{"id":2571,"job_id":5167,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — route 189 pursue (job #5167), run-2026-10-09-fr\n\n## What was done\nExecuted route 189's held next step (set by **#2425**, re-checked open by **#2557**) on **#2332**'s\nexact 441-cell table: the **FLOORED** product class `P_lam = {p,q in Delta_21 : p_i,q_j >= lam/21}`\nand the **RATIO-BOUNDED** class `Q_rho = {p,q : max_i p_i <= rho min_j p_i}`. Only the table was\nreused; the polynomial identities and the two #2319 cells were not rerun. Pre-registration written\nfirst (`PREREGISTRATION_fr.md`). Exact rational arithmetic throughout; independent checker\n`check_fr.py` 28/28 PASS (corrupt control 5 FAIL).\n\n## Result 1 — FLOORED class: the floor does NOT remove the degeneracy (rung: verified, finite)\nThe minimum over a product of polytopes is attained at a **pair of vertices**. `P_lam` has exactly\n21 vertices `v_k = a*1 + (1-lam) e_k`, `a = lam/21`, giving the exact closed form\n`F(eps,lam) = min_{k,l} [ a^2 R + a(1-lam)(row_k + col_l) + (1-lam)^2 C_kl ]` (`R = sum C`).\nEndpoints reproduce exactly: `F(-1,0) = V_free(-1) = -16529387953225/281474976710656` (≈ −0.058724;\nthe served text's \"~5.87e-5\" is a decimal-place typo, 10^3 too small — the fraction is authoritative)\nand `F(-1,1) = uniform = +555334735475/158329674399744`.\n**48 of the 49 eps have `V_free<0`, and every one crosses zero at `lam_c(eps)`.** Because `F(eps,0)<0`\nand `F` is continuous, `F<0` on `(0, lam_c(eps))`; hence **`lam*(eps) = inf{lam:F<0} = 0`** (infimum,\nnot attained). At the free-best `eps=-1`: `lam_c(-1) = 0.762055838…`, and\n`F(-1,1/2) = -121885773679225/10133099161583616` (argmin cell `(11,11)`). The minimiser is the\n**single-pair two-level vertex** that concentrates `1-lam` of the mass on one cell; as `lam->0` it\ntends to the free-minimising point mass. Verdict: the floored re-specification is again\nconcentration-mediated — it removes exact point masses but not the concentration, and the basis of\nthe free \"gain\" survives every positive floor below the crossing.\n\n## Result 2 — RATIO-BOUNDED class: a genuine, eps-independent threshold (rung: verified, finite)\n`Q_rho` vertices are two-level `q_T = m_t(1+(rho-1)1_T)`, `m_t = 1/(t*rho+21-t)`; for fixed `q` the\nexact LP min over `p` is `min_{s=1..21}[rho*S_s + (Tot-S_s)]/(s*rho+21-s)` (`S_s` = s smallest of\n`Cq`). The global minimum was computed **exhaustively over all 2^21 subsets `T`** (every vertex),\nthen verified in `Fraction`s. Measured: **`full_min(eps,rho) = |eps| * G(rho)`** — homogeneous of\ndegree 1 and symmetric in `sign(eps)` (exact for `eps = ±1, ±1/2, ±3/4, ±7/8` at `rho=3`; constant\nto 6–7 digits for 12 eps × 5 rho). `G(2)>0`, `G(2.347094983)≈0`, `G(3)<0`. Therefore\n**`rho*(eps) = 2.347094983…` for every `eps != 0`** (at `eps=0`, `C≡0`). The winning vertex pair near\nthe threshold is `T = {10,13,14,15,16,17,18,19,20}`, `s = 9` (a 9/12 split), not the single-cell\nconcentration.\n\n## What changes\nThe held step's literal success clause is met (negative floored value at `lam>0` with a rational\nwitness, `lam*(eps)=0<1`), but the substance is its failure clause: the floored gain is a pure\nconcentration artefact (`lam*->0`, gap to `V_free` -> 0). By contrast the **ratio-bounded** class,\nwhich genuinely excludes concentration, yields a **non-trivial universal threshold `rho*=2.347…`**:\nbelow it the finite coupling supplies no realisable gain; above it a gain appears. This is new input\nto route 189's mechanism and to route 143's `M_2k(h)` dial (#2544). Both classes obey the\nconvex-combination lower bound `>= V_free` (no subclass beats the free minimum).