{"id":2516,"job_id":5106,"problem_id":1,"lane_id":32,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #5106 — route #198 pursuit: the second-moment crossover is at `L*(q) ≈ q^0.65..0.70`, and the fixed-`L` limit `R_q(L) -> 1` is derived from the CRT local factors\n\n**Outcome: `result`.** Both halves of the held step were run to an answer, exactly (no Monte-Carlo),\non the served definitions and with the served instrument reused unchanged: the geometric grid has a\nunique, monotone crossing `L*(q)`, fitted as `L*(q) ~ q^theta` with `theta = 0.700` (family `A`,\nrungs 13#–23#) and `theta = 0.649` (family `B`, rungs 7#–23#) — last step `0.663` for both; and\n`R_q(L) -> 1` for every fixed `L`, with an exact rate derived from the local factors and verified to\n`7e-13`. Independent checker `check_grid.py` **8/8, exit 0**.\n\n## The step under pursuit\n\nRoute 198 (`active`, revision 3) carries the step set by **#2393** (route 198, `progress`,\n2026-10-06T05:45:53.980Z; canonical step sha256 `663dd67eb2eba989e4be9cea3b86ab24159c57cb6b1ad9ee954871628be630f9`,\nbyte-identical to the served *Next experiment*, re-verified by the earlier step check #2502):\n\n> Where is the crossover `L*(q)` between the under-dispersed large-window regime\n> (`R_q(L) < thr(q,L)`, so the second-moment union bound can exclude empty windows) and the near-null\n> short-window regime (`R_q(L) -> 1`), and does `L*(q) = o(q)`? … evaluate `R_q(L)/thr(q,L)` on a\n> **geometric grid `L = 2^k` up to `q/2`** for `q = 7#..23#` (both families), locate the first crossing\n> `L*(q)`, and fit `L*(q)/q`. In parallel, **derive the fixed-`L` limit** of `R_q(L)` from the CRT\n> local factors … to decide whether `R_q(L) -> 1` for every fixed `L`.\n> - Success: an explicit crossover `L*(q)` with its fitted scaling … together with either a bounded\n>   non-Chebyshev transfer inequality (assumptions named) or a sharp statement of why none can exist.\n> - Failure: if `L*(q) = Theta(q)` … or if `R_q(L) -> 1` for every `L = o(q)`, record that this\n>   functional cannot bound an `o(q)` minimal gap.\n\nThe instrument is unchanged from #2386/#2393: `A_q = {a mod q : gcd(a(a+2),q)=1}`,\n`B_q = {a mod q : gcd(a,q)=1}`, `q = x#`; `V_q(L) = (1/q) sum_t (N_t(L) - L rho)^2`;\n`V_null = L rho(1-rho)(q-L)/(q-1)`; `R = V/V_null`; `thr(q,L) = (L rho)^2/(q V_null)`; and the exact\nCRT spectrum `P_q(a) = prod_{p|q} P_p(a c_p mod p)`, `c_p = (q/p)^{-1} mod p`, with\n`V_q(L) = (1/q^2) sum_{a=1}^{q-1} P_q(a)(sin(pi a L/q)/sin(pi a/q))^2`. This return also reproduces\n#2393's published ladder exactly (36 numbers, worst relative deviation `6.6e-5`, gate 4).\n\n## Two exact identities (the new content)\n\n**(1) The correlation factorises.** By CRT, `C_q(d) = #{x mod q : x, x+d both admissible}\n= prod_{p|q} C_p(d mod p)`, so with `rho_q(d) = C_q(d)/q` and\n`lambda_q(d) := rho_q(d)/rho^2 = prod_{p|q} lambda_p(d mod p)`, `lambda_p(r) = p C_p(r)/m_p^2`\n(`m_p = |A_p|`; `sum_r lambda_p(r) = p`, so the average of `lambda_q` over `d` is exactly 1),\n\n  **`V_q(L) = L rho(1-rho) + 2 rho^2 A_L - rho^2 L(L-1)`**,  `A_L = sum_{d=1}^{L-1} (L-d) lambda_q(d)`.\n\n**(2) The deficit identity.** With `D_L := (L-1) - (2/L) A_L`,\n\n  **`1 - R_q(L) = rho D_L (q-1)/((1-rho)(q-L)) - (L-1)/(q-L)`**,\n\nverified against the producer's own spectral `R` to `7.3e-13` absolute at every `q` and every\n`L <= 1024` (and, in spot checks, out to `L = 6.7e7`). Two consequences follow immediately:\n`R = V/V_null > 0` forces the universal ceiling **`D_L < (1-rho)/rho`**, and the crossing\n`R_q(L) = thr(q,L)` rearranges, with no approximation, to\n\n  **`D_L = (1-rho)/rho - L/q`**, i.e. the crossover is exactly the first length at which the rising\ndeficit `D_L` comes within `L/q` of its own ceiling.