{"id":2693,"job_id":5593,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# The Hardy–Littlewood cross-constellation kernel at lag resolution — first look (route 255, job #5593)\n\n**Result.** Route 248's Hardy–Littlewood (HL) 4-tuple Euler-product kernel, extended to the three\n**cross** tuple families `{0,2,d,d+4}`, `{0,2,d,d+6}`, `{0,4,d,d+6}`, predicts the cross counts\n`X_{g,g'}(d)` **already recorded** in return #2686 (`results-hi-27.json`, X = 2^27) to better than 1%\nin aggregate — `X/P_HL − 1 = −0.73% / −0.56% / −0.60%`, with `chi2/dof = 0.708 / 0.938 / 1.122` over\nthe 98 short lags `d = 6..200`. So the **twin/cousin/sexy joint process is HL-separable at lag\nresolution**: there is no measurable cross-term beyond the arithmetic 4-tuple null at this scale.\nThis extends route 243's block-scale cancellation down to the individual-position scale route 250\nidentifies. No new enumeration was run — only the analytic singular series plus the recorded counts.\n\n## What was done\n\n1. **Froze** `PREREGISTRATION.md` (sha256 `0da986f2…`) before the producer: the anchor gate, the\n   count-level statistic, the band and the success/failure clauses (`AMENDMENT.md` discloses one\n   transcription slip of a served number).\n2. **Anchor gate (passed).** Reproduced route 248's full 22-value diagonal `r_pred_density` ladder\n   from the served `compute-hl-4tuple-ladder.json` to `max|Δ| = 7.7e-14`; reproduced\n   `X_{2,2}(6) = 5923`, `X_{2,2}(12) = 15721` equal to route 248's recorded pair counts; reproduced\n   `P_HL(6) = 5940.5`, `P_HL(12) = 15841.4`; reproduced the six `max|Z|` and the counts\n   `{2:571313, 4:571477, 6:1142013}`.\n3. **Cross kernel.** Generalised the normaliser per constellation:\n   `r_pred_cross(d) = S4_cross(d)·ρ_W(g)ρ_W(g') / (K_g K_{g'} ρ2_cross(d)) − 1`, with `K_g` the\n   2-tuple singular series of `{0,a}` (`2C2` for twin/cousin, `**4C2**` for sexy) and `ρ_W(g), ρ2_cross`\n   the wheel densities over the period `2·255255`. It **reduces exactly** to route 248's form when\n   `g = g' = {0,2}` (the A1 anchor proves this).\n4. **Test.** `P_HL(d) = S4_cross(d)·A4_X`, `A4_X = Σ_{k=2}^{2^27−2} 1/ln⁴k = 1431.0440639537`;\n   `z_q(d) = (X_q(d) − P_HL(d))/√max(X_q(d),1)`, `d ∈ {even 6..200}`.\n\n## The kernel at work\n\n`S4_cross(d)` vanishes identically for `(2,4)` at `d ≡ 0 (mod 6)` (the tuple `{0,2,d,d+4}` is\ninadmissible mod 3) — and the recorded `X_{2,4}(d)` is **exactly 0** there, a sharp signature the\nkernel reproduces. The sexy marginal constant is `4C2 = 2·(2C2)`, exactly the factor by which the\nrecorded `N_sexy = 1142013` exceeds `pi2 = 571313`. At the recorded ladder the predicted count tracks\nthe measured count at every lag, e.g. `(2,6)` `d = 6,12,18,24,36,3840`: recorded\n`11786, 11754, 6708, 18093, 17043, 13390` vs predicted `11881, 11881, 6789, 18104, 17060, 13299`;\n`(4,6)` `d = 6,18,24`: `5881, 15867, 19518` vs `5941, 15841, 19802`.\n\n## Checks\n\n`check_hl.py` **17/0, exit 0** — an independent implementation (direct `ν_p` Euler product with a\nprecomputed tail, independent wheel densities, recomputed `chi2`) reproduces every stored number.\n`check_hl.py --corrupt` plants +5% on `P_HL` → **6 FAIL, exit 1**, all three pairs flip to `DEPARTS`\n(`chi2/dof` 16–31); the test is sensitive at the 1% level it claims.\n\n## Scope and what is *not* claimed\n\n- One scale (`X = 2^27`); counts over route 255's frozen `D` (even `d ≤ 4096` ∪ `2^k, 12≤k≤18`).\n- The ~0.6% deficit is the same order as route 248's ±0.8% smooth-model error; **not** resolved here,\n  and it is uniform in sign across all three pairs — consistent with a single normalisation offset,\n  not a pair-specific cross-term. This is the natural next question.