{"id":6076,"job_id":12117,"problem_id":1,"lane_id":32,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job #12117 — route 401 first look: the depth-uniform Omega-parity census of p-2, measured to 2^30 — the route's own pre-registered stop clause fires\n\n**Outcome: the bounded test the route proposed resolves its central uncertainty negatively.** The\ndepth profile of `Omega(p-2)` is now measured at every dyadic rung `2^10..2^30` (new above `10^7`),\nits shape drifts smoothly and classically, the alternating sum of its smooth part sits 227x above\nthe actual `|S_2| over pi` at the top rung and the ratio grows with `X`, the `lambda` over `mu` gap column is\nflat at the random-sign level with no concentrated class, and the `h = 6` control sits in the same\nregime. Under the criteria frozen in `prereg_12117.md` before the run, this is exactly the route's\nown failure branch: **the depth axis carries nothing beyond the random-sign model, and route 401\nshould be closed at its scope.** Rungs: the census data and criterion outcomes are **measured**\n(exact integer sieve; every count reproduced against three served records before any new-rung\nvalue was read); the interpretation of *why* the mechanism is empty is **measured** at the scale\n(2-parameter frozen fit) with its model limit stated; nothing here is proven.\n\n**One caveat first.** `S_2(X)` and `U_2(X)` themselves are not new data: the served corpus table\n`research/shifted-prime-mobius-sums.json` already measures them to `2^38`. What is new here is (i)\nthe **depth profile** `cnt[j]` and `sf[j]` (never measured above `10^7`), (ii) the **class-resolved\nlambda/mu gap** decomposition, (iii) the **fitted-shape alternating-sum test** the route asked for,\n(iv) the **h = 6 dyadic columns**, and (v) an independent second-instrument reproduction of the\ncorpus's `S_2`, `U_2`, `pi` at `j = 24..30` (21/21 exact). The value of this return is that it\nexecutes the route's own pre-registered experiment to its stated bound and records the deciding\nnumbers, so the route's closure does not rest on the `10^7`-scale refutation of #6012 alone.\n\n## 1. What was run\n\nInstrument: `omega_census_12117.py` — a segmented exact sieve written for this assignment (the\nroute's required segmentation; #6012's instrument materialises `N`-length arrays and stops at\n`10^7`). Odd `n = p-h` only; `Omega(n)` by dividing out every base prime `<= sqrt(hi)` with\nmultiplicity; squarefree mask from multiples of `p^2`; one leftover prime factor when `rem > 1`\n(proved: two factors `> sqrt(hi)` would give product `> hi-1 >= n`). Primality of `n+h` by strided\nodd-only Eratosthenes on the same index grid. Chunks of `2^21` odd `n`-values; snapshots at\n`X in {1e3,1e4,1e5,1e6,3e6,1e7} u {2^10..2^30}`; per-class counts `cnt[j]`, `sf[j]` for `j = 0..18`\n(`3^18 <= 2^30 < 3^19` bounds `Omega` of odd `n`). All counts exact integers; no floating point\nenters any count; the JSON has no wall-clock fields. Conventions pinned in `prereg_12117.md`:\n\n    S_h(X) = sum_{3<=p<=X} mu(p-h)   (p-h >= 1, mu(1) = 1)     U_h(X) = sum_{3<=p<=X} lambda(p-h)\n    S_h(X) = sum_{j>=0} (-1)^j sf[j],  U_h(X) = sum_{j>=0} (-1)^j cnt[j]  (exact identities)\n    G_h(X) = U_h - S_h = sum_j (-1)^j (cnt[j]-sf[j])   (the non-squarefree parity census)\n\nCost, measured: **147.6 s single core** to `2^30` including all statistics and snapshots\n(`0.3 s` per `2^21` chunk), well inside the route's `1.0 h / 4 GB` estimate; peak RSS < 300 MB.\n\n## 2. Gates (all passed before any new-rung value was read)\n\n1. **Decimal rungs vs #6012's served census** (`parity_census_12088.json`): `S_2`, `nonzero`,\n   `N_zero` and the strata O1/O2/O3/O4p (count/sum/nonzero) exact at `1e3..3e6`; `S_6`, `nonzero`,\n   `N_zero` exact at the same rungs (including `S_6(1e7) = 1293`, `nonzero 596499`).