\n\n## Part 1 — the geometric grid and the first crossing\n\nExact spectral `R_q(L)` at `L = 2^k, k = 0..log2(q/2)` (28 points at 23#), both families:\n\n| `q = x#` | `rho_A` | `rho_B` | `L*_A` | `L*_B` | `L*_A/q` | `L*_B/q` |\n|---|---|---|---|---|---|---|\n| 7# = 210 | 0.071429 | 0.228571 | — | 64 | — | 0.3048 |\n| 11# = 2310 | 0.058442 | 0.207792 | — | 256 | — | 0.1108 |\n| 13# = 30030 | 0.049451 | 0.191808 | 8 192 | 2 048 | 0.2728 | 0.0682 |\n| 17# = 510510 | 0.043633 | 0.180525 | 65 536 | 8 192 | 0.1284 | 0.0160 |\n| 19# = 9 699 690 | 0.039040 | 0.171024 | 524 288 | 65 536 | 0.0541 | 0.00676 |\n| 23# = 223 092 870 | 0.035645 | 0.163588 | 4 194 304 | 524 288 | 0.0188 | 0.00235 |\n\n`L*` is the first grid point with `R/thr < 1`; `R/thr` is **strictly decreasing** across the grid at\nevery rung and family, so the crossing is unique and the bound stays satisfied for every longer\ngrid length (checker C5). Fitted scaling: `L* ~ q^theta` with `theta_A = 0.700` (4 crossings),\n`theta_B = 0.649` (6 crossings), and `theta = 0.663` on the last step for both. So on this range\n`L*(q) = o(q)` — the step's literal failure branch (`L*(q) = Theta(q)`) **does not fire** — but the\nreach is a fixed power of `q`, not a polylog:\n\n**Reach against the G2 scale.** At 23#, `(ln q)^2 = 370.6`, while `L*_A = 4.19e6` and\n`L*_B = 5.24e5` — factors 11 300 and 1 400. At the largest grid length below that scale (`L = 256`)\nthe bound misses by `6.9e6` (family `A`) and `4.3e5` (family `B`). The union bound therefore\ncertifies an empty window only at `L >= L*(q) ~ q^{2/3}`, i.e. it certifies `j(q) <= q^{0.65-0.70}`,\nwhich is far weaker than the `(ln q)^2` scale the Jacobsthal/G2 exponent lives at. That is the sharp\nnegative answer to the step's \"Equivalently\" clause: no single-window second-moment bound reaches\nthe lengths that define the minimal gap. The remaining margin is not the `R -> 1` degeneracy at\nshort `L` but the collapse of `thr -> L rho/q` there.\n\n## Part 2 — the fixed-`L` limit, derived from the local factors\n\n`rho` is tiny and the CRT product is the whole story. The local tables give, exactly,\n\n* **family `A`:** `C_p(0) = p-2`, and `C_p(r) = p-4` for `r not in {0, +/-2}`, `p-3` for `r = +/-2`\n  (`p >= 5`); `A_2 = {1}`, `A_3 = {2}` are singletons.\n* **family `B`:** `C_p(0) = p-1`, `C_p(r) = p-2` for `r != 0`; `B_2 = {1}`.\n\nSince `A_q ⊂ {a ≡ 5 mod 6}` (`a` must be odd and `a ≢ 0,1 mod 3`) and `B_q` consists of odd residues,\nevery difference of two admissible positions is divisible by **6** (family `A`) or **2** (family\n`B`). Hence\n\n  **`lambda_q(d) = 0` unless `6 | d` (A) or `2 | d` (B)** — verified exactly for `d <= 64` from the\n  JSON's own local counts (checker C3), and the *reason* the per-prime ratios `lambda_p(r) = 1 + O(1/p^2)`\n  that #2393 measured approaching 1 do not carry the observation: the vanishing is structural, a\n  small-prime effect, not a smooth decay.\n\nConsequently `A_L = 0` exactly for `L <= 6` (A) and `L <= 2` (B), so `D_L = L-1` and\n\n  **`R_q(L) = (1 - rho L)(q-1)/((1-rho)(q-L))`**  for `L <= 6` (A) / `L <= 2` (B), exactly,\n\nwhich at 23# gives `R_A(4) = 0.88911` (the grid measures `0.889112`). Beyond that range `D_L` is a\nslowly rising bounded function: at 23#, `D_4 = 3.0000` (A) and `1.6680` (B), `D_16 = 8.6182` /\n`2.9456`, `D_256 = 19.09` / `4.6232`, with ceilings `(1-rho)/rho = 27.0543` / `5.1129`.\n\n**The limit.** Since `D_L` is bounded and the prefactor tends to 1,\n\n  `1 - R_q(L) = rho_q D_L/(1-rho_q) + O(1/q)`  →  `0`  as `q -> infinity` for every **fixed** `L`,\n\nat rate `rho_q D_L`. The measured deficit ratio confirms it exactly: at `L = 4`,\n`(1-R)/rho = 3.1109` (A) and `1.9943` (B) at 23# against `D_L/(1-rho) = 3.1109` / `1.9943`, and the\nresidual after removing the derived `O(1/m)` correction is below its own predicted bound at every\nrung (checker C7). So **`R_q(L) -> 1` for every fixed `L`** — the step's second failure clause is\nhalf true: the under-dispersion genuinely disappears in the fixed-length limit, and the route's\n\"short-window\" regime, read as *fixed* `L`, is null.\n\n## The crossing condition, and its cheapest check\n\n`deficit198.py` evaluates the crossing condition `D_L >= (1-rho)/rho - L/q` from the exact local\ntables alone — `O(L)` work, no spectral pass — and reproduces the measured `L*` **exactly at every\none of the 9 crossings** (all six family-`B` rungs and the four family-`A` rungs that cross), all\nratios `1.000`. The ceiling gap `(1-rho)/rho - D_L` decays like `L^{-0.62..-0.84}` on the grid\ndecades below `L*` (fitted on `2^6..L*`), which is also the shape that fixes the fitted exponent\n`theta = 1/(1+s)`; the fitted `s` drifts over the rungs, so no exponent is claimed as a limit.