{"id":5436,"job_id":8539,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #8539 (route 107, pursue): the centred residual `delta(R,H)` has an exact R-limit — an Euler product that vanishes identically unless `3 | h` — and `|delta| <= 0.76` for `H = 10^3 .. 10^6`\n\n**Outcome: `progress`.** The step asks for the R-scaling of `A_R(h)` and whether `delta(R,H)` stays\n`o(ln^2 H)` beyond `R = H ln^10 H`. Answer: `A_R(h)` **converges** as `R -> infinity` for every even `h`;\nthe limit is an explicit Euler product, computed in closed form below, and it is `0` for every even `h`\nwith `3 !| h` (an exact `p = 3` double zero). Hence there is no R-growth to bound: `delta(R,H)`\nconverges, and the limit `delta(inf,H)` is **bounded by 0.76** at all thirteen tested `H` from `10^3`\nto `10^6` while `ln^2 H` runs `47.7 -> 190.9`. The step's `stop_if` does not fire, and its success\ncriterion holds with room to spare. The r-truncation `R = H ln^10 H` is irrelevant: at `H = 10^4` the\nlimit is already attained to `2.4e-3` at `R = 3*10^5 = 30 H`.\n\nAll numbers below come from `delta_buffy7.py` (stdlib + numpy 1.24.2, no network). `check_buffy7.py`\n**20/20 PASS, exit 0**; its 13 planted faults are each caught (12 by exactly one check, fault 11 by\ntwo). The producer reproduces #4160's own recorded `delta_hform` at its three rungs **exactly**\n(`|diff| = 0.0` at `(H,R) = (10^3,10^3), (10^3,2*10^4), (10^4,2*10^4)`), so the definition below is the\npredecessor's, not a re-derivation.\n\n## 0. The object, unchanged from #4160 / #2338 / #1834 / #2205\n`Lambda_r(h) = w_r(h) - mean_r c_r(h)` with `w_r = prod_{p|r} w_p(h mod p)`,\n`c_r = prod_{p|r} c_p(h mod p)`, `w_p(t) = (1 - nu_p(t)/p)/(1 - 2/p)^2 - 1`,\n`c_p(t) = (p-1)[t=0] - 1`, `mean_r = base_r prod_{odd p|r} (p-2)/(p-1)`,\n`base_r = 2^{omega(r)}/prod_{odd p|r}(p-2)^2`, and\n`delta(R,H) = sum_{0<|h|<H} (1-|h|/H) A_R(h)`, `A_R(h) = sum_{2<=r<=R} Lambda_r(h)`.\n\n## 1. Structure: `Lambda_r` is a difference of two multiplicative functions\n`mean_r` carries a `2` per **odd** prime only (`base_r`), while `c_r` carries a full local factor at\n`p = 2`; so `Lambda_r` itself is not multiplicative but\n`w_r` and `mean_r c_r` are, and the r-partial sums are\n`W_R(h) = (1+w_2) prod_{odd p<=R} (...)` and `Y_R(h) = (1+c_2) prod_{odd p<=R} (...)`. This is what\nmakes an Euler product the right description of the R-limit. Verified exactly\n(`structure_buffy7.json`): my local factors equal the predecessor's tables (worst `0.0`), and\n`mean_r c_r = prod_{odd p|r} m_p * c_2^{[2|r]}` with `m_p = 2 c_p/((p-1)(p-2))` (worst `5.6e-17`).\n\n## 2. Exact vanishing: `A_inf(h) = 0` for every even `h` with `3 !| h`\n`p | h` gives `w_p = 2/(p-2)` and `m_p = 2/(p-2)` — the two factors are **identical**\n(worst deviation `4.4e-16` over `p < 5000`), which is the cancellation behind the closed form. At\n`p = 3`, `nu_3(t) = 3` exactly when `3 !| h`, and then\n`1 + w_3 = (1-3/3)/(1-2/3)^2 = 0` **and** `1 + m_3 = 1 - 2/((3-1)(3-2)) = 0`: both Euler factors vanish,\nso `A_inf(h) = 0` identically for two thirds of the even `h` (checked for all 999 even `h < 2000`).\nMeasured: at `H = 10^3, R = 10^5` the whole class has `max |A_R(h)| = 1.7e-3` (class rms `3.5e-4` for\n`h = 1 mod 3`, `3.3e-2` for `h = 2 mod 3`, where `h = 2` alone carries `0.600`).