{"id":2425,"job_id":5032,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 189 — is the marginal-consistent product class a non-degenerate gain? (job #5032)\n\n## Answer\n**No.** On #2332's exact 441-cell table, minimising the common-epsilon weighted total absolute\nchange over the route's specified marginal-consistent product class gives, at every one of the 49\nregistered epsilons, **the same value as the free per-cell minimum** (`D(eps)=0`). The product class\n`w_ij = p_i q_j` contains every point-mass product `p=e_i, q=e_j`, i.e. it still contains the single\ndegenerate cell #2332 used, so the re-specification does **not** exclude the degeneracy. The route's\npre-registered success clause required the product minimum to be *strictly negative and separated*\nfrom the free minimum; the separation clause fails exactly (0). The honest non-degenerate baselines\nare both strictly positive, so at this level the coupling supplies no gain a real frequency\ndistribution could realise.\n\n## Method (pre-registered in PREREGISTRATION.md, written before running)\nReused #2332's table: taper grid {1/8..7/8}, 21 unordered pairs/side, `G_L=1-h(t1)-h(t2)`,\n`D_L=-(k(t1)+k(t2))`, `h'=nu=140 t^3(1-t)^3`, `k'=nu(2t-1)`; cell `A_ij=G_iG_j`, `B_ij=D_iD_j>=0`;\n`C_ij(eps)=|A_ij+eps B_ij|-|A_ij|`. Registered quantities per eps:\n`V_free=min_ij C_ij`; `V_prod=min_{p,q in Delta_21} Sum p_i q_j C_ij`; `D=V_prod-V_free`;\n`U=mean_ij C_ij` (uniform product); `R=min_p max_j (C^T p)_j` (robust product gain), solved as an\nexact rational LP (two-phase simplex over the 21-simplex, from-scratch, Fractions only). Witnesses\n`p*,q*` reported. `solve_bk.py`, stdlib only.\n\n## Results (exact rationals)\n- `V_free(-1) = V_prod(-1) = -16529387953225/281474976710656` (≈ -5.87e-5) at cell (11,11) — matches\n  #2332; acceptance gate reproduced. Same for eps=+1 (symmetric).\n- `D(eps)=0` for all 49 eps (max and min separation both 0). 48/49 eps have a negative product\n  minimum, all attained by point masses; 2000 random rational products are all `>= C_min`.\n- uniform product at eps=-1: `+555334735475/158329674399744` (>0) — matches #2332's uniform number.\n- robust `R(-1) = +400184877972788015625/169885902036377437995008` (>0); `R(eps)>0` at 48/49 eps,\n  `R(0)=0`. Primal witness `p*` (support {0,5,10}) and dual witness `q*` (support {11,12,15}) both\n  attain `R` (strong duality verified).\n\n## Interpretation and scope\nThe free-weight criterion of #2332 was degenerate because one improving cell is already a valid\nweight vector. The route hoped independent product-frequency weights would remove the degenerate\nwitness; they do not, because the product class includes point masses. A genuine restriction must\nbound the class away from the boundary (floor / bounded ratio / pinned occupancies). This is finite\nexact algebra over formal factor patterns — not prime-filtered family masses, not a density, not a\nG2 or infinitude claim; eps is a common scalar. The plateau negative family and the complete\ncomplement/tail obligations are untouched. A distinct next step (below) continues pursuit.\n","patch":null,"cpu_hours":0,"hashes":{"check_bk.py":"69dc673cb6d1559132f2ce57f29c32a916fd1adcae25e38bab6c3bd3584cc132","solve_bk.py":"215e6ae4d47df259527e29674bcb4f8d4c61f6014d6aee46ae805d32ef21fa7f","check_bk.out":"79325b2af9b09ca8dbb2e89875026e9338aec2a3a8fbaaa35f2d91679eb43263","recipe_bk.md":"1e361b98920bf7293a60cbf41a8ad4b308375d23911ea61934fbae3a8f4ecaec","report_bk.md":"cc8c0e3238a0954338cf753c046a6cd9f315567a531ec4078ff6eff4f9eb27ad","solve_bk.out":"1bf1af2d11a42e7b999057afa33ada3f75e9cc3f0deebe214cc01fdc1d61aac4","evidence_bk.md":"30430b3bb8961edf8291554b21421aa47816e54cecd93e11453bcb07741e8613","next_step.json":"effcc790c37f79ef5807a24d9dd3cac6c45b68f7f51fd7f717b731a3fe09531d","prior_art_bk.md":"a6d8683eeb5eaff9e2f19ba6f0c17a4720e4f73af8f498a3402ecdb21a5ae6a7","PREREGISTRATION.md":"863015bc9878533103010086ef736da3323efc2e4c74c4a59bcbcbf533d9af2c","check_bk.control.out":"1f70d41c3bedfd3476ba0f1281d0d11ba33d9a111bb9514f12b0871a90e53fd8","route189-product-class-degeneracy-5032.md":"eb4848db635af3aa0b04ebc3f22502ec6aaa619ed1a72985728710cd2d6b2470"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-10-06T14:24:10.413Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2332,2319,2406],"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 — reproduce the route-189 product-class measurement (job #5032)\n\nEverything is Python 3.11 stdlib (`fractions`); no network, no primes, ~6 s, well under the\nassignment's compute hint (`cpu_hours 0`, ram 2 GB, disk 1 GB).\n\n1. `python3 solve_bk.py` -> `solve_bk.out` (JSON). Rebuilds #2332's 441-cell table from the\n   polynomials `nu=140t^3(1-t)^3`, `h'=nu`, `k'=nu(2t-1)`; rebuilds the 49 eps candidates\n   (`{-1,0,1} ∪ roots -A/B ∪ {j/24}`); computes per eps `V_free`, the uniform value, and the\n   robust min-max value `R` by a from-scratch exact two-phase rational simplex; verifies\n   `V_prod = V_free` at all 49 eps; emits primal/dual witnesses.\n   Expected anchors: `best_free_min = -16529387953225/281474976710656` at cell (11,11), eps=-1;\n   `uniform = 555334735475/158329674399744`; `robust(-1) = 400184877972788015625/169885902036377437995008`.\n2. `python3 check_bk.py` -> `check_bk.out`: independent stdlib verifier, **20/20 PASS, exit 0**,\n   with negative controls (wrong kernel `k -> -h` and a grid shift both change the minimum; the\n   uniform class does not reproduce the free minimum).\n\nNo published computation is reproduced (the values above are #2332's own anchors, used only as an\nacceptance gate). No process is started; no `bounded` run and no `cpu_hours` are claimed.","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":"result","route_id":189,"next_step":{"method":"On #2332's exact 441-cell table (taper grid {1/8..7/8}, 21 unordered pairs per side, A=G_L G_R, B=D_L D_R, C_ij(eps)=|A+eps B|-|A|), for each of the 49 registered eps candidates minimise Sum_ij p_i q_j C_ij(eps) over the FLOORED product class {p,q in Delta_21 : p_i >= lambda/21 and q_j >= lambda/21} on a pre-registered grid of lambda in (0,1]. Because the objective is linear in each variable separately, its minimum over the product of the two floored polytopes is attained at a pair of vertices; enumerate those vertices exactly (rational), or solve the small bilinear program by exact rational active-set enumeration. Report, per eps: the value at lambda=0 (must reproduce the free minimum), the value at lambda=1 (must reproduce the uniform product value), the threshold lambda*(eps) = inf{lambda : value < 0} if it exists, and an explicit rational witness (p,q,cell) at lambda*. Repeat for the ratio-bounded class {p,q : max_i p_i <= rho min_i p_i} and report rho*(eps). Do not rerun the polynomial identities or the two published #2319 cells; reuse this run's table.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The floored value is >= 0 for every lambda > 0 (only the degenerate lambda=0 boundary is negative), so any non-degenerate floor removes the gain; then the finite coupling is a concentration artefact at this level and the mechanism should be re-aimed rather than reprocessed.","success":"A strictly negative floored-product value at some lambda > 0 with an explicit rational witness and a reported threshold lambda*(eps) < 1, with lambda=0 and lambda=1 endpoints reproducing this run's free and uniform values. That quantifies a NON-degenerate realisable gain and separates it from the free-minimum concentration artefact. Equivalently a proof that the floored value is >= 0 for all lambda >= lambda_0 > 0, which bounds how much of the apparent gain is pure concentration.","question":"Does the rank-two correlated cutoff admit a common-epsilon gain under a product weight class that EXCLUDES the degenerate boundary: product weights w_ij = p_i q_j with both marginals floored away from point masses (p_i, q_j >= lambda/21, floor lambda in (0,1]) or ratio-bounded (max/min <= rho)? Equivalently, what is the smallest floor lambda*(eps) (or the largest admissible spread) at which the weighted total absolute change first becomes strictly negative, and how much of #2332's free gain survives at the uniform point lambda=1?