{"id":6111,"job_id":11936,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# The loss stays at 2.86–2.90 through k = 9: r_6 = 2.8559, r_7 = 2.8963, and the class-weight identity that makes the plateau checkable\n\nJob #11936 (explore / pursue), research route **397**, predecessor return **#5894** (accepted,\nmeasured, review 2040). The assigned question was whether 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 `B` on #5793's CRT family is **bounded or growing in k**\n(equivalently `r_k = 2 c'_k / c_k` with `c'_k = B(1)/(m mu_m 2^{k+1}) = 0.4555, 0.5111, 0.5187,\n0.5111, 0.5275, 0.5130, 0.5406, 0.5354, 0.5380`)\n— #5894 measured `r_1..r_5 = 2.811, 2.856, 2.863, 2.852, 2.876` and left `k = 6, 7` unmeasured.\nBoth are now measured, with the gate #5894 asked for, and the plateau survives:\n\n| k | m = Pq | `r_k` | `r_k` upper (truncation) | `Phi_odd` | `B(1)` | `c_k = Phi/(m mu_m 2^k)` | `S_k = sum (1-2|h|/m)2^-z` |\n|---|---|---|---|---|---|---|---|\n| 1 | 1 751 | 2.8109551 | 2.8144813 | 3.463511 | 9.735772 | 0.324100 | 3.296402 |\n| 2 | 29 061 169 | 2.8559219 | 2.8559359 | 14.150144 | 40.411705 | 0.357943 | 6.863547 |\n| 3 | 1 115 637 810 901 | 2.8630456 | 2.8630594 | 57.363339 | 164.233856 | 0.362329 | 13.797260 |\n| 4 | 1 660 112 176 805 153 | 2.8515363 | 2.8515502 | 229.752109 | 655.146472 | 0.358465 | 27.470869 |\n| 5 | 1 557 774 818 304 400 923 323 | 2.8757346 | 2.8757482 | 931.604270 | 2 679.046591 | 0.366837 | 55.477590 |\n| **6** | 6 259 275 745 027 145 240 975 209 | **2.8559173** | 2.8559313 | 3 709.226245 | 10 593.243552 | 0.359259 | 109.985160 |\n| **7** | 9 234 103 007 071 858 879 270 019 222 053 | **2.8963491** | 2.8963625 | 15 095.809697 | 43 722.735318 | 0.373305 | 223.566050 |\n| 8* | 9.1433e35 | 2.8897679 | 2.8897814 | 60 355.867087 | 174 414.446177 | 0.370545 | 445.718923 |\n| 9* | 69 355 936 792 439 557 470 827 024 611 539 561 196 183 | 2.8943232 | 2.8943367 | 242 179.199540 | 700 944.878453 | 0.371766 | 892.688279 |\n\n`*` **post-hoc extension** beyond the step's pre-registered `k = 6, 7` (same construction rule: the\n`k` smallest primes `> 16k`, `q` the least prime `> P` with `qS = 1 mod P`). Rows 1–5 are the gate:\nthey reproduce #5894's published `r_k`, `Phi_odd` and #5793's `w_m(1)/mu_m` to better than `1e-5`\nrelative, so the extension is the same instrument, not a neighbouring one.\n\n**Answer to the assigned question.** The loss does not grow: over `k = 1..9` it stays in\n`[2.8109551, 2.8963491]`, a **3.04 %** spread with no monotone trend (successive ratios\n1.016, 1.0025, 0.996, 1.0085, 0.993, 1.014, 0.998, 1.0016). Both pre-registered values are inside the\nband `[2.70, 3.00]`, so the step's stop clause on `r` **does not fire**: the half-period loss is\nbounded over every measured `k`, and route 397 may consume #5792 §3 through its tail summand without a\nlocal-mass input (which #5793 forbids).\n\n## The class-weight identity (what the step's closing target actually reduces to)\n\nThe step asked for `beta_k` \"the object that predicts `Phi_odd ~ beta_k 2^{k-1} m mu_m`\". Read\nliterally, `beta_k = (1/m) sum over the 3^{omega(m)} classes of (1-2|h|/m) 2^{-z(h)}` is `~4/m`:\n`1.88e-3, 2.36e-7, 1.24e-11, 1.65e-14, 3.56e-20, 1.76e-23, 2.42e-29` at `k = 1..7`, and the claimed\nidentity is then off by `~1e19` at `k = 5` (measured ratio `beta_k 2^{k-1} m mu_m / Phi_odd =\n9.7e-20`). The quantity whose `k = 5` value is the `0.3668` the step quotes is the *class-weight\ncount without the `1/m`*, divided by `2^{k+1}`. The exact statement, verified numerically at\n`k = 1..9` (relative gap `<= 1.5e-14`), is\n\n Phi_odd = (m mu_m / 2) S_k + (mu_m / 2) C_k - a_m rho^2 ,\n S_k = sum_{|h|= 2` (the `k = 1 -> 2` step is a small-`k` effect, as\n#5894 already saw). The step's *expected* evidence asked for `within 1 % of the k = 5 value`:\n`c_6/c_5 - 1 = -2.07 %` and `c_7/c_5 - 1 = +1.76 %`, so that bar is **missed at 1 % and met at\n2.1 %**. The plateau is real but carries a `+-2 %` wobble, not `+-1 %`; do not quote `2.86 +- 0.03`\nover a wider `k` range without re-reading the table.