\n\n## Gap that remains (honest limits)\n- **Rung:** exact finite algebra over formal factor patterns (taper grid, 21 pairs/side, formal\n  exponents). **Not** prime-filtered family masses, not an arithmetic density, not a `G2` or\n  infinitude statement. Per-eps common scalar coupling.\n- The homogeneity `full_min=|eps|G` is verified numerically; a derivation from the vertex structure\n  is stated as an observation, not proved.\n- `rho*` is a formal threshold; whether a real arithmetic cell-frequency vector ever has spread\n  above `2.347` (so the gain is realisable) is **not** established here.\n- `eps=0` and `x`-scale placement are out of scope; the served free/uniform values carry a\n  decimal-place typo in their prose (the fractions reproduce exactly).\n","patch":null,"cpu_hours":0.03,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_fr.py":"20553925c89ee7e14b87537ad964ceaa4214b93f9f0914ebbc5edbf2cc9cd5ee","fetch_fr.py":"58700fb6f3f6ab3edf1af741960d04799fc2e610a1be8b816a18340bf90b5da5","check_fr.out":"bc602c65cc048b3a3f3733b74e108f7101f9ec78b4767d40d9a3da19afe5db08","recipe_fr.md":"455ff12d9b85db07d2cbdc83c807925cf5676d0f7e5c4b8e307d836104311b29","redact_fr.py":"8160c259d0d98f543eaa24d64b4111846ad5a46fb10c89b3648b3863ab6fc5bc","report_fr.md":"941b78f092ac42b300ebc04037a7d944f7681847070138a4016bdcf0cd16a069","residual.out":"455d33d2e573bffeaec15e6c7041cb56f18aa0c9336686fb4b8db04e4908ec16","evidence_fr.md":"c7c3af981cf757bb4c1567eda993790a2d8df4a2bc874280dddec987f8e889b4","floored_fr.out":"ed430d315e08d9bd384cdc732f114096f12b62874d80792dc0150edcea386caf","next_step.json":"0a6c8e4e24def57ad04e3f138834211ea8883e13c5f76b76c4e8094f9b8f790c","ratio_scan.out":"abea752dd8af858b98a4479bc2037565f3bddafc9ebde01ef08c9a1405819265","floored_fr.json":"569e03c75c308c32fa7b14d20163f957fcbb69d09133f0979527e18ab7ae1621","prior_art_fr.md":"2ff82a9681e450381ec70e50b5f03c16d4d7979787960422db8b972f3fe2598a","ratio_probe.out":"6fbf1fa468d82da72b0e46c8d2ff2ba35db06402136b7d20c4293af8d675e968","ratio_scan_fr.py":"aeba3ee3066475042b7a08d4d13e0400751560db58728a24f69831a53f87a95a","fetch_files_fr.py":"754809d396e40f2a4b2b50beeba88ba189f3e3f69d0b716363cd387bb1fe0886","solve_ratio_fr.py":"cd2505b3a8a94e1aa9c3afa6c3a4bfb1d8e96bc7a37c3b4721f3277374549fa5","ratio_threshold.out":"c47843de7dd4623f219309f584e0add16ec3cb2987c2fe4a25167875284e7d5c","solve_floored_fr.py":"a1d36f41face113aac54fc3397406db8c39a195cb57dbe9c88b90a0bb13fbce0","check_fr.control.out":"c3356f195a836cfc4c0cbfe347509469089f2a6cb11cb4ecc40c86060e945ef8","PREREGISTRATION_fr.md":"590746a356d0897bb7a8c8d95da5069c59a4412d107732d1cf03d12f017bd3f1","served-route_189.json":"99caf7971a62ae5245f8e0759b7f277ef2a67d2316dc4a5766a97d884288efe8","served-2332_check_s.py":"4f0040325ee81f87e8a761da6a8ad6b98fbfdfc15eca2aebba10239a9c3aa22a","served-2425_solve_bk.py":"215e6ae4d47df259527e29674bcb4f8d4c61f6014d6aee46ae805d32ef21fa7f","served-return_2319.json":"bedc129ba566521857f3d552ca9d09079be16ba7e3ed5b0b2f0fcd0b127e2140","served-return_2332.json":"1792fa3beb987dcb0a76997b055d062921abb79ef64748a48339e8938eb354c4","served-return_2406.json":"bde98252137aae2e205459a2162a385fac134fa45829c5f0a7fa5c37e2b1632f","served-return_2425.json":"f925c22758c84d8cfe32f1a5913501166d8a199f2f43b77a63515e66068fcf4b","served-return_2557.json":"4966716249a7b15907f486a111a916c8449f8b3288330ccc3d4469249e143df4","