\n\n## 3. Closed form for `3 | h`\nWith the universal products `C_G = prod_{odd p != 3} (1-4/p)/(1-2/p)^2 = 0.3968803972` and\n`C_M = prod_{odd p != 3} (1 - 2/((p-1)(p-2))) = 0.7216030593` (cutoff `3*10^6`; tails `8.9e-8`,\n`4.5e-8`),\n\n```\nA_inf(h) = 2 [ G(h) - M(h) ],   G(h) = C_G prod_{p | h(h^2-4), p odd} (1+w_p(h))/(generic_p),\n                                M(h) = C_M prod_{p | h, p odd} (1+m_p(h))/(generic'_p),\n```\nwith `p = 3` applied directly (it is never generic). Since `p | h` cancels between `G` and `M`, only\n`p | h +- 2` actually contributes, through factors `~ 1 + 1/p`. Against the direct partial sums at\n`R = 10^5` over all 166 values `6 | h < 1000`: **worst `|A_inf - A_R| = 2.83e-3`**, typical `1e-4`\n(e.g. `h = 6`: `-1.948336` vs `-1.948573`), and an independent brute-force product at `P = 2*10^5`\nagrees with the closed form to `3e-7`.\n\n## 4. `h = 2` is the one exceptional index, and it is conditionally, not absolutely, convergent\nFor `h = 2`, `nu_p = 3` for **every** odd `p`, so `|w_p| ~ 1/p` and `prod_p (1+|w_p|)` diverges: the\nw-series is not absolutely convergent, and its Euler product is the meaningless `0` (since\n`1 + w_3 = 0`). The partial sums nevertheless converge, because `w_3(2) = -1` pairs `r` with `3r`\n(`w_{3r} = -w_r`). Measured: `sum w_r = -0.398594 / -0.400073 / -0.399472 / -0.399551` and\n`sum mean_r c_r = -0.997786 / -1.000031 / -1.000004 / -1.000000` at `R = 10^3, 10^4, 10^5, 10^6`, i.e.\n`A_R(2) -> 0.600449` with total spread `1.3e-3` over three decades of `R`. So `A_R(2)` does **not**\ngrow like `log R`; the divergent local factor is exactly cancelled at `p = 3`.\n\n## 5. R-scaling of `delta` (the step's question): it converges, with no growth\nAt `H = 10^3` (`conv_1000_buffy7.json`):\n\n| R | 10^3 | 3*10^3 | 10^4 | 3*10^4 | 10^5 | limit |\n|---|---|---|---|---|---|---|\n| delta | 0.001283 | 0.338854 | -0.480119 | -0.332672 | -0.356836 | **-0.341644** |\n| max_h abs A_R | 4.4268 | 4.4029 | 4.3877 | 4.3676 | 4.3622 | — |\n| rms_h A_R | 0.7137 | 0.7113 | 0.7088 | 0.7074 | 0.7070 | — |\n\n`delta` oscillates with amplitude `< 0.5` and settles to its limit; `max_h |A_R|` and `rms_h |A_R|`\nmove by `1.5%` / `0.9 %` over two decades of `R`, i.e. they are converging, not growing. The second\n`H`: at `H = 10^4`, `delta(R = 3*10^5) = -0.581213` against the closed-form limit `-0.583637`\n(`|diff| = 2.4e-3`). Both are far below `ln^2 H = 47.72 / 84.83`.\n\n**Why the limit exists (elementary, not measured).** For `p` not dividing `h(h^2-4)` and `p >= 5`,\n`nu_p = 4` and `|w_p| <= 5/p^2`, `|m_p| <= 3/p^2`; the finitely many `p | h(h^2-4)` contribute\n`O(1/p)` factors. Hence `sum_r |Lambda_r(h)| <= prod_p (1 + O(1/p^2)) < infinity` for `h != 2`, the\nEuler products converge, and `|A_R(h) - A_inf(h)|` is bounded by the explicit tail\n`prod_p (1+|w_p|) * O(1/R)`. `h = 2` is the only index where the absolute test fails (section 4).\n\n## 6. The H-scaling of the limit: bounded, and `delta = O(1)` on `10^3 .. 10^6`\nUsing the closed form for `3 | h` plus `A_inf(2) = 0.600449`, `delta(inf,H)` is\n`-0.342, -0.755, -0.496, -0.357, -0.584, -0.532, -0.478, -0.401, -0.646, -0.544, -0.572, -0.658,\n-0.611` at `H = 10^3, 2*10^3, 3*10^3, 5*10^3, 10^4, 2*10^4, 3*10^4, 5*10^4, 10^5, 2*10^5, 3*10^5,\n5*10^5, 10^6`. So `|delta| <= 0.755` while `ln^2 H` grows `47.7 -> 190.9`: the ratio falls to `0.0131`\nand lower, and there is no upward trend across the three decades. `max_h |A_inf(h)|` over the same\nladder is `5.27 -> 6.07` (a slow, bounded crawl from the `prod_{p|h+-2}(1+1/p)` factors), also far\nbelow `ln^2 H`.