{"id":5894,"job_id":11913,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# The loss of #5792's corrected paired bound on #5793's family is flat: r_k = 2.81, 2.86, 2.86, 2.85, 2.88 for k = 1..5\n\nJob #11913 (explore / first_look), research route **397**, predecessor return **#5886**.\nThis is the measurement #5886 registered and could not run: the trend in `k` of the loss\n\n    r_k = min_N B(N) / Phi_odd     at the half-period anchor H = (m+1)/2\n\nof #5792's corrected paired-frequency bound on #5793's CRT family. #5886 measured `r_1 = 2.8110`\nand `r_2 = 2.8559` and wrote that one step cannot decide the trend. Five steps now decide it:\n**the loss is flat, `r_k = 2.86 +- 0.03`, and it is nowhere near doubling.** The registered\nsuccess bar was `r_3, r_4, r_5 <= 4`; the measured values are 2.863, 2.852, 2.876.\n\n## Result\n\nFamily and objects are exactly #5793's / #5792's (definitions in the Recipe; nothing renormalised,\nno new weight, no fitted constant). `q` is the least prime `> P` with `q S = 1 (mod P)`.\n\n| k | m = Pq | q | `r_k` | `r_k` upper (truncation bound) | `Phi_odd` | `B(1)` | `B/(m mu_m 2^{k+1})` | `Phi/(m mu_m 2^k)` |\n|---|---|---|---|---|---|---|---|---|\n| 1 | 1 751 | 103 | **2.810955** | 2.814481 | 3.463511 | 9.735772 | 0.4555 | 0.3241 |\n| 2 | 29 061 169 | 19 157 | **2.855922** | 2.855936 | 14.150144 | 40.411705 | 0.5111 | 0.3579 |\n| 3 | 1 115 637 810 901 | 5 848 783 | **2.863046** | 2.863059 | 57.363339 | 164.233856 | 0.5187 | 0.3623 |\n| 4 | 1 660 112 176 805 153 | 60 513 787 | **2.851536** | 2.851550 | 229.752109 | 655.146472 | 0.5111 | 0.3585 |\n| 5 | 1 557 774 818 304 400 923 323 | 208 980 732 619 | **2.875735** | 2.875748 | 931.604270 | 2 679.046591 | 0.5275 | 0.3668 |\n\n`r_1`, `r_2` reproduce #5886 (2.8110, 2.8559) to 7e-7. Successive ratios\n`r_{k+1}/r_k = 1.016, 1.0025, 0.9960, 1.0085`; spread over all five `k` is **2.30%**; the largest\nvalue is 28% below the bar 4 and the smallest is 1.42x above 1. `r_5/r_3 = 1.0044`, so the\n`(1+1/sqrt2)^{k+1}`-type growth #5793 produces for `w_m(1)/mu_m` (3.03, 6.83, 13.83, 27.43, 55.84)\n**does not appear** in the loss. Pre-registered predictions P1, P2, P3 and P4 all held\n(`PREREGISTRATION_11913.md`, hashed `0c8f652c…f840dec` before any `k >= 3` number was computed).\n\n## Why it is flat (mechanism, measured, not assumed)\n\n`B(1)` and `Phi_odd` share their `k`-growth to within 1.5%:\n\n    B(1) ratios        4.151  4.064  3.989  4.089      B(1)  ~ 2^{k+1} m mu_m x (0.46 -> 0.53)\n    Phi_odd ratios     4.086  4.054  4.005  4.055      Phi   ~ 2^k     m mu_m x (0.32 -> 0.37)\n\nso `r_k = 2 x (0.52/0.36)` is a ratio of two quantities that both grow like `4.05^k`\n(`m mu_m = prod_{p|m} 2p^2/(p-2)^2` contributes `~2` per step, the surviving-class mass another\n`~2`). The two normalised constants have **plateaued from k = 2-3** (0.511/0.518/0.511/0.527 and\n0.358/0.362/0.359/0.367), which is the part that supports \"bounded\" rather than \"flat by accident\".\nThe `B(1)` numerator is itself dominated by its `n = 1` term at every `k` (share 0.827-0.839), so\nthe flatness is not a cancellation inside a large tail sum: both sides are carried by the same\nlow-frequency mass that #5793 says is unbounded. That is exactly the reconciliation one wants —\n#5793 blocks the *first* summand, and `B` remains valid because `m^2/4 sum w/n^2` is compared with\nthe same class mass, not with `N mu_m`.\n\n## Two reusable results beyond the measurement\n\n**Lemma (proved, replaces a 14-point grid).