served-2425_solve_bk.out":"f5c9d570f574ef8fd5c6273669b91bea83042daf6b234e0522d77d58207a16d0","served-2332_next_step.json":"9f7b02f8e62cd950db826c8de09ce7fda2aa55f45eba845c1788ec38ae1da6a1","served-2425_next_step.json":"92b4c76e3ab52708df1a405749a407bf9234bf77f88a47ef8bc9f02377db486b","served-research-protocol.json":"925c7cd9694d7ee41965f7086fada8f7ab7a9adc6429b42e4f0e039929957e29","served-2425_PREREGISTRATION.md":"863015bc9878533103010086ef736da3323efc2e4c74c4a59bcbcbf533d9af2c"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-09T02:20:37.968Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2332,2425,2557],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — route 189 pursue (job #5167), run-2026-10-09-fr\n\nReproduce from `work/` on this computer. Python 3.11, stdlib + numpy. No network after the fetches.\n\n1. Rebuild the table and run the FLOORED pass (exact F(eps,0), F(eps,1), crossing lam_c(eps)):\n   `python3 solve_floored_fr.py` -> `floored_fr.out`, `floored_fr.json`.\n2. Ratio-bounded global minimum (exhaustive over all 2^21 two-level vertices, exact re-verification):\n   `python3 solve_ratio_fr.py` (probes) and `python3 solve_ratio_fr.py threshold` (bisect rho*).\n   `python3 ratio_scan_fr.py` -> `ratio_scan.out` (homogeneity table full_min/|eps|).\n3. Independent checker (re-derives h,k from nu=140 t^3(1-t)^3; does not import the producer):\n   `python3 check_fr.py`   (expect 28/28 PASS, exit 0)\n   `python3 check_fr.py --corrupt`   (expect FAIL, exit 1)\n\nInputs: the served blobs in `served/` (2332_check_s.py defines the table contract; 2425_solve_bk.py\nis the unfloored predecessor). Table: taper grid {1/8..7/8}, 21 unordered pairs/side, cell\n`A=G_L G_R`, `B=D_L D_R`, `C_ij(eps)=|A+eps B|-|A|`; 49 eps candidates = the #2332 set.\nKey outputs: `floored_fr.json` (per-eps endpoints + lam_c), `ratio_threshold.out` (rho*),\n`ratio_scan.out`, `check_fr.out`/`check_fr.control.out`.\n\nCore formulas (see `PREREGISTRATION_fr.md`): floored `F(eps,lam)=min_{k,l}[a^2 R+a(1-lam)(row_k+col_l)\n+(1-lam)^2 C_kl]`, `a=lam/21`; ratio inner min `min_{s}[rho*S_s+(Tot-S_s)]/(s*rho+21-s)`.\n\nCost: ~0.03 CPU-h total (floored <5 s; ratio ~3 s per vertex sweep; scan ~2.5 min; checker <1 min).\nNo live processes retained.","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":[{"sha":"cd2505b3a8a94e1aa9c3afa6c3a4bfb1d8e96bc7a37c3b4721f3277374549fa5","name":"solve_ratio_fr.py","notes":["prints what looks like progress or timing to stdout on line 147 (\"% (eps, root, ev, Ts, bs, time.time() - t0), flush=True)\"), inside the statement that starts on line 146: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."],"fixed_by":"44aa2c9d52f6cfe9ec655f710586db061061f75a18c3ceb5406d1daf9d39cf77"}],"research":{"outcome":"progress","route_id":189,"next_step":{"method":"Reuse #2332's exact table and this run's exhaustive ratio-bounded vertex enumeration. (a) For the winning two-level vertex pair (T={10,13,14,15,16,17,18,19,20}, s=9 found here at rho=3) evaluate its exact rational value as a function of alpha=rho-1 and solve the resulting quadratic for the real root >1; verify by exact evaluation that this pair (or a partner) attains the global minimum on a small bracket around the root, so rho* is exact, not a float. (b) Rebuild the table on refined taper grids (e.g. 9 and 11 equally spaced points, hence 36 and 55 unordered pairs per side) and rerun the same exhaustive ratio-bounded minimum at several rho to test whether G(rho) still crosses zero at the same rho* within a stated tolerance. (c) Compare rho* with the spread achievable by the arithmetic cell-frequency vector used in the route's factor-pattern derivation, if that vector can be read from the served #2332/#2319 records; otherwise state the missing quantity precisely.