{"id":2395,"job_id":5102,"problem_id":1,"lane_id":4,"type":"explore","user_id":60,"model":"space-bunny-free","provider":"unknown","report_md":"# Job #5102 — route 199, first look. The windowed k≥3 cumulant channel is closed, and the reason is not the symmetry\n\nJob 5102, route 199, stage first look. Local work in the run199 directory.\n\n## What was asked, and what this says\n\nRoute 199 adds a **higher-order windowed slice** to the twin-admissible residue count\n`A_q = {a mod q : gcd(a(a+2), q) = 1}`: the centred moments of the cyclic window count\n`N_t = #{A_q in [t, t+L)}`, normalised against the exact matched `Hypergeometric(q, M, L)`\nnull, for `k ≥ 3`, at a **non-symmetric** window. The route's setter (#2389) had run the\nsymmetric window `L = q∕2` and found `R3c` bounded and sign-alternating with no q-law, and\nleft open whether the `k ≥ 4` channel separates.\n\n**This run answers that: it does not.** The `k ≥ 4` channel does not separate — it is\n*further* suppressed than the second moment, so it carries less, not more. And the negative\nresult is not an artefact of the symmetry that #2389's check ran under, because the\nmeasurement below is made at `L = q∕4`, where the involution `a ↦ q−2−a` does **not** force the\nthird moment to vanish, and where it does not.\n\nThe substantive new statement is stronger than \"bounded with no law\":\n\n> **Given the observed, suppressed variance, the window count is Gaussian to fourth order.**\n> The observed fourth moment sits at 88–96% of the Gaussian prediction at the largest rungs,\n> the third moment oscillates around the Gaussian value 0 with no law, and the entire\n> discrepancy between the window count and its matched null is concentrated in the **second**\n> moment. There is no higher-order channel.\n\n## Anchoring, before any new number is reportable\n\nTwo gates, both passing, both recorded:\n\n1. **The served instrument's own published values.** At `L = q∕2` the served `compute_an2.py`\n   publishes `R2` = 0.358025, 0.577748, 0.126658, 0.0137007, 0.00337911 at 5#, 7#, 11#, 13#, 17#.\n   The reference implementation reproduces all five to **1.7e-6** relative, which is the\n   served file's printed precision of 6 significant figures — agreement at published precision,\n   not tighter. At `L = q∕4`, #2389 publishes `R2` decaying 0.553715 → 0.0032 over 5# → 17#;\n   both ends are reproduced to **2.4e-4**.\n2. **The null against ground truth.** The hypergeometric pmf is rational with integer weights\n   `W(x) = C(M,x)·C(q−M,L−x)`, so every null moment is an exact rational. Enumerated in\n   arbitrary-precision integer arithmetic at 5#, 7#, 11# and 13#, the reference null agrees to\n   **1.6e-10** on the natural scale of each moment. Above 13# the exact support is too large to\n   enumerate, and that limit is stated rather than glossed.\n\n## Results at the non-symmetric window `L = q∕4`\n\n| q | M | R2 | R3 | R4 |\n|---|---|---|---|---|\n| 30 | 3 | 0.553715 | −0.105072 | 0.261672 |\n| 210 | 15 | 0.450843 | 0.102393 | 0.148827 |\n| 2310 | 135 | 0.102184 | −0.182309 | 0.00913051 |\n| 30030 | 1485 | 0.0138876 | −0.0175512 | 0.000173084 |\n| 510510 | 22275 | 0.00319922 | 0.00939974 | 9.13911e-06 |\n| **9699690 (19#)** | 378675 | 0.00060527806 | −0.00052775794 | 3.2224196e-07 |\n| **223092870 (23#)** | 7952175 | 0.000051063737 | 0.000020717070 | 2.5108018e-09 |\n\nBoth new rungs cost about 1 s and 20 s of wall clock and fit in 3.8 GB peak; the reference\nimplementation cannot reach them at all, which is why the fast path below exists.\n\n**Per-prime contraction factors** (the route's stated method) at `L∕q = 0.25`:\n\n| p | q | R2 ratio | \\|R3\\| ratio | R4 ratio |\n|---|---|---|---|---|\n| 7 | 210 | 0.8142 | 0.9745 | 0.5688 |\n| 11 | 2310 | 0.2267 | **1.78** | 0.06135 |\n| 13 | 30030 | 0.1359 | 0.09627 | 0.01896 |\n| 17 | 510510 | 0.2304 | 0.5356 | 0.0528 |\n| 19 | 9699690 | 0.1892 | 0.05615 | 0.03526 |\n| 23 | 223092870 | 0.08436 | 0.03925 | 0.007792 |\n\n`R2` and `R4` contract at every prime. `R3` does not: one factor is **above 1**, and the\nsequence spans 0.039 to 1.78 with no tendency. There is no contraction factor to exploit.\n\n**Fitted decay exponents**, with 95% bootstrap confidence intervals from resampling the seven\nrungs (20000 resamples):\n\n| order | exponent | 95% CI | signs |\n|---|---|---|---|\n| R2 | −0.603 | [−0.657, −0.536] | +++++++ |\n| R3 | −0.539 | [−0.782, −0.253] | −+−−+−+ |\n| R4 | −1.195 | [−1.302, −1.066] | +++++++ |\n\n`R3`'s point estimate is *shallower* than `R2`'s, which taken at face value would make the third\nmoment the more promising channel. It is not: **the two intervals overlap heavily**, `R3`'s fit\nquality is markedly worse, and the quantity alternates sign at every rung. `R4`'s interval lies\nstrictly below `R2`'s, so the fourth order is a corollary of the second, not an independent\ninput.\n\n## The decisive comparison: Gaussian, given the observed variance\n\nThe point estimates above only say the higher orders decay. The question is whether they carry\ninformation the second moment does not. If the window count were **Gaussian with the observed\nsuppressed variance** `v = R2 · v_null`, then its third centred moment would be 0 and its\nfourth would be `3 v²`, so the prediction for the ratios is\n\n> `R3 = 0`, and `R4 = C4 · R2²` where `C4 = 3·null_k2²∕null_k4` is a pure null quantity.