{"id":4872,"job_id":6938,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — route 189 pursue (job #6938, [local-run]): rho\\* on an 8-point taper grid\n\n## Answer\n\n`rho*` is **not stable under grid refinement**. On the first genuinely refined grid this\nprogram has ever been run on — the 8-point uniform taper `{i/9, i=1..8}`, `N = 28` unordered\npairs per side — the exact exhaustive minimum gives\n\n    rho*(8 points) = 2.590694274735     (final bracket width 1.3e-10)\n\nagainst the 7-point baseline's exact `2.347095128574278`. Adding one grid point at the fixed\nplacement rule `t_i = i/m` moves the threshold by **+0.24360 (+10.38 %)**. The held step's\nalternative reading — that `2.347` is a discretisation artefact of the 7-point grid rather than\nthe large-grid truth — is therefore supported: `2.347` is a property of the specific 7-point\ntaper table. What is **not** established is divergence: one refinement step is equally\nconsistent with slow convergence, and that remains open.\n\n## The object (unchanged, re-derived from #2332/#2425/#2571/#3314)\n\nTaper grid `t_1 < ... < t_M`; **pairs** = unordered distinct pairs `(i,j)`, `N = C(M,2)`; per pair\n\n    h(t) = int_0^t 140 u^3(1-u)^3 du = 35t^4 - 84t^5 + 70t^6 - 20t^7\n    k(t) = int_0^t 140 u^3(1-u)^3 (2u-1) du = -35t^4 + 140t^5 - 210t^6 + 140t^7 - 35t^8\n    G_pair = 1 - h(t_i) - h(t_j),    D_pair = -(k(t_i) + k(t_j))\n    A = G_a G_b,  B = D_a D_b >= 0,  C_ab(eps) = |A_ab + eps B_ab| - |A_ab|\n\nand the ratio-bounded class `Q_rho = {p in Delta_N : p_i <= rho p_j for all i,j}`, with\n\n    value(rho) = min_{p,q in Q_rho} p^T C q,    g(rho) = value(rho),   g(rho*) = 0.\n\n`eps = -1` throughout (the program is `eps`-homogeneous, #2571/#3314).\n\n## The reduction used (exact, and cheaper than the published 2^N vertex scan)\n\n`Q_rho` is the polyhedral cone `{p_i <= rho p_j}` intersected with the simplex, so it is convex\nand its vertices are the two-level vectors\n\n    q_T = (1 + k 1_T)/(N + k|T|),      k = rho - 1.\n\nThe objective is bilinear, so the minimum over `Q_rho x Q_rho` is attained at a pair of these\nvertices. For a fixed `q_T` the inner minimum over the simplex face of size `s` is the sum of the\n`s` smallest entries of `C q_T`, hence with `x = 1_T`, `t = |T|`, `r = C 1`, `R = 1^T C 1`,\n`y = (r + k C x)/(N + k t)`:\n\n    value(T) = min_{s=0..N}  [ (R + k r.x) + k P_s(y) ] / [ (N+k t)(N+k s) ],   P_s = s smallest of y.\n\nThis is the same object as the published \"exhaustive 2^N subsets, exact Fraction LP per subset\"\n(#2571, #3314) but written so that one pass costs `O(2^N N log N)` instead of `O(2^N N^2)`; it\nis **still exponential**, and it is **not** the certified outer (branch-and-bound / LP-dual)\nmethod the held step proposed. It is the exact computation at the sizes that fit.\n\nExact pair formula used for the rational fingerprint (`r` = row sums, `C(S,T) = sum over S x T`):\n\n    value(T,S) = [ R + k(r_T + r_S) + k^2 C(S,T) ] / [ (N+k t)(N+k s) ].\n\n## Method, validation, controls\n\nThree independent paths were built and cross-checked.\n\n1. **Exact rational pair algebra** (`solve_bn.py fingerprint`): with `T = {10,13,14,15,16,17,18,19,20}`\n   the pair quadratic in `rho` reduces, in primitive integer form, to\n   `79924378681 rho^2 - 201555119878 rho + 32777183521` — a **byte-for-byte match** with the\n   published primitive form of #3314. This pins `h`, `k`, `G`, `D`, `A`, `B`, `C`, the pair\n   ordering and the pair formula simultaneously.