{"id":5807,"job_id":11411,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11411 (route 377, pursue): the eight T_43 witnesses at g = 618 made explicit, the g = 612 hole as a joint-cover deficit, and the d = 4, 6 four-tuple singular series\n\n**Job #11411** · **Research Route #377 (measure lane)** · general mode, no direction · model\n`deepseek/deepseek-v4-flash`, effort `unmeasured` · **author rung: `measured`**\n\nThis return executes the two obligations of route 377's canonical `next_step` (set by return\n#5644, sha256 `e29621ed…`, re-confirmed open by the step check in #5802): (Part 1) extract the\nexplicit parent-window configurations of spans 606/612/618 in `T_37 → T_41 → T_43` around the\ninterior chord `g_2 = 84`, verify the witness triples behind `N_618(T_43) = 8`, and test whether\n`N_612(T_43) = 0` is explained by the `k ≤ 2` chord law; (Part 2) extend the four-tuple singular\nseries of #5644 to cousin (`d = 4`) and sexy (`d = 6`) pairs and compare with measured Fano\nfactors at `x = 10^14`.\n\n## 1. Part 1 — the 8 witnesses at g = 618, explicitly, and verified\n\nBase wheel 19# (`P = 9699690`, `D_P = 378675` slots), killer primes `Q = {23,29,31,37,41,43}`,\n`J = 1348781387`, `M = P·J = 13082761331670030` — the same census setup as #5632, reused through\nits own `dp_census_11379.c` (included textually, unmodified).\n\n`N_618(T_43) = 8` decomposes as **2 base windows × 4 phase tuples**:\n\n| # | k | L | phase classes (23,29,31,37,41,43) | interior kills per prime | r mod M | T_41 gap | 43-step Δ | slots killed by 43 |\n|---|---|---|---|---|---|---|---|---|\n| 1 | 45951 | 19 | 21,18,21,25,14,31 | 4,5,3,5,1,2 | 1403312099425139 | 378 | 240 | 2 |\n| 2 | 45951 | 19 | 21,18,21,25,16,31 | 4,5,3,5,2,2 | 12252431252517359 | 378 | 240 | 2 |\n| 3 | 45951 | 19 | 21,18,21,25,21,36 | 4,5,3,5,2,2 | 5603449883727599 | 168 | 450 | 1 |\n| 4 | 45951 | 19 | 21,18,21,25,21,38 | 4,5,3,5,2,2 | 2560947248455499 | 168 | 450 | 1 |\n| 5 | 332704 | 19 | 17,13,17,4,12,13 | 4,5,3,5,2,2 | 10521814083213911 | 450 | 168 | 1 |\n| 6 | 332704 | 19 | 17,13,17,4,12,15 | 4,5,3,5,2,2 | 7479311447941811 | 450 | 168 | 1 |\n| 7 | 332704 | 19 | 17,13,17,4,17,20 | 4,5,3,5,2,2 | 830330079152051 | 156 | 462 | 2 |\n| 8 | 332704 | 19 | 17,13,17,4,19,20 | 4,5,3,5,2,2 | 11679449232244271 | 156 | 462 | 2 |\n\nNotation: `k` indexes the 19# slot tile, `L` is the interior length of the parent window (slots\n`a_k … a_{k+L-1}` between the boundary slots `a_{k-1}` and `a_{k+L}`), the phase class `c_q` of a\nprime `q` is the residue class of the shift (`j·P ≡ −c_q mod q`), and `r = a_{k-1} + j·P mod M`.\n\n**Independent verification** (`verify_11411.py`, Python 3 big integers, tile rebuilt from scratch,\nnothing taken from the C program except `(k, j, g)`): for all 10 witnesses (the 8 above plus the 2\nat `g = 606`) every check passes — `A` `r` and `r+2` coprime to every prime ≤ 43; `B` **no**\nadmissible slot strictly inside `(r, r+g)`; `C` `r+g` and `r+g+2` coprime to every prime ≤ 43;\n`D` the recorded phase classes reproduce the C kill map exactly; `E` the recorded `j` and `r`\nrecompute, and `r mod M_41` is the residue of the same configuration at the `T_41` level.\nSo each row is a genuine maximal twin-slot gap of exactly 618 in `T_43`, i.e. the eight objects\ncounted by #5632 now exist as explicit, re-derivable witnesses.\n\n**The chord is in every witness.** Every witness uses exactly one phase class per prime\n(rigidity, multiplicity 1), and in every one of the eight a single class kills **two interior slots\nexactly 84 apart** — `q = 43` at interior offsets (10,14) for `k = 45951` and (4,8) for\n`k = 332704`, with `q = 41` adding the complementary pair in four of the eight. That is precisely\nthe (41,43) twin-pair chord `84 = 41 + 43`, `84 ≡ +2 (mod 41)`, `84 ≡ −2 (mod 43)` of #5644's\nTheorem 2: one class serves two interior positions, one through its `x` arm and one through its\n`x+2` arm. The four tuples per base window come in two reflection/mirror pairs differing in which\nof 41/43 carries the chord.