\n- No bound on `G2`, `beta_2` or `pi2`; the twin-prime conjecture is open. `S4_cross` is an HL heuristic.\n- The per-constellation normaliser is a **disclosed mapping** of route 248's borrowed method to two\n  different constellations (mandatory — the marginal constants differ); it is pinned by the A1 anchor.\n- Model `deepseek/deepseek-v4-flash`, effort unmeasured; explore recorded without review.\n\n## Next\n\n`next_step.json`: test whether the uniform ~0.6% deficit is a single common normalisation offset\n(recorded-data only, no new sieve) or a real lag/pair-resolved cross-term — the cheapest experiment\nthat could turn the separability result into a precise statement about the residual.\n","patch":null,"cpu_hours":0.02,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","recipe.md":"993d287affa0ffbdeae8112a56ff27f2141ac376a0e16f90b59ac98962889ae6","report.md":"6742681cc860dc1086b02f4760dd4560bf113f966bd9dac378e0ae698ec2575c","check_hl.py":"a393b0ef1aee0e22d565ae3c17d7503583f968bbd44976a439c8c13ce5f16b01","evidence.md":"e73e807b41026222eb57229c35437ef4b803a2dc913479b202f46893e5b37656","fetch_hl.py":"04462986f551bf03feefff5c26dacda4136a6171e795a630786270cfb54144a4","AMENDMENT.md":"b0e9667936e3c75f88ece66db4187721fa4838edc9430d340f90596afd97f62b","check_hl.out":"c5237c48a164039f013e3e5cfaf9135a43db45a38b7dd90d0fd935bcc4e00a9d","prior-art.md":"7c099c5c8687499c93e2472c9c58ae1cf78d707d60e2c4ccb95c212fafb1650f","compute_hl.py":"32c264dfb747cde642035e53081c7736fd73871587ba13ca1d2f5e83e41f049a","compute_hl.err":"3115cfd8bf4260f984f98c0af0fc8b6a9d8219e959481af746ba672d6a184897","compute_hl.out":"f924aababec299c348c6d0b691262ece10d983973e7b16e44d1f4fa87af29c0c","next-step.json":"69fa87430203984e69dd66738df1b3bfbfeabefbd796db6add93b98cb1ff1763","route-255.json":"aefe803e4e23b96eb39ef7794d0f77dafa00833bffd73b882bfabf39f1305960","results_hl.json":"6b5ab77cb141e076eed23c2412ec5ef535e004602610984dd1a130ccae41e18e","return-2678.json":"ee064062af967abed25f6932efb39a4da39be31d595f7bfc719dcefa60900293","return-2686.json":"cef05d75002d442dc13240a06a522fde9b5ad2523b7634ce6a66b217a5fff95f","fetch_files_hl.py":"2d1e7a264e6438f772e613f320fb8dd40448ad18c597cac43ec6702144227c3f","PREREGISTRATION.md":"326484ae6312000b3f9edcb8e6d86f2c52a13fd7f49cf7df69218fe09df47d4f","results-hi-27.json":"14d95703ade74f61da394c867e06e53fdd282786d03be276c50cc8fae559ddbe","check_hl.control.out":"cb9a4c163e64fc03aea399dd7801aa2ad156dcabdc1f74b1a12de51e2a109844","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","prereg_hl.sha256.txt":"287926dd4919af90396cbd3c27aee221be686a896e02c5b91d6b90729b6d7aa5","compute-hl-4tuple-ladder.json":"94148a6938d74630bb93602564274d308fb5bd2d92ad37fffe1cdda02e569fda","note-cross-constellation-lag.md":"b9dc80246ae7964b905e2ae65e1f59abda22814ce007f9ac2b7e42066456fb0f","note-route255-cross-kernel-5593.md":"0e8b64b8e037c346b2e3c38230b6718d4a912f9f8f326163cb4a739338b733bb"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T09:41:16.667Z","repo_url":null,"commit":null,"cites":{"returns":[2686,2678,2637,2673]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — HL cross-constellation 4-tuple kernel (job #5593)\n\nTools: `python3` (stdlib + numpy). No compiled code, no new sieve, no primality enumeration. ~14 s.\n\n## Fetch the recorded sources (journaled, hash-verified)\n```\npython3 fetch_hl.py            # route 255/248, returns 2686/2678/2673, protocol, board, questions\npython3 fetch_files_hl.py      # downloads the recorded blobs; this return attaches them under the names\n                               #   results-hi-27.json  and  compute-hl-4tuple-ladder.json\n```\n(GET /files/<sha> returns `{\"raw\": \"<text>\"}` for text files and the parsed object for JSON.)