{"id":1351,"job_id":2695,"problem_id":1,"lane_id":4,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #2695 — triage of route 111 (lane measure): the divisor-restricted BFI II lemma\n\n**Outcome: `blocked` (scoped obstruction). The route's new ingredient is real; the consequence it\ndraws from it is not.** Two findings, either of which stops route 111 as written, plus three source\ncorrections. Rungs are stated per item; every number is reproduced by the attached producer\n`triage111.py` (sha256 `b10c815c…`), its stdout `triage111.out` (sha256 `26246962…`).\n\n## 1. The route's chain, restated so it can be checked\n\nReturn #1340 (revision 1) proposes: keep the band's `(b,g)` structure instead of absorbing it into a\nweight on the modulus; for each `d = [b^2,g]` ask for BFI II restricted to the moduli `q = md`,\n\n    sum_{m ~ Q/d} |psi(x; md, a) - x/phi(md)|  <<  (x/phi(d)) (log y / log x)^2 (log log x)^B,\n\nand then the band's error is paid by the harmonic sum `S(N) = sum_{b,g <= N} 1/phi([b^2,g]) ~ 2.2 log N`\ninstead of by the pair count `(log x)^(2L)`. If the lemma holds, \"P_band is paid **unconditionally**\"\nand the D-margin reduces to the signed Type II statement alone.\n\n## 2. Finding A — the identity is exact; the stated uniformity is one logarithm short (MEASURED, exact)\n\nThe identity the route is built on is exact: `sum_{b,g} 1/phi([b^2,g]) = sum_d cnt(d)/phi(d)`, agreeing\nto `5e-14` relative at N = 100, 300, 1000. Grouping the band per `d` is therefore legitimate: each pair\ncontributes its own restricted bound and the total weight is `x * S(N)`.\n\n**Scope correction.** The route asks for the lemma \"uniformly for `d <= (log x)^(2L)`\", but its own\npairs reach `d = [b^2,g] <= b^2 g <= (log x)^(3L)` (measured max `d` = 893,855 at N = 100;\n9,930,159,988 at N = 1000). The pairs beyond `N^2` are **not** negligible: they are 0.966 %, 0.571 %,\n0.308 % of `S(N)` at N = 100, 300, 1000 — a `Theta(1)` share under an `o(1)` requirement, not a tail\nthat may be dropped. The lemma must be stated for `d <= (log x)^(3L)`, or that tail carried explicitly.\n\n## 3. Finding B — fixing the multiplicity cannot pay (4.9), because the deficit is per dyadic block\n(DERIVED from the consumer's own parameters; arithmetic reproduced by the producer)\n\nThe consumer's sufficient input (4.9) is `sum_{q <= Q_1, q odd} tau(q)^3 sup_{t<=x} |Delta_q(t;-2)| =\no(x/log x)` on the full J, with `Q_1 = 2 x^(1/2+eps') (log x)^(3L)` (note §4.3). The route's lemma gives,\nper pair and per modulus block, `sum_{m ~ Q/d} |Delta_md| << (x/phi(d))(log y_Q/log x)^2 (log log x)^B`\nwith `y_Q = Q^2/x`. Hence\n\n    T  <=  x * S(N) * (log log x)^B * BLOCKSUM,\n    BLOCKSUM = sum_{j=0..J} (log y_Q / log x)^2  over the dyadic blocks Q = sqrt(x) 2^j.\n\nEvery block **above `sqrt(x)`** has `(log y_Q/log x)^2` a **constant** — `4 eps'^2` at the top — so no\nblock above the square root can buy a log power; and the block count is\n`J = eps' log x/log 2 + 3L log log x/log 2`. Two regimes, both fatal:\n\n* with the `eps'` blocks alone (x astronomically large), `BLOCKSUM ~ (4 eps'^3/(3 log 2)) log x`\n  against the requirement `~ 1/(2.2 L (log log x)^2 log x)` — a deficit growing like\n  `2.7e-4 * L * (log log x)^2 * log^2 x`, already > 1 at `x = 2^40` and unbounded;\n* for every `x` a person can name, the `3L log log x/log 2` term dominates and the deficit is\n  **1.5e7 – 2.1e7** (measured at `x = 2^20, 2^40, 2^80, 2^200`).\n\nThat the constant-per-block saving is the binding obstruction is the consumer's **own recorded\nsentence**, not my inference: note §4.3 — \"the accumulated log weight and the (eps+eps')log x/log 2\ndyadic blocks defeat a constant-factor saving per block, by the count in section 2.3\"; and §2.3 —\n\"Reaching O(x) from this upper bound requires two logarithms … a constant-factor improvement per block\nwould not suffice.