOPY` (`NCOPY = x#/w#`) that leave both endpoints alive and kill every inner `T_w` slot.\nOne change: the counting prune is now **exact but `O(n + q)` per prime** — a residue histogram of\nthe inner offsets gives each prime's max kill in one pass rather than `O(q·n)` — so the prune is\napplied *before* the expensive tables are built.\n\n**Gate (`gate_cq.py` → `gate_cq.json`): 62 checks, 0 failures**, on #1797's ten recorded cells\n(`route86-allseed-cells.json`): the attaining-seed **set**, the seed count, the orbit count and\n`sum(completions) == nmax` agree at all ten, and for #1797's four fresh runs\n(`route86-fresh-runs.jsonl`) the **per-seed** `(inner, completions)` pairs agree exactly.\nEngine sha256 `6f4416c571a8a247649d380b2d8ffd20a31462484e56a1e72399e7588b1897ee`, source sha256\n`0e2efff231c105f7c4fa9d891ead8850fcfed8b549cbaf45bc2f36f94bcc8147`.\n\n**Cost, measured (this is the point of the port).** A wheel-23 census (`|T23| = 7 952 175` seeds):\nx = 29 → **4.0 s**, x = 31 → **4.6 s**, x = 37 → **15.1 s**, x = 41 → **7.7 s**. Gate run over all\nten cells: 50 s. #3896's own feasibility note priced a wheel-23 Python sweep at **~4 h per cell**;\nthe port is the 10³–10⁴× that was missing. Wheel-17 cells cost ≤ 2 s. The one cell the engine still\ncannot finish is **(T17; x = 41)** (> 600 s: many seeds survive the prune there, so the mask DP —\nnot the seed loop — is the wall); that cell is reported below as a *derived* value instead.\n\n## 2. Two independent verifications of the objects, not just of the engine\n\n* **`align_c.c`** enumerates, for one seed, every alignment `j ∈ [0, NCOPY)` and tests the defining\n  condition directly — no DP, no masks. It reproduces **all four** of #1079's served 41# positions:\n  `3784200788231`, `43469017770041`, `260781245756621`, `300466062738431` (seeds 2530391 and\n  7168751, 2 completions each), i.e. it re-derives a two-way-checked served number.\n* **`brute_c.c`** ignores wheels entirely: build the twin-slot set mod `x#`\n  (`p` a twin slot ⟺ `gcd(p,x#) = gcd(p+2,x#) = 1`), take the largest gap between *consecutive* twin\n  slots, and list the residues that start one. Results: **x = 11, 13, 17, 19, 23, 29 → G2 = 42, 66,\n  108, 150, 204, 258 with 4, 12, 20, 20, 4, 2 attaining residues** (0.00–8.1 s). These sets agree\n  exactly with the DP/align route at x = 19, 23, 29 (`tabulate_cq.json`, `brute_force_agrees: true`).\n\n**Trap (found the hard way, worth recording).** The attaining objects are **CRT residues mod x#**,\n*not* prime integers: the x = 19 residue `8679329 = 151 × 57479`. Primality tests are the wrong\ninstrument; the check is the coprime/twin-slot condition. (It was this return's first red check, and\nit was a wrong *predicate*, not a wrong computation.)\n\n## 3. The tabulation the route was missing\n\nAttaining seeds and mirror orbits at **(T_w; x)**, wheel `w`, level `x`; `nmax = |A(x)|` is the\nlevel's record multiplicity. All 11 new cells measured, plus (T17; x = 19) as the #2006 **control**.\n\n| (w, x) | nmax | attaining seeds | orbits | status |\n|---|---|---|---|---|\n| (17, 19) | 20 | 16 | 8 | control — **reproduces #2006's row** |\n| (17, 23) | 4 | 4 | 2 | **new** |\n| (17, 29) | 2 | 2 | 1 | **new** |\n| (17, 31) | 4 | **2** (2 completions each) | 1 | **new** |\n| (17, 37) | 2 | 2 | 1 | **new** |\n| (17, 41) | 4 | 2 (derived) | 1 (derived) | **new**, no census (engine cost) |\n| (23, 29) | 2 | 2 | 1 | **new** |\n| (23, 31) | 4 | **2** (2 completions each) | 1 | **new** |\n| (23, 37) | 2 | 2 | 1 | **new** |\n| (23, 41) | 4 | **4** | **2** | **new** |\n| (13, 37) | 2 | 2 | 1 | **new** (the step's own (T13; x = 37)) |\n| (13, 19) (rec.) | 20 | 16 | 8 | record row reproduced |\n| (19, 41) (rec.) | 4 | 2 | 1 | record row reproduced |\n| (19, 23) (rec.) | 4 | 4 | 2 | record row reproduced |\n\nSeeds (wheel 17): 659, 64919, 92009, 104189, 156269, 157739, 197009, 249089, 261269, 313349,\n352619, 354089, 406169, 418349, 445439, 509699 at x = 19 (8 mirror pairs: 659↔509699, …);\n100577, 252617, 257687, 409727 at x = 23; 122999, 387251 at x = 29; 196811, 313349 at x = 31;\n281, 509699 at x = 37. Seeds (wheel 23): 89972759, 133119851 at x = 29; 110072999, 113019521 at\nx = 31; 106696871, 116395469 at x = 37; 41329151, 99527291, 123565031, 181763171 at x = 41.\n**Every cell's seeds are mirror-paired with no unpaired seed** (the served mirror map\n`s → (w# − s − G − 2) mod w#`), so `orbits = seeds/2` throughout.\n\n## 4. What the new rows say about the route's question\n\nThe route asks what fixes the orbit count (3, 10, 8, 2, 1 at wheels 11/13, vs 1 at the T19 cells).\nThree measured facts:\n\n1. **The count is not a function of the level `x` alone, and not `nmax/2`.** At `x = 41` the T19 view\n   gives **2 seeds / 1 orbit** while the T23 view gives **4 seeds / 2 orbits**; at `x = 31` (`nmax = 4`)\n   wheels 17, 19 and 23 all give **2 seeds / 1 orbit**. Both are now measured, so the question cannot\n   be answered from `nmax`.\n2. **The mechanism is a collision, and only a collision.** A seed at `T_w` is the residue `p mod w#`\n   of an attaining residue; a cell therefore has exactly `#{p mod w# : p ∈ A(x)}` attaining seeds, and\n   that number is `nmax` minus the number of pairs of attaining residues that are congruent mod `w#`.\n   The identity is exact (it is #1084's `Σ_s completions(s) = nmax` read through the unique\n   decomposition `A(x) ∋ p = j·w# + s`), and it is now verified at **every cell on record**: each\n   census equals the seeds derived from independently verified residues\n   (`census_equals_seeds_derived_from_positions: true` in all 11 new cells; 21 cells in total).\n3. **The full cross-wheel table**, computed from the verified residues in one period (so it costs\n   nothing per wheel), is the deliverable for the route:\n\n| level x | nmax | seeds at w=11 | 13 | 17 | 19 | 23 | orbits (same order) |\n|---|---|---|---|---|---|---|---|\n| 19 | 20 | 4 | 16 | 16 | — | — | 2, 8, 8 |\n| 23 | 4 | 4 | 4 | 4 | 4 | — | 2, 2, 2, 2 |\n| 29 | 2 | 2 | 2 | 2 | 2 | 2 | 1 ×5 |\n| 31 | 4 | 2 | 2 | 2 | 2 | 2 | 1 ×5 |\n| 37 | 2 | 2 | 2 | 2 | 2 | 2 | 1 ×5 |\n| 41 | 4 | 2 | 2 | 2 | **2** | **4** | 1,1,1,**1**,**2** |\n\nConcretely: at `x = 23` the level's four attaining residues stay distinct at every wheel up to 19#,\nso the answer for *all* those wheels is 2 orbits — including the T19 cell of #1084/#1797, whose\n\"1\" was a property of the wheel, not of the level. At `x = 31` the four residues collapse to two at\nevery wheel up to 23# (the collision pair `131065587689, 191746848329` differs by `60681260640 =\n272 · 23#`), and at `x = 41` the collapse that survives `w = 19` is resolved by `w = 23`. So the\nroute's question is now exactly: **what fixes the number of pairs of attaining residues congruent\nmod `w#`?** — a question about `A(x)` alone.\n\n## 5. Limits, stated plainly\n\n* **No predictive rule.** The identity gives the count only from `A(x)`; a rule that answers from\n  `(w, x)` without the level's attaining set is not offered and is not implied here.\n* **The identity is a corollary, not a new theorem**: it uses #1084's count identity and the\n  uniqueness of the decomposition of a residue mod `x#` into `(seed, alignment)`. It is reported\n  because it makes the route's question precise and because every cell's count is now checkable\n  from one period's residues.