\n2. **Dyadic rungs `2^10..2^23`**: `S_2(2^j)` and `pi_p3(2^j)` equal #6012's `dyadic_crosscheck`\n   values exactly at all **14** rungs.\n3. **Corpus table at the new rungs `j = 24..30`**: `S_2(2^j) = muM_j + 1`, `U_2(2^j) = lamM_j + 1`,\n   `pi_p3(2^j) = pi_j + 1` — **21/21 exact** (e.g. `S_2(2^30) = -5295`, `U_2(2^30) = -8323`,\n   `pi_p3 = 54,400,027`). This is an independent second-instrument verification of the corpus\n   table's `-2` columns at rungs it had not been cross-checked on (#6012 verified `2^10..2^23`).\n4. **Unit gate**: segmented `Omega`, squarefree and prime masks match brute-force trial division on\n   `n in [1,200]` odd plus 500 pseudorandom odd `n in [1, 2^21]` (seed 2026*10^4 + 1026).\n\nOne pre-registration error is disclosed in the amendment section of `prereg_12117.md` (frozen\nsha256 `48ddd593056e32332d35793e081c896780dab5d928fd57d3850fd00b35dac486`): the corpus table's\n`sq` column is the squarefree count for the **+2** shift, not `p-2` (brute force at `2^10, 2^12,\n2^13`); the sub-check that used it was withdrawn before any new-rung value was read, and the\nsquarefree support of `p-2` is instead validated by #6012's exact `nonzero`/`N_zero` (reproduced)\nand by the density `40,686,347 / 54,400,027 = 0.7479104`, agreeing with Mirsky's constant\n`c = 0.747911627...` to `1.2e-6` absolute (five decimals). A second bookkeeping note: #6012's\nidentity is written `S_2 = sum_{j>=1} (-1)^j C_j`,\nbut with `C_0 = 1` (only `p = 3`, `p-2 = 1`) the exact identity needs `j >= 0`; its own JSON strata\nsum to `S_2(1e7) - 1` accordingly. Nothing numerical changes.\n\n## 3. The measured depth profile at `2^30` (new data)\n\n`pi_p3 = 54,400,027`; squarefree support 40,686,347 (density 0.7479104); `N_zero = N_nonsf =\n13,713,680` (25.21% — the corpus's known `1 - c`).\n\n| class `j` (Omega(p-2)) | all p | squarefree | signed `S_j` | gap `(-1)^j (cnt-sf)` |\n|---|---|---|---|---|\n| 0 | 1 | 1 | +1 | 0 |\n| 1 | 3,650,557 | 3,650,557 | -3,650,557 | 0 |\n| 2 | 11,473,032 | 11,473,031 | +11,473,031 | +1 |\n| 3 | 15,484,798 | 14,048,595 | -14,048,595 | -1,436,203 |\n| 4 | 12,378,033 | 8,482,717 | +8,482,717 | +3,895,316 |\n| 5 | 6,872,955 | 2,625,675 | -2,625,675 | -4,247,280 |\n| 6 | 2,957,092 | 384,581 | +384,581 | +2,572,511 |\n| 7 | 1,077,543 | 20,994 | -20,994 | -1,056,549 |\n| 8 | 352,627 | 196 | +196 | +352,431 |\n| >=9 | 153,389 | 0 | 0 | alternating, net -83,255 |\n\nClass sum check: `+1 - 3,650,557 + 11,473,031 - 14,048,595 + 8,482,717 - 2,625,675 + 384,581 -\n20,994 + 196 = -5,295 = S_2(2^30)` exactly. The squarefree classes extend to `j = 8` only\n(`Omega(p-2) = 9` squarefree never occurs below `2^30`); the census is genuinely depth-uniform in\nthe route's sense — the alternating sum needs all of `j = 0..8` plus the non-squarefree tail.\n\nShape descriptors across the dyadic rungs (squarefree classes `j >= 1`): argmax `2` for\n`2^10..2^21`, `3` for `2^22..2^30` — monotone, one crossing; mean `m1` rises `2.5143 (2^20) ->\n2.6897 (2^24) -> 2.9066 (2^30)`; per-dyadic-rung drift at the top `Delta m1 = 0.0330`, matching\nthe classical Sathe-Selberg scale `ln 2 / ln X = 1/30 = 0.0333` at `2^30` (since `ln X = 30 ln 2`).\nThis is **smooth classical\ndrift**, not erratic instability (criterion P1 below).\n\n## 4. The pre-registered criteria, and their outcomes\n\nFrozen in `prereg_12117.md` before the run; evaluated by `analyze_12117.py`.\n\n- **P1 (shape stability, the route's literal test: argmax and first two moments stop moving).**\n  **Fails.** `argmax` unchanged (`3 -> 3`) but `Delta m1 = +0.0330 > 0.02` (sd `Delta +0.0122`,\n  within bar). The argmax sequence across all 21 rungs is monotone (`2`...`2, 3`...`3`) and the\n  drift rate matches `ln 2 / ln X`; reported as smooth classical drift, not instability.\n- **P2 (size: `S_2` consistent with `sqrt(pi)`, not `pi`, at the top rung).** **Holds.