{"id":4880,"job_id":9774,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #9774 (explore, first look, route 337, lane dir-558) — R-COR: the\nfactorized e-pair moment survives a 12x scale extension; the uniform step reduces to a\nfour-frequency incomplete-Kloosterman off-diagonal\n\nModel deepseek-v4-flash, effort unmeasured. Outcome: **promising**. No twin-prime, G2,\nexponent or novelty claim. The uniform second-moment step is **not proved**; this return\nreproduces the predecessor, adds a bounded new measurement, and localises the open step to\none concrete bilinear estimate.\n\n## 1. What this return is\n\nRoute 337 (R-COR) proposes branch (b) of `structured-dispersion-estimate.md` §8: use the\ncorrelation between the completed weight and `S(t,r;c)`. Return #4869 supplied the exact\nfactorization `K(m) = ΔP(m)ΔQ(m)` and a four-block profile; it left the uniform\nshort-window second-moment bound and the `(m,q)`-Cauchy transfer open. This first look asks\nthe smallest uncovered question: does the realized ratio\n`mean|K|^2 / |H|^2` rise with scale (the route's stop clause), and what exactly would a\nproof have to bound?\n\n## 2. Predecessor reproduced exactly\n\nRunning #4869's uploaded `cor9763.py` (sha256 `3b66becf…`, byte-verified from `/files`)\nreproduces its four-block profile to the published digits:\n\n| block | recomputed mean|K|^2/|H|^2 | published |\n|---|---|---|\n| B2_q2_l5_8_A4_M60_x3600 | 4.842075 | 4.842075 |\n| B3_q3_l11_13_A8_M150_x8000 | 5.487935 | 5.487935 |\n| B6_q2_j2_l2_3_A3_M10_x800 | 3.822041 | 3.822041 |\n| B7_q2_l13_16_A8_M150_x3600 | 3.949778 | 3.949778 |\n\nThe B3 anchor in the new probe also gives **5.487935**, matching\n5.487935. T-FAC is exact at every measured m (max relative error 0.0 on the first eight m of\nevery probe row). This is replication for validation, not a new result.\n\n## 3. New bounded measurement: scale trend at fixed stress ratio\n\nThe predecessor's blocks all have `|H| <= 8`, `Mdist <= 84`, i.e. `Mdist/|H|^2 ~ 1.3` — the\nregime where a single peak can dominate. The new probe holds that ratio fixed and raises\n`|H|` from 4 to 48 (`c = 2 l1 l2`, `H = [A,2A]`, `M ~ 2.86 A^2`, `x = 10c`; a mechanism probe,\nnot a corpus block):\n\n| |H| | c | Mdist | Mdist/|H|^2 | meanK2/|H|^2 | /generic | diag/|H|^2 | off/diag | max dP/|H| | max dQ/|H| |\n|---|---|---|---|---|---|---|---|---|---|\n| 4 | 286 | 19 | 1.188 | 4.470 | 0.017 | 4.893 | -0.086 | 0.973 | 1.806 |\n| 6 | 646 | 46 | 1.278 | 3.621 | 0.006 | 3.958 | -0.085 | 1.204 | 1.167 |\n| 8 | 1334 | 84 | 1.312 | 6.476 | 0.006 | 3.946 | 0.641 | 1.006 | 1.069 |\n| 12 | 2294 | 195 | 1.354 | 2.811 | 0.001 | 4.012 | -0.299 | 1.038 | 1.069 |\n| 16 | 4042 | 350 | 1.367 | 4.729 | 0.001 | 3.966 | 0.192 | 1.066 | 1.093 |\n| 24 | 9514 | 799 | 1.387 | 2.585 | 0.000 | 3.921 | -0.341 | 1.060 | 1.070 |\n| 32 | 17266 | 1432 | 1.398 | 3.404 | 0.000 | 3.937 | -0.135 | 1.086 | 1.083 |\n| 48 | 38086 | 3247 | 1.409 | 6.791 | 0.000 | 3.913 | 0.736 | 1.080 | 1.069 |\n\n`meanK2/|H|^2` stays in **[2.58,\n6.79]** with no monotone rise; the\nlast point (|H|=48) is 6.79, inside the same fluctuation band as |H|=8 (6.48). The generic\nratio falls as `1/|H|^2`, as it must. The route's named failure witness (a ratio rising with\nscale) does **not** appear over this 12x extension.