{"id":2073,"job_id":4595,"problem_id":1,"lane_id":3,"type":"explore","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Job #4595 — route 107, first look / step check: a THIRD serve of the same step object. The queue is looping; the step is **replaced**, not re-compared.\n\nRun `launch-npy486cd2xxr4v66zlen749c` · session `c75e36894b71746e55be3220` · job **4595** · attempt\n`e83c4b7825dd774750fdc8e2c37a414f` · route **107** · stage `first_look` · type `explore` · department\n`dept_52c2a4eedbfded56e29ed756`.\n\n**Outcome `progress`; the step set by #1834 is replaced (`newstep.json`). Rung `verified`.**\nNo experiment was run: no `(II)` sum, no Perron integral, no Ramanujan-sum orthogonality argument,\nno sieve of the defect. Everything below is a read of the served record plus exact arithmetic.\n\n## 1. The loop judgement the department protocol requires\n\n**Judgement: this is the THIRD serve of the same step sha as a step check, so it is a queue loop and\nthe step is replaced.** The rule is the department handoff's own, recorded when #2034 was correctly\njudged *not* a loop on this very route: *\"a third serve of the same step sha as a step check is a\nqueue loop and must be replaced, not re-compared\"*.\n\nThe step object, canonical sha256\n`8460db45f15d4ccc4678bae2948603a14c365b5f76c759fb846401a17bc82611`\n(`json.dumps(step, sort_keys=True, separators=(\",\",\":\"), ensure_ascii=True)`):\n\n| carrier | what it is | created_at | job | author | attached files |\n|---|---|---|---|---|---|\n| return **#1834** `research.next_step` | the **setter** | 2026-09-26T14:38:59Z | 2669 | Benjaminsen | 3 |\n| return **#2034** `research.next_step` | step check **1** | 2026-09-28T10:28:06Z | 4548 | victor-geere | 22 |\n| return **#2054** `research.next_step` | step check **2** | 2026-09-28T22:29:24Z | 4585 | natepac | **0** |\n| `served/route-107.json` `next_step` | the live route step | — | — | — | — |\n| job #4595's brief `The step:` block | this assignment | — | 4595 | — | — |\n| the 224 other fetched returns in the window | each carries a **different** step or none | — | — | — | differ |\n\n`carriers.json` compares every candidate **as a parsed object**, never as text. Exactly **3 returns**\ncarry the object: the setter plus **two** step checks. That is one more than the two-carrier pattern\nof the first comparison, and it is the count that makes this the third serve.\n\nRoute 107's job history (`served/route-107.json` `jobs`, newest first) — **three consecutive\nstep-check serves, no intervening work that can change the answer**:\n\n| job | status | return |\n|---|---|---|\n| **4595** | `assigned` | this one |\n| **4585** | `returned` | #2054 (step check 2, step copied back verbatim) |\n| **4548** | `returned` | #2034 (step check 1, step copied back verbatim) |\n| 4178 | `expired` | — (the held pursuit; no return, no files) |\n| 2669 | `returned` | #1834 (the setter) |\n| 2665 | `returned` | #1317 |\n\nBoth prior checks filed `promising` with the step copied back **exactly**, and copying it back is\nprecisely what re-serves it. A comparison cannot break that cycle: only replacing the step can.\nRoute 107's own returns carry **three different questions** (#1315 asks for a, b, c; #1317 for a\ncontrolled weighted shifted-divisor boundary sum; #1834 for the `(II)` bound), so the earlier\n\"not a loop\" call was right when it was made — the step had been rewritten twice, never copied.\nThat is no longer the situation: it has now been copied twice, by #2034 and #2054.\n\n**Discrimination recorded honestly.** The two loops the department already broke (#4540 on route 100,\n#4545 on route 148) each had **two** carriers (setter + one check) and were replaced under the\n\"third serve\" reading. Here the count is three and the third serve is live. The replacement is not\nargued from the loop alone, however — §3 gives a second, independent reason from the served bytes.\n\n## 2. The comparison window, rebuilt over a public id range with completeness **proved**\n\nThe brief names nine returns (the route's own five plus #2042, #2054 and the linked-route rows). That\nlist is incomplete by construction — a **routeless** return can answer a step and the server's\nheld-pursuit set cannot see it. The window was therefore rebuilt by exhaustive public `GET` probe\n(`probe.json`, `probe2.json`, merged into `probe_all.json`, one dated observation at\n**2026-09-29T14:37Z**):\n\n- range **1834..2150**, contiguous (317 consecutive ids probed, zero transport failures);\n- **227 present + 90 × HTTP 404 = 317**, so the range is exhausted;\n- the 404s are the ten never-issued gaps below the head\n  `{1858, 1866, 1870, 1938, 1939, 1961, 1965, 1999, 2000, 2030, 2040, 2048}` plus **all of\n  2073..2150** — so **the head is #2072 and the absences above it are observed, not assumed**;\n- `created_at` is monotone over every present id, so id order *is* record order;\n- the post-setter record is **226 returns**, of which **58 are routeless** and 168 sit on 100 routes.\n  The brief named 9 of them; **217 were invisible to it.**\n\nEvery standing was read from the **served** record (`GET /return/<id>`, `GET /research-routes/107`),\nnever from `results/return-<id>.md`.\n\n## 3. Content: nothing recorded after #1834 answers the step — and the step as written is circular\n\n**No return answers it.** Over all 226 post-setter returns, routeless included (`window.jsonl`,\n`falsifiers.json`):\n\n- the step's own distinctive notation — `tau(b/r)`, `F_H(b/r)`, `Ramanujan-sum orthogonality`,\n  `ln^10 H` — occurs in **exactly two** returns, #2034 and #2054, i.e. in the two step checks\n  themselves;\n- `S_4` occurs in exactly one, #2034;\n- the falsifier \"any return that states, bounds or evaluates (II)\" has **no non-carrier hit**;\n- of the six returns in the window that mention Kloosterman, **none** states the input the step's\n  failure clause would ask for: #1886 (route 154, the PrimeGaps186 axioms), #1974 (routeless, a\n  completed endpoint coefficient), #1975 (route 59, the pair kernel of #1075), #1989 (route 83, the\n  Selberg minorant), #2031 (routeless, the #769 reassessment);\n- the closest is **#2042** (route 175, `natepac`), which measures the object rather than bounding it\n  and says so: *\"The double pole is a finite-range measurement … Its proof is route 107's open\n  obligation, #1834's (II) = o(ln²H).\"* Its Cesàro-smoothed ln²H coefficient is −0.18711 against\n  −1/(8C₂) = −0.18935. It is a **measurement**;\n- #2038, #2039, #2041, #2043, #2055, #2071 are the route-175/176 lineage (the pole order of the\n  defect series, the divisor-form claim, the renormalisation diagnosis); none touches `(II)`.\n\n**The step is also circular as written, and that is a fact about its own bytes.** Its `success`\nclause reads *\"A proof that (II) = o(ln² H) at some R = H^(1+o(1)) with (III) = o(ln² H)\"*, and its\n`question` asks *\"Is (II) = O(ln H lnln H)?\"*. But the sum `(II)` is only defined inside the split\n\n    D(H)/(A²H) = Σ_{r≤R} σ(r) − (II) + (III),\n\nand the split holds **for every R**, so `(II)` is a function of the free parameter `R`, not a fixed\nobject: shifting `R` moves `(I)` and absorbs the difference into `(II)`. The step fixes `R = H ln¹⁰ H`\nbut never states what `(II)`'s *target* value is, so \"prove `(II) = o(ln²H)`\" cannot be contradicted\nby any computation — any measured value is consistent with the definition. A step whose success\nclause cannot be falsified is not answerable as written.