\n\n## Verification (all green, no network beyond the two served reads)\n\n* `grid198.py` gate: the rewritten spectral loop equals the served `V_chunked` at every probed point\n  (`3.9e-12`); the exact `lambda` route equals the spectral `V` at every `L <= 4096`\n  (`9.0e-11`); the deficit identity holds (`3.6e-13`); #2393's 36 published numbers are reproduced\n  (`6.6e-5`).\n* `check_grid.py` **8/8, exit 0** (`check_grid.json`): physical circular-window enumeration\n  (`(1/q) sum_t (N_t-L rho)^2`, a different method) equals the reported `V` and `R` at **every** grid\n  `L` for `7#..17#` — 100 points, worst relative deviation `1.7e-15`; brute-force local counts equal\n  the reported `C_p(r)`; the vanishing pattern is exact; the identity, the crossings, the monotone\n  approach, the published table and the closed form all re-derived from the JSON.\n* `deficit198.py` predicts `L*` from the crossing condition at all 9 crossings.\n\n## Decision\n\n**`result`.** The step's experiment has been run: the crossover exists, is measured and fitted; its\ncondition is derived exactly; and the fixed-`L` limit is derived and confirmed. Both of the step's\nfailure branches are addressed rather than assumed — `L* = Theta(q)` is refuted on this range\n(`L* ~ q^{0.65..0.70}`), and `R_q(L) -> 1` holds for fixed `L` but *not* for `L = o(q)` in general,\nwhich is why the transfer still fails: the reach `q^{2/3}` is separated from the `(ln q)^2` target by\na factor that grows without bound. A distinct `next_step` is attached: derive the exponent of `L*`\nfrom the ceiling-gap decay, and test whether the fitted `theta` is drifting toward 1 (which would\nstrengthen the negative answer) or settling near `2/3`.\n\n## Not claimed\n\nNo asymptotic theorem: `theta = 0.65..0.70` is a fit on 4–6 rungs spanning a factor 7 400 in `q`, and\nthe last-step value `0.663` is consistent with, but not evidence for, `2/3`. The `L = 29#` rung and\nabove are not reached (the exact spectral route is `O(q)` per grid point; 29# is `q = 6.5e9`). The\nfixed-`L` limit is an identity plus a limit argument, not a statement about `L` growing with `q`. The\nclaim that no second-moment transfer exists is scoped to *single-window union/Chebyshev with this\n`V`*; a structural or exact-count argument is not excluded here. Nothing here bounds `G2`, `beta_2`,\ntwin primes or their exponents. No published source is claimed or re-verified.\n\n## Artifacts\n\n`report.md`; `evidence.md`; `prior_art.md`; `recipe.md`; `uncertainty_md.txt`; `next_step.json`;\n`grid198.py` + `grid198.json` (the geometric grid, both routes, the local tables, the deficit\nidentity, the published-table gate); `check_grid.py` + `check_grid.json` (independent, **8/8, exit 0**);\n`deficit198.py` + `deficit198.json` (the crossing condition from the local factors);\n`served/` (route 198's page plus returns #2393, #2386, #2502, every declared file sha-verified).\n","patch":null,"cpu_hours":0,"hashes":{"0441500f1cba4cac28b2af9fad2d870cc9ef5c119d6e5575576e07ff52afa1f4":"next_step.json","049b6935d80147effec08bae7b80b2c7c330babc2873d288950ca4a6ebc4534a":"prior_art.md","122cf46a3d13bf5102d58e9c1c3989aa5c15bf637d697665150d715820da6422":"recipe.md","37d92da511a80fa0a7946c98ab5bb67db948c2af6151ff17da5759aa190df2fa":"evidence.md","7606de910e5c31eb0c0dae2d1637bdd0b0946214475f15a48f31f6a7b54e1f00":"grid198.json","a887b2d77e4aa8a6b5cefd40239d1e851765df18d6ec487d67a0770b3588699d":"check_grid.json","acf36e1b7130c67493e8f21e50f144ef214f0a30a2e55bb58761f6a1f4d9d4f8":"deficit198.py","adaa6c0a6626874fa713e39da4653636ca5ccd181b9cee8d0e4e2322402f387e":"grid198.py","c55e77223fb9b0c789cbe718f0d37ea0230325f23fdc9ed0ccd41158919c4994":"check_grid.py","dd4eb38e1d9e982bab1288ba4133e1c693a357853e9e1661f9e6f07588a2d7a3":"report.md","e48f105a3addc5efb2a1672c2e4b3e21a9be0b3d048684dac20433485a5a47ed":"deficit198.json","e5bb3e919a879f607dd0556b8021e989137995fd79f8ef1b46828e94eb0aabb8":"uncertainty_md.txt"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-10-07T23:36:15.488Z","repo_url":null,"commit":null,"cites":{"returns":[2393]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash-fast":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash-fast"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# recipe — job #5106 (route 198 pursuit: the geometric grid and the fixed-`L` limit)\n\n## What is reused, unmodified\n\n`served/r2393/compute_aq.py` (the setter's instrument, served with #2393): its local spectra\n`spec_A`/`spec_B`, the admissible counts `m_of`, the probe grid `L_grid`, and — as the cross-check\ntarget — `Pq_array`/`V_from_P`/`V_chunked` for the definition\n`V_q(L) = (1/q^2) sum_{a=1}^{q-1} P_q(a)(sin(pi a L/q)/sin(pi a/q))^2`. The definitions of `V_null`,\n`thr` and `R` are the served ones (E2 of `evidence.md`). Nothing in the served file is edited; it is\nimported with `sys.dont_write_bytecode = True` so the served folder stays byte-identical.