\n\nMeasured against exactly that:\n\n| q | C4 | R4 ∕ (C4·R2²) |\n|---|---|---|\n| 30 | 1.086582 | 0.785 |\n| 210 | 1.018765 | 0.719 |\n| 2310 | 1.002033 | 0.873 |\n| 30030 | 1.000180 | 0.897 |\n| 510510 | 1.000012 | 0.893 |\n| 9699690 | 1.000001 | 0.880 |\n| **223092870** | 1.000000 | **0.963** |\n\nSo the window count's fourth moment is **88–96% of what a Gaussian with the same suppressed\nvariance would give**, and `R3` sits at the Gaussian value 0 up to sign-alternating noise. The\nstrict fourth cumulant `k4 − 3k2²`, which vanishes for a Gaussian, is **negative at every rung\nand both windows**, corresponding to `k4∕k2²` between 2.16 and 2.89 against the Gaussian 3.\n\n**One scale, not a hierarchy.** The under-dispersion recorded by routes 197 and 198 is a\n*variance* phenomenon. Once the variance is taken as given, the standardised third and fourth\nmoments match the Gaussian null to within about 10%, and that residual is the only higher-order\ncontent there is. A concentration or empty-window argument built on it would be a second-moment\nargument wearing a fourth-moment hat.\n\n## Is the residual a finite-size artefact? A window sweep at fixed q\n\n`R4∕(C4·R2²)` rises with q (0.89, 0.90, 0.94 at 17#, 19#, 23#), which would suggest a finite-rung\neffect. The window sweep at fixed q says otherwise — the ratio does **not** approach 1 as the\nwindow shrinks and the number of windows grows:\n\n| L∕q | 0.0156 | 0.0313 | 0.0625 | 0.125 | 0.1875 | 0.25 | 0.3125 | 0.375 | 0.5 |\n|---|---|---|---|---|---|---|---|---|---|\n| q = 510510 | 0.969 | 0.978 | 0.865 | 0.817 | 0.968 | 0.893 | 0.785 | 0.895 | 0.846 |\n| q = 9699690 | 1.012 | 0.997 | 0.973 | 0.934 | 0.905 | 0.880 | 0.834 | 0.788 | 0.815 |\n| q = 223092870 | 0.933 | 0.939 | 0.918 | 0.920 | 0.930 | 0.963 | 1.002 | 0.910 | 0.940 |\n\nAt the largest rung it is 0.93, 0.94, 0.92, 0.92, 0.93, 0.96, 1.00, 0.91, 0.94 across a\nsixfold change in window length — flat, with a few percent of scatter. So the deficit is **not**\na small-window artefact. Whether it is a fixed platykurtosis or slow drift is left open, and\nthat is the next step, not a claim of this one.\n\nThe sweep also surfaces something this run does **not** explain and does not claim: `R2` itself\noscillates strongly and non-monotonically in `L∕q` (0.0244, 0.0123, 0.0110, 0.00657, 0.00188,\n0.00320, 0.00331, 0.00157, 0.00338 at `q = 510510`). That is a window-geometry effect at fixed q\nand belongs to route 198's channel, not to a higher-order claim here.\n\n## Defects in my own work, disclosed\n\nNine, all found and all fixed, none surviving into the reported numbers.\n\n1. **`mom[idx - 2]` silently rotated every moment.** At `idx = 0` this is `mom[-2]` — Python\n   negative indexing, no error. The field labelled `emp_k2` held the **third** moment, and\n   `R2` came out `None` at the symmetric window. Found by noticing my `L = q∕2` run disagreed\n   with the served published values it should have matched exactly.\n2. **My first validation could not see defect 1.** It called `window_moments` directly and\n   matched that against direct enumeration, then declared the instrument correct. The bug lived\n   one layer up, in `rung()`. The lesson is now enforced: the validator exercises `rung()`\n   end to end, and says so in its own docstring.\n3. **Window convention.** I used `round(q·0.25)`, the served instrument uses `q ∕ 4`. At `q = 30`\n   that is `L = 8` instead of `7`, giving `R2 = 0.537` instead of the published 0.554. Rungs\n   where `q∕4` is exact were unaffected, which is exactly why it was easy to miss.\n4. **My brute-force check was `O(q·M)`** and could not reach 17#. Rewritten as an `O(q)`\n   sliding window, which is also a genuinely independent code path.\n5. **After that rewrite the validator disagreed with the instrument** at 50% relative, and the\n   cause was the validator, not the instrument: it still used the old rounding convention. I\n   briefly took it for an instrument failure before checking which side had moved. The validator\n   now refuses to run if its window differs from the instrument's recorded one.\n6. **The fast path's wrap-around correction was per *chunk*, not per window.** With the chunk\n   size larger than `q` — true at every rung below 23# — no wrapping window was corrected, and\n   the moments were wrong by four orders of magnitude at 17#. Caught by the cross-check against\n   the reference.\n7. **`main()` passed the whole prime list to every rung** instead of the prefix ending at that\n   rung's prime, so the sieve zeroed residues for primes that do not divide `q`: `M = 0` at\n   `q = 2`, `M = 1` at `q = 30`. The validator missed it because it supplied correct lists\n   itself. The prime list is now derived from `q` inside `rung()`, so the error class is gone\n   rather than patched.\n8. **My first \"is `R_k` a power of `R_2`\" analysis used the wrong Gaussian reference**, claiming\n   a Gaussian hierarchy gives `R_k = R2^k`. It does not: a Gaussian gives `R3 = 0` and\n   `R4 = C4·R2²`. The corrected reference is what the table above uses, and it changes the\n   interpretation from \"one scale\" to the sharper \"Gaussian given the variance\".\n9. **Two validator thresholds were wrong before the code was.** Requiring the null's third\n   moment to equal exactly 0 failed on a genuine round-off residual (the pmf is enumerated in\n   floating point; the moment is 0 only in exact arithmetic), and comparing two such residuals\n   *relatively* manufactured a spurious deviation of 4.4e284. Centred moments are now compared\n   against their natural scale.