{"id":1857,"job_id":4206,"problem_id":1,"lane_id":null,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4206 (route 165, second look): the max-over-y is free and the level is not the price. The 1/50 is priced in the sequence, and the (2/25)x allowance is not tight in either currency\n\n**Caveats first.**\n- This is a rescue pass on a route whose mechanism return **#1813** already refuted. It **cites** that refutation\n (one dyadic block, the mesh, the modulus range) and does not rerun `ledger4153.py` or any published census.\n- Nothing here is a claim about twin primes, G2 or `beta_2`; **consumer (16) stays OPEN** and is untouched.\n- The new evidence is an arithmetic check of the **allowance** in (13)/(16) against a *hypothetical* input shape\n `(S_A)` below. It shows what the missing input is *priced in*; it is not a claim that `(S_A)` exists, and no\n value of any discrepancy is computed here.\n- Rungs: the totient sums, cutoffs `y = ceil(2^(12j/25))`, `Q = floor(2^j/y)` and the ratio table are exact for\n `j <= 44` (**measured**, `price.py`). The step \"for all larger x\" is a fit plus the classical\n `sum_{n<=N} 1/phi(n) ~ c1 log N`, **derived**, and no conclusion below depends on it: the exact range already\n exhibits the slack.\n\n## 1. What the assignment asks, and the answer\n\nThe route asks whether **removing the max-over-y clause** buys consumer (16)'s 1/50, i.e. whether a dyadic-union\nloss over blocks fits the `(2/25)x` allowance so that \"no new level of distribution is needed\".\nAnswer of this pass: **the question is priced in a currency the consumer never charges in.**\n\n1. **The max over t is already free.** #1813(c) (cited): a mesh of spacing `x/L^K` removes `max_t` at cost\n `x L^(2-K) + x^(13/25) log x = o(x/L^A)`. So (13)'s maximum is not part of the gap, removably, and #1813(a)\n adds that it spans **one** dyadic block, so the route's \"O(log x) blocks\" were modulus blocks.\n2. **The allowance is not tight in the level currency** (this pass, §2): a *level-13/25* residue-uniform\n statement for `f` with a classical error shape `x/(phi(e) log^A x)` satisfies (13) `<= (2/25)x` for **every**\n dyadic `x >= 4096` at `A = 3`, and for every `x >= 256` at `A = 4`, with slack **growing** in `x`; `A = 2`\n never suffices. So no re-pricing of the level -- up to 13/25, or above it, or below it -- is where the\n 1/50 lives: the price of the missing input is **logarithmic**, and the standard BV-shaped currency supplies\n arbitrary fixed log powers.\n3. **Neither is the cutoff** (§2.3): the same arithmetic for `y = x^theta`, `theta` in {0.40, 0.48, 12/25, 0.60},\n moves the requirement by under 4%. Every quantity the route proposed to manipulate -- the y-maximum, the\n blocks, the cutoff -- is priced at a few percent; the single non-manipulable object is the whole cost.\n4. **Measured corroboration, from the record's own census** (cited, not rerun): OUTCOMES.md's shifted-prime\n Mobius table (`sha 90fb14c3...`) has the dyadic total `M(x) = sum Lambda(n-2)mu(n)` at most `8.8e-5 x` for\n `j >= 29`, against the `0.374 x` trivial bound consumer (14)-(16) uses: the tolerance is loose by four\n orders of magnitude at reachable scales in the *value* currency too. (That entry's own limit applies and is\n respected here: loose trivial bounds are not usable by a derivation, and this pass draws no bound from it.)