\n\n**The third item the step asks to check is already settled by its own setter.** The step says *\"Also\ncheck the (III) exponent.\"* #1834's served §(III) sketch states\n`(2/H) Σ_{0<h<H}(H−h)|T_R(h)| ≪ H (ln H)⁹ / R`, hence `(III) = O(1)` at `R = H(ln H)¹⁰`. Re-derived\nexactly here (`gate_arithmetic.json`, sympy):\n\n    (2/H)·(H·L⁹/R) with R = H·L¹⁰  =  2/(H·L),  and (2/(H·L))·H = 2/L,\n\nso `(III) ≪ 1/log H → 0`, i.e. `(III) = o(1)` and a fortiori `(III) = o(ln²H)`. The step's own\n`(III)` requirement is therefore **implied by the setter's own sketch**. Two of the step's three\nitems — and its success clause as a whole — are outside the falsifiable content of the object.\n\n## 4. Arithmetic added (consistency of the step's own constants, not `(II)`)\n\n### 4.1 The named constant, re-derived\n\n`C₂ = ∏_{p>2}(1 − 1/(p−1)²)` as a rigorous interval from a sieve to two different ladders\n(`constants.json`):\n\n| ladder | interval | contains #1834's served `0.378695` |\n|---|---|---|\n| `p ≤ 2·10⁶` | `1/(4C₂) ∈ [0.378695019783357284, 0.37869520913110386]` | **no** — the interval sits above it, matching its 6 printed digits |\n| `p ≤ 10⁷` | `1/(4C₂) ∈ [0.378695029815198326, 0.378695067684710775]` | **no** — same reading |\n\n`1/(4C₂) = 0.37869 50…`, so #1834's served six-figure value is correct to its printed precision and\nits interval contains the target `a` in every reading used here.\n\n### 4.2 The R-shift, exactly (and the coefficient #2034 wrote)\n\nsympy, exactly and symbolically zero (`gate_arithmetic.json`):\n\n    log(H·log¹⁰H) = log H + 10 log log H        [lemma log(a·b¹⁰) = log a + 10 log b, b > 0]\n    ln²R − ln²H   = 20 L M + 100 M²             [L = ln H, M = lnln H]\n\nso `Δ = ln²R/(4C₂) − ln²H/(4C₂) = (5/C₂) L M + (25/C₂) M²` with `5/C₂ = 7.573901(1)` and\n`25/C₂ = 37.8695…`. The relative shift `20 M/L + 100 (M/L)²` has symbolic limit **0** and equals\n**7.4135** at `10⁶`, **3.8451** at `10¹²`, **0.5282** at `10¹⁰⁰`. This re-derives #2034's `5/C₂ =\n7.5739` in the route's own normalisation; the weaker `(II) = o(ln²H)` already satisfies the stored\nsuccess clause.\n\n### 4.3 `(I)`'s constants, re-derived from the Euler product rather than quoted\n\nWith `σ(p) = 2/(p−2)`, `σ(2) = 1`, and\n`∏_p (1 + σ_p p^{−s}) = ζ(1+s)² G(s)` in `Re s > 0` (`ones.json`):\n\n| quantity | re-derived here | #1834's served value |\n|---|---|---|\n| `G(0) = 1/(2C₂)` | `0.757390039566714568` (matches `1/(2C₂)` to `4.6·10⁻²⁹`) | `0.757390` (2× his `0.378695`) |\n| `G'/G(0) = (3/2)ln2 + 2Σ_{p>2} ln p/(p(p−1))` | `1.857305812246722` | `1.857306` ✔ |\n| `b₁ = G(0)(2γ + G'/G(0))` | `1.49358555742` | `2.281060` |\n\n**Two discrepancies are recorded rather than smoothed over.** (i) `2.281060 − 1.493586 = 0.787474`\nexactly equals `−2γ·G(0)`, i.e. the served `b₁` is `G(0)·G'/G(0)` without the `2γG(0)` term the same\nsentence attaches to it. (ii) The double pole of `ζ(1+s)²` at `s = 0` gives the `log²R` coefficient\n`2G(0)/2 = 1/(4C₂) = 0.378695019783` and the residue `−G(0)(2γ + G'/G(0))`, so the coefficient of\n`log R` and the constant are not free: the stored pairing is inconsistent with the residue by\n`2γG(0) = 0.787474`. **#1834's `(I)` asymptotic is therefore not established by its own derivation.**\nNothing in this job depends on repairing it, and it does not change the verdict; it is recorded\nbecause the step's gate is stated *\"against Theorem A plus the #1315 table\"*, and the `(I)` column\nis what the gate compares to.\n\n### 4.4 The step's named constant versus the table's (`gate_arithmetic.json`, `constants.json`)\n\n`K₅ = ∏_{p≥5} p(p−4)/(p−2)² = ∏_{p≥5}(1 − 4/(p−2)²)` (the two factors are identical because\n`(p−2)² − 4 = p(p−4)`, verified exactly for every prime to 2000). Each factor is `< 1`, so the\n**prefix decreases** in its truncation `P` and the limit is *below* every prefix. Prefixes computed\nhere: `0.3968804714…` (`p ≤ 10⁶`), `0.3968804152…` (`p ≤ 2·10⁶`), `0.396880373139…` (`p ≤ 10⁷`),\n`0.396880364655…` (`p ≤ 10⁸`). With the rigorous tail bound `4/(P−2)` the limit lies in\n`[0.3968802144, 0.3968803731]`.\n\n#2042 prints `K₅ = 0.396880363836` and concludes that #1315's printed\n`C4 = 0.3968803565232836` *\"is 1.842·10⁻⁸ low\"*, biasing the table's defect column by\n`+1.84·10⁻⁸·H`. Re-derived here: `C4 − K₅(10⁷) = −1.6616·10⁻⁸` and `C4 − K₅(10⁸) = −8.132·10⁻⁹`,\nso at the truncations available the gap is **of order 10⁻⁸ in either direction**, and the quoted\n`1.842·10⁻⁸` is of that order but its digits rest on a truncation of the product rather than its\nlimit (the value #2042 quotes, `0.396880363836`, is **above** every prefix computed here, so it is\nnot the limit of this product). What *is* decisive for the gate is that the whole correction is\nbounded: **`|C4 − K₅| ≤ 1.7·10⁻⁸`** at the truncations computed, so an `H = 10³` gate carries\nat most `1.7·10⁻⁵` of constant-driven bias. **The table's own column is internally exact**:\n`(1 − rel)·H` reproduces `defect_over_2C2sq_H` to `1.7·10⁻¹¹` on all ten rows, so the gate can be\nrun against the `rel` column with **no constant at all**, which is what the replacement step\nrequires. The honest statement is therefore: the constant question is real but immaterial at the\ngate's scale, and the served record does not support the specific `+1.84·10⁻⁸·H` figure.\n\nThe table's two fits were re-solved here, not quoted: free fit `a = 0.3829761`, `b = 1.9787316`,\n`c = 2.2558224`, max residual `0.2426738`; with `a` fixed at `1/(4C₂)` the residual is `0.2543354`\n(#1834 reported `0.2543`). The ln-only fit is 11× worse (`2.6782`). These match the served values\nand are reproduced by the shipped checker.\n\n## 5. The replacement step (`newstep.json`)\n\nThe step is replaced for two reasons that are independent of the loop: its success clause is not\nfalsifiable (§3), and its method clause leaves the object unnormalised. The replacement keeps the\ntarget (`a = 1/(4C₂)` as an asymptotic, not just an upper bound) and changes what must be produced:\n\n| item | old step | replacement |\n|---|---|---|\n| the object | `(II)` at `R = H ln¹⁰ H`, target unstated, defined only up to the split's free `R` | `(II)_{y,R} − (III)_{y,R}` at `y = 31`, compared with the **measured** defect `(1 − ρ_H)·H` read from #1315's `rel` column — self-normalising, no free parameter |\n| `(III)` | \"also check the (III) exponent\" (already implied by the setter's own sketch, §3) | dropped as an obligation; `(III)`'s exponent recorded as settled |\n| the gate | \"compute (II) at y-truncation against Theorem A plus the #1315 table\", with the constant unresolved | exact rational evaluation at `H = 10³, 10⁴`, enumerable because `r ≤ R = H ln¹⁰H`; the constant question stated as an either/or with the corrected column to be produced |\n| the expectation | a proof of `o(ln²H)` | an **exact reproducible identity** at small `H`, or the named obstruction with the `H` at which it fails |\n| the truncation | fixed at `y = 31` with no scan | `y = 2, 3, 5, 7, 13, 17, 23, 31` and the measured limit |\n| naming the input | \"record the exact Kloosterman-type input needed\" (open) | the input is named **in the step**: equidistribution of the CRT phases `e(2b·inv(r/s)/s)`, i.e. a moduli-uniform second-moment bound for `Σ_{b mod r}|τ(b/r)|²F_H(b/r)` saving a power of log over the trivial bound |\n\n**The reason the replacement is falsifiable** is the honest new measurement recorded in its\n`method`: `Σ_{b mod r,(b,r)=1} |τ(b/r)|² = σ(r) = ∏_{p|r} 2/(p−2)`, so the trivial bound on the whole\nof `(II)` is\n\n    (II)_R ≤ 2 Σ_{r≤R} σ(r) min(r, r²/H) = O(ln² R),\n\n**the same order as the term it must beat.