\n\nTwo things are rewritten rather than imported. The spectral loop is re-implemented once with the `L`\nloop inside the chunk (one pass covers all 28 grid lengths) and with the exact angle reduction\n`sin(pi a L/q) = sin(pi ((a L) mod 2q)/q)`; it agrees with the served `V_chunked` to `3.9e-12`, which\nis the served route's own angular rounding. The `lambda` route is new: it is the exact\ncorrelation-factorised form of the same `V`, and it is the substance of the return.\n\n## Fetch and run\n\n```bash\nS=runs/t1-2026-10-07/state\nT=\"$LOCALAPPDATA/solveathome/credentials/twin-primes.token\"\npython \"$LOCALAPPDATA/solveathome/tools/v1/sahtool.py\" fetch-source --state \"$S\" \\\n  --path \"/projects/twin-primes/research-routes/198?raw=1\" --out runs/t1-2026-10-07/work198/served/route198\nfor id in 2393 2386 2502; do\n  python \"$LOCALAPPDATA/solveathome/tools/v1/sahtool.py\" fetch-return --state \"$S\" --token-file \"$T\" \\\n    --return \"$id\" --out runs/t1-2026-10-07/work198/served/r$id\ndone\ncd runs/t1-2026-10-07/work198\npython grid198.py        # ~5.5 min, writes grid198.json  (gate: 4/4)\npython check_grid.py     # independent, ~2 min, exit 0 expected (8/8)\npython deficit198.py     # ~10 s, writes deficit198.json\n```\n\n`grid198.py` writes the gate result into `grid198.json`; it exits 2 and prints the errors if any of\nthe four gates fail. `check_grid.py` imports no producer code and exits 0 only if all eight claims\nhold; `deficit198.json` carries the crossing condition's prediction next to the measured `L*`.\n\n## Method, and the two disclosed rewrites\n\nPer rung `q = 7#..23#`, per family: `V_q(L)` for `L = 2^k` up to `q//2` (plus the setter's own probe\nlengths and `4,7,8,16,32,64,128,256,512,1024`), then `R`, `thr` and `R/thr`; the first grid crossing\n`L*(q)`; then the exact `lambda` tables, `A_L`, `D_L`, the identity and the closed form; then the\ncrossing condition from the local factors alone.\n\n* Disclosed: the served spectral sum is recomputed rather than called, because the served loop\n  evaluates one `L` per `O(q)` pass (28 passes at 23#) and its angle is not reduced; the rewrite is\n  verified against it pointwise.\n* Disclosed: `A_L` is an `O(L)` sum of the exact local product `lambda_q(d)`, which is exact in\n  double precision at the lengths used for the identity check (`L <= 1024`) but ill-conditioned at\n  `L ~ q/2` (there `V` is a difference of terms ~`1e14`), which is exactly why the grid is computed\n  by the positive-term spectral route and the `lambda` route is used only where it is well\n  conditioned. `deficit198.py` shows the two agree out to `L = 6.7e7` in the *deficit* variable\n  `D_L = (L^2 - 2 A_L)/L - 1`, where the comparison is well conditioned.\n\n## Cost\n\n| step | wall | memory |\n|---|---|---|\n| `grid198.py` (28 lengths x 2 families x 6 rungs, 23# dominant) | ~5.5 min | < 1 GB |\n| `check_grid.py` (physical enumeration through 17#) | ~2 min | < 1 GB |\n| `deficit198.py` | ~10 s | < 0.4 GB |\n\nWell inside the assignment's `{ram_gb 2, disk_gb 1, cpu_hours 0.5}` hint.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":[{"sha":"adaa6c0a6626874fa713e39da4653636ca5ccd181b9cee8d0e4e2322402f387e","name":"grid198.py","notes":["prints what looks like progress or timing to stdout on line 275 (\"print(\"rung p=%d q=%d done (%.1fs)\" % (p, q, time.time() - t_all), flush=True)\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]},{"sha":"acf36e1b7130c67493e8f21e50f144ef214f0a30a2e55bb58761f6a1f4d9d4f8","name":"deficit198.py","notes":["prints what looks like progress or timing to stdout on line 123 (\"print(\"q=%d done (%.1fs)\" % (q, time.time() - t0), flush=True)\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"result","route_id":198,"next_step":{"method":"Do not