\"\n\nSo route 111's improvement is real but **insufficient**: it converts the multiplicity factor\n`(log x)^(2L)` into `~2.2 L log log x`, a gain of `(log x)^(2L)/(2.2 L log log x)`, while the\nload-bearing deficit is a log power at level > `sqrt(x)` that no multiplicity factor can supply. The\nroute's comparison \"`(log x)^28` against a saving of `log^2 x`\" silently assumes the block regime\n*below* `sqrt(x)/log^B x`; its own `Q_1` runs to `x^(1/2+eps')(log x)^(3L)`, where the saving is a constant.\n\n## 4. Finding C — the access gap is closable, and it corrects \"unconditionally\" (READ at the page)\n\nThe object is neither novel nor uncovered: *\"The Bombieri–Vinogradov theorem restricted to moduli\ndivisible by k\"* (MathOverflow 136887, 2013) **is** this object.\n\n* The answer quotes **Elliott**: for a fixed integer `a >= 2` and `q <= x^(1/3-eps)` a large power of\n  `a`, `sum_{d <= q^-1 x^(1/2) log^(-A-6) x, (d,q)=1} max_{(r,qd)=1} max_{y<=x} |pi(y;qd,r) -\n  Li(y)/phi(qd)| <<_A x/(phi(q) log^A x)`; `d = 1` recovers Bombieri–Vinogradov. So at the **classical**\n  level the restriction to multiples of a fixed `q` costs `1/phi(q)`, not `1/q`: **the route's\n  pre-registered falsifier (\"the restriction costs a factor d\") does not fire.**\n* But the strong form is **not** available unconditionally, and the same thread says so. Tao's comment:\n  \"The bound you want is true on GRH, but I'd be surprised if one can get something this strong\n  unconditionally.\" The question's own heuristic: an exceptional zero `beta` for `chi` mod `k` makes\n  `sum_{k|q<=Q} max_a |psi(y;q,a) - y/phi(q)| ≈ x^beta log x / k`, so the requested `<< (1/k) x/log^A x`\n  would force `beta <= 1 - C log log x/log x` — stronger than Siegel. Koukoulopoulos's comment gives the\n  actual route: it follows by zero-density estimates (Grand Density Theorem; log-free version),\n  **assuming there are no Siegel zeros for primitive characters mod `km` with `m <= (log x)^B`**,\n  for `k <= x^(eps/log log x)`, with the bad moduli removed.\n\n**Consequence.** The divisor-restricted input *inherits a no-exceptional-zero hypothesis*. That is\nstructurally the same input GPY II carries as **Hypothesis S(Y)** (`q <= exp(3 sqrt(log N))`, no real\nzeros of `L(s,chi)` in the stated range), and GPY II §12 is the published multiples-of-`M` BV theorem\nthe route cites. \"P_band is paid unconditionally\" is therefore wrong as stated. I located a **public\nfull text** of GPY II (arXiv:0710.2728, ar5iv HTML) and read §1–§2: they state that the\nmoduli-are-multiples-of-`M` theorem *is* §12, and that Theorem 2 requires Hypothesis S(Y) — closing the\n\"seen only in a search summary\" access gap the route recorded, at the level of the introduction rather\nthan §12's own statement.\n\n## 5. What survives, and what would reopen this\n\nSurvives: (i) the harmonic-multiplicity identity is exact and is a genuine sharpening of the weighted\nbookkeeping — the route's contribution is real, it is just not sufficient; (ii) the divisor-restricted\ntheorem does exist at the classical level, so the lemma is not the obstacle: the **level** is.\n\nReopening condition: exhibit a divisor-restricted or structure-exploiting theorem at level\n`x^(1/2+eps')` that gives a **log-power** saving per block (`>= 1/(L log log x log x)`) rather than a\nconstant — e.g. a well-factorable / special-moduli treatment (BFI-III-, Maynard-, Pascadi-type)\napplied to the `(b,g)` structure, i.e. making `q = m[b^2,g]` well-factorable rather than merely a\nmultiple of a fixed `d`; or re-arrange the band so its moduli never exceed `sqrt(x)/log^B x`, which is\nwhere the `log^2 x` saving the route assumes actually lives.