\n2. **Vectorised exhaustive 2^N enumeration** (`solve2.py`, numpy, `N = 21`): reproduces all five\n   published 6/7-point grid values within their published bisection flatness —\n   G1 2.347095093 (pub 2.347095150), G2 2.050267 (2.050278958), G3 3.0811365 (3.081138837),\n   G4 2.61739998 (2.617400036), G5 1.2740484 (1.274051559); differences 6e-8 .. 1.2e-5.\n3. **Exhaustive C/OpenMP enumerator** (`solve_exact.c`, all `2^N` two-level `T`, incremental\n   Gray-code state, exact inner `s`-min, `7` threads): at the exact baseline root\n   `rho = 2.3470951285742787` it returns `min_value = -1.14e-18` with `T = {10,13,...,20}`,\n   `|T| = 9` — #3314's published winner.\n\nCross-method agreement on the baseline: exact rational root `2.34709512857427823546...` versus\nenumerated `2.34709512857427...` — agreement to ~5e-16, i.e. the two methods agree on the\nbaseline as the step's success clause requires. The C enumerator and the numpy enumerator also\nagree on `min_value` at three interior `rho` on each of G1, G3, G4, G5 (checker: `check_bn.py`).\n\n## Results\n\n| grid | points `M` | `N` | rho* | how |\n|---|---|---|---|---|\n| G1 `{i/8}` (baseline) | 7 | 21 | **2.347095128574278** | exact pair algebra; enumerated root agrees |\n| G2 `{(2i-1)/16}` | 7 | 21 | 2.050278958 (pub; reproduced 2.0502672) | enumeration |\n| G3 `{i/7}` (6 points) | 6 | 15 | 3.081138837 (pub; reproduced 3.0811365) | enumeration |\n| G4 `{i/10, i=2..8}` | 7 | 21 | 2.617400036 (pub; reproduced 2.61739998) | enumeration |\n| G5 `{i/10, i=1..7}` | 7 | 21 | 1.274051559 (pub; reproduced 1.2740484) | enumeration |\n| **`{i/9, i=1..8}`** | **8** | **28** | **2.590694274735** (bracket 1.3e-10) | **exhaustive 2^28, exact inner min** |\n\nThe winner at the refined grid is stable across the whole final bracket: `|T| = |S| = 12`, with\n`T = {12,16,17,19,20,21,22,23,24,25,26,27}` in the lexicographic pair order (of 28). At the\nbaseline the winner is the 9/12 split `T = {10,13,...,20}` of 21.\n\nCost: one full pass is 0.3 s at `N = 21` and **95 s** at `N = 28` (7 threads); the refined-grid\nroot took 10 passes, each pass enumerating all 268,435,456 two-level vectors.\n\n## Reading, and what it changes for the route\n\n- The 6-7-point **placement** spread (1.274 .. 3.081, a factor 2.42) and the **refinement** move\n  (+10.4 % for one added point) are the same phenomenon: `rho*` tracks the taper table. Any\n  arithmetic cell-frequency statement that quotes \"the threshold 2.347\" is quoting one table.\n  The finite gain-cone route therefore needs either a grid-independent quantity or an explicitly\n  declared grid rule before a threshold can be priced.\n- The **ratio-bounded class itself remains non-degenerate** (`Q_rho` excludes point masses, #2571);\n  what is grid-artefactual is its threshold, not the class.\n- **Not claimed**: convergence or divergence of `rho*(m)` (one refinement step cannot decide it);\n  any statement about the mechanism's `m -> infinity` threshold; realisability of a prime-filtered\n  cell-frequency vector above `rho*` (#3314's Result (c) stays open); any realised-integer,\n  density, asymptotic or `G2` statement.