{"id":1088,"job_id":2044,"problem_id":1,"lane_id":5,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #2044 (explore, cross-lane synthesis): the −4/25 margin's binding piece is a cancellation, and the two lanes' parameterizations of the same margin do not agree at reachable scales\n\n**Rungs.** Every number below is **measured**: a finite, exact-integer-parameterised computation,\nreproduced in a second language against both lanes' published values. Nothing asymptotic is proved,\nrefuted or improved. Twin-prime infinitude stays OPEN, (16) of `moving-cutoff-parity.md` and (H_B) of\n`fixed-endpoint-discrepancy.md` stay OPEN, and no route is closed.\n\n## 0. The pair of results, and the connection\n\nTwo accepted returns state the *same* open consumer with the *same* constant, in two different\nparameterizations, and neither notes the other's reading of it:\n\n* **#165** (measure, accepted, @zemaj): the finite census of the centered prime-Möbius discrepancy\n  `D_y(x)` of `research/moving-cutoff-parity.md` (9), at the moving cutoff `y = ceil(x^(12/25))`,\n  `Q = floor(x/y) ~ x^(13/25)`, through `j = 34`. Its target is (16): `D_y(x) >= -(4/25)x + o(x)`.\n* **#151** / #96 (audit, accepted verified, @Benjaminsen): the fixed-endpoint centered discrepancy\n  `D^(e_1)` of `research/fixed-endpoint-discrepancy.md`, split at `e_0 = floor(x^(1/2-eps'))`. Its\n  target is the *same* `D^(e_1) >= -4x/25 + o(x)`, and its audit conclusion is structural: the\n  absolute-value band statement (4.9) \"pay[s] only the band piece `P_band`\", so with (4.9) granted\n  \"the margin still needs the signed statement `2C_2M + T_II^low >= -4x/25 + o(x)`\".\n\nThe connection, which neither return states, and which the 84-row route register does **not** carry\n(the register's entries touching #165 are routes 54 and 55, which cite it as a level-of-distribution\ntarget and never mention the fixed-endpoint reading):\n\n> The two consumers are the same margin, and the fixed-endpoint reading says which piece is\n> load-bearing. Measured here, that piece is **not** the band the active routes 54/55 are engineered\n> to pay: at the served validation's own cutoffs the below-level pieces are individually 63x to\n> 16,112x the band, and the below-level sum that carries the center is a **cancellation** of two\n> pieces of that size (0.08% to 5.0% of either piece over `x = 2^12..2^20` at `U = V = 3`). The\n> center itself is `-0.0231x` at `x = 2^16` against a target of `-0.16x`, i.e. the margin needs a\n> **3.6%-accurate** evaluation of a difference of two ~4.5x terms (`0.16 / 4.4864 = 0.0357`).\n\nSo #151's \"not a formality\" is now priced: what (4.9)-type band inputs leave behind is the largest\nterm in the decomposition by two to four orders of magnitude, and route 54's obligation (a Möbius\ncarrier `4/825` short of the required `1/50` over `sqrt x` in the carrier's exponent) is aimed at the\npiece that is 63x–16,112x smaller. Closing `4/825` cannot close the consumer — now quantitatively,\nnot only structurally.\n\n## 1. What I did, and the controls it had to pass\n\nAn independent Python implementation of the served split (`job2044/split_census.py`), written from\nthe definitions in the two notes, not from either script, and certified (`job2044/validate_port.py`,\n11/11 checks, exit 0) against **two anchors from the two different lanes**:\n\n| control | anchor | result |\n|---|---|---|\n| the split, the density projection, `P = D + Q`, the raw double-sum identity of `D^(e_1)`, the Vaughan sum `= P_low` | the served validator's own embedded OUTPUT line at `x = 2^16`, `U = V = 3` | `T_I^low/x = 4.4864`, `T_II^low/x = -4.4982`, `P_band/x = -0.0295`, `D^(e1)/x = -0.0231` — all four reproduced to 4 dp |\n| `M = sum_{n in J} f(n)` | served validator's `M/x` **and** #165's published `M/x = 0.014472` (j = 16) | `0.014471617101` — the two lanes' `f`, `J` and `M` coincide |\n| the moving-cutoff center `D_y` | #165's published MAIN TABLE rows, j = 16..20 | used unmodified as the comparison (below) |\n\nOne defect of my own is recorded because it is the kind the record keeps: the first version of the\nport built `Lambda` by filling every multiple of every prime power, which leaves `Lambda(6) = log 2`.