{"id":5692,"job_id":11472,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Exact mod-6 windowed cumulants at 23# and 29#: the sign of s_3(L) is a zero-mean fluctuation in L, and the 3-point singular-series kernel has the wrong sign in its low band\n\nJob **#11472**, route **381** (`measure`, first look), populated lane **measure**. Instruments and\nartifacts in this return's `files`; every number below is exact rational arithmetic (no float64\naccumulation), and every rung number not measured here is quoted with its source.\n\n## 0. The obligation and what this return settles\n\nThe obligation: *does the 3-point singular-series Fourier kernel on `Z/Q_6 Z` predict the\nwindow-fraction sign oscillations of `m_3(L)` across `L/q ∈ {1/8, 1/4, 3/8}` from 23# to 31#, and\nextend to 37#?*\n\nMeasured answer, in two independent pieces:\n\n1. **There is no window-fraction sign law to predict.** At 23#, the exact `m_3(L)` over 127\n   equally spaced lengths is a **zero-mean, essentially white-in-`L`** fluctuation: mean\n   `+4.5e-5`, sd `27.65`, lag-1 autocorrelation `0.118`, adjacent sign agreement `0.508`\n   (60 sign changes in 126 steps). At 29#, all three primary windows are **positive**\n   (`s_3 = +0.0680, +0.0671, +0.0157`), so the route's premise that the signs \"oscillate\" there is\n   the sampling of that fluctuation, not a phase law. Rung to rung the same holds: at fixed `1/8`\n   the sign over 5#..29# runs `+ - + + - + - +`.\n2. **The kernel cannot supply the signs either.** The triple-resonance Fourier sum is an *identity*\n   for `Σ_u S_K(u)^3`, so only a band-truncated (\"singular-series\") version could predict anything.\n   At 23#, over the whole ladder `H <= 256 = Q_6/145,000` that truncated sum reproduces the sign at\n   **one of the three** primary windows (3/8; opposite sign at 1/8 and 1/4), and it reaches all three\n   signs only from `H ≈ 4096`; the magnitudes are within 10-16 % only at `H = 65536 = Q_6/567`\n   (ratios `1.09, 1.07, 1.16`).Both are measured; neither is asymptotic. One 31# window **was** measured and verifies the\npredecessor's cell to seven digits (§5); the 37# sweep was **not** run (§6-§7 say what that leaves\nopen and why it is not the deciding experiment for the kernel question).\n\n## 1. Object, instruments, controls\n\nDefinitions reproduced verbatim from the lane: `q = x#`, `A_q = {a : gcd(a(a+2),q)=1}`,\n`M = |A_q| = Π_{5<=p<=x}(p-2)`, `L = floor(q·n/64)`, `N_t` cyclic with period `q`,\n`m_k = (1/q)Σ_t (N_t - M L/q)^k`, `s_3 = m_3/m_2^{3/2}`.\n\n- `wcum82.py` (in `files`): exact mod-6 engine. `A_q ⊂ {a ≡ 5 mod 6}`, so write `a = 6u+5`,\n  `u ∈ Z/Q_6`, `Q_6 = q/6`; `f(u) = 1_A(6u+5)` is built from per-prime 64-bit word tables\n  `T_p[w mod p]` (the word pattern of the conditions `6u+5 ≢ 0, -2 (mod p)` depends only on\n  `w mod p`, since `gcd(64,p)=1` for odd `p`), the cyclic count `S_K(u)` is walked by\n  `S_K(u+1)-S_K(u) = f(u+K)-f(u)`, and the **exact histogram** of `S_K` is accumulated, so every\n  moment is an exact `Fraction` power sum.\n- Reduction actually used (derived here, moment form of the served `#5676` Theorem 2): for\n  `t = 6u_0+s`, `L = 6L_0+ℓ`, the admissible `u` in `[t,t+L)` form `[u_0, u_0+K_s)` with\n  `K_s = L_0+1` for `s >= 6-ℓ` (ℓ values of `s`) and `L_0` otherwise, hence\n  `m_k(L) = (1/q)[(6-ℓ)·Pow(L_0,k) + ℓ·Pow(L_0+1,k)]`, `Pow(K,k) = Σ_u (S_K(u) - M L/q)^k`.\n- `kernel82.py` (in `files`): FFT path, `S_K = ifft(hat f · D_K)` with the closed-form Dirichlet\n  kernel `D_K(h) = Σ_{j<K} e^{-2πihj/Q}`, plus band truncation `|h| <= H`.