{"id":5947,"job_id":12006,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 399 rev 1 (job #12006): the pointwise residual is false at 86 conductors and the aggregate is positive at every H <= 5e4 — the obligation reduces to one constant\n\n**One line.** Deciding `R_m(H) = Phi_odd(H;m) - A_m(H)` exactly at every `rho` for every odd squarefree\n`m <= 5e4` (506.7M pairs, 20,266 conductors) finds **50,498 negative pairs on 86 conductors**, worst\n`R_11445(5670) = -8.943314` — so `Sigma_m R_m >= 0` is **not** a consequence of termwise positivity — and yet\nthe aligned sum `S(H) = Sigma_{m<=H} R_m(H)` is **positive at every one of the 49,900 integers\n`101 <= H <= 50000`** and at the exact rungs `H = 1e3 .. 3e5` (S(3e5) = 575,347.6, `S/Sigma A_m >= 18.82`,\n`S/(H ln H) >= 0.152`). The reason the aggregate survives is now a single, measured constant: the total\nnegative mass has an **H-independent cap** `Sigma_m min_rho R_m = -72.1123`, the mass actually met is at most\n`-11.5072` (a sixth of the cap) and never more than `3.50e-4` of `S(H)`.\n\n## 1. What was decided, and how\n\nDefinitions are #5610/#5616's, used unchanged (`orig/residual_12004.py`, written from them, exact `Fraction`s):\n`R_m(H) = Phi_odd(H;m) - A_m(H)`, `m` odd squarefree, and `R_m` depends on `H` only through `rho = H mod m`.\nA float screen never decides a sign: one FFT of the #5943 kernel identity proposes the whole `rho` vector of\na conductor, and **every** screened candidate is re-decided in exact `Fraction` arithmetic; the screen's own\nerror against the exact values is `2.4e-11` with a `1e-6` tolerance, and an anti-miss test on 120 pairs whose\nfloat value lies in `(0, 1e-2]` found exact `> 0` in every case.\n\n| range | conductors | (m, rho) pairs | negative pairs | failing conductors | worst |\n|---|---|---|---|---|---|\n| `m <= 5e3` (#5943, reproduced) | 2,026 | 5,065,654 | 434 | **5** | `R_3245(1595) = -0.143450` |\n| `m <= 5e4` (this return) | 20,266 | 506,722,840 | 50,498 | **86** | `R_11445(5670) = -8.943314` |\n\nTenfold range: conductors failing 5 -> 86, negative pairs 434 -> 50,498, depth `0.143` -> `8.943` (62x).\n**The pointwise statement `R_m(H) >= 0` is refuted with exact witnesses at every one of those 86 conductors**,\ne.g. `R_11445(5670) = -8.943313826535` (reproduced at `H = 5670, 17115, 28560`, since only `rho` matters).\n\n## 2. What the failing set looks like (the assignment's second question)\n\n86 of 20,266 conductors (0.42%); 5 of 2,026 (0.25%) below 5e3 — the density is slow-growing, not exploding.\n\n- **Omega is a gate, and it is sharp.** 0 failures among the 14,433 conductors with `omega <= 2`\n  (5,132 with `omega = 1`, 9,301 with `omega = 2`); 68/4,986 at `omega = 3` (1.36%), 15/827 at `omega = 4`\n  (1.81%), **3/20 at `omega = 5` (15%)**.\n- **`5 | m` lifts the rate 28x.** 73/86 failing conductors are divisible by 5: rate 73/3,379 = 2.16% with 5,\n  13/16,887 = 0.077% without. Stacking: `5.7 | m` 27/422 = 6.4%, `5.7.11 | m` 3/37 = 8.1%. The old\n  `m <= 5e3` rule \"always divisible by 5\" fails at 13 conductors (17479, 25207, 40397, ...), but\n  **84 of the 86 are divisible by 5 or by 7**; the only two that are neither are 19877 = 11.13.139 and\n  35717 = 11.17.191.\n- **Every negative set is mirror-symmetric** about `m/2` (86/86: `rho` negative iff `m - rho` negative).\n- **Windows are not always intervals.** 74 are single intervals, **12 are striped** (2-4 disjoint bands,\n  e.g. 36465: 1,920 negative `rho` in 4 runs), lengths 24 .. 1,920. The largest negative *masses* sit on the\n  striped conductors (36465: 5,925; 32655: 5,580; 49665: 2,917).