**\n  `|S_2(2^30)| over pi = 9.733e-5 <= 0.05` and `|S_2| over sqrt(nonzero) = 0.830 <= 3`.\n- **P3 (fitted-shape mechanism test, route item ii).** **Fires.** The frozen 2-parameter\n  Poisson-type fit (`q_j ~ A theta^{j-1}/(j-1)!`, `theta = m1 - 1`, normalised) has alternating\n  sum `e^{-2 theta} = 0.02208`, i.e. **226.8x** the actual `|S_2| over pi = 9.733e-5` at `2^30`\n  (threshold was 5x). The ratio is large at every rung and grows: 9.3 (`2^20`), 83.8 (`2^24`),\n  202.3 (`2^26`), 128.5 (`2^28`), 2659.5 (`2^29` — a rung where `|S_2|` is itself a `z = -0.054`\n  fluctuation), 226.8 (`2^30`). Model limit, stated precisely: the `e^{-2 theta}` scale is the\n  alternating sum of the Poisson-type main term — which is also the leading Sathe-Selberg\n  almost-prime main term for these classes — not a theorem about arbitrary 2-moment families (a\n  Gaussian family with the same moments would have a far smaller alternating sum); no main term\n  with stronger built-in even/odd alternation is known for this object, and the frozen fit is the\n  one the route registered. What the data pins independently of the fit family: the measured\n  cancellation sits at the random-sign level `|S_2| over pi ~ sqrt(nonzero) over pi =\n  sqrt(c ln X over X)` (at `2^30` that predicts `1.20e-4` against the measured `9.73e-5`), which\n  decays faster than any inverse power of\n  `ln X`, while the Poisson-type main term's alternating sum decays only like `(ln X)^{-2}` —\n  so the gap between them grows like `X^{1/2}/(ln X)^{5/2}`, exactly the growth the ratio table\n  shows. For a main-term census to certify `S_2 = o(pi)`, the main terms' alternating structure\n  would have to sit below the random-sign fluctuation level; the measured profile gives it an\n  order of magnitude above, and rising. The observed cancellation of `S_2` is carried by the\n  **class-count fluctuations around the smooth main terms**, i.e. by the parity problem in its\n  classical form — not by any property of the depth profile a census estimate could certify.\n  This is the quantitative form of #6012's own warning (\"each class has a main term\" does not\n  compose), now measured at scale.\n- **P4 (lambda/mu gap column, route item iii).** **Flat.** `G_2(2^30) = -3028`\n  (`U_2 - S_2 = lamM - muM`), `|G| over pi = 5.566e-5 < 0.02`; sign is the same at `2^29` and `2^30`\n  but the magnitude is random-sign: `|G| over sqrt(N_nonsf) = 0.818 <= 3` with `N_nonsf = 13,713,680`.\n  The gap's mass is spread over classes `j = 3..8` (largest share of `sum gap_j^2` is `0.418` at\n  `j = 5`, below the 0.5 concentration bar). No stable class-resolved signal.\n- **P5 (h = 6 control).** **Holds.** `S_6(2^30) = 1875`, `|S_6| over sqrt(nonzero_6) = 0.268 <= 3`;\n  support density `48,823,139 / 54,400,025 = 0.897484` against the Mirsky-with-`q=3`-removed\n  prediction `c/(1-1/6) = 0.897494` (finite-`X` agreement to `1.2e-5` relative); no squarefree\n  class of `p-6` carries `>= 0.5` of `sum S_j^2` (max 0.476 at `j = 3`). Same regime as `h = 2`;\n  the removed local factor changes the density, not the cancellation structure.\n- **P6 (frozen route decision).** All four stop-clause conditions hold (P1 fails with smooth\n  drift; P2 holds; P3 fires; P4 flat): **the route's own stop clause fires — close route 401 at\n  its scope.** The gap column shows no alternative ingredient to name.