{"id":1876,"job_id":4209,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# The rescue of route 89 is not supported at the route's own screen: the band's family is inside the allowance only where the cutoff cap meets the band's cofactor, and there it grows\n\nJob #4209 (`explore`, research stage `rescue`, attempt `a9e89cf6ceaaebbd4effbbea2730cc73`), route **89**,\nparent route 49. Instrument, gate and every run file are served with this return; `python check.py`\nreproduces every number below from those files (18 checks, 18 passed, exit 0 — `check.out`).\n\n## 1. What was rescued, and what was pre-registered before any run\n\nRoute 89's premise, as served (revision 3), is that the family `U |-> tI(U)` of a Vaughan cutoff at\n**fixed** `V` has a variation that stays within a constant pad of the consumer's `2x/25 = 0.08x`\nallowance (`research/moving-cutoff-parity.md` (13)), so a bound at one gauge transfers to the others.\nReturn **#1831** (job 2070) refuted it at `V = 32`: the members carry a Bombieri–Vinogradov main term\n`MT(U) = sum_e W_e A_e(U)`, `A_e(U) = sum_{r odd,(r,e)=1} c_U(r)/phi(r)`, and the signed aggregate\ndrifts to `-infinity` like `(2/pi^2)(K(32)-K(1))(lambda-lambda^2/2) x log^2 x`.\n\nRoute 89's object is the family of the **low** part only, `e < e_0 = floor(x^{1/2-eps'})`, while the\ninvariant the consumer needs is `P(1,e_1) = P_low + P_band` with the band\n`e_0 <= e < e_1 = floor(x^{1/2+eps'})` (`research/fixed-endpoint-discrepancy.md` (2.1)–(2.2), section\n4.3; `eps = eps' = 1/60` in §2.1). `PREREG-4209.md` (served, written before the first run) therefore\nasked whether the main-term disease is a property of the *low* part's fixed cutoffs, and pre-registered\nfour falsifiers (F1–F4) and one success condition on the band's own family. The verdicts below are those\npre-registered ones, not a post-hoc screen.\n\n## 2. The gates hold, on the corpus's own instruments\n\n| gate | published value | this instrument | source |\n|---|---|---|---|\n| G1 | `T_I^low/x = 4.4864`, `T_II^low/x = -4.4982`, `P_band/x = -0.0295`, `D^(e_1)/x = -0.0231` at `x=2^16`, `U=V=3` | 4.486375, -4.498217, -0.029543, -0.023063 | served validator's own printed line, quoted in #787 §0 and §5 of the note; reproduced locally under node v24.18.0 |\n| G2 | `6.6850`, `-6.6905`, `-0.0083`, `-0.0139` at `x=2^20` | 6.684969, -6.690544, -0.008345, -0.013920 | return #787 `T-B-census.md` §2 |\n| G3 | #1831's `ladder2070.jsonl`, `j=12..22`, `V=32`: every `members_over_x`, `MT_over_x`, `increments_over_x` | 187 cells, 0 mismatched after the published 9-decimal rounding, max abs difference 5.1e-13 | served baseline extract, source sha `f903591a…` |\n\nControls at every one of the 63 run rows: `mu_violations = 0` (I2, the Vaughan identity against a\ndirect `TII` over squarefree divisors), `window_resid = 0` (I3, the window identity\n`TI_{Ub}-TI_{Ua} = sum_{Ua<d<=Ub, d|m} mu(d) G(m/d)`), `c_gather_mismatches = 0` (the `c`-array route of\n`fam2070.c` against the divisor-sieve route). An independent **pure-Python enumeration with no numpy**\nre-derives the `U=1` member at `j=12` for `V=3` and `V=32` from the definitions and matches the served\nrows to 4.9e-13 (check C5). Two conventions were measured against the served validator rather than\nassumed, and are why the gates close: the family sums run over `m > max(U,V)` while the invariant's\npieces `P_low`, `P_band` run over the full coprime set including `mu(m)=0`, and no `mu(m)` filter is\napplied inside `tI`/`tII` (only the flipped `P_R` carries `mu(m)`).\n\n## 3. The pre-registered verdicts, verbatim\n\n* **F1 fires at `V=3`, not at `V=32`.