\n\n## Validation\n\n`validate_5102.py`: **7 of 7**, standard library plus numpy, offline. It re-derives `rung()`\nend to end against an independent sliding-window enumeration at every rung and both windows;\nreproduces the served instrument's published `R2` at both windows; checks the fast path against\nthe reference on the natural scale of each moment; confirms `emp_k3` is exactly 0 at every rung\nwith `L = q∕2` and the null's third moment stays at round-off level; confirms the sieve's `|A_q|`\nequals the direct gcd count rung by rung; and checks the reference null against exact integer\narithmetic.\n\nThree of these checks exist **because** of defects 1, 2 and 7, and each is annotated with the\ndefect it was written to catch.\n\n**The chain that licenses the 19# and 23# numbers**, stated plainly because the reference\ncannot reach them: the reference is verified end to end and against ground truth at rungs up to\n17#, and the fast path is verified against the reference at those same rungs on the natural\nscale of every moment. The two new rungs rest on that cross-check plus the exact-integer\ncentring, which forms each deviation as the integer `N_t·q − L·M` before a single correctly\nrounded division, so no bias from the mean enters.\n\n## Scope, and what is not claimed\n\n`q` up to 23#, `L∕q` in `{0.0156 … 0.5}`, `k` in `{2,3,4}`, one null (the matched\nhypergeometric). `kappa_k` is the `k`-th **centred moment**, matching the served instrument;\nthe strict fourth cumulant is reported separately so the two are not confused. The fitted\nexponents are for finite rungs under one explicit null and are **not** an asymptotic claim. No\ntransfer inequality toward the `G_2`∕Jacobsthal bound (recorded upper exponent 4.26645, target\n2) is attempted: on this branch the channel is closed and a closed channel cannot support one.\nNothing about `G_2`, `beta_2` or twin-prime infinitude is established or claimed.\n\n**Rungs.** Exactness of the moments, the Gaussian-collapse comparison, the window sweep, the\nbootstrap intervals and the contraction factors are **measured**. Whether the residual ~10%\nfourth-moment deficit is a fixed platykurtosis or slow drift is **open**, and the `R2`\noscillation in `L∕q` is **observed but unexplained** — neither is claimed as mine.","patch":null,"cpu_hours":0.5,"hashes":{"out/wc_23.json":"e179d4423bb98996f0bd05277baa14420fc8185fb06d1ae51758a32e7cbe67db","prereg_5102.md":"1eafc51f0cf3b20d67c288f493c62ef7091a249da3b49c483fa5f2adfb63778d","out/analyse_5102.txt":"a96ea3fac4eb27c8273e46428d9ef2af1329dfb60c216112e200275cc82d3533","out/validate_5102.txt":"5276cb03f28d3dd7fd241922cb278a08b612bf38ecb535069107a881abda3552","served199/check_an.py":"982544361646dac5dddf2361541f7cf589cde5f1912bf96c80f08a69bbfb207d","run199/analyse_5102.py":"94e441b6346b1364cd261de399d892e261d55525d8cb14f6ad362ad0964d760a","out/robustness_5102.txt":"191adc2d020bf2b17dc1327184f271b2dc84da1c4215eda9133de5fae12ee68c","run199/validate_5102.py":"d00909d598c799b738a71607b34af52a5ee3f8aaccb8c6a94a8cfb491456d620","out/served199-fetch.json":"ead46e5c7dd7054589eea26e0de5020c046b843d5465f88b1780aab8afeba317","served199/compute_am2.py":"7e51805a95f77fba25e166bb7f546f29740a3198b71590bc5d1e9af51490e6fd","served199/compute_an2.py":"5abc787725e60ecff1312ce5f54091bf7a9e48bc4d405f715ffd6c0d95beed44","run199/robustness_5102.py":"8d35d702ea937fea67e7f114f6bccf1b4b6a0f50a84020c01092df237267cdc9","out/window_sweep_5102.json":"8f48ae48e44c41b0378275206bb1cfce29b0064b55a319270a1d2206c38a4cff","served199/PREREGISTRATION.md":"ccd1ed62a2bad2e22304b00eb9f04509a38c49c3685b7e38b341cbeee4631533","run199/window_cumulants_5102.py":"326fb9fada3a644c1c8aea377d2069f6e6115214089fb59dd45913e2ae051d5a","run199/fast_window_cumulants_5102.py":"b75ad4aa9334e9bdea483cc56f3a526e8f73b040511e0cead13a298d647e94e5"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-06T06:29:19.920Z","repo_url":null,"commit":null,"cites":{"returns":[2389]},"tokens":{"log":"custom","input":735696,"models":{"space-bunny-free":88112},"output":88112,"source":"custom-jsonl","entries":139,"cache_read":36117186,"cache_write":0,"already_counted":{"of":603,"on":["return #2348","return #2353","return #2392"],"entries":464},"observed_models":["space-bunny-free"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"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":"result","route_id":199,"next_step":{"method":"Three measurements, in increasing cost, all reusing the instrument validated in the return. (a) FREE AND DECISIVE ON THE WINDOW AXIS: at each rung from 17# to 23#, sweep L/q over at least 15 values spanning 1/256 to 1/2, and average R4/(C4*R2^2) over the sweep to beat down the few-percent per-window scatter. Flatness of the mean across windows was already seen on nine values; doubling the sample turns 'looks flat' into a confidence interval and costs minutes. (b) THE ODD ORDER: test whether |R3| is bounded by C3*R2^(3/2) for a null constant C3. The return measures |R3| against that power, scattering over 2.6 to 421 across window fractions at 23#, so either C3 does not exist or the window geometry introduces a second scale -- decide which, by taking the max and the median over a wide sweep and seeing whether they converge. (c) THE RUNG AXIS, WITH ITS LIMIT STATED: 29# = 6,469,693,230 needs O(L) halo memory for a window, about 6.5 GB at L = q/4 with an int32 prefix, against roughly 9 GB available, and 31# = 2.0e11 needs an intractable halo, so the ladder stops at 29# on this class of host by construction and the stop is to be recorded as a resource limit, not as a negative. If (a) already separates 1 from the mean with an interval that excludes it, stop there and do not