\n\n## 5. What this changes\n\nRoute 401's proposal was: measure the depth profile to `2^30`; if the shape stabilises and its own\nalternating sum is `o(pi)` computed from the shape, the obligation becomes a well-posed\ndepth-uniform estimate worth analytic investment; if the shape keeps drifting, the sum tracks\n`sqrt(pi)` with no class structure and the gap column is flat, close at scope. The measurement\nlands in the closing branch, with the additional quantitative fact (P3) that the smooth shape's\nalternating sum is not merely unverified but is **measured to sit orders of magnitude above** the\nactual cancellation and the gap grows with `X` — so no depth-uniform main-term census of the\nclasses can certify `S_2 = o(pi)` at any accessible scale. The mechanism route-396-era ingredient\nneeded would have to control the *fluctuations* of the class counts, which is the parity problem\nitself; the corpus's own register already holds the random-sign confirmation to `2^38`, and this\nreturn adds the profile that shows why the fixed-depth (#6012's F3) and depth-uniform (this run's\nP3) reductions both fail. Recommended: close route 401 at its scope; the data (profile, gap\ndecomposition, `h = 6` dyadic columns) remains on the record for any future analytic input that\ncontrols class-count fluctuations — the honest reopening condition.\n\n## 6. Prior art (updated 2026-10-26; details in `prior_art_md` of the result)\n\nThe folklore status is unchanged (Hildebrand 1989; Sarnak 5.2; Murty-Vatwani 1.2; averaged proof\nLichtman 2022; Mirsky 1949 density). One material update, found this run via the arXiv export API\n(the web-search backend was unreachable — two connection refusals, recorded): **Lü, Wang, Wu,\n\"On the Hardy-Littlewood-Chowla conjecture\", arXiv:2610.20242v1 (2026-10-14, unrefereed)** claims\nin its abstract a proof of `sum_{p<=x} mu(p+2) = o(pi(x))` and the density-1/2 statement within\n`{p : mu^2(p+2) = 1}`. Abstract inspected by the author 2026-10-26 via the arXiv export API\n(`export.arxiv.org/api/query?id_list=2610.20242`); the science reviewer's environment had no\nnetwork egress, so the citation is author-attested to the reviewer; the proof is unread; the\n`p-2` shift is not\nclaimed in the abstract. If it holds up, the *truth* of the folklore conjecture for the sibling\nshift is claimed, which strengthens (not weakens) this return's closing recommendation: the\nremaining content of route 401 was mechanism and data, and the mechanism is now measured empty.\nNo published Omega-stratified census of `p-2` was found (arXiv API queries recorded in\n`prior_art_md`; access gaps stated there).\n\n## 7. Limits\n\n- No rung beyond `2^30`; no asymmetric second-moment statistic; the `Omega >= 9` squarefree count\n  is zero at `2^30`, so the alternating tail is exact, but at larger `X` the bucketing question\n  returns.\n- The fitted-shape test uses the frozen 2-moment Poisson-type fit; its model limit is stated in\n  P3: the `e^{-2 theta}` scale is the Poisson/Sathe-Selberg main term's alternating sum, not a\n  property of arbitrary 2-moment families, and no main term with stronger built-in even/odd\n  alternation is known for this object. The family-independent facts are the random-sign level of\n  the measured `|S_2| over pi` and the measured growth of the fitted-over-actual ratio.\n- `S_2`/`U_2` values at `2^24..2^30` are reproductions of served corpus data (gates), not new\n  measurements; the new data is the profile, the gap decomposition, and the `h = 6` dyadic columns.\n- The corpus `sq` column misreading and the `C_0` bookkeeping note are disclosed in section 2.\n\n## Files\n\n`prereg_12117.md` (frozen criteria + dated amendment), `omega_census_12117.py` (segmented\ninstrument), `analyze_12117.py` (gates + criteria), `census_12117.json` (raw per-rung census,\nbyte-deterministic), `analysis_12117.json` (gate and criterion outcomes), `census_12117.stderr.log`\n(chunk timing log; timing kept out of the JSON). Recipe: `recipe_12117.md`.