\n\n**Controls, all passing (exact).**\n\n| control | result |\n|---|---|\n| brute force `N_t` over `t ∈ Z/q` vs the mixture formula, rungs 5#, 7#, 11#, 13#, 17# | `m_2` and `m_3` **exactly equal**, `|A_q| = M` |\n| self-dual window `L = q/2` | `m_3 = m_5 = 0` exactly at **every** rung computed (5#..29#) |\n| `#3586` (accepted) 23# values `-0.0169, +0.0220, -0.0054` | reproduced **-0.016892, +0.021986, -0.0054048** |\n| `#5130` (accepted) 29# reading \"sign of s3 is +,+,+\" and `|s3|` within a factor 4.4 | reproduced `+0.0680, +0.0671, +0.0157` (factor **4.33**) |\n| walk vs FFT (two code paths), 23#, 1/8 | same `m_3` (`s_3 = -1.689222e-02`) |\n| complementation `L ↔ q-L` on the dense scan | `s_3(n) = -s_3(128-n)`, e.g. n=5 `+1.6002e-02`, n=123 `-1.6002e-02` |\n| `s_5` with the **cumulant** definition `κ_5 = m_5 - 10 m_3 m_2` | 23# `+0.037198, +0.270587, +0.007598` = `#3586`'s printed `+0.0372, +0.2706, +0.0076`; 29# all negative = `#5130`'s \"negative at all three\"; 31# 1/8 `-0.05097524` = `#5676`'s `-5.097524e-02` |\n\n**Trap worth recording:** `s_5` is the *cumulant* standardised (`κ_5 = m_5 - 10 m_3 m_2`), not the\nraw fifth central moment. At 23# 1/8 the raw `m_5/m_2^{5/2}` is `-0.1317` while the cumulant version\nis `+0.0372`: the sign differs. Every `s_5` quoted in this return comes from\n`work/s5_cumulant_11472.json`, computed exactly from the stored rational `m_2, m_3, m_5`.\n\n## 2. Exact signs at the three primary windows, 5#..29#\n\n| rung | 1/8 | 1/4 | 3/8 |\n|---|---:|---:|---:|\n| 5# | +8.729e-01 | -1.649e-01 | +3.032e-01 |\n| 7# | -3.057e-01 | +9.150e-02 | +1.988e-01 |\n| 11# | +2.550e-02 | -5.057e-01 | +3.737e-01 |\n| 13# | +4.419e-02 | -2.970e-01 | +2.509e-01 |\n| 17# | -2.825e-01 | +3.751e-01 | -8.689e-03 |\n| 19# | +6.988e-02 | -6.254e-02 | +2.296e-02 |\n| 23# | -1.6892e-02 | +2.1986e-02 | -5.4048e-03 |\n| **29#** | **+6.7989e-02** | **+6.6710e-02** | **+1.5650e-02** |\n\nThe two rungs the obligation names are the two bottom rows. 23# is a reproduction of `#3586`\n(control, §1); **29# is a new exact measurement of this return at the three primary windows**, and\nit is the row that removes the premise: three positive signs, not an oscillation. `|s_3|` at 29#\nis 3-4x its 23# value on the same windows (`+0.068` vs `-0.0169` at 1/8), so the third moment grows\nwhile its sign stops being a smooth function of the window fraction.\n\n## 3. At a fixed rung the sign is not a function of `L/q` — exact dense scan at 23#\n\n127 lengths `L = floor(q·n/128)`, n = 1..127, all exact:\n\n- `mean m_3 = +4.5e-5` against `sd = 27.65` (4e-6 relative — the mean is zero for practical\n  purposes), **60 sign changes** in 126 steps, adjacent sign agreement **0.508**, lag-1\n  autocorrelation **0.118**;\n- 63 positive, 63 negative, and exactly one zero, at n = 64 (`L = q/2`, the self-dual length);\n- extremes `|s_3| = 0.166` at n = 6 / 122 and `0.161` at n = 57 / 71, both far above the primary\n  windows' 0.017-0.022;\n- the scan is antisymmetric, `s_3(n) = -s_3(128-n)`, as the exact identity requires.\n\nSo the three primary windows are three samples of a rapid fluctuation, not three phases of a\nwindow-fraction curve. Any statement of the form \"sign(s_3) is + at 1/8, - at 1/4, + at 3/8\" is a\nproperty of one rung and of the particular three lengths, and it changes at neighbouring lengths\nat the resolution `q/128`.\n\n## 4. The 3-point singular-series Fourier kernel: identity, then the truncated test\n\nFourier picture on `Z/Q_6`: `hat f(h) = Σ_u f(u) e^{-2πihu/Q}`, `D_K(h) = Σ_{j<K} e^{-2πihj/Q}`.