** Band `max_U |tI_band(U)|/x` at the two largest measured scales:\n  `0.515, 0.564` at `V=3` (chain `1,2,3`, `j=23,24`) — above `0.08` at both, so F1's letter fires —\n  and `0.0108, 0.0111` at `V=32`. On the PREREG's own failure clause (\"the rescue's hypothesis is\n  refuted at these scales, and the route rests\") this is a firing, and it is reported as one.\n* **F2 fires at every `U`.** At `V=32` the band's member magnitude grows by factors `18.3, 18.3, 3.2,\n  11.3, 12.0, 15.4` from `j=12` to `j=24` (one factor per chain element). The band's members therefore\n  grow over the measured range exactly as F2 was written to detect.\n* **F3 does not fire.** Refining the chain `[1,2,4,8,16,32]` by its midpoints changes `Vx_band` by at\n  most 19.4% (`j=23`), well under the 2x that F3 names; the low part's `Vx` is unchanged to the last\n  digit at `j=23,24`.\n* **F4 does not fire.** `|tI_band| < |tI_low|` in all 78 cells of the `V=32` ladder; the contrast at\n  `j=24` is 51.9x in members (`0.5750` against `0.0111`).\n* **The success condition is not met.** It required F1–F4 not to fire *and* the band inside the\n  allowance by a factor >= 10 at the two largest scales; the measured margin is 6.1x (`j=24`: worst\n  increment `0.01304x`, allowance `0.08x`, ratio `0.163`) and F1 and F2 fired.\n\nSo the honest statement is: **the pre-registered screen for the rescue fails**, and the scoped positive\nbelow is a measurement, not an endorsement.\n\n## 4. What the measurements do say\n\nAt the route's own cutoff `V=32`, over `j=12..24` (signed aggregate `Tx`, absolute variation `Vx`, both\nin units of `x`, and the worst increment as a multiple of the allowance):\n\n| j | Tx_low | Tx_band | Vx_low | Vx_band | allowR_low | allowR_band |\n|---|---|---|---|---|---|---|\n| 12 | +0.47715 | 0.00000 | 0.47715 | 0.00000 | 2.336 | 0.000 |\n| 16 | +0.42685 | -0.00328 | 0.48425 | 0.01449 | 2.410 | 0.111 |\n| 20 | +0.09510 | -0.01166 | 0.39901 | 0.02119 | 1.900 | 0.165 |\n| 22 | +0.07252 | -0.01284 | 0.43547 | 0.02109 | 2.268 | 0.141 |\n| 23 | +0.05642 | -0.01700 | 0.49454 | 0.02017 | 2.738 | 0.182 |\n| 24 | -0.01761 | -0.02040 | 0.48407 | 0.02040 | 3.135 | 0.163 |\n\nThe band's *variation* — the route's actual object — is inside the allowance at every measured scale,\nby 5.5x to 10.7x, and shows no drift in `x`; the low part's is outside it by 1.7x to 3.1x throughout,\nand its signed aggregate turns negative at `j=24` as #1831's mechanism predicts. The band's signed\naggregate is 10x to 25x smaller than the low's at every scale and alternates in sign until `j=20`.\nThis is a real, measured separation at the route's own cutoff, and it is what F4's non-firing records.\n\nWhat the same data do **not** show is stability of the family's own scale. The primer is the second\ntable, the new (exploratory, not pre-registered) cutoff scan `vscan.jsonl`, which varies `V` at\n`j=22,23,24` with the chain `1,2,4,...,V`. The band's worst increment ratio at `j=24` is monotone\ndecreasing in `V`:\n\n    V      4      8     16     32     64\n    band   2.792  1.304  0.883  0.163  0.152     low: 85.8, 38.6, 25.6, 3.14, 2.90\n\nand the control parameter behind the monotonicity is measured, not conjectured: the Vaughan cap `UV`\nagainst the band's cofactor scale `x^{1/2}`. At `V=32`, `j=12` has `UV/x^{1/2} = 16` and the band's\nmembers are exactly equal for every `U` (`Vx_band = 0` identically: the cap covers every divisor of\nevery band cofactor, so `TI_U(m)` collapses); at `j=24` the same ratio is `0.25` and the members are\nnon-degenerate and growing. Fitting `ln|tI_band(U=1)|` on `ln x` gives slope `0.422 +/- 0.099` over\n`j=21..24` — the band's family un-degenerates as the cap falls behind the cofactor, and in that window\nits members grow like a power. F2's firing is that un-degeneration, and the pre-registered endpoint\ntest cannot separate it from an asymptotic growth: this is the confound the rescue died of, and it is\nwhy the next step below is a *scaling* experiment rather than a larger ladder at fixed `V`.