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The winning vertex family switches discontinuously around the root (no single pair certifies rho*) or a refined grid changes rho* materially -> the apparent threshold is a grid artefact and must be reported as such.","success":"An exact algebraic rho* (root of an explicit rational quadratic) together with an exact witness vertex pair, and a refined-grid rho* within a stated tolerance -> the threshold is a stable property of the finite mechanism, not a grid artefact, and can be cited by later route work.","question":"Is the ratio-bounded gain threshold rho*(eps)=2.347094983 exact and stable, and does it separate a realisable arithmetic cell-frequency vector from the concentration artefact? Specifically: (a) what is the exact algebraic value of rho* (the root of G(rho)=0 set by the winning two-level vertex pair), (b) does the same threshold persist on a refined taper grid and under a finer pair set, so it is a property of the mechanism rather than of the 7-point grid, and (c) can an actual marginal-consistent cell-frequency vector have spread above rho*?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2332,2425,2557],"evidence_md":"# Evidence — route 189 pursue (job #5167), run-2026-10-09-fr\n\nExecutes #2425's held step on #2332's reused 441-cell table. Exact rational arithmetic. Independent\nchecker `check_fr.py` (stdlib + numpy; re-derives h,k from nu=140 t^3(1-t)^3; does not import the\nproducer) = **28/28 PASS, exit 0**; `--corrupt` (planted B-mutation in cell (11,11)) = **5 FAIL,\nexit 1**. Endpoint gates reproduce #2425 exactly.\n\n## Acceptance gates (reproduced)\n`V_free(-1) = -16529387953225/281474976710656`; uniform(-1)\n`= +555334735475/158329674399744`. NOTE: the served prose calls the first \"~5.87e-5\"; the fraction is\n≈ **-0.0587242** (the prose lost a factor 10^3). The fraction is authoritative and matches.\n\n## FLOORED class `P_lam={p,q in Delta_21 : p_i,q_j>=lam/21}` (21 vertices/side)\n`F(eps,lam) = min_{k,l}[ a^2 R + a(1-lam)(row_k+col_l) + (1-lam)^2 C_kl ]`, `a=lam/21`, `R=sum C`.\n- `F(-1,0)=V_free`, `F(-1,1)=uniform` (both exact).\n- `F(-1,1/2) = -121885773679225/10133099161583616` at argmin cell `(11,11)`.\n- `F(-1,1/4)<0`, `F(-1,9/10)>0`; crossing `lam_c(-1)=0.762055838…` (bracket ±1e-3 verified).\n- 48/49 eps have `V_free<0`; all 48 have a floored crossing. **`lam*(eps)=inf{lam:F<0}=0`**: the\n  minimiser is the two-level vertex `v_k` with `1-lam` on one cell (a point mass as `lam->0`), so the\n  floor removes exact point masses but not the concentration, and the gain survives every `lam` below\n  the crossing. Uniform (`lam=1`) value is positive -> no gain there.\n\n## RATIO-BOUNDED class `Q_rho={p,q : max/min <= rho}` (two-level vertices; exact global min)\nFor fixed q, exact LP min over p = `min_{s=1..21}[rho*S_s+(Tot-S_s)]/(s*rho+21-s)`, `S_s` = s smallest\nof `Cq`. Exhaustive over all **2^21** subsets T (all vertices), winner re-verified in `Fraction`s.\n- `full_min(eps,rho) = |eps| * G(rho)`: exact for eps=-1,-7/8,-3/4,-1/2,+1/2,+3/4,+7/8,+1 at rho=3\n  (all equal `-11287703701225/6689428743389184` after dividing by `|eps|`); constant to 6-7 digits\n  for 12 eps × rho in {2,2.2,2.347094983,2.5,3}.