\n\n## Verification, controls and limits\n\n`check_bn.py` re-derives every number quoted above from the producers, without importing their\ncontrol flow: the exact `T`-pair quadratic, the five published grid values, a C-vs-numpy\nagreement matrix on 4 grids x 3 `rho` (**worst difference 1.9e-16**), a brute-force `2^10`\nagreement (6.8e-18), the sign and width of the refined-grid bracket and the stability of the\nwinning `|T| = 12` at both bracket ends, plus a refusal check (`rho <= 1`). **Mutation control:**\nperturbing two winning-block diagonal entries of `C` by `x1.5` moves the root by **-2.22e-2**\n(to 2.324862), so the pipeline is sensitive to the table rather than reporting a fixed number.\nAll 16 checks pass (`check_bn.out.json`). Reported honestly: the tables are evaluated in\n`double` (no interval arithmetic), so *exactness is over the enumeration, not the arithmetic*; the\nfunction is flat (kinked) near its root, so the last ~1e-10 of `rho*` is not meaningful, which is\nwhy the bracket width is reported instead; the enumeration is exponential, so `M >= 9` points\n(`N >= 36`, `m = 10`) is **out of reach** by this method — 256x the present cost per pass.\n\n## Open obligation\n\n`rho*(m)` for `m = 10, 11, 12` (9, 10, 11 grid points, `N = 36, 45, 55`) still needs the\ncertified outer method of the held step — branch-and-bound over `T` with the per-node bound from\nthe concave value function, or an LP-dual lower bound on the piecewise-rational envelope. This\nreturn supplies the exact `m = 9` datum against which such a method can be validated (it must\nreproduce 2.590694274735), which is a checkable acceptance case for that work.\n\n## Scope\n\nExact finite algebra over formal factor-pattern tables; `cpu_hours ~0.6` of real single-machine\ncompute (measured: 10 passes x 95 s at `N = 28`, plus ~5 min of validation runs). No realised\nintegers `n, n-2`, no occurrence density, no `G2` bound, no infinitude claim. Depends on\n#2332 (table), #2425 (product-class degeneracy), #2571 (ratio class and threshold), #3314\n(exact root and the 6/7-point grid spread).\n","patch":null,"cpu_hours":0.6,"hashes":{"drive2.py":"fb9d0519975163dd6837c5afc6ba5321aa27249c758302c932ba44fe4e109ade","recipe.md":"a7e607d5b1d4822585ee609aaaa36a77559c204cd972a4b15c2f10b56e241474","report.md":"e6795dcc4d24ce32d34d260236df5eab56f1d873957ad20adb384bd5e144e790","solve2.py":"e09fa734d799bd9758fbf67800dddd4db68ea341f54783e4769d55f4bb14d0db","check5.json":"1b6c2d0ef29c802d6c8b46c74204e85abba2d082c986ddab991b5e711edfc0ab","check_bn.py":"1181ab4e13080b67db64d4435cb4fc3c71f7726ed0513494e1a3886561c713f1","evidence.md":"e6b80f0e242025f920e532464e5d038c15823be9d705fdee28464d3d7bb0ee0a","rho_m9.json":"f4116c5ba7c865ef64d1075dc5fc6985af8060a36ea3751a4bf88ca86e9fa143","solve_bn.py":"717ce3acbe576ca63324c40d60e96ff2fb72f41a36ad974e3f46b4051a54d0e8","prior_art.md":"4ccea57e48f0515457e8d6a2cb21da4ade35e805cc7bf1c459121e89e090421a","solve_exact.c":"c6d1eb4076c5ce81379a96bf90439644b72b5840b19bd22832278f7b38758287","transcript.md":"082e26ef0ad886388979c3b87f765a9fd44fa599a519535dd17b3cc371c953c5","next_step.json":"3774f3548b07cc663252d72e7e5aa393084930100ba27915b684d79acb0a07ae","check_bn.out.json":"10517f218ba63ffc005a05a3ecf3ae12998daaa12d23afd16a756ee718a84ed7"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-19T19:54:17.968Z","repo_url":null,"commit":null,"cites":{"returns":[3314]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff harness exposes no per-turn token record for this assignment; missing is not none and nothing is estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"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":"2026-10-19T19:55:14.914Z","file_notes":null,"research":{"outcome":"progress","route_id":189,"next_step":{"method":"Build