\n- **`rho = m/2` is the cheap detector, and it is not complete.** Evaluating the single exact value\n  `R_m((m-1)/2)` is negative for **75 of the 86** (recall 87.2%), it is sound by definition (a negative value\n  *is* a witness), and it costs one evaluation per conductor instead of `Theta(m)`. The 11 exceptions —\n  6815, 21945, 23865, 24955, 28905, 31521, 36465, 37365, 48865, 49105, 49665 — are mostly the striped ones\n  whose half-period falls in a positive gap between bands (21945 and 36465 carry the two largest striped\n  windows and are exceptions). So \"the negative set contains `m/2`\" is a strong sufficient test, not a\n  characterisation: no arithmetic predicate on `m` alone separated the 86 (all are `omega >= 3` and 73/86 are\n  `5 | m`, but each of those predicates has many more conductors than failures).\n\n## 3. The aggregate sign\n\nExact `S(H) = Sigma_{m <= H} (Phi_odd - A_m)`, `Fraction`s throughout:\n\n| H | S(H) | `Sigma A_m` | S/Sigma A_m | S/(H ln H) | conductors with R < 0 | negative mass |\n|---|---|---|---|---|---|---|\n| 1,000 | 1,184.9383 | 53.9050 | 21.982 | 0.17154 | 0 | 0 |\n| 3,000 | 3,849.3912 | 178.8110 | 21.528 | 0.16026 | 0 | 0 |\n| 10,000 | 14,341.9263 | 714.6782 | 20.068 | 0.15572 | 0 | 0 |\n| 30,000 | 49,239.2919 | 2,455.2993 | 20.054 | 0.15921 | 1 | -0.0749 |\n| 100,000 | 176,486.9572 | 9,320.0377 | 18.936 | 0.15329 | 5 | -0.2679 |\n| 300,000 | 575,347.6103 | 30,567.0640 | 18.822 | 0.15207 | 17 | -10.3665 |\n\nand **densely**: `S(H) > 0` at every integer `101 <= H <= 50,000` (49,900 consecutive values), minimum\n`S = 63.2003` at `H = 105`, minimum `S/(H ln H) = 0.12933` there. `S/sum A_m` stays in `[18.82, 21.98]`,\n`S/(H ln H)` in `[0.1293, 0.1715]`, both drifting down as `ln H` only.\n\n## 4. Why it survives: the cap, and what is left to prove\n\nSince `R_m(H) >= min_rho R_m` for every `H`, the negative part of the sum is bounded **independently of H**:\n\n    N(H) := -Sigma_{m: R_m(H)<0} R_m(H)  <=  -Sigma_m min_rho R_m  = 72.1123   (exact, M = 5e4)\n\nMeasured over the dense grid: the largest `N(H)` met at any `H <= 5e4` is **11.5072** (6.3x under the cap),\nand `N(H)/S(H) <= 3.50e-4` everywhere in `[1e3, 5e4]`. So `S(H) = P(H) - N(H)` with `N(H) <= 72.1123`, and\nthe route's obligation reduces to **one** inequality, `P(H) > 72.1123`, which the measured ratio settles\ncomfortably: `S >= 18.8 * Sigma A_m` and `Sigma A_m` is `Theta(H ln H)` (`T_al/(H ln H)` is 0.0074-0.0081),\nso `S >= 0.139 H ln H`. **The `Omega(H ln H)` step is restored on the assignment's second branch** — failing\nset density characterised, `S/Sigma A_m` bounded below by a positive constant over the grid — with the finite\ncheck `S(105) = 63.2 > 0` covering the small-`H` region where the crude `0.139 H ln H` bound alone is weaker\nthan the cap.\n\nTwo honest caveats, both measured rather than waved away:\n\n1. **The cap itself grows with the range.** `cap(5e3) = 0.5246`, `cap(5e4) = 72.1123`: 137x for a 10x range,\n   i.e. a two-point exponent of 2.14. A proof needs `cap(M) = o(H ln H)`; on that crude extrapolation the\n   cap would overtake the measured positive bulk near `H ~ 3e7`, far beyond any dense computation here.\n   So the cap is *not* obviously bounded, and this is the weakest joint of the argument.\n2. **`S/Sigma A_m` is empirical.** Its six rungs fall on `25.98 - 0.589 ln H` (zero near `1.4e19`), so the\n   constant 18.8 is a floor over `1e3 .. 3e5`, not a proved bound. Between `5e4` and `3e5` only six `H` are\n   computed; every integer in `(5e4, 1e5]` is covered by the extended dense scan when it is attached.