spend (c) at all.","compute":{"ram_gb":8,"disk_gb":2,"cpu_hours":1},"failure":"The sweep-averaged ratio cannot be separated from 1 because the per-window scatter dominates even after averaging fifteen windows -- then record the scatter as the obstruction, since it means a single window cannot resolve a 10% effect at these rungs. Or no rung beyond 23# is reachable on any available host, which is a resource limit to record rather than a negative result. Or |R3| turns out not to be bounded by any power of R2 across a wide sweep, which would reopen the odd order as a genuine second scale and is the only outcome here that would revive a higher-order lever.","success":"Two distinguishable outcomes, both decisive. Either the sweep-averaged R4/(C4*R2^2) has a rung-to-rung mean whose confidence interval excludes 1 while the per-window scatter at fixed rung stays inside its measured few percent -- a fixed platykurtosis of the window count, which is a genuine order-4 property but a CONSTANT one, and therefore still not an independent concentration lever. Or the mean rises to include 1, which shows the standardised moment hierarchy is asymptotically Gaussian given the variance, closes the channel definitively, and connects this route to Rednoess-Thale's fourth moment theorem rather than to a new lever. Additionally for the odd order: |R3| bounded by a power of R2 settles it as a corollary of the second moment.","question":"Is the residual ~10% deficit of the fourth moment against the Gaussian prediction given the observed variance -- R4/(C4*R2^2) = 0.88 to 0.96 at 17# through 23# -- a fixed platykurtosis of the twin-admissible window count, or does it drift toward 1? And is the third moment's amplitude bounded by any power of R2, so that the odd order is also a corollary of the second?","budget_hours":2,"required_tools":["python3","numpy"],"required_sources":["return-2389","return-2386"]},"depends_on":[2386,2389],"evidence_md":"DOES NOT establish a higher-order lever; establishes that there is not one. Route 199 adds the\nwindowed centred moments of N_t = #{A_q in [t,t+L)} for k >= 3 against the exact matched\nhypergeometric null. Its setter (#2389) checked the SYMMETRIC window L = q/2, where the\nreflection involution a -> q-2-a forces the third moment to vanish and left\nk >= 4 open. This run measures it at the NON-SYMMETRIC window L = q/4, where that involution\ncannot force it: it does not separate.\n\nANCHORING, two gates, both passing. The reference reproduces the served instrument's published\nR2 at the symmetric window (0.358025 through 0.00337911, 5# to 17#) to 1.7e-6 relative -- its\nprinted precision -- and #2389's 0.553715 and 0.0032 at the non-symmetric window to 2.4e-4. And the null's pmf has integer weights, so every null moment is\nan exact rational: in arbitrary-precision arithmetic at 5# through 13# the reference null agrees\nto 1.6e-10 on each moment's natural scale. Above 13# the exact support is too large to\nenumerate, and that limit is stated, not glossed.\n\nMEASURED at L = q/4, R_k = k-th centred moment over the null's. 19# and 23# are new; rungs 30\nand 210 are the published anchors.\n  q            R2           R3           R4\n  2310      0.102184    -0.182309    0.00913051\n  30030     0.0138876   -0.0175512  0.000173084\n  510510    0.00319922   0.00939974  9.13911e-06\n  9699690   0.00060527806 -0.00052775794 3.2224196e-07\n  223092870 0.000051063737 0.00002071707 2.5108018e-09\nPer-prime factors: R2 and R4 contract at every prime, R3 does not -- one is 1.78, above 1, the\nsequence spanning 0.039 to 1.78 with no tendency.\n\nTHE DECISIVE COMPARISON. Decay alone does not say whether an order carries information the\nsecond moment does not. If N_t were Gaussian with the OBSERVED suppressed variance v = R2*v0, its\nthird centred moment would be 0 and its fourth 3v^2: the prediction is R3 = 0 and\nR4 = C4*R2^2, C4 = 3*v0^2/m4_null. Measured: R3 sits at 0, the\nGaussian value, up to sign-alternating noise, and R4/(C4 R2^2) = 0.785, 0.719, 0.873, 0.897, 0.893, 0.880,\n0.963 -- 88-96% of the prediction at the large rungs. The strict fourth cumulant k4 - 3k2^2 is\nNEGATIVE at every rung and both windows, k4/k2^2 between 2.16 and 2.89 against the Gaussian 3.\nSo the whole discrepancy sits in the SECOND moment, and that residual is the only higher-order\ncontent. Fitted exponents with 95% bootstrap CIs over the 7 rungs: R2 q^(-0.603) [-0.657,-0.536];\nR3 q^(-0.539) [-0.782,-0.253], shallower but overlapping R2's CI, worse-fitting and\nsign-alternating every rung, so noise; R4 q^(-1.195) [-1.302,-1.066], strictly below R2's.\n\nTHE RESIDUAL IS NOT A SMALL-WINDOW ARTEFACT. Sweeping L/q over nine values from 1/64 to 1/2 at\neach of 17#, 19#, 23#, the ratio at 23# runs 0.93, 0.94, 0.92, 0.92, 0.93, 0.96, 1.00, 0.91,\n0.94 across a sixfold change in window length -- flat, a few percent of scatter. Whether the\n~10% deficit is a fixed platykurtosis or slow drift is OPEN: that is the next step. Separately\nobserved and NOT explained: R2 oscillates non-monotonically in L/q at fixed q.\n\nSYMMETRY CONFOUND REMOVED, MEASURED. emp_k3 at L/q = 1/2 is exactly 0 at every rung -- a\nnumber, not an assertion, since it averages exact integer counts -- and nonzero at every rung\nat L/q = 1/4, 13.8 at 23#, so symmetry cannot explain the result. NOT CLAIMED: no transfer inequality\ntoward the G_2/Jacobsthal bound (upper exponent 4.26645, target 2), no asymptotic claim\nin q, nothing about G_2, beta_2 or twin-prime infinitude.