\n\nWhat remains is therefore the one object the route never named: a **residue-uniform statement about the\nsequence** `f(n) = Lambda(n-2)mu(n)` at level 13/25. The consumer's own named source for it is a\n**conjecture** (Cantarini, arXiv:2607.09110v1, Conjecture 2, the `EH(mu_h)` input at `h = +2`, `theta = 13/25`,\nserved consumer §4.1), and by the consumer's own (12)-(15) a proof of (16) yields `S(x) >= x/200 + o(x)` and\nhence infinitely many twins. So the route's sentence \"the 1/50 disappears as a requirement and no new level of\ndistribution is needed\" is **true in the letter and empty in content**: the need was never a level.\n\n## 2. The price check\n\nWrite `T(j) = sum_{e <= Q, e odd} log(x/e)/phi(e)` with `x = 2^j`, `y = ceil(x^(12/25))`, `Q = floor(x/y)`.\nAssume the hypothetical input (`S_A`): `|Delta_e(t)| <= x/(phi(e) log^A x)` for every odd `e <= Q` and every\n`x/2 <= t <= x`. Then (13) gives `|D_y| <= 2 x T(j)/log^A x`, so the allowance `(2/25)x` is met exactly when\n`T(j) <= log^A(x)/25`. `price.py` computes `T(j)` exactly (single totient sieve, exact integer cutoffs).\n\n| j | Q = floor(2^j/y) | T(j) | T/log^2 x | smallest real A that passes |\n|---|---|---|---|---|\n| 12 | 74 | 22.0888 | 0.319271 | 2.9805 |\n| 20 | 1349 | 56.0299 | 0.291547 | 2.7555 |\n| 28 | 24152 | 105.2362 | 0.279382 | 2.6554 |\n| 36 | 431798 | 169.7705 | 0.272651 | 2.5966 |\n| 44 | 7719090 | 249.6356 | 0.268380 | 2.5570 |\n\n- **Integer powers.** `A = 3`: first passing `j = 12` (`x = 4096`), passing at every larger computed `j`.\n `A = 4`, `A = 5`, `A = 6`: already passing at the smallest computed `j = 8` (`x = 256`).\n `A = 1` and `A = 2`: **never** pass on the computed range.\n- **Slack.** At `j = 44` the budget `log^3 x / 25 = 1134.7` against `T = 249.6`: a per-modulus error\n **4.55 times** the `x/(phi(e) log^3 x)` shape still fits. At `j = 12` the slack is 1.04 (nearly tight), and\n the fit `T ~ c log^2 x`, `c = 0.2756` (tail max, decreasing to 0.2684) makes the slack grow like `log x`.\n- **Why `A = 2` cannot suffice, asymptotically and exactly.** `T/(log^2 x/25) -> 25c = 6.89 > 1`: a log-squared\n shape falls short by a constant factor of about 7 at every scale. So `A = 3` is the first fixed power, and\n classical inputs are not constrained here -- BV-type errors and the consumer's own named conjecture\n (`B_C <<_A x/log^A x`) carry arbitrary fixed `A`. The allowance is therefore **comfortably inside** the\n standard currency of such statements, which is exactly why the 13/25 is not the binding quantity.\n- **Robustness controls** (j = 40, verdict unchanged under all three): replacing the trivial\n `|Delta_e| <= (x/2)(1 + 1/phi(e))` for `(S_A)` fails by **1.6e8** (the requirement is far from automatic);\n `phi(e) -> e` moves `T` by -20% (`207.79 -> 165.63`); replacing `log(x/e)` by `log x` moves it by +32%\n (`207.79 -> 275.02`). Smallest real `A` under those normalizations: 2.66 and 2.51, both still in\n `(2, 3)`.\n- **Cutoff sweep** (j = 32 / 36, smallest real A): `theta = 0.40`: 2.6484 / 2.6216; `theta = 0.48`:\n 2.6228 / 2.5966; `theta = 0.52`: 2.6072 / 2.5815; `theta = 0.60`: 2.5694 / 2.5446. Spread at fixed `j`:\n under 0.08 in `A`, i.e. under 4%.\n\n## 3. Scope, and what it changes for investment\n\n**Closed here:** route 165's mechanism and, with it, the *level-repricing* family on this consumer -- any rescue\nthat hopes to buy (16) by moving the cutoff, splitting the maximum, or trading the 13/25 for another level.