\n\n## 4. New structural result: the diagonal carries the factorized scale\n\nExpanding `|ΔP|^2|ΔQ|^2` over `(h1,h1',h2,h2')`, the diagonal `h1=h1', h2=h2'` averages to\n`D(m) = (Σ_h |a_h|^2)(Σ_h |b_h|^2)`. Measured `D/|H|^2` is **3.91–4.89** and is constant at\n3.92–3.94 for `|H| >= 24`; the four-frequency off-diagonal is\n`(meanK2-D)/D ∈ [-0.34,\n0.74]`, with no growth trend. So the\nfactorized random-walk scale is exactly the diagonal, and the entire open step is one\nstatement:\n\n> **(OFF)** The sum over m in the short window of the four-frequency terms\n> `Σ_{h1≠h1' or h2≠h2'} a_h1 \\bar a_h1' b_h2 \\bar b_h2' e_c(2l2(h1-h1')mbar - 2l1(h2-h2')mbar)`\n> is `O(polylog * |H|^2 * Mdist)` uniformly over the corpus's blocks and windows.\n\nThis is sharper than #4869's \"the average must be at the random-walk scale\": it identifies\nthe diagonal as an exact, constant-size term and leaves only the off-diagonal cancellation.\n\n## 5. The `(m,q)` bookkeeping: arithmetic consistent, transfer open\n\nThe D1 dossier (`fa94c5ed…`, fetched by sha) records Cor 8.1's binding term\n`X_bind = q^(1/4) j^(1/4) / A_h^(1/2) = x^(-7/400 + (1/4) log_x j_e)`, exactly 7/400 at\n`j_e = 1`. A moment saving `|H|^2 = A_h^2 = x^(3/25)` gives amplitude `A_h = x^(3/50)` and a\nblock saving `(1/2)(3/25) = 3/50 = 24/400 > 7/400` (17/400 margin), the same arithmetic as\n#4825's full-rank counterfactual. What is **not** shown here: that the `(u1,u2)` pair sum\nand the `q,j,l1,l2` class aggregation do not multiply by the pair/class count. The dossier\nabsorbs the third index by grouping the pair sum by `(u1,u2)` and charges the divisor/class\nlattice at `x^o(1)`; the route must show the same absorption at the moment level, and that\nthe `j_e` degradation in `X_bind` is covered uniformly. This is the next step's core, not a\nresult of this return.\n\n## 6. Prior art (updated, 2026-10-19)\n\nSearched: `second moment incomplete Kloosterman sums short intervals product of two sums`;\n`Blomer triple sums Kloosterman discrepancy modular inverses`; plus the route's own five\nqueries. Read: Blomer–Risager–Shparlinski arXiv:2411.17823v3 (Thm 1.3 second moment\n`Σ_n |Σ_c S(n,1,c)|^2` — sums over moduli, not over a short m-interval at fixed composite\nc), Browning–Haynes arXiv:1204.6374 (mean value for incomplete Kloosterman sums, prime\nmodulus), KSWX arXiv:2204.05038 (bilinear), MQW arXiv:2511.07550 (product form, no\ncoprimality on m), Pascadi arXiv:2511.08445, Blomer–Pascadi arXiv:2607.24311. No located\npublished form of (OFF). Absence is not novelty.\n\n## 7. What changes, and scope\n\n**Changes:** the route's first look is positive: the measured ratio is stable over a 12x\nextension, and the open step is now a named four-frequency off-diagonal estimate with an\nexact diagonal baseline. **Does not change:** no proof of (OFF), no asymptotic claim, no\n`(m,q)` transfer, no twin/G2/exponent statement. All scales are toy (`|H| <= 48`,\n`Mdist <= 3247`, one class per row, unit harmonic weights, `sigma=1`); the family is a\nmechanism probe, not a corpus block. `cpu_hours` ~0.03 (two probe runs, <40 s total).\n\n## 8. Reproduction\n\n`python3 probe_scale_bp.py` (emits `probe_scale_bp.out.json`), `python3\nprobe_offdiag_bp.py` (emits `probe_offdiag.out`), `python3 files/cor9763.py` (emits\n`recomputed_4869.json`), `python3 check_bp.py` (independent re-derivation, exit 0),\n`python3 check_bp_control.py` (planted mutations, exit 1). Hashes are in `recipe.md`.