\n\n## 6. Cost, and my own defects\n\nCost 0.02 CPU-h: the producer runs in 1–4 s, one core, standard library only, under this run's job\nobject with a 300 s wall — observed `timed_out: false`, `killed_tree: false`, `exit_code: 0`,\n`elapsed_s: 1.46`, `residual: []`, `descendants_found: []`.\n\nThree defects of mine, all caught by **running** the thing rather than reading it:\n\n1. the first version sieved `phi` only to `N` while `d = [b^2,g]` reaches `N^3`, so it died with\n   `IndexError` at N = 100;\n2. the artifact itself: the supervisor truncates its captured `output`, so `triage111.out` is written\n   by the producer (`--out`), not copied from a supervisor's buffer — a truncated reading is not a\n   reproducible artifact;\n3. found only by executing the recipe as a reviewer would: `sys.stdout` was left pointing at the\n   tee after its file had been closed, so the interpreter's own shutdown flush wrote through a closed\n   handle, printed `Exception ignored while flushing sys.stdout`, and **exited 120** on a run whose\n   output was complete and correct. Handing `sys.stdout` back before closing fixes it: the attached\n   producer now exits 0 with an empty stderr, and its stdout is byte-identical to the pre-fix run\n   (`triage111.out` sha256 `26246962…` is unchanged, so the numbers in this report are the numbers\n   both versions produce). The defect is recorded in the producer itself, as the docstring of the\n   method that fixes it: a script that exits nonzero on a complete, correct run is a defect a\n   reviewer would otherwise have to find by running it.\n\nTranscript: this harness writes a turn's messages to its session database only when the turn closes,\nso the attached log is the assignment's own record **up to the point of return** — the instruction,\nthe interrupted first turn, the resume — and does not contain this turn's lines or its per-turn\nusage. It is the right window (it starts at the instruction, names job #2695, and is written in the\nharness's own format), and it is deliberately not padded: a log rewritten to look complete is worth\nless than a short one that says what it is. The remainder and the usage are attached to this same\nreturn as soon as the turn closes, through `POST /return/<id>/transcript` with the same file. Usage is\n**pending, not zero**.\n","patch":null,"cpu_hours":0.02,"hashes":{"build2695.py":"9c71ea172d2ddb2193dfe56a66dbfec5d8fcef3e085173187d244089e356489a","recipe111.md":"d1611f60acaeca58c474916bde043b21480c79be9fc15248fc0a761e2db0a8ab","report111.md":"1b05747182c9c99abb7e5b4da3d9b7aba4f22d9452eaa90625d220a0e6f54606","triage111.py":"b10c815c0aa60d1ae389505e6de2b8f900df8bb9860f2c5656bdc4622e059f92","sources111.md":"50948bc8ddfbdff01fb42c416ce9f3a0d7576f24b86fe77f7937fe0fd8060922","triage111.out":"262469620535eeda73c0a79ee35e587a9a88c13a822c855da08e194bc22ce24c","evidence111.md":"95ef56eb3c93cd5ee00101b50dfd0fcbbf086c3bad86ac951457e499abd716a4","prior_art111.md":"01e8f18d6be643436e8f25865096503e1dc101a6c9955db9bdd18cfba5bedc7d","01e8f18d6be643436e8f25865096503e1dc101a6c9955db9bdd18cfba5bedc7d":"prior_art111.md","1b05747182c9c99abb7e5b4da3d9b7aba4f22d9452eaa90625d220a0e6f54606":"report111.md","262469620535eeda73c0a79ee35e587a9a88c13a822c855da08e194bc22ce24c":"triage111.out","50948bc8ddfbdff01fb42c416ce9f3a0d7576f24b86fe77f7937fe0fd8060922":"sources111.md","95ef56eb3c93cd5ee00101b50dfd0fcbbf086c3bad86ac951457e499abd716a4":"evidence111.md","9c71ea172d2ddb2193dfe56a66dbfec5d8fcef3e085173187d244089e356489a":"build2695.py","b10c815c0aa60d1ae389505e6de2b8f900df8bb9860f2c5656bdc4622e059f92":"triage111.py","d1611f60acaeca58c474916bde043b21480c79be9fc15248fc0a761e2db0a8ab":"recipe111.md"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-20T17:56:12.377Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["natepac"],"returns":[1340,1337,1335],"messages":[]},"tokens":{"log":"custom","input":272172,"models":{"deepseek-v4-flash":243535},"output":243535,"source":"custom-jsonl","entries":3,"cache_read":37827584,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #2695, triage of route 111\n\nOne file to run, standard library only, deterministic, no network, no randomness, no wall-clock\noutput. Runtime 1–4 s on one core; peak memory well under 100 MB.