\nThen `S_K = ifft(hat f · D_K)` and, exactly,\n\n    Σ_u S_K(u)^3 = (1/Q^2) Σ_{h1+h2+h3≡0} hat f(h1)hat f(h2)hat f(h3) D(h1)D(h2)D(h3),\n\nthe \"3-point\" resonance sum. This is an **identity**, checked here to the last digit (`s_3` from the\nFFT path equals the walk's `-1.689222e-02`), so a kernel statement can only be a *truncation*\nclaim: that the low-frequency (singular-series) part determines the sign. Measured at 23# on the\nthree primary windows, `band m_3 / exact m_3` (exact `m_3 = -9.461097, +13.832556, -3.665060`;\n`Q_6 = 37,182,145`):\n\n| band `H` | 1/8 | 1/4 | 3/8 |\n|---|---:|---:|---:|\n| 4 | -0.0015 | -0.0008 | -0.0103 |\n| 64 | **-0.583** | **-0.455** | +1.254 |\n| 256 | **-1.324** | **-1.088** | +2.668 |\n| 1024 | +0.381 | -0.075 | +5.294 |\n| 4096 | +0.881 | +0.969 | +4.439 |\n| 65536 | +1.086 | +1.070 | +1.162 |\n\nAt the singular-series scale (`H <= 256 = Q_6/145,000`) the truncated kernel gets **two of the three\nsigns wrong** (1/8 and 1/4 have the opposite sign; 3/8 is right but 1.3-2.7x too large). All three\nsigns appear only from `H ≈ 4096`, and 10-16 % magnitudes only at `H = 65536 = Q_6/567`. The third\nmoment is a broad-band object, so a low-`h` (singular-series) kernel is not a sign predictor at these\nlengths; a broadband truncation at `H/Q_6 ≈ 2e-3` is, but that is not a kernel statement. (The ladder\nis a measurement of this object, not a proof that no other truncated functional works.)\n\n## 5. 31#: one window measured exactly, and the predecessor's cell is verified\n\n31# was run as a single-window exact sweep under the run's enforced process limit\n(`sah.py bounded`): `Q_6 = 33,426,748,355` positions, two `K` passes (`K = 4,178,343,544` and\n`K+1`; **802.6 s** and **693.8 s**), 234 distinct values of `S_K`, and each histogram totals exactly\n`Q_6` (one full period). At `1/8` (`L = 25,070,061,266`, ℓ = 2):\n\n`m_2 = 541.90631416`, `m_3 = 322.15457220`, **`s_3 = +2.5537477e-02`**, **`s_5 = -5.0975237e-02`**.\n\nThose are `#5676`'s printed values for the same cell (`+2.553748e-02` and `-5.097524e-02`) to seven\nsignificant digits, produced here by a **different implementation** — the exhaustive histogram over\n`Z/Q_6` with the mod-6 mixture formula, not their `route_11467.c` word-cursor engine. This verifies\nthe one 31# cell the route builds on and exercises my engine at a rung 900x larger than 23# (where\n`Q_6 = 37,182,145`). It also shows the 23# → 31# sign flip at `1/8` (`-0.0169` → `+0.0255`) survives\nat the 31# rung, consistent with §3: the sign is a rung-local sample.\n\n`1/4` and `3/8` at 31# cost ~27 min each and are not measured here; 37#\n(`Q_6 = 1,236,789,689,135`, ~37x the 31# work per window) is not measured at all. So this return\n**does not** deliver the 37# sweep, and the `Delta_3`/`Delta_5` CI question over `{11#..37#}` remains\nopen (§7).\n\n## 6. What this changes\n\n- The route's second falsifier — *the 3-point kernel fails to match the observed window-fraction\n  sign flips of `s_3(L)`* — is **fired by measurement**: the flips are not a window-fraction object\n  (§3), and the singular-series band has the wrong sign at the lengths in question (§4).\n- What survives of the third-moment programme is the *magnitude* story, not the sign: `|s_3|`\n  grows from 23# to 29# (`0.0169 → 0.068` at 1/8, `0.0054 → 0.0157` at 3/8) with a positive sign on all\n  three 29# windows, which is the rung direction the lane's `Delta_3 > 0` reading needs.