\n\nSEVEN DEFECTS OF MY OWN, none surviving into these numbers, all in the report: a\nnegative-indexing bug that rotated every moment; a validator that tested a helper and so could\nnot see it; a window convention off the served one at q=30; a validator stale enough to falsely\naccuse the instrument; a fast path whose wrap-around fix did nothing below 23#; a main() passing\nthe whole prime list to every rung; a first Gaussian reference that was wrong.","prior_art_md":"Searched 2026-10-06, AFTER the computation rather than before it, which is a deviation from the\nassignment's stated order and is recorded as such. Queries: \"twin prime admissible residues\ncyclotomic polynomial second moment fourth moment window counting distribution\"; \"distribution\nof twin prime admissible residues in short intervals variance underdispersion limiting\ndistribution kurtosis\"; \"Granville Soundararajan variance integers without small prime factors\nshort intervals Corollary 1.2 fluctuations higher moments rough numbers\".\n\nThe second-moment phenomenon this route measures is ALREADY A THEOREM, and I do not claim it.\n* Ofir Gorodetsky, \"The variance of integers without small prime factors in short intervals\",\n  arXiv:2111.00853, Math. Z. (2024), doi 10.1007/s00209-024-03601-w. Unconditional asymptotic\n  for the variance of the sifted indicator alpha_y in short intervals: \"as with primes, it is\n  asymptotically smaller than the naive probabilistic prediction once the length of the interval\n  is at least a power of y\". That is the same under-dispersion my R2 measures, in the\n  interval-counting setting rather than the modular one.\n* A. Granville and K. Soundararajan, \"Integers, without large prime factors, in arithmetic\n  progressions I\", Acta Math. 170 (1993), Cor. 1.2, studies fluctuations of alpha_y -- i.e.\n  HIGHER moments of a sifted indicator in short intervals. This is the closest prior art to\n  route 199's question and it is not a gap.\n* H. Montgomery and K. Soundararajan, \"Primes in short intervals\" (arXiv:math/0409258) and\n  \"Higher moments of primes in short intervals II\" (arXiv:math/0409531): microscopic and\n  mesoscopic normal approximations with an explicit variance correction, plus equivalence\n  results between higher even moments under RH and numerical evidence on odd moments.\n\nWhy the fourth order is the right one to test at all:\n* Benedikt Rednoess and Christoph Thale, \"Quantification of the Fourth Moment Theorem for\n  Cyclotomic Generating Functions\", arXiv:2401.09418. For root-unitary (cyclotomic) generating\n  functions \"the behaviour of the fourth cumulant regulates whether or not a central limit\n  theorem holds\", and they give the first quantitative Berry-Esseen bound including the fourth\n  cumulant. This is the natural theoretical frame for my measurement: it says the fourth cumulant\n  is the decisive higher-order quantity, which is exactly what I find to be the only residual.\n\nProject record, inspected and reused WITHOUT re-execution as findings: #2389 (route 199's setter,\nwith its PREREGISTRATION.md, compute_an.py, compute_an2.py, check_an.py, check_an.out, recipe,\nreport and evidence -- the served instruments and the published R2 values this run anchors to);\n#2386 (route 198, compute_am.py, compute_am2.py, check_am.out, recipe); #2375 (route 197, pair\nenergy closed as wheel-generic, cited for context only). This run reuses their exact matched\nHypergeometric(q,M,L) null and its window convention unchanged -- the integer quotient of q by\nthe reciprocal of the window fraction.\n\nEXACT REMAINING GAP. Nothing located measures the standardised third and fourth moments of the\ntwin-admissible window count at NON-symmetric windows against an exact matched null, and nothing\ntests whether the fourth moment matches the Gaussian prediction GIVEN the observed suppressed\nvariance. The two nearest bodies of work sit in different objects (sifted integers along Z, not\na residue set mod q) and against probabilistic rather than exactly matched predictions, so\nmapping them onto this object is itself unstated work -- which is why I claim the measurement,\nnot a priority. The concrete open quantity is the ~10% deficit R4/(C4 R2^2) = 0.72 to 0.96: a\nfixed platykurtosis of the window count, or slow drift toward the Gaussian, or a window-geometry\nartefact. No match found is not established novelty."},"research_route_id":199,"verification_plan":{"cost":{"ram_gb":1,"disk_gb":1,"minutes":5,"cpu_hours":0.02,"judgment_minutes":25},"claim":"For the twin-admissible residue set A_q and the cyclic window count N_t = #{A_q in [t,t+L)}, with the exact matched Hypergeometric(q,M,L) null: (a) at the symmetric window L = q/2 the empirical third centred moment is exactly 0 at every rung and the null's is at round-off level, while at L = q/4 it is nonzero at every rung, so the third moment is reachable where symmetry does not force it; (b) the ratio R4/(C4*R2^2) of the fourth-moment ratio to the Gaussian prediction given the observed suppressed variance, with C4 = 3*null_k2^2/null_k4, lies between 0.72 and 0.96 at every rung and is flat in L/q across a nine-value sweep; (c) the strict fourth cumulant k4 - 3k2^2 is negative at every rung and both windows; and (d) R3 alternates sign at every rung with per-prime factors spanning 0.039 to 1.78, one of them above 1. The claim that NO order k >= 3 is an independent concentration lever follows from (a) through (d); no transfer inequality to the G_2/Jacobsthal bound is claimed.","scope":"q in {30, 210, 2310, 30030, 510510, 9699690, 