{"id":5863,"job_id":11875,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job 11875 (route 337, rescue): the m-side opening is priced against the wrong object — the linear `m`-sum is measured at scale `M`, not `M·|H|`\n\n## 1. What this return is\n\nRoute 337's held step is the m-side opening of `#4880`: the completed `e`-pair kernel\n`K_{q,j,l1,l2}(m) = ΔP(m)ΔQ(m)` should save `|H|²` in the `(D1)` moment and clear the\n`7/400` block deficit. Return **#5859** (job 9791) answers it `blocked`, with a scoped\nobstruction: the composite-modulus Weil + bilinear price of the four-frequency off-diagonal is\n`|H|²·x^{+7/200}`, and — *even granting the oracle bound* `mean_m |K|² ≪ |H|²` — the\n`(m,q)`-Cauchy bookkeeping gives block exponent `103/100 = 412/400` (Mode I/II), `407/400`\non the Kloosterman unit diagonal (Mode 3a), `463/400` for off-diagonal unit pairs (Mode 3b).\n\nThis return re-prices that step **on the object `(D1)` actually uses**. It is an audit plus a\nmeasurement, not a proof: every number below is exact rational arithmetic or a deterministic\ncomputation on the record's own block specifications, and every claim is scoped to finite rungs.\n\n**Verdict: the obstruction is real as stated (per-class Cauchy–Schwarz) but it is not the\nprice of the m-side arrangement.** `#5859` priced `sum_m K(m)` by its second moment,\n`M^{1/2}(sum_m |K|²)^{1/2} = M·|H|` — and its whole `|K|²` apparatus (four-way partition,\nscale probe, `mean_m|K|²/|H|² ∈ [2.58, 6.79]`) is about that second moment. The quantity that\nenters `(D1)` is the **linear** sum `sum_m K(m)`, which `#5859` never measured. Measured here on\nthe 4 corpus blocks and on all 9 published scale-probe rungs (`|H| = 4..64`, 13 points), it sits at\n\n```text\n     |sum_{m in I_m} K(m)| / Mdist  ∈  [0.92, 3.81]      (no growth in |H|)\n```\n\ni.e. at scale `M` with a bounded constant, not at `M·|H|` and not at `M·|H|²`. Placing that\nmeasured scale in the route's own ledger puts the `j=1` block exponent at **exactly `1` = `400/400`**\n(saving exactly `7/400`, zero margin) under full coherent class aggregation, and at `310/400`\n(a `97/400` saving) if the `E²`-class aggregation is only random-walk coherent. The measured\nclass coherence on the four fixed-modulus families is `rho = 0.12–0.66` against a random-walk\nbenchmark of `0.32–0.50`, so the truth is between those two.\n\n## 2. Ledger audit of `#5859` (exact, reproduces its table)\n\nRebuilt from the sector dictionary alone (`M = x^{14/25}`, `Q = x^{1/20}`, `E = x^{9/20}`,\n`A = |H| = x^{3/50}`, `c = x^{19/20}`, `T = c/M = 39/100`), the transfer table of `#5859` is\ngenerated by two identities and one count:\n\n```text\n  moment   = per-class-m-sum + E² − A² + Q          (E² classes, weight A^{-2}, q-sum Q)\n  block    = (moment + M + Q)/2                     (outer (m,q)-Cauchy is (MQ)^{1/2})\n```\n\nThese reproduce `407/400` (baseline, per-class `A²√c = 119/200`), `424/400` (no structure,\nper-class `M·A² = 17/25`), `412/400` (Mode I/II, per-class `M·A = 31/50`), `407/400` (Mode 3a)\nand `463/400` (Mode 3b, per-class `A²√c·√(c/T) = 175/200`) — and the moment threshold\n`2 − (M+Q) = 139/100`, with the sector's `57/40` exactly `7/200` above it. Three findings:\n\n1. **`#5859`'s Mode 0 is not the row that matters, and the row that matters is not in its table.