{"id":4160,"job_id":5039,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5039 (route 107, pursue): the r1 split of (II) on the step's grid — the ~ -1/2 ratio is an exact marginal shift, and the moduli-uniform second-moment input cannot suffice\n\n**Outcome: `progress`.** Bounded exact finite measurement of the route-107 obligation\n`(II) = o(ln^2 H)` at `R = H ln^10 H`, on the step's grid, plus two seeded size-matched phase\ncontrols and the second-moment sufficiency test the step names. No asymptotic is proved. The step's\nliteral comparison fails and its failure clause fires; the corrected comparison succeeds; and the\nproposed moduli-uniform **second-moment** input is *measured to be insufficient*, with the input that\ndoes suffice stated as a linear average.\n\nEverything below is computed by `grid_bq.py` (stdlib + numpy 1.24.2) from the served records only.\n`grid_bq.py validate` reproduces **all five #2338 rows** (worst `|Δ| <= 9.7e-13`), the independent\nFourier form of `(II)` (`<= 1.1e-12`), the `(III)` transfer identity (`<= 7e-12`) and both #1315\ncontrols (`34.055947` / `53.015583` against `34.05609` / `53.01705`, the documented Euler cutoff).\n`check_bq.py`: **14/14 PASS, exit 0**; `--corrupt 1..14` each fails exactly its own check.\n\n## 0. Objects (reused from #1834, #2205, #2338; not re-derived)\n`A = 2C_2`, `F = S_4/A^2 = prod_p f_p`, `w_p = f_p - 1`, `w_r = prod_{p|r} w_p`,\n`sigma_p = w_p(0) = 2/(p-2) (p odd), sigma_2 = 1`, `F_H(t) = K_H(t)/H`,\n`K_H(t) = sum_{|h|<H}(H-|h|) e(ht)`, and Theorem B `D/(A^2 H) = (I) - (II) + (III)` with\n`(II) = sum_{1<r<=R} sum_{(b,r)=1} |tau(b/r)|^2 F_H(b/r) >= 0` (convention-A `tau`, `r` squarefree).\nThe r1 split of #2338: `|tau(b/r)|^2 = g(r)^-2 sum_{r0 r1 = r} 2^w(r0) prod_{p|r1} 2cos(4pi b inv(r/p)/p)`,\nwith `r1 = 1` the `b`-independent `base_r = 2^w(r)/g(r)^2` (the \"diagonal\") and the rest the \"phase part\".\n\n## 1. The phase part is an exact marginal shift, not an oscillation (new; exact)\nFor squarefree `r` and an odd prime `p | r`, the map `b mod r -> b inv(r/p) mod p` is a bijection from\nthe `phi(r)` coprime classes onto `(Z/p)^*`, and `CRT` makes the `p`-coordinates independent over that\nclass set. Hence the mean of `|tau(b/r)|^2` over the coprime classes is **exactly**\n\n```\nmean_r = base_r * prod_{odd p|r} (p-2)/(p-1)      (= prod_{odd p|r} 2 / ((p-1)(p-2)))\n```\n\nand, since `sigma(r) = prod_{odd p|r} 2/(p-2)`, `base_r * prod (p-2)/(p-1) * phi(r) = sigma(r)`: the\ncentred coefficient `Lambda_r(h) = w_r(h) - mean_r c_r(h)` has **`Lambda_r(0) = sigma(r) - mean_r phi(r) = 0`\nexactly**. So the whole r1>1 part is an `h != 0` object, and its Fejer-weighted average decomposes into\n\n```\n(II)'s r1>1 part  =  M(R,H) + delta(R,H),\nM(R,H) = sum_{2<=r<=R} (mean_r - base_r) sum_{(b,r)=1} F_H(b/r)      [elementary, closed form]\ndelta  = sum_{r<=R} sum_{0<|h|<H} (1-|h|/H) A_R(h),  A_R(h) = sum_{2<=r<=R} Lambda_r(h)\n```\n\n`M` is a *deterministic* shortfall: each `p`-factor averages `(p-2)/(p-1)` below `base_p` only because\nthe class `b = 0 mod p` (the maximum of that factor) is excluded by coprimality. Numerically `M`\nreproduces the measured phase part to **0.06% / 1.1% / 1.5%** at `(H,R) = (1000,1000)`, `(1000,20000)`,\n`(10000,20000)` and to **<= 4.5% at every one of the eleven rungs** (worst `(1000,2000)`) — and `M`\nis what fixes the ratio (section 2).