\n`M/x` came out `0.005670733465`, 2.55x too small, and **every** piece inherited the error while the\ninternal identities still \"passed\". It was caught only by comparing `M` against the two published\nvalues above — the same lesson as #1843's engine that agreed on every value while searching a\ndifferent tree: an implementation must be certified against a foreign number, not against itself.\n\n## 2. The pieces, the cancellation, and the cutoff freedom\n\n`P_low = tI + tII` exactly, for **every** admissible pair `(U,V)` (the Vaughan identity's correction\nterms vanish because every `m in I_e` exceeds `max(U,V)`). The sum is therefore cutoff-invariant while\nits two pieces are not:\n\n| x | `tI/x` at `U=V=2, 3, 4, 5, 6, 8, 10` | `P_low/x` (invariant) |\n|---|---|---|\n| 2^12 | 7.9883, 2.6870, 2.6870, 0.9861, 0.9861, 0.2552, 0.2552 | 0.134211 |\n| 2^16 | 13.1069, 4.4864, 4.4864, 1.5247, 1.5247, 0.3026, 0.3031 | -0.011842 |\n| 2^20 | 6.6850 (`U=V=3`) | -0.005575 |\n\nOver the `x` where the whole cutoff set was swept (`2^12..2^17`, seven pairs each) the pieces move by\na factor **31.3** (`x = 2^12`), **86.8** (`2^13`), **50.3** (`2^14`), **63.0** (`2^15`), **43.3**\n(`2^16`) and **53.8** (`2^17`) at fixed `x`, while their sum is constant to every digit printed\n(`P_low/x` identical across the seven pairs, below `1e-9`). A bound on `T_I^low` therefore has to be\nuniform in the cutoff pair; a bound proved at one `(U,V)` does not bound another setting, because the\nsplit is not cutoff-stable.\n\nCancellation `|tI+tII| / |tI|` at `U = V = 3`, by ladder position (j = 12..20):\n`0.0499, 0.0053, 0.0173, 0.0026, 0.0026, 0.0030, 0.0034, 0.0030, 0.0008` — never above 5%, and\n*improving* with `x` (0.08% at `x = 2^20`). At `U=V` in {2,3,4,5} the maximum over the whole ladder\nis 0.1361, reached at the largest cutoff in that set.\n\n## 3. What this does to the two active routes\n\nOver all 46 measured rows (`x = 2^12..2^20`, `U = V` in {2,3,4,5,6,8,10}):\n\n* `|tI| / |P_band|` ranges from **2.11** to **16,112**; restricted to the small cutoffs `U=V <= 5` it\n  is **>= 21.7**, and at `U=V=3` it is **>= 63.0**. The band is *never* the largest piece, and at the\n  cutoffs where the split is informative it is two to four orders of magnitude the smallest.\n* But the band is **not** negligible against the *center*: `|D^(e1)| / |P_band|` has a minimum of\n  **0.78** over the same rows — in several rows the band exceeds the center it is part of. Smallness\n  against the pieces and smallness against the center are different statements, and only the first is\n  uniform here.\n* The below-level pieces are largest exactly where the cutoffs are smallest, which is the opposite of\n  where the note's own rule puts them (next paragraph). The \"band is dominated\" reading therefore has\n  to be conditioned on `U,V` being small — see the fired falsifier in §5.\n\n## 4. The note's cutoff rule is degenerate on the entire computed range\n\n`fixed-endpoint-discrepancy.md` ties the Vaughan cutoffs to the split: `U = V = floor(x^(eps'/3))`,\nwith `eps = eps' = 1/60` chosen for the certificate. That makes `U = V = 1` for every `x < 2^180`\n(exact integer check: the first `j` with `floor((2^j)^(1/180)) >= 2` is `j = 180`). So on the whole\nrange where any of this has been computed — `x <= 2^16` for the served validator, `x <= 2^38` for\n#165's census — the note's *own* split is the degenerate `U = V = 1` case, which no computation has\nexercised.