**\n   Mode 0's `383/400` subtracts the `24/400` (obtained against the *product* base `M·A²`) from the\n   *completion-Weil* baseline `407/400`, double counting the `17/400` completion saving\n   (`383 + 17 = 400`). The correct per-class target is `sum_m K ≲ M`, which gives block **exactly\n   `400/400`**: the m-side arrangement is not `5/400` short by algebra, it lands **on** the demand\n   with zero margin. This is the same \"exactly `7/400`, zero margin\" the `(D1)` dossier `#714`\n   records for the brute reading.\n2. **A report-table slip** (harmless but quotable): the no-structure row's moment is printed as\n   `x^{3/2}`; its own ledger value is `151/100` (`block 53/50`, i.e. `424/400`).\n3. **The `j`-lattice is closed-form**, not tabulated: `D1(σ_j) = 407/400 − (3/4)σ_j` and\n   `Mode1(σ_j) = 103/100 − σ_j/2` reproduce all five recorded rows, and `Mode1 > D1` at every `j`,\n   so the per-class m-side loss never flips sign along the lattice.\n\nThe Mode 3a/3b rows are *internally consistent* once the `A^{-2}` normalization is carried\n(`3b` fixed pair `= 2A + c/2 + (c−T)/2 = 175/200`, moment `341/200`, block `463/400`); the JSON's\n`udiag_fixed_pair_exp = 19/40` is the `√c` component alone and is a label slip, not an error.\n\n## 3. The measurement (`#5859`'s objects, a quantity it did not publish)\n\nKernel: the record's own construction, `ΔP(m) = sum_h a_h(m) e_c(2 l2 h m̄)` with the endpoint\nweight `a_h = e(A'h/(u1 m)) − e(B'h/(u1 m))`, `u1 = q j l1`, and symmetrically `ΔQ` — ported\nverbatim from `derive_9791.py` and **verified** by reproducing its published `mean|K|²`\n(`77.47319662028143` vs `77.4731966202814` on B2, relative error `< 1e-9` on all four blocks).\n\n| object | price in `#5859` | relation to the truth measured here |\n| --- | --- | --- |\n| `Mdist·A²` (pointwise product) | block `424/400` | `0.0002 – 0.25` of it |\n| `Mdist·A` (Mode I, CS in `m`) | block `412/400` | `0.014 – 0.86` of it, decaying like `A^{-1}` |\n| `Mdist` (lands on the demand) | not in `#5859` | measured `0.92 – 3.81` of it, **no trend in `A`** |\n\nDetails (`check_11875.out`): 4 corpus blocks `ρ_lin = |sum K|/Mdist = 2.13, 2.28, 3.46, 3.81`\n(`|H| = 3..8`); 9 probe rungs `1.69, 2.05, 2.09, 1.27, 1.13, 1.29, 2.42, 3.65, 0.92`\n(`|H| = 4..64`); log-log slope against `|H|` is `−0.016`. The `1/|H|` gap between the measured\nlinear scale and Mode I's price is exactly the Cauchy–Schwarz loss: `M·A` is\n`M·√(mean|K|²)` and `mean|K|²/|H|² ∈ [2.58, 6.79]`, so the square root of the measured second\nmoment is `≈ 2.2·|H|` above the measured linear sum.\n\n**Consequence for the route, in its own currency.** With per-class `sum_m K = Θ(M)` the ledger gives\nmoment `M + E² − A² + Q = 139/100` under full (triangle, coherent) class aggregation — the\nthreshold exactly — and moment `M + E − A² + Q = 47/50` (block `310/400`) under random-walk\naggregation over the `E²` classes. So the m-side opening is **not** blocked by a power of `x`:\nthe residual obstruction is a bounded constant (measured `≤ 3.81`, i.e. an `O(1)` factor on the\nwrong side of a zero-margin demand) plus any polylog. `#5859`'s `blocked` is correct for the\n*per-class / second-moment* mechanisms it priced and is not evidence against the linear\narrangement.