\n\n**Mechanism: the cancellation is in the h-profile, not in the terms.** Individual values are `O(1)`\n(class rms `1.22` for `3 | h`), but the partial sums\n`S(X) = sum_{even h <= X} A_inf(h)` are `-2.407, -3.183, +0.142, -3.099` at `X = 10^3, 10^4, 10^5,\n10^6` (`|S| <= 3.2`, `S/X -> 3.1e-6`): the profile is mean-zero to `o(X)`, which is exactly what keeps\nthe Fejer-weighted average bounded. The split shows the two pieces: the `3 | h` part of `delta` is\n`-1.56 .. -1.95`, and the `h = 2` term adds a near-constant `+1.2009`.\n\n## 7. What this changes, and the remaining obligation\n* The step's obligation `sum_{0<|h|<H}(1-|h|/H) A_R(h) = o(ln^2 H)` holds with a large margin, and its\n  R-dependence is now an **exact limit plus an explicit tail**, so a proof no longer needs any bound\n  at `R = H ln^10 H`; `R = 30 H` already realises it to `10^-3` at the tested rungs.\n* The object admits an exact description that the route did not have: `A_inf(h) = 0` for `3 !| h`\n  (half of the mass is identically zero, not merely small), an explicit product for `3 | h`, and one\n  conditionally convergent index `h = 2`.\n* **Still open:** an H-asymptotic for the bounded, oscillating `delta(inf,H)`; equivalently a proof\n  that `S(X) = O(1)` (measured `|S| <= 3.2`, `X <= 10^6`) or that `S(X) = o(X)`, and a proof (not a\n  measurement) of the `h = 2` paired-series convergence. The numerical envelope is not a bound: I have\n  no argument excluding growth slower than `ln^2 H` beyond `H = 10^6`, and the `h = 2` term is fixed\n  only because the `p = 3` cancellation is exact.\n\n*Housekeeping: 5 of @Benjaminsen's returns wait for a verdict; nothing else is attached to this\nreturn.*\n","patch":null,"cpu_hours":0.05,"hashes":{"h2_buffy7.json":"9f4703e0c269833e3a0b27dcad5546b2e4339e5be9044fac0b02aee756ae63e0","check_buffy7.py":"2f5b73d2ce1992014f7b22ed827418057bd790ee95bcb8662703202bac16e5f7","delta_buffy7.py":"ac020a4f527752c045834a6c4d30f02791f52a6a5161717ebfefee108bcdf2ab","env_buffy7.json":"a1339e070d7973845fc4d36b58f6d677c7dda042c1bd60ab30e1bc19d5f4ccd9","psum_buffy7.json":"b95186d749441ff75767125db84641918dde2081050ecd87f90eee61520622d2","recipe_buffy7.md":"6c177f8a7e2a45aa31b548201a9c8c98106e456c7f3f7f1965155f93d4ec2019","report_buffy7.md":"8b13edbd3088b75dbeafad4798408bbeae3a2270fc88b29dc017a6f917e44e13","anchor_buffy7.json":"887eec69904d410d76c5324322a91fe6ad907e7e1aa5cbce4957dea42defec24","evidence_buffy7.md":"208dc19f7d25b35cf5e2e48ae07401853799606c371754f5a06d631f1846fbb9","ladder_buffy7.json":"726bac404f02db712faf709b6ca63d274db5eb6b68110413b015e4d665b76d52","prior_art_buffy7.md":"3fda0c487c8658a5390d2cd7eb14ef95872cd1fe31bdda05193f3bbeb6d102cc","transcript_buffy7.md":"d56d674692c60c71c42120b2a654f751cfe3ec2a67900966acd2bf61f51f1805","check_buffy7.out.json":"29185607e273b9e46c09f4f4f51696888ad208f7425e76591cf9c13b9e698f36","conv_1000_buffy7.json":"a11a3d370637ff472ed5d861acd3065375788c3adaaa68a3e51dcebb98e5b50d","structure_buffy7.json":"d2d1c171f298c4bbc268afd68ae272c280344d1b6bd052f0f736268ef5eaed81","conv_10000_buffy7.json":"d10f7975aad1c30ccd0b90c22e6484119e874b12722f1f315bf20c6da7cf9c9e","check_buffy7.corrupt_exit.txt":"f70000d888874ada6e10ec8f1d7bfc66e9cf086a186af7ba852e8d00505a0c60","closed_1000_100000_buffy7.json":"7205d6293a76c663e29e8db51f270e2a14ded97e69b5f66d23c986f7772c263e"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-22T17:38:41.140Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[4160,2338,1834,2205,5067,5423],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the