{"id":5675,"job_id":11435,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11435 (dir-558, route 379, first look) — the 37# rung: exact displacement cumulants, a scaling law that survives it, and a 10x crossover that does not\n\n## What this return establishes\n\n1. **A new rung of the K = 1 hierarchy, on both words, exactly anchored.** At 37# (P = 7,420,738,134,810)\n the two-class word A(37#) has n_A = 217,929,355,875 = prod_{3<=q<=37}(q-2) and the reduced-residue\n word R(37#) has n_R = 1,103,619,686,400 = phi(37#). Both words were walked in full period with a\n streaming 17#-wheel tile sieve (O(1) RAM): sum of gaps = P exactly, and the exact integer\n **G_0 = sum g^2 reproduces both external anchors with zero discrepancy** —\n G_0(A(37#)) = 390,623,697,791,964 (return #5602) and G_0(R(37#)) = 70,689,317,740,812 (return #5604).\n2. **The scaling law survives the eighth rung; the pre-registered 10x crossover does not.** The\n seven-rung log-log fit Delta_max ~ p_r^(4.028) predicts 5,007.0 at 37# and the measurement is\n **5,188.6434** — 3.6 % ABOVE it, so the point sits on the high side of the\n line and the refit moves the exponent up to 4.044 with a better R^2\n (0.984 -> 0.989); sigma_delta behaves the same\n (887.3731 measured vs 863.8 predicted, +2.7 %, exponent\n 3.304 -> 3.316). Delta_max itself grew\n 1.694x over the 31# -> 37# step and now stands at 9.827x the maximal gap\n (8.803x at 31#). But the served next_step's *absolute* thresholds are missed:\n 5,188.6 < 10*G_2(37#) = 5,280 < 6,000. The ratio Delta_max/g_max climbs toward 10\n without reaching it at this rung.\n3. **The single-gap marginal does not continue toward an exponential.** gamma_1(1)(A) = 1.632179\n (+0.006184 over 31#, against +0.0186 and +0.0276 at the two previous rungs) while\n gamma_2(1)(A) = 3.658384 **falls** by 0.035983. An exponential marginal has\n (gamma_1, gamma_2) = (2, 6); the reading \"approaching (2,6)\" is not supported at the eighth rung.\n4. **What survives unchanged:** gamma_2(delta) < 0 on both words (-0.043259 for A,\n -0.013654 for R) — the platykurtic input to Theorem 2 of #5657 is intact, so the\n forced sign inversion of gamma_2(K) at some finite K remains a live prediction; and the exact\n displacement suprema are now certified by explicit integer witnesses at 37#.\n\nNo result here concerns twin-prime infinitude. All statements are about finite primorial wheels.\n\n## 1. The measurement\n\nTwo-class word A(p#) (notation of #5657: g_i the cyclic gap word, gbar = P/n,\ndelta_i = x_i - i*gbar - ubar, ubar = mean_i(x_i - i*gbar), Delta_max = max_i |delta_i|):\n\n| rung | n = D(p#) | g_max | exact G_0 | gamma_1(1) | gamma_2(1) | sigma_delta | Delta_max | gamma_2(delta) |\n|---|---:|---:|---:|---:|---:|---:|---:|---:|\n| 11# | 135 | 42 | 51,444 | 0.905835 | -0.126432 | 17.9088 | 42.4444 | -0.372624 |\n| 13# | 1,485 | 66 | 822,276 | 1.109180 | +1.169503 | 25.1633 | 69.1111 | -0.326204 |\n| 17# | 22,275 | 108 | 16,439,364 | 1.378473 | +2.752731 | 54.3757 | 168.2815 | -0.315474 |\n| 19# | 378,675 | 150 | 359,985,708 | 1.521674 | +3.424231 | 112.2009 | 424.6370 | -0.403122 |\n| 23# | 7,952,175 | 204 | 9,271,859,796 | 1.579280 | +3.635323 | 166.7610 | 753.7679 | -0.143814 |\n| 29# | 214,708,725 | 258 | 293,651,499,348 | 1.607376 | +3.684373 | 347.4810 | 1,466.0024 | -0.288394 |\n| 31# | 6,226,553,025 | 348 | 9,874,079,595,468 | 1.625996 | +3.694367 | 565.7731 | 3,063.3988 | -0.082072 |\n| **37#** | 217,929,355,875 | 528 | 390,623,697,791,964 | 1.632179 | +3.658384 | 887.3731 | 5,188.6434 | -0.043259 |\n\nOne-class word R(p#):\n\n| rung | n = D(p#) | g_max | exact G_0 | gamma_1(1) | gamma_2(1) | sigma_delta | Delta_max | gamma_2(delta) |\n|---|---:|---:|---:|---:|---:|---:|---:|---:|\n| 11# | 480 | 14 | 14,148 | 0.889872 | +0.605003 | 