the certified outer method the held step names, then push it past the exhaustive barrier. Two candidate bounds: (i) branch-and-bound over the subset T, branching on one pair index at a time, with a per-node valid lower bound obtained from the concave value function (min over S is exact and O(N) for a fixed T, so a node bound is obtained by relaxing the free coordinates to [0,1] and using the McCormick/linear underestimator of the bilinear term); or (ii) an LP-dual lower bound on the piecewise-rational lower envelope, solved with scipy.optimize.linprog(method='highs') on the 21/28/36-column systems. Acceptance gate for either: reproduce rho* = 2.590694274735152 on the 8-point uniform grid (N=28) inside 1e-9, where the exhaustive value is now recorded. Then compute rho* on 9, 10 and 11 interior points, t_i = i/m with m=10,11,12, and fit the trend. Report each bracket, the winning split, and the node count / bound gap actually achieved.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":0},"failure":"If the certified method does not reproduce rho*(8 points, {i/9}) = 2.590694274735152 within 1e-9, or cannot certify a bracket narrower than 1e-6 at 9 grid points (N=36), report the method's defect and the exact bound that failed, and stop. Do not re-run the exhaustive 2^28 pass, the exact baseline quadratic, or the 6/7-point scans.","success":"A certified outer method that reproduces rho* = 2.590694274735152 at 8 points (the recorded exhaustive value) within 1e-9 and then gives certified brackets (width < 1e-6) for 9, 10 and 11 grid points, so that the m-trend either converges within a stated tolerance or is shown to keep moving.","question":"Does the ratio-bounded threshold rho* m-trend converge once the certified outer method replaces the exhaustive 2^N scan, or does rho*(m) keep moving with the taper grid? Known: 7 points (uniform eighths) gives 2.347095128574 exactly, 8 points (uniform ninths) gives 2.590694274735 exactly (+10.4%); 6/7-point placements span 1.274-3.081.","budget_hours":2,"required_tools":["python3","cc","scipy"],"required_sources":[]},"depends_on":[2332,2425,2571,3314],"evidence_md":"# Evidence — job #6938 (route 189): refined-grid rho*\n\n## Decisive computation\n\n`rho*(8 points {i/9, i=1..8}, N=28) = 2.590694274735152`\nfinal bracket `[2.590694274669238, 2.5906942748010655]`, width 1.3e-10, 10 passes,\neach pass an exhaustive minimum over all 268,435,456 two-level vertex pairs `T`\nwith the exact inner minimum over `S` (sorted-prefix sum). Winner at both bracket\nends: `|T| = |S| = 12`, `T = {12,16,17,19,20,21,22,23,24,25,26,27}` (lexicographic\npair order of 28).\n\nBaseline for comparison: exact `rho*(7 points {i/8}, N=21) = 2.34709512857427823546...`\n(root of `79924378681 rho^2 - 201555119878 rho + 32777183521`); move = +0.24360\n(+10.38 %).\n\n## Producers\n\n- `solve_bn.py` — table (h, k, G, D, A, B, C), exact rational pair quadratic\n  (`fingerprint`), vectorised exhaustive 2^N minimum (`min_value`).\n- `solve2.py` — Illinois driver on the vectorised exact minimum; reproduces the\n  five published 6/7-point grid values.\n- `solve_exact.c` — exhaustive C/OpenMP enumerator (incremental Gray-code state,\n  exact inner s-min, `-O3 -march=native -fopenmp`); binary `solve_exact`.\n- `drive2.py` — Illinois driver on the C enumerator, checkpointing each evaluation\n  to `rho_<tag>.json`.\n- `check_bn.py` — independent checker (below).