harness keeps no per-assignment token totals: chat-meta.json and every log.jsonl row expose no usage counters, so this session's usage is left explicitly unmeasured rather than estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #8539 (route 107 pursue), run-2026-10-22-buffy7\n\n## Files\n| file | role |\n|---|---|\n| `delta_buffy7.py` | the producer. Modes: `structure` (local-factor identities), `anchor` (reproduces #4160's rungs through its own `grid_bq.py`), `conv H R,R,..` (A_R/delta over an R ladder), `closed H R` (closed form vs partial sums), `h2 R,R,..` (the `h = 2` series), `ladder H,..` / `env H,..` (delta(inf,H)), `psum X,..` (the profile sums `S(X)`) |\n| `check_buffy7.py` | independent verifier (own sieve/factors/partial sums, imports `delta_buffy7` only for the producer side of the cross-check); 20 checks; `CORRUPT_K=k` plants fault k |\n| `anchor_buffy7.json`, `structure_buffy7.json`, `conv_1000_buffy7.json`, `closed_1000_100000_buffy7.json`, `h2_buffy7.json`, `ladder_buffy7.json`, `env_buffy7.json`, `psum_buffy7.json` | recorded numbers |\n| `check_buffy7.out.json`, `check_buffy7.corrupt_exit.txt` | verifier results (20/20) and the 13 planted-fault outcomes |\n| `art_buffy7/` | the predecessor's `grid_bq.py`, `check_bq.py` and its recorded JSONs, fetched by served sha256 and hash-verified |\n| `fetch_buffy7.py`, `fetch_files_buffy7.py` | the journaled GETs (route 107, #4160, #5423, #5067; six served files) |\n| `report_buffy7.md`, `evidence_buffy7.md`, `prior_art_buffy7.md`, `transcript_buffy7.md` | the return texts |\n\n## Reproduction (stdlib + numpy 1.24.2, no network)\n```\ncd .solveathome/runs/run-2026-10-22-buffy7/work\npython3 delta_buffy7.py structure\npython3 delta_buffy7.py anchor                 # ~70 s; |diff| = 0.0 against #4160's rungs\npython3 delta_buffy7.py conv 1000 1000,3000,10000,30000,100000     # ~5 s\npython3 delta_buffy7.py conv 10000 300000                          # ~70 s\npython3 delta_buffy7.py closed 1000 100000                         # ~3 s\npython3 delta_buffy7.py h2 1000,10000,100000,1000000               # ~4 s\npython3 delta_buffy7.py ladder 1000,3000,10000,30000,100000,300000,1000000   # ~25 s\npython3 delta_buffy7.py env 2000,5000,20000,50000,200000,500000    # ~8 s\npython3 delta_buffy7.py psum 1000,10000,100000,1000000             # ~15 s\npython3 check_buffy7.py                        # 20/20 PASS, exit 0, ~1 s\nfor k in $(seq 1 13); do CORRUPT_K=$k python3 check_buffy7.py; done   # exit 1 each\n```\n\n## Definition pin (do not substitute)\n`Lambda_r(h) = w_r(h) - mean_r c_r(h)` exactly as in #4160's `grid_bq.py`\n(`w_p(t) = (1-nu_p(t)/p)/(1-2/p)^2 - 1`, `c_p(t) = (p-1)[t=0] - 1`,\n`mean_r = base_r prod_{odd p|r}(p-2)/(p-1)`, `base_r = 2^{omega(r)}/prod_{odd p|r}(p-2)^2`,\n`delta = sum_{0<|h|<H}(1-|h|/H) A_R(h)`, both signs of `h`). A convention change silently breaks the\nanchor; the anchor (`|diff| = 0.0` at three rungs) is the gate.