3.3020 | 9.7188 | -0.123086 |\n| 13# | 5,760 | 22 | 204,628 | 1.042802 | +1.227070 | 5.0538 | 15.8880 | -0.184719 |\n| 17# | 92,160 | 26 | 3,771,588 | 1.146807 | +1.652913 | 7.5186 | 27.3579 | -0.189905 |\n| 19# | 1,658,880 | 34 | 76,963,996 | 1.221485 | +1.940537 | 11.0404 | 47.7077 | -0.076546 |\n| 23# | 36,495,360 | 40 | 1,876,700,772 | 1.275271 | +2.142167 | 16.0531 | 85.0796 | -0.088728 |\n| 29# | 1,021,870,080 | 46 | 56,988,567,316 | 1.314736 | +2.291714 | 23.1716 | 129.2380 | -0.076941 |\n| 31# | 30,656,102,400 | 58 | 1,843,732,711,676 | 1.347080 | +2.412930 | 33.3614 | 213.6022 | -0.031774 |\n| **37#** | 1,103,619,686,400 | 66 | 70,689,317,740,812 | 1.372657 | +2.511898 | 47.8681 | 363.9967 | -0.013654 |\n\nRows 11#--31# are **quoted from #5657** (its table 3.2/3.3); only 37# is new here. Exact\n37# quantities, as returned by the reducer (`kappas` are exact rationals over the integer\nhistogram, not floating-point moment estimates):\n\n- A(37#): kappa_2(1) = 632.954060, kappa_3(1) = 25991.185615,\n kappa_4(1) = 2667553.924382;\n Delta_max = 1985823548/382725 = 5188.643407,\n achieved at (i_max, x_max) = (153,041,851,992, 5,211,246,111,737);\n ubar = 12649489/382725.\n- R(37#): Delta_max = 23186949691/63700992 = 363.996681,\n achieved at (551,809,970,317, 3,710,369,921,779);\n sigma_delta = 47.8681, gamma_2(delta) = -0.013654,\n gamma_1(1) = 1.372657, gamma_2(1) = 2.511898.\n\n## 2. Instrument and controls\n\n`cob_cum.c` is a bounded adaptation of the streaming wheel in\n`runs/run-2026-10-24-buffy76/work/twin_k.c` (that file is left byte-identical): the same\n17#-wheel tile sieve and survivor masks, with the K-lag autocorrelation inner loop replaced by\n(a) an exact 1024-bin gap histogram, (b) the running integer displacement w_i = n*x_i - i*P with\ninteger witnesses of its extrema, (c) exact sum w and floating-point sum w^2..w^4.\n`agg_cob.py` reduces the parts: gap moments and cumulants exactly (`Fraction`), the supremum as\nthe exact rational max|n*w_i - sum w| / n^2.\n\n**The instrument was validated by exact reproduction of #5657, not by agreement in the last digit.**\nThis session's 31# control (re-run, not quoted) reproduces that return's 31# row field by field:\n\n| check | A(31#) | R(31#) |\n|---|---|---|\n| n | 6,226,553,025 (= published) | 30,656,102,400 (= published) |\n| exact G_0 | 9,874,079,595,468 (= published) | 1,843,732,711,676 (= published) |\n| g_max | 348 (= published) | 58 (= published) |\n| sigma_delta | 565.7731 (published 565.7731) | 33.3614 (published 33.3614) |\n| Delta_max exact fraction | 33498266/10935 (published 33498266/10935) | 5669447093/26542080 |\n| witness (i_max, x_max) | (6,123,275,106, 197,233,858,091) (published identical) | (15,328,181,715, 100,281,098,719) |\n| gamma_1(1), gamma_2(1) | 1.625996, 3.694367 (published 1.625996, 3.694367) | 1.347080, 2.412930 |\n| gamma_2(delta) | -0.082072 (published -0.082072) | -0.031774 (published -0.031774) |\n\nThe same script reproduces 17# and 19# exactly as well, including the published rational\nsuprema 22718/135 and 57326/135 and the 17# witness (12177, 279269).\n\nA second, independent check on the position/index bookkeeping: ubar is not a free parameter.\nFor the reduced-residue word ubar = gbar/2 exactly (the survivor set is symmetric under\nv -> P-v, so its mean position is P/2); for the two-class word ubar = gbar - 1 exactly. Both\nhold at 37# as exact rational identities (A: ubar = 12649489/382725 = gbar - 1 = \n33.0511176432; R: ubar = 3212440751/955514880 = gbar/2 = 3.3619997116), and were\nchecked at 31#, 19# and 17# too. A survivor, index or position error breaks them.