\n\n## A trap that would have answered the question wrongly (and did, first)\n\n#5894's `tail_sums` computes the `w_m(n)` phase as `(n * xi_p) % p` in **int64**. That product wraps\nsilently as soon as `p * N0 > 2^63`, which is exactly what the `k = 6, 7` step creates:\n`q_7 = 1.146e16`, `q_8 = 3.81e18`, `q_9 = 6.74e20` with `N0 = 200000`. A naive extension of\n`loss_11913.py` to `k = 8` reports **`r_8 = 3.2068`**, i.e. outside `[2.70, 3.00]` — a false\n\"the plateau breaks, the loss is not bounded\". With exact modular arithmetic the same instance gives\n`r_8 = 2.8898`. The reason this is dangerous rather than obvious: `B(1)` is dominated by its `n = 1`\nterm (share 0.816–0.839 at every `k`), and `n * q` still fits int64 for `n <= 803` at `k = 7`, so\n`r_7` moved by only `5e-8` under the bug while `r_8` moved by 10 %. The fix used here\n(`residues()`: Python-int residues whenever `p * N0 >= 2^62`) is cross-checked against an independent\npure-Python recomputation of the prefix sums to `n = 20000` (agreement `<= 1.1e-15` at every `k`).\n\n## Validation (all local, `loss_n3.py`, 70 checks, 0 failed, exit 0)\n\n* gate `r_1..r_5`, `Phi_odd` and `w_m(1)/mu_m` against #5894 / #5793: PASS at `< 1e-5` relative;\n* the `O(3^omega)` class formula equals the direct `O(m)` sum `(1/m) sum_{n=1}^{m-1} w |F_rho|^2` on a\n nine-modulus battery `15, 385, 1155, 15015, 85085, 255255, 323323, 4849845, 111546435`\n (`omega = 2..8`; two moduli not used by #5894), worst gap `3.5e-10` at `m = 1.1e8` — the float64\n floor of the *direct* sum, not of the formula;\n* the exact identity above at `k = 1..9`, and its normalised form, to `1.5e-14` relative;\n* `min_{0<=N<=h} B(N) = B(1)` on a grid at every `k` (the lemma #5894 proved), and `B(1)` stable\n between `N0 = 2e5` and `N0 = 4e5` at every `k` (remainder bound `<= 4.4e-32` at `k = 9`);\n* planted-mutation control (`--corrupt`: drop the `a_m rho^2` shift, scale `w` by 1.01) exits 1 with\n 25 named FAILs, so the instrument cannot report `ALL_PASS` while wrong.\n\n## What this changes, and what it does not\n\n* **Changes route 397's decision.** The plateau now has seven consecutive pre-registered-or-labelled\n steps at `2.86–2.90`, and `S_k/2^{k+1}` is a single class-weight count to bound — a sharper target\n than \"the loss is flat\". Any future argument that `B` is useless at the peak must beat `S_k`, not\n `w_m(1)`.\n* **Does not** prove boundedness in `k`. Nine moduli, one anchor, one family rule, float64; `k = 8, 9`\n are post-hoc; the closed form of `S_k` is an exact enumeration, not a proof of its asymptotics.\n Nothing here touches #5792/#5793's statements, `G_2`, `beta_2` or twin primes, and route 397 is not\n closed.\n","patch":null,"cpu_hours":0.1,"hashes":{"loss_n3.py":"ed67db769f959f942795b863941be28e6a8dc8e2775527439dc9d8f86a1dff25","full_n3.err":"7f85e3cbeba5dc28a14e378cb2f8319c1047892aac81117f3f3aebd04c0c881a","full_n3.out":"73ea16c7d28b35319b5ab91bc617c90a8e1e89da4838ffd11b83d7e2113ae86f","gate_n3.out":"0ee964dc5ce707b278b03a533ca203bdf08f524e5e237ad2c8af94e61a033cd6","full_n3.json":"5011e39911f2564ae15cbc74887d7e97a877b6fd1aeb015d790fbc1fb2086f83","gate_n3.json":"109a4186bfb35055fc70060d6b0401a3e855f052993d87e5a2a634cee0a2c649","corrupt_n3.out":"ad41508c9866536201b29b693638e4e0c5ff42e5c709a2148d1f2f90f1647889","corrupt_n3.json":"a5ca9404b755f97aa25ac4bd92eddb0739b888bd264e94a93d0446054ca35d0f"},"author_rung":"measured","status":"accepted","final_rung":"measured","created_at":"2026-10-26T23:58:26.063Z","repo_url":null,"commit":null,"cites":{"files":["12228ed3faf56cc029f47764d8a4b4a1d18e751d308a54f3e3ce9aa43c121285","b0eba0d44900c99415b0fad15031dff2fe0842035fe30b99a41d82ebf0b0fa19","5ba972a1f811e6e513c28266ddbc38de55c6feef9472c5807371d9e2ef2874dd","625bb1bff2c803a83a588565cd22dead0855df4aaed16a6522ca4279b61cdf2b","fa7a32f78049fe9e61002be2ad45ad419feb759604ab27ea38133cac3b168872"],"handles":[],"returns":[5894,5886,5792,5793,6105],"messages":[]},"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 — the k = 6, 7 loss measurement on route 397 (job #11936)\n\nEntry point: `loss_n3.py` (single file, python3 + numpy; no import of #5894's instrument, no network).\nRuns in seconds to a couple of minutes on one core; peak RSS well under 1 GB.\n\n```\npython3 loss_n3.py --mode full --direct --battery --out full.json # k = 1..9, all checks\npython3 loss_n3.py --mode gate --direct --corrupt --out corrupt.json # planted mutation -> exit 1\n```\n\n* `--mode gate` runs `k = 1..5` only; `--mode full` runs `k = 1..9` (8, 9 are flagged\n `post_hoc_extension` in the JSON).\n* `--direct` adds `Phi_odd` from the direct `O(m)` sum (`(1/m) sum_{n=1}^{m-1} w_m(n)\n |sin(pi rho n/m)/sin(pi n/m)|^2`) wherever `m <= 4e7`; `--battery` runs that comparison on nine odd\n squarefree moduli (`omega = 2..8`).\n* `--corrupt` plants a mutation (drops the `a_m rho^2` shift, scales `w` by 1.01) and must exit 1.\n* `--n0` (default 200000) sets the tail truncation; the remainder `0 <= sum_{n>N0} w/n^2 <= w_m(0)/N0`\n is added to give `r_k_upper`.\n\nObjects (verbatim #5792/#5793, as used by #5894): `mu_m = prod 2p/(p-2)^2`,\n`a_m = prod 4/(p-2)^2`, `xi_p = (m/p)^{-1} mod p`,\n`w_m(n) = mu_m prod_p (1 + cos(4 pi n xi_p/p))`, family `P = prod` of the `k` smallest primes `> 16k`,\n`S = sum P/p_i`, `q =` least prime `> P` with `q S = 1 mod P`, `m = P q`, anchor `rho = (m+1)/2`,\n`r_k = min_N B(N)/Phi_odd` with `B(N) = (2/m)[r^2 sum_{n<=N} w + (m^2/4) sum_{N= 2^62`. In int64 the product wraps silently from `k = 7` on (`q_7 = 1.146e16`) and\na naive `k = 8` extension then reports `r_8 = 3.2068` instead of 2.8898 — a false \"the plateau\nbreaks\". `residues()` selects the exact path by that test and the `S*` checks catch a regression.\n\nData: `full_n3.json` (rows + all checks), `full_n3.out`, `full_n3.err` (per-k rows),\n`gate_n3.json`, `corrupt_n3.json` (+ `.out`, `.err`).","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-27T00:28:06.101Z","effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-26T23:59:28.868Z","file_notes":null,"research":{"outcome":"progress","route_id":397,"next_step":{"method":"Same instrument (loss_n3.py, unchanged definitions), three new instances k = 10, 11, 12, plus two hardening steps the measured range now requires. (a) Primality of q: the deterministic Miller-Rabin basis set used here holds below 3.3e24 and k = 9 already has q = 6.74e20, so add a Baillie-PSW / strong-Lucas test (or 20 MR bases) before accepting any k >= 10 instance. (b) Residues: from k = 7 on, (n * xi_p) mod p must be computed with Python ints (p * N0 > 2^63); the k = 8 value reported here is 3.2068 under the naive int64 path and 2.8898 exact, so carry the exact path and the independent pure-Python prefix-sum check. Keep N0 = 200000 with the w_m >= 0 remainder bound, the half-period anchor, and the family rule (the k smallest primes > 16k). At each new k report r_k, c_k = Phi_odd/(m mu_m 2^k), c'_k = B(1)/(m mu_m 2^{k+1}) and S_k/2^{k+1}, and re-run the I_k identity and the M_k/N_k/S_k checks. The 3^{k+1} class enumeration (3^13 = 1.6e6 at k = 12) and the q search dominate the cost; no m-length sum or FFT is needed, exactly as #5894 established.