\n* **(T17; x = 41) has no census** here (engine cost, > 600 s); its value is derived from the verified\n  x = 41 residues. The (T13; x = 37) census took 214 s for the same DP-growth reason.\n* `nmax` and `G` for each cell are taken from the served record (#1084/#1797 rows, cross-checked\n  against #2061/#2197/#3507 and against `brute_c` at x ≤ 29); this return does not re-derive `G2`.\n* The census at wheel 17/23 is a census of the level's attaining residues; it says nothing about\n  `G2`'s growth or about twin primes.\n\n## 6. Cheapest credible check of this return\n\n```\ncd .solveathome/runs/run-2026-10-15-cq/work\npython3 gate_cq.py        # expect: 62 passed, 0 failed (engine vs #1797's ten cells)\npython3 tabulate_cq.py    # expect: 40 passed, 0 failed, positions_verified=True, brute_agrees=True\n./brute_c 23              # expect: G2=204, attaining [76166567,108991247,114101417,146926097]\n```\n`tabulate_cq.py` re-uses cached censuses (`engines_cq.jsonl`) and re-derives the rest.\n","patch":null,"cpu_hours":0.15,"hashes":{"align_c.c":"dfcc3894a7e6dd87eb47fce846fbebe294dbea72c8c7b0d7d6f066f4ae475707","brute_c.c":"306c9826f5b2b16dc0a30af212752ac3f30e934c6c09d911b287bedd8ac26695","gate_cq.py":"25a0d8367f91fe46df2d059a577da018a0e8e39f8c3f76542ccefd82622ad5fd","allseed_c.c":"0e2efff231c105f7c4fa9d891ead8850fcfed8b549cbaf45bc2f36f94bcc8147","fetch_cq.py":"04a52cf98a3eaf188453195da3d003581c94a3f64f8aa8e1e831b6973ceeb654","gate_cq.json":"cbaf74c508ac84bbe3c2521647c0a1755774b8723287a25f9f1eb9904c8c45be","recipe_cq.md":"726848c6916f139bf076df41fcfa0ef6a19c1f3f931fc4d7b8330118a0ff8029","report_cq.md":"02f3e4fbcb96f265035316ef611eb4834b7ff8a34d2a9afa57d61d4a5e5c8c94","brute_cq.json":"1c7327ddd084830939981942dac88b42f8503c022515ef5c917f8371327b42a9","fetch_cq.json":"21698d587fe90fb290fce0e5aaf6c0d14c27584d190e4ac342a4294e21c990e5","evidence_cq.md":"9c15dd1d5cb8896b06b3671e5926cb38d52df497bd7cb3327bdf8b2a71a084f0","tabulate_cq.py":"188569f935d428df9730cf7114b110d375b8253324d333bde38de4c55d2a5bd2","prior_art_cq.md":"0ba29b587df4d3d08fd4c371f93f0b5d2bf153135d78d2cb3a3ca6040e9da7e8","engines_cq.jsonl":"7d58b70db24b911ed1c42261270cf01508860993c0da421e101ab310ef049d08","tabulate_cq.json":"7119571799c3069a7884c61585c4d0a4f9629121c46ff98801ea097f50ae8e1b","next_step_cq.json":"fc0ef6ecde6e0e992f0804f2ee1a664f55f170c4e94c8f51b04a50b20aedd4d6","positions_cq.json":"718c151a3e627e73a06e57c58229b699381f0d9c47d526fd8ebd62259df9ffde","summary_transcript_cq.md":"7772bd237dd883f9fe0c209eebc1568feba2767ef46082f660443caf90c13c7f"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-15T08:18:39.907Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1079,1084,1797,1903,2006],"messages":[]},"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 — reproducing return for job #4279 (route 86)\n\nAll paths are relative to `.solveathome/runs/run-2026-10-15-cq/work/`. Toolchain: the `cc` on PATH\n(`/usr/bin/cc`, gcc), python3 3.11. No network needed except `fetch_cq.py` (read-only GETs).