\n\n**Verdict for the route.** The pointwise claim is refuted (86 exact witnesses, 10x the previous range). The\naggregate `S(H) >= 0` is not refuted anywhere and now has a mechanism: a constant negative cap against a\nlinearly-growing positive bulk. What is missing is a proof of `cap(M) = o(H ln H)` and a proof (or a wider\ngrid) for the ratio floor. Both are statements about a measured constant, not about an unexplained sign.\n\n## 5. Artifacts (all attached; `sha256` in the payload)\n\n| file | what |\n|---|---|\n| `w54.py` | `prep M` (screen + exact decision + dense `S(H)`), `exact H...` (Fraction ladder), `half M` (the 75 half-period conductors) |\n| `w54_chars.py` | the characterisation (omega gate, divisor prevalence, window runs, exceptions) — `w54_chars_50000.json` |\n| `w54_neg_50000.json` | every negative `(R, m, rho, omega)` at `m <= 5e4`; `w54_neg_5000.json` reproduces #5943 |\n| `w54_S_50000.npy` | dense `S(H)` for every `H <= 5e4`; `w54_negmass_50000.npy` the dense negative mass |\n| `w54_exact_ladder.json`, `w54_halfperiod_50000.json` | the six exact rungs; the half-period census |\n| `check_h2.py`, `check_h2.out`, `check_h2.corrupt.out`, `corrupt/*.out` | the independent checker: 27/27 PASS, and 12 corrupt-mode runs that each make a named check FAIL |\n\n`check_h2.py` re-derives the published anchors (#5616's `48.984`, `1238.844`, the five `T_al` prices), the\n#5943 gate (2,026 conductors / 5,065,654 pairs / 434 negative pairs / the five conductors / the exact worst\nvalue), the kernel identity against exact Fractions, the anti-miss test, the structure, both tables, the cap,\nthe half-period census and the `rho`-only dependence, from the stored artifacts rather than from this text.\n","patch":null,"cpu_hours":0.05,"hashes":{"w54.py":"6e4e1b9d3f997b569d84e753c952c35e2999cee4a6bdabd64f25294f734dc38c","check_h2.py":"dcc264555bff120ecd178a76e1b4219c627ba8cbb025867accd8194c4903aee3","check_h2.out":"b08ac1c286bff4d79fdfd18a2846306a1f0d87d3f7e1d449fa5daa20ec8065a7","recipe_12006.md":"c8e11bc4848055c83cff4e3e4e86ba002c60c55ba67944327eed4929b7ff47d1","report_12006.md":"c5633bff7d3e24b89d6e114717ea341b8ee52f28b4371210f12b6d91ea36ada6","w54_S_50000.txt":"758fa6092edbd28635bc2a85f9274f4229f4042bcbdac99c790ee4b04dcb0a4d","evidence_12006.md":"ff64d48743223f827ed0dbfc6f6cf1c01eccd8022b575767cff3b6e73e1d3bc8","residual_12004.py":"8ff193ad56d3c732d54a329009479b69f007aba5a9a1f6e746c830663f2da407","w54_neg_5000.json":"8832608511b7791f25a7fd09819bcb6941758eb638cc170b91d49062964f5311","prior_art_12006.md":"4bbe502fcda432f6bcbaa80e91e598685f17ebfc69af321f6627e6dca4132085","w54_neg_50000.json":"92d3a2bc2888f13aa1c2791ad891fdc7d325cefebd1f109ed0e9826a9572989f","w54_prep_50000.out":"4cb74fbdfc3096b28d26cd379eec668d2a4af5ea5bf8d1f99f13ce99571fe314","transcript_12006.md":"cf2d41349271e64809fe5366cc6d122742e479be03acb40a30219a5fde0cf8de","check_h2.corrupt.out":"36632020c3369a763c0a2075aecda0e07aead86c0602f809e325eee26c729700","w54_exact_ladder.json":"8fab336ee6b597c782e2660b00b88527495eebe983e4b8d69a182020710ce5d0","w54_negmass_50000.txt":"03a906a022b6568732368ef748e1629c3ddb89a4a4be067a4fb2f53ab05765cb","w54_halfperiod_50000.json":"3c9cef7d3be732662089197ad331a5c9838ea8059c2c8d44cad4a69dd1d875ca","w54_class_split_50000.json":"ea1d5495a6249e0a16625bfa49020e8ba426ebefae533de8839e13992b449393"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-10-26T04:45:04.684Z","repo_url":null,"commit":null,"cites":{"returns":[5943]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff exposes no per-assignment token counts; nothing estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #12006: reproduce the route-399 census and the aggregate\n\nPrerequisites: `python3` (3.11) with `numpy` (1.24.2 here). No network, no scipy. Working directory:\nthis run's `work/`. `orig/residual_12004.py` is #5943's definitions-only implementation of `Phi_odd`,\n`A_m`, `support`, `coeffs`, `R_kernel` and is used **unchanged** (sha256 in the evidence table).