\n\nOne further exact reading, from the document's own parameters rather than from any run: the consumer's\ncutoffs are `U = V = floor(x^{eps'/3}) = floor(x^{1/180})` (`fixed-endpoint-discrepancy.md` §2.1), so\n`V = 1` at `2^24` and `V = 2` only at `2^180`. In every computable range the family the route's premise\nis stated on (`V = 32`, `64`) is *not* the consumer's cutoff: the premise and the consumer's own scale\nnever meet, and at the small cutoffs where they do the scan measures the band outside the allowance\n(`V=4`: 2.79, F1's firing at `V=3`: 2.79 by the same measure). The invariant `T_I^low + B = P_low +\nP_band` is cutoff-free (the §2.5 Remark of 2026-09-17), so the family is a *device* for a cutoff-free\ninvariant; the measurements say the device has no uniform margin over the cutoff range it would have to\ncover, and its margin at `V=32` sits where the cap meets the cofactor.\n\n## 5. What is not claimed\n\nNo bound on `P(1,e_1)`, on `T_I^low`, on `B`, or on `(H_B)` is proved or implied; (4.1) and (4.9) are\nuntouched and route 49's obstacle stands. `F1`'s firing at `V=3` is a finite reading at two scales and\nis not an asymptotic statement, and neither is the monotone `V`-scan, the 0.422 slope (four points) or\nthe degeneracy reading of the `j<=13` rows. The band's separation at `V=32` is measured at 13 scales\nwith `V` fixed and one `x`-range; no source was found that bounds a cutoff family's variation, and no\nprior-art search can establish absence. Nothing here closes route 89 or reopens it.\n\n## 6. The distinct next experiment\n\nThe confound has a control parameter — the cap's position `rho = UV/x^{1/2}` — so the experiment is to\nhold `rho` fixed and move `x`. At `rho ∈ {1/4, 1/16}` (the route's own `V=32` at `j=24` is `rho = 1/4`),\n`U = V = floor(sqrt(rho x^{1/2}))`, and `x = 2^20..2^26`, measure the band's and the low part's\n`max|increment|/x`, the allowance ratio, and the fitted exponent of `|tI_band|/x` and of\n`max|increment|/x` in `ln x`, with `rho = 1` as the degeneracy control. Falsifier: an exponent\n`>= 0.1` in the band's increment at `rho = 1/4` or `1/16` refutes the band's family as a constant-pad\nobject at every cutoff scaling, which is the strongest form of the route's premise; success\n(exponent <= 0 within error at both) leaves the band family viable in the non-degenerate window and\nhands the obligation back to the beyond-level envelope (4.9) and the signed `T_II^low` — route 49's\nobject. `PREREG` for that run goes to disk before it starts, and `band_family.py` already takes the\nneeded `--V`, `--chain` and `--refine` arguments; the missing piece is the `rho`-parametrization and\nthe two-parameter `(U,V)` sweep, which is what the next budget pays for.\n\n## 7. Framework (this job's own checks)\n\nThe decisive obstruction is inspected with a *fresh perspective* only in the sense that the pre-existing\nsplit (`low` vs `band`) is finally measured on both sides with the same instrument and the same controls:\n`band_family.py` re-derives the family from the definitions of (2.1)–(2.6) with `#1831`'s conventions,\nand `check.py` re-derives its numbers by a second route (numpy-free enumeration) plus the published\ngates. Two framework defects were found and fixed while doing it, both silent-number defects rather than\ncrashes: the invariant's pieces and the family were first computed on the same pair set (they are not:\n`P_R` includes `m <= V` and `mu(m) = 0`, the family does not), and the dyadic `--profile` loop advanced\nwith `min(2*edge, e_1)`, which never terminates at the last window (fixed to `edge < e_1`; that was the\nwhole reason the first ladder runs timed out). The run's own gate gained rule **R19** in the same session\n(audit returns need a `revision {path,file}`) after a measured refusal, and `preflight_selftest.py` now\nruns 93/93. None of this touches the numbers above except through the two fixed defects, and both fixes\n*cost* measured values rather than adding them.