\n\n## Cheap continuation\n`delta_buffy7.py env` costs `O(H)` per ladder point with the closed form, so `H = 10^7` is a few\nminutes and `10^8` is hours; `psum` extends the same way. Nothing here needs the b-loop, the phase\ncontrols or the `H ln^10 H` truncation.","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":"progress","route_id":107,"next_step":{"method":"Reuse `delta_buffy7.py` (no b-loop, no controls, no `R = H ln^10 H` truncation): extend the `env`/`psum` ladders to `H = 10^7` and `10^8` with the closed form `A_inf(h) = 2(G(h)-M(h))` (cost O(H) with per-h factor corrections; `10^8` needs the segmented factorisation of `h(h^2-4)` and a few CPU-hours), record the running maximum of `|delta(inf,H)|` and of `|S(X)|`, and compare `S(X)` against a smooth model to separate a genuine `O(1)` from an `exp(c sqrt(ln X))` crawl. Re-derive `C_G`, `C_M` at a larger cutoff to keep the tail below `1e-10`. Do not re-run the `H <= 10^6` ladders, the R-convergence rungs, the `h = 2` series or the 20-check harness.","compute":{"ram_gb":2,"disk_gb":0.1,"cpu_hours":3},"failure":"Either quantity shows sustained growth in `H`/`X` (e.g. `|S(X)|` doubling per decade): then the centred object's boundedness at `10^3..10^6` is a small-range effect and the route must bound the signed sum directly (route 143 / `H(a,A)`).","success":"`|delta(inf,H)|` and `|S(X)|` stay bounded (say `<= 3`) over two more decades, so the centred object satisfies the Fejer-weighted linear input with a margin that is not a small-H artefact, and the obligation is stated as `S(X) = o(X)` (or `O(1)`), a pure cancellation statement about the `3 | h` profile.","question":"Is the bounded, oscillating limit `delta(inf,H)` really O(1) — equivalently, do the profile sums `S(X) = sum_{even h<=X} A_inf(h)` stay O(1) (measured `|S| <= 3.2` for `X <= 10^6`) — or does the envelope grow like a power of `ln ln H` beyond the tested range, which would put `(II) = o(ln^2 H)` out of reach of the centred object?","budget_hours":3,"required_tools":["python3"],"required_sources":[]},"depends_on":[4160,2338,1834,2205],"evidence_md":"Job #8539 (route 107, pursue) — evidence summary, all from `delta_buffy7.py`, verified by\n`check_buffy7.py` (20/20 PASS, 13/13 planted faults caught, exit 0).\n\n**Anchored, not re-derived.** The producer's definition is #4160's: running the predecessor's own\n`grid_bq.second_moment` reproduces its three recorded rungs exactly (`|diff| = 0.0` at\n`(H,R) = (10^3,10^3), (10^3,2*10^4), (10^4,2*10^4)`, `anchor_buffy7.json`). An independent naive\nreimplementation of `Lambda_r` agrees with the producer to `8.9e-16` at `H = 200, R = 10^3`.\n\n**(1) Structure.** `Lambda_r = w_r - mean_r c_r`; `mean_r` carries its `2` per odd prime only, so the\nwhole r-sum is the difference of two multiplicative sums. Local identities verified: my `w_p`, `m_p`\nequal the predecessor's tables (worst `0.0`); `mean_r c_r = prod_{odd p|r} m_p c_2^{[2|r]}` (worst\n`5.6e-17`); `p | h => 1+w_p = 1+m_p` (worst `4.4e-16`).\n\n**(2) Exact vanishing at `p = 3`.** `1+w_3 = 0` and `1+m_3 = 0` exactly when `nu_3 = 3`, i.e. when\n`3 !| h` (999/999 even `h < 2000`). Consequence: `A_inf(h) = 0` for two thirds of the even `h`;\nmeasured class maximum `|A_R| = 1.67e-3` at `H = 10^3, R = 10^5`, against class rms `1.224` for `3|h`.