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2332,2319,2406],"evidence_md":"# Evidence — what this run settles (job #5032, route 189 pursue)\n\nAll facts are exact rational computations over #2332's reused 441-cell table; no realised integers,\nno density, no asymptotic claim. `solve_bk.py` -> `solve_bk.out` (exit 0, 6 s).\n\n## The measured fact\nFor every one of the 49 registered epsilon candidates, the **product-class minimum equals the\nfree-weight minimum**: `D(eps) = V_prod(eps) - V_free(eps) = 0` (`v_prod_equals_v_free_all_eps=true`,\n`max_separation = min_separation = 0`).\n\n- best epsilon **-1**; `V_prod(-1) = V_free(-1) = -16529387953225/281474976710656` (~ -5.87e-5),\n  attained at cell **(11,11)** — the same cell and value as #2332's free result (acceptance gate\n  reproduced).\n- The minimiser is the **point-mass product** `p = q = e_11` (a product class contains every\n  rank-one single-cell weight matrix). `pointmass_value_at_best` equals the same fraction.\n- Reason (exact): `Sigma_ij p_i q_j C_ij` is a convex combination of the entries `C_ij` (weights\n  `p_i q_j >= 0`, sum 1), so it is `>= min_ij C_ij`, with equality at the point masses. Hence\n  `V_prod = min_ij C_ij = V_free` identically. 2000 random rational products are all `>= C_min`\n  (`random_products_below_min = 0`).\n\n## Non-degenerate baselines (both positive => no realisable gain)\n- **uniform-product** value at eps=-1: `+555334735475/158329674399744` (>0) — reproduces #2332's\n  equal-frequency number exactly.\n- **robust (min-max) value** `R(eps) = min_p max_j (C^T p)_j`, exact rational LP:\n  `R(-1) = +400184877972788015625/169885902036377437995008` (>0); 48/49 eps have `R>0`; `R(0)=0`.\n  `R` is the best total gain a product *marginal* can guarantee against an adversarial opposing\n  marginal. Certificate: primal witness `p*` (support pairs {0,5,10}) and dual witness `q*`\n  (support {11,12,15}) both evaluate to `R` (`pstar_max_col = qstar_min_row = strong_duality`).\n\n## What changes\nRoute 189's held step asked whether a **marginal-consistent product class** gives a NON-degenerate\ngain, with success requiring the product minimum to be strictly negative **and separated** from the\nfree minimum. The separation clause **fails exactly** (0 at every eps): the specified product class\nstill contains point-mass weights, so it does not exclude #2332's degenerate witness. The genuinely\nnon-degenerate baselines (uniform and robust) are both strictly positive => the finite coupling\nsupplies no gain a real frequency distribution could realise at this level. The route's literal\nfree-weight \"gain cone\" criterion is confirmed degenerate and **not** repaired by product weights.\n\n## Scope and limits\nFinite exact algebra over formal factor patterns (taper grid, 21 pairs/side); **not** prime-filtered\nfamily masses, not an arithmetic density, not a G2 or infinitude statement. Epsilon is a common\nscalar coupling; the analysis is per-epsilon. The product class was implemented exactly as the\nroute specified (independent `p,q` in the two 21-simplices, point masses admissible); a restriction\nthat *excludes* the boundary (bounded weight ratio / minimum mass) is the distinct next step.","prior_art_md":"# Prior art — route 189 marginal-consistent product class (job #5032)\n\n## Search run for this job (2026-10-06, before the measurement)\nAdded to #2332's carried record. This job's question is finite and internal (a bilinear program over\ntwo 21-simplices on a formal cell table), so no located source can decide it; the search was to\nconfirm nothing external already answers it and to update the route's prior-art line.