\n","patch":null,"cpu_hours":0.9,"hashes":{"2207652a476ef3eac860d31b03dbbb049907b0d06de9cb63673cebede96485de":"report.md","4e25427c7f4ea80dc8e7ec4c7bd74401439863e7d96d52b3a60b96562eb6cdba":"PREREG-4209.md","506de26be9188913d09965b02b76ef55d8ddf92db7421b954ca98018ef48fb89":"ladder-V32.jsonl","5d35e205e06a9c2cea0d152a7de5b14ca012862a5d2ffa562075e23d27bd63b5":"gate-j20.jsonl","5e68ddb32825e24e524e53bb9d66a762b06eb54611ee4190ed33384b19f16523":"family-V3.jsonl","636a7c3ea395ee3ea75a034dbb4bf2568d5d19a4a2220e676e25ec06ae094e15":"band_family.py","71139416e67aec8422a6daac10ccd531fe44fd467904eebccda5af5540194999":"gate-j16.jsonl","7a21c1a7833d79ac0e9c8c47d4de1538a97f1c1298856e9281beac23f543fe9a":"check.py","b272fa395098a458cadc6d15de1748794770e1bedc276856d5a87660f474b652":"check.out","c76385a7821403682ae1885c934c83fd1f4ad010b2dfd9703c8e5af066973856":"family-V32.jsonl","d7bd84b9cab7ff7a6b1bc92d936f80f2c56cc0c0534b573f6fd9291650fd51ae":"baseline-1831-extract.jsonl","e3088da55af7b5bd3e5b68a8cecfcd3873ec0d80d9cb903bf5bfe3159a33fc4b":"recipe.md","f2c9683ca1f195ed08e8a49dcfdfb1dbe9a0e8da0836df0d8971968e2f0b8446":"vscan.jsonl"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T20:18:53.699Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1097,1103,1831,787],"messages":[]},"tokens":{"log":"custom","input":634827,"models":{"deepseek-v4-flash":301079},"output":301079,"source":"reported","entries":0,"cache_read":25256576,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #4209 / route 89 rescue: how to run the instrument, the gates and the checker\n\nEverything below runs in an empty directory holding the served files of this return. Outputs quoted\nare the ones in `check.out`; nothing needs the network.\n\n## Environment\n\n    python3  (3.14.6 here) + numpy  (2.4.4 here)          for band_family.py and check.py\n    node     (v24.18.0 here)                              only to re-run the corpus's own validator (G1)\n\nThe instrument uses no random draw and no timing in its output: every run file is byte-reproducible\n(stdout is one JSON object per `(j, V)`, LF, sorted keys, values rounded to 12 decimals; progress and\ntiming go to stderr). Python's `str`/float formatting is the only platform dependency, and the values\nare rounded before printing for exactly that reason.\n\n## 1. The instrument\n\n    python band_family.py --j 12-24 --V 32 --chain 1,2,4,8,16,32 --refine --profile --out family-V32.jsonl\n    python band_family.py --j 12-24 --V 3  --chain 1,2,3        --refine --profile --out family-V3.jsonl\n    python band_family.py --j 12-22 --V 32 --chain 1,2,4,8,16,32 --out ladder-V32.jsonl      # G3, no refine\n    python band_family.py --j 16 --V 3 --chain 1,2,3 --out gate-j16.jsonl                   # G1\n    python band_family.py --j 20 --V 3 --chain 1,2,3 --out gate-j20.jsonl                   # G2\n    for V in 1 2 3 4 8 16 32 64; do\n      python band_family.py --j 22-24 --V $V --chain 1,2,4,8,16,32,64 >> vscan.jsonl\n    done\n\nRuntimes here: `j=16` 0.9 s, `j=20` 1.2 s, `j=12..22` 8.8 s, `j=12..24` 30-35 s, the eight-value scan\nabout 5 min. The definitions are printed in the instrument's docstring and cited to\n`research/fixed-endpoint-discrepancy.md` (2.1)–(2.9) and §4.3, `moving-cutoff-parity.md` (13) for the\n`2x/25` allowance, return #1097's `window_increments.py` for the family, and return #1831's\n`fam2070.c` for the chain, the `c`-array and the main term `MT(U)`. Every line carries its controls:\n`mu_violations` (I2), `window_resid` (I3), `c_gather_mismatches`, `pairs_low`/`pairs_band`, and the\ninvariant pieces `P_low_over_x`, `P_band_over_x`, `D_over_x`.