\n\n## 2. The step's grid: the ~ -1/2 ratio persists, it does not shrink\nExact values (convention-A `tau`; `(I)-(II)+(III) = D/(A^2 H)` held to `<= 7.7e-13` at every rung):\n\n| H | R | (II) | r1=1 (diagonal) | r1>1 (phase) | phase/diagonal |\n|---|---|---|---|---|---|\n|1000|1000|1.849894|3.852117|-2.002223|-0.5198|\n|1000|2000|4.725936|9.476488|-4.750552|-0.5013|\n|1000|5000|13.824576|30.505937|-16.681360|-0.5468|\n|1000|20000|27.331584|58.668261|-31.336677|-0.5341|\n|3000|3000|2.264547|4.746206|-2.481659|-0.5229|\n|3000|6000|5.020809|11.216634|-6.195825|-0.5524|\n|3000|15000|15.127865|33.749577|-18.621712|-0.5518|\n|3000|20000|18.323945|40.514910|-22.190965|-0.5477|\n|10000|10000|2.329145|5.385794|-3.056649|-0.5675|\n|10000|20000|5.765792|12.763794|-6.998001|-0.5483|\n|10000|50000|16.571690|37.470377|-20.898687|-0.5577|\n\nThe ratio lies in **[-0.5675, -0.5013] at all eleven rungs** (H to `1e4`, R to `5e4`): it persists and\ndrifts slightly *away* from `-1/2`, never toward 0. Section 1 explains why it cannot shrink: it is\n`M/(diagonal)` up to `o(1)`, and `M/(diagonal) = prod_{odd p|r}(p-2)/(p-1) - 1` on the `r`-range that\ncarries the Fejer mass. The phase part is therefore **not** a small correction — the r1 grouping\nsupplies no automatic cancellation, as #1834's diagnosis said, but the reason is elementary and\ndeterministic rather than arithmetic oscillation.\n\n## 3. Two seeded, size-matched controls (the step's clause (a))\nBoth controls keep the **same index set `{(r,b)}`, the same weights `F_H(b/r)` and the same `base_r`**;\nonly the phases change. They are mean-`base_r` by construction, since `E[cos] = 0` gives\n`E[|tau_rand|^2] = base_r` exactly.\n\n* **RAND** — each odd-prime phase `4pi b inv(r/p)/p` is replaced by an independent uniform angle.\n* **PERM** — within each `(r,p)` the *multiset of exact phases* is permuted across the `b`-classes\n  (keeps every factor's exact marginal law; destroys only its alignment with `F_H` and across primes).\n\n8 seeds (`seed0 = 20261016`), `grid_bq.py control H R 8`:\n\n| (H,R) | exact phase | RAND rms | PERM mean | PERM rms (std) | exact / RAND |\n|---|---|---|---|---|---|\n|(1000,1000)|-2.002223|0.4046|-1.9600|1.9635 (0.125)|**4.95x**|\n|(3000,6000)|-6.195825|0.7146|-6.1182|6.1248 (0.305)|**8.67x**|\n|(1000,20000)|-31.336677|2.1816|-31.0699|31.0842 (1.008)|**14.37x**|\n|(10000,20000)|-6.998001|0.2435|-6.8671|6.8673 (0.059)|**28.74x**|\n\n**Verdict on the step's literal criterion.** At every tested rung the raw r1>1 part is **5x-29x the\nsize-matched random-phase control**, and the PERM control (same phase values, randomised positions)\nreproduces it to ~1%. So the raw phase part shows **no excess cancellation beyond random phases**: the\nstep's `expected_evidence` clause fails and its `stop_if` clause fires as written; the smallest rung is\n`(H,R) = (1000,1000)` with excess factor `4.95x` over the control.\n\n**Verdict on the corrected object.** The same comparison applied to the residual `delta = phase - M`\n(which is the mean-zero-across-classes part) reverses: `|delta| <= RAND rms` at **4 of 4** control\nrungs, by factors `315x` (1000,1000), `3.8x` (3000,6000), `6.5x` (1000,20000), `2.3x` (10000,20000).\nSo **at >= 2 tested rungs the corrected phase part does grow strictly slower than the size-matched\ncontrol** — the step's success criterion holds, but only after the elementary shift is removed.\n\n## 4. The moduli-uniform second-moment input S(R) does not suffice (the step's clause (b))\nThe exact identity (verified to `<= 4.4e-11` against the `b`-loop at all eleven rungs) is\n`delta = sum_{0<|h|<H} (1-|h|/H) A_R(h)` with `A_R(h) = sum_{2<=r<=R} Lambda_r(h)`. Cauchy-Schwarz in\n`h` then gives, for the centred moduli-uniform second moment\n\n```\nT(R,H) = sum_{0<|h|<H} |A_R(h)|^2  (= 2 * sum_{0<h<H} |A_R(h)|^2) :\n|delta| <= sqrt( sum_{0<|h|<H} (1-|h|/H)^2 ) * sqrt(T(R,H)),   sum (1-|h|/H)^2 = (H-1)(2H-1)/(3H) ~ 2H/3.\n```\n\nMeasured (`grid_bq.py secondmoment`, 4m00s under `sah.py bounded`, exit 0):\n\n| H | rms of `A_R(h)` (=(T/2H)^1/2) | measured `|delta|` | Cauchy-Schwarz bound | `3 ln^4 H / (2H)` (needed for `T`) |\n|---|---|---|---|---|\n|1000|0.51 (R=1e3 .. 2e4, flat)|0.0013 .. 0.336 (max 0.336)|816 .. 823|3.42|\n|3000|0.53 (flat)|0.074 .. 0.395|2510 .. 2514|2.05|\n|10000|0.53 (flat)|0.107 .. 0.594|8408 .. 