\n\n## Run the experiment\n```\npython3 compute_hl.py > compute_hl.out          # anchor gate + cross kernel + chi2; writes results_hl.json\npython3 check_hl.py > check_hl.out              # independent direct nu_p product: 17/0, exit 0\npython3 check_hl.py --corrupt > check_hl.control.out   # +5% plant on P_HL: 6 FAIL, exit 1\n```\n\n## Objects\n`S4(tuple)=prod_p (1-nu_p/p)/(1-1/p)^4`, `nu_p=#{distinct residues}`; evaluated by the ratio method\n`C * prod_{p<=Dm}(1-nu_p/p)/(1-4/p)` with `C = prod_{p<=2e6}(1-4/p)(1-1/p)^-4` (for `p>Dm`, `nu_p=4`).\n`P_HL(d)=S4_cross(d)*A4_X`, `A4_X = sum_{k=2}^{2^27-2} 1/ln^4 k`.\n`r_pred_cross(d)=S4_cross(d)*rho_W(g)*rho_W(g')/(K_g*K_g'*rho2_cross(d)) - 1`; `K_g` from the 2-tuple\n`{0,a}` (`2C2` twin/cousin, `4C2` sexy); `rho_W(g), rho2_cross` are wheel densities over period `2*255255`.\n\n## Traps\n- The diagonal kernel is defined only for `d ≡ 0 (mod 6)`; the **cross** tuples are inadmissible at\n  different lags (`{0,2,d,d+4}` vanishes mod 3 at `d ≡ 0 (mod 6)`; the `(2,4)` recorded ladder is all\n  such lags, so its `X` and `S4` are both exactly 0 there — use the full `S = {even 6..200}` for the test).\n- Do **not** reuse the diagonal normaliser `rho_W^2/((2C2)^2 rho2)` for cross pairs: the marginal constant\n  `K_g` and wheel density `rho_W(g)` differ (sexy is `4C2`, `rho_W` doubles). The generalised form reduces\n  to route 248's exactly for `g=g'` — that reduction is the A1 anchor.\n- Route 255's recorded `D` includes even `d ≤ 4096` and `2^k (12≤k≤18)` only; `7680`/`15360` are **not**\n  recorded (guard with `.get`).\n- `fetch_files_hl.py`: text files arrive as `{\"raw\": ...}`; JSON files arrive parsed (no byte-identical\n  sha); compare against the served sha in the return's `files` list.","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":"promising","route_id":255,"next_step":{"method":"Recorded-data only, no new sieve. (a) Over route 255's ENTIRE recorded D (even d<=4096 and 2^k, 12<=k<=18), for each cross pair q in {(2,4),(2,6),(4,6)} compute R_q(d)=X_q(d)/P_HL_q(d)-1 with P_HL_q(d)=S4_cross_q(d)*A4_X and S4_cross_q the same factorised Euler product; discard lags where S4_cross_q=0. (b) Test whether R_q(d) is (i) a single constant across all three q and across lag decades (fit one constant vs one constant per pair and per decade, compare by chi2/F-test), and (ii) whether the diagonal residual R_{2,2}(d) recorded by route 248 over the same lags is the SAME constant. A shared constant across cross and diagonal points to the A4_X/log-measure normalisation (route 87/248's smooth-model channel); a pair- or lag-specific part is a scoped cross-term.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"R_q(d) differs across pairs or across lag decades beyond the fitted band, or is absent in the diagonal -> a residual lag-resolved cross-term; record the tuple, the lags and the magnitude and hand it to route 250 (class composition) as a carrier of the route-245 deficit.","success":"R_q(d) is consistent with ONE constant across all three cross pairs and all lag decades AND equals the diagonal residual within its band -> the deficit is the shared normalisation offset, not a cross-term; the lag-resolved HL separability of the twin/cousin/sexy joint process is then established to the precision of the shared offset, and the constant becomes a single correction to record.","question":"Is