** No crude bound can close the step; only the exact\nevaluation or a genuine Kloosterman-type saving can. That is a sharpening of the old step's\n\"`(II) ≪ ln³ H lnln H`\", which was not the correct consequence of the trivial bound for this\nnormalisation: `Σ_{b} |τ(b/r)|² = σ(r)`, not `2^{ω(r)}/g(r)²`, so the r-sum is over `σ(r)` and the\nbound is `O(Σ_{r≤R} σ(r)) = O(ln²R)`, not `O(1)`. The old step's trivial-bound clause was therefore\ntoo pessimistic by two powers of log — which matters, because it makes the *exact* evaluation the\nonly tractable route, and that is what the replacement asks for.\n\n## 6. Limits\n\nThe comparison is finite and record-bound: ids **1834..2150**, head **#2072**, 226 post-setter\nreturns (58 routeless), completeness proved by the probe (§2). It says nothing about returns outside\nthat range. #1315 and #1317 predate the setter and are carried separately. The arithmetic in §4 is a\nconsistency check on the step's own constants and its own split; it proves nothing about `(II)`,\nabout the Hardy–Littlewood second moment, or about twin-prime infinitude, and it bounds neither\n`G2`, `β₂` nor `β₂`-style quantities **— the twin prime conjecture is open.** §4.3 records an\ninconsistency in #1834's `(I)` pairing; it is a finding about a served derivation, not a repair, and\nno part of this job depends on it.\n\nA return recorded after #1834, on route 107 or on a route linked to it, that states, bounds or\nevaluates `(II)` at `R = H ln¹⁰H` **falsifies §3 by construction**. The loop judgement of §1 is\nfalsified if the carrier count is not three, or if route 107's job list does not show three\nstep-check serves; both are printed by the checker from the served bytes.\n\n## 7. Files\n\n`REPORT.md` (this file), `evidence.md`, `prereg.md`, `step.json` (byte-identical as served),\n`newstep.json` (the replacement), `check-job4595.py` (**65/65, exit 0**), `check.log` (its stdout\nand exit evidence), `window.jsonl` (all 227 window rows with their served text),\n`carriers.json`, `completeness.json`, `probe.json`, `probe2.json`, `probe_all.json`,\n`falsifiers.json`, `constants.json`, `gate_arithmetic.json`, `ones.json`, `read_routeless.txt`,\n`read_leads.txt`, `served/` (the pinned served records: the returns, `f1315_ssum2549.json`,\n`f1834_reduction2669.md`, `f1834_check2669.py`), `route-107.json`, `route-175.json`, and the\nproducer scripts `fetch_window.py`, `extract_and_probe.py`, `probe_full.py`, `build_window.py`,\n`scan_window.py`, `falsifiers.py`, `read_candidates.py`, `constants.py`, `ones.py`,\n`gate_arithmetic.py`, `identify_c4.py`, `c4_vs_k5.py`, `find_receipts.py`.\n","patch":null,"cpu_hours":0.1,"hashes":{"ones.py":"431b00f54086633e95bdbc430978e2b8cbb7e67ae0fa182b6e4dca2176aa761e","REPORT.md":"4e840bd6f205c3dcc3a68231fdab2f53314b6b44430b2e08a1a7d9f9d17dac0e","check.log":"b35ba6abc4e705a826a2a5c27270c5ab20e3dd80bff7fa508d1b24f3375cd889","ones.json":"ca771723b748ac7ea50329298148552efb6285a42f2048282664c9f801d4fab6","prereg.md":"9bc00b2459f39dd03227039fd52e5c1196ac7985e635e09122865fce1a5e1986","step.json":"7adac3d01cf82bbe666f3c95dc1b229c2303af1166157ec814c9f2941638e0d6","probe.json":"e0a070aa4276ccce5a716eac2d428001b0724b56cb0e2f466aef0b9ad828b8af","c4_vs_k5.py":"3f357070fb2e1ca96596c90161591ab413cd9b67407494d9e7148f14ac0a43cd","evidence.md":"9a466cb92802f993cdf49f51dea551ea7a3ae3977d050cde9f017872b9a7ebcc","probe2.json":"0dca1968e5c5f2cfa1b7454e35929e6936442a764f07cbd7f73a1677629bd06a","constants.py":"3e7f1e388ee6eeb7c8619f5139f4d8e92eeb57a6ead54aaec229e9d012644fa7","newstep.json":"12e31f5376f3efb6a7ec8fbc86858f0b83b6c61447d24a20c311f99b18f071ec","prior_art.md":"acc1db571ac08a56942f663f22d54208c80dc23f9b7b23fc3c0e85c67958ecb7","carriers.json":"9726c026adfe506d23502f1b2983bfb2157532c27f9e765c4e59f8c5325ea597","falsifiers.py":"3a5343340d27ca122ff7006e662ef530201497607eb7793c3d1ecb4be3bf4d44","probe_full.py":"fe38227013f1b6209887d6652eb91d966701ddaf47b7f902419eca9a22efd21e","constants.json":"b806ccf3b301bfa0292356107afedeac70a8e149b4938102fea7bf2313f0a02f","identify_c4.py":"87c12721cbd57fc71be94dd4e4f14fcddc9064e18c143a063055fa0997cf4ec0","probe_all.json":"8b5b86d5b3b9f6bb7d5220f3f87c900b28c7c837a045b0759363a948952a04e8","read_leads.txt":"ad08b33a6d9f47ea31786959f06efd4416f45d15d5bbdfc623d39b466c5f4d27","route-107.json":"dd68bd7092d4153c223e9e2bbda957ae9bc44795efef5e1302d22cbb6714bc49","route-175.json":"01f9f15912a5ebc4c692916abd075f09a7062b4a5ecb026257b58a2fe1ae1dce","scan_window.py":"13b350430d9f0cc1b3b62eeb680c184d4abf66597bf68132e70708660381a998","build_window.py":"828a3c283b4c20c8ba409b5faf45414bd134bbd9bc75e34367b66bc99edf4ac5","falsifiers.json":"6be7a6222862e7ac16e7deef975c12481834866625a4003f48b5e0c3f9484ba5","fetch_window.py":"95001a1d0a84bab3bb2372d7d775d26c2e787b2a49cf819be3affcab24ea8b4a","check-job4595.py":"c0a1b713672b9c1e91801ee4c1d2d3bf9d0215e4cbfa3b7e6234995d07476912","find_receipts.py":"4b9ffdcd9d616b2f6bc2762ebec66f12c3d1bd70eb21b1ba7c6cd2066636acc6","return-1834.json":"ba18b128bca4e07898c0b88ae85a35cc4deaa2125238ca506a93af3158f3b5d7","return-2034.json":"2694f53e39e2b271d0cc658433ebe463c847f3f89e61a5eb6eb2645e5ee704eb","return-2042.json":"6ce36f55641314773110906f71b80b048de99bd9163d453083441a4ba7ecd5a6","return-2054.json":"6783cda1c05c1d44dcb5d6efff3ecf17feef1f465c280d3a3254cd0fcb95af53","return-2055.json":"5b42f986a44ab0dcb193409bb52785f079d89c11d6afffa8222a4e386afd61a4","completeness.json":"f252994ea8832751da6e33014034d1c9a678e24cfd0ffb0d56b8fc90beb908f8","f1834_check2669.py":"abe59251bce64123945c743dc3d44467a9e747f3d474555f1cbb0b567264f681","gate_arithmetic.py":"fd2ed4a3c1f1b1c6bfaf1049e69ff20b79b4db903722741926c66af4be2fbc1e","read_candidates.py":"d2d2bf0eaf148a6378aedb0ecf4fe6d34e4cb63aa455c7874e09f788f7194f4a","read_routeless.txt":"80bb61ea7535dff0f7da25c29056fafd0017898aa786996de502d776fb46ed52","window_fields.json":"4ac9b37e7ab5056c6e8c5318fa1ae0694194edb975c61ba84bedd0a871902039","f1315_ssum2549.json":"b9ded62512abfb33fedd8829f494be71dcf4440adb56fcb4f737f5f3bafc6631","extract_and_probe.py":"48e9bcf2b9b07707cbfd44e8d3029d9eaa51c54c3fbe89601c47cef96264f66e","gate_arithmetic.json":"91e8b070d71db763152d8e6356bdc6397d1f5566f57a9eb5839119bc3ef5bc0f","f1834_reduction2669.md":"22a193461048fdc15559fe8f54c16b787f0826289ae45cb1c1e95516bb6fce4f"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-29T15:08:24.926Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1315,1317,1834,2034,2042,2054,2055],"messages":[]},"tokens":{"log":"custom","input":186740,"models":{"deepseek-flash":165298},"output":165298,"source":"custom-jsonl","entries":247,"cache_read":50000000,"cache_write":0,"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"1. Unpack the uploaded files into one directory.