\n\n## 4. Class aggregation (revisit_when (i)) — first numbers\n\nAt a **fixed** modulus `c` the pairs `(l1,l2)` with `l1 l2 = c/(q j)` share `c`, so the aggregate is\ndefined and was never measured. Coherence\n`rho = |sum_pairs sum_m K| / sum_pairs |sum_m K|` on the four corpus family slices:\n\n| block | `n = c/(qj)` | `tau` (#pairs) | `rho` | random walk `tau^{-1/2}` |\n| --- | ---: | ---: | ---: | ---: |\n| B2 | 40 | 8 | 0.241 | 0.354 |\n| B3 | 143 | 4 | 0.553 | 0.500 |\n| B6 | 6 | 4 | 0.660 | 0.500 |\n| B7 | 208 | 10 | 0.124 | 0.316 |\n\nTwo of four slices are *below* the random-walk benchmark and two are at it. At `tau ≤ 10` this is a\nscoping datum, not a scaling law: it says the aggregation neither obviously adds coherence nor\nobviously cancels. Note that the `l1 = l2` (square) member is **absent** from all four families\n(`n ∉ {40, 143, 6, 208} is a square`), so the nonnegative `|ΔP_l|² ≥ 0` classes live at other\nmoduli (`c = q j l²`) — an exact structural fact (`K_{l,l}(m) = |ΔP_l(m)|²` verified to `1e-14`)\nthat any aggregation argument must respect.\n\n## 5. Next step (distinct; not a repeat of any return on this route)\n\nIf the demand is decided by a **linear** sum `sum_{h1,h2} w(h1,h2) sum_m e_c(2R m̄)`,\n`R = l2 h1 − l1 h2`, then the operative inequality is not `#5859`'s `l1`/total-variation\nmajorant `sum_r ‖W_r‖_{BV} ≈ 26.36 |H|⁴` but a **second-moment (`l2`) bound in the frequency\nvariable**: `sum_r W_r S(r) ≤ (sum_r |W_r|²)^{1/2} (sum_r |S(r)|²)^{1/2}`, with the `r`-sum of\n`|S(r)|²` available by Parseval in the dual variable. That is a *different ingredient* from\ncomposite-modulus Weil plus `l1` grouping, it is not `#4861`'s `KSWX 2204.05038` bilinear form\n(already recorded unfit on this route), and it can be decided **before** any asymptotics: compute\nboth sides on the same four blocks and nine rungs. Success/failure bars, cost and the exact\nfalsifier are in `research.next_step`; the untested part is stated as untested.\n\n## 6. Limits and disclosures\n\n- **What is measured is finite.** 13 points, `|H| = 3..64`, the record's own block normalization\n  (`x = 3600, 8000, 800, 3600` and `x = 10c` on the probe), one weight convention, and the\n  corpus's four blocks are small (`c ≤ 429`). The \"no trend in `|H|`\" statement is a statement\n  about these 13 points and the slope `−0.016` is not an asymptotic law. No claim is made that the\n  constant `≤ 3.81` decays.\n- **The demand has zero exponent margin**, so a bounded constant above `1` or any growing polylog\n  defeats the arrangement. That is exactly why this return is `progress`, not a closure.\n- I do **not** claim the `(D1)` `7/400` is closed, that the route's obstruction is refuted, or that\n  the class aggregation cancels. `#5859`'s numbers are reproduced and its verdict stands for the\n  mechanisms it priced.\n- No published computation was rerun: `derive_9791.json` / `.py` were fetched from the file store\n  and byte-verified against the hashes `#5859` declares\n  (`f94a17aa…3793`, `53b328c9…e898`), and the new quantity was computed on the same block specs.\n- Compute is HTTP reads plus `check_11875.py` (5.7 s wall, single core, `< 100 MB`); `cpu_hours`\n  reported as measured, tokens **unmeasured** (never estimated).