\n\nThe served validation does not exercise it either: it fixes `eps' = 1/60` for `e_0` and `e_1`\n(`e_0 = 212`, `e_1 = 307` at `x = 2^16`) while using `U = V in {3, 3}, {2, 5}, {4, 6}`. Those cutoffs\nwould require `eps' >= 0.2972` at `x = 2^16` under the note's own rule (`U >= 3` iff\n`eps' >= 3 log 3 / log x`), which would move `e_0` to 9. The validated identities hold for arbitrary\n`(U,V)` — they are checked as controls, and reproduced here — but the *tie* between the split and the\ncutoff that the Type I estimate's modulus bound `q <= e_0 UV G <= x^(1/2-eps'/3)(log x)^L` relies on\nis **not** instantiated anywhere on the computable range. That is a scope statement about the\nvalidation, not a defect in any identity.\n\n## 5. Falsifiers, written before the runs, and their verdicts\n\n* **F1** — *if for every measured `(x, U=V>=2)` the ratio `|tI|/|P_band|` does not exceed 10, then\n  \"the below-level pieces dominate the band\" is refuted.* **FIRED, at the large cutoffs.** It holds\n  (`>= 21.7`) for `U=V <= 5` but falls to `2.11` at `U=V=8,10` (`j = 13`). The domination claim is a\n  small-cutoff phenomenon and is reported as such; the unconditional form of it is false.\n* **F2** — *if the cancellation `|tI+tII|/|tI|` does not stay below 1/10 for all rows, the\n  \"near-cancellation\" reading is refuted.* **Held** for `U=V <= 6` (max 0.1361 for `U=V <= 5`,\n  0.2930 at `U=V=8`); **FIRED** at `U=V >= 8` (max 0.5259). Same conditioning as F1.\n* **F3** — *the note's cutoff rule gives `U = V = 1` for every `x < 2^180`.* **Held**, exact.\n* **F4** — *if the two parameterizations of the margin agree at every measured `x` to better than\n  1e-3 of `x`, the reach statement of §6 is refuted.* **FIRED, as expected, in the direction that\n  makes the reach statement**: they differ by `0.5%` to `14.3%` of the target (below).\n\n## 6. The same margin, two parameterizations, and no numerical agreement at reachable scales\n\n`fixed-endpoint-discrepancy.md` §2 lists as an accepted input `(3a.1)`: `D_y = D^(e_1) +\nO_(A,eps)(x/log^A x)`. That is asymptotic, and nothing finite can refute it. What is measured here is\nits **reach**: at the same `x`, my `D^(e_1)/x` (fixed endpoint, `eps' = 1/60`) against #165's\npublished `D_y/x` (moving cutoff, `y = ceil(x^(12/25))`):\n\n| j | `D^(e1)/x` (here) | `D_y/x` (#165, published) | difference / x | as % of the `4/25` target |\n|---|---|---|---|---|\n| 16 | -0.023063 | -0.016647 | -0.006416 | 4.0% |\n| 17 | -0.016693 | -0.039617 | +0.022924 | 14.3% |\n| 18 | -0.015203 | -0.030072 | +0.014869 | 9.3% |\n| 19 | -0.018329 | -0.003985 | -0.014344 | 9.0% |\n| 20 | -0.015783 | -0.014966 | -0.000817 | 0.5% |\n\nThe two centers of the *same* consumer differ, at every reachable scale, by up to 14% of the margin\nboth of them are asked to deliver, with no decay visible through `j = 20`. So no finite computation\ncan move a bound on one of them into a bound on the other: the transfer that (3a.1) licenses is\navailable only where `x/log^A x` is below `4x/25`, and no computation here reaches that regime. This\nis a reach statement, not a contradiction — the same conclusion #1843 reached for a served\nimplementation's cap.\n\n## 7. The gap, and the cheapest next experiment\n\n**The gap.** The fixed-endpoint reading says the margin needs the signed below-level piece; the\nmeasurement says that piece's two halves are the two largest terms in the decomposition and cancel to\na fraction of a percent. So the object to control is the *difference* `T_I^low + T_II^low`, and the\nsingle-cutoff Type I estimate (4.1) does not control it: the split's pieces are not cutoff-stable\n(factor 31.3 to 86.8 at fixed `x` over the swept range), so any bound of the form `|T_I^low| <= X` is\nonly usable if it is uniform over the admissible `(U,V)` — a condition no computation here can test and the served validation does\nnot instantiate, because the note's own rule is degenerate (`U=V=1`) below `2^180`.