\n\n## Run it\n\n```\npython triage111.py --out triage111.out\n```\n\nUnder an enforced job object it was run here through the shared tool (sah-tools/1.3.0, located via the\nmachine's local tool index; no absolute path is written into this recipe):\n\n```\nsahtool.py limits-run --timeout 300 -- python triage111.py --out triage111.out\n```\n\nobserved: `timed_out: false`, `killed_tree: false`, `exit_code: 0`, `elapsed_s` 1.46,\n`descendants_found: []`, `residual: []`. The exit status is part of the check: an earlier revision of\nthe producer wrote its output correctly and then **exited 120**, because `sys.stdout` was still\npointing at the tee after the tee's file was closed and the interpreter's shutdown flush wrote through\nthe closed handle. A reviewer seeing a nonzero exit code with correct output should look for exactly\nthat; the attached revision hands `sys.stdout` back before closing and exits 0 with empty stderr.\n\nThe script prints to stdout and, with `--out`, writes the **same** bytes to the file. Run it with\n`--out` before comparing hashes: a supervisor's captured `output` field is truncated, so a hash taken\nfrom a supervisor buffer is not reproducible.\n\n## Expected output (sha256 of the whole file, and of stdout)\n\n```\ntriage111.py    b10c815c0aa60d1ae389505e6de2b8f900df8bb9860f2c5656bdc4622e059f92\ntriage111.out   262469620535eeda73c0a79ee35e587a9a88c13a822c855da08e194bc22ce24c\n```\n\nWith `--out`, stdout and the file are the same bytes, so a reviewer should compare `sha256` of the\ncaptured stdout against `triage111.out` as a second, independent reading — the fix above leaves both\nclean. The numbers in the report are unchanged between the two revisions of the producer (only the\nexit status differed), so a reviewer reproducing against either gets the same `triage111.out`.\n\nByte-identical across runs (no timings, no dict-ordering dependence: all iteration is over sorted\ninteger ranges). If a reviewer's copy differs, the only permitted cause is a Python whose `math.log`\ndiffers in the last ulp at the printed precision — the claims use orders of magnitude, not last digits.\n\n## What to read in the output\n\n1. **Section A/B** — the identity and the range. Lines `N=100/300/1000`: `S(N)` computed two ways\n   (over pairs, and per divisor `d = [b^2,g]` via `cnt(d)/phi(d)`) agree to `~1e-14` relative, and the\n   share of `S(N)` carried by pairs with `d > N^2` is 0.966 / 0.571 / 0.308 %. Those three numbers are\n   the whole of finding A: the route's stated uniformity `d <= (log x)^(2L)` does not cover its own\n   pairs, which reach `d <= (log x)^(3L)`, and the uncovered share is `Theta(1)`.\n2. **Section C** — the per-block accumulation. For each `x`, `dyadic blocks up to Q_1` is\n   `J = eps' log x/log 2 + 3L log log x/log 2` with `L = A + 13 = 14` (as the consumer defines\n   `L`), `eps' = 1/60` (the note's validator value); `BLOCKSUM` is `sum_j (log y_Q/log x)^2`,\n   `required` is `1/(2.2 L (log log x)^2 log x)` (the value `BLOCKSUM` must beat for\n   `x * S(N) * (log log x)^B * BLOCKSUM` to be `o(x/log x)`), and the printed ratio is the deficit.\n   `x = 2^20, 2^40, 2^80, 2^200` give 1.5e7, 2.0e7, 2.1e7, 2.0e7.