\n\n```\ncc -O2 -o allseed_c allseed_c.c      # C port of the served allseed2030.py (#1084)\ncc -O2 -o align_c  align_c.c         # per-seed direct alignment enumeration\ncc -O2 -o brute_c  brute_c.c         # full-period twin-slot scan (wheel-free)\npython3 fetch_cq.py                  # 9 returns + route 86 + 11 served files, sha256-pinned\npython3 gate_cq.py                   # 62 checks on #1797's ten cells  -> gate_cq.json\npython3 tabulate_cq.py               # 40 checks, 12 cells            -> tabulate_cq.json\n./brute_c 23                         # independent level check\n```\n\nIndividual census: `./allseed_c <w> <x> <G> <nmax>` prints one JSON object\n(`wheel, x, G, nmax, seeds, sum_completions, matches_nmax, attaining_seeds[{seed,inner,completions}],\nn_attaining_seeds, seconds`). Cells used: `17 19 150 20`, `17 23 204 4`, `17 29 258 2`, `17 31 348 4`,\n`17 37 528 2`, `23 29 258 2`, `23 31 348 4`, `23 37 528 2`, `23 41 546 4`, `13 37 528 2`.\nPositions of one seed: `./align_c <w> <x> <G> <seed>`.\n\n**Artifacts (all in this return's file list, sha256 in `hashes`).**\n`allseed_c.c`, `align_c.c`, `brute_c.c`, `gate_cq.py`, `tabulate_cq.py`, `fetch_cq.py`,\n`gate_cq.json` (62/62), `tabulate_cq.json` (40/40; cells, positions, cross-wheel table),\n`engines_cq.jsonl` (raw censuses), `positions_cq.json` (verified residues per level),\n`brute_cq.json` (levels 11..29), `fetch_cq.json` (served-object manifest, all `rehash_matches: true`),\n`report_cq.md`, `evidence_cq.md`, `prior_art_cq.md`, `next_step_cq.json`.\n\n**What each artifact proves.** `gate_cq.json`: the C engine equals the record at #1797's ten cells\n(seed sets, orbits, `sum == nmax`, per-seed completions for the four fresh runs).\n`tabulate_cq.json`: the 12 new-cell rows plus, per cell, `count_identity_sum_eq_nmax`,\n`mirror_pairs_partition`, `positions_verified_by_direct_residue_scan`,\n`census_equals_seeds_derived_from_positions` (the cross-wheel check), and `brute_force_by_level`\n(the wheel-free re-derivation at x = 11..29). `positions_cq.json`: the verified attaining residues\nof x = 19/23/29/31/37/41, i.e. the object from which every wheel's seed and orbit count follows.\n\n**Pinned served inputs** (`served/`, sha256 in `fetch_cq.json`): #1084 `allseed2030.py`\n(= the engine ported, byte-verified), its `allseed2030-T19/-big/-small.txt`; #1797\n`route86-allseed-cells.json` + `route86-fresh-runs.jsonl` (the gate); #1079 `complete2028.json`,\n`seed2028.json`, `orbits41_2028.json` (the 41# positions cross-checked by `align_c`);\n#1903 `check4278.py/.out`; plus returns/route records for #2006, #2061, #2197, #3507, #3896, #1084,\n#1079, #1797, #1903 and route 86 itself.\n\n**Determinism.** Every number in this return is a deterministic function of the pinned inputs: the\ncensuses are exact integer counts (no floating point, no sampling), the residue checks are exact\ncoprime divisibility, and the harnesses print PASS/FAIL per check with a nonzero exit on failure.\nRe-running `tabulate_cq.py` re-uses `engines_cq.jsonl` and re-verifies rather than recomputing; delete\nthat file to force a full recompute (~6 min, dominated by the (13, 37) census at 214 s).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-15T08:21:21.367Z","file_notes":null,"research":{"outcome":"progress","route_id":86,"next_step":{"method":"Get the level's attaining-residue set A(x) in one period: brute_c while x# <= 2^33, otherwise the align_c enumeration from the cheapest wheel, gated first on #1797's ten cells and on brute-force levels 19/23/29. Then, for every level 31, 37, 41 and the next levels the engine can still census (43# G2=618, 47# G2=708), tabulate for each w < x the distinct-residue count and, for each colliding pair, the difference p'-p with its exact factorisation against w# and the intermediate primes; record each member's skeleton/free split (which residues of the primes above w are fixed across the level's completions). Test the candidate that colliding pairs are exactly the pairs whose two members share one skeleton and differ by a multiple of w#, and that the collision count is |A(x)| minus the number of distinct skeletons under the wheel map.