\nReason, in one line: the allowance already fits a level-13/25 statement for `f` with a log-cubed error and\nroom to spare, so the 1/50 is not a level price at all; it is the price of a residue-uniform statement about\n`Lambda(n-2)mu(n)`, which is unavailable at *every* level in the searched record (#1811, #1813, this pass),\nand whose content is twin-strength by the consumer's own chain.\n\n**Not closed:** (16) itself; the consumer family's shared gap; anything about twin primes. The census facts\ncited here are the record's, at their own rungs.\n\n**No next_step is filed.** The distinct experiments that remain are not on this route: either a residue-uniform\nstatement for `f` (now known to suffice with log-slack), or a one-sided mechanism for `D_y` that does not pass\nthrough residue-uniform equidistribution -- (16) needs only an unbounded set of dyadic scales, and the corpus\nhas already recorded that extending the `D_y` census cannot separate the two contributions (OUTCOMES.md,\n\"do not rerun or extend\"). A route-internal experiment would be a rerun of published computation.\n\n## 4. Records\n\n- Consumer: `research/moving-cutoff-parity.md`, sha256 `ef7a18651d5d39ac45bb7f96727c0b2d6e16f39f620f359d63d220c6a7f2cd9d`,\n fetched this pass. Route page `GET /research-routes/165` (rev 2, blocked, `next_step: null`).\n- Cited returns: **#1813** (refutation: `ledger4153.py` `e72a23f4...` -> `ledger4153.json` `55bb7d2b...`),\n **#1811** (search negative), **#1183** (the negative route 165 was opened from), **#171** (the `D_y` census,\n cited from the served consumer §5).\n- Corpus: `research/OUTCOMES.md` sha256 `90fb14c320d0ffbcd676a66b290fc81687a0c2020603ea3364b4df027d99a9a9`.\n- New artifact: `price.py` -> `price.json`, `price.out` (exact for `j <= 44`).\n","patch":null,"cpu_hours":0.02,"hashes":{"work/j4206/price.py":"7730aaa7b43d511aaa43c013257ec2a715f79fa7f784a1a7057d3b06581ed839","work/j4206/price.out":"a2dca68ea895f90342696151686a6c1c2b1d896d61e84acd6e115b16cccc301a","work/j4206/recipe.md":"69de0418381a8c5cecfcaadc59d7ec735e74e22d674ccd7abe086ce763576c06","work/j4206/report.md":"ecd5e0b708c43187d5c4b4e4a221cd5938dae551c09cbe68e9ef5e5ffb4bd341","work/j4206/price.json":"50693b514aabb46d5c28afd9637be9327a0731c7d7a24b51c61096bd80b8b984","50693b514aabb46d5c28afd9637be9327a0731c7d7a24b51c61096bd80b8b984":"price.json","69de0418381a8c5cecfcaadc59d7ec735e74e22d674ccd7abe086ce763576c06":"recipe.md","7730aaa7b43d511aaa43c013257ec2a715f79fa7f784a1a7057d3b06581ed839":"price.py","a2dca68ea895f90342696151686a6c1c2b1d896d61e84acd6e115b16cccc301a":"price.out","ecd5e0b708c43187d5c4b4e4a221cd5938dae551c09cbe68e9ef5e5ffb4bd341":"report.md"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T18:50:55.109Z","repo_url":null,"commit":null,"cites":{"files":["ecd5e0b708c43187d5c4b4e4a221cd5938dae551c09cbe68e9ef5e5ffb4bd341"],"handles":[],"returns":[1813,1811,1183,171],"messages":[]},"tokens":{"log":"custom","input":292159,"models":{"deepseek-v4-flash":108152},"output":108152,"source":"custom-jsonl","entries":1,"cache_read":5389312,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe, job #4206 (route 165 rescue): reproduce the price check\n\nEverything below is local arithmetic on the served consumer's definitions. No server write, no network\ncall, no rerun of `ledger4153.py` or of any census.\n\n1. **Consumer**, verbatim: fetch `GET /docs/research/moving-cutoff-parity.md` and check\n `sha256 = ef7a18651d5d39ac45bb7f96727c0b2d6e16f39f620f359d63d220c6a7f2cd9d` (`work/j4206/fetch_doc.py`\n does this and prints `local_sha256`). The three objects used: (3) `x = 2^j`, `y = ceil(x^(12/25))`,\n `Q = floor(x/y)`, `f(n) = Lambda(n-2)mu(n)`; (13) `|D_y| <= 2 sum_{e<=Q, e odd} log(x/e) max_t|Delta_e(t)|`;\n the tolerance remark under (16) that the first weighted absolute sum in (13) being `<= 2x/25` suffices.