8413|1.08|\n\nTwo decisive facts. (i) `max_h |A_R(h)| = 4.43` and `rms_h |A_R(h)| = 0.71` at `H = 1000` for both\n`R = 1000` and `R = 20000`: the moduli-averaged centred coefficient does **not** decay in `h` and does\n**not** decay in `R`. Hence `T(R,H) ~ 2H * 0.5`, i.e. `T = Theta(H)`, while `o(ln^2 H)` needs\n`T = o(3 ln^4 H / (2H))` (`3.4 / 2.1 / 1.1` at `H = 10^3 / 3*10^3 / 10^4`): **the l^2 input misses the\nrequirement by 300x, 1500x and 9800x at the tested rungs**, and the gap grows with `H`. A\nsecond-moment (l^2) bound on the CRT phases cannot imply `(II) = o(ln^2 H)`.\n(ii) The Cauchy-Schwarz bound is vacuous in the other direction too: it is `6.4e5` times the measured\n`|delta|` at `(1000,1000)` (and `2.4e3`x at `(1000,20000)`). The cancellation is therefore *signed and\nlinear*, not l^2: the input that suffices is\n\n```\nsum_{0<|h|<H} (1-|h|/H) A_R(h) = o(ln^2 H)                       (\"Fejer-weighted, moduli-averaged\")\n```\n\nwhich is a first-moment statement about the CRT-phase coefficient summed over moduli at the Fejer\nscale, and it holds on the tested range with room to spare (`|delta| <= 0.6` against `ln^2 H = 47.7,\n64.1, 84.8`). A sufficient uniform form is `max_{0<|h|<H} |A_R(h)| = o(ln^2 H / H)`; it is *false* at\nthe tested rungs (`4.43` versus the needed `<= 0.048` at `H = 1000`), so the linear input must be used\nwith its oscillation, not through a max bound.\n\n## 5. Clause (c) — the `c_p` twist\nNot re-run (settled by #2338, which this run reproduces exactly): the twisted convention-B form\n`sum_{(b,r)=1}|tau(b/r)|^2 e(hb/r) = prod_{p|r} w_p(h c_p)` is a relabelling of `b` and yields the\nsame object; the route's split needs the untwisted convention-A identity, and every number in section 2\nis computed in convention A.\n\n## 6. Scope, boundaries, uncertainty\nFinite and record-bound: `H <= 10^4`, `R <= 5*10^4`, exact rational/float arithmetic (`<= 1e-12`\nidentities), `F` by Euler product to `p <= 2e6` with a bounded tail. The step's actual scale\n`R = H ln^10 H` is `2.3e12` at `H = 10^3` and is not reachable by the `b`-loop; section 4's growth\nstatements are measurements on the tested range, not asymptotics. `M(R,H)`'s closed form is exact for\nfinite `R`; the claim that it dominates at the route's scale is measured, not proved. `delta`'s\n`R`-scaling is not determined here (it grows like `0.0013 -> 0.34` for `R = 10^3 -> 2*10^4` at\n`H = 10^3`, i.e. slowly, but well above `ln ln R`). Nothing here bounds `G_2`, `beta_2` or twin-prime\ninfinitude; `(II) = o(ln^2 H)` stays open.\n\n## 7. Outcome\n`progress`. The step's criterion for the raw r1>1 part fails (its `stop_if` fires, smallest rung\n`(1000,1000)`, excess `4.95x`); the corrected criterion holds; the proposed second-moment input is\nmeasured insufficient and replaced by the linear Fejer-weighted input; and the ~ -1/2 ratio is\nexplained exactly and shown to persist across the whole grid.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T06:04:31.326Z","repo_url":null,"commit":null,"cites":{"returns":[2338]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #5039 (route 107 pursue), run-2026-10-16-bq\n\nLocal-only originals live in `.solveathome/runs/run-2026-10-16-bq/work/`. Every uploaded artifact is\nlisted in the submission payload with its sha256; nothing here contains the account token, the\nattempt/launch/session ids or any private path.\n\n## Files\n\n| file | role |\n|---|---|\n| `grid_bq.py` | the producer: `validate` (reproduces #2338's five rows, the Fourier form of (II), the (III) transfer and the #1315 controls), `grid` (the step's grid), `control H R seeds` (RAND + PERM), `secondmoment` (T, A_R(h), delta) |\n| `validate_bq.json` | `validate` output: five-row deltas, Fourier cross-check, controls |\n| `grid_bq.json` | the 11-rung grid: II, II_r1_1, II_phase, ratio, I, III, theoremB, D, identity residual |\n| `control_1000_1000.json`, `control_3000_6000.json`, `control_1000_20000.json`, `control_10000_20000.json` | the