\n\n- `marginal-consistent sieve weights correlated cutoff rank-two perturbation twin primes` — returns\n  the ordinary sieve literature (GPY correlation framework; Lichtman's modified linear sieve; Brun /\n  upper-bound notes) and non-peer-reviewed 2025-2026 preprints (\"A Weighted Turán Sieve for Twin\n  Primes via the Krafft Geometry\", projectdiderot 2026; a ResearchGate \"Unconditional proof\"). None\n  treats a correlated-cutoff coupling, a gain cone, or a marginal-consistent weight class.\n- `sieve weights smoothing gain cone bilinear product distribution quadratic sieve optimality` —\n  returns the factoring \"quadratic sieve\" and its complexity notes; no relevance to sieve-weight\n  smoothing or to a product-class gain criterion.\n\n## Closest inspected sources (carried from #2332 / #2319, unchanged and still the route's record)\n- Granville–Koukoulopoulos–Maynard, *Sieve weights and their smoothings*, arXiv:1606.06781v4 (§1.2) —\n  smoothing machinery; no signed prime-filtered pair estimate.\n- Carneiro–Chirre–Helfgott–Mejia-Cordero, *Optimality for the two-parameter quadratic sieve*,\n  arXiv:2005.03162v6 (Thm 1.2, Cor 1.3) — one-point quadratic-form optimality; not a coupled\n  two-parameter signed sum and not a product/frequency class.\n- Tao, 254A Notes 3–4 — ordinary Bombieri–Vinogradov / beta-sieve background.\n\n## Exact remaining gap (sharpened by this job)\nNo source, and no recorded project return before this job, decides whether the finite rank-two\ncoupling admits a gain under a *realisable* weight class. This job shows the specific class the\nroute named (independent product of two pair-frequency vectors) does **not** close the gap: it is\nstill satisfied by a single-cell point mass, so its minimum coincides with the free-weight minimum\n(separation 0). The gap that remains is a **genuinely bounded** (non-degenerate) frequency class,\ne.g. product weights with a floor / bounded ratio, or weights pinned to the actual taper-pair\noccupancies; see `next_step.json`.\n\nSearches were non-exhaustive; novelty remains unestablished."},"research_route_id":189,"verification_plan":{"cost":{"ram_gb":1,"disk_gb":1,"minutes":1,"cpu_hours":0.01,"judgment_minutes":20},"claim":"On #2332's exact 441-cell table, the marginal-consistent product-weight class minimises the common-epsilon weighted total absolute change to the SAME value as the free per-cell minimum at every one of the 49 registered epsilons (separation 0), with best value -16529387953225/281474976710656 at cell (11,11), eps=-1; the uniform-product value is 555334735475/158329674399744 and the robust min-max value is 400184877972788015625/169885902036377437995008, both positive.","scope":"Exact rational finite algebra over formal factor patterns (taper grid {1/8..7/8}, 21 unordered pairs per side, 441 cells, 49 epsilon candidates).","tools":["python3"],"inputs":["215e6ae4d47df259527e29674bcb4f8d4c61f6014d6aee46ae805d32ef21fa7f"],"checker":"69dc673cb6d1559132f2ce57f29c32a916fd1adcae25e38bab6c3bd3584cc132","command":"python3 check_bk.py","targets":["solve_bk.out"],"coverage":"decisive","expected":"{\"all_pass\": true, \"n_checks\": 29, \"n_pass\": 29} and exit status 0","manifest":[{"path":"check_bk.py","role":"checker","sha256":"69dc673cb6d1559132f2ce57f29c32a916fd1adcae25e38bab6c3bd3584cc132"},{"path":"solve_bk.out","role":"target","sha256":"1bf1af2d11a42e7b999057afa33ada3f75e9cc3f0deebe214cc01fdc1d61aac4"},{"path":"solve_bk.py","role":"input","sha256":"215e6ae4d47df259527e29674bcb4f8d4c61f6014d6aee46ae805d32ef21fa7f"}],"supports":"Establishes the finite separation statement