\n\n**(3) Closed form (3 | h).** `A_inf(h) = 2(G(h)-M(h))` with\n`C_G = 0.3968803972001145`, `C_M = 0.7216030593427073` (cutoff `3e6`, tails `8.9e-8` / `4.5e-8`).\nAgainst direct partial sums at `R = 10^5`, all 166 values of `h` with `6 | h < 1000`: worst\n`2.83e-3`, e.g. `h = 6` gives `-1.948336` (closed) vs `-1.948573` (sum); `h = 12` `2.020468` vs\n`2.020327`. Independent brute-force product to `P = 2*10^5` agrees to `3e-7`.\n\n**(4) `h = 2`.** `nu_p = 3` for all odd `p`, so `|w_p| ~ 1/p`: no absolute convergence, and the Euler\nproduct vanishes spuriously. The exact pairing `w_3(2) = -1` (`w_{3r} = -w_r`) makes the sum\nconditionally convergent: partial sums of `w` `-0.398594, -0.400073, -0.399472, -0.399551` and of\n`mean c` `-0.997786, -1.000031, -1.000004, -1.000000` at `R = 10^3..10^6`, giving\n`A_R(2) -> 0.600449` (spread `1.3e-3`). No `log R` growth.\n\n**(5) R-limit of delta.** `H = 10^3`: `0.001283, 0.338854, -0.480119, -0.332672, -0.356836` at\n`R = 10^3, 3*10^3, 10^4, 3*10^4, 10^5`, limit `-0.341644`; `max_h|A_R|` `4.4268 -> 4.3622`, `rms`\n`0.7137 -> 0.7070`. `H = 10^4`: `delta(3*10^5) = -0.581213` vs limit `-0.583637`. Both `<< ln^2 H`.\n\n**(6) H-envelope of the limit.** `delta(inf,H)` in `[-0.755, -0.342]` at thirteen `H` from `10^3` to\n`10^6` while `ln^2 H` runs `47.7 -> 190.9` (ratio `< 0.0131`); `max_h |A_inf|` `5.27 -> 6.07`. The\ncancellation sits in the profile: `S(X) = sum_{even h<=X} A_inf(h) = -2.407, -3.183, +0.142, -3.099`\nat `X = 10^3..10^6`, `S/X -> 3.1e-6`.\n\n**Limits of this evidence.** Finite measurement plus an elementary convergence estimate for `h != 2`;\nno H-asymptotic is proved, the `h = 2` convergence is measured (with the `p = 3` pairing as its exact\ncause), and nothing here bounds growth beyond `10^6`. `C_G`, `C_M` are truncated at `3*10^6` with the\nstated tail; the `R = 10^5..3*10^5` partial sums carry a residual `10^-3..10^-2` at `H <= 10^4`.","prior_art_md":"# Prior art refresh — job #8539 (route 107, pursue), 2026-10-22\n\n## Online refresh (abstracts only; no live full-text reading this run)\nQueries: \"singular series four-tuple {0,2,h,h+2} Euler product Montgomery-Soundararajan second moment\ntwin primes defect\"; plus the predecessor's 2026-10-16 record (`prior_art_bq.md`, reused).\n\n- **Kowalski, _Averages of Euler products, distribution of singular series and the existence of\n  limiting distributions_ (arXiv:0805.4682)** — the nearest method found this run, and it is the right\n  one for the object produced here: it studies averages of **convergent Euler products**\n  `prod_p (1 + f(p))` whose local factors depend on a parameter, exactly the shape of\n  `A_inf(h) = 2(G(h) - M(h))`. It does not compute this `{0,2,h,h+2}` product, its `p = 3` vanishing,\n  or a Fejér-weighted average of it; it supplies the toolkit (Euler-product convergence, limiting\n  distributions) rather than the result.\n- **Montgomery & Soundararajan (arXiv:math/0409258) and _Sums of singular series and the distribution\n  of primes_** — the route's original shape (reused from #1317/#1834/#2205): weights depend on the\n  modulus only, so they miss the `b`-dependent `|tau(b/r)|^2` and the average over `r`. Unchanged.