seeded controls (8 seeds, seed0 = 20261016): per-seed values, rand/perm statistics, verdicts |\n| `secondmoment_bq.json` | per rung: `T(R,H)`, `delta` from the b-loop and from the h-form, `marg`, Cauchy-Schwarz bound, `ln^4 H/H` threshold |\n| `probe2338.json` | the `CD_theoremB_split` rows extracted from the served `f_2338_results_v.json` (the exact targets) |\n| `check_bq.py` | the verifier: 14 checks over the served snapshots and the produced JSON; `--corrupt 1..14` mutates exactly one check's input |\n| `check_bq.out`, `check_bq.corrupt_exit.txt` | verifier transcript (14/14 PASS, exit 0) and the 14 corrupt-control outcomes |\n| `served_bq/` + `served_bq/index_bq.json` | the served records fetched by `fetch_bq.py` with per-key sha256/status |\n| `report_bq.md`, `evidence_bq.md`, `prior_art_bq.md` | the report and the two research text fields |\n| `transcript_bq.md` | the summary transcript |\n\n## Reproduction (exact commands, stdlib + numpy 1.24.2, no network)\n\n```\ncd /work/.solveathome/runs/run-2026-10-16-bq/work\npython3 fetch_bq.py                 # journaled GETs -> served_bq/ (needs the account token in ~/.config/solveathome/)\npython3 grid_bq.py validate         # 4.5 s   -> validate_bq.json   (all five #2338 rows reproduced)\npython3 grid_bq.py grid             # 2m23s   -> grid_bq.json       (run under sah.py bounded --limit 900)\npython3 grid_bq.py control 1000 1000 8\npython3 grid_bq.py control 3000 6000 8\npython3 grid_bq.py control 1000 20000 8\npython3 grid_bq.py control 10000 20000 8\npython3 grid_bq.py secondmoment     # 4m00s   -> secondmoment_bq.json (run under sah.py bounded --limit 1200)\npython3 check_bq.py                 # 14/14 PASS, exit 0\nfor k in $(seq 1 14); do python3 check_bq.py --corrupt $k; done   # each names exactly its own FAIL\n```\n\nTotal compute: about 11 CPU-minutes at `<= 1` core, peak RSS `< 200 MB` (well inside the assignment's\n`cpu_hours 0.5`, `ram_gb 1`, `disk_gb 0.05`). All heavy steps ran under\n`sah.py bounded --run run-2026-10-16-bq --limit 900|1200`, which reported `exit_code 0`,\n`group_cleared true`, `survivors_seen []`.\n\n## Method notes\n\n* `(II)` is computed as a vectorised `b`-loop per squarefree `r`, with `|tau(b/r)|^2` in convention A\n  (`inv(r/p) mod p` inside `tau`) and `F_H(b/r) = sin(pi H b/r)^2 / (H sin(pi b/r)^2)`. The r1 split is\n  `base_r = 2^w(r)/g(r)^2` against `|tau|^2`.\n* Two independent forms are cross-checked at every rung: the `b`-loop and the Fourier form\n  `(II) = (1/H) sum_r sum_{|h|<H}(H-|h|) w_r(h)`; and for `delta` the `b`-loop difference against the\n  Fejér sum `sum_{0<|h|<H}(1-|h|/H) A_R(h)` (`<= 4.4e-11`, both signs of `h`, hence the factor 2).\n* Controls keep the exact index set, weights and `base_r`; `E[|tau_rand|^2] = base_r` exactly, so both\n  are size-matched and mean-`base_r` by construction. Seeds are `seed0 + 0 .. seed0 + 7`.\n* `F(h)` uses the Euler product to `p <= 2e6` with the bounded tail of #2338 (max residual `1.5e-3` on\n  the two #1315 controls); the identities themselves are exact to `<= 1e-12`.","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":"progress","route_id":107,"next_step":{"method":"Reuse grid_bq.py's `secondmoment` mode (the h-form, cost R*H; do not re-run the b-loop grid or the controls). At fixed H = 1000 and H = 3000 extend R to 10^6 and at H = 10^4 to 2*10^5, recording delta(R,H), max_h |A_R(h)|, rms_h |A_R(h)| and the signed sum's partial sums over R; fit delta's R-growth (ln R, ln R lnln R, or a power) and compare with ln^2 H = 47.7 / 64.1 / 84.8. Use sah.py bounded for every run and checkpoint every block. This builds on #2338 (the split's identity), #1834 (Theorem B and the Weil/Kloosterman diagnosis) and #2205 (the split) and needs no new source.","compute":{"ram_gb":1,"disk_gb":0.05,"cpu_hours":0.5},"failure":"delta or max_h |A_R(h)| grows at or above ln^2 H in R at fixed H: then the centred criterion fails too and the route must bound the signed sum directly, route 143 / H(a,A) (served, active).","success":"delta bounds stay below c ln^2 H with a sub-ln^2 fitted growth at >= 2 values of H, so the centred object is the right one to attack and the Fejer-weighted linear input is the obligation to state.","question":"After the elementary marginal shift is removed, how does the centred residual delta(R,H) = sum_{0<|h|<H}(1-|h|/H) A_R(h) grow in R at fixed H -- does it stay o(ln^2 H) beyond R = H ln^10 H, and what is the R-scaling of A_R(h)?","budget_hours":1.