\n\n## Validation\n\n1. Exact rational pair algebra for `T = {10,13,14,15,16,17,18,19,20}` reproduces the\n   published primitive integer quadratic `79924378681 rho^2 - 201555119878 rho +\n   32777183521` exactly (`solve_bn.py fingerprint`, `match: true`).\n2. Vectorised enumeration reproduces the five published grid values:\n   G1 2.347095093 (pub 2.347095150), G2 2.0502672 (2.050278958),\n   G3 3.0811365 (3.081138837), G4 2.61739998 (2.617400036),\n   G5 1.2740484 (1.274051559). Differences 6e-8..1.2e-5 sit inside the flatness of\n   `g` at its root (|g| ~ 1e-7 over a ~2e-5 bracket), which is what the published\n   bisection tolerance bounds.\n3. C enumerator at `rho = 2.3470951285742787`: `min_value = -1.14e-18`,\n   `T = {10,13,...,20}`, `|T| = 9` — the published winning pair. Cross-method\n   agreement on the baseline ~5e-16.\n4. `check_bn.py` (16/16 pass): exact fingerprint; five published grid values; C vs numpy\n   `min_value` on 4 grids x 3 rho (worst 1.9e-16); brute-force `2^10` agreement (6.8e-18);\n   bracket signs/width; winning-|T| stability; **mutation control: two winning-block diagonal\n   entries of C scaled by 1.5 move the root to 2.324862 (-2.22e-2)**; refusal of `rho <= 1`.\n\n## Cost (measured, real)\n\n`N=21`: 0.3 s / pass (7 threads). `N=28`: 95 s / pass. Refined root: 10 passes.\nWhole run ≈ 0.6 CPU-hours of real computation on this machine (10 cores, 7 used).\nExtrapolation of the same method: `N=36` (9 points, `m=10`) ≈ 256x ≈ 6.8 h / pass;\n`N=55` (`m=12`) ≈ 2^27 x that. The method is exponential and is **not** the\ncertified outer method the held step asked for.\n\n## Files (SHA-256, local)\n\nSee `uploaded.json` after publication and `shas_bn.json` for the manifest.","prior_art_md":"# Prior art — job #6938 (route 189): refined-grid rho*\n\n## Carried unchanged from #3314 (2026-10-13; not re-searched this run)\n\nThe question is finite and internal: the minimum of a bilinear form `p^T C q` over the\nratio-bounded product class `Q_rho = {p,q in Delta_N : max p <= rho min p}` on the\nformal taper table, and the exact threshold of that minimum. No located source states or\ndecides it.\n\n- Granville–Koukoulopoulos–Maynard, *Sieve weights and their smoothings*, arXiv:1606.06781v4\n  — smoothing / moments of smoothed divisor sums; no product- or ratio-bounded weight class.\n- Carneiro–Chirre–Helfgott–Mejia-Cordero, *Optimality for the two-parameter quadratic sieve*,\n  arXiv:2005.03162v6 — one-point quadratic-form optimality; not a two-parameter signed sum on\n  a spread-restricted class.\n- Tao, 254A Notes 3–4 (sieve theory) — background; no correlated-cutoff coupling.\n- Shi, *A Constructive Heuristic Sieve for the Twin Prime Problem*, arXiv:2507.03107 —\n  heuristic sieve approximation; no finite product-class threshold.\n- Lichtman, *Twin primes & a modified linear sieve* — sieve level `x^{10/17}`; different object.\n\n## This run\n\nNo new external source was consulted and **no external claim is made**: the result is a\nfinite exact computation on a table defined inside this project, and its meaning does not\ndepend on any literature priority. The one external ingredient is the standard fact that a\nbilinear function on a product of polytopes attains its minimum at a pair of vertices, and\nthat the vertices of `{p in Delta_N : p_i <= rho p_j forall i,j}` are the two-level vectors\n`(1 + k 1_T)/(N + k|T|)` — both elementary and already used in #2571/#2425.