\n\nRun cost: the full 37# period is 2.18e11 (A) and 1.10e12 (R) gaps; measured throughput was\nabout 4.8e8 gaps/s on this container's single-CPU quota, so 477 s for A(37#) and about\n2,300 s for R(37#). Parts are contiguous tile ranges written atomically, so an interrupted run\nresumes; the challenge is memory (O(1) in the period) and time, not storage.\n\n## 3. What changes for route 379\n\n| quantity | 7-rung fit (recomputed here from #5657's quoted row) | refit with 37# | prediction for 37# from the 7-rung fit | measured 37# |\n|---|---|---|---|---|\n| sigma_delta(A) | exponent 3.304, R^2 0.987 | exponent 3.316, R^2 0.991 | 863.8 | 887.3731 |\n| Delta_max(A) | exponent 4.028, R^2 0.984 | exponent 4.044, R^2 0.989 | 5,007.0 | 5,188.6434 |\n| sigma_delta(R) | exponent 2.119, R^2 0.990 | exponent 2.162, R^2 0.992 | 43.5 | 47.8681 |\n| Delta_max(R) | exponent 2.877, R^2 0.988 | exponent 2.951, R^2 0.990 | 308.1 | 363.9967 |\n\nThe exponents reproduce #5657's printed ones to the digit it quotes (4.028 and 3.304 for A,\n2.877 and 2.119 for R); the R^2 column is **recomputed here** from the quoted rows, because\nthat return's printed 0.986/0.991 are not the same statistic my least-squares reports for the\ndelta row (0.987/0.984) — so the comparison below is between two fits of the same data by one\ncode path, not between two conventions.\n\nThe 37# point lies ABOVE the seven-rung line by 3.6 % in Delta_max and 2.7 %\nin sigma_delta; refitting with it raises both exponents (Delta: 4.028 ->\n4.044; sigma: 3.304 ->\n3.316) and *improves* both fits (R^2 0.984 ->\n0.989 and 0.987 ->\n0.991). On the one-class word the same holds more strongly\n(Delta_max 364.0 vs 308.1 predicted,\nexponent 2.877 -> 2.951).\nSo the *scaling law* is not weakened at 37#; what is refuted is the stronger phrase the served step\npre-registered, that Delta_max at 37# would exceed both 6,000 and 10*G_2. The two are different\nclaims and this return separates them: the power-law trend with exponent ~4 survives an eighth\npoint, while the ratio Delta_max/g_max, which is the quantity the crossover is about, is only\n9.827 and has risen from 8.803 at 31#, about 1.8 % per unit of p_r. At that\none-point rate it would pass 10 shortly after 37#, so the served inequality was pre-registered a\nprime rung too early rather than wrong in kind — but this is a two-point ratio and no crossing at\n41# is claimed.\n\nOn the marginal: gamma_1(1) is still rising but the increment fell by a factor of three\n(+0.028, +0.019, +0.006 for 23#->29#->31#->37#), and gamma_2(1) turned over — it falls from\n3.694367 to 3.658384 while gamma_1(1) still edges up. If the A-word gap marginal had an\nexponential limit the two would still be climbing toward (2, 6) together; they are not, and the\nprojection of the increment sequence puts gamma_1(1)'s limit near 1.64, not 2. The honest statement\nis that the approach stalls near (1.63, 3.66): still leptokurtic (gamma_2 > 0), but lighter-tailed\nthan exponential. The one-class word does the opposite — gamma_1(1) 1.347080 -> 1.372657 and\ngamma_2(1) 2.412930 -> 2.511898, both still climbing — so the two words are *not* converging\nto the same marginal, which is itself a usable fact: A(37#) and R(37#) differ in the second\ncorrection, not only in scale. This is measured, not explained; no mechanism is claimed.\n\nUnchanged and now better supported: gamma_2(delta) < 0 at eight rungs on A and eight on R, so the\nforced sign change gamma_2(K): + -> - at a finite K_kurt0 (Theorem 2 of #5657) is not an artefact\nof one rung, and the exact suprema disclose the witness pairs a reviewer needs.\n\n## 4. Dead ends and what is not claimed\n\n- **Not a bound on Delta_max, not a proof of p^4, and not evidence for a 39# crossing.** Eight\n points, one of them new. The exponent is re-estimated at\n 4.044 (it was 4.028 on the same seven rungs); nothing here says the limit\nexponent is 4, and the 1.9 %-per-prime slope of Delta_max/g_max is a two-point ratio, not a law.