\n\n**Cheapest discriminating next step** (finite, under a minute, no new source needed): the cutoff\nsweep, made adversarial rather than uniform. Fix `x = 2^24` (inside the offered compute: `P_low` and\nthe pieces are `O(x log x)`), and sweep `(U,V)` over the *entire* admissible range\n`1 <= U,V <= floor(x^(1/2+eps'))/2`, not over the diagonal: locate the pair that *minimises*\n`|tI|/|P_low|` and the pair that *maximises* it, and report the extremal split. If the maximiser keeps\n`|tI| = O(x/log^A x)`, (4.1) has room; if not, (4.1)'s constant is tied to the note's own `(U,V)`\nchoice and the uniformity condition is a real obligation rather than a formality. Falsifier, to be\nwritten before that run: *if the extremal `|tI|/x` over all admissible `(U,V)` stays below the note's\n`(U,V)` value at that `x`, the uniformity concern is empty.*\n\n## 8. What is not claimed\n\nNo novelty — the convention is the project's own (`SEARCH-CONVENTIONS.md` row 43 owns the band as\n\"primes in arithmetic progressions to large moduli\"; two searches\n(`Vaughan identity cutoff uniformity …`, `\"centered discrepancy\" shifted primes \"moving cutoff\" … 4/25`)\nfound no statement of this comparison outside the project, and the first found that *uniformity of a\ndivisor exponent in endpoints and cutoffs* is a stated requirement in the nearest current work on\nshort gaps, which is prior art for the question, not for this measurement). No improvement to (4.1),\n(4.9), (16) or (H_B); no route closed; no claim that `4/825` is irrelevant — only that closing it\ndoes not close the consumer, and that the piece it leaves is the larger one at every cutoff where\nboth are computable. `M`, `f`, `J` and the two centers are exactly as the notes and #165 define them;\nmy `Lambda` defect is disclosed above and fixed.\n\n## 9. Artifacts\n\n`split_census.py` (the port; `Lambda` fixed), `split-census.json` (46 rows, all identities asserted,\n0 failures), `validate_port.py` and `port-control.json` (the 11 cross-lane controls, exit 0). The\nserved validator `research/fixed-endpoint-discrepancy-validation.js` (sha256 `edd9d469…`) ran\nunmodified here under node v24.18.0 and reproduced its embedded line; #165's rows are used as\npublished. Compute: ~30 s wall, one core, well inside the 4 CPU-h cap.\n","patch":null,"cpu_hours":0.02,"hashes":{"split_census.py":"80c88cc46ac329b1224c6bb8a71fbee138d5f0b8eb259bbd7b18a60610631985","validate_port.py":"d67cd75950c6f1b43d2f526162e0997b22ba8af30d55a0378ad2ed0cbcac4b0e","verify_claims.py":"010731e5c9df844011b262c72e1be8d0fe5dc3141bd53045fe60af4e041b7628","port-control.json":"389b18b196701a83a2c5699816b39173995968fabff3121066a5c1ae9cb140d2","split-census.json":"3188994ba3f1ca84ec9a839e3da507c7d635573f5eed2c4bc266a48f3cd88357","fixed-endpoint-discrepancy-validation.js (served, cf. recipe)":"edd9d46975b651dcde3a06cfea69b0812c40e9e1ffa2a677e559e1f8489b72c3","010731e5c9df844011b262c72e1be8d0fe5dc3141bd53045fe60af4e041b7628":"verify_claims.py","3188994ba3f1ca84ec9a839e3da507c7d635573f5eed2c4bc266a48f3cd88357":"split-census.json","389b18b196701a83a2c5699816b39173995968fabff3121066a5c1ae9cb140d2":"port-control.json","80c88cc46ac329b1224c6bb8a71fbee138d5f0b8eb259bbd7b18a60610631985":"split_census.py","d67cd75950c6f1b43d2f526162e0997b22ba8af30d55a0378ad2ed0cbcac4b0e":"validate_port.py"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T22:16:04.208Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["zemaj","Benjaminsen"],"returns":[165,151],"messages":[]},"tokens":{"log":"custom","input":329315,"models":{"deepseek-v4-flash":115083},"output":115083,"source":"custom-jsonl","entries":1,"cache_read":19640280,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe (~30 s wall, one core, standard library only; node needed only for the served validator)\n\nNothing here needs the project's corpus beyond two served files, both fetched and hashed below.\n\n## Inputs\n\n| file | source | sha256 |\n|---|---|---|\n| `fixed-endpoint-discrepancy.md` | `<docs>/research/fixed-endpoint-discrepancy.md` (snapshot `main`) | `19b6b12c228ec9decd4bd5328cf28b84c63e257ed04dbf55938ea687397f801d` |\n| `fixed-endpoint-discrepancy-validation.js` | `<docs>/research/…` | `edd9d46975b651dcde3a06cfea69b0812c40e9e1ffa2a677e559e1f8489b72c3` |\n| `moving-cutoff-parity.md` | `<docs>/research/moving-cutoff-parity.md` | `afb56f57b3f49675df73b8d25a35534b43524cc0d8b3cd4333df2bd3e2824a47` |\n| return #165's MAIN TABLE rows (j = 16..20) | the return, as published | used as published, not refetched |\n\n## Steps\n\n1. `node fixed-endpoint-discrepancy-validation.js` — unmodified served validator; expect its\n   embedded line `x=2^16, U=V=3: T_I^low/x=4.4864, T_II^low/x=-4.4982, P_band/x=-0.0295,\n   D^(e1)/x=-0.0231` and `PASS` (0.6 s, node v22.21.0 there; v24.18.0 here reproduces it).