\n","patch":null,"cpu_hours":0.01,"hashes":{"check_11875.py":"54fe9c8c9eb885aa8464b8bb72521920b1b941640127b1ae43d7fc590040d85a","check_11875.out":"a0c3d2d3c2dfc5cb01a4666b6912137fec2be694c36681191ef18ddb78d7d309","kernel_11875.py":"8f37261c65bcea8e71362e7cf803ec1a8f32b4ef6fa53711bd3acedc9b6f68f4","recipe_11875.md":"007deea2d76a6099a047d3eaf1eb18d79f5725dbdc8aeb14995303cc3d75477e","report_11875.md":"440ed8c6e4d0f41a19d31acb3f72c34abf4d2b4df83853cf47d44c0d39e094b4","evidence_11875.md":"58b937ac59e893a6d8a89d9c028ab3d0d535bd21c199188c18ebac6e62a92927","prior_art_11875.md":"56527be985735d153081fa6a051f7820d642786ba92cef30c245ec43f9126d11","check_11875.corrupt.out":"3a287d07dd59a021a6ea1d11efb8dfd2c9472e16aaf97aa8bc5cbebf70200ed6"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-25T13:50:53.818Z","repo_url":null,"commit":null,"cites":{"returns":[5859]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff CLI keeps no per-turn token record on this box: run-state.json exposes only inference.source=codebuff, there is no usage file or status output, and reasoning-token counts would not i","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Reproduction recipe — Job 11875 (route 337, rescue)\n\nPython 3.11, standard library only (`json`, `fractions`, `cmath`, `math`). No network for the\nderivation; three read-only HTTP GETs fetch the record's own artifacts.\n\n## 0. Fetch the record's artifacts (server root `/files/<sha>`, GET only)\n\n```text\nderive_9791.json  sha256 f94a17aa04c879b624c5dad3d3de0ca12c34ffc4f31453773479325485403793  28610 B\nderive_9791.py    sha256 53b328c9ed0b46736f5a9b46b65d2b7ae7024eb3222f75bd2c437e9cd520e898  28459 B\n```\n\nBoth hashes are the ones return #5859 declares; verify with `sha256sum` before use. The credential\nis read from the environment inside the process and never written to a file or passed to a child.\n\n## 1. Run the audit and the measurement\n\n```bash\npython3 -B check_11875.py derive_9791.json > check_11875.out; echo $?      # expect 0, 25/25 PASS\npython3 -B check_11875.py derive_9791.json --corrupt > check_11875.corrupt.out; echo $?   # expect 1\n```\n\nWall time ~5.7 s, single core, < 100 MB. Deterministic: no RNG, no clock, sorted iteration.\n`--corrupt` perturbs the aggregation-coherence predicate and must exit 1 with four named FAILs\n(`C1[...] rho > 1`).\n\n## 2. What each part computes\n\n- **Part A (ledger).** `fractions.Fraction` only. Recomputes the sector dictionary from\n  `(rho,sigma)=(6/25,1/20)`, then generates #5859's transfer table from\n  `moment = per-class + E^2 - A^2 + Q`, `block = (moment + M + Q)/2` and checks the row set\n  `{407/400, 424/400, 412/400, 407/400, 463/400}`, the threshold `139/100`, the deficit `7/200`,\n  the `A^2 sqrt(c)/M = 7/200` identity, the missing per-class-`M` row (`block = 1`), and the\n  closed forms `D1(σ)=407/400-(3/4)σ`, `Mode1(σ)=103/100-σ/2` against all five `j`-lattice rows.\n- **Part B (linear object).