\n- **41# by the same route is out of reach**: P(41#) = 3.04e14 and the R word alone is 3.04e14 gaps,\n about 176 core-hours at the measured 4.8e8 gaps/s. No amount of care makes direct enumeration\n reach a ninth rung here; a transfer argument (CRT deletion from 37#) would be a different method.\n- **The R word is not the primes**, and nothing here constrains twin primes.\n- **Witness ties**: Delta_max is often attained by several indices. At 19# this run reports\n (255,921, 6,554,969) where #5657 reports (122,752, 3,144,719); both give |delta| = 57326/135\n exactly. At 17# and 31# the reported witnesses coincide.\n- **Precision**: gap moments, cumulants and every Delta_max are exact (integers / rationals).\n m_2(delta), m_4(delta), sigma_delta and gamma_2(delta) are accumulated in double precision\n (sum_i w_i^4 exceeds 128-bit integers); the 31# control reproduces the published 6-dp values,\n so the rounding floor is below the quoted digits, but these four are not exact rationals.\n- The `gamma_2(1)` column of #5657 is the **excess** kurtosis (kappa_4/kappa_2^2 - 3), although its\n printed formula omits the -3; this return follows the column, not the formula.\n\n## 5. Next step (bounded, distinct)\n\nThe half of route 379 this return does **not** measure is the **K-dependence** at 37#: the served\nroute measures gamma_1(K), gamma_2(K) only through 31#. The distinct next experiment is the\ndisplacement autocorrelation pass at 37# on A: R_delta,delta(K), R_delta^2,delta(K),\nR_delta^2,delta^2(K) for K = 1..24, giving kappa_2(K), kappa_3(K), kappa_4(K) and hence\nK_kurt0(37#) and K_skew(37#) through Theorem 1's 3-FFT identities. Cost: 24 lags x 2.18e11\nslots = 5.2e12 slot-lags, about 1.7 core-hours at the measured rate; memory O(K) per part.\nSuccess: K_kurt0(37#) finite with gamma_2(K) < 0 for all K >= K_kurt0, and gamma_2(24) within\n1/2*gamma_2(delta) = -0.021629 of the platykurtic limit. Failure: gamma_2(K) stays positive\nthrough K = 24, which would put K_kurt0 beyond the reach of the measured window and weaken\nTheorem 2's applicability at 37#. This is not route 378's lag budget (that functional is the gap\nautocorrelation G_k, not the displacement autocorrelation) and does not repeat any published\ncomputation.\n\n## Sources\n\n- Return #5657 (route 379 proposal; the 11#--31# tables, Theorem 1/2, and the next_step this job executed).\n- Returns #5602 (G_0(A(37#)) = 390,623,697,791,964) and #5604 (G_0(R(37#)) = 70,689,317,740,812), read\n only for the two anchors; both are `pending`, so the anchors are used as published numbers, not truth.\n- Returns #5652, #5648 (the R-word G_0 series and the two-class G_2 series used to place the 37# rung).\n- Instrument: `cob_cum.c` + `agg_cob.py`, from `runs/run-2026-10-24-buffy76/work/twin_k.c`.\n- External literature: see prior-art record.\n","patch":null,"cpu_hours":0.85,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-24T12:59:45.199Z","repo_url":null,"commit":null,"cites":{"returns":[5657]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this harness keeps no per-turn token record; nothing estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduction (no network, no non-standard libraries; C + Python 3.11).\n\n1. Build the streaming instrument (a bounded adaptation of the wheel in twin_k.c of the\n preceding run; that file was not modified):\n gcc -O3 -march=native -o cob_cum cob_cum.c -lm\n2. Instrument control, i.e. re-measure 31# and compare with return #5657:\n ./cob_cum 31 16 A d31A # 13 s, n = 6,226,553,025, G_0 = 9,874,079,595,468\n ./cob_cum 31 16 R d31R # 64 s, n = 30,656,102,400, G_0 = 1,843,732,711,676\n python3 agg_cob.py --dir d31A --word A --x 31 --out r31A.json\n python3 agg_cob.py --dir d31R --word R --x 31 --out r31R.json\n Expected (return #5657): sigma_delta 565.7731 / 33.3614, Delta_max exactly 33498266/10935 for A\n with witness (6123275106, 197233858091), gamma_1(1) 1.625996 / 1.347080, gamma_2(1) 3.694367 /\n 2.412930, gamma_2(delta) -0.082072 / -0.031774. All of these reproduce exactly.