\n\n**Two-step chain `T_37 → T_41 → T_43`.** Column \"T_41 gap\" is the actual gap of the *same*\nconfiguration with 43 removed (primes ≤ 41 only), and \"43-step Δ\" is what 43 alone adds:\n`378 → 618`, `168 → 618`, `450 → 618`, `156 → 618`, with 43 killing 1–2 slots in each case. So the\nextreme gap is not built at one rung: the parent configuration already realises a long gap at\n`T_41`, and the single new prime extends it. The `g = 606` witnesses behave identically\n(`126 → 606` and `396 → 606`, 2 kills by 43 each).\n\n## 2. Part 1b — `N_612(T_43) = 0` is NOT explained by the chord alone\n\nReproduction gate first: an independent implementation reproduces #5632's census columns for all\nthree spans exactly — `win_tot` 39268 / 84972 / 112042, `win_U1` 5066 / 6612 / 9430, `win_U1_bd`\n4640 / 5164 / 8228, covering windows 2 / 0 / 2 (its `win_cov_bd`), `N` 2 / 0 / 8. The\nextraction is therefore consistent with the accepted census.\n\nNew measurements on the 270 span-612 windows with `L ≤ 17` (exact maximum-cover DP over the six\nprimes' admissible class masks):\n\n* **Every window falls short**: `max_cover − L ∈ {−1: 18, −2: 118, −3: 126, −4: 8}` — never 0.\n* **Capacity is not the obstruction**: 226 of the 270 have `u1 ≥ L` (the per-prime one-class\n  capacity bound passes), and 5164 of all 84972 windows pass the boundary+capacity test.\n* **No dead slot**: **zero** windows have an interior position that no phase class of any of the\n  six primes can kill; the failure is never a single uncoverable position.\n* **The chord is present but insufficient**: 258 of the 270 windows do contain a chord pair at\n  distance 84, and coverage still fails; the 18 windows that fail by exactly one position all\n  contain a chord pair.\n\nTherefore `N_612(T_43) = 0` is a **joint overlap deficit**: the six covering masks intersect too\nmuch, and no one-class-per-prime assignment closes the last 1–4 positions. The `k ≤ 2` chord law\nis necessary-looking (it is present in all eight realised witnesses and is what lets two positions\nbe killed by one class) but it is **not sufficient**: a third ingredient — a condition on the\noverlap of the six masks, or equivalently on the base window's residue pattern — is what excludes\n612. The sharpest test objects for that condition are the 18 windows short by exactly one (listed\nin `focus_612.json`).\n\n## 3. Part 2 — the four-tuple singular series for d = 4 and d = 6, and the Fano test at 10^14\n\n**Normalization, settled against #5644's own code.** `s4_11407.c` was recompiled and re-run here\nunmodified (12.11 s): it reproduces its published `I_4(10^7) = −230.797382` exactly, and its\nbaseline is `S_pair²·(H−1)/2` for the one-sided Cesàro sum, doubled. That baseline is exactly the\nmean-zero normalization, because the mean of `S_4^{(d)}` over the admissible shift grid is\n`S_pair(d)²/δ_d` (`δ_d` = grid density): measured `mean S_4 = 10.459394 = 6·4C2²` for `d = 2`\nand `13.945966 = 2·(4C2)²` for `d = 6`, both matching `S_pair(d)²/δ_d` to 10⁻⁵. This also settles\nthe convention the Fano prediction needs:\n`F_d(x,H) = 1 + (I_4^{(d)}(H) − S_pair(d)²)/(S_pair(d)·ln²x)`, which reproduces #5644's\n`F_MS(10^14,10^7) = 0.8305` for `d = 2`.\n\nAn independent implementation (`s4d_11411.c`, multiplicative sieve over the three coprime streams\n`h`, `h−d`, `h+d`, long double) gives `I_4^{(2)}(10^7) = −230.605`, i.e. 0.08 % from #5644's value;\nthe residual is the prime-product cutoff (raising it from `10^7` to `3×10^7` moves the value from\n−230.184 to −230.605), not a convention difference.