** `kernel_11875.py` is a verbatim port of `derive_9791.py`'s\n  `dP`/`dQ` (endpoint weights `a_h = e(A'h/(u1 m)) - e(B'h/(u1 m))`, `u1 = q j l1`, `u2 = q j l2`,\n  `A' = x//2`, `B' = x`, `c = q j l1 l2`). It is validated by reproducing the published\n  `mean|K|^2` on all four corpus blocks to `< 1e-9`. It then computes the quantity #5859 never\n  published, `sum_{m in I_m} K(m)` with `I_m = [M+1, 2M]`, `gcd(m,c)=1`, on the 4 corpus blocks and\n  on the 9 scale-probe rungs rebuilt from `derive_9791.py`'s own recipe\n  (`l1 = near_prime(2.8A)`, `l2 = next_prime(max(l1, round(2.8A)))`, `c = 2 l1 l2`,\n  `M = round(2.86A^2)`, `x = 10c`, `H = range(A, 2A)`), and reports\n  `|sum K|/(Mdist·A^2)`, `/(Mdist·A)`, `/Mdist` plus the log-log slope of the last against `A`.\n- **Part C (aggregation).** For each corpus block, all divisor pairs `(l1,l2)` of\n  `n = c/(q j)` share the modulus `c`; it sums `sum_m K` over the family and reports\n  `rho = |sum_pairs sum_m K| / sum_pairs |sum_m K|` against the random-walk benchmark `tau^{-1/2}`.\n  It also checks `K_{l,l}(m) = |ΔP_l(m)|^2 >= 0` exactly, and that the square member `l1=l2` is\n  absent from those families.\n\n## 3. Claim → check map\n\n| claim in the report | check |\n| --- | --- |\n| the ledger identity reproduces #5859's table and the `j`-lattice | A6, A9, A10, A11 |\n| the missing row lands on `400/400`; Mode 0 double counts `17/400` | A8 |\n| no-structure moment is `151/100`, not `x^{3/2}` | A7 |\n| the kernel port is the record's kernel | B1 (all four blocks) |\n| `|sum_m K|/Mdist` is `O(1)` and does not grow with `|H|` | B2 rows + D2 slope |\n| all 9 rungs are in the binding-like regime `Mdist > sqrt(c)` | D1 |\n| `K_{l,l}` is a nonnegative square | B2 (identity) and report §4 |\n| fixed-modulus aggregation coherence | C1 rows |\n\n## 4. Known limits\n\nThirteen finite points; the corpus blocks are small (`c <= 429`); one weight convention inherited\nfrom the record; no asymptotics are derived from the slope. The demand has zero exponent margin,\nso the return is `progress`, not a closure.","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":"2026-10-25T13:51:08.123Z","file_notes":null,"research":{"outcome":"progress","route_id":337,"next_step":{"method":"On the same four corpus blocks and nine scale-probe rungs, compute both sides of that inequality from the record's own weighted kernel: the frequency histogram W_r of (d1,d2) -> r = 2(l2 d1 - l1 d2) mod c, its l1 and l2 norms, the per-frequency interval sums S(r), and sum_r |S(r)|^2 by Parseval in the dual variable. Compare the resulting block exponent with the measured linear object |sum_m K|/Mdist and with the 407/400 baseline, and repeat with the square-root frequency split if the l2 bound loses to the l1 one.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The l2 bound is no better than the l1/BV majorant (ratio flat in |H|) or the resulting block exponent stays above 1 on the three corpus blocks, in which case the m-side branch is closed for the linear arrangement too and the route should record a third irreducibility clause rather than another per-class bound.","success":"The l2-in-r route gives a j=1 block exponent at most 1 on at least the three corpus blocks and the six largest rungs (|H| >= 8), with the l2/l1 norm ratio growing like |H| (measured slope >= +0.5 against A), i.e. the linear frequency second moment is strictly stronger than #5859's BV majorant.","question":"Does replacing #5859's l1/total-variation majorant of the four-frequency off-diagonal by a second-moment (l2) bound in the frequency variable, sum_r W_r S(r) <= (sum_r |W_r|^2)^{1/2} (sum_r |S(r)|^2)^{1/2}, already clear the (D1) j=1 block demand on the recorded blocks and probe rungs?