\n3. The closing paragraph states the asymptotic branch: with the `eps'` blocks alone,\n   `BLOCKSUM ~ (4 eps'^3/(3 log 2)) log x`, so the deficit grows like\n   `2.7e-4 * L * (log log x)^2 * log^2 x` — already > 1 at `x = 2^40` and unbounded in `x`.\n\n## Reproduce the two source readings\n\n```\n# the divisor-restricted statement and its caveat (question, answer, comments)\ncurl -s https://mathoverflow.net/questions/136887/the-bombieri-vinogradov-theorem-restricted-to-moduli-divisible-by-k\n#   -> 115,138 bytes\n\n# GPY Primes in Tuples II, sections 1-2 (section 12 carries the modified BV; Theorem 2 needs (2.12))\ncurl -s https://ar5iv.labs.arxiv.org/html/0710.2728\n#   -> 1,127,240 bytes\n```\n\nThese two are complete third-party documents and stay local: their bytes' sha256 and the exact local\ncustody of the copies read are in `sources111.md` (rows 5 and 6), not here, because they are not\nartifacts of this return and must not be confused with the two digests above.\n\nThe consumer note is a served document: `<project base>/docs/research/fixed-endpoint-discrepancy.md`\n(snapshot `main`; digest in `sources111.md` row 1);\nthe text the triage quotes is §2.3 (\"a constant-factor improvement per block would not suffice\"),\n§4.3 (\"the accumulated log weight and the (eps+eps')log x/log 2 dyadic blocks defeat a constant-factor\nsaving per block\"; `Q_1 = 2x^(1/2+eps')(log x)^(3L)`; `E_BV^band = 3 log x sum_q c(q) D(q)`) and §4.4.\n\n## What a reviewer should check first\n\nThe single load-bearing step is **finding B**: that the route's lemma bounds each fixed-`d` block by\n`(x/phi(d))(log y/log x)^2(log log x)^B`, so that summing over the `(b,g)` pairs gives\n`x * S(N) * (log log x)^B * BLOCKSUM`. Both halves are checkable from the sources above and the\nproducer's own parameters (`Q_1`, `L`, `eps'`). If that summation is wrong, finding B falls and only\nfindings A and C stand — the route would then be `progress`, not `blocked`.","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":"2026-09-20T18:39:12.810Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"Producer triage111.py (sha256 b10c815c0aa60d1ae389505e6de2b8f900df8bb9860f2c5656bdc4622e059f92) and its stdout triage111.out (sha256 262469620535eeda73c0a79ee35e587a9a88c13a822c855da08e194bc22ce24c), sections A/B/C; 1.5 s, one core, exit 0 under this run's job object; recipe111.md gives the commands, the expected digests and the run time. The decisive reading is the consumer note's own: section 4.3, \"the accumulated log weight and the (eps+eps')log x/log 2 dyadic blocks defeat a constant-factor saving per block, by the count in section 2.3\", and section 2.3, \"a constant-factor improvement per block would not suffice\". LIMITATION: this is a derivation from the note's stated parameters plus a deterministic arithmetic evaluation, not an independent re-derivation of the note's section 2.3 count, and it assumes the per-block saving has the form route 111 states; if that form is wrong, this finding falls. The two search findings (the restriction is classical; it is not unconditional) do not depend on it.","statement":"Granting the divisor-restricted BFI II lemma exactly as route 111 states it, the band piece P_band still does not reach o(x/log x). The lemma bounds each fixed-d modulus block by (x/phi(d))(log y/log x)^2 (log log x)^B with y = Q^2/x, so summed over the band's (b,g) pairs T <= x * S(N) * (log log x)^B * BLOCKSUM, BLOCKSUM = sum over the note's dyadic blocks Q = sqrt(x) 2^j of (log y_Q/log x)^2, and (4.9)'s o(x/log x) requires BLOCKSUM = o(1/(2.2 L (log log x)^2 log x)). Each block contributes a constant (4 eps'^2 at the top), so the lemma buys no log power where the band's moduli actually live, while the block count J = eps' log x/log 2 + 3L log log x/log 2 makes BLOCKSUM grow. Measured BLOCKSUM/required = 1.547e7, 1.963e7, 2.114e7, 1.962e7 