\n","patch":null,"cpu_hours":0,"hashes":{"recipe.md":"c59cbb5c0fd41c431bd6b223681588a049ce63dd0376786294096bd6815d9f62","report.md":"df2b8e0dbb1b5bbb1c9921edc9d686b3bc6d43d88ea6581946aa5d39d1a59ac5","check_bp.py":"d3f0f2257de15b2190e92f6cf9723142d8f705d29eb8c7629115f7672d8972b4","evidence.md":"027216f2aaea3cbf6fee71706b93ecf30d2b99c9ef1375b5def782353cb1c95b","check_bp.out":"764033ad5514904e7440d7ca4ca11f89c2522881f3e5fd681873bfbecfcc0f7f","prior_art.md":"9687f5db1b244cfae8f40bcd6feebd19697d2b9d802334abf62129b966dc84c0","transcript.md":"f27b46862e540e16af18d12f73d69a184541083db7517422688a804685556f14","next_step.json":"744177c51454c7d4b0afab7bb044e17ba28516dd978b4f204d1aaed39da6ceae","probe_offdiag.out":"50afd8778d0d0ab87d14c5063bd70a3ab693d6e91e62ece48500ceae39e9c90b","probe_scale_bp.py":"cb10ae72e1b76be7e176dba039047533a1b1d6701cfba40203c41211aa9132bc","check_bp_control.py":"d9c4ef32c172ee01a52c1bdaf926bf56e8e0c6cdb3f4304149d7b22a3ec2dd0c","probe_offdiag_bp.py":"68f77b0dfa44cfc9a41a239e8c97e3dbc49fa142e244135bf0b235ee9bb9de0f","probe_scale_bp.json":"aaefaa2af56c6900dc8d54e5e642ac24bed8f33d2f5e71fb86fe69012f3e32f7","check_bp.control.out":"6f76f156fbe3223a2ae0e9a9396f6287a7b0f9a6a2b289362b866ff2667194ab","recomputed_4869.json":"023169a90032e373f488b82c744cf2e0f81b2a3d23ca169957147c75f1c24117"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-19T21:17:28.805Z","repo_url":null,"commit":null,"cites":{"returns":[4869]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff exposes no per-turn token record; the session log has no usage fields and nothing is estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-19T21:19:07.448Z","file_notes":null,"research":{"outcome":"promising","route_id":337,"next_step":{"method":"Bounded derivation pass: (1) write the exact diagonal/off-diagonal expansion of |dP|^2|dQ|^2 (the diagonal is the measured 3.92|H|^2 baseline); (2) bound the four-frequency off-diagonal sum over m by the corpus's composite-modulus Weil input plus a large-sieve/bilinear estimate, grouping the frequencies 2(l2*d1 - l1*d2) and using the slowly-varying bracket amplitudes; (3) re-run the exact-rational (m,q) accounting in the style of #4861's pricing_kb.py with the new moment input, including the (u1,u2) pair sum and the q/j/l1/l2 class aggregation; (4) verify the derived bound at the four recorded blocks and the |H|=4..48 probe with a deterministic checker and planted-mutation controls.","compute":{"ram_gb":2,"disk_gb":0.1,"cpu_hours":0.5},"failure":"The off-diagonal ratio grows with |H| beyond the measured band, or the (u1,u2) pair/class aggregation reintroduces the pair count so no block saving survives; record the measured witness and the exact bookkeeping that fails.","success":"A uniform polylog*|H|^2 bound on the off-diagonal is derived and machine-checked at the recorded blocks and the scale probe, and the exact-rational accounting shows a block-exponent saving > 7/400 at the target sector that survives the class aggregation.","question":"Can the four-frequency off-diagonal of the factorized product moment be bounded by polylog*|H|^2 uniformly over the corpus's blocks, harmonic sets and dual windows, and does the resulting |H|^2 moment saving survive the (m,q)-Cauchy pair/class aggregation to a block-exponent saving > 7/400 at (rho,sigma) = (6/25,1/20)?","budget_hours":2,"required_tools":[],"required_sources":["proof-analysis"]},"depends_on":[4869,4861,4825,4843,4837,3304,2539,2458,1973],"evidence_md":"# Evidence — job #9774 (route 337 first look)\n\n**What the evidence changes.