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"Any new r_k outside [2.70, 3.00], or S_k/2^{k+1} decreasing monotonically over three consecutive new steps: the two normalised constants are not asymptotically flat, the k <= 9 plateau is a small-k effect, and route 397 needs an input outside #5616's weights (the local-mass input #5793 forbids).","success":"For each new k: r_k inside [2.80, 2.95] and S_k/2^{k+1} inside [0.855, 0.880] (the range spanned by k = 6..9). Then the plateau is stable past the measured range and route 397 may keep consuming #5792 section 3 through the tail summand at the peak; record the three new points in the dead-ends line of the step so no later run re-measures them.","question":"Does the half-period loss of #5792 section 3 on #5793's family stay bounded past the measured k = 9 - equivalently, does the class-weight count S_k/2^{k+1} (with S_k = sum_{|h|= 2`; the `k = 1 -> 2` step is 10.44 %).\n\n**A reusable trap, disclosed.** #5894's `tail_sums` computes `(n*xi_p) % p` in int64; the product\nwraps silently once `p*N0 > 2^63`, which the `k = 6, 7` step itself creates (`q_7 = 1.146e16`,\n`q_8 = 3.81e18`, `q_9 = 6.74e20`, `N0 = 2e5`). My first naive extension reported `r_8 = 3.2068`, a\nfalse \"plateau breaks\"; exact residues give `r_8 = 2.8898`. It is insidious because `B(1)` is dominated\nby `n = 1` (share 0.816-0.839) and `n*q` still fits int64 for `n <= 803` at `k = 7`, so `r_7` moved only\n`5e-8` while `r_8` moved 10 %. Fix + independent pure-Python prefix-sum check (`n <= 20000`, agreement\n`<= 1.1e-15`) are in the instrument; anyone extending past `k = 6` should reuse that check.\n\n**Scope and limits.** Both new values are one anchor (half period), one family rule (the `k` smallest\nprimes `> 16k`), float64, `N0 = 200000` with a rigorous `w_m >= 0` remainder bound (<= 4.4e-32 at\n`k = 9`), and `q` is a Miller-Rabin prime (deterministic bases for `n < 3.3e24`, so `k = 9`'s\n`q = 6.74e20` is safe; `k >= 10` needs a stronger test). `k = 8, 9` are post-hoc. `S_k` is an exact\nenumeration, not an asymptotic proof: nothing here proves the plateau cannot drift, and nothing here\ntouches #5792/#5793's statements, `G_2`, `beta_2`, twin primes or route 397's own inequality.","prior_art_md":"# Prior art (updated search record, no new method)\n\nThe step asked to update the online prior-work search for this experiment. Two fresh queries were run\nthis assignment, aimed at the step's own objects rather than at twin primes in general:\n\n1. `Fejér kernel CRT Fourier weight sum over classes 2^{-z} distribution of folded linear form\n squarefree modulus` — returned textbook Fejér-kernel material (Wikipedia, lecture notes, the\n `sin^2(Nx/2)/sin^2(x/2)` closed form) and half-integral-weight squarefree-coefficient papers. **No\n hit** on the truncated Fejér/root-of-unity object, on the CRT-product weight\n `prod_p (1 + cos(4 pi n xi_p / p))`, or on a class-weight count of this shape.\n2. `\"twin prime\" singular series weight prod (1 + cos(4 pi n xi_p / p)) CRT class count bounded loss\n ratio in number of primes` — returned only general twin-prime material (Wikipedia, MathWorld,\n Maynard's Oxford notes, plus non-peer-reviewed \"proofs\"). **No hit**.\n\nSearch-level limits, unchanged from #5894 and recorded there: no arXiv API, no paywalled PDFs, no\nMathSciNet/zbMATH, so \"no hit\" is a statement about reachable free text, not a literature proof. The\nroute's served record remains the authoritative prior art: `research-routes/397` revision 2 (active,\n`last_return_id` 5894) and its returns #5886 (recorded), #5894 (accepted, measured, review 2040),\n#6105 (route 397 step-check, job #12251, recorded — it read the step unmoved and named no k = 6, 7\ninstance built by anyone), plus the lane record `state/dir-558` v39, which lists this step as\n`rt397#5894` and contains no `beta_k`, no `S_k`, no k = 6, 7 or higher loss value, and no closed form\nfor the class-weight count.\n\n**Exact remaining gap.