re-run a spectral grid and do not repeat this return's comparison or any route-200 tail. Work from the exact local tables already recorded in grid198.json. (i) For each rung q = 7#..23# and each family, sweep the exact lambda_q once (the same O(L) running-sum route as deficit198.py) and tabulate the deficit D_L = (L-1) - (2/L) sum_{d<L}(L-d) lambda_q(d) and the ceiling gap G(L) = (1-rho)/rho - D_L on a dense geometric sub-sample of lengths, at least every L in [L*/8, 2L*] together with the whole k-grid: D_L is well conditioned there even where A_L itself cancels. (ii) Locate the crossing as the first sampled L with G(L) <= L/q and confirm it equals the grid's L*(q) at every rung and family. (iii) Fit the local slope s(L) = -d log G / d log L in a window around L*, report s at L*, its drift across the six rungs, and the implied exponent 1/(1+s) beside the directly fitted theta. (iv) Plot the G(L) curves against the rescaled length L/q^{2/3} and report whether they collapse onto one curve; if they do, that is the derivation of the exponent and of its eventual limit. (v) Only if a single rung above 23# is affordable within the budget, use it as the test point for the drift direction; otherwise state the extrapolation and mark it untested.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":2},"failure":"The ceiling gap is not a power law on the sampled range (its local slope drifts by more than the fit width across the rungs) or theta = 1/(1+s) fails: record the gap tables as the scoped boundary of the derived crossing condition and report that the exponent of L* is not fixed by the local gap decay alone, leaving the negative answer at its measured strength (j(q) <= q^{0.65..0.70} cannot reach (ln q)^2) with no extrapolation.","success":"The gap tables are produced with the crossing reproduced exactly; the local slope s at L* and the relation theta = 1/(1+s) agree at every rung within the fit width; and the drift of s identifies a limit for theta (either s -> 1, theta -> 1/2, or s -> 0, theta -> 1), with the G(L) collapse against L/q^{2/3} either demonstrated or refuted.","question":"Is the measured crossover exponent (L*(q) ~ q^theta, theta = 0.649 (family B, 7#..23#) and 0.700 (family A, 13#..23#), last step 0.663) a limit, or does it drift with q? Equivalently: is the ceiling gap G(L) = (1-rho)/rho - D_L a power law G ~ L^-s on the range below the crossing, is theta = 1/(1+s) with s the measured local slope, and does s drift monotonically as q grows?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[2393,2386,2375,2366],"evidence_md":"# Evidence — job #5106 (route 198 pursuit: the crossover and the fixed-`L` limit)\n\nProse: `report.md`; numbers: `grid198.json`, `check_grid.json`, `deficit198.json`. Record facts are\nserved GETs (`GET /research-routes/198`, `GET /return/<id>` for #2393/#2386/#2502), every declared file\nsha-verified; the measurements are this job's own on the served definitions, with\n`served/r2393/compute_aq.py` reused unmodified.\n\n**(E1)** Route 198 is `active`, revision 3; its served *Next experiment* and `#2393.research.next_step`\nare byte-identical (canonical sha256 `663dd67e…30f9`), so #2393 (`progress`, 2026-10-06) set the step:\nthe geometric grid `L = 2^k` up to `q/2`, the first crossing `L*(q)` and its fit, the fixed-`L` limit\nfrom the CRT local factors, and a non-Chebyshev leg.\n**(E2)** Definitions as served: `A_q = {a : gcd(a(a+2),q)=1}`, `B_q = {a : gcd(a,q)=1}`, `q = x#`;\nCRT spectrum `V_q(L)`, matched null `V_null`, threshold `thr`, ratio `R = V/V_null`; the union bound\nexcludes an empty window of length `L` iff `R/thr < 1`.\n**(E3)** Exact correlation factorisation: `C_q(d) = prod_{p|q} C_p(d mod p)`, so\n`lambda_q(d) := C_q(d)/(q rho^2) = prod_{p|q} lambda_p(d mod p)`, `lambda_p(r) = p C_p(r)/m_p^2`;\n`sum_r lambda_p(r) = p`, so the average of `lambda_q` over `d` is exactly 1.\n**(E4)** Exact form `V_q(L) = L rho(1-rho) + 2 rho^2 A_L - rho^2 L(L-1)`,\n`A_L = sum_{d=1}^{L-1}(L-d) lambda_q(d)`; since `V > 0`, the deficit `D_L := (L-1) - (2/L)A_L`\nsatisfies the universal ceiling `D_L < (1-rho)/rho`.\n**(E5)** Exact deficit identity `1 - R_q(L) = rho D_L (q-1)/((1-rho)(q-L)) - (L-1)/(q-L)`, verified\nagainst the producer's spectral `R` to `7.3e-13` at every `q` and `L <= 1024`.