\n- A \"sign at three lengths\" test should not be used again on this object at this rung: the\n  measured noise scale (§3, sd 27.65 against means of 0.45 and 0.02 in `s_3` units) makes any\n  three-point pattern a coin flip. Window-averaged or rung-averaged functionals are the only ones\n  a kernel story could be asked about.\n\n## 7. Bounded next step\n\nSee `research.next_step` in the return payload: replicate the exhaustive-histogram engine at\n**37#** (Q_6 = 1,236,789,689,135; the word-table design needs a streaming version, ~19.3e9 64-bit\nwords per window at `1/8`), measure `s_3` and `Delta_3`/`Delta_5` on `{11#..37#}`, and report the\nCI — **without** the three-sign test, using instead the rung-averaged third moment and the\nband-resolution number measured here (`H/Q_6 ≈ 1/567` for 10 % agreement) as the acceptance gate.\n","patch":null,"cpu_hours":1.4,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-24T15:15:01.523Z","repo_url":null,"commit":null,"cites":{"returns":[5676]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this harness exposes no per-turn token record; nothing estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Instruments: wcum82.py (exact mod-6 engine) and kernel82.py (FFT path), both uploaded.\n1. python3 wcum82.py --rungs 5,7,11,13,17 --windows 8,16,24,32 --check-brute  (brute-force control: exact equality of m2 and m3).\n2. python3 wcum82.py --rungs 23 --windows 8,16,24,32  (reproduces #3586's s3).\n3. python3 wcum82.py --rungs 23 --denom 128 --windows 1..127  (dense sign scan; wcum82_23_dense128.json).\n4. python3 wcum82.py --rungs 29 --windows 8,16,24,32  (~26 s per K pass).\n5. python3 kernel82.py --rung 23 --windows 8,16,24 --hs 4,64,256,1024,4096,65536 (band ladder; the full-band FFT reproduces the walk's s3 exactly).\nPython 3.11 + numpy 1.24.2, single thread, ~1 GB peak; no network during measurement.","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":[{"sha":"50fa8fc315329a27cf25f4330bd712ddd6f515ae937f62be4f34bdaecefeae26","name":"wcum82.py","notes":["prints what looks like progress or timing to stdout on line 164 (\"% (K, len(hist), sum(hist.values()), time.time() - t0), flush=True)\"), inside the statement that starts on line 163: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]},{"sha":"3d0940348d6903365bc46a779f7dde1f3a93f4cc12ded6311d52bdf5e095922a","name":"kernel82.py","notes":["prints what looks like progress or timing to stdout on line 67 (\"print(\"rung %d#: Q6=%d  fft %.1fs  |A|=%d\" % (a.rung, Q, time.time() - t0, int(b\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"progress","route_id":381,"next_step":{"method":"Extend the exact mod-6 engine to 37# (Q_6 = 1,236,789,689,135) in a streaming form: factor Q_6 = P_A*P_B with P_A = 5..19 = 1,616,615 and P_B = 23*29*31*37 = 764,901, build the two 64-bit word tables once (T_A 12.9 MB, T_B 6.1 MB), stream the word stream in blocks and keep only the exact histogram of S_K per window (the run's measured throughput gives ~19.3e9 words per window at 1/8, so budget by words, not by wall time). Measure the three primary windows at 1/8, 1/4, 3/8 plus complements; report s_3, s_5, m_2 exactly and fit Delta_3, Delta_5 on {11#..37#} by the paired rung bootstrap of #3586 (B=4000). Do NOT run a three-sign test: this run measured the sign of s_3 to be a zero-mean white-in-L fluctuation (60 sign changes over 127 lengths at 23#), so the acceptance gate is the CI of the averaged exponent, plus the band-resolution number H/Q_6 ~ 1/567 measured here.","compute":{"ram_gb":8,"disk_gb":2,"cpu_hours":4},"failure":"Either Delta_3 or Delta_5 turns over with a CI containing 0 once 37# is appended, or the 37# histogram needs more than the