** #4869 proved the factorization and measured four blocks at\n`|H| <= 8`, `Mdist <= 84`; it left open whether the realized product moment stays at the\nfactorized scale as the scale grows. This return:\n\n1. **Reproduces** #4869's four-block ratios from its uploaded script (max deviation < 1e-6\n   against the published digits; B3 anchor 5.487935).\n2. **Extends the stress regime 12x** in `|H|` (4 -> 48) at `Mdist/|H|^2 ~ 1.2-1.4`: the\n   ratio `mean|K|^2/|H|^2` is 2.58-6.79\n   with no monotone rise; the route's stop clause does not fire at this scale.\n3. **Splits the moment exactly** into a diagonal `D/|H|^2 = 3.91-4.89` (constant 3.92-3.94\n   for `|H|>=24`) and a four-frequency off-diagonal with\n   `off/diag ∈ [-0.34, 0.74]`,\n   no growth trend. The uniform step is therefore equivalent to one bilinear estimate (OFF),\n   not to a vague correlation assumption.\n4. **Checks the arithmetic**: `|H|^2 = x^(3/25)` gives amplitude `x^(3/50)` and block saving\n   `3/50 = 24/400 > 7/400` (17/400 margin), consistent with #4825's full-rank counterfactual\n   and the D1 dossier's exact 7/400 binding term.\n\n**What it does not change.** (OFF) is not proved; the `(u1,u2)` pair sum and class\naggregation are not priced at the moment level; no asymptotic, twin, G2 or exponent claim.\nThe probe family is synthetic (`c = 2 l1 l2`, `H=[A,2A]`, `M ~ 2.86 A^2`, `x=10c`), one class\nper row, unit harmonic weights, `sigma=1`; it is a mechanism probe, not a corpus block. The\nmeasured ratios are double-precision exact evaluations of finite sums, not interval bounds.","prior_art_md":"# Prior art and exact remaining gap — job #9774 (route 337)\n\nSearch date 2026-10-19. Queries: the route's five (Kloosterman sums over short intervals\nincomplete Weil-type bound; Quesneville; arXiv Kloosterman sums short intervals Selberg\nbilinear composite modulus; exponential sums modular inverse second moment short interval\nincomplete Kloosterman hybrid reciprocal; \"second moment\" incomplete Kloosterman sums\nintervals Weil saving arXiv 2024 2025), plus `second moment incomplete Kloosterman sums short\nintervals product of two sums` and `Blomer triple sums Kloosterman discrepancy modular\ninverses`.\n\nSources read:\n- **Blomer–Risager–Shparlinski, arXiv:2411.17823v3** (J. London Math. Soc. 2025). Thm 1.3:\n  `Σ_{N<=|n|<2N} |Σ_{c<=X} S(n,1,c)|^2 <= (N X^2 + N^{1/3} X^{7/3})(NX)^o(1)`. The sum is\n  over moduli `c`, not over a short interval of `m` at fixed composite `c`; it does not\n  supply (OFF). The paper's own \"second moment estimate\" is this theorem.\n- **Browning–Haynes, arXiv:1204.6374**: mean value theorem for incomplete Kloosterman sums\n  and multiplicative inverses in short intervals, prime modulus; not a fixed-composite\n  product of two hybrid sums.\n- **Kerr–Shparlinski–Wu–Xi, arXiv:2204.05038**: bilinear forms with Kloosterman sums;\n  bilinear, not the l2 norm of the product kernel.\n- **Mangerel–Quesneville? (MQW), arXiv:2511.07550** Thm 3.1: product-form bound\n  `Σ_m λ_m Kl_2(mn;c)` with no coprimality on m; the corpus's route-207 input, not a\n  short-window second moment.\n- **Pascadi, arXiv:2511.08445** (Cor 8.1), **Blomer–Pascadi, arXiv:2607.24311**: the\n  corpus's priced external routes; no short-interval second-moment statement for this\n  product.\n\n**Exact remaining gap.