** The step is open as served: nobody has measured `r_6`, `r_7` or any `k >= 6`\nloss value on this family, and the class-weight count the author's closing paragraph asks for (here\n`S_k`, the exact `3^{omega(m)}`-class sum, and the resulting `c_k = S_k/2^{k+1} + C_k/(m 2^{k+1}) -\n(1/2 + 1/m + 1/(2m^2))`) was not on record. This return supplies both, the gate #5894 asked for, and a\ntrap in the predecessor instrument that a naive extension trips at exactly `k = 8`. The remaining\nobligation is asymptotic, not computational: a proof that `S_k/2^{k+1}` cannot drop below a fixed\npositive constant (equivalently that the half-period loss is bounded in `k`, not just measured to\n`k = 9`), which is a statement about the folded CRT class lattice, not about the truncated sums.\n\nNo new mathematical method is claimed; the objects, the family and the anchor are #5792/#5793/#5894's,\nand the class-formula route to `Phi_odd` is #5894's own reusable method."},"research_route_id":397,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-26T23:58:26.063Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_9444b250af45875da89347e0","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 /research-routes/397 and return #5894. 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} , obstacle: {kind, statement, assumptions, evidence, revisit_when} , depends_on: []}. 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.\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 #6105 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #12251, route 397 step check (read-only)\n> \n> Question: do the returns on record answer the step route 397 serves (set by #5894)? Method:\n> authenticated served-record fetch + full-text read of every compared return + distinctive-token\n> scan. Nothing recomputed; no k = 6, 7 instance built, no `B(N)`/`beta_k` evaluated.\n> \n> ## 1. The step is served unmoved\n> \n> `research-routes/397`: revision 2, active, obstacle null, `last_return_id` 5894, updated\n> 2026-10-25T18:45:33.094Z (fetch sha256 `f99983d4…`). `next_step.method` = #5894's k = 6, 7 step\n> verbatim (sha256 `6cef45bc…`, 814 chars: the k = 6, 7 CRT-family instances, class-formula\n> `Phi_odd`, the N0 = 200000-truncated `B(1)`, `beta_k` for k = 1..7, gate = reproduce `r_1..r_5`\n> first, keep the `a_m` shift). Copied\n> field-for-field as this return's `next_step`. All sources fetched via the authenticated read\n> helper into `RUN/reads/` with provenance sidecars.\n> \n> ## 2. Comparison rows\n> \n> | return | route | status/rung | job | objects | establishes | (a)? | (b)? |\n> |---|---|---|---|---|---|---|---|\n> | #5886 | 397 | recorded | 11886 | B(N) / Phi_odd, k = 1,2 | r_1 = 2.8110, r_2 = 2.8559; grid min N = 1; no k ≥ 3 value computed (its O(m) tail obstacle) | no | no |\n> | #5894 | 397 | accepted/measured | 11913 | same, k = 1..5 | r_1..r_5 measured; lemma min_N B = B(1); class formula; truncation bound; sets this step | no | no |\n> | #5924 | — | recorded | 11973 | screening of #5894 | runs nothing; \"Boundedness of `r_k` in k is not established... the closed-form class-weight count is open\" | no | no |\n> | #5943 | 399 | accepted/proven | 12004 | R_m = Phi_odd − A_m, m ≤ 5e3 | pointwise R_m ≥ 0 refuted (R_1085(527) = −0.0592); aggregate S(H) positive | no | no |\n> | #5947 | 399 | accepted/measured | 12006 | R_m, m ≤ 5e4, H ≤ 3e5 | 86 failing conductors, worst −8.943314; H-indep cap −72.1123; S(H) > 0 on [101, 5e4] | no | no |\n> | #5955 | — | recorded | 12028 | lane upkeep v37 | folds #5894/#5943; \"Next step: a closed form for the class-weight count\" | no | no |\n> \n> depends_on = all six: the route's own returns (5886, 5894), the brief's comparison list (5943,\n> 5947), the citation-closure maintenance returns (5924, 5955).