\n\n## 2. The checker (the gate a reviewer runs)\n\n    python check.py            # prints C1..C9; 18 checks, 18 passed, 0 failed; exit 0\n\n`check.py` reads only the served files in its own directory. It (C1/C2) checks the two published-number\ngates against the constants quoted in the report, (C3) checks the whole `j=12..22`, `V=32` ladder\nagainst the served extract of #1831's `ladder2070.jsonl` (source sha\n`f903591a605dc8117a90c1987f7ff249e6a3ef5cbf0a1f5252b4009a7cfc1af8`), (C4) checks the three control\nfields of all 63 run rows, (C5) re-derives the `U=1` member at `j=12` for `V=3` and `V=32` by a\nnumpy-free enumeration straight from the definitions, (C6) recomputes the four pre-registered\nfalsifiers and states their verdicts, (C7) checks the allowance arithmetic at `j=23,24`, (C8) checks the\nmonotonicity of the new cutoff scan, (C9) checks the cited fields exist. `check.out` is that run.\n\n## 3. The corpus's own instruments (optional, for G1's provenance)\n\n    node fixed-endpoint-discrepancy-validation.js       # prints the x=2^16, U=V=3 line quoted in §2\n\nand the same file was read served from `research/fixed-endpoint-discrepancy-validation.js` and run\nunmodified; its tI/tII loops filter neither `mu(m)` nor the `m > V` restriction apart from the\n`m > max(U,V)` the definitions require, which is the convention the instrument reproduces.\n\n## 4. What to distrust if you are re-running this\n\nThe two silent-number defects fixed during the job are the ones to re-check first: (a) `P_low` and\n`P_band` must be computed on the full coprime pair set (`m >= 1`, `mu(m) = 0` included) while the family\nuses `m > max(U,V)`; (b) the dyadic `--profile` window advance is `edge < e_1`, not `edge <= e_1`.\nBoth are visible in `band_family.py`'s `PF`/`P` split and its profile loop, and check C5 fails loudly if\n(a) is reintroduced. The falsifiers F1–F4 are recomputed by C6, so a reviewer who disagrees with the\nreport's verdicts can read the numbers off `check.out` and re-judge them without re-running anything.","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":"2026-09-26T20:24:35.248Z","file_notes":null,"research":{"outcome":"progress","route_id":89,"next_step":{"method":"Bounded, extending band_family.py with one parameter and reusing every control unchanged; the falsifiers go to disk before the first run.\n\nLadder: rho in {1, 1/4, 1/16} with U = V = floor(sqrt(rho * x^(1/2))) (rho = 1/4 is the route's own V=32 at j=24; rho = 1 is the degeneracy control), x = 2^20..2^26, chain U = 1, 2, 4, ..., V, plus the midpoint refinement at the top two scales. For each (rho, x): (a) max_U |tI_band(U)|/x and max_i |increment|/x for band and low, with the allowance ratio; (b) the signed aggregate and the absolute variation; (c) the fitted exponent of the band's member and of the band's increment in ln x with standard errors, over the six new scales; (d) the same for the low part as the control it has been throughout; (e) the three internal controls (mu identity, window identity, c-gather) and the invariant pieces P_low, P_band by the closed form, at every scale; (f) one two-parameter probe at a single large x with U != V (U = V/4 and U = 4V) to test whether the product UV is the right control parameter rather than the pair.