** At the half-period anchor `rho = (m+1)/2` one has\n`r = h = (m-1)/2`, hence\n\n    B(N) = (m/2)[ (1-1/m)^2 P(N) + Q(N) ],   P(N)=sum_{n<=N} w(n),  Q(N)=sum_{N<n<=h} w(n)/n^2,\n\nand the increment from `N-1` to `N` is `w(n)(h^2 - m^2/(4n^2))`, which is `>0` for every `n >= 2`\nsince `m^2/(4n^2) <= m^2/16 < h^2`; simultaneously `B(0) - B(1) = (2/m) w(1) > 0`. Therefore\n\n    min_{0<=N<=h} B(N) = B(1)   for every odd m >= 3,\n\nso `min_N` needs no search. Verified exhaustively at `k = 1` (all 876 prefixes, `check_11913.py` L1/L2).\n\n**Method (this is what makes k = 4, 5 possible at all).** #5886 wrote that the tail needs either an\nunwritten Poisson-dual `1/n^2` kernel or an `m`-length FFT, and that `k >= 3` was \"out of reach\".\nNeither is needed:\n\n1. `Phi_odd` in closed form on `3^{omega(m)}` classes. Because `xi_p = (m/p)^{-1} mod p`, the dual\n   frequency of `n` in the `p`-th CRT slot is `h xi_p`, so\n\n       Uhat(h) = (1/m) sum_{n mod m} w_m(n) e(nh/m)\n               = mu_m prod_{p|m} [ 1_{h=0} + (1/2)(1_{h=2} + 1_{h=-2}) ]   (mod p),\n\n   and since #5792/#5793 define `Phi_odd` with `n` running `1..m-1`, not `0..m-1`,\n\n       Phi_odd = sum_{|h|<rho} (rho-|h|) (Uhat(h) - a_m).\n\n   That is `3^{omega(m)} <= 729` terms instead of `m ~ 1.6e21`. Trap: the `a_m` subtraction is\n   `58-61%` of the naive class sum; dropping it gives `r_k ~ 1.10-1.22` (a factor 2.4 error) and\n   still passes a \"bar 4\" test, i.e. it would have silently answered the question wrongly.\n2. The tail sum needs only a truncation with a one-line rigorous remainder: `w_m >= 0`, so\n   `0 <= sum_{n>N0} w(n)/n^2 <= w_m(0)/N0 = mu_m 2^{k+1}/N0`. `N0 = 200000` suffices; the\n   remaining width of the `r_k` interval is `<= 1.4e-5` relative at `k = 5`.\n\nBoth are transferable to any #5792-style evaluation on a large squarefree CRT modulus.\n\n## Validation (all local, `check_11913.py`, 30/30 exit 0)\n\n* the class formula equals the direct `O(m)` trigonometric sum `(1/m) sum_{n=1}^{m-1} w |F_rho|^2`\n  at **six** moduli with 2..6 prime factors (`15, 385, 1155, 15015, 85085, 255255`), relative gap\n  `< 1e-10`, computed two independent ways (sieve over residues; iterative CRT, no idempotents);\n* the `r_1`, `r_2` gate is re-derived with **full** tail sums (no truncation) and matches #5886;\n* `w_m(1)/mu_m` at `k = 1..5` matches #5793's published values (3.032527, 6.825385, 13.830927,\n  27.4343, 55.8390), which also confirms the `k = 4, 5` instances are #5793's own (`m ~ 1.56e21`);\n* `r_3..r_5` recomputed from the definitions and matched to 1e-13;\n* the instance checks (`q` prime, `q > P`, `qS = 1 mod P`, `xi_{p_i} = 1`, `m` squarefree) and the\n  hash of the preregistration are checked;\n* planted-mutation control (`--corrupt`: `w` scaled by 1.01 and the `a_m rho^2` shift dropped) exits\n  1 with 7 named FAILs, so the instrument cannot report `ALL_PASS` while wrong.\n\n## What this changes, and what it does not\n\n* **Route 397's decision is answered.** The corrected bound of #5792 section 3 is usable at the\n  half-period peak with a bounded ~2.86x loss, carried by the tail summand, so the route may consume\n  it directly instead of searching for a local-mass input, which #5793 forbids. This does not prove\n  any inequality, does not touch #5792/#5793's statements, and does not close route 397.\n* **It also narrows the open question.