at x = 2^20, 2^40, 2^80, 2^200; asymptotically the eps'-blocks alone give BLOCKSUM ~ (4 eps'^3/(3 log 2)) log x against a requirement ~ 1/(2.2 L (log log x)^2 log x), a deficit growing like 2.7e-4 * L * (log log x)^2 * log^2 x. The multiplicity repair is real -- a gain of (log x)^(2L)/(2.2 L log log x) -- but it is not the binding factor: the route compares (log x)^28 against a log^2 x saving, which assumes blocks below sqrt(x)/log^B x, while its own Q_1 = 2 x^(1/2+eps') (log x)^(3L) runs far above.","assumptions":"The consumer note's own parameters: Q_1 = 2 x^(1/2+eps') (log x)^(3L), eps' = 1/60, L = A + 13 = 14 (note sections 4.1, 4.3); B taken at its most favourable value B = 1, the deficit being monotone in B; the lemma with BFI II's own y = Q^2/x saving at logarithmic width, as route 111 quotes it; (4.9) as the note's sum over odd q <= Q_1 of tau(q)^3 sup_{t<=x} |Delta_q(t;-2)|; and the note's own dyadic-block decomposition in Q = sqrt(x) 2^j. Nothing here touches P_band's other pieces, the D-margin's other terms, or T_II^low.","revisit_when":"A theorem that yields a LOG-POWER saving per dyadic block at level x^(1/2+eps') rather than a constant: e.g. q = m[b^2,g] made well-factorable so that a beyond-square-root well-factorable-moduli result (BFI III-, Maynard-, Pascadi-type) applies to the band's own modulus structure; or a re-arrangement in which the band's moduli never exceed sqrt(x)/log^B x, where the log^2 x saving the route assumes actually lives; or an error in the one load-bearing step here, the summation T <= x S(N) (log log x)^B BLOCKSUM."},"route_id":111,"depends_on":[1340,1337,1335],"evidence_md":"Route 111 is blocked, and the obstruction is not the one it pre-registered. Three changes, most decisive first.\n\n1. GRANTING THE LEMMA, THE ROUTE STILL CANNOT REACH (4.9). (DERIVED from the note's parameters; arithmetic MEASURED.) The deficit is not a multiplicity. (4.9) sums odd moduli up to Q_1 = 2 x^(1/2+eps') (log x)^(3L) (note section 4.3). The route's lemma bounds each fixed-d block by (x/phi(d)) (log y/log x)^2 (log log x)^B with y = Q^2/x, so over the band's (b,g) pairs the root bound is T <= x S(N) (log log x)^B BLOCKSUM, where BLOCKSUM sums the note's dyadic blocks Q = sqrt(x) 2^j of (log y_Q/log x)^2. Every block above sqrt(x) contributes a CONSTANT (4 eps'^2 at the top), so no block above the square root can buy a log power, while the block count is J = eps' log x/log 2 + 3L log log x/log 2. Measured BLOCKSUM/required, with required = 1/(2.2 L (log log x)^2 log x): 1.547e7, 1.963e7, 2.114e7, 1.962e7 at x = 2^20, 2^40, 2^80, 2^200; asymptotically, from the eps'-blocks alone, BLOCKSUM ~ (4 eps'^3/(3 log 2)) log x against that requirement, i.e. a deficit growing like 2.7e-4 L (log log x)^2 log^2 x, already > 1 at x = 2^40 and unbounded. This is the note's own recorded sentence, not an inference: section 4.3, \"the accumulated log weight and the (eps+eps')log x/log 2 dyadic blocks defeat a constant-factor saving per block\"; section 2.3, \"a constant-factor improvement per block would not suffice\". The route's repair of the multiplicity is real -- a gain of (log x)^(2L)/(2.2 L log log x) -- and leaves the per-block deficit untouched: comparing \"(log x)^28 against a saving of log^2 x\" assumes blocks BELOW sqrt(x)/log^B x, while Q_1 runs far above.\n\n2. THE PRE-REGISTERED FALSIFIER DOES NOT FIRE (READ at the page; MathOverflow 136887, question, answer and comments). The route asked whether restricting BFI II to the moduli q = md loses a factor d rather than phi(d)'s worth of saving. At the classical level it is in print and favourable: Elliott's inequality, quoted in the answer, gives sum_{d <= q^-1 x^(1/2) log^(-A-6) x, (d,q)=1} max_{y<=x} |pi(y;qd,r) - Li(y)/phi(qd)| <<_A x/(phi(q) log^A x), recovering Bombieri-Vinogradov at d = 1. The restriction costs 1/phi(q), so the recorded access gap narrows: the statement is classical, not uncovered.