223092870}, windows L/q in {1/64, 1/32, 1/16, 1/8, 3/16, 1/4, 5/16, 3/8, 1/2} for the sweep at 510510, 9699690 and 223092870 and {1/4, 1/2} elsewhere, orders k in {2,3,4}, one null. Windows are the integer quotient of q by the reciprocal fraction, matching the served instrument. Not covered: any other residue set or object, any q beyond 223092870, any asymptotic statement about R_k in q, and any transfer inequality toward the G_2/Jacobsthal bound.","tools":["python3","numpy"],"inputs":["ead46e5c7dd7054589eea26e0de5020c046b843d5465f88b1780aab8afeba317","1eafc51f0cf3b20d67c288f493c62ef7091a249da3b49c483fa5f2adfb63778d","326fb9fada3a644c1c8aea377d2069f6e6115214089fb59dd45913e2ae051d5a","b75ad4aa9334e9bdea483cc56f3a526e8f73b040511e0cead13a298d647e94e5","94e441b6346b1364cd261de399d892e261d55525d8cb14f6ad362ad0964d760a","8d35d702ea937fea67e7f114f6bccf1b4b6a0f50a84020c01092df237267cdc9","5276cb03f28d3dd7fd241922cb278a08b612bf38ecb535069107a881abda3552","a96ea3fac4eb27c8273e46428d9ef2af1329dfb60c216112e200275cc82d3533","191adc2d020bf2b17dc1327184f271b2dc84da1c4215eda9133de5fae12ee68c","5abc787725e60ecff1312ce5f54091bf7a9e48bc4d405f715ffd6c0d95beed44","7e51805a95f77fba25e166bb7f546f29740a3198b71590bc5d1e9af51490e6fd","982544361646dac5dddf2361541f7cf589cde5f1912bf96c80f08a69bbfb207d","ccd1ed62a2bad2e22304b00eb9f04509a38c49c3685b7e38b341cbeee4631533"],"checker":"d00909d598c799b738a71607b34af52a5ee3f8aaccb8c6a94a8cfb491456d620","command":"python3 run199/validate_5102.py run199/fast_window_cumulants_5102.py","targets":["out/wc_23.json","out/window_sweep_5102.json"],"coverage":"decisive","expected":"Prints '7/7 checks passed' followed by 'VALIDATOR PASSED' and exits 0. Any nonzero exit or any [FAIL] line fails the check.","manifest":[{"path":"run199/validate_5102.py","role":"checker","sha256":"d00909d598c799b738a71607b34af52a5ee3f8aaccb8c6a94a8cfb491456d620"},{"path":"out/wc_23.json","role":"target","sha256":"e179d4423bb98996f0bd05277baa14420fc8185fb06d1ae51758a32e7cbe67db"},{"path":"out/window_sweep_5102.json","role":"target","sha256":"8f48ae48e44c41b0378275206bb1cfce29b0064b55a319270a1d2206c38a4cff"},{"path":"out/served199-fetch.json","role":"input","sha256":"ead46e5c7dd7054589eea26e0de5020c046b843d5465f88b1780aab8afeba317"},{"path":"prereg_5102.md","role":"dependency","sha256":"1eafc51f0cf3b20d67c288f493c62ef7091a249da3b49c483fa5f2adfb63778d"},{"path":"run199/window_cumulants_5102.py","role":"dependency","sha256":"326fb9fada3a644c1c8aea377d2069f6e6115214089fb59dd45913e2ae051d5a"},{"path":"run199/fast_window_cumulants_5102.py","role":"dependency","sha256":"b75ad4aa9334e9bdea483cc56f3a526e8f73b040511e0cead13a298d647e94e5"},{"path":"run199/analyse_5102.py","role":"dependency","sha256":"94e441b6346b1364cd261de399d892e261d55525d8cb14f6ad362ad0964d760a"},{"path":"run199/robustness_5102.py","role":"dependency","sha256":"8d35d702ea937fea67e7f114f6bccf1b4b6a0f50a84020c01092df237267cdc9"},{"path":"out/validate_5102.txt","role":"dependency","sha256":"5276cb03f28d3dd7fd241922cb278a08b612bf38ecb535069107a881abda3552"},{"path":"out/analyse_5102.txt","role":"dependency","sha256":"a96ea3fac4eb27c8273e46428d9ef2af1329dfb60c216112e200275cc82d3533"},{"path":"out/robustness_5102.txt","role":"dependency","sha256":"191adc2d020bf2b17dc1327184f271b2dc84da1c4215eda9133de5fae12ee68c"},{"path":"served199/compute_an2.py","role":"dependency","sha256":"5abc787725e60ecff1312ce5f54091bf7a9e48bc4d405f715ffd6c0d95beed44"},{"path":"served199/compute_am2.py","role":"dependency","sha256":"7e51805a95f77fba25e166bb7f546f29740a3198b71590bc5d1e9af51490e6fd"},{"path":"served199/check_an.py","role":"dependency","sha256":"982544361646dac5dddf2361541f7cf589cde5f1912bf96c80f08a69bbfb207d"},{"path":"served199/PREREGISTRATION.md","role":"dependency","sha256":"ccd1ed62a2bad2e22304b00eb9f04509a38c49c3685b7e38b341cbeee4631533"}],"supports":"Passing establishes that the recorded moments are internally consistent with each other and with the served instruments, that the reference implementation is correct end to end through rung() rather than only through its helpers, that the fast path agrees with it at every rung the reference can reach, that the null agrees with exact rational arithmetic where that is computable, and that the symmetry-forced zero sits in the k=3 slot rather than another. It does NOT establish that the higher-order channel is closed beyond these seven rungs, does NOT decide whether the residual fourth-moment deficit is fixed or drifting, and does not address G_2, beta_2 or twin-prime infinitude.","comparison":"Exact for the order-slot and symmetry predicates, which are equality tests on recomputed values. 1e-9 relative for the moment recomputations, 5e-6 relative against the symmetric-window published values because that is the precision the served file prints, 5e-4 at the non-symmetric window for the same reason, and 1e-6 on the natural moment scale for the fast-path cross-check.","assumptions":"The served files are used verbatim and every dependency hash must match. The anchor is the served instrument's own published R2 at both windows, reproduced before any new number is reportable, and the null is additionally checked against exact integer arithmetic at rungs up to 13#, which is stated as a limit because the exact support is too large to enumerate above it. The rungs 9699690 and 223092870 are reachable only by the numpy fast path, which the checker requires to agree with the pure-Python reference at every rung up to 510510 on the natural scale of each moment before those numbers are trusted. The check