\n\n2. **Run it**: `python price.py` (~10 s; writes `price.json`, prints the table to `price.out`). Exact pieces:\n `y = ceil_root(12j, 25)` is an exact integer `25`-th root bound, `Q = 2^j // y` is integer division, and\n `phi` comes from a single sieve up to `max_j Q`.\n\n3. **Check these, in this order** (all in `price.out` / `price.json`):\n - `T(j) = sum_{e<=Q, e odd} log(x/e)/phi(e)` for `j = 8..44`, and the integer-power thresholds:\n `A = 3` first passes at `j = 12` (`x = 4096`); `A = 4,5,6` pass at the smallest computed `j = 8`\n (`x = 256`); `A = 1, 2` never pass.\n - slack at `j = 44`: budget `log^3 x/25 = 1134.9` vs `T = 249.6`, i.e. an error `4.55x` the\n `x/(phi(e) log^3 x)` shape still fits.\n - the fit `T ~ c log^2 x` with `c = 0.27559` as the tail maximum (falling to 0.26838 by `j = 44`), and hence\n `T/(log^2 x /25) -> 25c = 6.89 > 1`: a log-squared shape fails by a constant factor near 7 at every scale.\n - cutoff sweep at `j = 32, 36` for `theta` in {0.40, 0.48, 12/25, 0.60}: smallest real `A` moves by under\n 0.08 (under 4%).\n - controls at `j = 40`: trivial `|Delta_e| <= (x/2)(1 + 1/phi(e))` fails by `1.6e8`; `phi(e) -> e` gives\n 165.63 and a flat `log x` weight gives 275.02, against 207.79 for the stated shape (smallest real `A`\n 2.51 and 2.66: the verdict `A = 3` passes, `A = 2` fails, is not delicate).\n\n4. **What the check does and does not do.** It prices a *hypothetical* input shape `(S_A)` against the\n consumer's own allowance. It does not assert that `(S_A)` holds for `f`, does not compute `Delta_e`, `D_y`\n or any discrepancy value, and does not touch (16). The step \"and so for all larger `x`\" is a fit plus the\n classical `sum_{n<=N} 1/phi(n) ~ c1 log N`; the exact range `j <= 44` is already enough for the conclusion\n used (slack exists and grows over that range).\n\n5. **Independent reuse.** The same table can be rebuilt in any language from the two integer definitions\n (`y`, `Q`), a totient sieve and the double inequality `2 T(j)/log^A x <= 2/25`; nothing else from this\n run is needed, and no file here declares a revision.","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":"2026-09-26T18:55:10.948Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"claim_refuted","evidence":"work/j4206/price.py -> price.json and price.out (exact odd-e totient sums and exact integer cutoffs y = ceil(2^(12j/25)), j <= 44, with cutoff sweep and three controls); return #1813's ledger4153.py (e72a23f4...) -> ledger4153.json (55bb7d2b...), cited not rerun; served consumer sha ef7a1865...; research/OUTCOMES.md sha 90fb14c3... (shifted-prime Mobius table: |M(x)| <= 8.8e-5 x for j >= 29 against the 0.374x trivial bound); return #1811's search negative.","statement":"The implication 'if the dyadic-union loss over y-blocks fits the (2/25)x allowance, consumer (16)'s 1/50 disappears as a requirement and no new level of distribution is needed' is refuted twice over. (i) #1813: the max in (13) spans one dyadic block and is removable at o(x) cost by a mesh, so there is nothing to split, while the level 13/25 is the modulus range e <= x/y and is untouched by any split in t; the blockwise input the route named is for mu, not for Delta_e's sequence Lambda(n-2)mu(n). (ii) This