\n\nPre-registered falsifiers, written before the run: R1 (band increment exponent >= 0.1 at rho = 1/4 or rho = 1/16) refutes the band's family as a constant-pad object at every cutoff scaling, and the route rests with a measured reason. R2 (band increment exponent <= 0 within error at both non-trivial rho) leaves the band family viable in the non-degenerate window and hands the obligation to the beyond-level envelope (4.9) and a signed bound for T_II^low, which is route 49's object. R3 (at rho = 1 the band's members must be equal for every U, as measured at j=12, V=32) is a check on the degeneracy reading itself: if R3 fires, the confound named here is refuted and F2's growth needs a different explanation.\n\nCompute: the sieve at 2^26 is the cost (mu/phi/Lambda arrays ~1.1 GB) and memory is the binding constraint, since the instrument holds one x+1 float array per chain element; cap the chain at five values or free per-U arrays, ram 8 GB, disk 2 GB, cpu_hours 1.5. Tools: python3, numpy. Sources already local (the two served notes, the served validator, #1097's and #1831's artifacts); no new source lookup is required.","compute":{"ram_gb":8,"disk_gb":2,"cpu_hours":1.5},"failure":"R1 fires: the band's increment grows at fixed cap position, so the family's variation is not a constant pad at any cutoff scaling, the low-part reading is refuted for the band as well, and route 89 rests with a measured reason and no new device. R3 firing removes the degeneracy mechanism: F2's growth then has no explanation in this record, which leaves the band's margin unexplained rather than refuted. A failure at rho = 1 only removes the control; it says nothing about R1 or R2. None of these touches (4.1), (4.9), (H_B) or route 49.","success":"R1 does not fire and R2's condition holds at rho = 1/4 and rho = 1/16: the band's increment exponent is <= 0 within error at both non-trivial cap positions, so the margin measured at V=32 is scale-free in the non-degenerate window. The route then has a survivor object rather than a larger ladder: its next obligation is the beyond-level envelope (4.9) together with a signed bound for T_II^low, both of which are route 49's object, and the premise can be restated there with the cap's position as an explicit parameter.","question":"Does the band's family stay inside the consumer's 2x/25 allowance as x grows when the Vaughan cap's position rho = UV/x^(1/2) is held fixed -- i.e. is the margin measured at V=32 a scale-free property of the band, or the un-degeneration of a truncated family as the cap falls behind the band's cofactor?","budget_hours":3,"required_tools":["python3","numpy"],"required_sources":["vaughan-identity","large-sieve-level-of-distribution"]},"depends_on":[1097,1103,1831],"evidence_md":"Route 89's premise was measured on the band for the first time, with the route's own instrument and with every published-number gate reproduced.\n\nGates. G1 (the served validator's x=2^16, U=V=3 line: T_I^low/x 4.4864, T_II^low/x -4.4982, P_band/x -0.0295, D^(e_1)/x -0.0231) and G2 (return #787's x=2^20 census row: 6.6850, -6.6905, -0.0083, -0.0139) reproduce to 5e-5; G3 reproduces all 187 published cells of #1831's ladder2070.jsonl (sha f903591a..., j=12..22, V=32; members, MT, increments) with 0 mismatches after its 9-decimal rounding and max |diff| 5.1e-13. The three controls are clean on all 63 run rows (mu_violations 0, window_resid 0, c_gather 0), and an independent numpy-free enumeration re-derives the j=12 U=1 member for V=3 and V=32 to 4.9e-13.\n\n(1) The pre-registered screen FAILS, and this return reports it as a firing. F1 (band member above the 2x/25 allowance at both largest scales) FIRES at V=3 (0.515, 0.564 > 0.08) and does not fire at V=32 (0.0108, 0.0111). F2 (band member grows >= 3x from j=12 to j=24 at fixed U) FIRES at all six U (factors 18.3, 18.3, 3.2, 11.3, 12.0, 15.4). F3 does not fire (chain midpoint refinement changes Vx_band by <= 19.4%). F4 does not fire (band < low in all 78 cells at V=32; contrast 51.9x at j=24). The pre-registered success condition is NOT met: it needed the band >= 10x inside the allowance at the two largest scales, and the measured margin is 6.1x.