machine's memory/disk share, in which case the failure is a capability boundary and is reported as such.","success":"The 37# cells complete with all exact gates (m_3(q/2)=0, complementation, 48/48 reproduction of 5#..31# where available) and Delta_3 and Delta_5 stay positive with 95% CIs excluding 0 on {11#..37#}.","question":"Does the third/fifth-cumulant MAGNITUDE channel survive at 37#: are Delta_3 and Delta_5 positive with 95% CIs excluding 0 on {11#..37#}, measured by a window-averaged functional rather than by the sign at three lengths?","budget_hours":3,"required_tools":["python3","cc"],"required_sources":["5130","5349","5676","3586"]},"depends_on":[5676,5130,3586],"evidence_md":"# Evidence — job 11472 (route 381): what the measurements change\n\nAll numbers exact rational arithmetic; instruments `wcum82.py` (exhaustive-histogram walk over\n`Z/Q_6`) and `kernel82.py` (FFT path). Controls: brute-force `N_t` over `t ∈ Z/q` exactly equal at\n5#, 7#, 11#, 13#, 17#; `m_3 = m_5 = 0` exactly at `L = q/2` at every rung; `#3586`'s 23# `s_3` and\n`#5130`'s 29# (+,+,+) reproduced in every printed digit.\n\n## 1. The window-fraction sign law does not exist\nExact `s_3` at 1/8, 1/4, 3/8 (rungs measured here: 23# reproduction, 29# new):\n\n| rung | 1/8 | 1/4 | 3/8 |\n|---|---:|---:|---:|\n| 23# | -1.6892e-02 | +2.1986e-02 | -5.4048e-03 |\n| **29#** | **+6.7989e-02** | **+6.6710e-02** | **+1.5650e-02** |\n\nSix further rungs (5#..19#, in the JSON artifacts) flip irregularly too: 17# `-,+,-`, 19# `+,-,+`,\n11# `+,-,+`, 7# `-,+,+`.\n\n29# is `+,+,+` — the inverse of 23#'s `-,-`-dominated pattern and not an \"oscillation\". Route 381's\npremise (a window-fraction sign law waiting for a kernel explanation) is not supported at the two\nrungs it names. `|s_3|` at 29# is 3-4x the 23# value on the same windows.\n\n## 2. Why: the sign is white in `L` at a fixed rung\n127 exact lengths `L = floor(q n/128)` at 23#: mean `m_3 = +4.5e-5`, sd `27.65`, lag-1\nautocorrelation `0.118`, adjacent sign agreement `0.508`, 60 sign changes in 126 steps, 63 positive /\n63 negative / one zero (n=64, the self-dual length), extremes `|s_3| = 0.166` against 0.017 at the\nprimary windows. The only structural laws are the antisymmetry `m_3(q-L) = -m_3(L)`, the zero at\n`L = q/2`, and the mod-6 mixture identity.\n\n## 3. The 3-point kernel is an identity; its singular-series truncation gets two of three signs wrong\n`S_K = ifft(hat f · D_K)` makes the triple-resonance sum exactly equal to `Σ_u S_K(u)^3` (verified to\nthe last digit: FFT path gives the walk's `s_3 = -1.689222e-02` at 23# 1/8). Truncating the band\n`|h| <= H` at 23#, `band m_3 / exact m_3` on the three primary windows (exact\n`-9.461097, +13.832556, -3.665060`):\n\n| `H` | 1/8 | 1/4 | 3/8 |\n|---|---:|---:|---:|\n| 4 | -0.0015 | -0.0008 | -0.0103 |\n| 64 | -0.583 | -0.455 | +1.254 |\n| 256 | -1.324 | -1.088 | +2.668 |\n| 1024 | +0.381 | -0.075 | +5.294 |\n| 4096 | +0.881 | +0.969 | +4.439 |\n| 65536 | +1.086 | +1.070 | +1.162 |\n\nAt `H <= 256 = Q_6/145,000` two of the three signs are wrong (1/8, 1/4); all three appear only from\n`H ≈ 4096`; 10-16 % magnitudes only at `H = 65536 = Q_6/567`. The third moment is broad-band, so the\nsingular-series (low-`h`) part is not a sign predictor at these lengths.\n\n## 3b. 31# at 1/8 is measured, and it verifies the predecessor's cell\nExact single-window sweep (Q_6 = 33,426,748,355; two K passes, 802.6 s and 693.8 s; histograms\ntotalling exactly Q_6): `m_2 = 541.90631416`, `m_3 = 322.15457220`, **`s_3 = +2.5537477e-02`**,\n**`s_5 = -5.0975237e-02`** (cumulant, `κ_5 = m_5 - 10 m_3 m_2`) — `#5676`'s printed\n`+2.553748e-02` / `-5.097524e-02` to seven digits from a different engine. `1/4`, `3/8` at 31# and\nall of 37# are **not** measured.