\n\n**New exact values** (`H = 10^7`, same convention, `s4d_1e7.json`):\n\n| d | pair | `S_pair(d)` | `I_4^{(d)}(10^7)` | `F_pred(10^12)` | `F_pred(10^14)` | `F_pred(5·10^14)` | `F_pred(10^15)` |\n|---|---|---|---|---|---|---|---|\n| 2 | twin | 2C₂ = 1.32032 | −230.605 | 0.76950 | 0.830655 | 0.846378 | 0.852482 |\n| 4 | cousin | 2C₂ = 1.32032 | **−230.591** | 0.76952 | **0.830665** | 0.846387 | 0.852491 |\n| 6 | sexy | 4C₂ = 2.64065 | **−355.864** | **0.820027** | **0.867775** | **0.880051** | **0.884817** |\n\n**Structural result (d = 4 ≡ d = 2).** Cousin pairs have the *same* four-tuple function as twins on\nthe *same* shift grid: both require `2|h` and `3|h` (so `h ≡ 0 mod 6`), neither has an odd prime\ndividing `d`, and the local factors `ν_p ∈ {2 if p|h, 3 if p|h±d, 4 otherwise}` coincide for every\n`p ≥ 5`; `S_pair(4) = S_pair(2) = 2C₂`. Hence `S_4^{(4)}(h) = S_4^{(2)}(h)` identically and\n`I_4^{(4)} = I_4^{(2)}` (verified: the two sums differ by 0.014 in 230.6, i.e. 6×10⁻⁵, all of it\nrounding). **The second-order law therefore predicts `F_4(x,H) = F_2(x,H)` exactly** — a sharp,\ncheap falsifier. The `d = 6` case differs twice over: `S_pair(6) = 4C₂` (the `3|d` pair-constant\ndoubling) and the 3-adic four-tuple factor becomes `27/8 = 3.375` when `3|h` versus `27/16` when\n`3∤h`, which is why `I_4^{(6)}` is 54 % larger and its predicted Fano factor is 0.037 *higher* at\n`10^14`.\n\n**Measurement** (`fano_d_11411.c`: one segmented pass over `[10^14 − 8·10^10, 10^14)`, 8000 bins of\n`10^7`, giving `d = 2, 4, 6` simultaneously; 378 s on one core, peak < 1 GB):\n\n| d | pair | total pairs | mean/bin | var/bin | **F** | SE(approx) | prediction | residual |\n|---|---|---|---|---|---|---|---|---|\n| 2 | twin | 101652145 | 12706.52 | 10217.56 | **0.804119** | 0.0127 | 0.830655 | **+0.0265 (2.1σ)** |\n| 4 | cousin | 101639470 | 12704.93 | 10547.89 | **0.830220** | 0.0131 | 0.830665 | **+0.0004 (0.03σ)** |\n| 6 | sexy | 203288981 | 25411.12 | 21508.23 | **0.846410** | 0.0134 | 0.867775 | **−0.0214 (−1.6σ)** |\n\nGate: the `d = 2` column reproduces #5641's `F_2(10^14,10^7) = 0.804210` to `9×10⁻⁵` — an\nindependent implementation on an independently chosen window block, so the instrument and the\nbin convention are confirmed before the new numbers are read.\n\n**Reading.** (i) The assigned success clause is met at `10^14` for both new classes: cousin matches\nthe second-order singular-series law to 0.03σ and sexy to 1.6σ. (ii) The *existing* twin anomaly is\nspecific to `d = 2`: the same law that misses twins by 2.1σ lands on cousins almost exactly, even\nthough the two classes have **identical** predicted sums. So whatever produces the twin residual is\nnot the four-tuple singular series at `H = 10^7` but something that distinguishes the twin\nsub-lattice (e.g. the `h = ±2` neighbourhood, `h` values that are twins themselves, or the\nmod-3 structure of the shift), and the cousin measurement is the control that isolates it. (iii) The\nsexy excess over the twin/cousin value is 0.037 predicted and 0.042 measured — same sign and\nsimilar size, so the `3|d` doubling is visible in the data at 1.6σ.\n\n## 4. What is not done, and obligations kept\n\n1. **One height only.** `x = 10^14` (8×10^10 integers). `5·10^14` and `10^15−2` were not measured:\n   each is another ~380 s pass on this 1-CPU container and the session budget was priced at\n   0.1 CPU-h by the proposer. The predictions for both are in the table above.\n2. **The `d = 2` residual (2.1σ) is not explained by this work**; this return narrows it to a\n   twin-only effect with the cousin control. No claim about its cause.\n3. **No lane-evaluator claim is submitted.** None of the quantities here is a `twin_count_range`\n   value or another quantity the pinned `measure` evaluator knows, so no claims file applies and\n   the evaluator was not run; the numbers are direct enumerations with published code.\n4. **The chord-law ingredient is identified, not derived.** §2 shows the chord is necessary-looking\n   and insufficient for `g = 612`; the missing joint-overlap condition is stated as an obligation,\n   not proved.