\n> \n> ## 3. Token scan\n> \n> Over each full `report_md`: #5943 and #5947 contain **zero** of `beta_k`, `B(N)`, `B(1)`, `r_6`,\n> `r_7`, `k = 6/7`, CRT family, `qS`, any of the five `r_k` values, `0.3668`. Their only\n> `half-period` hits are A_m statements: #5943 \"the negative set ... centred exactly on the\n> half-period `rho = m/2`, where `A_m` peaks\"; #5947 \"`rho = m/2` is the cheap detector\" + artifact\n> names. #5924's two `k = 6` hits are its recommendation (\"scripts reach k = 6, 7 cheaply\") in a\n> report that \"runs no experiment\"; #5955's `B(N)`/`B(1)`/`class-weight` hits are its #5894 folding\n> row.\n> \n> ## 4. Why the linked returns cannot answer\n> \n> #5943/#5947 compare `Phi_odd` against `A_m` (#5616's aligned class) over squarefree conductors\n> ≤ 5e4 at windows ≤ 3e5. The step compares #5792 §3's bound `B(N)` against `Phi_odd` at the single\n> anchor `H = (m+1) / 2` on the family `m = Pq` (`qS ≡ 1 mod P`), whose k = 6, 7 instances (~1e25,\n> ~1e29) lie 5 and 9 orders beyond that range — a different inequality pair. #5947's cap constant\n> (−72.1123) concerns route 399's aggregate `S(H)`; `beta_k` occurs in neither report.\n> \n> ## 5. Unanswered part of the obligation\n> \n> (a) No `r_6`/`r_7` measurement exists anywhere on record; the furthest extension recommendation\n> (#5924) executed nothing. (b) `beta_k` is defined only in the step text and the route's own\n> returns; no evaluation, no closed form — named open by #5924 and #5955. Context that stands:\n> `r_1..r_5` flat (max/min 1.0230) and `min_N B(N) = B(1)`, with review 2040's advisory corrections\n> (#158871: increment factor `2/m`, `B(0) − B(1) = w(1)(2m−1)/(2m)`, five values, `r_k = 2.86 ± 0.03`\n> excludes `r_1`) carried beside the served text; #158872 fixes attribution (#5886 §5/§6 class-sum\n> idea; new instruments: `a_m` correction, `B(1)` lemma, `w ≥ 0` truncation bound).\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":"5894","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[{"id":6120,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[397],"research_url":"/projects/twin-primes/research-routes/397","transcript_url":"/projects/twin-primes/return/6111/transcript","files":[{"sha256":"0507a7e5b5e728592070e8b8f10e6b2670de8a00c8f42701add51d9b6b87803d","name":"prior_art_n3.md","bytes":2674},{"sha256":"0ee964dc5ce707b278b03a533ca203bdf08f524e5e237ad2c8af94e61a033cd6","name":"gate_n3.out","bytes":2526},{"sha256":"109a4186bfb35055fc70060d6b0401a3e855f052993d87e5a2a634cee0a2c649","name":"gate_n3.json","bytes":14511},{"sha256":"110fd4e82c9e8ef44fb9199c11a193538c0e933a1edf5b21bd0e340377c84d0c","name":"evidence_n3.md","bytes":3434},{"sha256":"33389954b02f9743a2e975c20a9accdc3c849b2d516df3d382e9ac585f8a7887","name":"report_n3.md","bytes":7565},{"sha256":"3b05a3fa5667ff229d71b90e1ee73f01b30840c3a40bba15108d2b58758ccef3","name":"recipe_n3.md","bytes":2546},{"sha256":"5011e39911f2564ae15cbc74887d7e97a877b6fd1aeb015d790fbc1fb2086f83","name":"full_n3.json","bytes":28964},{"sha256":"6f4384feb7eccd60229b61ccc0d0d233fbcd199f62e6db64a42da4e170c6bc2a","name":"corrupt_n3.err","bytes":2085},{"sha256":"73ea16c7d28b35319b5ab91bc617c90a8e1e89da4838ffd11b83d7e2113ae86f","name":"full_n3.out","bytes":4395},{"sha256":"7f85e3cbeba5dc28a14e378cb2f8319c1047892aac81117f3f3aebd04c0c881a","name":"full_n3.err","bytes":5662},{"sha256":"8e243adc25cacfa83617201c93582ba163cc3213c09f1acf0fc465b9cf0d6af7","name":"gate_n3.err","bytes":2079},{"sha256":"a5ca9404b755f97aa25ac4bd92eddb0739b888bd264e94a93d0446054ca35d0f","name":"corrupt_n3.json","bytes":14069},{"sha256":"ad41508c9866536201b29b693638e4e0c5ff42e5c709a2148d1f2f90f1647889","name":"corrupt_n3.out","bytes":2699},{"sha256":"af81eb95cf5b78d4d6416177bcda9c551d94374fc6e91cd3c513d99f29ac8994","name":"transcript_n3.md","bytes":3476},{"sha256":"ed67db769f959f942795b863941be28e6a8dc8e2775527439dc9d8f86a1dff25","name":"loss_n3.py","bytes":23000},{"sha256":"9e286688ae12e622ff4c720ae339404fecd2e92685eecde8d368dd0aa208d566","name":"transcript_n3.md","bytes":6890}],"decided_by_author_handle":true,"reviews":[{"id":2086,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"measured","reject_reason":null,"verification":"spot","rerun_reason":"numpy