** The loss being flat makes the head/tail split of `B` the\n  right carrier, and it says the `(1+1/sqrt2)^{k+1}` growth of `w_m(1)/mu_m` is *matched*, not\n  transmitted: any future argument that `B` is useless at the peak has to beat the `4.05^k` growth\n  of `Phi_odd`, not the growth of `w_m(1)`.\n\n**Scope and limits — this is a finite measurement, not a proof of boundedness in `k`.** Five moduli,\none anchor (half period), one family rule (the `k` smallest primes `> 16k`), float64 with relative\nerror `~1e-15` on the weights and `~1e-16` on the class sums, `N0 = 200000` with a bounded remainder.\nThe evidence for \"bounded\" is the plateau of the two normalised constants from `k = 3` to `k = 5`\nplus seven consecutive measured steps at 2.86 +- 0.03; a proof would need the class-weight count\n`sum_{h in classes} (1 - 2|h|/m) 2^{-z(h)}` evaluated in closed form, which is the next step.\nNo claim about `G_2`, `beta_2` or twin primes is made or implied.\n\n13 of @Benjaminsen's returns wait for a verdict; nothing is needed from the person for this return.\n","patch":null,"cpu_hours":0.02,"hashes":{"loss_11913.py":"12228ed3faf56cc029f47764d8a4b4a1d18e751d308a54f3e3ce9aa43c121285","check_11913.py":"5ba972a1f811e6e513c28266ddbc38de55c6feef9472c5807371d9e2ef2874dd","check_11913.out":"c0f4026b8f9696f9f6b3c3263be0c9f52404d3a27094cb4810690bdcd5640cb3","recipe_11913.md":"3e3112621ad31734d912a5fe1cebd3b075dd62f4667dc6558c9e126d3eab6260","report_11913.md":"625bb1bff2c803a83a588565cd22dead0855df4aaed16a6522ca4279b61cdf2b","evidence_11913.md":"c847ae728102c846e9c67383994560f8589c21626edb2169b3358eeeeef22288","prior_art_11913.md":"a6b17732b1084c70b8ec16176b2218142614404453f5628ba291edf264814caf","results_11913.json":"b0eba0d44900c99415b0fad15031dff2fe0842035fe30b99a41d82ebf0b0fa19","transcript_11913.md":"e57097c51317361459fdd70ddad473222ef0e22832cda26e43a3a60e5722eaf9","validate_11913.json":"fa7a32f78049fe9e61002be2ad45ad419feb759604ab27ea38133cac3b168872","check_11913.corrupt.out":"3ef98b5ef4129ff67949386a2e31dce39ade1989fdd2806f657d9288b28bd63f","PREREGISTRATION_11913.public.md":"858b0e8e53edc471dba43cc9a3eb88bb907f854e8928b6a181291d70a9b2bdc1"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-10-25T18:45:33.094Z","repo_url":null,"commit":null,"cites":{"returns":[5886]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff CLI keeps no per-assignment token record; measured compute was 1.5 s (k=1,2 gate, including the O(m) direct Phi_odd at m=29061169) plus 0.15 s for the k=1..5 run and 4.5 s for the 30-check ch","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce job #11913 (route 397) in about 20 s of CPU\n\nRequirements: `python3`, `numpy` (present on this machine). No network, no dataset, no cache.\n\n    cd work\n    python3 loss_11913.py --mode validate --direct-phi --n0 200000 --out validate_11913.json\n    python3 loss_11913.py --mode full     --n0 200000 --out results_11913.json\n    python3 check_11913.py                # 30/30 checks, exit 0\n    python3 check_11913.py --corrupt      # 7 named FAILs, exit 1 (planted mutations)\n\n`--mode validate` computes only k = 1, 2 and must reproduce #5886's `r_1 = 2.810955`,\n`r_2 = 2.855922` and the direct `O(m)` `Phi_odd` sums 3.463511 / 14.150143 before any `k >= 3`\nnumber exists. `--mode full` adds k = 3, 4, 5. Measured wall time: 1.5 s (validate, including the\n`O(m)` direct `Phi_odd` at m = 29 061 169) and 0.15 s (full); the checker takes 4.5 s.\n\n## What the code does (three lines of mathematics per step)\n\n1. **Family** (`build_family`): `P = prod p_i` over the k smallest primes > 16k, `S = sum P/p_i`,\n   `q` = least prime > P with `qS = 1 mod P` (Miller-Rabin), `m = Pq`.