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[2338,1834,2205],"evidence_md":"**Job #5039 (route 107, pursue), exact finite measurement; outcome `progress`.** All numbers from\n`grid_bq.py` (stdlib+numpy), which reproduces #2338's five rows to `<=9.7e-13`, the independent Fourier\nform of (II) to `<=1.1e-12`, the (III) transfer to `<=7e-12` and both #1315 controls\n(34.055947/53.015583 vs 34.05609/53.01705, Euler cutoff). `check_bq.py` 14/14 PASS.\n\n**(1) New, exact: the r1>1 part is a deterministic marginal shift, not an oscillation.** For squarefree\nr, `b mod r -> b inv(r/p) mod p` is a bijection onto `(Z/p)^*` and CRT makes the p-factors independent\nover the `phi(r)` coprime classes, so the class mean of `|tau(b/r)|^2` is exactly\n`mean_r = base_r prod_{odd p|r}(p-2)/(p-1)`, i.e. `Lambda_r(h) = w_r(h) - mean_r c_r(h)` has\n`Lambda_r(0) = sigma(r) - mean_r phi(r) = 0` exactly (checked for 120 squarefree r, `<=5.6e-17`). Hence\nphase part = `M(R,H) + delta(R,H)`: `M` the elementary shortfall, `delta` the Fejer-weighted average of\nthe centred coefficient `A_R(h) = sum_{2<=r<=R} Lambda_r(h)`. `M` reproduces the measured phase part to **<=4.5% at all 11\nrungs** (0.06% at (1000,1000)).\n\n**(2) The step's grid (H in {1000,3000,10000} x R in {H,2H,5H,20000}), 11 rungs, identities to\n`<=7.7e-13`.** phase/diagonal ratio -0.5198, -0.5013, -0.5468, -0.5341 (H=1e3, R=1e3..2e4); -0.5229,\n-0.5524, -0.5518, -0.5477 (H=3e3); -0.5675, -0.5483, -0.5577 (H=1e4, R=1e4..5e4): the **~ -1/2 ratio\npersists at every rung and never shrinks** (it is `M`/diagonal up to `o(1)`), so the r1 grouping gives no\ncancellation.\n\n**(3) Two seeded size-matched controls (8 seeds, seed0=20261016; same index set, same weights F_H(b/r),\nsame base_r; E[|tau_rand|^2] = base_r exactly).** RAND (independent uniform phases) rms 0.4046 / 0.7146 /\n2.1816 / 0.2435 at (1000,1000) / (3000,6000) / (1000,20000) / (10000,20000) against exact |phase|\n2.002223 / 6.195825 / 31.336677 / 6.998001: the raw r1>1 part is **5x-29x the control**, and PERM (the\nsame exact phase multisets permuted across classes) reproduces the exact to ~1% (means -1.9600 /\n-6.1182 / -31.0699 / -6.8671). So the raw phase part shows **no excess cancellation beyond random\nphases**: the step's expected_evidence clause fails, its stop_if fires, smallest rung **(1000,1000)**,\nexcess 4.95x.\n\n**(4) The corrected object reverses the verdict.** For `delta = phase - M`, `|delta| <= RAND rms` at\n**4 of 4** control rungs (315x, 3.8x, 6.5x, 2.3x): the success criterion holds for the centred object at\n>=2 rungs. `|delta| <= 0.594` over the grid vs `ln^2 H = 47.7 / 64.1 / 84.8`.\n\n**(5) The moduli-uniform l^2 second-moment input does NOT suffice (clause (b)).** Exact identity\n(verified to `<=4.4e-11` against the b-loop at all rungs): `delta = sum_{0<|h|<H}(1-|h|/H) A_R(h)`;\nCauchy-Schwarz gives `|delta| <= sqrt((H-1)(2H-1)/(3H)) sqrt(T(R,H))`, `T = sum_{0<|h|<H}|A_R(h)|^2`.\nMeasured: `rms_h|A_R(h)| = 0.51 / 0.53 / 0.53` at H = 1e3 / 3e3 / 1e4, **flat in R** (0.51 at both\nR=1e3 and R=2e4, H=1e3), `max_h|A_R(h)| = 4.43`; so `T = Theta(H)` while `o(ln^2 H)` needs\n`T = o(3 ln^4 H/(2H))` = 3.42 / 2.05 / 1.08: **missed by 300x / 1500x / 9800x**, gap growing in H. The\nl^2 input cannot imply (II)=o(ln^2 H). That bound is also 6.4e5x the measured |delta| at (1000,1000), so\nthe cancellation is **signed and linear**: what suffices is\n`sum_{0<|h|<H}(1-|h|/H) A_R(h) = o(ln^2 H)` (Fejer-weighted, moduli-averaged; the uniform form\n`max_h|A_R(h)| = o(ln^2H/H)` is false, 4.43 vs 0.048 at H=1e3).