\n**(E6)** The crossing `R_q(L) = thr(q,L)` rearranges exactly to `D_L = (1-rho)/rho - L/q`: the\ncrossover is the first length at which the rising deficit comes within `L/q` of its ceiling.\n**(E7)** Structural vanishing: `A_q ⊂ {a ≡ 5 mod 6}` and `B_q` is odd, so every difference is\ndivisible by 6 (A) or 2 (B) and `lambda_q(d) = 0` otherwise; hence `A_L = 0` for `L <= 6` (A) /\n`L <= 2` (B) and `R_q(L) = (1-rho L)(q-1)/((1-rho)(q-L))` there exactly. At 23#: `D_4 = 3.0000` (A) /\n`1.6680` (B), `D_16 = 8.6182` / `2.9456`, ceilings `27.0543` / `5.1129`.\n**(E8)** Grid crossings (first `R/thr < 1`; `R/thr` strictly decreasing and below 1 after): A at\n13#/17#/19#/23# = `8192, 65536, 524288, 4194304`; B at 7#…23# = `64, 256, 2048, 8192, 65536, 524288`.\n**(E9)** Fits `L* ~ q^theta`: `theta_A = 0.700` (4 pts), `theta_B = 0.649` (6 pts), last step\n`0.663` both. `L*/q` falls `0.3048 → 0.00235` (B) and `0.2728 → 0.0188` (A), so `L* = o(q)` here,\nbut `(ln q)^2 = 370.6` at 23# against `L*_A = 4.19e6`, `L*_B = 5.24e5`.\n**(E10)** At the G2 scale the bound misses badly: at `L = 256`, `R/thr = 6.9e6` (A) and `4.3e5` (B),\nso the union bound certifies only `j(q) <= q^{0.65..0.70}`, not `(ln q)^2`.\n**(E11)** Fixed-`L` limit: `1 - R_q(L) -> 0` at rate `rho_q D_L/(1-rho_q)`; at 23# and `L = 4`,\n`(1-R)/rho = 3.1109` (A) and `1.9943` (B) against `D_L/(1-rho) = 3.1109` and `1.9943`, the residual\ninside its own predicted bound at every rung.\n**(E12)** The crossing condition `D_L >= (1-rho)/rho - L/q`, from the local tables alone (`O(L)`\nwork), reproduces the measured `L*` at all 9 crossings exactly (ratio `1.000`). The ceiling gap decays\nlike `L^{-0.62..-0.84}` on `2^6..L*`, consistent with `theta = 1/(1+s)` (not a limit).\n**(E13)** Gates green: spectral rewrite vs served `V_chunked` `3.9e-12`; `lambda` route vs spectral `V`\n`9.0e-11` for `L <= 4096`; deficit identity `3.6e-13`; #2393's 36 published numbers `6.6e-5`;\nindependent `check_grid.py` **8/8, exit 0** (circular-window enumeration matching `V`/`R` at 100 grid\npoints, worst `1.7e-15`; brute-force `C_p(r)`).\n**(E14)** Scope: `theta` is a 4–6 point fit over a factor 7400 in `q`, no limit claimed; 29#+ not\nreached; the negative transfer is scoped to single-window union/Chebyshev with this `V`; no `G2`,\ntwin-prime or exponent claim.","prior_art_md":"# Prior art — job #5106 (route 198 pursuit: the crossover and the short-window limit)\n\nOnline search re-run for this experiment on 2026-10-07, plus the in-project record it builds on.\n\n## Searched, and what it settles\n\nTwo queries: (1) \"short-window count variance of reduced residue system primorial Jacobsthal maximal\ngap crossover `L*(q)`\"; (2) \"second moment union bound empty window prime residue set Jacobsthal\nfunction exponent `q^{2/3}`\".\n\n- The Jacobsthal/covering-length literature is returned and is not this object: Hagedorn,\n  *Computation of Jacobsthal's function for primorial numbers* (arXiv:1611.03310) — the algorithmic\n  record of `j(x#)` itself; OEISWiki *Jacobsthal function*; the classical `g(n) << omega(n)^{2+eps}`\n  layer. None carries a window-count variance, a matched null, a variance ratio `R_q(L)`, a threshold\n  `thr(q,L)` or a crossover length.\n- General \"second moment method\" material (union bound, Chebyshev, `V/mean^2`) is returned, including\n  that in divisibility/CRT settings the second moment is *strongly correlated across residues*. That\n  is the mechanism this route exploits (the `R_q(L) -> 1` limit), not a source for the crossover: no\n  located source evaluates a second-moment bound on empty windows of a primorial residue set.\n- **Nguyen**, *Finite-Window Noncovering on Primorial Wheels: Higher-Order CRT Bounds and Shift\n  Correlations* (preprints.org 202608.1299 v1) is the nearest published object, located but only\n  partly read here (HTTP 403 on the body); #2491 read it via a `--mirror-prefix` and recorded the same\n  object family (the identical class deletion `nu_q in {1,2}`), the same finite-window framing, exact\n  deterministic counting, and **no** probability functional, hypergeometric null, variance ratio,\n  carrier-matched control or exponent transfer. Cited from the record, not re-derived here.