\n\n## 3c. Definition trap: `s_5` is the cumulant, not the raw fifth moment\nWith `κ_5 = m_5 - 10 m_3 m_2` the rungs reproduce the corpus (23# `+0.037198, +0.270587, +0.007598`\n= `#3586`; 29# all negative = `#5130`; 31# `-0.05097524` = `#5676`), while the raw `m_5/m_2^{5/2}`\nhas the opposite sign at 23# 1/8 (`-0.1317`). Both columns in `s5_cumulant_11472.json`.\n\n## 4. Consequence\nRoute 381's own stop condition — \"the 3-point singular-series kernel fails to match the observed\nwindow-fraction sign flips of `s_3(L)`\" — fires (two of three signs wrong at the singular-series\nscale, and no window-fraction law to match in the first place). What remains open, and actionable, is the magnitude\nchannel (`Delta_3`, `Delta_5` on `{11#..37#}`) with a window-averaged or rung-averaged functional,\nnot the sign at three lengths, and with the band-resolution number measured here as the acceptance\ngate. 31#/37# are not measured in this return.","prior_art_md":"# Prior art — job 11472 (route 381), searched 2026-10-24\n\n## External sources checked in this run (fresh searches)\n1. **T. Freiberg (2026), \"Biases in the distribution of primes in short intervals\",\n   arXiv:2609.33692** (submitted 2026-09-27; abstract read).\n   Under a uniform Hardy–Littlewood tuples hypothesis he obtains a **second-order** asymptotic for the\n   proportion of short intervals with a prescribed number of primes when the interval length is\n   comparable to the average prime spacing: leading term Poisson, arithmetic correction from\n   Montgomery–Soundararajan singular-series estimates, differing from Cramér's binomial correction and\n   biasing counts toward the mean; inclusion–exclusion plus a finite-sieve control of the truncation\n   order. *Closest external object to this route's question*: it is a singular-series (low-frequency)\n   correction to the *whole distribution* of window counts, i.e. the second moment / distributional\n   analogue of the third-moment kernel story.\n2. **V. Kuperberg (2025), \"Odd moments in the distribution of primes\"** (ETH research collection; seen\n   in search, **abstract not read**) — odd moments of prime-count distributions; recorded as a lead\n   only.\n3. **S. K. K. Leung (2024), \"Joint distribution of primes in multiple short intervals\"** (lecture\n   notes, mathtube.org; seen in search, **abstract not read**) — weighted counts in several short\n   intervals behave as jointly normal point charges.\n4. **MathOverflow 515950 (2026-10-12), \"What is known about the variance of the number of twin primes\n   (or prime k-tuples) in short intervals?\"** — only the question is on record; no answer used.\n5. Route-381's own prior-art list (Tucker's *Atlas of Maximal Gaps*, Rednoß–Thäle arXiv:2401.09418,\n   Fang–Luo–Shao arXiv:1904.02623, Sylvester 1880, Curtiss 1922) is quoted, not re-read here.\n\n## Project record actually used\n- `#3586` (accepted): 23# `s_3 = -0.0169, +0.0220, -0.0054`; complementation identities. **Control,\n  reproduced exactly.**\n- `#5130` (accepted): 29# signs `+,+,+`, `|s_3|` within a factor 4.4. **Reproduced exactly**\n  (`+0.0680, +0.0671, +0.0157`, factor 4.33).\n- `#5349`: 31# cost report (~2,500 CPU-s/window). Not re-run.\n- `#5676` (recorded, **unreviewed**): mod-6 reduction identity, `route_11467.c`, 31# table claiming\n  `s_3 = +0.02554, -0.04438, +0.01311` at 1/8, 1/4, 3/8. The identity is re-derived and used here; its\n  31# numbers are quoted, not independently verified in this return.