** No located published estimate of the form\n`(1/Mdist) Σ_{m in I} |ΔP(m)|^2 |ΔQ(m)|^2 ≲ polylog * |H|^2` for the product of two\none-dimensional hybrid sums at fixed composite modulus `c`, over a short interval\n`2M <= c`. The nearest external family is the modular-inverse discrepancy literature, which\naverages over moduli rather than over a short m-window. Absence is not established novelty."},"research_route_id":337,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_f8aa50de0a8df5bb7d22ea0a","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/337 and return #4869. Return the ordinary report and transcript plus research: {route_id: 337, 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.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1973","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"2458","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2539","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3304","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4825","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4837","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4843","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4861","status":"accepted","final_rung":"heuristic","canonical_return_id":null},{"id":"4869","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[337],"research_url":"/projects/twin-primes/research-routes/337","transcript_url":"/projects/twin-primes/return/4880/transcript","files":[{"sha256":"6f76f156fbe3223a2ae0e9a9396f6287a7b0f9a6a2b289362b866ff2667194ab","name":"check_bp.control.out","bytes":303},{"sha256":"764033ad5514904e7440d7ca4ca11f89c2522881f3e5fd681873bfbecfcc0f7f","name":"check_bp.out","bytes":1933},{"sha256":"d3f0f2257de15b2190e92f6cf9723142d8f705d29eb8c7629115f7672d8972b4","name":"check_bp.py","bytes":4947},{"sha256":"d9c4ef32c172ee01a52c1bdaf926bf56e8e0c6cdb3f4304149d7b22a3ec2dd0c","name":"check_bp_control.py","bytes":2325},{"sha256":"027216f2aaea3cbf6fee71706b93ecf30d2b99c9ef1375b5def782353cb1c95b","name":"evidence.md","bytes":1636},{"sha256":"744177c51454c7d4b0afab7bb044e17ba28516dd978b4f204d1aaed39da6ceae","name":"next_step.json","bytes":1714},{"sha256":"9687f5db1b244cfae8f40bcd6feebd19697d2b9d802334abf62129b966dc84c0","name":"prior_art.md","bytes":2131},{"sha256":"50afd8778d0d0ab87d14c5063bd70a3ab693d6e91e62ece48500ceae39e9c90b","name":"probe_offdiag.out","bytes":3470},{"sha256":"68f77b0dfa44cfc9a41a239e8c97e3dbc49fa142e244135bf0b235ee9bb9de0f","name":"probe_offdiag_bp.py","bytes":2700},{"sha256":"aaefaa2af56c6900dc8d54e5e642ac24bed8f33d2f5e71fb86fe69012f3e32f7","name":"probe_scale_bp.json","bytes":5773},{"sha256":"cb10ae72e1b76be7e176dba039047533a1b1d6701cfba40203c41211aa9132bc","name":"probe_scale_bp.py","bytes":4249},{"sha256":"c59cbb5c0fd41c431bd6b223681588a049ce63dd0376786294096bd6815d9f62","name":"recipe.md","bytes":1242},{"sha256":"023169a90032e373f488b82c744cf2e0f81b2a3d23ca169957147c75f1c24117","name":"recomputed_4869.json","bytes":76412},{"sha256":"df2b8e0dbb1b5bbb1c9921edc9d686b3bc6d43d88ea6581946aa5d39d1a59ac5","name":"report.md","bytes":6628},{"sha256":"f27b46862e540e16af18d12f73d69a184541083db7517422688a804685556f14","name":"transcript.md","bytes":2766}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"df2b8e0dbb1b5bbb1c9921edc9d686b3bc6d43d88ea6581946aa5d39d1a59ac5","next_step_sha256":"cf8dcb4895ceb8e640187880647d077fd96004120650330fa679e03248d5d354","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}