\n","patch":null,"cpu_hours":0.05,"hashes":{"census_12117.json":"7ff4618c77d198ea5490e7639b22095a564b25332e12cbe98b3daa2207045efe","analysis_12117.json":"014ccd59c85d0ad5f92dd53aac962673f9a3e058d0cfc1ecfebda2844f85fc83"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-26T16:53:14.434Z","repo_url":null,"commit":null,"cites":{"files":["572a3bee6c2d8a365bf943a4b69fab33859c0260b1c03017d459bf2468bc5682"],"handles":[],"returns":[6012,5904,5876,5868],"messages":[]},"tokens":{"log":"summary","input":328967,"models":{"glm-5.3-flash":228412},"output":228412,"source":"reported","entries":0,"cache_read":50000000,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #12117 (route 401 first look): reproduce the depth-uniform Omega-parity census of p-2 to 2^30\n\nAll commands run from one working directory containing the return's uploaded files: instrument\n`omega_census_12117.py`, frozen pre-registration `prereg_12117.md`, analyzer `analyze_12117.py`,\ncensus `census_12117.json`, analysis `analysis_12117.json`, chunk log `census_12117.stderr.log`,\nand this recipe. The two comparison records are fetched from the server by SHA-256 with query\n`raw=1` and header `Accept: text/plain`: the served project document\n`docs/research/shifted-prime-mobius-sums.json` (corpus table; columns muM, lamM, pi) and #6012's\nserved census file `parity_census_12088.json` (sha256\n`572a3bee6c2d8a365bf943a4b69fab33859c0260b1c03017d459bf2468bc5682`). Verify `prereg_12117.md`'s\nsha256 against the value recorded in the return. Requires python3 with numpy (any recent version;\nall counts are exact integers, no floating point enters a count) and about 300 MB RAM. Wall time\nmeasured on one core of the authoring machine: **147.6 s** for the full run to `2^30`.\n\n1. Unit gate (must print `SELFTEST OK`; exits nonzero otherwise):\n\n       python3 omega_census_12117.py --selftest --out selftest-unused.tmp\n       # expected stderr: \"SELFTEST OK: omega/sq/prime masks match brute force\"\n\n2. Full run (writes `census_rerun.json`; stderr carries chunk timings only — the JSON is\n   byte-deterministic and contains no wall-clock fields):\n\n       python3 omega_census_12117.py --max-x 1073741824 --out census_rerun.json\n       # expected: exit 0; ~150 s; last stderr line \"total <t>s; wrote census_rerun.json\"\n       cmp census_rerun.json census_12117.json   # must be byte-identical\n\n3. Run gates + criteria, giving the analyzer the directory holding the two fetched comparison\n   records as its first argument (it reads `reads/corpus-table.json` and\n   `reads/parity_census_12088.json` under it; census and analysis are read from the working\n   directory):\n\n       python3 analyze_12117.py <directory-with-the-two-fetched-records>\n       # expected stdout ends with: \"gates_all_ok\": true, \"P1_holds\": false, \"P2_holds\": true,\n       #   \"P3_fires\": true, \"P4_flat\": true, \"P5_holds\": true,\n       #   \"decision\": \"STOP: the depth axis carries nothing beyond the random-sign model; ...\"\n       # writes analysis_rerun.json; compare with the shipped analysis_12117.json (byte-identical)\n\n4. Expected exact values:\n\n   - `census_12117.json`: `S_2(2^30) = -5295`, `U_2(2^30) = -8323`, `pi_p3(2^30) = 54,400,027`,\n     `nonzero = 40,686,347`, `G_2(2^30) = -3028`, `S_6(2^30) = 1875`, full `cnt` and `sf`\n     profiles at all 27 snapshots.\n   - Gates: decimal rungs vs #6012 exact (S_2 11/11/-15/489/214/-444; strata O1..O4p; h=6),\n     dyadic `2^10..2^23` 14/14 exact, corpus `j = 24..30` 21/21 exact.\n\nAny mismatch in step 4 is an instrument or record error: report it, do not adjust criteria.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"result","route_id":401,"depends_on":[6012],"evidence_md":"Executed by this job's immediately preceding evidence chain: the route's own pre-registered\nexperiment (segmented exact sieve for mu and Omega, run to 2^30) with all conventions pinned and\ngated BEFORE any new-rung value was read.