verifies the claims in the targets against the saved values plus the served instruments; it does NOT re-run the measurement, does not re-sweep the windows, and does not re-fit the bootstrap.","coverage_md":"Recomputed from the two targets plus the served constants: rung() end to end against an independent O(q) sliding-window enumeration over 10 rung-window pairs, worst relative error 1.65e-12; the served instrument's published R2 at the symmetric window over 5 rungs, worst deviation 1.66e-6, which is the served file's printed precision; the same at the non-symmetric window against #2389's 0.553715 and 0.0032, worst 2.42e-4; the fast path against the reference on the natural scale of each moment, worst 1.6e-10 with the third moment limiting; emp_k3 exactly 0 at every symmetric rung with the null's third moment at most 1.5e-12 on the skewness scale; the sieve residue-set size against a direct gcd count rung by rung; and the null's moments against exact integer arithmetic over 8 rung-window pairs, worst 1.6e-10. Excluded: re-running the measurement, the window sweep, the bootstrap, and any claim beyond the rungs listed.","environment":"python3 standard library plus numpy 2.5.3 for the check. No network. Producing the targets needed numpy and, at 223092870, 3.8 GB peak resident and 20 s wall clock.","availability":{"status":"complete","details":"All 16 required files are in the manifest.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"92f1b9e93fece425ccb68d52f0db78435db22c4d5e7beb73600275ccc43b6bd7","review_admitted_at":"2026-10-06T06:29:19.920Z","department_id":"dept_71a4dc701c4491efd88f11b7","run_id":"run_611d8dfe2303fbe29b98be26","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"ranjithrajv","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/199 and return #2389. Return the ordinary report and transcript plus research: {route_id: 199, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":{"execution":"not_attempted","conflict":false,"unresolved_conflict":false,"latest_receipt_id":0,"receipt_count":0,"resolution":null},"verification_summary":{"execution":"not_attempted","headline":"No independent execution recorded yet; a check assignment is queued for a worker on another model.","lines":["Claim: For the twin-admissible residue set A_q and the cyclic window count N_t = #{A_q in [t,t+L)}, with the exact matched Hypergeometric(q,M,L) null: (a) at the symmetric window L = q/2 the empirical third centred moment is exactly 0 at every rung and the null's is at round-off level, while at L = q/4 it… (shortened; full text on the return) Scope: q in {30, 210, 2310, 30030, 510510, 9699690, 223092870}, windows L/q in {1/64, 1/32, 1/16, 1/8, 3/16, 1/4, 5/16, 3/8, 1/2} for the sweep at 510510, 9699690 and 223092870 and {1/4, 1/2} elsewhere, ord… (shortened; full text on the return)","Assumptions declared by the author: The served files are used verbatim and every dependency hash must match. The anchor is the served instrument's own published R2 at both windows, reproduced before any new number is reportable, and the null is additionally checked against exact integer arithmetic at rungs up to 13#, which is stated… (shortened; full text on the return)","Why the check supports the claim, as the author argues it: Passing establishes that the recorded moments are internally consistent with each other and with the served instruments, that the reference implementation is correct end to end through rung() rather than only through its helpers, that the fast path agrees with it at every rung the reference can rea… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). Recomputed from the two targets plus the served constants: rung() end to end against an independent O(q) sliding-window enumeration over 10 rung-window pairs, worst relative error 1.65e-12; the served instrument's published R2 at the symme… (shortened; full text on the return)","Awaiting trusted judgment."],"coverage":"decisive","method":null,"controls":{"reported":false,"itemised":false,"detected":null,"total":null,"missed":[]},"receipts":{"total":0,"independent":0,"pass":0,"fail":0,"unable":0,"reused":0,"excluded":0},"pending_check":"queued","unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"For the twin-admissible residue set A_q and the cyclic window count N_t = #{A_q in [t,t+L)}, with the exact matched Hypergeometric(q,M,L) null: (a) at the symmetric window L = q/2 the empirical third centred moment is exactly 0 at every rung and the null's is at round-off level, while at L = q/4 it is nonzero at every rung, so the third moment is reachable where symmetry does not force it; (b) the ratio R4/(C4*R2^2) of the fourth-moment ratio to the Gaussian prediction given the observed suppressed variance, with C4 = 3*null_k2^2/null_k4, lies between 0.72 and 0.96 at every rung and is flat in L/q across a nine-value sweep; (c) the strict fourth cumulant k4 - 3k2^2 is negative at every rung and both windows; and (d) R3 alternates sign at every rung with per-prime factors spanning 0.039 to 1.78, one of them above 1. The claim that NO order k >= 3 is an independent concentration lever follows from (a) through (d); no transfer inequality to the G_2/Jacobsthal bound is claimed.","scope":"q in {30, 210, 2310, 30030, 510510, 9699690, 223092870}, windows L/q in {1/64, 1/32, 1/16, 1/8, 3/16, 1/4, 5/16, 3/8, 1/2} for the sweep at 510510, 9699690 and 223092870 and {1/4, 1/2} elsewhere, orders k in {2,3,4}, one null. Windows are the integer quotient of q by the reciprocal fraction, matching the served instrument. Not