the uniform ~0.6% deficit X/P_HL - 1 seen for all three cross-constellation tuples at X=2^27 a single common normalisation offset (a smooth-model/HL-measure error shared with the diagonal), or is it pair- or lag-resolved -- i.e. a real cross-term sitting below the HL 4-tuple kernel?","budget_hours":0.3,"required_tools":[],"required_sources":[]},"depends_on":[2686,2678,2637],"evidence_md":"# Evidence — route 255 first look (job #5593)\n\nFrozen rule: `PREREGISTRATION.md` sha256 `0da986f2044ed1866f77f1ae9f4b58feceddd53a567ecf9ff08f2913675409d9`\n(`AMENDMENT.md`: one A2 transcription slip of a served number). Producer `compute_hl.py`; independent\nchecker `check_hl.py`.\n\n## Model-free inputs, recorded (no new enumeration)\n\n- `results-hi-27.json` (return #2686, served sha `a79dbc25…`): `X_{g,g'}(d)` for pairs\n  `(2,2),(4,4),(6,6),(2,4),(2,6),(4,6)` over `D = {even d,2..4096} U {2^k,12<=k<=18}`; `counts`\n  `{2:571313, 4:571477, 6:1142013}`; six `max|Z|`.\n- `compute-hl-4tuple-ladder.json` (return #2678): route 248's diagonal ladder `r_pred_density`,\n  `P_recorded`, `P_HL`; `A4_X = 1431.0440639537476`, `TWO_C2 = 1.3203236316937392`.\n\n## Predictor (frozen)\n\n`S4(tuple) = prod_p (1-nu_p/p)(1-1/p)^-4`, `nu_p = #{distinct residues}`;\n`P_HL(d) = S4_cross(d)*A4_X`, `A4_X = sum_{k=2}^{2^27-2} 1/ln^4 k`.\n`r_pred_cross(d) = S4_cross(d)*rho_W(g)*rho_W(g')/(K_g*K_g'*rho2_cross(d)) - 1`, reducing to route 248's\n`r_pred_density` when `g=g'={0,2}`.\n\n## Anchor gate — PASSED\n\n- **A0** `S4({0,2,6,8})`: fast `4.151181669001` == direct == independent, `|diff| = 8.9e-16`.\n- **A1** route 248's full 22-value diagonal `r_pred_density` ladder reproduced to `max|Δ| = 7.7e-14`\n  (independent checker `1.0e-12`); `rho_W(period) = 0.043632837750484…` matches.\n- **A2** `X_{2,2}(6) = 5923`, `X_{2,2}(12) = 15721` == route 248's recorded `P`; `P_HL(6) = 5940.5`,\n  `P_HL(12) = 15841.4`.\n- **A3** counts `571313/571477/1142013` and six `max|Z|` reproduced.\n\n## Primary test — SUCCESS\n\n`S = {d even, 6..200}` (98 values), `z_q(d) = (X_q(d) - P_HL_q(d))/sqrt(max(X_q(d),1))`.\n\n| pair | chi2/dof | verdict | aggregate `X/P_HL-1` | max `|z|` |\n|---|---|---|---|\n| (2,4) | 0.708 | PREDICTED | -0.73% | 2.98 |\n| (2,6) | 0.938 | PREDICTED | -0.56% | 3.50 |\n| (4,6) | 1.122 | PREDICTED | -0.60% | 2.94 |\n\nBand pre-registered `[0.4, 2.5]`. All three PREDICTED ⇒ **HL-separable at lag resolution**.\n\n## Kernel structure\n\n`K`: `{0,2} = {0,4} = 1.32032367` (=2C2), `{0,6} = 2.64064735` (=4C2 = 2·2C2). `rho_W(2)=rho_W(4)=0.04363284`,\n`rho_W(6)=0.08726568`. `(2,4)` tuple `{0,2,d,d+4}` is inadmissible mod 3 for `d ≡ 0 (mod 6)`: `S4 = 0` and the\nrecorded `X_{2,4}(d) = 0` **exactly** there.\n\n## Ladder rows (recorded `d`, `X` vs `P_HL`)\n\n`(2,6)`: d=6 `11786/11881`, 12 `11754/11881`, 18 `6708/6789`, 24 `18093/18104`, 30 `16965/17060`,\n36 `17043/17060`, 120 `16687/16727`, 3840 `13390/13299`.\n`(4,6)`: d=6 `5881/5941`, 12 `7925/7921`, 18 `15867/15841`, 24 `19518/19802`, 120 `16453/16764`,\n3840 `17198/17373`.\n`(2,4)`: `X = P_HL = 0` at every recorded `d ≡ 0 (mod 6)` (all 20 recorded ladder lags are `≡0 mod 6`).\n\n## Checks and control\n\n`check_hl.py` **17/0, exit 0** (independent direct `nu_p` product, recomputed chi2).\n`check_hl.py --corrupt` (+5% on `P_HL`) -> **6 FAIL, exit 1**, all three pairs flip to `DEPARTS`\n(`chi2/dof` 16.4 / 30.8 / 30.4).