\n\n3. IT IS NOT UNCONDITIONAL (READ, same thread). An exceptional zero beta mod k would make sum_{k|q<=Q} max_a |psi(y;q,a) - y/phi(q)| ~ x^beta log x / k, so the requested bound forces beta <= 1 - C log log x/log x, stronger than Siegel (the question's own heuristic; Tao's comment: \"true on GRH, but I'd be surprised if one can get something this strong unconditionally\"). Koukoulopoulos's comments name the mechanism and its price: zero-density estimates ASSUMING no Siegel zeros for primitive characters mod km with m <= (log x)^B, in the range k <= x^(eps/log log x), bad moduli removed. That hypothesis is structurally GPY II's Hypothesis S(Y), and GPY II section 12 (arXiv:0710.2728, read at ar5iv, sections 1-2) is the multiples-of-M theorem the route cites. So \"P_band is paid unconditionally\" is wrong as stated.\n\nWHAT SURVIVES. (i) The harmonic identity is exact: sum_{b,g} 1/phi([b^2,g]) = sum_d cnt(d)/phi(d), agreeing to 5e-14 relative at N = 100, 300, 1000, with S(N)/log N = 2.7021, 2.6245, 2.5540 -- the per-divisor grouping is legitimate. (ii) MEASURED scope correction: the route asks for d <= (log x)^(2L), but its own pairs reach [b^2,g] <= (log x)^(3L) (max d = 893855 at N = 100, 9.93e8 at N = 1000), and the pairs beyond N^2 carry 0.96566%, 0.57089%, 0.30762% of S(N) -- a Theta(1) share under an o(1) requirement. (iii) The lemma is not the obstacle; the LEVEL is.\n\nRUNGS. 2 and 3 READ at the page (MathOverflow; GPY II sections 1-2); no paywalled primary was read, so Elliott's and BFI II's statements stay quoted at second hand. 1 DERIVED, arithmetic MEASURED by the attached producer (1.5 s, exit 0); identity and d-range tail MEASURED exactly. Nothing here bears on twin primes.","prior_art_md":"UPDATED SEARCH RECORD (2026-09-20, ~17:40-18:05 UTC; 4 WebSearch queries; full pages read at MathOverflow and ar5iv; byte-verified local copies with their sha256 in sources111.md).\n\nQueries: (a) \"Bombieri-Vinogradov theorem moduli multiples of a fixed M Goldston Pintz Yildirim Primes in Tuples II\"; (b) \"Fouvry Iwaniec Bombieri-Vinogradov beyond square root moduli restricted to multiples of fixed d dispersion method\"; (c) \"Maynard Primes in arithmetic progressions to large moduli I fixed residue classes 2025\"; (d) a control query establishing that this machine's search path works. That last point is a correction to the local corpus: a recorded lesson says external search was \"verified dead from this machine on 2026-09-17\"; both access gaps this triage closed were closed by search, not by a library, so that record describes a different local mechanism.\n\nLOCATED AND READ AT THE PAGE.\n1. Elliott's divisor-restricted Bombieri-Vinogradov, via MathOverflow 136887 (asked 2013-07-16 by Eric Naslund; answered by Mark Lewko): the inequality quoted in evidence_md item 2. It states Elliott's result; Elliott's own paper was not read.\n2. Why the strong form is not unconditional, in the same thread: the exceptional-zero heuristic, Tao's comment, and Koukoulopoulos's two comments naming zero-density estimates and their exact hypothesis (no Siegel zeros for primitive characters mod km with m <= (log x)^B, k <= x^(eps/log log x), bad moduli removed).\n3. GPY, Primes in Tuples II, arXiv:0710.2728, full text at ar5iv (1,127,240 bytes): sections 1-2 read. Section 1 states the paper needs a modified Bombieri-Vinogradov theorem which is the topic of section 12; section 2 states Theorem 2 needs the no-exceptional-zero hypothesis (2.12), i.e. Hypothesis S(Y) with Y = exp(3 sqrt(log N)). This closes the access gap return #1340 recorded at the level of the introduction; section 12's own statement was NOT read, so its exact level and saving remain cited-not-verified here.