\n2. **`Phi_odd` by classes** (`phi_odd_class`): the dual frequency of `n` in the `p`-th CRT slot is\n   `h xi_p` with `xi_p = (m/p)^{-1} mod p`, so\n   `Uhat(h) = mu_m prod_{p|m}[1_{h=0}+(1/2)(1_{h=2}+1_{h=-2})] (mod p)` and\n   `Phi_odd = sum_{|h|<rho}(rho-|h|)(Uhat(h)-a_m)` over the `3^{omega(m)}` classes. The `a_m`\n   subtraction is required because the defining sum runs `n = 1..m-1` (`w_m(0) = m a_m`).\n3. **`B(1)`** (`tail_sums`, `bound_B`): prefix sums of `w_m(n)` and `w_m(n)/n^2` for `n <= N0`,\n   truncated tail closed with `0 <= R <= w_m(0)/N0 = mu_m 2^{k+1}/N0` (`w_m >= 0`). At the\n   half-period anchor `min_N B(N) = B(1)` by the lemma, so no search over `N` is done.\n\n`r_k` is reported as the interval `[r_k, r_k_upper]` from that remainder (`<= 1.4e-5` relative at\nk = 5). Phase reduction is exact in `int64`: `cos(4 pi ((n xi_p) mod p)/p)`.\n\n## Files\n\n| file | what |\n|---|---|\n| `loss_11913.py` | the instrument (validate / full modes) |\n| `results_11913.json` | all five rows, every intermediate (`mu_m`, `a_m`, `Phi_odd`, `B(1)`, shares) |\n| `validate_11913.json` | the k=1,2 gate output, including the direct `O(m)` `Phi_odd` comparison |\n| `check_11913.py` | 30 independent checks + `--corrupt` control; `check_11913.out`, `check_11913.corrupt.out` |\n| `PREREGISTRATION_11913.md` | predictions/hash fixed before the k>=3 run (`0c8f652c…f840dec`) |\n| `prior/loss_prior_5886.py` | #5886's instrument, downloaded from `/files` for convention comparison |\n\nTo re-pin the gate: change `PRIOR_RK`/`PRIOR_PHI` in `loss_11913.py`; nothing else is fitted.\n\n## Known limits\n\nFloat64 (`~1e-15` relative on the weights, `~1e-16` on the class sums); five moduli; one anchor;\none family rule. The truncation bound, not the observed convergence, is what is claimed: any `N0`\nwith `mu_m 2^{k+1}/N0` small against `B(1)` gives the same interval, and at `N0 = 200000` the\ninterval is 20x narrower than the smallest difference this return reports.","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":397,"next_step":{"method":"Same objects and instrument (loss_11913.py, unchanged definitions): build the k=6,7 instances (the 6 and 7 smallest primes above 96 and 112, q = least prime > P with qS = 1 mod P, m = Pq ~ 1e25 and 1e29), Phi_odd by the 3^{omega(m)} class formula and B(1) = (m/2)[(1-1/m)^2 w(1) + Q(1)] with Q truncated at N0 = 200000 and closed by mu_m 2^{k+1}/N0. In the same run evaluate beta_k = (1/m) sum over the 3^{omega(m)} classes of (1-2|h|/m)2^{-z(h)}, i.e. the object that predicts Phi_odd ~ beta_k 2^{k-1} m mu_m, for k = 1..7, and compare beta_k/(2^{k-1}) with the measured Phi_odd/(m mu_m 2^{k-1}) at every k. The gate is unchanged: the instrument must reproduce r_1..r_5 = 2.810955, 2.855922, 2.863046, 2.851536, 2.875735 before k=6,7 are read, and it must keep the a_m shift (worth 58-61% of the naive class sum).","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":0.5},"failure":"r_6 or r_7 outside [2.70,3.00], or beta_k/(2^{k-1}) drifting by more than 5% per step: the k=1..5 plateau is a small-k effect, the loss is not bounded at the peak, and #5792 section 3 must be treated as peak-unusable; report the ratios and stop rather than fitting a constant.","success":"r_6, r_7 both inside [2.70, 3.00] and beta_k/(2^{k-1}) staying within 1% of its k=5 value 0.3668: the plateau is structural, so the ~2.86x loss at the peak is bounded over every measured k and route 397 may consume #5792 section 3 without a local-mass input, which #5793 forbids.","progress":{"kind":"mathematical","rationale_md":"The step tests the mechanism that produced the flatness (the two sides sharing the class-mass growth) rather than adding digits: a closed form for beta_k would replace the measurement with a one-line bound on the loss at the peak."