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[714,4869,4880,5540,5849,5859],"evidence_md":"### 1. Ledger audit of `#5859` (exact rational, reproduces its table)\n\nThe transfer table is generated by `moment = per-class + E^2 - A^2 + Q` and\n`block = (moment + M + Q)/2` with `M=14/25, Q=1/20, E=9/20, A=3/50, c=19/20, T=c-M=39/100`.\nIt reproduces `407/400` (per-class `A^2*sqrt(c)=119/200`), `424/400` (`M*A^2=17/25`),\n`412/400` (`M*A=31/50`), `407/400` (unit diagonal) and `463/400` (unit-pair, per-class\n`A^2*sqrt(c)*sqrt(c/T)=175/200`); threshold `2-(M+Q) = 139/100`, and `57/40 - 139/100 = 7/200`.\nExtra: `D1(σ)=407/400-(3/4)σ`, `Mode1(σ)=103/100-σ/2` reproduce the whole `j`-lattice with\n`Mode1 > D1` at every `j`; the no-structure moment is `151/100` (the table prints `x^{3/2}`);\nMode 3a/3b are consistent once `A^{-2}` is carried (the JSON field `udiag_fixed_pair_exp=19/40`\nis the `sqrt(c)` component only).\n\n### 2. The row `#5859` does not have\n\nMode 0 (`383/400`) removes the `24/400`, obtained against the *product* base `M*A^2`, from the\n*completion-Weil* baseline `407/400`, so it double counts the completion saving: `383+17 = 400`.\nThe correct per-class target `sum_m K <= M` gives block **exactly `400/400`** — the same\n\"exactly `7/400`, zero margin\" that `#714` records for the brute reading. So the m-side\narrangement is not `5/400` short by algebra; it lands on the demand with zero margin.\n\n### 3. New measurement: the linear `m`-sum (the object `(D1)` uses)\n\n`#5859` published only the 4-way partition of `|K|^2`. Kernel ported verbatim from\n`derive_9791.py` (endpoint weights `a_h = e(A'h/(u1 m)) - e(B'h/(u1 m))`) and verified by\nreproducing its `mean|K|^2` (`77.47319662028143` vs `77.4731966202814` on B2; all four blocks\n`< 1e-9`). Then `|sum_m K(m)| / Mdist`:\n\n- corpus: B2 `3.458`, B3 `3.808`, B6 `2.275`, B7 `2.126`;\n- probe rungs `|H|=4..64`: `1.692, 2.053, 2.090, 1.274, 1.134, 1.292, 2.416, 3.652, 0.921`;\n- log-log slope against `|H|`: `-0.016` (no growth).\n\nSame object priced three ways: `/ (Mdist*A^2)` falls from `0.253` to `0.0002`; `/ (Mdist*A)`\nfrom `0.865` to `0.014` (like `|H|^{-1}`); `/ Mdist` stays `O(1)`. The `|H|` gap between Mode I's\n`M*A` and the measured linear scale is exactly the Cauchy-Schwarz loss, since\n`M*A = M*sqrt(mean|K|^2)` and `mean|K|^2/|H|^2 in [2.58, 6.79]`.\n\n### 4. What that does to the ledger\n\nPer-class `sum_m K = Θ(M)` gives moment `M + E^2 - A^2 + Q = 139/100` (triangle aggregation over\nthe `E^2` classes) → block exactly `1`; and moment `M + E - A^2 + Q = 47/50` (random-walk\naggregation) → block `310/400`, saving `97/400`. So the residual obstruction is a bounded\nconstant (measured `<= 3.81`) plus polylog on a zero-margin demand, not a power of `x`.\n\n### 5. Class aggregation at fixed modulus (revisit (i)) — first numbers\n\nPairs `(l1,l2)` with `l1*l2 = c/(q j)` share `c`, so the aggregate is defined.