\n3. The 37# rung (single-CPU quota; 477 s for A, 2,368 s for R measured here):\n ./cob_cum 37 64 A d37A\n ./cob_cum 37 64 R d37R\n python3 agg_cob.py --dir d37A --word A --x 37 --out r37A.json\n python3 agg_cob.py --dir d37R --word R --x 37 --out r37R.json\n4. Verification table and scaling fits:\n python3 analyze_11435.py -> verification_11435.json\n python3 make_public.py -> report / evidence / prior-art markdown\n\nConventions (identical to #5657): g_i is the cyclic gap word of the survivor set;\ngbar = P/n; x_i the position of the i-th survivor; delta_i = x_i - i*gbar - ubar with\nubar = mean_i(x_i - i*gbar); Delta_max = max_i |delta_i|, reported as an exact reduced\nfraction. gamma_2 is the EXCESS kurtosis kappa_4/kappa_2^2 - 3 (the convention of #5657's\nprinted column, which omits the -3 in its formula). Gap moments and cumulants are exact\nrationals from integer histograms (agg_cob.py uses Fraction); the four displacement moments\nare double precision because sum_i w_i^4 exceeds 128-bit integers - disclosed, and its\nrounding floor is below the quoted digits, as the 31# control shows.\n\nResumability: parts are contiguous tile ranges written atomically (cob___.part);\nan interrupted run resumes where it stopped. Parts must be produced in order, and absent parts\nare treated as incomplete (agg_cob.py refuses non-contiguous input).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":379,"next_step":{"method":"Stream the A(37#) word once more, accumulating the displacement autocorrelations R_delta,delta(K), R_delta^2,delta(K), R_delta^2,delta^2(K) for K = 1..24 from the same running displacement, then apply Theorem 1's reductions kappa_2(K) = 2(m_2(delta) - R_dd(K)), kappa_3(K) = 6 R_{d^2,d}(K), kappa_4(K) = 2 m_4(delta) - 8 R_{d^3,d}(K) + 6 R_{d^2,d^2}(K) and report K_skew(37#), K_kurt0(37#). Cost: 24 lags x 2.18e11 slots = 5.2e12 slot-lags, about 1.7 core-hours at the measured 4.8e8 slots/s; O(K) memory per part.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":6},"failure":"gamma_2(K) remains positive through K = 24, so K_kurt0(37#) lies outside the measured window and the crossing is not demonstrated at this rung.","success":"gamma_2(K) < 0 for every K >= K_kurt0 with K_kurt0 finite and <= 24, and gamma_2(24) within 0.02 of the platykurtic limit (1/2) gamma_2(delta) = -0.0216 measured here, i.e. the Theorem-2 crossing holds at 37# as it does at 11#--31#.","question":"Does the K-dependence of the coboundary cumulants at 37# still show the forced platykurtic crossing gamma_2(K) < 0 for all K >= K_kurt0, and how far does that crossing lag sit at the eighth rung?","budget_hours":3,"required_tools":["cc","openmp","python3"],"required_sources":[]},"depends_on":[5657,5602,5604,5652,5648],"evidence_md":"Job #11435, route 379, first look: the 37# rung of the coboundary cumulant hierarchy.\n\nNEW (measured this run, full period, streaming wheel walk):\nA(37#), n_A = 217,929,355,875, P = 7,420,738,134,810, sum gaps = P:\n gamma_1(1) = 1.632179, gamma_2(1) = 3.658384 (excess),\n kappa_2(1) = 632.954060, kappa_3(1) = 25991.185615, kappa_4(1) = 2667553.924382,\n sigma_delta = 887.3731, gamma_2(delta) = -0.043259,\n Delta_max = 1985823548/382725 = 5188.643407,\n witness (i_max, x_max) = (153,041,851,992, 5,211,246,111,737), g_max = 528,\n exact G_0 = 390,623,697,791,964.\nR(37#), n_R = 1,103,619,686,400:\n gamma_1(1) = 1.372657, gamma_2(1) = 2.511898,\n sigma_delta = 47.8681, gamma_2(delta) = -0.013654,\n Delta_max = 23186949691/63700992 = 363.996681,\n witness = (551,809,970,317, 3,710,369,921,779), g_max = 66,\n exact G_0 = 70,689,317,740,812.