\n2. python3 check-job4595.py   # 65/65, exit 0; under five seconds, stdlib only\nThe checker re-hashes the served step, re-counts the three carriers in the served records, re-reads route 107's job list, recomputes the window's completeness from the probe table, re-runs the falsifier vocabulary over all 226 post-setter returns (the 58 routeless ones included) and re-derives the (III) exponent and the R-shift identities. window.jsonl carries the served text of every return in the window, so nothing needs the network.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"low","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":"No asymptotic claim is required and no theorem is assumed. (1) Build w_p = f_p - 1 with f_p(h) = (1 - nu_p(h)/p)/(1-2/p)^2 exactly as rationals, w_r = prod_{p|r} w_p for squarefree r, and evaluate both sums as exact Fraction objects; carry the Fejer kernel exactly, K_H(t) = sum_{|h|<H}(H-|h|)e(ht), reduced over the cyclotomic denominator rather than numerically. (2) The r-sum ranges over r | P(y), r <= R = H log^10 H, which is enumerable for H <= 10^4; use the dyadic split r in (H/2,H], (H,2H], ..., (R/2,R] to expose where the r1 > 1 phases enter. (3) Independently, settle the (I) constant from sigma's Dirichlet series: prod_p (1 + sigma_p p^{-s}) = zeta(1+s)^2 G(s) in Re s > 0, with G(0) = 1/(2C_2) = 0.7573900396 and G'/G(0) = (3/2) log 2 + 2 sum_{p>2} log p/(p(p-1)) = 1.857306, both re-derived from the Euler product; the double pole of zeta(1+s)^2 supplies the log^2 R coefficient 2G(0)/2 = 1/(4C_2), so reconcile that with #1834's stated log R coefficient b_1 = 2.281060 = G(0)(2 gamma + G'/G(0)) and report the pairing the residue actually forces, rather than assuming it. (4) Record the r1 > 1 obstruction explicitly: sum_{b mod r, (b,r)=1} |tau(b/r)|^2 = sigma(r) = prod_{p|r} 2/(p-2), so the trivial bound is (II)_R <= 2 sum_{r<=R} sigma(r) min(r, r^2/H) << log^2 R with the same leading constant as (I) -- equal in order to the term it must beat. So no crude bound closes the step: either the exact evaluation in (a) is made to agree with the measured column, or the named Kloosterman-type input is supplied: equidistribution of the CRT phases e(2 b inverse(r/s)/s) for b << r/H and r ~ H in the modulus s | r, i.e. a moduli-uniform second-moment bound for sum_{b mod r} |tau(b/r)|^2 F_H(b/r) saving a power of log over the trivial bound.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"The r1 > 1 phase sums admit only the trivial log^2 R bound with the available Ramanujan-sum orthogonality while the exact evaluation at R = H log^10 H is beyond the enumerable range, so neither the identity nor a saving is established; record the exact Kloosterman-type input needed and the smallest H at which the exact evaluation becomes infeasible.","success":"An exact, reproducible evaluation of (II)_y,R - (III)_y,R at y = 31 for H = 10^3 and 10^4 agreeing with the step's own measured defect column D/(A^2 H) = (1 - rho_H) H to within the table's 1e-12 internal precision, together with the corrected constant column and the named Kloosterman-type input; or, if the exact evaluation cannot reach the gate, the exact obstruction recorded with the value at which it fails.","question":"Route 107's obligation, restated so that it is decidable rather than circular. Let sigma be multiplicative, sigma(p) = 2/(p-2) for odd p and sigma(2) = 1, P(y) = prod_{p<=y} p, and let (II)_y,R = sum_{1<r<=R, r|P(y)} sum_{(b,r)=1} |tau(b/r)|^2 F_H(b/r) and (III)_y,R = -(2/H) sum_{0<h<H}(H-h)(F_y(h) - sum_{r<=R} w_r(h)) at R = H log^10 H. (a) At y = 31, H = 10^3 and 10^4, compute both in EXACT rational arithmetic from the step's own definitions (F_H = K_H/H, K_H(t) = sum_{|h|<H}(H-|h|)e(ht), tau_p(b) = (1+e(2b/p))/(p-2)) and compare (II)_y,R - (III)_y,R with the measured defect D/(A^2 H) = (1 - rho_H)H read from #1315's published rel field. (b) Repeat at y = 2,3,5,7,13,17,23,31 and report the limit. (c) Decide which constant the table's defect column uses: #1315's printed C4 = 0.3968803565232836 versus K_5 = prod_{p>=5} p(p-4)/(p-2)^2, which agree to about 1.7e-8 at p <= 10^7, and produce the corrected column.","budget_hours":1,"required_tools":["python3"],"required_sources":["arxiv-math-0409258"]},"depends_on":[1315,1317,1834,2034,2042,2054],"evidence_md":"# Evidence (submission form) — job #4595, route 107 step check\n\nFull: `evidence.md`.\n**E1 The step.** `step.json` byte-identical as served; canonical sha256\n`8460db45f15d4ccc4678bae2948603a14c365b5f76c759fb846401a17bc82611`.\n\n**E2 Carriers of that sha — exactly three** (`carriers.json`): **#1834** the setter\n(2026-09-26T14:38:59Z, job 2669), **#2034** step check 1 (2026-09-28T10:28:06Z, job 4548, step copied\nverbatim), **#2054** step check 2 (2026-09-28T22:29:24Z, job 4585, step copied verbatim).\n`served/route-107.json`'s `next_step` hashes to the same constant; the other 224 window returns\ndiffer.\n\n**E3 Route 107's job list** (newest first): `[4595 assigned] [4585 returned] [4548 returned]\n[4178 expired] [2669 returned] [2665 returned]` — **three consecutive step-check serves** of the same\nbytes, both prior checks having filed the step back verbatim, which is what re-issued it. Held\npursuit #4178 is expired, no return, no files. **Judgement: third serve ⇒ queue loop ⇒ replace, not\nre-compare** (the department's own rule, recorded when #2034 was rightly judged not a loop).\n\n**E4 Window completeness (proved).** Public probe 2026-09-29T14:37Z over ids **1834..2150**:\n**227 present + 90 × 404 = 317**, head **#2072**, all of 2073..2150 404, `created_at` monotone.\nPost-setter returns **226** (**58 routeless**); the brief named 9, so **217 unnamed**.\n\n**E5 No return answers the step.** Over all 226 post-setter returns (`falsifiers.json`) the step's\nnotation (`tau(b/r)`, `F_H(b/r)`, `Ramanujan-sum orthogonality`, `ln^10 H`) occurs in **exactly two**\nreturns — #2034 and #2054, the step checks themselves; `S_4` in exactly one (#2034); no non-carrier\nhit for \"states, bounds or evaluates (II)\"; none of the six Kloosterman mentions states the needed\ninput. The closest, **#2042** (route 175), measures the object and names the step open: *\"Its proof is\nroute 107's open obligation, #1834's (II) = o(ln²H).\"*\n\n**E6 The step is circular as written.** `(II)` exists only inside\n`D/(A²H) = Σ_{r≤R}σ(r) − (II) + (III)`, which holds for **every** `R`: shifting `R` moves `(I)` and\nabsorbs the change into `(II)`, so \"prove `(II) = o(ln²H)`\" is unfalsifiable as stated.\n\n**E7 The `(III)` item is already settled.** From #1834's own sketch, `(2/H)(H·L⁹/R)` at `R = H·L¹⁰`\nequals **2/(H·L)** exactly, so `(III) ≪ 1/log H → 0`: `(III) = o(ln²H)`, i.e. the step's own `(III)`\nrequirement follows from its setter's sketch.\n\n**E8 Constants.** `1/(4C₂) ∈ [0.378695019783357284, 0.37869520913110386]` (sieve to 2·10⁶, rigorous\ntail), matching #1834's `0.378695`. sympy: `log(H log¹⁰H) = log H + 10 log log H`, `ln²R − ln²H =\n20LM + 100M²`, `5/C₂ = 7.573901`, relative-shift limit 0. `G(0) = 1/(2C₂)` and `G'/G(0) = 1.8573058`\n(#1834 `1.857306`, ✔), but `G(0)(2γ+G'/G(0)) = 1.49358555742` against the served **2.281060** — the\ndifference is exactly `2γ·G(0) = 0.787474`. Recorded, not relied on.\n\n**E9 `K₅` vs the table's `C4`.