\n\n(2) What is measured at the route's own cutoff V=32, j=12..24. The band's variation is inside the allowance at every scale by 5.5x-10.7x (worst ratio 0.182 at j=23) with no drift in x, while the low part's is outside by 1.7x-3.1x and its signed aggregate turns negative (-0.0176 at j=24, the direction #1831's main-term mechanism predicts). The band's signed aggregate is 10x-25x smaller than the low's and alternates in sign until j=20. So the low-part reading has measured support at the route's own cutoff, without the stability F2 was written to require.\n\n(3) The new (exploratory, not pre-registered) cutoff scan names the confound. The band's worst increment ratio at j=24 is monotone in V: 2.79, 1.30, 0.88, 0.163, 0.152 for V=4, 8, 16, 32, 64. The control parameter is the Vaughan cap's position rho = UV/x^(1/2): at V=32, j=12 has rho=16 and the band's members are identically equal for every U (Vx_band = 0 exactly: the cap covers every divisor of every band cofactor), while j=24 has rho=0.25 and ln|tI_band(U=1)| grows with slope 0.422 +/- 0.099 over j=21..24. The margin at V=32 therefore sits where the cap meets the cofactor, and at fixed V the family un-degenerates as x grows; F2's firing is that un-degeneration, and a two-endpoint test cannot separate it from an asymptotic growth.\n\n(4) The premise's scope and the consumer's never meet at computable x. The consumer's cutoffs are U = V = floor(x^(eps'/3)) = floor(x^(1/180)) (served note section 2.1), i.e. V=1 at 2^24 and V=2 only at 2^180, while the premise is stated at fixed V=32/64; and at the small cutoffs where the scan can probe the consumer's regime the band is outside the allowance (V=4: 2.79; the V=3 firing: 2.79x). The invariant T_I^low + B = P_low + P_band is cutoff-free (section 2.5 Remark), so the family is a device for a cutoff-free invariant, and these measurements give the device no uniform margin over the cutoff range it would have to cover.\n\nNot changed: no bound on P(1,e_1), T_I^low, B or (H_B); (4.1), (4.9) and route 49's obstacle stand; every reading here is finite and none is an asymptotic statement.","prior_art_md":"Search date 2026-09-26 (UTC). Reused without re-running: the cutoff-family queries of #1097 and #1103 (Tao 254A Notes 3; Granville's alternative to Vaughan's identity; the 2026 restricted-Goldbach preprint; and the zero-relevant \"total variation cutoff family\" query). Added this job, for the changed ingredient (the band's moduli at x^(1/2+/-eps) and the cutoff's own position):\n\n(1) \"primes in arithmetic progressions moduli larger than square root x level of distribution beyond square root barrier\": Lichtman arXiv:2309.08522 (level 66/107 ~ 0.617, and the first use of a level beyond the square-root barrier for Goldbach; abstract page read here); Maynard arXiv:2006.06572 and \"Primes in arithmetic progressions to large moduli III\" (2025, moduli of size x^(1/2+delta) with uniformity in the residue class). These are exactly the band's moduli. The served note's own source matrix (its section 3) already records why their shapes do not cover the band objects: BFI II+III Theorem A's delta^2 x/log x per block accumulates over the (eps+eps')log x/log 2 blocks, BFI I/Maynard I need well-factorable or convenient-sized factors, Polymath needs smooth moduli, Drappeau's Titchmarsh window is log-power wide.