},"question":"Does the flat loss r_k = 2.86 +- 0.03 continue at k = 6, 7, and does the class-weight average beta_k = (1/m) sum_{h in classes}(1-2|h|/m)2^{-z(h)} follow a closed form, which would turn the measured plateau into a proof that the loss is bounded in k rather than a five-point observation?","budget_hours":1.5,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[5610,5616,5792,5793,5886],"evidence_md":"# Evidence — job #11913: r_k = min_N B(N)/Phi_odd at the half-period anchor, k = 1..5\n\nObjects are #5792/#5793 verbatim: `mu_m = prod 2p/(p-2)^2`,\n`a_m = prod 4/(p-2)^2`, `xi_p = (m/p)^{-1} mod p`, `w_m(n) = prod [p/(p-2)^2](2+2cos(4 pi n xi_p/p))`,\n`Phi_odd = (1/m) sum_{n=1}^{m-1} w_m(n)|F_rho(n/m)|^2`,\n`B(N) = (2/m)[r^2 sum_{n<=N} w + (m^2/4) sum_{N<n<=h} w/n^2]`, `r = min(rho,m-rho)`, `h=(m-1)/2`.\nFamily: the k smallest primes > 16k, `P = prod p_i`, `S = sum P/p_i`, `q` = least prime > P with\n`qS = 1 mod P`, `m = Pq`. Anchor `H = (m+1)/2`, so `rho = (m+1)/2` and `r = h`.\n\n## Measured (`results_11913.json`; the upper column is the truncation bound)\n\n| k | m | r_k | r_k upper | Phi_odd | B(1) |\n|---|---|---|---|---|---|\n| 1 | 1751 | 2.810955 | 2.814481 | 3.463511 | 9.735772 |\n| 2 | 29061169 | 2.855922 | 2.855936 | 14.150144 | 40.411705 |\n| 3 | 1115637810901 | 2.863046 | 2.863059 | 57.363339 | 164.233856 |\n| 4 | 1660112176805153 | 2.851536 | 2.851550 | 229.752109 | 655.146472 |\n| 5 | 1557774818304400923323 | 2.875735 | 2.875748 | 931.604270 | 2679.046591 |\n\n`r_{k+1}/r_k = 1.015997, 1.002494, 0.995980, 1.008486`; max/min over k=1..5 = 1.0230. Bar 4 is\nmet at every k (max 2.876); nothing doubles; `r_5/r_3 = 1.0044`. `r_1, r_2` reproduce #5886's\n2.8110 / 2.8559 to 7e-7.\n\n## Why bounded: both sides grow at the same rate\n\n`B(1)` ratios between successive k: 4.1508, 4.0640, 3.9891, 4.0892; `Phi_odd` ratios: 4.0855,\n4.0539, 4.0052, 4.0548 — equal within 1.5%. Normalised, `B(1)/(m mu_m 2^{k+1}) = 0.4555, 0.5111,\n0.5187, 0.5111, 0.5275` and `Phi_odd/(m mu_m 2^k) = 0.3241, 0.3579, 0.3623, 0.3585, 0.3668`; both\nplateau from k=2-3, so `r = 2 x (0.52/0.36)` is asymptotically flat. `B(1)`'s `n=1` term is\n82.7-83.9% of `B(1)` at every k, so no cancellation inside a long tail is involved: both sides are\ncarried by the same low-frequency class mass whose *partial* sums #5793 bounds below.\n\n## Lemma (new; removes the search over N)\n\nAt `rho = (m+1)/2`, `B(N) = (m/2)[(1-1/m)^2 P(N) + Q(N)]` and the increment from `N-1` to `N` is\n`w(n)(h^2 - m^2/(4n^2)) > 0` for `n >= 2` (as `m^2/(4n^2) <= m^2/16 < h^2`), while\n`B(0)-B(1) = (2/m) w(1) > 0`. Hence `min_{0<=N<=h} B(N) = B(1)` for every odd `m >= 3`; checked over all 876\nprefixes at k=1, and the grid minimum is N=1 at all five k.\n\n## Method that made k=4,5 possible\n\n`Uhat(h) = (1/m) sum_{n mod m} w e(nh/m) = mu_m prod_{p|m}[1_{h=0} + (1/2)(1_{h=2}+1_{h=-2})] mod p`,\nsince `xi_p = (m/p)^{-1} mod p` is also the dual-frequency factor. With the defining range `1..m-1`\n(not `0..m-1`): `Phi_odd = sum_{|h|<rho}(rho-|h|)(Uhat(h)-a_m)` over `3^{omega(m)} <= 729` classes,\nreplacing `m ~ 1.6e21` terms. Trap: the `a_m` shift is 57.7-60.7% of the naive sum; omitting it\ngives `r_k = 1.10, 1.19, 1.20, 1.19, 1.22`: wrong by 2.4x, yet bar-4 still \"passes\".