pass: the (2/25)x allowance is not tight in the level currency at all -- a level-13/25 residue-uniform statement for f with a classical error shape x/(phi(e) log^A x) satisfies (13) at every dyadic x >= 4096 for A = 3 and every x >= 256 for A = 4, with slack growing in x, while A = 2 never suffices (by a constant factor near 7), and the cutoff sweep moves the price by under 4%. So no repricing of the y-maximum, the blocks or the cutoff can be the operative gap: the whole cost is the existence of a residue-uniform statement about the sequence f = Lambda(n-2)mu(n), unavailable at ANY level in the searched record, and twin-strength by the consumer's own (12)-(15).","assumptions":"The consumer is as served in research/moving-cutoff-parity.md (3), (9), (13), (16) at sha256 ef7a18651d5d39ac45bb7f96727c0b2d6e16f39f620f359d63d220c6a7f2cd9d, with the fixed cutoff y = x^(12/25). The price check is arithmetic on the allowance against a HYPOTHETICAL input shape (S_A); it is not a claim that (S_A) holds for f, and it computes no discrepancy value. Scope: this route's mechanism and the level/cutoff currency of this consumer only; (16) is untouched and stays OPEN, and nothing here bears on twin primes, G2 or beta_2.","revisit_when":"A residue-uniform statement for Lambda(n-2)mu(n) (or a class containing it) at the consumer's level 13/25 -- which by this job's arithmetic then suffices with log-slack -- or a ONE-SIDED mechanism for D_y that does not pass through residue-uniform equidistribution (e.g. a second-moment or density argument over dyadic scales, since (16) needs only an unbounded set of scales). Level repricing, cutoff choice or removal of the max over t are each now known NOT to be reasons to revisit this route."},"route_id":165,"depends_on":[1813],"evidence_md":"**Outcome `blocked`.** Route 165's question is answered and its premise refuted a second time: **the max-over-y clause was never the price, and neither is the level.**\n\n**What the route assumed.** That consumer (16)'s 1/50 is bought by a *level* -- a max-form Mobius-Bombieri-Vinogradov at 13/25, #1183's unbuyable object -- and that removing the max over y instead makes the 1/50 disappear. #1813 (cited, not rerun) showed the max in (13) spans one dyadic block, is removable at `o(x/L^A)` cost by a mesh, that the 13/25 is the modulus range `e <= x/y`, and that the published 1/2-o(1) input is for `mu`, not for `Delta_e`'s sequence.\n\n**What this pass adds -- the price check.** Assume the hypothetical input shape `(S_A)`: `|Delta_e(t)| <= x/(phi(e) log^A x)` for every odd `e <= Q`, `x/2 <= t <= x`, i.e. a level-13/25 residue-uniform statement for `f(n) = Lambda(n-2)mu(n)` with a *classical* error shape. Then (13) gives `|D_y| <= 2 x T(j)/log^A x` with `T(j) = sum_{e<=Q, e odd} log(x/e)/phi(e)`, so the consumer's allowance `(2/25)x` is met exactly when `T(j) <= log^A(x)/25`. `price.py` computes `T(j)` exactly, `j <= 44`, with exact integer cutoffs `y = ceil(2^(12j/25))`.\n\n- `A = 3` passes from `j = 12` (`x = 4096`) and at every larger computed `j`; `A = 4,5,6` pass from the smallest computed `j = 8` (`x = 256`); `A = 1,2` never pass.\n- Slack grows: at `j = 44`, `T = 249.6` against the budget `log^3 x/25 = 1134.7`, so an error `4.55x` the `x/(phi(e) log^3 x)` shape still fits; the fit `T ~ c log^2 x` (`c = 0.2756` tail max, falling to 0.2684) makes the slack grow like `log x`.\n- `A = 2` cannot suffice at any scale: `T/(log^2 x/25) -> 25c = 6.89 > 1`, a constant factor near 7 short.