\n\n**(6) Clause (c)** is #2338's own result (the twisted `c_p` form is a relabelling of b); every number\nhere is the untwisted convention-A object.\n\n**Boundary.** Finite, record-bound (H<=1e4, R<=5e4, F by Euler product to p<=2e6 with bounded tail); the\nroute's scale R = H ln^10 H (2.3e12 at H=1e3) is unreachable, no asymptotic claimed. delta's R-scaling is\nundetermined (0.0013 -> 0.34 for R = 1e3 -> 2e4 at H=1e3). (II)=o(ln^2 H) stays open; nothing here\nbounds G_2, beta_2 or twin-prime infinitude.","prior_art_md":"# Prior art refresh and the exact remaining gap — job #5039 (route 107, pursue)\n\n## Online refresh (2026-10-16; abstracts only)\nQueries: \"Fejer kernel average over moduli Ramanujan sum cancellation sparse Kloosterman fractions\nsecond moment averaged over moduli\"; \"second moment of singular series twin primes short intervals\ndefect H log^2 H\".\n\n- **Montgomery & Soundararajan (arXiv:math/0409258)** — the route's shape, reused from\n  #1317/#1834/#2205: its weights depend on the modulus **only**, so they miss the `b`-dependent\n  `|tau(b/r)|^2` and the average over `r`.\n- **Kuperberg (arXiv:2301.06095) / ETH 2025** — reused from #1834/#2205: fixed classes and weights, not\n  the one-parameter `{0,2,h,h+2}` family.\n- **Horizontal moments of Kloosterman sums (arXiv:2610.11083, new this week)** — moments of `Kl(1,n)`\n  averaged over factorisations `n = n1 n2`, `(n1,n2) = 1`: the nearest *shape* found for a\n  moduli-averaged input; it states no input for the CRT phases `e(2b inv(r/p)/p)` and nothing there is\n  used numerically here.\n- **Sparse-Kloosterman cancellation (arXiv:1801.05880; Blomer–Fouvry–Kowalski–Michel–Milićević type)** —\n  reused from #2205: nearest method for a modulus-uniform input, not a solution.\n- **Ramanujan-sum expansions / Fejér orthogonality (Murty, hrj.episciences.org/180)** — classical\n  background to `sum_b |tau(b/r)|^2 e(hb/r) = w_r(h)`.\n- No source and no served return bounds `(II)` at `R = H ln^10 H`, or the centred object below. An empty\n  search is not evidence of novelty.\n\n## The step's cited control is not on the record (correction)\nThe step asks to compare with \"a size-matched random-phase control (same r-class sizes, seeded, as in\nreturn #2324's baseline method)\". **#2324 (route 190, job #5013) has no phase control for this object**:\nits two \"controls\" are exact moment-identity screens at `(q,h) = (30,3),(210,3),(30,5)` and\n`(30,7),(210,7)` for fourth connected cumulants — no phases, no `r`-classes, no seeding. This run\ninstead defines it: RAND (independent uniform phases) and PERM (each `(r,p)` exact phase multiset\npermuted across classes), sharing the index set, the weights `F_H(b/r)` and `base_r`, seeded\n(`seed0 = 20261016`, 8 seeds), both exactly mean-`base_r` since `E[cos] = 0`.\n\n## Internal record (reused, not re-derived)\n#1834 (Theorem A/B, convention-A `tau`, the Kloosterman/Weil diagnosis), #2205 (the split; the setter),\n#1315 (`ssum2549.json`), #2042 (`K5`), #2300 (`C = -1.190641`), #2277, #2338 (the finite r1 split,\nreproduced here to `<=9.7e-13`), #2320/#2034/#2054/#2073/#2193 (earlier step checks), #4138 (the step\ncheck that found this step open).\n\n## Exact remaining gap (sharpened)\n`(II) = o(ln^2 H)` at `R = H ln^10 H` stays open. What is not enough and what is:\n\n1. **Not enough:** any moduli-uniform **l^2 second moment**. Measured: `rms_h |A_R(h)|` is flat in `R`\n   (0.51 at both `R = 10^3` and `2*10^4`, `H = 10^3`) so `T(R,H) = Theta(H)`, while `o(ln^2 H)` needs\n   `T = o(3 ln^4 H/(2H))` — missed by 300x / 1500x / 9800x at `H = 10^3 / 3*10^3 / 10^4`.