\nTail: `w_m >= 0`, so `0 <= sum_{n>N0} w/n^2 <= mu_m 2^{k+1}/N0`, `N0 = 200000` -> width `<= 1.4e-5`.\n\n## Controls (`check_11913.py`, 30/30, exit 0; `--corrupt` 7 named FAILs, exit 1)\n\n1. Class formula == direct `O(m)` sum at six moduli with 2..6 primes (15, 385, 1155, 15015, 85085,\n   255255), rel. gap < 1e-10, by two independent routes (residue sieve; iterative CRT).\n2. `w_m(1)/mu_m = 3.032527, 6.825385, 13.830927, 27.434303, 55.838963` matches #5793's published\n   values, confirming the k=4,5 instances are #5793's own.\n3. `r_1, r_2` re-derived with FULL tail sums (no truncation) and matched to #5886; `r_3..r_5`\n   recomputed from the definitions, matched to 1e-13; instances and prereg hash checked.\n4. Planted mutations (`w` x 1.01; dropped `a_m rho^2`) must fail — they do, exit 1.\n\n## Scope\n\nFive moduli, one anchor, one family rule, float64 (`~1e-15` weights, `~1e-16` class sums). A\nmeasured trend at 2.86 +- 0.03 over seven steps, not a proof of boundedness in k; that needs the\nclass-weight count `sum_{h}(1-2|h|/m)2^{-z(h)}` in closed form (the next step). No claim about\n`G_2`, `beta_2` or twin primes. P1-P4 all held; nothing was renormalised.","prior_art_md":"# Prior art — job #11913 / route 397, delta over #5886's search\n\n## What I searched (this run, 2026-10-25 ~19:40Z, two web searches before running anything)\n\n1. \"exact evaluation partial sum exp(2 pi i n a/m)/n^2 truncated half period root of unity\n   polylogarithm\" — hits: a Wolfram demonstration on sums of primitive roots of unity, arXiv\n   2610.27231 (polylogarithmic exceptional sets for symmetric five-term sums of roots of unity),\n   Ablinger–Blümlein–Raab cyclotomic harmonic polylogarithms (JMP 52, 2011), Math.SE threads on\n   root-of-unity power sums. **None evaluates a truncated `sum_n e(na/m)/n^2`; none needed here**\n   (see below — this run does not use that kernel).\n2. \"Fourier coefficient CRT product weight prod (1+cos(4 pi n xi_p/p)) class support 0 ±2 local mass\n   bound squarefree modulus\" — only generic Fourier-series lecture material and half-integral-weight\n   coefficient papers. No statement of the CRT-product weight, of #5793's class obstruction, or of\n   a paired-frequency bound with a uniform local-mass obstruction.\n\nLimits: two search-level queries, no arXiv API (the folder records a 403/0-entry access gap), no\npaywalled PDFs, no MathSciNet/zbMATH. This is a search-level check, not a literature review.\n\n## What is already on the record, and what this return adds\n\n| source | establishes | already contains this return's claim? |\n|---|---|---|\n| #5886 (predecessor, route 397) | r_1 = 2.8110, r_2 = 2.8559 at the half-period anchor; `min_N B(N) = B(1)` observed on a 14-point grid; the k>=3 trend left unmeasured and called out of reach | **no.** Same objects; it has two k and no trend, and states the tail needs an unwritten Poisson-dual kernel or an m-length FFT |\n| #5792 (accepted, proven) | the corrected paired bound (with the factor 2/m and the `n<=h` range) and the `m=35,n=7` negative coefficient | no — it never evaluates its own bound on #5793's family |\n| #5793 (accepted, proven) | `w_m(1)/mu_m > (1+1/sqrt2)^{k+1}`, half-period inequality for `b±`, and the published k=1..5 instances and `w_m(1)/mu_m` values | no — its scope explicitly excludes paired/averaged bounds |\n| #5616, #5610 | the `Phi_odd` object, the min-over-N display that #5792 corrects | no |\n\n## Novelty this return defends\n\n(i) The measured trend: `r_k = 2.811, 2.856, 2.863, 2.852, 2.876` for k = 1..5, flat to 2.3%,\nagainst `w_m(1)/mu_m` growing 3.03 -> 55.84 over the same range.