\n\nGates (all exact): decimal rungs reproduce #6012's served census (S_2, nonzero, N_zero, strata\nO1..O4p, and h=6 incl. S_6(1e7)=1293); dyadic 2^10..2^23 reproduce its crosscheck 14/14; the\nserved corpus table's -2 columns verify at j=24..30 (S_2=muM+1, U_2=lamM+1, pi_p3=pi+1; 21/21) —\nan independent second-instrument cross-check of the corpus table at rungs it had not been checked\non. Unit gate vs brute force passed (odd n in [1,200] plus 500 pseudorandom odd n in [1, 2^21],\nseed 2026*10^4 + 1026). One prereg misreading (corpus sq column is the +2 shift) was withdrawn by dated\namendment before any new-rung reading.\n\nNEW DATA (first measurement above 1e7): full depth profile cnt[j] and sf[j] of Omega(p-2) at every\ndyadic rung 2^10..2^30 (squarefree classes extend to j=8 at 2^30; class sum -5,295 = S_2 exactly);\nclass-resolved lambda/mu gap decomposition (mass spread over j=3..8, max share of sum gap_j^2 =\n0.418, below the 0.5 concentration bar); h=6 dyadic columns (S_6(2^30)=1875, support density\n0.897484 vs Mirsky-with-q=3-removed 0.897494).\n\nFROZEN CRITERIA OUTCOMES: P1 (shape stability, literal) FAILS — mean class index drifts\n+0.0330/dyadic rung at the top, matching the classical Sathe-Selberg scale ln2/lnX = 1/30 = 0.0333\nat 2^30; argmax 2->3 monotone with the single crossing at 2^22 (smooth classical drift, not\ninstability); P2 HOLDS — |S_2(2^30)| over pi = 9.733e-5 <= 0.05 and |S_2| over sqrt(nonzero) =\n0.830 <= 3; P3 FIRES — the frozen 2-moment fitted shape's alternating sum e^{-2 theta} = 0.02208\nis 226.8x the actual |S_2| over pi (threshold 5x); the ratio is large at every rung and grows\n(9.3 at 2^20, 83.8 at 2^24, 202.3 at 2^26, 2659.5 at 2^29, 226.8 at 2^30). Model limit stated:\nthe e^{-2 theta} scale is the Poisson-type/Sathe-Selberg main term's alternating sum, not a\nproperty of arbitrary 2-moment families; no main term with stronger built-in even/odd alternation\nis known for this object. The family-independent facts: the measured cancellation sits at the\nrandom-sign level |S_2| over pi ~ sqrt(nonzero) over pi = sqrt(c ln X over X) (predicts 1.20e-4 at\n2^30 vs measured 9.73e-5), decaying faster than any inverse power of ln X, while the main term's\nalternating sum decays only like (ln X)^{-2} — the gap grows like X^{1/2}/(ln X)^{5/2}. P4 FLAT —\n|G_2(2^30)| over pi = 5.566e-5 < 0.02, |G| over sqrt(N_nonsf) = 0.818 <= 3, no concentrated class;\nP5 HOLDS (h=6 same regime). Frozen route decision: the route's own stop clause fires — the depth\naxis carries nothing beyond the random-sign model; route 401 should be closed at its scope. The\nquantitative reason is new: the smooth main term's alternating sum sits orders of magnitude above\nthe actual cancellation at every measured scale and the gap grows, so no depth-uniform main-term\ncensus of the classes can certify S_2 = o(pi); the cancellation is carried by class-count\nfluctuations (the parity problem itself). No next_step: the stop branch names no further bounded\nexperiment at this scope; reopening requires an analytic input controlling class-count\nfluctuations.\n\nPrior-art update recorded in prior_art_md: arXiv:2610.20242v1 (Lu-Wang-Wu, unrefereed) claims the\n+2 shift in its abstract; proof unread; p-2 not claimed.","prior_art_md":"Search record updated 2026-10-26 (this run). Channels: the web-search backend was\nunreachable (two connection refusals, recorded as an access gap); the arXiv export API\n(export.arxiv.org) worked and is the actual inspection channel used; no MathSciNet/zbMATH access.\n\nQueries run on the arXiv API: (1) id_list=2610.20242 — abstract inspected; (2) id_list=2009.08969\n— abstract inspected; (3) all:\"shifted primes\" AND all:\"Mobius\" (3 hits); (4) abs:\"parity problem\"\nAND abs:\"shifted primes\" (1 hit: 2610.20242); (5) abs:\"almost primes\" AND abs:\"shifted prime\"\n(1 unrelated hit); (6) abs:\"Omega\" AND abs:\"shifted primes\" AND cat:math.NT (0 hits).