\n- Cutoff sweep `y = x^theta`, `theta` in {0.40, 0.48, 12/25, 0.60}: smallest real `A` moves by under 0.08 (under 4%).\n- Controls at `j = 40`: the trivial bound `(x/2)(1 + 1/phi(e))` fails by `1.6e8` (the requirement is not automatic); `phi(e) -> e` gives `T = 165.63` and a flat `log x` weight gives `275.02` against `207.79` (smallest real `A` 2.51 / 2.66), so `A = 3` passing and `A = 2` failing is not delicate.\n\n**Why it matters.** A BV-shaped statement for `f` at the consumer's *own* level 13/25 satisfies (13) with logarithmically growing slack, while classical inputs (and the consumer's own named source, `B_C <<_A x/log^A x`) carry arbitrary fixed log powers. So the 1/50 is not a level price at all; every quantity the route proposed to manipulate -- the y-maximum, the blocks, the cutoff -- is priced at a few percent, and the whole cost is the existence of a residue-uniform statement about the *sequence* `Lambda(n-2)mu(n)`, which the searched record lacks at **every** level (#1811, #1813, this pass). By the consumer's own (12)-(15) a proof of (16) gives `S(x) >= x/200 + o(x)` and hence infinitely many twins, so the object is twin-strength. The route's sentence 'no new level of distribution is needed' is true in the letter and empty in content.\n\n**Measured corroboration, cited not rerun.** `research/OUTCOMES.md` (sha `90fb14c3...`, shifted-prime Mobius table) has the dyadic total `M(x) = sum Lambda(n-2)mu(n)` at most `8.8e-5 x` for `j >= 29`, against the `0.374 x` trivial bound used by (14)-(16): the tolerance is loose by four orders at reachable scales in the *value* currency too. That entry's own limit is respected: loose trivial bounds are unusable by a derivation, and no bound is drawn from it here.\n\n**Scope.** Consumer (16) stays OPEN and is untouched; nothing here touches twin primes, G2 or `beta_2`. The refutation is of the route's mechanism and of the level-repricing family on this consumer. No `next_step` is filed: the experiments that remain are not on this route (residue-uniform statement for `f`, now known to suffice with log-slack, or a one-sided mechanism for `D_y` not passing through residue-uniform equidistribution), and the corpus has recorded that extending the `D_y` census cannot separate the two contributions.","prior_art_md":"**Search date 2026-09-26**, on the changed ingredient: a *statement about the sequence* `f(n) = Lambda(n-2)mu(n)` in residue classes, at any level.\n\n**Reused, not rerun** (the route's own recorded search, plus #1811's): Granville-Shao arXiv:1706.05710v1 Thm 1.1(b) + uniform Thm 2.1 (Mobius BV at exactly `1/2-o(1)`, uniform in classes, with the max over `y <= T` in their hypothesis (2.1)); Shao-Teravainen arXiv:2006.05954v2 Rem. 1.8 (`1/4-eps`/`1/3-eps` max-over-classes, `1/2-eps` only for well-factorable `lambda_d`); Tao 254A Notes 3 Ex. 21-22; and the standard statement that adding or removing the max over `y` never changes the level (Kedlaya notes ch. 17; the corpus's own mesh in shifted-prime-decomposition.md §2). Nothing in it supplies `f` at any level.\n\n**New queries this job (two shapes).** (1) \"Cantarini arXiv 2607.09110 Bombieri-Vinogradov Mobius shifted conjecture level of distribution\": arXiv:2607.09110v1 is *Averages of diagonal Elliott-Halberstam problem twisted by Mobius function with Sobolev and Holder-Zygmund weights*; its own text recalls the Elliott-Halberstam conjecture **twisted by `mu(n)`** as the conjectural input `sum_{q <= N^theta} max_{y<=N} max_{(a,q)=1} ...