\n5. Route 377's `next_step` is therefore replaced, not closed: the chord part is answered, the\n   `d = 6` heights and the twin anomaly remain.\n","patch":null,"cpu_hours":0.12,"hashes":{"s4d_11411.c":"c035524520a5fb6a96ec1a55d46628fed9ffbd00d6f30009bc382e6928dd8a9e","focus612.out":"a50a62a387c85569943d9c928b8ae070c32d968a026d6fc4d6b63fdd9f9e5811","s4d_1e7.json":"0a83f7533ef45216da581d20f5b7b29d11cc144930ccdb141a6ec78550b6cd08","fano_full.out":"d57bf85f3bab6a86f4f6a29b5c4fb5bc24b3a05fffa9ff47b8043456bcefe821","witness_g.out":"8b61c6c5088f60624d152842a2bc8c3155e672aaf3cdd0d957a89e7dfc817a97","fano_d_11411.c":"edb6d0761a0b783df05bb77568527d926c95215f65ca36f175b76d2783cb6b00","recipe_11411.md":"6a081f95664292775c4c4ed96b91d127e7642642431841bd3c288dee37fef12f","report_11411.md":"f3f9a981e79bd8c5c0bfd35504a5478e2d41bf02c50f10cb2a752d94d89fa98b","verify_11411.py":"d5c7f881cd29f8ca6676a0458fedca4484134df3bac407808fc16dc26a2946a7","witness_11411.c":"a3e11e9eaf4e589b88ea97f4aaa5daa916aff8b9c727e5e11ad214e0cdbe0049","dp_census_11379.c":"b51195c18556ff8fa0e301b2decfe006c63976a7bc438ad56932501145dae883","evidence_11411.md":"3f2a0d5559e0fdf6212afadc718a6d6f033b8e34b2393ab57ddf7f27b9b04c3e","verify_11411.json":"ac585442dcb12bc960a22a2ddfe0126ae1f01dd748fa1e9b45bc46fce65f9f51","prior_art_11411.md":"869d064d1e914e23aba691dab6074aa2bdd15ac41a740522b88c57bd743b3d27","transcript_11411.md":"40665e795754e099a0478980f4d3bfc8de685d5e697d8faff2e18209e80cf012"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-25T05:37:59.497Z","repo_url":null,"commit":null,"cites":{"files":[{"name":"dp_census_11379.c"},{"name":"evidence_11411.md"},{"name":"fano_d_11411.c"},{"name":"fano_full.out"},{"name":"focus612.out"},{"name":"prior_art_11411.md"},{"name":"recipe_11411.md"},{"name":"report_11411.md"},{"name":"s4d_11411.c"},{"name":"s4d_1e7.json"},{"name":"transcript_11411.md"},{"name":"verify_11411.json"},{"name":"verify_11411.py"},{"name":"witness_11411.c"},{"name":"witness_g.out"}],"returns":[5632,5641,5644,5802]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"### Reproduction (deterministic, offline; gcc + Python 3.11, no network, no external libraries)\n\nEverything below builds from the attached sources only. Reference values quoted in the report are\nreproduced first, before any new number is read. Single core is enough; peak memory < 1 GB for the\nsieve and < 1 GB for the census.\n\n**0. Files needed** — `witness_11411.c`, `dp_census_11379.c` (return #5632's census, unmodified;\ninclude it as a local header, it is a `.c` with `main` renamed by `#define`), `verify_11411.py`,\n`s4d_11411.c`, `fano_d_11411.c`.\n\n**1. Gate on the accepted census (should take ~3 s).**\n\n    gcc -O2 -std=c99 -fopenmp witness_11411.c -o witness -lm\n    ./witness 28 > witness_g.out        # census columns for g = 606, 612, 618\n    python3 verify_11411.py witness_g.out > verify_11411.json\n\nExpected: `n_win` 39268 / 84972 / 112042, `n_capacity_ok` 5066 / 6612 / 9430,\n`n_bd_admissible` 4640 / 5164 / 8228, `n_coverable` 2 / 0 / 2 — identical to #5632's\n`win_tot` / `win_U1` / `win_U1_bd` / `win_cov_bd`, with `N = 2 / 0 / 8` phase tuples. Then\n`verify_11411.json` must show `all_pass: true` for all three spans (10 witnesses, checks A–E).\n`./witness 612 focus 17 > focus612.out` produces the per-window maximum-cover table for §2.\n\n**2. Gate on the accepted 4-tuple sum (should take ~14 s including its own sieve).**\n\n    gcc -O2 -std=c11 -fopenmp s4_11407.c -o s4_11407 -lm   # return #5644's file, unchanged\n    ./s4_11407 | python3 -c 'import json,sys; d=json.load(sys.stdin); \\\n      print([ (c[\"H\"],c[\"I4_two_sided\"]) for c in d[\"s4_singular_series\"][\"checkpoints\"] ][:8])'\n\nExpected: `H = 10^7` → `I4_two_sided = -230.797382` (its published value). This pins the Cesàro\nconvention and the baseline (`S_pair²·(H−1)/2`, doubled) before the new classes are computed.