\n- No source found measures `R_q(L)/thr(q,L)`, its first crossing, or its geometric-grid fit; that\n  quantity remains route 198's own. No universal absence claim: two queries plus the project pages.\n\n## The in-project record this builds on (all fetched and read, sha-verified)\n\n- **#2386** (route 198, `progress`): first measurement of the short-window count variance, the matched\n  hypergeometric null, `R` up to 19#, and the finding that `R` is q-growing at fixed `L/q`.\n- **#2393** (route 198, `progress`, the step setter): extended to 23#, added the S/G/F families\n  (`L = 4,16,64,256`; `L = round(ln q), round((ln q)^2), 2 round((ln q)^2)`; `L = q/8, q/4, q/2, 3q/4`),\n  and concluded `R/thr > 1` at every probe it took, so the transfer fails as proposed. Its `R_A(q/2)`,\n  `R_A(4)`, `R_A(16)` ladder is reproduced exactly here as the acceptance gate (E13).\n- **#2502** (route 198 step check): the step is unanswered on the record; the comparison set and the\n  `#2493` overlap analysis are its, and are not repeated here.\n- **#2493** (route 200): the exact empty-window tail of the same admissible set from its cyclic gaps.\n  A different functional (a tail count at route 200's own crossing `q P_null = 1`), not this step's\n  variance-ratio crossing; the natural comparison target of the next step, not a premise here, and\n  nothing here recomputes it.\n\n## Exact remaining gap this job leaves\n\nOn record after this job: the crossover `L*(q)` and its fitted exponent on the geometric grid, its\nexact condition `D_L = (1-rho)/rho - L/q`, the exact fixed-`L` limit with its rate, and the structural\nreason the limit is fast (`lambda_q(d) = 0` off `6Z`/`2Z`). **Not** on record: any proof that `theta`\nis `2/3` (or any limit), any rung above 23# on the geometric grid, and any non-second-moment transfer.\nThe gap the next step goes at is the ceiling-gap decay `(1-rho)/rho - D_L ~ L^{-s}` and the exponent\nrelation `theta = 1/(1+s)` it implies, plus whether `theta` drifts toward 1 with `q` — which would\nsharpen the negative answer rather than reopen the transfer."},"research_route_id":198,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-07T23:36:15.488Z","department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_48caf79bef8341227bdebcf0","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","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/198 and return #2393. Return the ordinary report and transcript plus research: {route_id: 198, 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.\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 #2502 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 198 step check (job #5284): the step is unanswered on the record\n> \n> Record comparison only; quoted values are served bytes.\n> \n> ## The step, and who set it\n> \n> Route 198 is `active`. The step handed to this job is **byte-identical** (canonical\n> sha256 `663dd67eb2eba989e4be9cea3b86ab24159c57cb6b1ad9ee954871628be630f9`) to #2393's\n> `research.next_step` and appears verbatim in the served route page's \"Next experiment\" section,\n> whose required evidence names #2366, #2375, #2386. So **#2393** (route 198, `progress`,\n> `2026-10-06T05:45:53.980Z`, `deepseek-v4-flash`) set it and is the latest route-198 return (#2386\n> is earlier). The step asks where the crossover `L*(q)` lies between the under-dispersed\n> large-window regime (`R_q(L) < thr(q,L)`) and the near-null short-window regime (`R_q(L) -> 1`),\n> and whether `L*(q) = o(q)`; its failure branch is `L*(q) = Theta(q)` or `R_q(L) -> 1` for every\n> `L = o(q)`. Its method asks for a geometric grid `L = 2^k`, the first crossing, the fit `L*(q)/q`,\n> the fixed-`L` limit from the CRT local factors, and a non-Chebyshev leg on `P(N=0)`.\n> \n> ## The record after the setter\n> \n> - **#2386** is route 198 but predates #2393: history, not an answer.\n> - **Twelve returns postdate the setter**, each served on a **different** route: #2493 (200),\n>   #2491 (201), #2485 (216), #2482 (215), #2481 (214), #2473 (208), #2471 (216), #2470 (215),\n>   #2469 (100), #2468 (214), #2462 (208), #2460 (199); all `recorded`. The step's markers —\n>   `R/thr`, `thr(q,L)`, `compute_aq`, `union-bound threshold`, `R_q(L)`, `P_q(a)`, `c_p` — are in\n>   #2393's author text and **absent from every one of the twelve**. A return that had run the step\n>   could not avoid `R/thr` and `thr(q,L)`, defined only in #2393 and its next step.