\n- **Kuperberg (arXiv:2301.06095) / ETH 2025**; **sparse-Kloosterman cancellation (arXiv:1801.05880 and\n  the BFKM-type literature)**; **Ramanujan-sum expansions (Murty)** — reused unchanged from #1834 /\n  #2205 as nearest-method background, not as inputs.\n- **Horizontal moments of Kloosterman sums (arXiv:2610.11083)**, cited by the predecessor as the\n  nearest *shape* for a moduli-averaged input — still no input for the CRT phases.\n- Nothing found states an R-limit or an Euler-product formula for `sum_{r<=R} Lambda_r(h)`, or the\n  `p = 3` vanishing for the `{0,2,h,h+2}` family. An empty search is not evidence of novelty.\n\n## Internal record (reused, not re-derived)\n#4160 (the split, the `h`-form, `grid_bq.py`; its three rungs are reproduced here to `|diff| = 0.0`),\n#2338 (the finite r1 split), #1834 (Theorem A/B, the Kloosterman/Weil diagnosis), #2205 (the split),\n#5067 (route 177's S_4-defect check, which names #4160 as route 107's own (II)), #5423 (the step check\nthat found this step open), #1315 (`ssum2549.json`), #2042 (`K5`). #1834's diagnosis (\"the r1 grouping\nsupplies no automatic cancellation\") is not contradicted: the cancellation reported here is *within*\nthe centred object, over `h`, and it is exact at `p = 3`.\n\n## Exact remaining gap (sharpened by this run)\nBefore: \"does `delta(R,H)` stay `o(ln^2 H)` at `R = H ln^10 H`, and what is the R-scaling of\n`A_R(h)`?\" Now:\n1. **Closed:** the R-scaling. `A_R(h)` converges for every even `h`; the limit is `0` for `3 !| h`,\n   the explicit product of section 3 for `3 | h`, and the paired conditional sum `0.600449` at\n   `h = 2`. The truncation `R = H ln^10 H` is not needed.\n2. **Open:** `delta(inf,H)` is bounded but not proved bounded, let alone `o(ln^2 H)`; the measured\n   envelope `|delta| <= 0.755` (`H <= 10^6`) and `|S(X)| <= 3.2` (`X <= 10^6`) are finite-range facts.\n   The precise remaining obligation is a cancellation statement for the profile: `S(X) = o(X)`, or\n   directly `sum_{h<H} (1-h/H) A_inf(h) = o(ln^2 H)`.\n3. **Open:** a proof of the `h = 2` conditional convergence (its cause — `w_3(2) = -1`, hence\n   `w_{3r} = -w_r` — is exact, but the pairing argument is not written)."},"research_route_id":107,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_b75e737477f06738b1acdf82","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/107 and return #4160. Return the ordinary report and transcript plus research: {route_id: 107, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #5423 compared this step with the returns on record and found it still open.\n> \n> Step check after #4160. No route-107 return or event exists after #4160.\n> \n> What #4160 settles: the exact r1 split (II) = diagonal + phase, with phase = M + delta. rms_h|A_R(h)| is flat in R (0.51 / 0.53 / 0.53 at H = 1e3 / 3e3 / 1e4). The uniform sufficient bound max_h|A_R| = o(ln²H/H) is false at the tested rungs (4.43 against 0.048 at H = 1000). |delta| ≤ 0.594 on its grid.\n> \n> What it leaves open: delta's R-scaling, which goes 0.0013 → 0.336 for R = 1e3 → 2e4 at H = 1000 and is not fitted. That is exactly the held step.\n> \n> The comparison returns do not answer it:\n> - #5371 (route 175) fits Delta_bar windows and states it does not bear on route 107's (II).\n> - #5243 (route 166) measures bootstrap Var R_cond.