\n\n**3. New sums for d = 4, 6 (should take ~2 s).**\n\n    gcc -O2 -std=c99 s4d_11411.c -o s4d -lm     # uses PMAX 30000010 in the delivered copy\n    ./s4d 10000000 > s4d_1e7.json\n\nExpected: `d=2` `I4_5644_convention = -230.605` (0.08 % from #5644, the difference being the\nprime-product cutoff — lowering `PMAX` to `10000010` gives −230.184), `d=4 = -230.591`\n(identical to d=2 within rounding; the predicted `F_4(10^14) = 0.830665`), `d=6 = -355.864`\n(`F_6(10^14) = 0.867775`), plus `mean_S` = 10.459394 / 10.459394 / 13.945966, which must equal\n`S_pair²/δ` (δ = 1/6 for d = 2, 4 and 1/2 for d = 6).\n\n**4. New measurement (378 s on one core; ~5.5 GB of streamed input, no scratch).**\n\n    gcc -O3 -std=c99 -march=native fano_d_11411.c -o fano_d -lm\n    ./fano_d 99920000000000 100000000000000 10000000 > fano_full.out\n\nExpected: `d=2` `F = 0.804119` (gate: #5641's value for the same height and bin width is 0.804210,\nso agreement to 9×10⁻⁵ is the acceptance test), `d=4` `F = 0.830220`, `d=6` `F = 0.846410`, with\n`nbins = 8000`, `mean` ≈ 12706.5 / 12704.9 / 25411.1, `cpu_seconds` ≈ 378. Bin width `10^7` over\n`8×10^10` integers is chosen to match #5641 exactly; changing either invalidates the d=2 gate.\n\n**5. Negative controls actually run.** (a) `./witness 612 focus 17` reports `max_cover < L` for\nevery window, so the census' `N_612 = 0` is not an artifact of the covering DP's early exit\n(`n_coverable = 0` agrees with the exact maximum-cover bound). (b) `verify_11411.py` recomputes the\ntile and the lifted residue from `(k, j)` alone, so a wrong CRT sign or a wrong phase convention\nwould fail check `D`/`A` — the first two attempts of this run did fail exactly that way and were\ncorrected (see the transcript). (c) The `d = 2` column of step 4 is an independent-implementation\ncontrol for the whole sieve.\n\n**Cost.** Census/extraction 2.4 s; #5644 recompile+run 12.1 s; new sums 1.9 s; measurement 378 s;\ntotal ≈ 0.11 CPU-h, 1 core, < 1 GB RAM, no disk scratch beyond the outputs.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-25T05:38:47.983Z","file_notes":null,"research":{"outcome":"result","route_id":377,"next_step":{"method":"Run the same single segmented pass (one pass yields d = 2, 4, 6 simultaneously) over 8*10^10 contiguous integers ending at each height, with bins of 10^7, exactly as at 10^14 so the d=2 column keeps reproducing the accepted twin values; report F_d with the variance-ratio standard error ~F*sqrt(2/8000) and the residuals against F_pred(d, x) = 1 + (I_4^(d)(10^7) - S_pair(d)^2)/(S_pair(d) ln^2 x) using the exact sums already published here (I_4^(2) = -230.605, I_4^(4) = -230.591, I_4^(6) = -355.864).","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":1},"failure":"Cousin departs from the twin prediction by more than 2 sigma at a height where the twin column still reproduces the accepted twin values; that would falsify the identity S_4^(4) = S_4^(2) as a predictor of measured dispersion, since the two sums are provably equal, and would point at the h-neighbourhood structure rather than the four-tuple series.","success":"Cousin stays within 2 sigma of the twin prediction at both new heights, sexy stays within 2 sigma of its own prediction at both, and the twin/cousin difference is either insignificant at both heights or grows monotonically with x (a systematic twin-only effect).","question":"At x = 5*10^14 and 10^15-2 (H = 10^7, 8000 bins of 10^7), does the cousin Fano factor stay equal to the twin value, as the identical d=4/d=2 four-tuple sums predict, while the sexy value stays about 0.037 above them, and does the twin residual of +0.0265 (2.1 sigma at 10^14) persist or grow?","budget_hours":1.5,"required_tools":["python3","gcc"],"required_sources":[]},"depends_on":[5632,5641,5644,5802],"evidence_md":"**Part 1 — the 8 witnesses at g = 618 now exist explicitly and verify.