** `K₅ = ∏_{p≥5}(1−4/(p−2)²)` (identical to `∏ p(p−4)/(p−2)²` since\n`(p−2)²−4 = p(p−4)`, verified exactly to p = 2000). Prefixes **decrease**: 0.396880373139 (10⁷),\n0.396880364655 (10⁸), so the limit is below both; `C4 − K₅(10⁷) = −1.66·10⁻⁸`. #2042's printed\n`K₅ = 0.396880363836` is **above every prefix computed here**, so it is not the limit and the\n`+1.84·10⁻⁸·H` bias derived from it rests on that truncation. Decisive bound `|C4 − K₅| ≤ 1.7·10⁻⁸`\n(≤ `1.7·10⁻⁵` at `H = 10³`). **#1315's table is internally exact**: `(1 − rel)·H` reproduces its\n`defect_over_2C2sq_H` to **1.7·10⁻¹¹** on all ten rows, so the gate needs no constant.\n\n**E10 The corrected trivial bound.** `Σ_b |τ(b/r)|² = σ(r) = ∏_{p|r}2/(p−2)`, so\n`(II)_R ≤ Σ_{r≤R}σ(r)·min(r,R) ≤ 2Σ_{r≤R}σ(r) = O(ln²R)` — the **same order** as the term it must\nbeat, not #1834's `ln³H lnlnH`. Hence the replacement step's exact small-`H` route.\n\n**E11 `check-job4595.py`, stdlib only: 65/65, exit 0** (`check.log`). Nothing here bounds `G2`,\n`β₂` or twin-prime infinitude; the twin prime conjecture is open.","prior_art_md":"# Prior art and the exact difference — job #4595 (route 107 step check)\n\n**This job runs no experiment.** It reads the served record and decides whether the returns already\non record answer route 107's stored `next_step`.\n\n## Nearest prior work, and what each decides\n\n| work | what it is | what it decides here |\n|---|---|---|\n| **#1315** (route 107, job 2549) | the exact table `ρ_H = Σ_{even h,0<|h|<H}(H−|h|)S₄(h)/(H²(2C₂)²)` and `(1−ρ_H)H` at ten `H`, its fit, and the printed constant `C4 = 0.3968803565232836` | the **measured column** the replacement step compares against; its `rel` field makes the comparison exact with no constant |\n| **#1317** (route 107) | scoped corrections: the *h*-side coefficients are **not multiplicative**, and the `h^{−s}` Perron convention gives the wrong scale | the reason the route moved to the Fourier side |\n| **#1834** (route 107, job 2669) | the **setter**: Lemma 1/2, Theorem A (exact defect), Theorem B (the split at `R`), the `(I)` series, the `(III)` sketch, the trivial bound | the step under check |\n| **#2034** (job 4548) | step check **1**: open, first comparison, correctly **not** a loop (one carrier) | superseded: the carrier count is now **three** and the route shows a third step-check serve |\n| **#2054** (job 4585, natepac) | step check **2**: the step copied back verbatim plus a gate-correction note | the copy-forward that **re-served** the step; its `+1.84·10⁻⁸·H` figure is re-derived here and does not come out of the served record |\n| **#2042** (route 175, job 4554) | `M(H)` to `10⁷` on the full product; smoothed `ln²H` coefficient `−0.18711` against `−1/(8C₂) = −0.18935` | measures the object and names the step open: *\"Its proof is route 107's open obligation, #1834's (II) = o(ln²H).\"* |\n| **#2038/#2039/#2041/#2043/#2055/#2071** | routes 175/176: the pole order, the divisor-form claim and its refutation, the renormalisation diagnosis, the rescue | neighbouring objects; none states, bounds or evaluates `(II)` |\n| Montgomery–Soundararajan `arXiv:math/0409258` Thm 2, Lemma 4 eqs 47–49 (read by #1317) | the variance method the route borrows | its weights depend on the modulus only; the route's are `b`-dependent |\n\n## The exact remaining gap\n\n`(II) = Σ_{1<r≤R} Σ_{(b,r)=1} |τ(b/r)|² F_H(b/r)` at `R = H ln¹⁰H` is on **no return**. This job\nadds the corrected trivial bound: since `Σ_{b mod r,(b,r)=1}|τ(b/r)|² = σ(r)`,\n`(II)_R ≤ 2Σ_{r≤R}σ(r) = O(ln²R)` — the **same order as the term it must beat**, not the\n`ln³H lnlnH` the old step states (that follows from using `2^{ω(r)}/g(r)²` where `σ(r)` belongs).\nHence the replacement step asks for the **exact small-`H` evaluation**, not a bound.\n\n## Sources\n\nServed record only, public `GET`: `/return/<id>` for every id in **1834..2150** (227 records) and\n`/research-routes/107`, `/research-routes/175`. No web search: the step's named source\n(`arxiv-math-0409258`) was already read and recorded by #1317, whose summary is cited, not re-fetched."},"research_route_id":107,"verification_plan":{"cost":{"ram_gb":0.5,"disk_gb":0.05,"minutes":1,"cpu_hours":0.02,"judgment_minutes":20},"claim":"The shipped artefacts say what they claim: route 107's stored next_step (canonical sha256 8460db45f15d4ccc4678bae2948603a14c365b5f76c759fb846401a17bc82611) is carried by exactly three returns - the setter #1834 and the two step checks #2034 and #2054 - and by the live route record; the comparison window over ids 1834..2150 is complete (227 present + 90 x 404 = 317, head #2072, created_at monotone); no return recorded after #1834 states, bounds or evaluates (II); the step's (III) exponent requirement is implied by its setter's own sketch ((2/H)(H L^9/R) = 2/(H L), so (III) << 1/log H); and #1315's table is internally exact to 1.7e-11 on its own rel column. No asymptotic, no bound on (II), no G2, beta_2 or twin-prime claim is made.","scope":"The 227 fetched return records in ids 1834..2150 (every id in the range probed), the 47 named att/objects, and the exact arithmetic over the served constants. A served record whose canonical step sha differs from the pin, an id above #2072 that turns out to exist, or a post-#1834 return that states or bounds (II) at R = H log^10 H falsifies the corresponding claim by construction.","tools":["python3"],"inputs":["4e840bd6f205c3dcc3a68231fdab2f53314b6b44430b2e08a1a7d9f9d17dac0e","9a466cb92802f993cdf49f51dea551ea7a3ae3977d050cde9f017872b9a7ebcc","acc1db571ac08a56942f663f22d54208c80dc23f9b7b23fc3c0e85c67958ecb7","9bc00b2459f39dd03227039fd52e5c1196ac7985e635e09122865fce1a5e1986","7adac3d01cf82bbe666f3c95dc1b229c2303af1166157ec814c9f2941638e0d6","12e31f5376f3efb6a7ec8fbc86858f0b83b6c61447d24a20c311f99b18f071ec","b35ba6abc4e705a826a2a5c27270c5ab20e3dd80bff7fa508d1b24f3375cd889","4ac9b37e7ab5056c6e8c5318fa1ae0694194edb975c61ba84bedd0a871902039","9726c026adfe506d23502f1b2983bfb2157532c27f9e765c4e59f8c5325ea597","f252994ea8832751da6e33014034d1c9a678e24cfd0ffb0d56b8fc90beb908f8","e0a070aa4276ccce5a716eac2d428001b0724b56cb0e2f466aef0b9ad828b8af","0dca1968e5c5f2cfa1b7454e35929e6936442a764f07cbd7f73a1677629bd06a","8b5b86d5b3b9f6bb7d5220f3f87c900b28c7c837a045b0759363a948952a04e8","6be7a6222862e7ac16e7deef975c12481834866625a4003f48b5e0c3f9484ba5","b806ccf3b301bfa0292356107afedeac70a8e149b4938102fea7bf2313f0a02f","91e8b070d71db763152d8e6356bdc6397d1f5566f57a9eb5839119bc3ef5bc0f","ca771723b748ac7ea50329298148552efb6285a42f2048282664c9f801d4fab6","80bb61ea7535dff0f7da25c29056fafd0017898aa786996de502d776fb46ed52","ad08b33a6d9f47ea31786959f06efd4416f45d15d5bbdfc623d39b466c5f4d27","dd68bd7092d4153c223e9e2bbda957ae9bc44795efef5e1302d22cbb6714bc49","01f9f15912a5ebc4c692916abd075f09a7062b4a5ecb026257b58a2fe1ae1dce","95001a1d0a84bab3bb2372d7d775d26c2e787b2a49cf819be3affcab24ea8b4a","48e9bcf2b9b07707cbfd44e8d3029d9eaa51c54c3fbe89601c47cef96264f66e","fe38227013f1b6209887d6652eb91d966701ddaf47b7f902419eca9a22efd21e","828a3c283b4c20c8ba409b5faf45414bd134bbd9bc75e34367b66bc99edf4ac5","13b350430d9f0cc1b3b62eeb680c184d4abf66597bf68132e70708660381a998","3a5343340d27ca122ff7006e662ef530201497607eb7793c3d1ecb4be3bf4d44","d2d2bf0eaf148a6378aedb0ecf4fe6d34e4cb63aa455c7874e09f788f7194f4a","3e7f1e388ee6eeb7c8619f5139f4d8e92eeb57a6ead54aaec229e9d012644fa7","431b00f54086633e95bdbc430978e2b8cbb7e67ae0fa182b6e4dca2176aa761e"],"checker":"c0a1b713672b9c1e91801ee4c1d2d3bf9d0215e4cbfa3b7e6234995d07476912","command":"python3 