`. That is the same object the served consumer names as its exact source input (§4.1: Conjecture 2, `EH(mu_h)`, `h = +2`, `theta = 13/25`, cited as Cantarini arXiv:2607.09110v1). So the consumer's named input is a **conjecture**, and the paper supplies *averages* of that problem with weights, not the conjecture. No proved residue-uniform statement for `f` at any level was located. (2) \"residue-uniform equidistribution Lambda(n-2)mu(n) twisted von Mangoldt Mobius fixed shift progressions beyond 1/2\": nothing for the twisted sequence. Adjacent items, all different objects: Lichtman arXiv:2009.08969 (averages of `mu` on shifted primes: the folklore conjecture `sum_{p<=X} mu(p+h) = o(pi(X))` remains OPEN -- the corpus's own OUTCOMES.md says the same), Carella arXiv:2206.12956v3 (unrefereed, and an unweighted mean, already dispositioned in the route's search), Sedunova arXiv:1705.06660 (a logarithmic improvement in BV for the *ordinary* sequence at level 1/2), Matomaki-Radziwill-Tao-type average-over-shifts results (an average over shifts cannot select the fixed shift `h = 2`; the served consumer says so itself). Averages exist for `mu` and for `Lambda`; the *twist at a fixed shift* does not.\n\n**Access gap, recorded.** HAL hal-05725912v1 (2026-08-25), *A Moving-Cut Correction in the Huang-Li Conditional ...*: the host served a bot-protection interstitial (Anubis) and the abstract could not be read. It is the closest new item found to this consumer's subject (a moving-cut correction) and should be read if it becomes accessible; it is not used, cited or relied on here.\n\n**Exact remaining gap** (kind unchanged, price sharpened by this job): a residue-uniform bound `sum_{e <= Q, e odd} log(x/e) max_t |Delta_e(t)| <= (2/25)x` for `f(n) = Lambda(n-2)mu(n)` at level 13/25 -- equivalently, by this pass, *any* statement about `f` of classical BV shape at that level with at least three fixed logarithmic powers, since the allowance then holds with slack growing in `x`. No such statement is proved at any level in the searched record; the consumer's own named source is the EH-twisted-by-`mu` conjecture. Removing the maximum, moving the cutoff, or re-pricing the level are all now known not to be reasons to revisit."},"research_route_id":165,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_cf9d09664a5f57211c6d964b","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 /research-routes/165 and return #1813. Return the ordinary report and transcript plus research: {route_id: 165, 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} , obstacle: {kind, statement, assumptions, evidence, revisit_when} , depends_on: []}. 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":"1813","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/165","transcript_url":"/projects/twin-primes/return/1857/transcript","files":[{"sha256":"ecd5e0b708c43187d5c4b4e4a221cd5938dae551c09cbe68e9ef5e5ffb4bd341","name":"report.md","bytes":8484},{"sha256":"69de0418381a8c5cecfcaadc59d7ec735e74e22d674ccd7abe086ce763576c06","name":"recipe.md","bytes":2879},{"sha256":"7730aaa7b43d511aaa43c013257ec2a715f79fa7f784a1a7057d3b06581ed839","name":"price.py","bytes":6396},{"sha256":"50693b514aabb46d5c28afd9637be9327a0731c7d7a24b51c61096bd80b8b984","name":"price.json","bytes":8246},{"sha256":"a2dca68ea895f90342696151686a6c1c2b1d896d61e84acd6e115b16cccc301a","name":"price.out","bytes":1544}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}