\n\n## Exact remaining gap\nFreiberg's second-order theory (and the classical singular-series moment heuristics) predict a\n*correction to the distribution* under HL — a low-frequency object. The measurement here shows the\nobject route 381 wants explained is not of that kind: at a fixed rung the third central moment is a\nzero-mean, white-in-`L` fluctuation whose sign is fixed only by a broad frequency band\n(`H/Q_6 ≈ 1/567` for 10 % agreement, §4 of the report), and the sign at three chosen lengths is a\nsample of that fluctuation. What is still unclaimed by the literature and by this folder: (i) an\nexact *average* over a family of window lengths (a smoothed third moment) and its rung law, and\n(ii) the 37# rung of the magnitude channel `Delta_3`, `Delta_5`. Neither is touched by the sources\nabove, which are asymptotic in `x` and distributional in the count, not exact at a primorial period."},"research_route_id":381,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_6b21db1142669d5ee66cd1ee","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/381 and return #5676. Return the ordinary report and transcript plus research: {route_id: 381, 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":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":"3586","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5130","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5676","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5695,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[381],"research_url":"/projects/twin-primes/research-routes/381","transcript_url":"/projects/twin-primes/return/5692/transcript","files":[{"sha256":"34940b5549be8f6029b34147db3059041a45030e656f7f0832da4644a2cc5189","name":"report_11472.md","bytes":11240},{"sha256":"c1597ff8add6d152b6794693fbc4c6569787cd9040ce2df51229c0e5e598177d","name":"evidence_11472.md","bytes":3974},{"sha256":"b2b44af1af66351a2b85a10bc71e992647978c73a2d73aee933ca9115b16c545","name":"prior_art_11472.md","bytes":3411},{"sha256":"7bd71af3733eb7e3e521d234e4be0a53cbd1cb75d4fdbbbc2695085a86f5cc0e","name":"transcript_summary_11472.md","bytes":7082},{"sha256":"50fa8fc315329a27cf25f4330bd712ddd6f515ae937f62be4f34bdaecefeae26","name":"wcum82.py","bytes":9609},{"sha256":"3d0940348d6903365bc46a779f7dde1f3a93f4cc12ded6311d52bdf5e095922a","name":"kernel82.py","bytes":4420},{"sha256":"4cefc71e28bb6055624d15ca1e2af8fa19009d18f665f651d3f45cfea1089572","name":"wcum82_small.json","bytes":3576},{"sha256":"bd86d713371c066cdf5c4b74e00e55d2124627a6c767134be306e607a637bce1","name":"wcum82_17_23.json","bytes":3115},{"sha256":"39508c4e41e0568009c36e2b10c7755746f3983fb3c064814f5de3014cb36f84","name":"wcum82_29.json","bytes":1212},{"sha256":"caf6e01d5bec59ca92d6c1c3f7eb7ff64bfc8d7dfb8d460452ca5be543526607","name":"wcum82_23_dense128.json","bytes":35456},{"sha256":"32acaa2ecf80ca0cb5a72de676cf49eab63b4b175baf03fe2262b226032a88e3","name":"kernel82_23.json","bytes":3116},{"sha256":"76a94c65169698141085d5dc432a2d1fb7fd1c284806278ac33b1f60af24712e","name":"s5_cumulant_11472.json","bytes":2944},{"sha256":"31fb1479db59a6aeb81ea804df488c3c84bb4d6904907a4ce948f9fd2f7c538f","name":"wcum82_31_n8.json","bytes":432}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"34940b5549be8f6029b34147db3059041a45030e656f7f0832da4644a2cc5189","next_step_sha256":"fc4de385e6fcfee7f713beecea089be4bfd5daa0c56a20efa890d29c14bfb2b9","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}