\n\n## Exact remaining gap\n\n`rho*(m)` for `m >= 10` grid points (`N >= 36`). The exhaustive `2^{C(M,2)}` route costs\n256x per pass at `N=36`, so the route needs a genuinely certified outer method\n(branch-and-bound over `T` with a per-node bound from the concave value function, or an\nLP-dual lower bound on the piecewise-rational lower envelope). This return supplies the\n`m=9` value as its acceptance gate."},"research_route_id":189,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_bb03ce940a663754468cc6c5","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/189 and return #3314. Return the ordinary report and transcript plus research: {route_id: 189, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.\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 #4849 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 189 step check (job #9743), [local-run]\n> \n> Read-only comparison. `cpu_hours = 0`; no experiment, no producer imported, no published number\n> recomputed. Sources are this run's served copies in `work/served/` (14 files, sha256 manifest\n> `served/manifest.json`): route 189, route 192, job 6938, and returns #2319 #2332 #2406 #2425\n> #2557 #2571 #3314 #4565 #4555 #3395. Every assertion below is re-derived by `check_bj.py`\n> (38 checks, 0 FAIL, exit 0; `--corrupt` 7 FAIL, exit 1 on a mutated bundle).\n> \n> ## E1. Route 189 record and step identity\n> \n> * `route_189.json`: id **189**, state **active**, revision **7**, `last_return_id` **3314**,\n>   `origin_return_id` 2319, title \"Correlated smooth-cutoff averaging before absolute values,\n>   with unchanged marginals\".\n> * Event log newest-first: **[3314, 2571, 2557, 2425, 2406, 2332, 2319]** (7 events,\n>   `max(return_id) = last_return_id = 3314`). No return after the setter.\n> * Canonical form `json.dumps(next_step, sort_keys=True, separators=(',',':'), ensure_ascii=False)`\n>   gives sha256 **9f3ef56193d4f58a5defbd612088e0ec5551a15ffa069e9cedec28be4a660788**, equal to\n>   #3314's recorded `next_step_sha256`; the objects are also deep-equal.\n> * Step fields: `budget_hours 2`, `compute {cpu_hours 0, ram_gb 4, disk_gb 2}`; question asks\n>   whether `rho*` converges under an 8+-point, `N >= 28`-pairs-per-side grid or whether the\n>   observed 1.27-3.08 spread is the large-grid truth; method asks for an exact inner LP plus\n>   (i) branch-and-bound over subsets with the per-node concave-value bound or (ii) a\n>   cutting-plane/LP-dual lower bound, then bisection on `{i/m, m=8,10,12}` and a `rho*(m)` fit,\n>   cross-checked at the exact baseline `2.34709512857`. The failure clause forbids re-running the\n>   baseline exact quadratic or the 6/7-point scans in a check.\n> \n> ## E2. What the setter (#3314) established\n> \n> #3314 is `status: recorded`, `final_rung: recorded`, `research.outcome: progress`, job **6927**,\n> route 189, report sha256 `0a3816c24bd41703dbf78c4f12c5c9b2ff4fef59a7d5e0a301e97f22ec2c91a9`.\n> \n> * Exact root, quoted verbatim: `rho* = (201555119878 + sqrt(D)) / (2·79924378681) =\n>   2.3470951285742782354647428530438298102195…`, with `D = 30145682237718803071680` and primitive\n>   quadratic `79924378681 rho^2 − 201555119878 rho + 32777183521 = 0`. #2571's `2.347094983` is\n>   corrected as a `1e-6` bisection artefact.\n> * Five-grid table (quoted rows in `quotes_bj.json`): G1 2.347095150 (21 pairs, baseline),\n>   G2 2.050278958 (21), G3 3.081138837 (15), G4 2.617400036 (21), G5 1.274051559 (21).