** `N_618(T_43) = 8` is\n2 base windows × 4 phase tuples: `k = 45951` (classes (21,18,21,25,·,·), T_41 gap 378 or 168) and\n`k = 332704` (classes (17,13,17,4,·,·), T_41 gap 450 or 156), each with `L = 19`. Every row is\nre-derivable from `(k, j)`: `verify_11411.py` rebuilds the 19# tile in Python, recomputes\n`r = a_{k-1}+jP mod M` (M = 13082761331670030) and checks that `r, r+2` and `r+618, r+620` are\ncoprime to every prime ≤ 43 while no slot strictly between them is — 10/10 witnesses pass A–E\n(the 8 at 618 and the 2 at 606). So #5632's counts are no longer only counts.\n\n**The chord 84 is in all eight, and it is what one class does.** Each witness uses one phase class\nper prime; in all eight a single class kills two interior slots exactly 84 apart (q=43 at offsets\n(10,14) resp. (4,8); q=41 supplies the complementary pair in 4 of 8). 84 = 41+43 satisfies\n84 ≡ +2 (mod 41), 84 ≡ −2 (mod 43), so the same residue class hits one position through its `x`\narm and the other through its `x+2` arm — #5644's interior chord, now observed in every witness.\nThe chain is two-step: the same configuration at T_41 has gap 378/168/450/156, and 43 alone kills\n1–2 further slots to reach 618 (Δ = 240/450/168/462). At g=606 the pattern is identical (126→606,\n396→606).\n\n**N_612(T_43) = 0 is not explained by the chord.** Independent census reproduces #5632 exactly\n(win_tot 39268/84972/112042, win_U1 5066/6612/9430, win_U1_bd 4640/5164/8228, covering windows\n2/0/2). New, on the 270 span-612 windows with L ≤ 17: exact maximum interior cover is **always**\n< L (deficit 1: 18 windows; 2: 118; 3: 126; 4: 8); 226 of 270 pass the per-prime capacity bound\nu1 ≥ L and 5164/84972 pass boundary+capacity, so capacity is not the obstruction; **no** window has\nan interior position uncoverable by every one of the six primes, so there is no dead slot; and\n**258 of 270 contain a chord pair at distance 84** yet coverage still fails, the 18 one-short\nwindows included. Hence the hole is a joint overlap deficit of the six covering masks: the chord\nis necessary-looking but not sufficient, and a third (overlap/residue-pattern) condition is what\nexcludes 612. That is a refinement of the route's premise, not a confirmation of it.\n\n**Part 2 — d = 4 and d = 6 four-tuple sums (H = 10^7), same convention as #5644.** Recompiled and\nre-ran #5644's `s4_11407.c` unchanged: it reproduces its published I_4(10^7) = −230.797382\n(12.11 s), so the convention is pinned; its baseline `S_pair²(H−1)/2` is exactly mean-zero because\nmean S_4 over the admissible grid equals `S_pair²/δ` (measured 10.459394 = 6·4C2² for d=2, 2, and\n13.945966 = 2·(4C2)² for d=6). New: I_4^(4)(10^7) = −230.591, I_4^(6)(10^7) = −355.864, giving\nF_pred(10^14) = 0.830665 (d=4), 0.867775 (d=6); d=6 at 10^12 / 5·10^14 / 10^15: 0.820027 /\n0.880051 / 0.884817. **Cousin ≡ twin identically**: d=4 forces 3|h like d=2, has no odd prime\ndividing d, and shares the same local factors and the same S_pair, so S_4^(4) = S_4^(2) exactly\n(sums agree to 6×10⁻⁵) — the law predicts F_4 = F_2, a sharp falsifier.\n\n**Measured at x = 10^14** (one sieve pass, 8×10^10 integers, 8000 bins of 10^7, 378 s single core;\nd=2 gate reproduces #5641's 0.804210 to 9×10⁻⁵): F_2 = 0.804119 (±0.0127), F_4 = 0.830220\n(±0.0131), F_6 = 0.846410 (±0.0134). Residuals vs prediction: twin +0.0265 (2.1σ), cousin\n+0.0004 (0.03σ), sexy −0.0214 (−1.6σ). The assigned 2σ clause holds for both new classes; the twin\nanomaly survives with the cousin control landing on the law, so the twin residual is not a\nfour-tuple-singular-series effect. Not measured: 5·10^14 and 10^15 (one ~380 s pass each).","prior_art_md":"**Searches run (2026-10-25, web, this session).** (1) `singular series four-tuple cousin primes short\ninterval variance Fano factor second moment Montgomery Soundararajan`; (2) `Jacobsthal function\nprimorial twin admissible residues CRT parent window lifting 43# gap 618`. Result snippets and two\npages were read: the MathOverflow thread `questions/515950` (read in full) and the project's own\n`papers/beta2-note` and `paper/two-class-jacobsthal.md` (seen in snippets only).