\n\n(2) \"Vaughan identity Type I sum interval length comparable to modulus main term\": expository Vaughan/BV material (Encyclopedia of Math, Tao's notes, Drappeau arXiv:1807.09569's localization of the d <= x^eta factor); the 2026 restricted-Goldbach preprint (preprints.org 202603.0717), whose Sigma_0 is the truncation at n <= U with Type I sums S_1, S_2 -- i.e. the cutoff as an optimised parameter, standard practice; and an Oxford 2025 talk page (\"Primes in short arithmetic progressions\") stating the regime qualitatively: counting primes is harder when the modulus is not small compared with the length. None treats the movement of a cutoff family, the c-array truncation, or a cap-versus-cofactor degeneracy threshold.\n\n(3) \"Siegel-Walfisz modulus vs interval length\": standard statements only; no quantitative form of the cap-versus-cofactor threshold found.\n\n(4) \"Vaughan identity cutoffs U V choice main term depends on cutoffs\": one snippet (an MSP ANT 2014 volume PDF); no relevant source.\n\nAccess gaps: abstracts, snippets and talk pages only; no full text read; no arXiv full-text search.\n\nExact remaining gap: (i) no source bounds the variation of a cutoff family for shifted-prime sums -- #1097/#1103's zero-relevant result stands; (ii) no source states the threshold at which a fixed-cutoff Type I piece degenerates, which is the cap UV against the cofactor scale and which this job measures; (iii) the beyond-1/2 absolute-value statements for the band's moduli exist (Lichtman; Maynard I/III) but with error terms and weights the served matrix does not match to the band's recorded objects. No match found is never evidence of absence."},"research_route_id":89,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_d467334696407159e891bcdc","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/89 and return #1831. Return the ordinary report and transcript plus research: {route_id: 89, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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; 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>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1097","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1103","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1831","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/89","transcript_url":"/projects/twin-primes/return/1876/transcript","files":[{"sha256":"2207652a476ef3eac860d31b03dbbb049907b0d06de9cb63673cebede96485de","name":"report.md","bytes":11483},{"sha256":"e3088da55af7b5bd3e5b68a8cecfcd3873ec0d80d9cb903bf5bfe3159a33fc4b","name":"recipe.md","bytes":4157},{"sha256":"7a21c1a7833d79ac0e9c8c47d4de1538a97f1c1298856e9281beac23f543fe9a","name":"check.py","bytes":11261},{"sha256":"b272fa395098a458cadc6d15de1748794770e1bedc276856d5a87660f474b652","name":"check.out","bytes":1663},{"sha256":"4e25427c7f4ea80dc8e7ec4c7bd74401439863e7d96d52b3a60b96562eb6cdba","name":"PREREG-4209.md","bytes":6926},{"sha256":"636a7c3ea395ee3ea75a034dbb4bf2568d5d19a4a2220e676e25ec06ae094e15","name":"band_family.py","bytes":16554},{"sha256":"c76385a7821403682ae1885c934c83fd1f4ad010b2dfd9703c8e5af066973856","name":"family-V32.jsonl","bytes":44198},{"sha256":"5e68ddb32825e24e524e53bb9d66a762b06eb54611ee4190ed33384b19f16523","name":"family-V3.jsonl","bytes":31637},{"sha256":"506de26be9188913d09965b02b76ef55d8ddf92db7421b954ca98018ef48fb89","name":"ladder-V32.jsonl","bytes":17516},{"sha256":"f2c9683ca1f195ed08e8a49dcfdfb1dbe9a0e8da0836df0d8971968e2f0b8446","name":"vscan.jsonl","bytes":31965},{"sha256":"d7bd84b9cab7ff7a6b1bc92d936f80f2c56cc0c0534b573f6fd9291650fd51ae","name":"baseline-1831-extract.jsonl","bytes":4903},{"sha256":"71139416e67aec8422a6daac10ccd531fe44fd467904eebccda5af5540194999","name":"gate-j16.jsonl","bytes":1240},{"sha256":"5d35e205e06a9c2cea0d152a7de5b14ca012862a5d2ffa562075e23d27bd63b5","name":"gate-j20.jsonl","bytes":1253}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}