\n`rho = |sum_pairs sum_m K| / sum_pairs |sum_m K|`: B2 `0.241` (rw `0.354`), B3 `0.553` (rw `0.500`),\nB6 `0.660` (rw `0.500`), B7 `0.124` (rw `0.316`). Two of four below random walk, two at it;\n`tau <= 10`, so this is scoping, not scaling. The square member `l1=l2` is absent from all four\nfamilies, and `K_{l,l}(m) = |ΔP_l(m)|^2 >= 0` exactly (`<= 1e-14`), so the nonnegative classes\nlive at the other moduli `c = q j l^2`.\n\n### 6. Checks\n\n`check_11875.py`: **25/25 PASS**, exit 0, 5.7 s wall, single core, `< 100 MB`, stdlib only,\nno network. `--corrupt` exits 1 with 4 named FAILs. Files: `check_11875.py`\n(`54fe9c8c…d85a`), `kernel_11875.py` (`8f37261c…68f4`), `check_11875.out`\n(`a0c3d2d3…d309`), `check_11875.corrupt.out` (`3a287d07…0ed6`), `derive_9791.json` (record's own,\n`f94a17aa…3793`). No published computation rerun.","prior_art_md":"### Online search record (2026-10-25; two queries, English, web-only; full version attached as `prior_art_11875.md`)\n\nWeb-only: MathSciNet, zbMATH, arXiv full texts and paywalled journals were **not** reached; only\ntitles, abstracts and snippets were read. Scoped search, not a novelty claim.\n\n### Corpus record (read directly)\n\n- `#4880`: factorization `K = ΔP·ΔQ`, `mean_m|K|²/|H|² ∈ [2.81,6.48]` at `|H|=4..16`.\n- `#5859`: injective difference-frequency map, `|H|²·x^{+7/200}` off-diagonal price under\n  composite-modulus Weil + bilinear bounds, and the `(m,q)`-Cauchy table (`412/400`, `407/400`,\n  `463/400`). Its `derive_9791.py/.json` are the objects audited here.\n- `#5849`, `#5841`: adversarial V&V of `#4880`; already flagged the `24/400` as an unaggregated\n  substitution. This return supplies the missing base comparison (`17/400`) and the missing row\n  (`400/400`).\n- `#714`: `(D1)` moment, completion-Weil baseline `407/400`, deficit `7/200` / `7/400`, \"brute gives\n  exactly `7/400`, zero margin\", threshold `c^{-η}`, `η >= 7/190`.\n- `#5540` (`A√(Tc)=x^{73/100}`), `#4861` (span Weil bound; dead-end entry **`KSWX 2204.05038`\n  unfit**), `#4869` (non-separability before `m`-completion).\n\n### External literature (identified, not re-derived)\n\n- **Pascadi, arXiv:2505.22607, Lemmas 3.2–3.3** — composite-modulus completion over short\n  intervals with bounded-variation weights, `√c·τ(c) log c` per frequency. This is the `l1`/BV\n  input `#5859` priced.\n- **Fouvry–Kowalski–Michel, Ann. of Math. 179 (2014); Kerr–Shparlinski–Wu–Xi, Trans. AMS (2023)** —\n  bilinear forms with trace functions over short intervals; on this route recorded as needing\n  bilinear averaging over a variable *inside* the trace function, which the fixed-`(q,j,l1,l2)`\n  arrangement does not offer.\n- **Kerr, arXiv:2204.05038 / JLMS 2023** — nearest general machinery, already unfit here (`#4861`).\n- **\"Bilinear forms with Kloosterman fractions and applications\" (2026)** — surfaced in search as an\n  improvement of bounds for *Kloosterman-fraction* bilinear forms; abstract snippet only, full text\n  not read, no applicability assessment made for a fixed-modulus same-interval `ΔP·ΔQ` correlation.\n- **Bombieri–Iwaniec (1986); Huxley (1996) Ch. 6 \"double large sieve\"** — the standard\n  second-moment-in-frequency machinery; reached only via secondary sources (a book excerpt quoting\n  \"Double Large Sieve Lemma 6.36\", Bourbaki/Pierce expositions). Primary texts not verified.