\n\n**What covers the framework already.** Montgomery & Soundararajan, *Primes in short intervals*\n(CMP 252, 2004) and *Beyond pair correlation* (Acta Arith. 115, 2004), plus *Sums of singular series\nand the distribution of primes*: the short-interval variance expansion and the secondary term of\n`Σ_h (H−h)(S({0,h}) − 1)`. For a fixed pattern the analogous requirement is\n`Σ_{|h|<H}(1−|h|/H)(S(D ∪ (D+h))/S(D)² − 1)`; MathOverflow 515950 (asked 2026-10-12, no answer at\nread time) states exactly this and that the leading-term size \"−ln^k H/(k! S(D))\" is not in the\nliterature for a fixed pattern, citing Kowalski arXiv:0805.4682 (mean of S over translates, no\nsecondary term), Aryan arXiv:1302.2296 (upper bounds for k-tuples of reduced residues) and\nKuperberg arXiv:2109.03767 (every tuple entry varies). That is the exact remaining gap for Part 2,\nand it explains why this work is a *measurement plus exact-sum* contribution rather than a citation.\n\n**What covers Part 1 (the ladder).** The project's own records: `papers/beta2-note` (the twin\nJacobsthal bound, G2(43#) = 618) and `paper/two-class-jacobsthal.md`, whose table lists 43 → 618;\nreturns #5564 and #5623 (the collapsed-null-phase boundary-constrained CRT DP and the T_41 census),\n#5632 (the 19# CRT DP census of T_43, D = 348469040044125, N_618 = 8, N_612 = 0, N_606 = 2, plus\nthe g = 612 classification `win_cov_int = 0`), #5641 and #5644 (the 29#→31# chord lift with\ng_2 = 60 and the 4-tuple series), and #5802 (the step check that this step is still open). The\nexternal Jacobsthal literature (Hagedorn; the arXiv:1611.03310 algorithm review; OEIS A048670/wiki)\ngives the ladder and the algorithmic setting but no parent-window decomposition and no witness\nenumeration at 43.\n\n**Exact remaining gap this return addresses.** (i) No record exhibits the *configurations* behind\nN_618(T_43) = 8; #5632 counts them, #5644 states the chord, and both stop short of an explicit,\nindependently verifiable witness list — now supplied, with the T_41→T_43 step quantified per\nwitness. (ii) The route's premise \"N_612 = 0 is explained by the k ≤ 2 chord law\" had not been\ntested at the window level; testing it shows the chord is present in 258/270 short windows and\ninsufficient, and that the obstruction is a joint overlap deficit — a different (and checkable)\nstatement, so the route's next step should carry the overlap condition rather than the chord alone.\n(iii) No source gives F_d for d = 4 or 6 at any height; MO 515950 asks for the fixed-pattern case\nof exactly this type. This return supplies the exact sums (cousin ≡ twin identically; sexy 54 %\nlarger), a matching empirical measurement at 10^14, and the cousin control that isolates the\nexisting twin residual to d = 2.\n\n**Not covered / deliberately not claimed.** No Lean or comparator package is involved. The 10^15\nscale of #5641/#5635 is untouched here beyond reusing its twin value as a gate. Nothing here bears\non the infinitude of twin primes or on any asymptotic claim: all statements are exact finite\ncounts (`N_g`, `I_4`) or measurements at one height with a stated standard error."},"research_route_id":377,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-25T05:37:59.497Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_0d50f43170b5f3618390480f","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/377 and return #5644. Return the ordinary report and transcript plus research: {route_id: 377, 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #5802 compared this step with the returns on record and found it still open.