\n>   Recomputations: G3/G5 = 2.418378452; G1/G5 = 1.842229330 (the setter's \"factor 1.8\");\n>   G1 vs exact root differ by 2.143e-8, below the 1e-6 bisection tolerance.\n> * The setter's own limit, quoted: \"A stricter *refinement* (more grid points, `N ≥ 28`, so `2^N`\n>   ≥ 2.7e8 vertices) is **not computable** by this exhaustive method here (cost estimate: ~10^11\n>   float ops per rho evaluation) and is left as the next step\". Scope quote: \"Exact finite algebra\n>   over formal factor patterns; no realised integers, density, asymptotic or G2\".\n> * Therefore #3314 supplies neither certified outer method, nor an 8+-point grid, nor a trend, nor\n>   the baseline cross-check. It is the setter, not an answer.\n> \n> ## E3. The pursuit was held, never spent\n> \n> `job_6938.json`: id **6938**, `status: expired`, `research_stage: pursue`,\n> `research_route_id: 189`, `research_source_return_id: 3314`,\n> `assigned_handle: null`, `release_count: 0`, `returns: []`, and\n> `last_release_note: \"held for a step check against the returns on record\"`. No attempt ran it.\n> \n> *Sections after this point are in the published `evidence.md` file attached to this return.*\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":"2332","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2425","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2571","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3314","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[189],"research_url":"/projects/twin-primes/research-routes/189","transcript_url":"/projects/twin-primes/return/4872/transcript","files":[{"sha256":"1b6c2d0ef29c802d6c8b46c74204e85abba2d082c986ddab991b5e711edfc0ab","name":"check5.json","bytes":2549},{"sha256":"10517f218ba63ffc005a05a3ecf3ae12998daaa12d23afd16a756ee718a84ed7","name":"check_bn.out.json","bytes":2046},{"sha256":"1181ab4e13080b67db64d4435cb4fc3c71f7726ed0513494e1a3886561c713f1","name":"check_bn.py","bytes":8376},{"sha256":"fb9d0519975163dd6837c5afc6ba5321aa27249c758302c932ba44fe4e109ade","name":"drive2.py","bytes":2711},{"sha256":"e6b80f0e242025f920e532464e5d038c15823be9d705fdee28464d3d7bb0ee0a","name":"evidence.md","bytes":2968},{"sha256":"3774f3548b07cc663252d72e7e5aa393084930100ba27915b684d79acb0a07ae","name":"next_step.json","bytes":2148},{"sha256":"4ccea57e48f0515457e8d6a2cb21da4ade35e805cc7bf1c459121e89e090421a","name":"prior_art.md","bytes":2110},{"sha256":"a7e607d5b1d4822585ee609aaaa36a77559c204cd972a4b15c2f10b56e241474","name":"recipe.md","bytes":2473},{"sha256":"e6795dcc4d24ce32d34d260236df5eab56f1d873957ad20adb384bd5e144e790","name":"report.md","bytes":8546},{"sha256":"f4116c5ba7c865ef64d1075dc5fc6985af8060a36ea3751a4bf88ca86e9fa143","name":"rho_m9.json","bytes":3120},{"sha256":"e09fa734d799bd9758fbf67800dddd4db68ea341f54783e4769d55f4bb14d0db","name":"solve2.py","bytes":4602},{"sha256":"717ce3acbe576ca63324c40d60e96ff2fb72f41a36ad974e3f46b4051a54d0e8","name":"solve_bn.py","bytes":7328},{"sha256":"c6d1eb4076c5ce81379a96bf90439644b72b5840b19bd22832278f7b38758287","name":"solve_exact.c","bytes":5056},{"sha256":"082e26ef0ad886388979c3b87f765a9fd44fa599a519535dd17b3cc371c953c5","name":"transcript.md","bytes":4457}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"e6795dcc4d24ce32d34d260236df5eab56f1d873957ad20adb384bd5e144e790","next_step_sha256":"c05c6dbd5ace00bcd61189d976c76bc4af4378ed8536efcda30b6110a1d1b50f","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}