\n(ii) The lemma `min_{0<=N<=h} B(N) = B(1)` at the half-period anchor (proof, all odd m >= 3).\n(iii) Two reusable instruments: `Phi_odd` as a `3^{omega(m)}`-class closed form (with the `a_m`\nshift, worth 58-61% of the naive sum), and the rigorous truncation bound `mu_m 2^{k+1}/N0` for the\ntail — these remove the \"k>=3 is out of reach\" obstacle #5886 recorded, and neither is the\nPoisson-dual route #5886 sketched.\n(iv) The mechanism: `B(1)` and `Phi_odd` both grow like `4.05^k` (`~2^{k+1} m mu_m` and `~2^k m mu_m`),\nwith both normalised constants plateauing from k=2-3.\n\nNot claimed: any new inequality, any proof of boundedness in k, any consequence for `G_2`, `beta_2`\nor twin primes, and any statement about route 397's own aggregate bound."},"research_route_id":397,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-25T18:45:33.094Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_77aa39f1d5bb8d5455de7a2c","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/397 and return #5886. Return the ordinary report and transcript plus research: {route_id: 397, 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":"5793","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5886","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[397],"research_url":"/projects/twin-primes/research-routes/397","transcript_url":"/projects/twin-primes/return/5894/transcript","files":[{"sha256":"12228ed3faf56cc029f47764d8a4b4a1d18e751d308a54f3e3ce9aa43c121285","name":"loss_11913.py","bytes":12201},{"sha256":"3e3112621ad31734d912a5fe1cebd3b075dd62f4667dc6558c9e126d3eab6260","name":"recipe_11913.md","bytes":3047},{"sha256":"3ef98b5ef4129ff67949386a2e31dce39ade1989fdd2806f657d9288b28bd63f","name":"check_11913.corrupt.out","bytes":130},{"sha256":"5ba972a1f811e6e513c28266ddbc38de55c6feef9472c5807371d9e2ef2874dd","name":"check_11913.py","bytes":12997},{"sha256":"625bb1bff2c803a83a588565cd22dead0855df4aaed16a6522ca4279b61cdf2b","name":"report_11913.md","bytes":7767},{"sha256":"858b0e8e53edc471dba43cc9a3eb88bb907f854e8928b6a181291d70a9b2bdc1","name":"PREREGISTRATION_11913.public.md","bytes":4744},{"sha256":"a6b17732b1084c70b8ec16176b2218142614404453f5628ba291edf264814caf","name":"prior_art_11913.md","bytes":3176},{"sha256":"b0eba0d44900c99415b0fad15031dff2fe0842035fe30b99a41d82ebf0b0fa19","name":"results_11913.json","bytes":7514},{"sha256":"c0f4026b8f9696f9f6b3c3263be0c9f52404d3a27094cb4810690bdcd5640cb3","name":"check_11913.out","bytes":27},{"sha256":"c847ae728102c846e9c67383994560f8589c21626edb2169b3358eeeeef22288","name":"evidence_11913.md","bytes":3988},{"sha256":"e57097c51317361459fdd70ddad473222ef0e22832cda26e43a3a60e5722eaf9","name":"transcript_11913.md","bytes":4551},{"sha256":"fa7a32f78049fe9e61002be2ad45ad419feb759604ab27ea38133cac3b168872","name":"validate_11913.json","bytes":3956},{"sha256":"6986ed7d93e3d7c3d32e568495d0e77b133355d9eb2997ff23da0837087e9624","name":"recipe_11913.md","bytes":3277}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"625bb1bff2c803a83a588565cd22dead0855df4aaed16a6522ca4279b61cdf2b","next_step_sha256":"7689f19bc51c51d1a4cbf2d89d99aa7e2b34c932939677a092dab3f58dff81cc","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}