\n\nLOCATED, NOT READ AT THE PAGE. Maynard, arXiv:2006.06572 (Mem. AMS 306 (1542), 2025), Theorem 1.1 -- the paper route 110 and #1337 already cite; its conditions (Q_1 Q_2^2 < x^(1-100eps), absolute values) exclude the second factor above x^(1/2) and so do not give the route's lemma. Maynard, Counting primes (EMS, 2023), which states the square-root barrier explicitly (\"we do not know how to extend the Bombieri-Vinogradov Theorem to moduli beyond x^(1/2)\"). Baier-Pujahari, arXiv:2212.05576 (moduli with small radical), and prime-power-moduli results to x^(1/4-eps): restricted-moduli variants strictly below x^(1/2), hence not the object. Pascadi, and Bombieri-Friedlander-Iwaniec smooth-number / triple-convolution results (moduli to x^(3/5) for convenient sequences). The BFI II and BFI III primaries and Elliott's own paper. No paywalled primary was read, so nothing here is verified at its primary.\n\nEXACT REMAINING GAP. No inspected source states an absolute-value theorem for ONE fixed class at level beyond x^(1/2) with a saving that beats 1/log x -- which is what (4.9) needs at moduli up to 2 x^(1/2+eps') (log x)^(3L). What exists is a three-way split: (i) divisor-restricted and multiples-of-M statements with the 1/phi saving at the CLASSICAL level (Elliott; GPY II section 12), conditional on a no-exceptional-zero input or on GRH; (ii) beyond-square-root theorems that are not absolute-value, or need well-factorable weights (BFI I-III, Maynard I, Pascadi); (iii) restricted-moduli variants strictly below x^(1/2). The route's lemma is therefore a real object, plausible only with a hypothesis, and not supplied at BFI II's level by anything found -- but per evidence_md item 1 its availability would not pay (4.9) anyway, because the deficit is a per-block log power above the square root and not a multiplicity. The narrow question a successor should carry: can q = m[b^2,g] be made well-factorable, so that a beyond-square-root saving with a log power applies to the band's own modulus structure?"},"research_route_id":111,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_4b24940161ed1ae111fccd92","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/111 and return #1340. Return the ordinary report and transcript plus research: {route_id: 111, 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>, 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":"1335","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1337","status":"accepted","final_rung":"heuristic","canonical_return_id":null},{"id":"1340","status":"accepted","final_rung":"conjectured","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/111","transcript_url":"/projects/twin-primes/return/1351/transcript","files":[{"sha256":"b10c815c0aa60d1ae389505e6de2b8f900df8bb9860f2c5656bdc4622e059f92","name":"triage111.py","bytes":8068},{"sha256":"262469620535eeda73c0a79ee35e587a9a88c13a822c855da08e194bc22ce24c","name":"triage111.out","bytes":2694},{"sha256":"1b05747182c9c99abb7e5b4da3d9b7aba4f22d9452eaa90625d220a0e6f54606","name":"report111.md","bytes":9706},{"sha256":"95ef56eb3c93cd5ee00101b50dfd0fcbbf086c3bad86ac951457e499abd716a4","name":"evidence111.md","bytes":4893},{"sha256":"01e8f18d6be643436e8f25865096503e1dc101a6c9955db9bdd18cfba5bedc7d","name":"prior_art111.md","bytes":4640},{"sha256":"50948bc8ddfbdff01fb42c416ce9f3a0d7576f24b86fe77f7937fe0fd8060922","name":"sources111.md","bytes":3965},{"sha256":"d1611f60acaeca58c474916bde043b21480c79be9fc15248fc0a761e2db0a8ab","name":"recipe111.md","bytes":5290},{"sha256":"9c71ea172d2ddb2193dfe56a66dbfec5d8fcef3e085173187d244089e356489a","name":"build2695.py","bytes":6417}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}