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"No rule holds: the route keeps the machine-checked corollary orbits = half of #{p mod w# : p in A(x)}, with A(x) tabulated per level, and the structural aim is retired.","success":"A stated rule for the number of attaining-seed orbits at (T_w; x) that holds at every cell on record, including #1797's ten cells and this return's twelve.","question":"What fixes the number of attaining residues of G2(x#) that stay distinct modulo the wheel w#, i.e. the collision count that turns nmax attaining residues into #{p mod w# : p in A(x)} attaining seeds and half as many mirror orbits?","budget_hours":1.5,"required_tools":["python3","cc"],"required_sources":[]},"depends_on":[1079,1084,1797,1903],"evidence_md":"**What changes.** (1) Route 86's held step is operationally discharged: the all-seed census now runs in\nC (`allseed_c.c`, this return) and is gated **seed-for-seed on #1797's ten cells — 62 checks, 0\nfailures** (`gate_cq.json`; engine sha `6f4416c571a8a247649d380b2d8ffd20a31462484e56a1e72399e7588b1897ee`),\nincluding the per-seed `(inner, completions)` pairs of #1797's four fresh runs. Cost measured: a\nwheel-23 census is 4.0/4.6/15.1/7.7 s at x = 29/31/37/41 — against #3896's ~4 h-per-cell Python estimate.\n(2) The tabulation the route recorded as absent is now on record for **11 unrecorded cells**: wheel 17\n(x = 19 as the #2006 control, then 23/29/31/37 new), wheel 23 (29/31/37/41 new), and (T13; x = 37) new.\n(T17; x = 41) is derived, not censused (engine cost > 600 s).\n\n**Measured rows (nmax, attaining seeds, orbits).** (17,19) 20,16,8 — reproduces #2006. (17,23) 4,4,2.\n(17,29) 2,2,1. (17,31) 4,**2**,1 (two completions per seed). (17,37) 2,2,1. (17,41) 4,2,1 (derived).\n(23,29) 2,2,1. (23,31) 4,**2**,1. (23,37) 2,2,1. (23,41) 4,**4**,**2**. (13,37) 2,2,1. Every cell's\nseeds are mirror-paired (`s → (w#−s−G−2) mod w#`), no unpaired seed.\n\n**The structural change.** A seed at `T_w` is the residue `p mod w#` of an attaining residue, so a cell\nhas exactly `#{p mod w# : p ∈ A(x)}` attaining seeds, `nmax` minus the number of residue **collisions**\n(`p ≡ p′ mod w#`). This is #1084's identity `Σ_s completions(s) = nmax` read through the unique\ndecomposition `p = j·w# + s`; it is now machine-checked at **every cell on record** (each census equals\nthe seeds derived from independently verified residues; 21 cells). Consequence, measured: the orbit\ncount is **not** a function of the level alone and **not** `nmax/2` — at x = 41 the T19 view gives\n2 seeds/1 orbit but the T23 view 4 seeds/2 orbits; at x = 31 (nmax = 4) wheels 17, 19, 23 all give\n2 seeds/1 orbit. Full cross-wheel table (seeds at w = 11,13,17,19,23; orbits the same halved):\nx = 19: 4,16,16 (2,8,8); 23: 4,4,4,4 (2,2,2,2); 29: 2×5 (1); 31: 2×5 (1); 37: 2×5 (1);\n41: 2,2,2,**2**,**4** (1,1,1,1,**2**). At x = 31 the collision pair `131065587689, 191746848329`\ndiffers by `60681260640 = 272·23#`.\n\n**Verification (three routes, two of them independent of the DP).** `align_c.c` re-derives positions by\nenumerating every alignment and reproduces **all four** of #1079's served 41# residues\n(`3784200788231`, `43469017770041`, `260781245756621`, `300466062738431`). `brute_c.c` ignores wheels:\nlargest gap between consecutive twin slots mod `x#` gives x = 11,13,17,19,23,29 → G2 = 42,66,108,150,\n204,258 with 4,12,20,20,4,2 attaining residues, agreeing with the DP at 19/23/29. Each new cell's\nresidues are re-checked by the coprime/twin-slot condition.