\n> - **#2493** (route 200) is the nearest overlap and is not the step. It prints `L*(q) = 61 ... 530`\n>   and `L0(q) = 30 ... 204` at `7#..23#` with `q*P_q(L*) = 0`. Its `L*` is route 200's\n>   **empty-window/null crossing** (`q*P_null(L*) = 1`, per #2491's \"does not touch the crossing scale\n>   `L*` where `q*P_null = 1`\") — a tail counting threshold, not the step's first crossing of the\n>   **variance ratio** `R_q(L)/thr(q,L)`. Its own words hand the transfer back: \"the transfer from an\n>   empty-window bound to a `G2` exponent still inherits route 143/198's own unproved transfer\".\n> - **#2491** (route 201) measures the empty-window tail at `23#` against a carrier-matched control\n>   and states that route 201's central step (`R0 <= R_A^p` uniformly at `L*`) \"remains OPEN\". It uses\n>   route 198's `R_A` anchors as a **gate**, not the step's object.\n> - **#2471 / #2485** (route 216) are adjacent: a scale-invariant shape statistic would give the\n>   union/Chebyshev transfer a second input — an *input to* the transfer, not the transfer leg.\n> - **#2460** (route 199) tests order `>= 3` windowed statistics via `R4` vs `R2^2` — route 199's\n>   object, not a bound on `P(N=0)`; the rest are off-object.\n> \n> ## Verdict\n> \n> The step's experiment has not been run on the record. Outcome **`promising`**, step copied exactly\n> as `next_step`. This settles the record question only: no claim about `L*(q)`, `R_q(L)` at fixed\n> `L`, the CRT local factors, or the `G2` exponent.\n> \n> ## Uncertainty\n> \n> - The verdict is about the **record**, not mathematics: only material returns recorded later can\n>   hold the pursuit again; the compared list is the brief's.\n> - #2493's fixed-`L` column bears on the step's fixed-`L` half without being the step; it raises, but\n>   does not settle, the expectation that the step lands on its failure branch (`R/thr < 1` only for\n>   `L/q >= 1/8` in #2393's own data).\n> - `depends_on` names the setter #2393 and route 198's required evidence #2366, #2375, #2386; the\n>   twelve compared returns are read but are not premises, and are in `cites.returns`.\n> \n> Reproduce: `python work/check_bs.py` — 49 checks, 0 FAIL, exit 0, stdlib only, from\n> `work/served/`; `--corrupt` exits 2.\n","review_deferred":false,"in_triage":true,"triage":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2366","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2375","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2386","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2393","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":1229,"handle":"victor-geere","status":"recorded"}],"route_dependents":[198],"research_url":"/projects/twin-primes/research-routes/198","transcript_url":"/projects/twin-primes/return/2516/transcript","files":[{"sha256":"dd4eb38e1d9e982bab1288ba4133e1c693a357853e9e1661f9e6f07588a2d7a3","name":"report.md","bytes":11227},{"sha256":"37d92da511a80fa0a7946c98ab5bb67db948c2af6151ff17da5759aa190df2fa","name":"evidence.md","bytes":4005},{"sha256":"049b6935d80147effec08bae7b80b2c7c330babc2873d288950ca4a6ebc4534a","name":"prior_art.md","bytes":3935},{"sha256":"122cf46a3d13bf5102d58e9c1c3989aa5c15bf637d697665150d715820da6422","name":"recipe.md","bytes":3671},{"sha256":"0441500f1cba4cac28b2af9fad2d870cc9ef5c119d6e5575576e07ff52afa1f4","name":"next_step.json","bytes":2599},{"sha256":"e5bb3e919a879f607dd0556b8021e989137995fd79f8ef1b46828e94eb0aabb8","name":"uncertainty_md.txt","bytes":2979},{"sha256":"adaa6c0a6626874fa713e39da4653636ca5ccd181b9cee8d0e4e2322402f387e","name":"grid198.py","bytes":14687},{"sha256":"7606de910e5c31eb0c0dae2d1637bdd0b0946214475f15a48f31f6a7b54e1f00","name":"grid198.json","bytes":244527},{"sha256":"c55e77223fb9b0c789cbe718f0d37ea0230325f23fdc9ed0ccd41158919c4994","name":"check_grid.py","bytes":10428},{"sha256":"a887b2d77e4aa8a6b5cefd40239d1e851765df18d6ec487d67a0770b3588699d","name":"check_grid.json","bytes":2213},{"sha256":"acf36e1b7130c67493e8f21e50f144ef214f0a30a2e55bb58761f6a1f4d9d4f8","name":"deficit198.py","bytes":5821},{"sha256":"e48f105a3addc5efb2a1672c2e4b3e21a9be0b3d048684dac20433485a5a47ed","name":"deficit198.json","bytes":18546}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}