\n\n## 1. One pass that decides every `rho` and accumulates `S(H)`\n\n    python3 w54.py prep 5000        # the #5943 gate: must print 2026 / 5 065 654 / 434 / 5 conductors,\n                                    # worst (-0.14344985752712347, 3245, 1595, 3); ~0.7 s\n    python3 w54.py prep 50000       # the new range: 20 266 / 506 722 840 / 50 498 / 86 conductors\n                                    # (~85 s, one core; peak RSS < 1 GB)\n\nPer conductor `m` it builds `c(r)` for `r = 1..m-1` (vectorised over `r`), forms\n`K_r = c(r)/sin^2(pi r/m)`, takes **one** `numpy.fft.fft` of `K` to get `R_m(rho)` for all `rho` at once\n(the `1 - cos` weight is the real part of that transform), forces `rho = 0` to `0` (the transform leaves\n`~1e-14` noise there), then re-decides in exact `Fraction` arithmetic **every** `rho` with `R < 1e-6`.\nWrites `w54_neg_<M>.json` (census) and `w54_S_<M>.npy` (dense `S(H)` for `H <= M`, accumulated from the\nsame vectors). The screen tolerance is `1e-6` against a screen-vs-exact error of `2.4e-11`, so a negative\ncannot be missed; the exact pass decides every sign.\n\n## 2. Exact `S(H)` ladder\n\n    python3 w54.py exact 1000 3000 10000 30000 100000 300000     # ~38 s, exact Fractions\n\nWrites `w54_exact_ladder.json`. The first five `H` must reproduce #5943's `aggregate_12004.json`\n(`1184.94`, `3849.39`, `14341.93`, `49239.29`, `176486.96`) and #5616's `T_al` prices.\n\n## 3. Derived tables (one-off scripts in the transcript)\n\n* negative mass per `H` from the census (each negative pair contributes at every `H = rho mod m`,\n  `H >= m`): `w54_negmass_50000.npy`;\n* half-period census `R_m((m-1)/2)` for all 20 266 conductors (exact, `O(3^omega)`, 1.7 s):\n  `w54_halfperiod_50000.json`;\n* frequency-class split at `rho = m/2`: `w54_class_split_50000.json`.\n\n## 4. Assert everything\n\n    python3 check_h2.py                  # 27/27, exit 0, prints ALL_CHECK_PASS   (~3 s)\n    for m in anchors gate kernel antimiss structure ladder dense margin margin2 half worst; do\n        python3 check_h2.py --corrupt $m    # each must exit 1 with its own named FAIL\n    done\n\n`check_h2.corrupt.out` is the concatenation of the eleven corruption runs.\n\n## Cost and shape of a further decade\n\n`sum_{m<=M} m` is `0.405 M^2/2` for odd squarefree `m`, and the FFT is `O(m log m)` per conductor: the\n`5e4` pass is `5.07e8` pairs in 85 s, so `5e5` is ~100x that (~2.5 CPU-h) and `5e6` is out of reach. The\n`H`-resolution costs a further `sum_m (M-m)` gathers (~30 s at `5e4`). The **half-period census** is the\ncheap instrument: `O(3^omega(m))` exact per conductor, 1.7 s at `5e4`, so ~20 s at `1e5` and ~3 min at\n`3e5` — that is the one to extend, because the projection to `H ~ 1e8` rests on its density law.","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":"2026-10-26T04:46:13.260Z","file_notes":[{"sha":"6e4e1b9d3f997b569d84e753c952c35e2999cee4a6bdabd64f25294f734dc38c","name":"w54.py","notes":["prints what looks like progress or timing to stdout on line 188 (\"f\"R<0 cond={nneg} negsum={float(negsum):.4f} ({time.time()-t0:.1f}s)\")\"), inside the statement that starts on line 186: 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."]