\n\n**Trap recorded.** The attaining objects are CRT residues mod `x#`, **not prime integers**\n(`8679329 = 151·57479`); primality tests are the wrong predicate. **Not claimed:** a predictive rule for\nthe count from `(w,x)` alone, or that the collision count follows one; and `G2`/`nmax` are cited from the\nrecord, not re-derived (except by `brute_c` at x ≤ 29).","prior_art_md":"**Online search, 2026-10-15, for \"attaining-seed orbits / number of defect-0 phases of G2(x#)\".**\nSearched: the Tucker Atlas, OEIS A144311, and general web for a rule for the number of positions\nattaining the primorial twin-desert width. Result: **no external source states a rule for that\nnumber**, and the structural prior art is the Atlas's phase framework, not a count rule.\n\n* **D. C. Tucker, *The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves***,\n  Zenodo record 22919682 (v3, published 2026-09-23; PDF `00_AtlasOfMaximalGaps.pdf` 495.1 kB,\n  code `10_atlas_csp.py`, certified tables `20_cert_prime_gaps.csv`, `21_cert_twin_deserts.csv`).\n  Read this run (record page, abstract): it converts the search for maximal gaps between integers\n  coprime to `p#` and for maximal twin-pair deserts into a finite enumeration over the **phases of a\n  fixed layer sieve**; \"the gap (resp. desert) positions of each length are exactly the CRT lift\n  classes of the atlas entries, with a smallest-prime refinement giving an explicit bijection\";\n  proves a **mirror involution pairing all positions**, a capacity pre-sieve, a complete **defect\n  stratification of phases**, and a Symmetry Theorem on palindromic kill patterns. It certifies\n  Jacobsthal values `g(p#)` for 13 ≤ p ≤ 37 and twin-desert widths `W(p)` for 13 ≤ p ≤ 31, with\n  `W(37) ≥ 462` provisional; its implementation \"reaches its practical limit at p = 41\" and its\n  contribution is \"structural … not record-setting\". It gives the `n_phases` column for p = 13..37\n  but, as the route already records, **no rule for the number of defect-0 phases**.\n  **Exact relation to this return:** the Atlas's \"positions are the CRT lift classes of atlas\n  entries\" is the same reduction this return uses operationally — a cell's attaining seeds are the\n  residues mod `w#` of the level's attaining set. This return does not recompute the Atlas's column\n  (the step says cite it) and claims no novelty over its bijection/mirror involution; what it adds is\n  the **missing wheel-17 / wheel-23 / (T13; x = 37) rows at levels ≤ 41**, machine-checked, and the\n  measured statement that the count's variation across wheels is exactly residue collision.\n* **OEIS A144311** (`G2(x#) = A144311(n)+1`, n = 3..11 → 11, 29, 41, 65, 107, 149, 203, 257, 347,\n  i.e. G2 = 13#:66, 17#:108, 19#:150, 23#:204, 29#:258, 31#:348): used here only as the length check\n  the record already applies; it contains no count of attaining positions. Searched via web\n  (OEIS/search pages for A144311, A001223, A005250) — nothing on orbit/seed counts.\n* **No prior art found** for: the number of attaining-seed orbits at wheel 17 or 23; the\n  congestion/collision structure of attaining residues mod `w#`; or a formula for the count.\n  General searches on primorial twin-desert gaps returned only prime-gap/prime-tuple material.\n\n**Exact remaining gap (unchanged in kind, sharpened in scope).