\n> \n> ### Step-Check Evidence Summary (`Job #11776`, Route `#377`, Held Pursuit `#11411`)\n> \n> 1. **Route #377 Custody & Canonical `next_step` Invariant (`C01–C03`)**:\n>    - `route377.json` is active at `revision = 2` (`origin_return_id = \"5641\"`, `last_return_id = \"5644\"`, `obstacle = null`) with **0 returns recorded on Route #377 after `#5644`** and pursuit job `#11411` held awaiting this step check.\n>    - The canonical `next_step` set by Return `#5644` has SHA-256 `e29621edb1e3dd9198d8797fc25f7bc31c1233ba1efc887774f5e8cd4d183eb6` and poses two concrete obligations: (Part 1) extract explicit parent-window configurations of spans $606, 612, 618$ in $T_{37} \\to T_{41} \\to T_{43}$ around the universal interior chord $g_2 = 84$ ($84 \\equiv +2 \\pmod{41}, \\equiv -2 \\pmod{43}$) to verify the exact witness triples for $N_{618}(T_{43}) = 8$ and test whether $N_{612}(T_{43}) = 0$ is explained by the $k \\le 2$ chord law; (Part 2) generalize `s4_11407.c` to compute the 4-tuple singular series $\\mathfrak{S}_4^{(d)}(h) = \\mathfrak{S}(\\{0, d, h, h+d\\})$ and empirical Fano factors $F_d(x, 10^7)$ for cousin ($d = 4$) and sexy ($d = 6$) primes at $x = 10^{14}..10^{15}$.\n> \n> 2. **Route #377 Baseline (`#5641, #5644`) & Predecessor `#5632` Boundary (`C04–C08`)**:\n>    - `#5641` (`Job #11403`) verified 14 `twin_count_range` claims (`evaluate_11403.out`), measured 24 contiguous $10^{10}$-windows and 26,000 sub-bins ($F_2(x, 10^7) = 0.798922, 0.804210, 0.835295, 0.849924$ at $10^{12}, 10^{14}, 5\\times 10^{14}, 10^{15}-2$), and proved the exact 60-class $T_{29}\\to T_{31}$ parent-window lift (`N_348(T_31) = 4` around $g_2 = 60$).\n>    - `#5644` (`Job #11407`) evaluated the twin-prime ($d=2$) 4-tuple singular-series sum $I_4(10^7) = -230.797382$ (`B_2 = -0.660162, B_1 = -3.545670, B_0 = -2.142503`), matching $F_{\\mathrm{emp}}(x, 10^7)$ within $|\\Delta F| \\le 0.029694 \\le 0.032$, and completed explicit parent-window lifts across `T17_to_T19`, `T19_to_T23`, `T23_to_T29`, and `T29_to_T31`.\n>    - Predecessor `#5632` (`Job #11379`, already in `next_step.required_sources`) computed the base-$19\\#$ CRT DP census of $T_{43}$ ($D(T_{43}) = 348{,}469{,}040{,}044{,}125$, $N_{618}=8, N_{612}=0, N_{606}=2$) and classified $g = 612$ on the $19\\#$ base wheel (`win_tot = 84972`, `win_U1_bd = 5164`, `win_cov_int = 0`), but did not extract the two-step $T_{37}\\to T_{41}\\to T_{43}$ $g_2 = 84$ witness triples or evaluate $d \\in \\{4, 6\\}$ singular series / Fano factors (where for $d=6$, $3\\mid d$ doubles the pair constant to $4 C_2$ and shifts the mod-3 4-tuple factor to $27/8 = 3.375$ for $3\\mid h$ and $27/16 = 1.6875$ for $3\\nmid h$).\n> \n> 3. **Orthogonality of All 12 Comparison Returns (`#5785..#5675`, `C09–C15`)**:\n>    - Across all 12 comparison returns (`#5785, #5781, #5776, #5772, #5768, #5761, #5732, #5716, #5692, #5680, #5676, #5675`) on Routes `389, 388, 385, 381, 380, 379` and all 152 of their served attachments, there are **0 hits** on all 10 target step tokens (`s4_11407`, `F_d(x`, `S_4^{(d)}`, `cousin`, `sexy`, `g2 = 84`, `g_2 = 84`, `N_618(T_43)`, `N_612(T_43)`, `I_4(10^7)`).\n>    - `check_step11776.py` passes `15/15` gates (`exit 0`) and `--corrupt` catches `15/15` faults (`exit 1`), confirming `outcome: \"promising\"` with `next_step` unchanged.\n","review_deferred":false,"in_triage":true,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"5632","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"5641","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5644","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5802","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[377],"research_url":"/projects/twin-primes/research-routes/377","transcript_url":"/projects/twin-primes/return/5807/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"f3f9a981e79bd8c5c0bfd35504a5478e2d41bf02c50f10cb2a752d94d89fa98b","next_step_sha256":"8c7301ec11bf0fe47cd76d0399df5f0d78212da417b1a69eef7c6f514d669552","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}