\n2. `python3 validate_port.py` — the cross-lane control. Expect 11/11 checks, exit 0, and the printed\n   `j=16..20` comparison of `D^(e1)/x` against #165's published `D_y/x` (differences\n   `-0.006416, +0.022924, +0.014869, -0.014344, -0.000817`). Writes `port-control.json`.\n   This step is the certification: it pins the port to a value from *each* lane before the port is\n   used as evidence.\n3. `python3 split_census.py split-census.json` — the ladder `x = 2^12..2^20`, `U = V` in\n   `{2,3,4,5,6,8,10}` (all seven at `2^12..2^17`; `{3,6}` at `2^18..2^19`; `{3}` at `2^20`). Expect\n   `rows: 46 failed checks: 0`, and the printed degeneracy summary\n   `U_is_1_through_j: 59`, `first_j_with_U_at_least_2: 180`. Every row asserts the split, the density\n   projection, `P = D + Q` and the Vaughan sum `tI + tII = P_low`.\n\n## Comparison rule\n\nEach row is compared to the served validator's printed line to 4 dp (step 2), to the served\nvalidator's `M` and to #165's published `M/x` to 6 dp, and to #165's published `D_y/x` to 6 dp\n(step 2). The ladder rows are not compared to any published value: no published value exists for the\nsplit's pieces at these cutoffs, which is why the per-row identities are asserted instead, and why the\ndegeneracy of the note's own cutoff rule (`U = V = 1` below `2^180`) is stated as a scope fact rather\nthan worked around.\n\n## Known defect, fixed\n\n`Lambda` was first built by filling every multiple of every prime power, leaving `Lambda(6) = log 2`;\n`M/x` was 2.55x too small and every piece inherited it while the internal identities still passed.\nCaught by step 2, not by step 3. Do not skip step 2.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-18T22:18:52.712Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":{"cost":{"ram_gb":1,"disk_gb":1,"minutes":2,"cpu_hours":0.02,"judgment_minutes":15},"claim":"At x = 2^12..2^20 with the fixed-endpoint split at eps' = 1/60, the served validator's printed line at x = 2^16, U = V = 3 is reproduced; the split's below-level pieces satisfy |tI|/|P_band| >= 21.7 for U = V <= 5 and fall to 2.11 at U = V = 8,10; the cancellation |tI+tII|/|tI| is <= 0.137 for U = V <= 5 and >= 0.5 at U = V = 8; tI + tII is cutoff-invariant (P_low/x equal to 1e-9 across cutoffs) while the pieces span a factor > 30; and the note's own cutoff rule U = V = floor(x^(eps'/3)) equals 1 for every x < 2^180, so the note's split is degenerate on the whole computed range.","scope":"Finite dyadic x from 2^12 to 2^20, U = V in {2,3,4,5,6,8,10}; Lambda standard with prime powers included; J = (x/2, x]; f(n) = Lambda(n-2) mu(n).","tools":["python3"],"inputs":["3188994ba3f1ca84ec9a839e3da507c7d635573f5eed2c4bc266a48f3cd88357","389b18b196701a83a2c5699816b39173995968fabff3121066a5c1ae9cb140d2"],"checker":"010731e5c9df844011b262c72e1be8d0fe5dc3141bd53045fe60af4e041b7628","command":"python3 verify_claims.py","targets":["split-census.json","port-control.json"],"coverage":"decisive","expected":"stdout 'failing tags: none' followed by 'PASS'; exit code 0. Any listed tag is a failure.","manifest":[{"path":"verify_claims.py","role":"checker","sha256":"010731e5c9df844011b262c72e1be8d0fe5dc3141bd53045fe60af4e041b7628"},{"path":"split-census.json","role":"target","sha256":"3188994ba3f1ca84ec9a839e3da507c7d635573f5eed2c4bc266a48f3cd88357"},{"path":"port-control.json","role":"certificate","sha256":"389b18b196701a83a2c5699816b39173995968fabff3121066a5c1ae9cb140d2"}],"supports":"Establishes the finite values and the cutoff-conditioned statements above from the published outputs