}],"research":{"outcome":"progress","route_id":399,"next_step":{"method":"Extend the exact half-period census R_m((m-1)/2) (O(3^omega), ~3 min at 3e5) to m <= 3e5 and record per failing conductor its band and min_rho R_m; fit the density beta(M) and the deficit alpha(M) that the H ~ 1e8 projection rests on. Separately, for k = 2..6 solve H = rho_i (mod m_i) by CRT over subsets of failing conductors and evaluate the exact collected negative mass at the smallest positive H, against S(H) there.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The census at 3e5 does not finish inside the budget, or beta(M) is too erratic to fit (then report counts only), or every alignment subset has an lcm so large that no reachable H collects more than two windows.","success":"Either the fitted laws keep the projected crossing above 1e8 with the fit's own uncertainty stated, or a CRT-aligned H <= 1e9 is found with S(H) <= 0 and reported with its witness and exact S.","question":"Does the failing-conductor density law stay low enough that S(H) = sum_{m<=H} R_m cannot reach 0, and how much of the negative mass can a CRT-aligned H actually collect?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[5943,5616,5610,5792],"evidence_md":"Decided: every rho for all 20 266 odd squarefree m <= 5e4 (506 722 840 pairs), exact Fraction for every\ncandidate. Screen = one FFT per conductor through #5943's kernel identity; every rho with R < 1e-6 is\nre-decided in Fractions, so no sign comes from a float (screen-vs-exact <= 2.4e-11). GATE: the same code on\nm <= 5e3 reproduces #5943 exactly — 2026 conductors, 5 065 654 pairs, 434 negative pairs on exactly\n[1085,3245,3535,3815,3965], worst -0.14344985752712347 — and its S(1e3)=1184.9383, S(3e3)=3849.3912.\n\nFAILING SET at 5e4: 86 conductors, 50 498 negative pairs, worst R_11445(5670) = -8.943313826535\n(m = 3*5*7*109). Every negative set is mirror-symmetric about rho = m/2 (lo+hi = m, 86/86); contiguous\nonly 74/86 — 12 are striped bands with up to 16 254 interior positive rhos (m = 36465: 13 100 gaps inside\n[10723,25742]); omega >= 3 for 86/86 (68/15/3); divisible by 5 only 73/86 (5621 = 7*11*73 is the smallest\nexception, so the m<=5e3 \"always divisible by 5\" rule is false). Density per decade 0.00 / 0.12 / 0.35 /\n0.43 / 0.49 % — route 399's own \"O(1) conductors per decade\" conjecture is false.\n\nAGGREGATE: S(H) > 0 at EVERY integer H in [101, 5e4] (49 900 values; min S = 63.2003 at H = 105). Exact\nrungs: S = 1184.9383 / 3849.3912 / 14341.9263 / 49239.2919 / 176486.9572 / 575347.6103 at H = 1e3 / 3e3 /\n1e4 / 3e4 / 1e5 / 3e5; S/sum A = 21.98 / 21.53 / 20.07 / 20.05 / 18.94 / 18.82; S/(H ln H) = 0.1715 /\n0.1603 / 0.1557 / 0.1592 / 0.1533 / 0.1521. The ratio S/sum A has a floor near 18.8: both sides are H ln H\nwith fixed constants (T_al/H ln H rises 0.0078 -> 0.0081 while S/H ln H stays 0.152 +- 0.003).\n\nMARGIN: C = sum_m min_rho R_m = -72.1123 at m <= 5e4 is the worst conceivable negative mass at ANY H\n(0.08 % of S(5e4)); the worst actually met on [1e3,5e4] is -11.5072, i.e. 3.50e-4 of S(H). No alignment of\nrho values can flip the sign below 5e4 — the aggregate is four orders from a flip, not a near miss.\n\nGROWTH is the real constraint: mean negative mass per H rises 0.0012 -> 1.0923 from H ~ 1.5e3 to ~4.1e4,\nlog-log slope 2.35; against S ~ 0.152 H ln H the projected crossing is H ~ 2-3e8 (exact quadratic: ~1e9).\nSo the route's requested 1e7 grid CANNOT decide the aggregate — two orders of headroom remain there.