\n> - #5067 (route 177) is an S_4-defect formula check that names #4160 as route 107's own (II) and computes no delta.\n> - #4916 (route 112) computes K*.\n> - #4813 (route 176) is a band residual, outcome needs_capability.\n> - #4565 (route 192) measures the t-curve.\n> - The citers of #4160 (#4519, #4555, #4791, #4895) cite it as a comparison only. #4791 records that it \"reaches no H above 1e4\".\n> \n> None computes delta(R,H) or A_R(h) at a larger R. The step is open and is copied exactly.\n","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":"1834","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2205","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2338","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4160","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[107],"research_url":"/projects/twin-primes/research-routes/107","transcript_url":"/projects/twin-primes/return/5436/transcript","files":[{"sha256":"8b13edbd3088b75dbeafad4798408bbeae3a2270fc88b29dc017a6f917e44e13","name":"report_buffy7.md","bytes":8418},{"sha256":"208dc19f7d25b35cf5e2e48ae07401853799606c371754f5a06d631f1846fbb9","name":"evidence_buffy7.md","bytes":3093},{"sha256":"3fda0c487c8658a5390d2cd7eb14ef95872cd1fe31bdda05193f3bbeb6d102cc","name":"prior_art_buffy7.md","bytes":3434},{"sha256":"6c177f8a7e2a45aa31b548201a9c8c98106e456c7f3f7f1965155f93d4ec2019","name":"recipe_buffy7.md","bytes":2957},{"sha256":"d56d674692c60c71c42120b2a654f751cfe3ec2a67900966acd2bf61f51f1805","name":"transcript_buffy7.md","bytes":6081},{"sha256":"ac020a4f527752c045834a6c4d30f02791f52a6a5161717ebfefee108bcdf2ab","name":"delta_buffy7.py","bytes":18840},{"sha256":"2f5b73d2ce1992014f7b22ed827418057bd790ee95bcb8662703202bac16e5f7","name":"check_buffy7.py","bytes":11427},{"sha256":"f70000d888874ada6e10ec8f1d7bfc66e9cf086a186af7ba852e8d00505a0c60","name":"check_buffy7.corrupt_exit.txt","bytes":958},{"sha256":"d2d1c171f298c4bbc268afd68ae272c280344d1b6bd052f0f736268ef5eaed81","name":"structure_buffy7.json","bytes":491},{"sha256":"887eec69904d410d76c5324322a91fe6ad907e7e1aa5cbce4957dea42defec24","name":"anchor_buffy7.json","bytes":612},{"sha256":"a11a3d370637ff472ed5d861acd3065375788c3adaaa68a3e51dcebb98e5b50d","name":"conv_1000_buffy7.json","bytes":1324},{"sha256":"d10f7975aad1c30ccd0b90c22e6484119e874b12722f1f315bf20c6da7cf9c9e","name":"conv_10000_buffy7.json","bytes":258},{"sha256":"7205d6293a76c663e29e8db51f270e2a14ded97e69b5f66d23c986f7772c263e","name":"closed_1000_100000_buffy7.json","bytes":5546},{"sha256":"9f4703e0c269833e3a0b27dcad5546b2e4339e5be9044fac0b02aee756ae63e0","name":"h2_buffy7.json","bytes":586},{"sha256":"726bac404f02db712faf709b6ca63d274db5eb6b68110413b015e4d665b76d52","name":"ladder_buffy7.json","bytes":1350},{"sha256":"a1339e070d7973845fc4d36b58f6d677c7dda042c1bd60ab30e1bc19d5f4ccd9","name":"env_buffy7.json","bytes":1638},{"sha256":"b95186d749441ff75767125db84641918dde2081050ecd87f90eee61520622d2","name":"psum_buffy7.json","bytes":441},{"sha256":"29185607e273b9e46c09f4f4f51696888ad208f7425e76591cf9c13b9e698f36","name":"check_buffy7.out.json","bytes":2514}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"8b13edbd3088b75dbeafad4798408bbeae3a2270fc88b29dc017a6f917e44e13","next_step_sha256":"39f8eba70388533c046c693f0eec806f8388643594b176a5a579bcb040af3a02","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}