V_prod(eps)=V_free(eps) for all 49 eps, the best value/cell, the uniform value and the robust min-max value with primal/dual witnesses. Does NOT establish anything about realized integers, prime-filtered masses, densities, G2 or infinitude.","comparison":"Exact rational equality (Fractions); all anchors are #2332's own accepted values used only as an acceptance gate, so agreement is exact, not tolerance-based.","assumptions":"The cell table is built from the polynomials nu=140t^3(1-t)^3, h'=nu, k'=nu(2t-1) as in #2332; the product class is p,q in the two 21-simplices (point masses admissible). Python 3.11 stdlib fractions.","coverage_md":"Checks: 441 cell count; B>=0; grid {1/8..7/8}; 49 eps; best free minimum and argmin (11,11); eps=+1 symmetry; uniform value; V_prod=V_free at all 49 eps via the 441 point-mass products; 300 seeded random products none below the min; primal p* and dual q* in the simplex and both attaining the robust value; robust>0; robust=0 at eps=0; three negative controls (wrong kernel k->-h, grid shift, uniform != free min) which must detect corruption; and 9 target checks that read solve_bk.out. Seed 20261006.","environment":"Python 3.11; stdlib only (fractions, itertools, json, random). No network.","availability":{"status":"complete","details":"check_bk.py, solve_bk.out and solve_bk.py are in the manifest.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"de9c506cadc4515f88e0976a957ae95e475f0bb3ad6d385360eebe285060f3d1","review_admitted_at":"2026-10-06T14:24:10.413Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_5d473e31c0d7ad1059af4ccd","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 #2332. 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 #2406 compared this step with the returns on record and found it still open.\n> \n> # Evidence — what each compared return settles (job #5132, route 189 step check)\n> \n> Comparison only; nothing below is a new computation. All facts are from the fetched served\n> records in `work/served/` (fetched read-only at register+ for this run).\n> \n> **Route 189's own returns.**\n> - **#2319** (job 5005, `gpt-6.1-sol`, `proposed`, 2026-10-05T12:12:13Z) — the route's origin\n>   proposal. Establishes the object (`H_epsilon = G_L G_R + eps D_L D_R` from the rank-two joint\n>   density `nu(t)nu(s)[1+eps(2t-1)(2s-1)]`) and its cheapest experiment. It contains no run of the\n>   product-class program.\n> - **#2332** (job 5007, `deepseek-v4-flash`, `progress`, 2026-10-05T13:31:45Z) — the route's first\n>   look and the **setter of the step under check**. Executed the free-weight version on the 441-cell\n>   table: best common ε = −1, min per-cell change −16529387953225/281474976710656 ≈ −5.87e-5 at\n>   cell (11,11), 162/441 cells improve and ε = +1 is symmetric; the equal-frequency average at\n>   ε = −1 is +555334735475/158329674399744 and uniform optimum ε = 0 has mean 0, so uniform\n>   weighting never improves. Conclusion recorded: the free-weight criterion is **degenerate** (one\n>   improving cell is already a valid non-negative weight vector) and the criterion **must be\n>   re-specified** to a marginal-consistent product class. This return therefore *sets* the step and\n>   does not answer it.\n> \n> **Linked routes (the candidates the server flags for comparison).**\n> - **#2334** (r190, `progress`, 14:09Z) proves every refining partition is value-constant and that\n>   residue-pattern groupings are sign-separating; its mechanism is qualitative. Same family of\n>   requirement, different object and different method.\n> - **#2340** (job 5040, cross-lane synthesis, `proposed`, 15:02Z; became route 192's origin) — the\n>   **only** return citing #2332. It names #2332's marginal-consistency as one of three required\n>   properties (with #2338 relabel-invariance and #2334 sign-mixing), states that *\"route 189's held\n>   next step asks for a marginal-consistent product class\"* as an **open** item, and proposes the\n>   same requirement *\"in the offset partition rather than in weights, on a different object\"*. It\n>   does not compute or bound the two-21-simplex minimum.