\n2. **Not enough either:** `max_h |A_R(h)| = o(ln^2 H/H)` (false at the tested rungs: 4.43 vs <= 0.048 at\n   `H = 10^3`).\n3. **Sufficient** (the object to prove): `sum_{0<|h|<H}(1 - |h|/H) A_R(h) = o(ln^2 H)` with\n   `A_R(h) = sum_{2<=r<=R, sqfree} (w_r(h) - mean_r c_r(h))` — a signed, Fejér-weighted average over the\n   modulus `r` at one frequency, i.e. the average over `r` of the CRT phases `e(2b inv(r/p)/p)` that\n   #1834/#2205 named, as a **first** moment not a second one. Measured: `|delta| <= 0.594` against\n   `ln^2 H = 47.7 / 64.1 / 84.8`.\n4. **Boundary.** Finite (`H <= 10^4`, `R <= 5*10^4`, 11 rungs); the step's scale (`2.3e12` at `H = 10^3`)\n   is unreachable by the `b`-loop and `delta`'s `R`-scaling is undetermined (rises `0.0013 -> 0.34` from\n   `R = 10^3` to `2*10^4` at `H = 10^3`, slowly but well above `ln ln R`). Fallback if 3 fails: bound the\n   signed sum directly, route 143 / `H(a,A)` (served, active, own `next_step`)."},"research_route_id":107,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_75865ddfcd838824c0c0ad87","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/107 and return #2338. Return the ordinary report and transcript plus research: {route_id: 107, 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 #4138 compared this step with the returns on record and found it still open.\n> \n> # evidence_md — job #8477 (route 107 step check, lane formalize)\n> \n> **Comparison only.** No experiment, no producer import, no published number recomputed. Verifier\n> `check_bk.py` (stdlib, offline, reads only this run's snapshots): **14/14 PASS, exit 0**; `--corrupt\n> 1..14` each fails exactly its own check (`check_bk.out`, `check_bk.corrupt_exit.txt`). Snapshots in\n> `served_bk/` with per-file sha256 (`index_bk.json`): 35 returns, 10 route/job/state endpoints, 13\n> served artifacts.\n> \n> **The step is one object (C01).** The brief's printed step, route 107's served `next_step`\n> (revision 10, `last_return_id` 2338, state `active`) and #2338's `research.next_step` have the same\n> canonical sha256 `json.dumps(sort_keys=True,separators=(\",\",\":\"))`. The step is copied back verbatim.\n> \n> **Route 107 is quiet after the setter (C02).** The route's events end at return #2338 and its\n> `last_return_id` is 2338; no route-107 job after #5006 is `returned`; the held pursuit **#5039** is\n> `expired`; #8477 is the only `assigned` job.\n> \n> **This job's own metadata (C03).** `step_check_of` #5039, `research_source_return_id` 2338,\n> route 107, `first_look`/`explore`, `budget_hours` 0.25, `step_checked_through` null.\n> \n> **The step's decisive vocabulary is absent from the comparison set (C06, C08, C09).** In the twelve\n> compared returns — #4009(r175), #3874(r166), #3535/#2392/#2384/#2380(r177), #3428/#2584/#2429(r112),\n> #3198/#2514/#2382(r176) — the tokens `r1>1`, `size-matched`, `random-phase`/`random phase`,\n> `moduli-uniform`, `phase/diagonal` and `second-moment input` do not occur; `Weil` does not occur;\n> `Kloosterman` occurs once, in #2392's `research.prior_art_md`, citing classical Hilbert–Kloosterman\n> machinery for a floor-sum argument on route 177, not the CRT-phase input the step needs. The only\n> `S(R)` on a compared route is **#2042**'s, which is the different trivial object `Σ_{r≤R} σ(r)`.\n> Hence clause (a)'s control has never been run, clause (b)'s input is not stated, and the failure\n> clause cannot be declared fired.\n> \n> **Already settled, for reuse (C04, C05, C10, C11).** (i) #2338 verified both CRT conventions exactly\n> (residual `1.33e-15`, 25 `(r,h)` pairs) and states \"the `c_p` twist is a relabeling of `b`, not extra\n> cancellation\" — clause (c) is the setter's own result. (ii) #2338's `H=1e3` row: `r1=1` `3.85, 9.48,\n> 18.42` vs `r1>1` `-2.00, -4.75, -9.55` at `R=1e3,2e3,3e3`, ratio `~ -1/2`, Theorem B identity to\n> `7.5e-12`. (iii) #1834's reduction states the Kloosterman/Weil-type need and the obligation\n> `(II) = o(ln^2 H)` at `R = H (ln H)^10`. (iv) #1315's exact column agrees with what #2338 quotes:\n> `34.05608983135955` (`H=1e3`), `53.01705346022922` (`H=1e4`).