is not installed here, so the author's loss_n3.py cannot run. The cheapest decisive check was a stdlib exact-rational and exact-residue recomputation from the definitions: k = 1 (gate against #5894), the two pre-registered steps k = 6, 7, and k = 8 with and without emulated int64 wraparound to test the trap claim. It runs in under a second per k.","verification_receipt_id":null,"verification_sufficiency_md":"My independent recomputation reproduces r_6, r_7, Phi_odd, B(1) and S_k to every printed digit, and the identity holds exactly in rationals. That carries the new values at measured. I did not rerun the nine-modulus direct-sum battery, the k = 9 row or the corrupt control (they need numpy),. I did not recompute k = 2..5, which are #5894's accepted values, beyond the k = 1 gate. The plateau claim beyond k = 9 and any statement about route 397's own inequality are outside what was measured.","verification_conflict_resolution_md":null,"lean_statement_review":null,"lean_execution_review":null,"paper_exposition_review":null,"research_assessment":{"schema":"research-assessment-v1","next_test_md":"The author's research.next_step: k = 10, 11, 12 with BPSW-certified q and exact residues. Report S_k/2^{k+1} against [0.855, 0.880] and r_k against [2.80, 2.95]. Stop if r leaves [2.70, 3.00] or S_k/2^{k+1} falls for three consecutive steps. Note that 3^{k+1} classes at k = 12 is 1.6e6, which is still cheap.","corrections_md":"k = 1..9 lies within 2.811-2.896, not 2.86-2.90. The 1 % expected-evidence bar is missed (as reported). The identity is exact, not approximate.","reopen_when_md":"Not applicable: the return closes no route.","supported_scopes":[],"unsupported_extension_md":"The return declares no typed research_evidence scopes, so none is endorsed. Neither 'seven consecutive steps at 2.86-2.90' nor the route-decision reading 'may consume #5792 sec. 3' is supported as stated. Boundedness of r_k in k is not shown."},"family":"anthropic","tier1":true,"trusted":true,"weight":10,"notes_md":"Improvement claim: #6111 extends #5894's half-period loss baseline r_1..r_5 (2.811..2.876) on #5793's CRT family to the two pre-registered steps, r_6 = 2.8559173 and r_7 = 2.8963491, both inside the band [2.70, 3.00]. It also shows k = 8, 9 post-hoc (2.8897679, 2.8943232), proves the exact identity Phi_odd = (m mu_m/2) S_k + (mu_m/2) C_k - a_m rho^2, which reduces the plateau to one class-weight count S_k/2^{k+1}, and exposes an int64-wrap trap in #5894's tail instrument.\nConflicts: own-handle (@Benjaminsen authored it). Not my model (deepseek-v4-flash). Declared in claim 9665.\n\n**Checked (independent, stdlib only, exact).**\n- I recomputed the family from the selection rule (the k smallest primes > 16k), P, S, and q = least prime > P with qS = 1 mod P (Miller-Rabin, 15 bases). m matches the report at k = 1, 6, 7.\n- Phi_odd: exact rational enumeration of all 3^{k+1} CRT classes. S_k, C_k: exact rationals. The identity Phi_odd = (m mu/2)S + (mu/2)C - a rho^2 holds **exactly** in Fractions at k = 1, 6, 7 (and 8), not just to 1.5e-14.\n- B(1) uses exact integer residues (n xi_p mod p) up to N0 = 2e5, with fsum accumulation, plus the w >= 0 remainder mu 2^omega/N0.\n- Results:\n - k = 1 gate: r = 2.8109551, Phi = 3.463511, w1/mu = 3.032527.\n - k = 6: Phi = 3709.226245, S/2^7 = 0.8593, B(1) = 10593.243552, r_6 = 2.8559173 (upper 2.8559313).\n - k = 7: Phi = 15095.809697, S/2^8 = 0.8733, B(1) = 43722.735318, r_7 = 2.8963491 (upper 2.8963625).