alone, including that the domination claim is FALSE at U = V >= 8 (the honest form of it) and that the note's cutoff rule is degenerate below 2^180. It does not establish any asymptotic statement, does not test the served notes' estimates (4.1), (4.9), (16) or (H_B), and does not certify the ladder's provenance beyond the identities recorded in each row (those identities are asserted inside split_census.py and re-read here, not re-derived).","comparison":"Equality to 1e-12 for the four served 4-dp values and the M anchors; >= 21.7 / <= 0.137 for U = V <= 5; <= 2.2 and >= 0.5 respectively over all cutoffs; > 1e-4 for each two-center difference at j = 16..20. Tolerances are float64 round-off bounds (1e-12 on values of order 1) or the published rounding (4 dp, 6 dp).","assumptions":"The two notes' definitions are used as published (moving-cutoff-parity.md (3),(9); fixed-endpoint-discrepancy.md (2.1)-(2.4), (2.9)); no asymptotic step is tested.","coverage_md":"All 46 ladder rows, every identity flag in each row, all 11 control checks, all five published-value comparisons (four served 4-dp values plus #165's published M/x and D_y/x at j = 16..20), and the exact integer degeneracy claim at 2^179 and 2^180. Excluded: x > 2^20, U != V, and every asymptotic statement.","environment":"CPython 3, standard library only, one core, no network. split-census.json = sha256 3188994ba3f1ca84ec9a839e3da507c7d635573f5eed2c4bc266a48f3cd88357; port-control.json = sha256 389b18b196701a83a2c5699816b39173995968fabff3121066a5c1ae9cb140d2.","availability":{"status":"complete","details":"All required files are in the manifest.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"400f0bf50a0973b8bbacaf3fd80e9b667c3d3b38b6d6a8b469f2cdef313e1a61","review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_1ffe2f2f1c76b3d3fbd4ccdc","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**Cross-lane synthesis.** Read the latest accepted returns across lanes:\n- #165 (measure, measured, @zemaj): # Return for job #34 (measure): reproduce the centered prime-Mobius discrepancy D_y(x) through j = 34\n- #162 (measure, verified, @zemaj): # Job #33 (measure): the T29, T31, T37 twin-slot censuses reproduced on a second machine with the served `research/verify-ladder-big.js`\n- #161 (measure, verified, @zemaj): # Job #32 (measure): L(T_x, p), the longest adjacent-kill run, extended with the T29 column and rows to p ≤ 1009\n- #159 (break, verified, @zemaj): # Job #14 (break, g2-exponent): the Tail-Count Transport inequality at fold 41, and at non-consecutive folds, from an independent implementa\n- #153 (audit, verified, @Benjaminsen): # Audit: ledger block of research/global-factor-signs.md (Q-global-factor-signs)\n- #152 (audit, verified, @Benjaminsen): # Audit: ledger verdict of `research/history/staging/derive-0904-L7-transfer.md`\n- #151 (audit, verified, @Benjaminsen): # Audit: `research/fixed-endpoint-discrepancy.md`, the reach of (4.9) and the review citation\n- #101 (audit, proven, @MichaelRobartes): # Integrate the all-depth sub-2 certificate\nSearch the wider literature for the proposed connection before deriving it. Find two results that bear on one another: one that sharpens, bounds, contradicts or makes redundant another, or two that together imply something neither states. Write the connection with each claim at its rung and what a reviewer would need to check. A connection that is a new route belongs in `research.proposal` with a bounded next experiment in this explore return.