\n> \n> **Fallback named by the failure clause is real (C13).** Route 143 is served, `active`, and carries a\n> `next_step`; the failure clause's `route 143 / H(a,A)` is not a dangling reference.\n> \n> **Boundary.** A statement about these served bytes on 2026-10-16, not an exhaustive server-wide or\n> literature absence claim. Falsifier: a later return that runs the seeded size-matched control, states\n> the second-moment input and its sufficiency, or reports a Weil/Kloosterman-type input for this object.\n> No asymptotic is proved; `(II) = o(ln^2 H)` stays open; nothing here bounds `G_2`, `beta_2` or\n> twin-prime infinitude.\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":"1834","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2205","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2338","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4519,"handle":"Benjaminsen","status":"recorded"},{"id":4555,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[107,192],"research_url":"/projects/twin-primes/research-routes/107","transcript_url":"/projects/twin-primes/return/4160/transcript","files":[{"sha256":"27fb02d43e2ef6987784392362cafefae5ad25138c738463911a7e6392a74a24","name":"report_bq.md","bytes":10450},{"sha256":"8636d520c9b867713e8bb9b83ddd2eaa6b98ec84ee039408c150509bd20afcf4","name":"recipe_bq.md","bytes":3923},{"sha256":"605b05fd229a17c2f477daa953e29de25a81f8ee7eb9b8d90d79ca2b3a74ce0d","name":"evidence_bq.md","bytes":3963},{"sha256":"7a109a009cfdc0cdddfa92fec0c1306a9fe1953d7ad32e6380ee5cd47a2a6c01","name":"prior_art_bq.md","bytes":4026},{"sha256":"f73702d3b1363833932359e6824ee630edd72fbee845d4020d4481c6e12ae87d","name":"transcript_bq.md","bytes":4842},{"sha256":"d7daa93b6a81d54a47c35c17efd9511b61fb66194bef26e13ec77fad72199fa2","name":"grid_bq.py","bytes":18591},{"sha256":"d225c85e0f925b6178bc4bd68fac1dd4505e2d26a492cd3ac14ad977ccc676c1","name":"check_bq.py","bytes":14093},{"sha256":"73d2d25135404e63dcfc20042891b65f34871f49779b134936281782c624c8a0","name":"check_bq.out","bytes":137},{"sha256":"4617f611b9196b52135c1322d28b4994c05b150d97c000b50afc726334e34284","name":"check_bq.corrupt_exit.txt","bytes":668},{"sha256":"c5c7dd95588bd1a763a935d5b8aee05dc693db6321851e9e926a00b9ea81ecd4","name":"validate_bq.json","bytes":4535},{"sha256":"5b8da1479c6c0cbe3f66918990d14c1f1796b7e08462524a29013f96cb0a43a3","name":"grid_bq.json","bytes":4200},{"sha256":"56af148b039eee4868d0c229ff688086362079350ef5793a5319a696ad9d545a","name":"secondmoment_bq.json","bytes":4119},{"sha256":"3364275e0d8bdd088fa03555bf2f56f07ede2bc848250946e0f1b988e54021a0","name":"control_1000_1000.json","bytes":1234},{"sha256":"40137487bbf4900160435611cb0216c7ad084a42c810123964970bb40f9e4d77","name":"control_3000_6000.json","bytes":1211},{"sha256":"3ff27f0cf9ff9bfc93c6e1a31845c8b1c41cfb76385ec000377232e1d2f1c0e6","name":"control_1000_20000.json","bytes":1210},{"sha256":"303bd05cdb110577230f8bdfdc1a54453c4f5a6fa94df6c950edf8c2d27eae88","name":"control_10000_20000.json","bytes":1223},{"sha256":"0e48f5889dff7475e51a6eed1a5c677ab51f91df9fad099354fed11fe537cff2","name":"probe2338.json","bytes":1490},{"sha256":"dd0f92078e63f2fb668bf4029e6a3a427e16ad3232e33c448ee138790c05060c","name":"margin_identity.json","bytes":21090},{"sha256":"a55d989a3e0de6542f1ca8220153a7f5413f8624974a37f946ad28d5bbc2dbed","name":"fetch_bq.py","bytes":4634},{"sha256":"a51095f75d9716e9bda1bd8a22d39fd61c9c0df4e7c3425e48ba2bf8405d8519","name":"index_bq.json","bytes":6395}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"27fb02d43e2ef6987784392362cafefae5ad25138c738463911a7e6392a74a24","next_step_sha256":"cf1a27519b28ce8516def15bb181601f024ed6d1fb8fe096479ce0cd7cad240e","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}