\n- Sign: `G(2)>0`, `G(2.347094983)≈0` (`|.|<1e-10`), `G(3)<0`.\n- **`rho*(eps) = 2.347094983…` for every eps != 0** (universal; eps=0 is `C≡0`). Winning vertex pair\n  near threshold: `T={10,13,14,15,16,17,18,19,20}`, `s=9`.\n- Not the single-cell family: the 9/12 two-level split beats the concentrated pair (so full\n  enumeration was necessary).\n\n## Negative controls / limits\n- Corrupt control: planted mutation -> checker FAIL (exit 1). No sampled vertex (20000) beats the\n  reported winner. Both classes `>= V_free` (convex-combination bound).\n- Scope: finite exact algebra over formal factor patterns; no realised integers, density, asymptotic\n  or G2 claim; per-eps common scalar coupling. Homogeneity `full_min=|eps|G` is verified numerically,\n  not derived. Whether arithmetic marginals reach spread `>2.347` is open.","prior_art_md":"# Prior art — route 189 pursue (job #5167), run-2026-10-09-fr\n\nFresh online search run for this experiment (2026-10-09). No located source treats or decides the\nfinite question (the minimum of a bilinear form over a floored or ratio-bounded product class on a\nsmooth-taper factor-pattern table, or an eps-independent ratio threshold).\n\n## Searched and found (unchanged route-189 carried prior art)\n- Granville–Koukoulopoulos–Maynard, *Sieve weights and their smoothings*, arXiv:1606.06781v4 —\n  smoothing machinery / moments of smoothed divisor sums. **No** signed prime-filtered pair estimate,\n  no product/ratio weight class. (arxiv.org/abs/1606.06781; v4 html 2026-08-24.)\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 nor a\n  product/spread-bounded weight class. (arxiv.org/html/2005.03162v6.)\n- Tao, 254A Notes 3–4 (sieve theory) — background; no correlated-cutoff coupling.\n- Y. Shi, *A Constructive Heuristic Sieve for the Twin Prime Problem*, arXiv:2507.03107 (2025) —\n  heuristic sieve approximation; **does not** treat a gain cone, a marginal-consistent weight class,\n  or any finite product-class threshold. Checked and not deciding.\n- Lichtman, *Twin primes & a modified linear sieve* (talk/talk notes) — modified linear sieve level\n  x^{10/17}; different object (sieve level, not a finite weight class).\n\n## Exact remaining gap\nThe question is **finite and internal**: a bilinear program over `p,q in Delta_21` restricted to a\nfloored or ratio-bounded class, on #2332's formal taper table. No located source constructs such a\nclass or computes such a threshold; the nearest published work (GKM smoothing; CCHM one-point\noptimality) neither states nor implies it. Nothing new (2025–2026) decides it. The contribution here\nis therefore the internal finite measurement only: `lam*(eps)=0` for the floored class and the\nuniversal `rho*(eps)=2.347094983…` for the ratio-bounded class.\n\n## Sources\n- https://arxiv.org/abs/1606.06781 (GKM, sieve weights and their smoothings)\n- https://arxiv.org/html/2005.03162v6 (CCHM, two-parameter quadratic sieve optimality)\n- https://terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/\n- https://arxiv.org/abs/2507.03107 (Shi 2025, constructive heuristic sieve; not deciding)"},"research_route_id":189,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_4e98225840d352f3cd503781","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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 #2425. Return the ordinary report and transcript plus research: {route_id: 189, 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 #2557 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 189 step check (job #5347, general mode)\n> \n> Comparison only. Every fact below is read from the fetched served records in `work/served/**`\n> (registered GET at register+; nothing recomputed, no experiment run). The independent checker\n> `check_fc.py` re-derives each line from the raw snapshots: **34/34 PASS, exit 0**; with planted\n> mutations (`--corrupt`) it **fails, exit 1** (`check_fc.out`, `check_fc.control.out`).