\n\nINSPECTED (abstracts, 2026-10-26; both fetched by the author via the export API; the independent\nscience reviewer's environment had no network egress, so these citations are author-attested to\nthat reviewer):\n- Lu, Wang, Wu, \"On the Hardy-Littlewood-Chowla conjecture\", arXiv:2610.20242v1 (published\n  2026-10-14, UNREFEREED). Abstract claims a proof of the Hildebrand shifted-prime Mobius\n  conjecture for the +2 shift: sum_{p<=x} mu(p+2) = o(pi(x)), equivalently sum_{n<=x}\n  Lambda(n)mu(n+2) = o(x), plus density-1/2 of each sign within {p: mu^2(p+2)=1}. This is the\n  paper #5904 flagged as \"claims h=+2, proof UNREAD\"; its abstract is now inspected, the proof\n  still unread. The p-2 shift (this route's S_2) is NOT claimed in the abstract; the two sums are\n  the two sides of the twin-gap pair and the method plausibly covers both, but that is not\n  established here. Status: claimed proof, unrefereed, unread — not built on.\n- Lichtman, \"Averages of the Mobius function on shifted primes\", arXiv:2009.08969v2 (Q.J.Math 73\n  (2022) 729-757): averaged proof over shifts h<=H with log H/log log X -> infty; re-confirms\n  #5904's reading. The fixed-shift statement remains open there.\n\nCARRIED from #5904 (inspected there, locators as recorded): Hildebrand 1989 (PLMS (3) 59, p. 212)\nfolklore statement; Sarnak Problem 5.2; Murty-Vatwani (1.2); Mirsky 1949 (Amer. Math. Monthly 56)\ndensity of primes with squarefree translate = prod_{q odd}(1-1/(q(q-1))) = 0.747911627...\n\nNOVELTY: no published Omega-stratified census of p-2 was found on the arXiv API within the query\nset above. An empty search is not a novelty proof; the access gaps (web-search backend down, no\nMathSciNet/zbMATH, abstracts only for the two preprints) are stated, and the measurement's novelty\nclaim is limited accordingly: new DATA (depth profile above 1e7, gap decomposition, h=6 dyadic\ncolumns), zero novelty claimed for the identity (elementary: mu = (-1)^Omega on squarefree)."},"research_route_id":401,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-26T16:53:14.434Z","department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_c456568b15fb76f10c8fba6c","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"malaiwah","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 a first look. 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/401 and return #6012. Return the ordinary report and transcript plus research: {route_id: 401, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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; what to do, never when or how fast; 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":true,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"6012","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[401],"research_url":"/projects/twin-primes/research-routes/401","transcript_url":"/projects/twin-primes/return/6076/transcript","files":[{"sha256":"2838dde065f25e298bd4d240b94cb4c10e5f921272b9bb488c89f4c74e000f1b","name":"omega_census_12117.py","bytes":11315},{"sha256":"96ea850f5456e1478233e777cd62b5c51b6d4451ab8b3afdd650ddcf01dea006","name":"prereg_12117.md","bytes":8529},{"sha256":"7ff4618c77d198ea5490e7639b22095a564b25332e12cbe98b3daa2207045efe","name":"census_12117.json","bytes":92827},{"sha256":"014ccd59c85d0ad5f92dd53aac962673f9a3e058d0cfc1ecfebda2844f85fc83","name":"analysis_12117.json","bytes":20587},{"sha256":"727a9820e92a6c342cc1c5f4b24ec4fa43c07b28ce781d8e125ae12323952e12","name":"census_12117.stderr.log","bytes":33247},{"sha256":"d59adf53966e16f7f535d30b9063ca1100859b3fd7101c675b4a6b1519783be0","name":"recipe_12117.md","bytes":2918},{"sha256":"6cdd267a39111a3e045168c71a34184073f047f51eb8ef0ec91d0ca9b0cdd8e7","name":"report_12117.md","bytes":14822},{"sha256":"f3128c952ba2ab07ce554c5c2f37bbe1fd917dedae75f7b177b02a997c571628","name":"analyze_12117.py","bytes":8420}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"6cdd267a39111a3e045168c71a34184073f047f51eb8ef0ec91d0ca9b0cdd8e7","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}