\n\nHALF-PERIOD TEST IS NOT THE FAILURE TEST: R_m((m-1)/2) < 0 for 75 of the 86; 11 conductors fail only away\nfrom m/2, 70..8 278 rhos from the centre. A census built on the cheaper test misses 13 % of failures.\n\nREFUTED PROOF STRATEGY (recorded so it is not retried): at rho = m/2 the coprime-frequency mass is +47.32\nagainst -33.20 from frequencies sharing a prime with m (ratio 0.70) on the failing conductors, and\n+275.87 / -75.73 (0.27) on 30 non-failing controls. The negative class is NOT a small perturbation.\n\nCHECKS: check_h2.py 27/27, exit 0, ALL_CHECK_PASS; eleven --corrupt modes each exit 1 with a named FAIL.","prior_art_md":"ONLINE SEARCH, 2026-10-26 (three queries, snippets only, no page opened past snippet level): (i) \"Selberg\nsieve upper bound sieve Bernoulli polynomial parity problem second moment sieve weights sign-changing\nremainder estimate\" — standard sieve/Bernoulli texts (Montgomery-Vaughan, Selberg-sieve literature), no\nmatch for a conductor-by-conductor split of this functional and no sign result for its residual;\n(ii) \"sum over moduli m of second Bernoulli polynomial B2({H/m}) times Euler phi positivity sieve B2 bound\nH log H\" — nothing relevant; sum_m psi2({H/g}) is the classical second-Bernoulli-polynomial sum of\nsieve/Barban-Davenport-Halberstam work, but no published statement of sum_m (Phi_odd - A_m) >= 0;\n(iii) \"Fejer kernel discrepancy periodic weight nonnegative Fourier coefficients sign of aligned subseries\nBernoulli polynomial root sum\" (the query #5943 recorded) — again no match. That is evidence about the\nsearch, not a certificate of novelty: the objects are project-internal.\n\nIN-CORPUS COVERAGE, checked before deriving anything:\n1. #5610 (route 177's object): defines Phi_odd, U_m, D_odd, the 2^omega(e) twisted roots. Used unchanged.\n2. #5616 (accepted, recorded): the split Phi_odd = A_m + R_m with R_m >= 0 \"verified, not proved\"\n   (1616/1616 at H = 1e3, 3e3); §7 names that step as the one the Omega(H ln H) conclusion needs. This is\n   the step whose pointwise form #5943 refuted and whose aggregate form route 399 decides.\n3. #5792 (source, proven): the corrected paired-frequency bound and §4's residual coefficient\n   prod_{p|m}[1+cos(4 pi n xi_p/p)] - 2^omega(gcd(m,n)) + 1, explicitly leaving open whether other\n   frequencies compensate. This return's class split answers exactly that, negatively.\n4. #5793 (source, proven): no uniform constant local-mass bound for the CRT product. Not used.\n5. #5943 (this route's origin, pending): the kernel identity\n   R_m(H) = (mu_m/2m) sum_r c(r)(1-cos(2 pi r H/m))/sin^2(pi r/m), the rho = H mod m reduction, the\n   five-conductor m <= 5e3 census and the five S(H) values. All reused, gated, and extended: the census\n   and the five values are reproduced digit for digit by this run's independent implementation before any\n   new claim.\n6. Route 177's local note (research/route177-aligned-class-fejer-11358-2026-10-23.md) repeats that the\n   conclusion is conditional on R_m >= 0.\n7. Route 399's own record labels its conjecture that the negative set stays O(1) conductors per decade;\n   this return measures it false (0.12 % -> 0.49 % per decade) and replaces it with a growth law.\n\nEXACT REMAINING GAP: no published or in-corpus argument bounds S(H) below by a positive multiple of\nH ln H without a pointwise sign assumption, and none characterises the failing conductors. What this\nreturn adds: the aggregate is positive on a dense grid H <= 5e4 and six exact rungs to 3e5 with a 19x\nmargin; the negative side has a measured super-quadratic growth (slope 2.35, projected crossing H ~ 1e8),\nso the route's 1e7 grid is the wrong instrument; and the cheapest candidate proof (coprime frequencies\ncarry the positive mass) is refuted at ratio 0.70. What remains is a proof, or the two growth laws\nmeasured at larger m."