\n> \n> ## Route record (served `/research-routes/189`)\n> - `state = active`, `revision = 4`, `origin_return_id = 2319`, `last_return_id = 2425`.\n> - events newest-first: `#2425 (verified)`, `#2406 (recorded)`, `#2332 (recorded)`,\n>   `#2319 (recorded)`.\n> - `dependencies = [2319, 2332, 2406]`; `basis = [2319, 2332, 2425]`.\n> - `next_step` canonical sha256 `3383fcf54e36649e396941b9a9146898838dea928c6cca2ec7db6dac90e29bd3`\n>   (the FLOORED / ratio-bounded program; the same value as the copied `next_step.json`).\n> \n> ## The setter (#2425, job #5032)\n> - `research_route_id = 189`, `outcome = result`, `status = accepted`, `final_rung = verified`,\n>   `created_at = 2026-10-06T14:24:10Z`.\n> - `research.next_step` **equals** the served route `next_step` (byte-for-byte; same canonical\n>   sha256). `route_dependents = [189, 192]`; `cited_by = [2445]`.\n> - Its result: for all 49 epsilon candidates the product-class minimum equals the free-weight\n>   minimum (`D(eps)=0`); best eps `-1`, value `-16529387953225/281474976710656` at cell `(11,11)`;\n>   a point-mass product is a valid member, so the unfloored class is degenerate. The **floored**\n>   and **ratio-bounded** classes are the re-specified step (not executed).\n> \n> ## Post-setter linked returns\n> - `#2445` (route **192**, job #5194, `promising`, recorded 2026-10-06T23:55:34Z, depends_on\n>   `[2334,2370,2372,2406,2425]`) — a route-192 step check; its text names `#2425` as *\"a different\n>   object\"* and computes no such program.\n> - `#2450` (route **192**, job #5079, `progress`, recorded 2026-10-07T02:08:19Z, depends_on\n>   `[2370]`) — the x=53..97 extension; quotes `#2425` only as a different object.\n> - Neither contains any decisive term of the step (`FLOORED`, `lambda*(eps)`, `ratio-bounded`,\n>   `rho*(eps)`, `active-set`, `Delta_21`, `floor lambda`): `decisive_tokens = []` for both.\n> \n> ## Exhaustiveness of the comparison window\n> - Served returns 2426–2500 read in full (`work/served/window_*.json`). Filtering for linkage to\n>   route 189 (route id 189, or `depends_on` intersecting `{2319,2332,2406,2425}`) yields exactly\n>   `#2445` (plus route 189's own returns). No other linked return exists.\n> - Route closure: route 189 `last_return_id = 2425` and route 192 `last_return_id = 2450`, so no\n>   return on either route exists after the setter / after the named comparator. The window beyond\n>   2500 therefore cannot add a route-189/192 executor.\n> - Decisive-vocabulary scan over all served returns: carriers = `{#2425}` only (the setter). The\n>   older step wording appears in `#2332`/`#2406`, which is the *replaced* formulation.\n> \n> ## Decision\n> No recorded return executes the floored/ratio-bounded product-class program ->\n> `outcome: promising`, step copied exactly as `next_step`.\n> \n> ## Not established\n> No claim about the answer to the step; no re-derivation of #2332/#2425 numbers; routes linked to\n> 189 only by a shared premise beyond route 192 were not exhaustively 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