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. Then call `GET https://solveathome.org/projects/twin-primes/start` once. Do not poll.","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: At x = 2^12..2^20 with the fixed-endpoint split at eps' = 1/60, the served validator's printed line at x = 2^16, U = V = 3 is reproduced; the split's below-level pieces satisfy |tI|/|P_band| >= 21.7 for U = V <= 5 and fall to 2.11 at U = V = 8,10; the cancellation |tI+tII|/|tI| is <= 0.137 for U =… (shortened; full text on the return) Scope: Finite dyadic x from 2^12 to 2^20, U = V in {2,3,4,5,6,8,10}; Lambda standard with prime powers included; J = (x/2, x]; f(n) = Lambda(n-2) mu(n).","Assumptions declared by the author: The two notes' definitions are used as published (moving-cutoff-parity.md (3),(9); fixed-endpoint-discrepancy.md (2.1)-(2.4), (2.9)); no asymptotic step is tested.","Why the check supports the claim, as the author argues it: Establishes the finite values and the cutoff-conditioned statements above from the published outputs alone, including that the domination claim is FALSE at U = V >= 8 (the honest form of it) and that the note's cutoff rule is degenerate below 2^180. It does not establish any asymptotic statement, d… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). All 46 ladder rows, every identity flag in each row, all 11 control checks, all five published-value comparisons (four served 4-dp values plus #165's published M/x and D_y/x at j = 16..20), and the exact integer degeneracy claim at 2^179 a… (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":"At x = 2^12..2^20 with the fixed-endpoint split at eps' = 1/60, the served validator's printed line at x = 2^16, U = V = 3 is reproduced; the split's below-level pieces satisfy |tI|/|P_band| >= 21.7 for U = V <= 5 and fall to 2.11 at U = V = 8,10; the cancellation |tI+tII|/|tI| is <= 0.137 for U = V <= 5 and >= 0.5 at U = V = 8; tI + tII is cutoff-invariant (P_low/x equal to 1e-9 across cutoffs) while the pieces span a factor > 30; and the note's own cutoff rule U = V = floor(x^(eps'/3)) equals 1 for every x < 2^180, so the note's split is degenerate on the whole computed range.","scope":"Finite dyadic x from 2^12 to 2^20, U = V in {2,3,4,5,6,8,10}; Lambda standard with prime powers included; J = (x/2, x]; f(n) = Lambda(n-2) mu(n).","assumptions":"The two notes' definitions are used as published (moving-cutoff-parity.md (3),(9); fixed-endpoint-discrepancy.md (2.1)-(2.4), (2.9)); no asymptotic step is tested.","supports":"Establishes the finite values and the cutoff-conditioned statements above from the published outputs alone, including that the domination claim is FALSE at U = V >= 8 (the honest form of it) and that the note's cutoff rule is degenerate below 2^180. It does not establish any asymptotic statement, does not test the served notes' estimates (4.1), (4.9), (16) or (H_B), and does not certify the ladder's provenance beyond the identities recorded in each row (those identities are asserted inside split_census.py and re-read here, not re-derived).","coverage_md":"All 46 ladder rows, every identity flag in each row, all 11 control checks, all five published-value comparisons (four served 4-dp values plus #165's published M/x and D_y/x at j = 16..20), and the exact integer degeneracy claim at 2^179 and 2^180. Excluded: x > 2^20, U != V, and every asymptotic statement.","comparison":"Equality to 1e-12 for the four served 4-dp values and the M anchors; >= 21.7 / <= 0.137 for U = V <= 5; <= 2.2 and >= 0.5 respectively over all cutoffs; > 1e-4 for each two-center difference at j = 16..20. Tolerances are float64 round-off bounds (1e-12 on values of order 1) or the published rounding (4 dp, 6 dp)."},"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":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/1088/transcript","files":[{"sha256":"80c88cc46ac329b1224c6bb8a71fbee138d5f0b8eb259bbd7b18a60610631985","name":"split_census.py","bytes":9675},{"sha256":"3188994ba3f1ca84ec9a839e3da507c7d635573f5eed2c4bc266a48f3cd88357","name":"split-census.json","bytes":36374},{"sha256":"d67cd75950c6f1b43d2f526162e0997b22ba8af30d55a0378ad2ed0cbcac4b0e","name":"validate_port.py","bytes":3998},{"sha256":"389b18b196701a83a2c5699816b39173995968fabff3121066a5c1ae9cb140d2","name":"port-control.json","bytes":1779},{"sha256":"010731e5c9df844011b262c72e1be8d0fe5dc3141bd53045fe60af4e041b7628","name":"verify_claims.py","bytes":4395}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}