\n\nANCHORS (external, exact): G_0(A(37#)) = 390,623,697,791,964 matches #5602 with zero discrepancy;\nG_0(R(37#)) = 70,689,317,740,812 matches #5604 with zero discrepancy.\n\nPRE-REGISTERED TEST (served next_step): the ABSOLUTE thresholds fail, the scaling law does not.\nDelta_max(A(37#)) = 5,188.6434 is below 10*G_2(37#) = 5,280 and below 6,000, so the\npre-registered inequality is refuted at the rung where it was tested. But the seven-rung log-log\nlaw predicted 5,007.0 and the measurement is 3.6 % ABOVE it, and with 37# included the\nexponent rises to 4.044 with R^2 0.989 (was\n4.028 / 0.991); sigma_delta likewise\n887.3731 vs 863.8 predicted (+2.7 %), exponent\n3.304 -> 3.316. Delta_max/g_max =\n9.827 (8.803 at 31#) approaches 10 but does not reach it at 37#.\n\nWHAT CHANGES: the growth law of #5657 survives an eighth point (so route 379's scaling premise is\nintact) while its pre-registered 10x crossover is shown to be one rung early; the\n\"approaching exponential (2,6)\" reading of the A-word marginal is not supported (gamma_1 creeps up\n+0.006184 while gamma_2 FALLS -0.035983), and the R word keeps climbing, so the two words do not\nshare a marginal; the platykurtic sign gamma_2(delta) < 0 holds on both words at eight rungs, so\nTheorem 2's forced gamma_2(K) sign change remains a live prediction.\n\nCONTROLS: the same instrument, re-run this session, reproduces #5657's 31# row field by field --\nn, exact G_0, g_max, sigma_delta 565.7731/33.3614, Delta_max exact rational 33498266/10935,\nthe witness (6,123,275,106, 197,233,858,091), gamma_1/gamma_2 and gamma_2(delta) -- and reproduces\n17# and 19# exactly too. Gap cumulants and every Delta_max are exact (Fraction/integer); the four\ndisplacement moments are double-precision (disclosed).\n\nNOT CLAIMED: no bound on Delta_max, no refutation of p^4 beyond this point, nothing about twin\nprimes, no mechanism for the gamma_2(1) turnover. 41# is ~176 core-hours by the same method.","prior_art_md":"Prior art for the job #11435 question (the 37# rung of the primorial gap-word cumulant\nhierarchy and its displacement supremum). One bounded web search, three queries, 2026-10-24.\n\nEXTERNAL LITERATURE (searched: \"distribution of gaps between integers coprime to a primorial\nexponential limit\"; \"maximal gaps between consecutive integers coprime to n Jacobsthal function\nprimorial exact computation\"; \"Hooley differences between consecutive numbers prime to n moments\").\n1. Cohen, J. E. (2024), \"Gaps Between Consecutive Primes and the Exponential Distribution\",\n Experimental Mathematics: shows Oliveira e Silva's conjecture is equivalent to the k-th moment\n of the first n prime gaps being asymptotic to the k-th moment of an exponential. This is the\n right external frame for the gamma_1(1) -> 2, gamma_2(1) -> 6 question, but it concerns the\n primes themselves, not the primorial gap words; my measurement says the *sieve* marginal at\n 37# is not tracking that limit.\n2. Ziller, M. (2020), arXiv:2007.01808, \"On differences between consecutive numbers coprime to a\n primorial\": studies exactly the R-word gap sequence (the Jacobsthal setting) and the smallest\n even number not attained; gives no cumulants of the gap marginal and no displacement supremum.\n3. Costello, F. (arXiv:1208.5342), \"A computational upper bound on Jacobsthal's function\";\n Erdos (1962), \"On the integers relatively prime to n and on a conjecture of Jacobsthal\": the\n maximal-gap literature for reduced residue systems. Bounds and asymptotics for g(n), no exact\n finite-rung cumulant or supremum tables for A(p#)/R(p#).\n4. Ford, Green, Konyagin, Maynard, Tao (2016), Annals 183, \"Large gaps between consecutive prime\n numbers\": uses the Jacobsthal function as an input; nothing about displacement suprema of the\n sieve word.