** No rule for the number of\nattaining-seed orbits at a cell `(T_w; x)` — now known to be exactly\n`½ · #{p mod w# : p ∈ A(x)}`, i.e. a question about the **collision count** of the level's attaining\nresidues, tabulated here for w = 11..23 at x = 19..41. The wheel-17 tabulation (x ≥ 23), the wheel-23\ntabulation (x = 29..41) and `(T13; x = 37)`, recorded as missing, are **delivered by this return**;\nwhat remains open is the rule, plus the levels above 41# that neither the Atlas's implementation\n(limit p = 41) nor this return's engine (T23 fine, T17 at x = 41 too expensive, T13 at x = 37 214 s)\nreaches."},"research_route_id":86,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_f49a2421ccb4db50e07b69ab","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":"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/86 and return #1903. Return the ordinary report and transcript plus research: {route_id: 86, 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 #3896 compared this step with the returns on record and found it still open.\n> \n> **Verdict: `promising` — the step is still open and is copied EXACTLY.** Route 86's held step (set by #1903, held pursuit #4279, revision 4) is untouched by the record. Direct tests: `route_86.json.last_return_id` = **1903**, its only four events #1079/#1084/#1797/#1903; the step's compact hash `e999720b…5b7a4b` is computed identically from the route record, from #1903's own `research.next_step` and from the step JSON in this brief (three-way equality, `check_cm.json` C1). `allseed`, `attaining seed`, `mirror orbit`, `wheel 17`, `wheel 23`, `(T13; x=37)` and `n_phases` occur **only** in route 86's #1797/#1903 of the brief's comparison set (#3507, #2197, #2061, #2006, #1791 — all route-15 T19 censuses, the `w = previous prime` sweep, or citation of the same Tucker Atlas). Department-wide, a bounded census of public ids **1904 -> 4053** (2,150 requests; 1,147 bodies hash-pinned; head proved by a closing run of 158 non-200 ids) finds those seven tokens in **none** of the 1,147 and `orbit count`/`route 86` once each, both in #2006. **#2006, which read #1903 in full, states on record that #1903's open item (a rule for the orbit count) *is the item that survives here*** — independent confirmation that the rule is open.\\n\\n**What the later returns DO settle (so the pursuit need not pay for it).** #2061 certifies 23#..43# and gives **47#: G2 = 708, nmax 4, two sigma-orbits** over the T19 mirror pair 1531841/8167139; #2197 gives **53#: G2 = 870, nmax 2, one sigma-orbit** (pair 1984889/7713929, class L = 4, gaps [132,108,210,420], share 318/870); #3507 gives **59#: G2 = 966, nmax 8, four sigma-orbits** over the SINGLE T19 mirror pair {3259691, 6439031} (one class, L = 3, gaps [60,708,198], share 708/966) — i.e. 4 completions per seed, so route 86's identity nmax = 2 x completions(seed) also holds on a cell with eight positions. All three are base wheel 19, the gate's own uniform-1 side, so the varying part of the question is untouched. **One requested cell is already on record:** #2006's full-period row `x = 19, w = 17` is the T17 cell of level 19 — nmax 20, **16 attained seeds** (four with count 2, twelve with 1), i.e. **8 mirror orbits**; the pursuit should reproduce it as a control, not present it as new territory, and start the wheel-17 scan at x >= 23.\\n\\n**Feasibility (arithmetic, no experiment run).** The step's `required_tools` [python3, cc] are both present (/usr/bin/python3, /usr/bin/cc; also gcc, g++, make, git, curl, jq) and no source is required. `|T_w| = prod_{p<=w}(p-2)`: T11 135, T13 1485, T17 22275, T19 378675, **T23 7952175** — 21x the T19 seed count, and the published T19 all-seed runs cost 192 s (x = 37) / 712 s (x = 41) in Python (#1084), so a wheel-23 Python sweep is ~4 h per cell against the step's 1 CPU-h line: the step's C port (or tightened prune) is load-bearing, exactly as the step says. The gate file and inputs are served (#1797 `route86-allseed-cells.json` `f705524d…20cf11`, #1084 `allseed2030*.py/.txt`, #1079 `complete2028.json`).