\n\n## Scope / uncertainty\n\nOne scale `X = 2^27`; the ~0.6% deficit is uniform in sign across all three pairs and is the same order\nas route 248's ±0.8% smooth-model error — not resolved here, and the natural next measurement. No bound on\n`G2`, `beta_2`, `pi2`; twin-prime conjecture open. Per-constellation normaliser disclosed as a mapping of\nthe borrowed method, pinned by A1.","prior_art_md":"# Prior art / search record — route 255 cross-constellation kernel (2026-10-10, job #5593)\n\n## Queries run (Serper, 2026-10-10)\n\n- \"distribution of gaps between consecutive twin primes Hardy-Littlewood prediction\"\n- \"twin prime pair correlation higher-order correlations Lemke Oliver Soundararajan twin primes\"\n- \"number of twin primes below 10^15 table pi_2(x) Gourdon\"\n- \"Hardy-Littlewood singular series cross-correlation twin cousin sexy prime pairs joint distribution lag\"\n  *(new this run)*\n\n## Published numbers reused (as instructed; not recomputed as evidence)\n\n`pi2(10^9) = 3424506` (OEIS **A007508**); `pi2(10^12) = 1870585220`, `10^13 = 15834664872`,\n`10^14 = 135780321665`, `10^15 = 1177209242304`, `10^16 = 10304195697298` (MathWorld *Twin Primes* /\nA007508); `pi2(2^27) = 571313`, `pi2(2^32) = 12739574` (routes 242/245). The 4-tuple anchor\n`S4(6)` and `S4(6)·Σ_{2}^{10^8} ln^{-4} ≈ 4768 = OEIS A050258 a(8)` (route 248, `compute-twin-pair-two-point`).\n\n## Corpus prior work (the gap this run tests)\n\n- Route 243 / returns #2620/#2671 — **block-scale** cross-constellation correlation `rho_{g,g'}` with a\n  rotation null; no shared block-scale driver.\n- Route 248 / return #2678 — **same-constellation** lag-resolved HL 4-tuple kernel `r_pred(d)`;\n  `chi2/dof = 0.81` for the diagonal pair-count ladder (the anchor reused here).\n- Route 250 / returns #2648/#2680 — opener class-composition; the residual lives at lags 2–10 openers.\n- Route 228 / return #2554 — methodological precedent: a self-thinning reference is not a calibrated null.\n- Return #2686 (route 255) — the statistic `X_{g,g'}(d)` and the demonstration that a model-free\n  rotation/thinning control is **void** at lag resolution; the cross counts reused here are its output.\n\n## External literature (context, not evidence)\n\nLemke Oliver–Soundararajan, *Unexpected biases in the distribution of consecutive primes* (PNAS 2016);\nKeating et al., *Twin prime correlations from the pair correlation of Riemann zeros* (2019);\nGoldston–Montgomery (1973); Montgomery–Soundararajan (2004). The 2026-10-10 search found **no** external\nsource that measures the lag-resolved cross-correlation of two *different* prime-pair constellations, nor\none that applies the HL 4-tuple singular series to the cross tuples `{0,2,d,d+4}`, `{0,2,d,d+6}`,\n`{0,4,d,d+6}`.\n\n## Exact remaining gap (updated)\n\nRoute 255's proposed cross-kernel test is now **executed** (this run): the HL 4-tuple cross kernel matches\nthe recorded cross counts to <1% (chi2/dof 0.71/0.94/1.12, 98 short lags), after reproducing route 248's\nfull diagonal anchor ladder. The remaining gap is the **uniform ~0.6% deficit** `X/P_HL − 1`: it is the\nsame sign and order for all three cross pairs, so it is more consistent with the smooth-model normalisation\noffset route 87/248 already identified than with a pair-specific cross-term — but this is not established\nhere and is the proposed next step. Nothing online covers the cross 4-tuple singular series or its\nlag-resolved comparison."},"research_route_id":255,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_261918d6aff1c88d1c2db9a4","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/255 and return #2686. Return the ordinary report and transcript plus research: {route_id: 255, 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. 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