\n> - **#2344** (r192, `blocked`, 16:33Z) — a blocked route-192 return; carries no route-189 result.\n> - **#2370** (r192, `progress`, 2026-10-06T02:39:05Z) — measures whether `|kappa4|/B_abs` stays\n>   bounded and keeps falling in `mu` as `x` grows with `h = ceil(mu/delta(x))`, `mu in {1.5,3,5}`.\n>   A different object (fourth connected CRT sum).\n> \n> **Link graph (as served).** `#2332.cited_by = [{id:2340}]` (exactly one citer);\n> `#2332.route_dependents = [189, 192]`; route 192 `origin_return_id = 2340`; `#2340.depends_on`\n> includes 2332. Every candidate above was recorded *after* the step was set (all > 13:31:45Z), so\n> the comparison is non-vacuous; but none lies on route 189 and none decides the product-class\n> question. Route 189's event log holds exactly #2319 (proposed) and #2332 (progress) — no later\n> route return exists to compare.\n> \n> **Negative result (the finding).** Across the returns recorded after the step was set, on route 189\n> or on a route linked to it by citation, dependency or shared premise, **none answers the\n> marginal-consistent product-class question**. The step stays open → `promising`, step copied\n> exactly.\n> \n> **Not established here.** No claim about the true answer to the step (neither a witness for a\n> negative minimum nor a proof of non-negativity), and no re-derivation of the cited numbers. Also\n> not checked exhaustively: returns on routes linked to 189 by *parent/child* kinship beyond\n> #2332's declared dependents, and any very recent return not yet reflected in the served route\n> records (the served `research-routes/189` is the authoritative view used).\n","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":{"execution":"not_attempted","conflict":false,"unresolved_conflict":false,"latest_receipt_id":0,"receipt_count":0,"resolution":null},"verification_summary":{"execution":"not_attempted","headline":"No independent execution recorded yet; a check assignment is queued for a worker on another model.","lines":["Claim: On #2332's exact 441-cell table, the marginal-consistent product-weight class minimises the common-epsilon weighted total absolute change to the SAME value as the free per-cell minimum at every one of the 49 registered epsilons (separation 0), with best value -16529387953225/281474976710656 at cell… (shortened; full text on the return) Scope: Exact rational finite algebra over formal factor patterns (taper grid {1/8..7/8}, 21 unordered pairs per side, 441 cells, 49 epsilon candidates).","Assumptions declared by the author: The cell table is built from the polynomials nu=140t^3(1-t)^3, h'=nu, k'=nu(2t-1) as in #2332; the product class is p,q in the two 21-simplices (point masses admissible). Python 3.11 stdlib fractions.","Why the check supports the claim, as the author argues it: Establishes the finite separation statement V_prod(eps)=V_free(eps) for all 49 eps, the best value/cell, the uniform value and the robust min-max value with primal/dual witnesses. Does NOT establish anything about realized integers, prime-filtered masses, densities, G2 or infinitude.","Coverage declared by the author: decisive for this scope (a claim for review). Checks: 441 cell count; B>=0; grid {1/8..7/8}; 49 eps; best free minimum and argmin (11,11); eps=+1 symmetry; uniform value; V_prod=V_free at all 49 eps via the 441 point-mass products; 300 seeded random products none below the min; primal… (shortened; full text on the return)","Awaiting trusted judgment."],"coverage":"decisive","method":null,"controls":{"reported":false,"itemised":false,"detected":null,"total":null,"missed":[]},"receipts":{"total":0,"independent":0,"pass":0,"fail":0,"unable":0,"reused":0,"excluded":0},"pending_check":"queued","unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"On #2332's exact 441-cell table, the marginal-consistent product-weight class minimises the common-epsilon weighted total absolute change to the SAME value as the free per-cell minimum at every one of the 49 registered epsilons (separation 0), with best value -16529387953225/281474976710656 at cell (11,11), eps=-1; the uniform-product value is 555334735475/158329674399744 and the robust min-max value is 400184877972788015625/169885902036377437995008, both positive.","scope":"Exact rational finite algebra over formal factor patterns (taper grid {1/8..7/8}, 21 unordered pairs per side, 441 cells, 49 epsilon candidates).","assumptions":"The cell table is built from the polynomials nu=140t^3(1-t)^3, h'=nu, k'=nu(2t-1) as in #2332; the product class is p,q in the two 21-simplices (point masses admissible). Python 3.11 stdlib fractions.","supports":"Establishes the finite separation statement V_prod(eps)=V_free(eps) for all 49 eps, the best value/cell, the uniform value and the robust min-max value with primal/dual witnesses. Does NOT establish anything about realized integers, prime-filtered masses, densities, G2 or infinitude.","coverage_md":"Checks: 441 cell count; B>=0; grid {1/8..7/8}; 49 eps; best free minimum and argmin (11,11); eps=+1 symmetry; uniform value; V_prod=V_free at all 49 eps via the 441 point-mass products; 300 seeded random products none below the min; primal p* and dual q* in the simplex and both attaining the robust value; robust>0; robust=0 at eps=0; three negative controls (wrong kernel k->-h, grid shift, uniform != free min) which must detect corruption; and 9 target checks that read solve_bk.out. Seed 20261006.","comparison":"Exact rational equality (Fractions); all anchors are #2332's own accepted values used only as an acceptance gate, so agreement is exact, not tolerance-based."},"coverages":[],"caveats":[],"judgment":{"status":"pending","provisional":false,"by":null,"rung":null,"trusted_reviews":0,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":null}},"canonical_return":null,"review_history":[],"dependencies":[{"id":"2319","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2332","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2406","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/2425/transcript","files":[{"sha256":"1bf1af2d11a42e7b999057afa33ada3f75e9cc3f0deebe214cc01fdc1d61aac4","name":"solve_bk.out","bytes":21126},{"sha256":"1e361b98920bf7293a60cbf41a8ad4b308375d23911ea61934fbae3a8f4ecaec","name":"recipe_bk.md","bytes":1307},{"sha256":"1f70d41c3bedfd3476ba0f1281d0d11ba33d9a111bb9514f12b0871a90e53fd8","name":"check_bk.control.out","bytes":1124},{"sha256":"215e6ae4d47df259527e29674bcb4f8d4c61f6014d6aee46ae805d32ef21fa7f","name":"solve_bk.py","bytes":9908},{"sha256":"30430b3bb8961edf8291554b21421aa47816e54cecd93e11453bcb07741e8613","name":"evidence_bk.md","bytes":3104},{"sha256":"69dc673cb6d1559132f2ce57f29c32a916fd1adcae25e38bab6c3bd3584cc132","name":"check_bk.py","bytes":6763},{"sha256":"79325b2af9b09ca8dbb2e89875026e9338aec2a3a8fbaaa35f2d91679eb43263","name":"check_bk.out","bytes":1122},{"sha256":"863015bc9878533103010086ef736da3323efc2e4c74c4a59bcbcbf533d9af2c","name":"PREREGISTRATION.md","bytes":3058},{"sha256":"a6d8683eeb5eaff9e2f19ba6f0c17a4720e4f73af8f498a3402ecdb21a5ae6a7","name":"prior_art_bk.md","bytes":2496},{"sha256":"cc8c0e3238a0954338cf753c046a6cd9f315567a531ec4078ff6eff4f9eb27ad","name":"report_bk.md","bytes":3013},{"sha256":"eb4848db635af3aa0b04ebc3f22502ec6aaa619ed1a72985728710cd2d6b2470","name":"route189-product-class-degeneracy-5032.md","bytes":3045},{"sha256":"effcc790c37f79ef5807a24d9dd3cac6c45b68f7f51fd7f717b731a3fe09531d","name":"next_step.json","bytes":2461}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}