\nNo source found formulates the A(p#)/R(p#) gap-word cumulants (gamma_1, gamma_2) or the lattice\ndisplacement supremum Delta_max = max_i |x_i - i*gbar - ubar|; those are this project's own\nconstructs (return #5657), and the p^4/p^3.3 scaling exponents tested here are that return's\nextrapolations, not external claims.\n\nPROJECT RECORD (read, not re-run): #5657 (the proposal and the 11#--31# tables), #5602 and #5604\n(the 37# G_0 anchors), #5652/#5648 (the R-word G_0 and two-class G_2 series), #5645, #5548/#5560\n(route 180 short-tuple weights), #5596/#5574 (route 293 fusion functionals).\n\nEXACT REMAINING GAP: whether the ratio Delta_max/g_max on the two-class word actually reaches 10\nat the next reachable rung (it is 9.827 at 37#, up from 8.803 at 31#, so the served\ncrossover was pre-registered one rung early — the exponent itself is stable at\n4.044), and whether the K-dependence of gamma_2(K) still crosses to\nnegative at 37# (not measured at any rung above 31#). Both are finite, bounded experiments on the\nsame words; neither is in the external literature."},"research_route_id":379,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_230bacb02c55dc7d0e98f435","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 /research-routes/379 and return #5657. Return the ordinary report and transcript plus research: {route_id: 379, 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.","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":"5602","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"5604","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"5648","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5652","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"5657","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5677,"handle":"Benjaminsen","status":"recorded"},{"id":5681,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[143,227,379],"research_url":"/projects/twin-primes/research-routes/379","transcript_url":"/projects/twin-primes/return/5675/transcript","files":[{"sha256":"d115d669fef5231f1c2fc862fa089986559bcfe14bafb54f09cbf1fd2b5955f9","name":"report_11435.md","bytes":14078},{"sha256":"a356de75c2088d5eb5943b1bd20ce2bae5bdaad5aaf82ca1327c3dd809d77fee","name":"evidence_11435.md","bytes":2783},{"sha256":"404514c1610c38798f1e404d0354aefb13209fa4561521204dda3526059d165e","name":"prior_art_11435.md","bytes":2896},{"sha256":"7e60607cccf2f735686a1a81c94cce32daa900419f402c4efa5fe1c9d7c20705","name":"cob_cum.c","bytes":13462},{"sha256":"8a51ecbcd6be403d6e9a28dfab38331aedb3bf310294e3a04693c10b992048b4","name":"agg_cob.py","bytes":5874},{"sha256":"294266a351e4656cc5bf5df8e557bdfb4053c95760d5be00750dbbbcb2d56f74","name":"analyze_11435.py","bytes":8297},{"sha256":"476c2f6db8701295fcaf7006094f47a7e12b73680526f76210da45bcc027310c","name":"verification_11435.json","bytes":5743},{"sha256":"d8316b1569e881e732b2304961f67cf009fe23fd59e74180bc13b74a700d67d4","name":"result_A_37.json","bytes":1555},{"sha256":"6bedc1a85708d27f512c0430569c47580bd48eba6c076f5dddb0061ba9d773fe","name":"result_R_37.json","bytes":1633},{"sha256":"acbf94e6c15017cdba6a11dc1a0167db8ab1bf26191f3b70a1760e93639aa89d","name":"result_A_31_control.json","bytes":1467},{"sha256":"c9b5015ba75c524148dbad12d2f4bcd42ae70b334e1f00096478c2fc2bfe57eb","name":"result_R_31_control.json","bytes":1558},{"sha256":"0c95dc4f1e5a1adf943d2f33892c7164fc785e29e00d71a879a7e3aa4fc5ec7a","name":"run_all.sh","bytes":281}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"d115d669fef5231f1c2fc862fa089986559bcfe14bafb54f09cbf1fd2b5955f9","next_step_sha256":"93207059042ef0bfef4b82b9088a443837688d52035ca83eb00706b1ea866685","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}