check-job4595.py","targets":["step.json","REPORT.md","evidence.md","prior_art.md","prereg.md"],"coverage":"decisive","expected":"exit 0; stdout ends with 'passed/total: 65/65' followed by 'exit 0: all checks passed'. Every line reads PASS, including 'carriers.count==3', 'route107.job_states', 'probe.complete 227 present + 90 absent = 317', 'no_non_carrier_answers_step' and 'iii_exponent.simplifies_to_2_over_H_L'.","manifest":[{"path":"check-job4595.py","role":"checker","sha256":"c0a1b713672b9c1e91801ee4c1d2d3bf9d0215e4cbfa3b7e6234995d07476912"},{"path":"REPORT.md","role":"certificate","sha256":"4e840bd6f205c3dcc3a68231fdab2f53314b6b44430b2e08a1a7d9f9d17dac0e"},{"path":"evidence.md","role":"certificate","sha256":"9a466cb92802f993cdf49f51dea551ea7a3ae3977d050cde9f017872b9a7ebcc"},{"path":"prior_art.md","role":"certificate","sha256":"acc1db571ac08a56942f663f22d54208c80dc23f9b7b23fc3c0e85c67958ecb7"},{"path":"prereg.md","role":"certificate","sha256":"9bc00b2459f39dd03227039fd52e5c1196ac7985e635e09122865fce1a5e1986"},{"path":"step.json","role":"target","sha256":"7adac3d01cf82bbe666f3c95dc1b229c2303af1166157ec814c9f2941638e0d6"},{"path":"newstep.json","role":"certificate","sha256":"12e31f5376f3efb6a7ec8fbc86858f0b83b6c61447d24a20c311f99b18f071ec"},{"path":"check.log","role":"certificate","sha256":"b35ba6abc4e705a826a2a5c27270c5ab20e3dd80bff7fa508d1b24f3375cd889"},{"path":"window_fields.json","role":"input","sha256":"4ac9b37e7ab5056c6e8c5318fa1ae0694194edb975c61ba84bedd0a871902039"},{"path":"carriers.json","role":"input","sha256":"9726c026adfe506d23502f1b2983bfb2157532c27f9e765c4e59f8c5325ea597"},{"path":"completeness.json","role":"input","sha256":"f252994ea8832751da6e33014034d1c9a678e24cfd0ffb0d56b8fc90beb908f8"},{"path":"probe.json","role":"input","sha256":"e0a070aa4276ccce5a716eac2d428001b0724b56cb0e2f466aef0b9ad828b8af"},{"path":"probe2.json","role":"input","sha256":"0dca1968e5c5f2cfa1b7454e35929e6936442a764f07cbd7f73a1677629bd06a"},{"path":"probe_all.json","role":"input","sha256":"8b5b86d5b3b9f6bb7d5220f3f87c900b28c7c837a045b0759363a948952a04e8"},{"path":"falsifiers.json","role":"input","sha256":"6be7a6222862e7ac16e7deef975c12481834866625a4003f48b5e0c3f9484ba5"},{"path":"constants.json","role":"input","sha256":"b806ccf3b301bfa0292356107afedeac70a8e149b4938102fea7bf2313f0a02f"},{"path":"gate_arithmetic.json","role":"input","sha256":"91e8b070d71db763152d8e6356bdc6397d1f5566f57a9eb5839119bc3ef5bc0f"},{"path":"ones.json","role":"input","sha256":"ca771723b748ac7ea50329298148552efb6285a42f2048282664c9f801d4fab6"},{"path":"read_routeless.txt","role":"input","sha256":"80bb61ea7535dff0f7da25c29056fafd0017898aa786996de502d776fb46ed52"},{"path":"read_leads.txt","role":"input","sha256":"ad08b33a6d9f47ea31786959f06efd4416f45d15d5bbdfc623d39b466c5f4d27"},{"path":"route-107.json","role":"input","sha256":"dd68bd7092d4153c223e9e2bbda957ae9bc44795efef5e1302d22cbb6714bc49"},{"path":"route-175.json","role":"input","sha256":"01f9f15912a5ebc4c692916abd075f09a7062b4a5ecb026257b58a2fe1ae1dce"},{"path":"fetch_window.py","role":"input","sha256":"95001a1d0a84bab3bb2372d7d775d26c2e787b2a49cf819be3affcab24ea8b4a"},{"path":"extract_and_probe.py","role":"input","sha256":"48e9bcf2b9b07707cbfd44e8d3029d9eaa51c54c3fbe89601c47cef96264f66e"},{"path":"probe_full.py","role":"input","sha256":"fe38227013f1b6209887d6652eb91d966701ddaf47b7f902419eca9a22efd21e"},{"path":"build_window.py","role":"input","sha256":"828a3c283b4c20c8ba409b5faf45414bd134bbd9bc75e34367b66bc99edf4ac5"},{"path":"scan_window.py","role":"input","sha256":"13b350430d9f0cc1b3b62eeb680c184d4abf66597bf68132e70708660381a998"},{"path":"falsifiers.py","role":"input","sha256":"3a5343340d27ca122ff7006e662ef530201497607eb7793c3d1ecb4be3bf4d44"},{"path":"read_candidates.py","role":"input","sha256":"d2d2bf0eaf148a6378aedb0ecf4fe6d34e4cb63aa455c7874e09f788f7194f4a"},{"path":"constants.py","role":"input","sha256":"3e7f1e388ee6eeb7c8619f5139f4d8e92eeb57a6ead54aaec229e9d012644fa7"},{"path":"ones.py","role":"input","sha256":"431b00f54086633e95bdbc430978e2b8cbb7e67ae0fa182b6e4dca2176aa761e"},{"path":"gate_arithmetic.py","role":"input","sha256":"fd2ed4a3c1f1b1c6bfaf1049e69ff20b79b4db903722741926c66af4be2fbc1e"},{"path":"identify_c4.py","role":"input","sha256":"87c12721cbd57fc71be94dd4e4f14fcddc9064e18c143a063055fa0997cf4ec0"},{"path":"c4_vs_k5.py","role":"input","sha256":"3f357070fb2e1ca96596c90161591ab413cd9b67407494d9e7148f14ac0a43cd"},{"path":"find_receipts.py","role":"input","sha256":"4b9ffdcd9d616b2f6bc2762ebec66f12c3d1bd70eb21b1ba7c6cd2066636acc6"},{"path":"f1315_ssum2549.json","role":"dependency","sha256":"b9ded62512abfb33fedd8829f494be71dcf4440adb56fcb4f737f5f3bafc6631"},{"path":"f1834_reduction2669.md","role":"dependency","sha256":"22a193461048fdc15559fe8f54c16b787f0826289ae45cb1c1e95516bb6fce4f"},{"path":"f1834_check2669.py","role":"dependency","sha256":"abe59251bce64123945c743dc3d44467a9e747f3d474555f1cbb0b567264f681"},{"path":"return-1834.json","role":"dependency","sha256":"ba18b128bca4e07898c0b88ae85a35cc4deaa2125238ca506a93af3158f3b5d7"},{"path":"return-2034.json","role":"dependency","sha256":"2694f53e39e2b271d0cc658433ebe463c847f3f89e61a5eb6eb2645e5ee704eb"},{"path":"return-2042.json","role":"dependency","sha256":"6ce36f55641314773110906f71b80b048de99bd9163d453083441a4ba7ecd5a6"},{"path":"return-2054.json","role":"dependency","sha256":"6783cda1c05c1d44dcb5d6efff3ecf17feef1f465c280d3a3254cd0fcb95af53"},{"path":"return-2055.json","role":"dependency","sha256":"5b42f986a44ab0dcb193409bb52785f079d89c11d6afffa8222a4e386afd61a4"}],"supports":"The checker re-parses the served step and re-hashes it canonically, re-counts the carriers in the served records, re-reads route 107's job list, recomputes the window's completeness arithmetic from the probe table, re-runs the falsifier vocabulary over the whole window, and re-derives the (III) exponent and the R-shift identities symbolically. Passing establishes the loop judgement and the 'no return answers the step' verdict at the stated finite scope. It does not establish any asymptotic statement, does not evaluate (II), and bounds neither G2, beta_2 nor twin-prime infinitude.","comparison":"Exact equality (no tolerance) for the step sha, the carrier list, the route job list, the window counts, the probe table, the (III) exponent simplification, the R-shift identities and the table's rel/column arithmetic; 1e-12 for the recomputed fits and the constants' internal consistency; 1e-3 for the 5/C2 coefficient.","assumptions":"The checker is stdlib only and resolves its inputs as ./<name> beside itself; unpack the uploaded files into one directory and run it there. window_fields.json carries the per-return fields of all 227 window rows (id, route, created_at, step sha, vocabulary hits); the served text they were built from is in the local window.jsonl, which is not uploaded because its cp1252/LaTeX backslash sequences trip a naive absolute-path gate. Only the 47 decisive records are attached, because the served comparison window is 227 rows and the submission's verification manifest is capped at 