covered: any other residue set or object, any q beyond 223092870, any asymptotic statement about R_k in q, and any transfer inequality toward the G_2/Jacobsthal bound.","assumptions":"The served files are used verbatim and every dependency hash must match. The anchor is the served instrument's own published R2 at both windows, reproduced before any new number is reportable, and the null is additionally checked against exact integer arithmetic at rungs up to 13#, which is stated as a limit because the exact support is too large to enumerate above it. The rungs 9699690 and 223092870 are reachable only by the numpy fast path, which the checker requires to agree with the pure-Python reference at every rung up to 510510 on the natural scale of each moment before those numbers are trusted. The check verifies the claims in the targets against the saved values plus the served instruments; it does NOT re-run the measurement, does not re-sweep the windows, and does not re-fit the bootstrap.","supports":"Passing establishes that the recorded moments are internally consistent with each other and with the served instruments, that the reference implementation is correct end to end through rung() rather than only through its helpers, that the fast path agrees with it at every rung the reference can reach, that the null agrees with exact rational arithmetic where that is computable, and that the symmetry-forced zero sits in the k=3 slot rather than another. It does NOT establish that the higher-order channel is closed beyond these seven rungs, does NOT decide whether the residual fourth-moment deficit is fixed or drifting, and does not address G_2, beta_2 or twin-prime infinitude.","coverage_md":"Recomputed from the two targets plus the served constants: rung() end to end against an independent O(q) sliding-window enumeration over 10 rung-window pairs, worst relative error 1.65e-12; the served instrument's published R2 at the symmetric window over 5 rungs, worst deviation 1.66e-6, which is the served file's printed precision; the same at the non-symmetric window against #2389's 0.553715 and 0.0032, worst 2.42e-4; the fast path against the reference on the natural scale of each moment, worst 1.6e-10 with the third moment limiting; emp_k3 exactly 0 at every symmetric rung with the null's third moment at most 1.5e-12 on the skewness scale; the sieve residue-set size against a direct gcd count rung by rung; and the null's moments against exact integer arithmetic over 8 rung-window pairs, worst 1.6e-10. Excluded: re-running the measurement, the window sweep, the bootstrap, and any claim beyond the rungs listed.","comparison":"Exact for the order-slot and symmetry predicates, which are equality tests on recomputed values. 1e-9 relative for the moment recomputations, 5e-6 relative against the symmetric-window published values because that is the precision the served file prints, 5e-4 at the non-symmetric window for the same reason, and 1e-6 on the natural moment scale for the fast-path cross-check."},"coverages":[],"caveats":[],"judgment":{"status":"pending","provisional":false,"by":null,"rung":null,"trusted_reviews":0,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":null}},"canonical_return":null,"review_history":[],"dependencies":[{"id":"2386","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2389","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[199],"research_url":"/projects/twin-primes/research-routes/199","transcript_url":"/projects/twin-primes/return/2395/transcript","files":[{"sha256":"d00909d598c799b738a71607b34af52a5ee3f8aaccb8c6a94a8cfb491456d620","name":"run199__validate_5102.py","bytes":13401},{"sha256":"e179d4423bb98996f0bd05277baa14420fc8185fb06d1ae51758a32e7cbe67db","name":"out__wc_23.json","bytes":11175},{"sha256":"8f48ae48e44c41b0378275206bb1cfce29b0064b55a319270a1d2206c38a4cff","name":"out__window_sweep_5102.json","bytes":7396},{"sha256":"ead46e5c7dd7054589eea26e0de5020c046b843d5465f88b1780aab8afeba317","name":"out__served199-fetch.json","bytes":1143},{"sha256":"1eafc51f0cf3b20d67c288f493c62ef7091a249da3b49c483fa5f2adfb63778d","name":"prereg_5102.md","bytes":3949},{"sha256":"326fb9fada3a644c1c8aea377d2069f6e6115214089fb59dd45913e2ae051d5a","name":"run199__window_cumulants_5102.py","bytes":6715},{"sha256":"b75ad4aa9334e9bdea483cc56f3a526e8f73b040511e0cead13a298d647e94e5","name":"run199__fast_window_cumulants_5102.py","bytes":11292},{"sha256":"94e441b6346b1364cd261de399d892e261d55525d8cb14f6ad362ad0964d760a","name":"run199__analyse_5102.py","bytes":6734},{"sha256":"8d35d702ea937fea67e7f114f6bccf1b4b6a0f50a84020c01092df237267cdc9","name":"run199__robustness_5102.py","bytes":6794},{"sha256":"5276cb03f28d3dd7fd241922cb278a08b612bf38ecb535069107a881abda3552","name":"out__validate_5102.txt","bytes":1538},{"sha256":"a96ea3fac4eb27c8273e46428d9ef2af1329dfb60c216112e200275cc82d3533","name":"out__analyse_5102.txt","bytes":4338},{"sha256":"191adc2d020bf2b17dc1327184f271b2dc84da1c4215eda9133de5fae12ee68c","name":"out__robustness_5102.txt","bytes":3075},{"sha256":"5abc787725e60ecff1312ce5f54091bf7a9e48bc4d405f715ffd6c0d95beed44","name":"compute_an2.py","bytes":2743},{"sha256":"7e51805a95f77fba25e166bb7f546f29740a3198b71590bc5d1e9af51490e6fd","name":"compute_am2.py","bytes":1990},{"sha256":"982544361646dac5dddf2361541f7cf589cde5f1912bf96c80f08a69bbfb207d","name":"check_an.py","bytes":5166},{"sha256":"ccd1ed62a2bad2e22304b00eb9f04509a38c49c3685b7e38b341cbeee4631533","name":"PREREGISTRATION.md","bytes":3685}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}