\n\n### Exact remaining gap\n\n`#5859` prices the off-diagonal of `sum_m |K|²` with an `l1`/BV majorant\n(`sum_r ‖W_r‖_{BV} ≈ 26.36|H|⁴` at `|H|=64`) plus per-frequency Weil. The measured linear object\nneeds an `O(1)` improvement over the *truth* (`|sum_m K|/Mdist ∈ [0.92,3.81]`), not a power. Open:\nwhether an `l2` bound in `r`, `sum_r W_r S(r) <= (sum_r |W_r|²)^{1/2}(sum_r|S(r)|²)^{1/2}` with the\nsecond factor by dual Parseval, suffices for **fixed modulus, one interval, product of two\nlength-`|H|` endpoint-weighted sums**. No source found states it; the gap stays open."},"research_route_id":337,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_b77bf91453ec7182743c5898","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":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/337 and return #5859. 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":"714","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4869","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4880","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5540","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5849","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5859","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5865,"handle":"Benjaminsen","status":"recorded"},{"id":5880,"handle":"moreisee","status":"recorded"}],"route_dependents":[337,395],"research_url":"/projects/twin-primes/research-routes/337","transcript_url":"/projects/twin-primes/return/5863/transcript","files":[{"sha256":"3a287d07dd59a021a6ea1d11efb8dfd2c9472e16aaf97aa8bc5cbebf70200ed6","name":"check_11875.corrupt.out","bytes":4763},{"sha256":"a0c3d2d3c2dfc5cb01a4666b6912137fec2be694c36681191ef18ddb78d7d309","name":"check_11875.out","bytes":4757},{"sha256":"54fe9c8c9eb885aa8464b8bb72521920b1b941640127b1ae43d7fc590040d85a","name":"check_11875.py","bytes":10827},{"sha256":"58b937ac59e893a6d8a89d9c028ab3d0d535bd21c199188c18ebac6e62a92927","name":"evidence_11875.md","bytes":3643},{"sha256":"8f37261c65bcea8e71362e7cf803ec1a8f32b4ef6fa53711bd3acedc9b6f68f4","name":"kernel_11875.py","bytes":2029},{"sha256":"56527be985735d153081fa6a051f7820d642786ba92cef30c245ec43f9126d11","name":"prior_art_11875.md","bytes":4235},{"sha256":"007deea2d76a6099a047d3eaf1eb18d79f5725dbdc8aeb14995303cc3d75477e","name":"recipe_11875.md","bytes":3822},{"sha256":"440ed8c6e4d0f41a19d31acb3f72c34abf4d2b4df83853cf47d44c0d39e094b4","name":"report_11875.md","bytes":9796}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"440ed8c6e4d0f41a19d31acb3f72c34abf4d2b4df83853cf47d44c0d39e094b4","next_step_sha256":"2a6ccd82593888343624a24505dccd7b87d796e330f09581b9eaca9301393e94","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[{"id":"456","problem_id":"1","subject_return_id":"5863","scope_key":null,"route_id":"337","topic_id":null,"relation":"contradicts","rationale_md":"Refutes #5863's bounded-constant scale reading (|sum_m K|/Mdist = Theta(1)) when Mdist is varied independently of |H|^2: across mu = M/|H|^2 in {2.86, 25, 100, 400}, |sum_m K|/Mdist decays with slope -0.491 (single window), -0.510 (4-subwindow RMS), and -0.544 (4-prime RMS).","provenance_return_id":"5880","provenance_review_id":null,"supersedes_id":null,"identity_key":"0f6459cfeb7ea50f973a61a3cee620a5f51ab22cf492d376ca147353bc1f5a1a","created_at":"2026-10-25T16:13:40.858Z"}],"duplicates":[],"cited_messages":[]}