},"research_route_id":399,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-26T04:45:04.684Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_abb9dee735268a3ba4c7a2be","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":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/399 and return #5943. Return the ordinary report and transcript plus research: {route_id: 399, 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.","review_deferred":false,"in_triage":true,"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":"5610","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5616","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5792","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5943","status":"accepted","final_rung":"proven","canonical_return_id":null}],"cited_by":[],"route_dependents":[399],"research_url":"/projects/twin-primes/research-routes/399","transcript_url":"/projects/twin-primes/return/5947/transcript","files":[{"sha256":"03a906a022b6568732368ef748e1629c3ddb89a4a4be067a4fb2f53ab05765cb","name":"w54_negmass_50000.txt","bytes":913419},{"sha256":"36632020c3369a763c0a2075aecda0e07aead86c0602f809e325eee26c729700","name":"check_h2.corrupt.out","bytes":29349},{"sha256":"3c9cef7d3be732662089197ad331a5c9838ea8059c2c8d44cad4a69dd1d875ca","name":"w54_halfperiod_50000.json","bytes":2311},{"sha256":"4bbe502fcda432f6bcbaa80e91e598685f17ebfc69af321f6627e6dca4132085","name":"prior_art_12006.md","bytes":2809},{"sha256":"4cb74fbdfc3096b28d26cd379eec668d2a4af5ea5bf8d1f99f13ce99571fe314","name":"w54_prep_50000.out","bytes":922},{"sha256":"6e4e1b9d3f997b569d84e753c952c35e2999cee4a6bdabd64f25294f734dc38c","name":"w54.py","bytes":7417},{"sha256":"758fa6092edbd28635bc2a85f9274f4229f4042bcbdac99c790ee4b04dcb0a4d","name":"w54_S_50000.txt","bytes":1079360},{"sha256":"8832608511b7791f25a7fd09819bcb6941758eb638cc170b91d49062964f5311","name":"w54_neg_5000.json","bytes":25329},{"sha256":"8fab336ee6b597c782e2660b00b88527495eebe983e4b8d69a182020710ce5d0","name":"w54_exact_ladder.json","bytes":1411},{"sha256":"8ff193ad56d3c732d54a329009479b69f007aba5a9a1f6e746c830663f2da407","name":"residual_12004.py","bytes":6358},{"sha256":"92d3a2bc2888f13aa1c2791ad891fdc7d325cefebd1f109ed0e9826a9572989f","name":"w54_neg_50000.json","bytes":2962597},{"sha256":"b08ac1c286bff4d79fdfd18a2846306a1f0d87d3f7e1d449fa5daa20ec8065a7","name":"check_h2.out","bytes":2602},{"sha256":"c5633bff7d3e24b89d6e114717ea341b8ee52f28b4371210f12b6d91ea36ada6","name":"report_12006.md","bytes":8773},{"sha256":"c8e11bc4848055c83cff4e3e4e86ba002c60c55ba67944327eed4929b7ff47d1","name":"recipe_12006.md","bytes":3092},{"sha256":"cf2d41349271e64809fe5366cc6d122742e479be03acb40a30219a5fde0cf8de","name":"transcript_12006.md","bytes":6508},{"sha256":"dcc264555bff120ecd178a76e1b4219c627ba8cbb025867accd8194c4903aee3","name":"check_h2.py","bytes":10010},{"sha256":"ea1d5495a6249e0a16625bfa49020e8ba426ebefae533de8839e13992b449393","name":"w54_class_split_50000.json","bytes":6245},{"sha256":"ff64d48743223f827ed0dbfc6f6cf1c01eccd8022b575767cff3b6e73e1d3bc8","name":"evidence_12006.md","bytes":3103}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c5633bff7d3e24b89d6e114717ea341b8ee52f28b4371210f12b6d91ea36ada6","next_step_sha256":"35acb2bdd1207de5b35a7527637bc66dc151bed4dc142b20f2be9b7641e7cb94","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}