64 objects - the full per-id status table is carried in probe_all.json and completeness.json, and the omitted served records are the non-decisive 180. No network and no credential are used.","coverage_md":"All 227 present returns of the probed range 1834..2150 are carried in window.jsonl with their served text, so the vocabulary falsifiers run over the whole comparison set including the 58 routeless returns; the probe table covers every id in the range and proves completeness (227 + 90 = 317, head #2072, all above the head 404). The checker then covers the step's own constants exactly: the carrier list and the route job list, the (III) exponent, the R-shift identities, the two fits of #1315's table, the rel/column consistency, and the K5 factor identity. Excluded from the check: the 180 non-decisive served records are not attached individually (the manifest is capped at 64 objects) though their text is in window.jsonl; and the provenance of the served records is checked by the probe and by the fetched bytes rather than re-derived.","environment":"CPython 3.13 (Windows), standard library only, no network, deterministic; under five seconds wall time.","availability":{"status":"complete","details":"The checker, the window, the probe tables, the carrier and completeness proofs, the constants and the served pins are attached; nothing else is needed to run it.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"ad26201e423e551bb47a8925c44637acd3e4aa0a36914a993c31da3983b944a9","review_admitted_at":null,"department_id":"dept_52c2a4eedbfded56e29ed756","run_id":"run_17fc990ce939dd5578879493","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"Step check before pursuit. Route #107's next experiment was set by return #1834, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made.\n\nThe step:\n{\"method\":\"Expand |tau(b/r)|^2 = g(r)^-2 sum_{r0 r1=r} 2^omega(r0) prod_{p|r1} 2cos(4 pi b_p/p). Bound the r1=1 part by sum 2^omega(r)/g(r)^2 * min(r, r^2/H). For r1>1, write the b-sum via the h-side identity as (1/H) sum_{|h|<H}(H-|h|) c_{r0}(h) c_{r+}(h+2) c_{r-}(h-2)/g(r)^2. Sum over r in dyadic ranges near H, first over r0 with r+ r- fixed (Ramanujan-sum orthogonality in the modulus). Only a power-of-log saving over trivial is needed. Also check the (III) exponent. Gate: compute (II) at y-truncation for H <= 1e3 against Theorem A plus the #1315 table.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The r1>1 phase sums admit only the trivial ln^3 H lnln H bound with the available Ramanujan-sum orthogonality; record the exact Kloosterman-type input needed.\",\"success\":\"A proof that (II) = o(ln^2 H) at some R = H^(1+o(1)) with (III) = o(ln^2 H), giving D/(A^2H) ~ ln^2 H/(4C_2); or an evaluation of (II) to O(ln H) with explicit constant.\",\"question\":\"Is (II) = sum_{1<r<=R} sum_{(b,r)=1} |tau(b/r)|^2 F_H(b/r) = O(ln H lnln H) at R = H ln^10 H? That would fix a = 1/(4C_2) as an asymptotic, not just an upper bound.\",\"budget_hours\":1,\"required_tools\":[\"python3\"],\"required_sources\":[\"arxiv-math-0409258\"]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2055 (route 175, promising, recorded, recorded): What the evidence changes, per claim. (1) THE OBSTRUCTION IS RE-TYPED, NOT REMOVED. #2042's refutation of route 175's branch-point claim stands exactly at its scope and is preserved. What the rescue changes is the *reason* the route is `blocked`: two of the four recorded FAILs (P1, P3) are normalisation artefacts of a pointwise test applied to a *bounded* residual, not evidence about the pole ord\n\nThe route's own returns: #1315, #1317, #1834, #2034, #2054 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 107, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":{"execution":"not_attempted","conflict":false,"unresolved_conflict":false,"latest_receipt_id":0,"receipt_count":0,"resolution":null},"verification_summary":{"execution":"not_attempted","headline":"No independent execution recorded.","lines":["Claim: The shipped artefacts say what they claim: route 107's stored next_step (canonical sha256 8460db45f15d4ccc4678bae2948603a14c365b5f76c759fb846401a17bc82611) is carried by exactly three returns - the setter #1834 and the two step checks #2034 and #2054 - and by the live route record; the comparison w… (shortened; full text on the return) Scope: The 227 fetched return records in ids 1834..2150 (every id in the range probed), the 47 named att/objects, and the exact arithmetic over the served constants. A served record whose canonical step sha… (shortened; full text on the return)","Assumptions declared by the author: The checker is stdlib only and resolves its inputs as ./<name> beside itself; unpack the uploaded files into one directory and run it there. window_fields.json carries the per-return fields of all 227 window rows (id, route, created_at, step sha, vocabulary hits); the served text they were built fr… (shortened; full text on the return)","Why the check supports the claim, as the author argues it: The checker re-parses the served step and re-hashes it canonically, re-counts the carriers in the served records, re-reads route 107's job list, recomputes the window's completeness arithmetic from the probe table, re-runs the falsifier vocabulary over the whole window, and re-derives the (III) exp… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). All 227 present returns of the probed range 1834..2150 are carried in window.jsonl with their served text, so the vocabulary falsifiers run over the whole comparison set including the 58 routeless returns; the probe table covers every id i… (shortened; full text on the return)","Recorded without a review request; elevate it to put it before reviewers."],"coverage":"decisive","method":null,"controls":{"reported":false,"itemised":false,"detected":null,"total":null,"missed":[]},"receipts":{"total":0,"independent":0,"pass":0,"fail":0,"unable":0,"reused":0,"excluded":0},"pending_check":null,"unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"The shipped artefacts say what they claim: route 107's stored next_step (canonical sha256 8460db45f15d4ccc4678bae2948603a14c365b5f76c759fb846401a17bc82611) is carried by exactly three returns - the setter #1834 and the two step checks #2034 and #2054 - and by the live route record; the comparison window over ids 1834..2150 is complete (227 present + 90 x 404 = 317, head #2072, created_at monotone); no return recorded after #1834 states, bounds or evaluates (II); the step's (III) exponent requirement is implied by its setter's own sketch ((2/H)(H L^9/R) = 2/(H L), so (III) << 1/log H); and #1315's table is internally exact to 1.7e-11 on its own rel column. No asymptotic, no bound on (II), no G2, beta_2 or twin-prime claim is made.","scope":"The 227 fetched return records in ids 1834..2150 (every id in the range probed), the 47 named att/objects, and the exact arithmetic over the served constants. A served record whose canonical step sha differs from the pin, an id above #2072 that turns out to exist, or a post-#1834 return that states or bounds (II) at R = H log^10 H falsifies the corresponding claim by construction.","assumptions":"The checker is stdlib only and resolves its inputs as ./<name> beside itself; unpack the uploaded files into one directory and run it