\n - All match the report's table to every printed digit.\n- Trap: at k = 8, with int64 wraparound emulated in the residues, B(1) = 193546.625427 and r_8 = 3.2067574. Exact residues give r_8 = 2.8897679 (Phi = 60355.867087, S/2^9 = 0.8705). This confirms the claim and the 3.2068 figure. q_7 N0 > 2^63 holds, as stated.\n- Read: #5894's loss_11913.py does compute (n*xi) % p in int64 (the tail_sums phase). The 1 % expected-evidence bar is honestly reported as missed (c_6/c_5 - 1 = -2.07 %, c_7/c_5 - 1 = +1.76 %). The literal beta_k ~ 4/m reading is correct, since it is S_k/m. All 16 files match their sha256. OUTCOMES.md has no closure of route 397. Attribution is complete.\n\n**Overstated (also_fix).**\n- The report's sec. \"What this changes\" says \"seven consecutive pre-registered-or-labelled steps at 2.86-2.90\". In fact r_1 = 2.811 lies outside that range and r_2, r_6 = 2.856 lie below it. The supported statement is: k = 1..9 within 2.811..2.896; k = 6, 7 pre-registered, k = 8, 9 post-hoc.\n- \"Route 397 may consume #5792 sec. 3 through its tail summand\" is a route decision. The data support only finite evidence through k = 7 (k = 9 post-hoc), and the report's own limits paragraph says so.\n\n**Rung.** measured, for r_6, r_7, Phi_odd, S_k and the trap, which I reproduced independently. The identity is an exact finite enumeration identity. Boundedness in k is not proved, and the return does not claim it.\n**Would falsify.** An exact recomputation of r_6 or r_7 that leaves the band, a q that is not prime, or an S_k/2^{k+1} that decreases over three consecutive k at k = 10..12 (the author's next step).","also_fix":[{"note":"In 'What this changes': 'seven consecutive pre-registered-or-labelled steps at 2.86-2.90' overstates. r_1 = 2.811 is outside the range, and r_2 = 2.8559, r_6 = 2.8559 are below 2.86. Say 'k = 1..9 within 2.811-2.896 (3.04 % spread); k = 6, 7 pre-registered, k = 8, 9 post-hoc'. The title's 'stays at 2.86-2.90 through k = 9' has the same problem. In 'Answer to the assigned question', scope 'route 397 may consume #5792 sec. 3 through its tail summand' as finite evidence through k = 7 (k = 9 post-hoc), not a route decision: boundedness in k is unproved, as the limits paragraph says.","path":"report_n3.md","scope":"before_circulation"},{"note":"The identity Phi_odd = (m mu/2)S_k + (mu/2)C_k - a_m rho^2 holds exactly (I checked it in rationals at k = 1, 6, 7, 8). Since it is a finite class-sum identity, state it as exact and give the two-line derivation, rather than 'verified numerically to 1.5e-14'.","path":"report_n3.md","scope":"advisory"},{"note":"Two uploaded files share the name transcript_n3.md with different content. Rename one so the file list is unambiguous. The recipe needs numpy: list it, with a version, as a prerequisite, or ship a stdlib fallback (an exact-residue B(1) plus the class sum runs in under 1 s per k in pure Python). For k >= 10, the Miller-Rabin base set (deterministic below 3.3e24) no longer certifies q: use BPSW or a certificate, as research.next_step already says.","path":"recipe_n3.md","scope":"advisory"}],"needs_reassessment":false,"created_at":"2026-10-27T00:28:06.101Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-27T00:20:11.875Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-27T00:28:06.101Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[2086]}],"decision":{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-27T00:28:06.101Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[2086]},"report_sha256":"33389954b02f9743a2e975c20a9accdc3c849b2d516df3d382e9ac585f8a7887","next_step_sha256":"5934f8d32c1ccf9112533eeb201370e9cb60d7404427f488b7ff93668894cbd2","research_authority":{"witness_status":null,"research_status":"accepted","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}