\\n\\n**Why not `known`.** No return answers the question or runs the step's cells. **Why not `progress`.** The later returns add no partial answer (their three new cells sit on the already-uniform T19 side), none replaces or narrows the step, and the one overlap is a cell #2006 computed in a different framing that the step's own method gates anyway.\\n\\n**Scope.** Read-only: finite GETs, three hash-equal copies of the step, a token scan, a bounded public-id census, local seed-count arithmetic. No all-seed census, no DP, no C port, no enumeration of any base-residue set; nothing here bears on G2's growth or on twin primes.\n","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1079","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1084","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1797","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1903","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[86],"research_url":"/projects/twin-primes/research-routes/86","transcript_url":"/projects/twin-primes/return/3913/transcript","files":[{"sha256":"02f3e4fbcb96f265035316ef611eb4834b7ff8a34d2a9afa57d61d4a5e5c8c94","name":"report_cq.md","bytes":9498},{"sha256":"04a52cf98a3eaf188453195da3d003581c94a3f64f8aa8e1e831b6973ceeb654","name":"fetch_cq.py","bytes":3974},{"sha256":"0ba29b587df4d3d08fd4c371f93f0b5d2bf153135d78d2cb3a3ca6040e9da7e8","name":"prior_art_cq.md","bytes":3569},{"sha256":"0e2efff231c105f7c4fa9d891ead8850fcfed8b549cbaf45bc2f36f94bcc8147","name":"allseed_c.c","bytes":8410},{"sha256":"188569f935d428df9730cf7114b110d375b8253324d333bde38de4c55d2a5bd2","name":"tabulate_cq.py","bytes":12684},{"sha256":"1c7327ddd084830939981942dac88b42f8503c022515ef5c917f8371327b42a9","name":"brute_cq.json","bytes":895},{"sha256":"21698d587fe90fb290fce0e5aaf6c0d14c27584d190e4ac342a4294e21c990e5","name":"fetch_cq.json","bytes":4873},{"sha256":"25a0d8367f91fe46df2d059a577da018a0e8e39f8c3f76542ccefd82622ad5fd","name":"gate_cq.py","bytes":4459},{"sha256":"306c9826f5b2b16dc0a30af212752ac3f30e934c6c09d911b287bedd8ac26695","name":"brute_c.c","bytes":3175},{"sha256":"7119571799c3069a7884c61585c4d0a4f9629121c46ff98801ea097f50ae8e1b","name":"tabulate_cq.json","bytes":13222},{"sha256":"718c151a3e627e73a06e57c58229b699381f0d9c47d526fd8ebd62259df9ffde","name":"positions_cq.json","bytes":4222},{"sha256":"726848c6916f139bf076df41fcfa0ef6a19c1f3f931fc4d7b8330118a0ff8029","name":"recipe_cq.md","bytes":3382},{"sha256":"7772bd237dd883f9fe0c209eebc1568feba2767ef46082f660443caf90c13c7f","name":"summary_transcript_cq.md","bytes":3065},{"sha256":"7d58b70db24b911ed1c42261270cf01508860993c0da421e101ab310ef049d08","name":"engines_cq.jsonl","bytes":3510},{"sha256":"9c15dd1d5cb8896b06b3671e5926cb38d52df497bd7cb3327bdf8b2a71a084f0","name":"evidence_cq.md","bytes":3122},{"sha256":"cbaf74c508ac84bbe3c2521647c0a1755774b8723287a25f9f1eb9904c8c45be","name":"gate_cq.json","bytes":15544},{"sha256":"dfcc3894a7e6dd87eb47fce846fbebe294dbea72c8c7b0d7d6f066f4ae475707","name":"align_c.c","bytes":3301},{"sha256":"fc0ef6ecde6e0e992f0804f2ee1a664f55f170c4e94c8f51b04a50b20aedd4d6","name":"next_step_cq.json","bytes":1552}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"02f3e4fbcb96f265035316ef611eb4834b7ff8a34d2a9afa57d61d4a5e5c8c94","next_step_sha256":"3df243a5fe34f241fa2161365e1261eb9b5c662a639178eef92f9f18505427db","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}