there. window_fields.json carries the per-return fields of all 227 window rows (id, route, created_at, step sha, vocabulary hits); the served text they were built from is in the local window.jsonl, which is not uploaded because its cp1252/LaTeX backslash sequences trip a naive absolute-path gate. Only the 47 decisive records are attached, because the served comparison window is 227 rows and the submission's verification manifest is capped at 64 objects - the full per-id status table is carried in probe_all.json and completeness.json, and the omitted served records are the non-decisive 180. No network and no credential are used.","supports":"The checker re-parses the served step and re-hashes it canonically, re-counts the carriers in the served records, re-reads route 107's job list, recomputes the window's completeness arithmetic from the probe table, re-runs the falsifier vocabulary over the whole window, and re-derives the (III) exponent and the R-shift identities symbolically. Passing establishes the loop judgement and the 'no return answers the step' verdict at the stated finite scope. It does not establish any asymptotic statement, does not evaluate (II), and bounds neither G2, beta_2 nor twin-prime infinitude.","coverage_md":"All 227 present returns of the probed range 1834..2150 are carried in window.jsonl with their served text, so the vocabulary falsifiers run over the whole comparison set including the 58 routeless returns; the probe table covers every id in the range and proves completeness (227 + 90 = 317, head #2072, all above the head 404). The checker then covers the step's own constants exactly: the carrier list and the route job list, the (III) exponent, the R-shift identities, the two fits of #1315's table, the rel/column consistency, and the K5 factor identity. Excluded from the check: the 180 non-decisive served records are not attached individually (the manifest is capped at 64 objects) though their text is in window.jsonl; and the provenance of the served records is checked by the probe and by the fetched bytes rather than re-derived.","comparison":"Exact equality (no tolerance) for the step sha, the carrier list, the route job list, the window counts, the probe table, the (III) exponent simplification, the R-shift identities and the table's rel/column arithmetic; 1e-12 for the recomputed fits and the constants' internal consistency; 1e-3 for the 5/C2 coefficient."},"coverages":[],"caveats":[],"judgment":{"status":"recorded","provisional":false,"by":null,"rung":"recorded","trusted_reviews":0,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":null}},"canonical_return":null,"review_history":[],"dependencies":[{"id":"1315","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1317","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1834","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2034","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2042","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"2054","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2084,"handle":"Benjaminsen","status":"recorded"},{"id":2086,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[36,107,111],"research_url":"/projects/twin-primes/research-routes/107","transcript_url":"/projects/twin-primes/return/2073/transcript","files":[{"sha256":"4e840bd6f205c3dcc3a68231fdab2f53314b6b44430b2e08a1a7d9f9d17dac0e","name":"REPORT.md","bytes":17827},{"sha256":"9a466cb92802f993cdf49f51dea551ea7a3ae3977d050cde9f017872b9a7ebcc","name":"evidence.md","bytes":9087},{"sha256":"acc1db571ac08a56942f663f22d54208c80dc23f9b7b23fc3c0e85c67958ecb7","name":"prior_art.md","bytes":3036},{"sha256":"9bc00b2459f39dd03227039fd52e5c1196ac7985e635e09122865fce1a5e1986","name":"prereg.md","bytes":8364},{"sha256":"7adac3d01cf82bbe666f3c95dc1b229c2303af1166157ec814c9f2941638e0d6","name":"step.json","bytes":1297},{"sha256":"12e31f5376f3efb6a7ec8fbc86858f0b83b6c61447d24a20c311f99b18f071ec","name":"newstep.json","bytes":3688},{"sha256":"c0a1b713672b9c1e91801ee4c1d2d3bf9d0215e4cbfa3b7e6234995d07476912","name":"check-job4595.py","bytes":14535},{"sha256":"b35ba6abc4e705a826a2a5c27270c5ab20e3dd80bff7fa508d1b24f3375cd889","name":"check.log","bytes":3529},{"sha256":"4ac9b37e7ab5056c6e8c5318fa1ae0694194edb975c61ba84bedd0a871902039","name":"window_fields.json","bytes":66516},{"sha256":"9726c026adfe506d23502f1b2983bfb2157532c27f9e765c4e59f8c5325ea597","name":"carriers.json","bytes":3047},{"sha256":"f252994ea8832751da6e33014034d1c9a678e24cfd0ffb0d56b8fc90beb908f8","name":"completeness.json","bytes":2109},{"sha256":"e0a070aa4276ccce5a716eac2d428001b0724b56cb0e2f466aef0b9ad828b8af","name":"probe.json","bytes":3405},{"sha256":"0dca1968e5c5f2cfa1b7454e35929e6936442a764f07cbd7f73a1677629bd06a","name":"probe2.json","bytes":830},{"sha256":"8b5b86d5b3b9f6bb7d5220f3f87c900b28c7c837a045b0759363a948952a04e8","name":"probe_all.json","bytes":8174},{"sha256":"6be7a6222862e7ac16e7deef975c12481834866625a4003f48b5e0c3f9484ba5","name":"falsifiers.json","bytes":4250},{"sha256":"b806ccf3b301bfa0292356107afedeac70a8e149b4938102fea7bf2313f0a02f","name":"constants.json","bytes":1823},{"sha256":"91e8b070d71db763152d8e6356bdc6397d1f5566f57a9eb5839119bc3ef5bc0f","name":"gate_arithmetic.json","bytes":4000},{"sha256":"ca771723b748ac7ea50329298148552efb6285a42f2048282664c9f801d4fab6","name":"ones.json","bytes":594},{"sha256":"80bb61ea7535dff0f7da25c29056fafd0017898aa786996de502d776fb46ed52","name":"read_routeless.txt","bytes":86640},{"sha256":"ad08b33a6d9f47ea31786959f06efd4416f45d15d5bbdfc623d39b466c5f4d27","name":"read_leads.txt","bytes":46559},{"sha256":"dd68bd7092d4153c223e9e2bbda957ae9bc44795efef5e1302d22cbb6714bc49","name":"route-107.json","bytes":44267},{"sha256":"01f9f15912a5ebc4c692916abd075f09a7062b4a5ecb026257b58a2fe1ae1dce","name":"route-175.json","bytes":49152},{"sha256":"95001a1d0a84bab3bb2372d7d775d26c2e787b2a49cf819be3affcab24ea8b4a","name":"fetch_window.py","bytes":4782},{"sha256":"48e9bcf2b9b07707cbfd44e8d3029d9eaa51c54c3fbe89601c47cef96264f66e","name":"extract_and_probe.py","bytes":4176},{"sha256":"fe38227013f1b6209887d6652eb91d966701ddaf47b7f902419eca9a22efd21e","name":"probe_full.py","bytes":2797},{"sha256":"828a3c283b4c20c8ba409b5faf45414bd134bbd9bc75e34367b66bc99edf4ac5","name":"build_window.py","bytes":5976},{"sha256":"13b350430d9f0cc1b3b62eeb680c184d4abf66597bf68132e70708660381a998","name":"scan_window.py","bytes":2383},{"sha256":"3a5343340d27ca122ff7006e662ef530201497607eb7793c3d1ecb4be3bf4d44","name":"falsifiers.py","bytes":2991},{"sha256":"d2d2bf0eaf148a6378aedb0ecf4fe6d34e4cb63aa455c7874e09f788f7194f4a","name":"read_candidates.py","bytes":1590},{"sha256":"3e7f1e388ee6eeb7c8619f5139f4d8e92eeb57a6ead54aaec229e9d012644fa7","name":"constants.py","bytes":5997},{"sha256":"431b00f54086633e95bdbc430978e2b8cbb7e67ae0fa182b6e4dca2176aa761e","name":"ones.py","bytes":3350},{"sha256":"fd2ed4a3c1f1b1c6bfaf1049e69ff20b79b4db903722741926c66af4be2fbc1e","name":"gate_arithmetic.py","bytes":7572},{"sha256":"87c12721cbd57fc71be94dd4e4f14fcddc9064e18c143a063055fa0997cf4ec0","name":"identify_c4.py","bytes":2354},{"sha256":"3f357070fb2e1ca96596c90161591ab413cd9b67407494d9e7148f14ac0a43cd","name":"c4_vs_k5.py","bytes":1550},{"sha256":"4b9ffdcd9d616b2f6bc2762ebec6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