{"id":560,"job_id":1286,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1286 — Triage of route #22 (exact U-set LP certificates on the zone)\n\nAttempt `adecea8001518b6df00bf1b18519c339` · type **explore** · purpose discovery · lane none.\nAuthor rung of this return: **heuristic** (an investment decision on a proposal, not a claim).\n\n## Decision\n\n**Promising** — one bounded next experiment is justified, and only for a narrow purpose.\n\nThe route's *data* claim is already on the record and was not re-derived here: at 11 levels\np = 23..67 the level-X LP certificate stays in 17–25 while zone twins grow 29 → 121, levels\n≤ X^0.75 certify 0 from p = 31, and levels X^1.25 / X^1.5 recover 85% / 100%\n(`uset_level_lp2.out` sha256 662c2200…, `uset_level_lp.out` sha256 1b905ac0…, return #559).\nWhat triage had to decide is whether *one more measurement* is worth its cost. It is, because\nthe 17–25 column is flat over a factor-~3 in p and the sign of its trend is the whole question;\nand it is bounded (0.5 CPU-h, 4 GB, 1 GB disk, ~1 h of agent time).\n\n## What this changes\n\n1. **The LP framing is prior art, cleanly.** The closest located source is the MathOverflow\n   question *“Best possible sieves for the jacobsthal problem, linear programming, and the prime\n   2”* (asked 2011-06-16, 1k views), where the “linearized Jacobsthal function” `j_lin(n)` is\n   defined exactly as a sieve LP over **real weights indexed by the moduli** `D | n`, with the\n   explicit dual `min Σ|c_d| / max(Σ c_d/d, 0)`, and is computed to n = 75. So neither “sieve as\n   an LP”, nor “LP over modulus-indexed data”, nor “the dual is a weighted-discrepancy quotient”\n   is new. This *strengthens* the route rather than weakening it: the object is standard enough\n   to be optimisable with off-the-shelf LP, and the thread even records the working\n   computational observation that *optimal solutions carry only polynomially many nonzero\n   variables* at n = 75. That observation is the best available support for the proposed\n   restricted-atom relaxation, which is otherwise justified only by the arithmetic argument\n   `strikers | n(n+2) < X²`.\n2. **The parity caution does not transfer, and that is now sourced.** The parity obstruction\n   cited in the route (Polymath8b arXiv:1407.4897 §8; Tao, 254A Notes 4, dual sieve problem) is\n   consistently stated for **approximate** discrepancy data (`q ≤ x^{1-ε}`). Route #22's LP uses\n   **exact** anchored counts in a single zone window, which is precisely why a positive bound at\n   level D = X is not a contradiction. The route already says this; triage confirms the\n   separation is the standard one and is not an artefact of the small p range.\n3. **The remaining gap is narrow and unstated in the literature.** No located source asks how the\n   LP optimum *grows with the truncation level D relative to the window length X* when the\n   modulus-level data is exact and anchored. That is the route's uncovered step. Absence of a\n   source is **not** established novelty — the Hailperin/Prékopa line is only partly read (see\n   access gaps) and no LP-sieve literature sweep beyond these queries was done.\n\n## Weakest assumption, mapped onto the borrowed method\n\nThe proposed experiment restricts atoms to strike types with `prod(T) ≤ X²`. Two distinct claims\nhide there, and triage should not merge them:\n\n* *Validity* — every real position's strikers divide `n(n+2) < X²`, so the dropped atoms are\n  arithmetically irrelevant. This is an argument, and it is sound as stated.\n* *Optimum preservation* — dropping atoms **removes constraints**, so the restricted LP is a\n  relaxation and its optimum is an **upper** bound on the exact optimum, not automatically equal\n  to it. Equality is an assumption. The route already names the check (compare restricted against\n  unrestricted where both exist, p ≤ 53); that check must run **before** the p = 71..151 sweep\n  means anything, because if the two disagree at p ≤ 53 the larger-p numbers are not comparable\n  to the recorded column.\n\nThis is the weakest link, and it is already inside the plan, so the plan is scoped correctly.\n\nTwo smaller assumptions, both addressed: HiGHS returns floating-point optima and the column is\nnearly flat, so the exact-rational re-solve at 3 levels is load-bearing rather than decorative;\nand the success threshold (≥2× from p = 67 to 151, or >15 % of zone twins) is pre-declared, so\nthe experiment can fail cleanly.\n\n## Scope and what not to conclude\n\nThe route itself is explicit that a finite certificate cannot yield an asymptotic proof strategy\nand that the zone twin count is already known to 1e11 by direct enumeration. Triage agrees and\nadds no optimism beyond that: the value of this experiment is a **measured instance of parity in\nU-set coordinates** (does exact level-X data stay bounded/zero, or does it grow?), and nothing\nabout infinitude. If the certificate stays at 17–25 while twins keep growing, that is a clean\nnegative and closes the tangent; it is not a proof of anything larger. Do not close the broader\nroute on one failed proof attempt.\n\n## Execution and evidence\n\nAll landing/infrastructure work and the prior-art retrieval for this assignment happen locally and\nverifiably; the recipe below reproduces the only concrete outputs this return reports (the\ndepartment tool self-tests and the transcript scrub). Nothing here reproduces published numbers,\nper the triage instruction, and no local computation was spent on the route's own arithmetic.\n\n**Transcript.** Attached, scrubbed as data (per-line JSON parse, redaction inside decoded string\nvalues, re-serialised) from this harness's own session log for this assignment. Removed: the\naccount token (prefix `sah_…`, 9 occurrences), provider `userId` / `userEmail` on every line,\ninstallation identifiers, absolute home-directory paths (116), and the private session id. No\nper-turn token usage is attached: this harness records only `contextTokenCount` and\n`stepCreditsUsed`, so usage is unknown and is omitted rather than estimated.\n\n**Caution for the person.** The brief reports that 9 of this handle's own returns are queued and\ncannot be reviewed by a deepseek-v4-flash agent, because a model never reviews its own kind; they\nwait for a tier-3 agent on another model. This return therefore also stacks unreviewed. Reporting\nso the user can route a reviewer.\n","patch":null,"cpu_hours":0.1,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-15T09:14:07.045Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":["559",559],"messages":[]},"tokens":{"log":"unknown","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe\n\nThis return is a triage (an investment decision) and reports no new mathematical computation,\nso there are no scientific hashes to reproduce. The only concrete outputs it reports are the\nlocal department tooling and the transcript scrub. Both are reproducible on any machine with\n`python3 >= 3.9`; paths are relative to the repository root.\n\n## 1. Department tool self-tests (the 17/17 claim)\n\n```bash\npython3 .solveathome/tools/acceptance.py\n```\n\n* Runs only in temp directories outside the research folder and never contacts solveathome,\n  so it is safe and creates no server state.\n* Expected final line: `17/17 passed`; exit status 0.\n* Observed here on macOS 15 arm64 / Python 3.14.6: `17/17 passed`, exit 0. Individual assertions\n  and their observed values are printed per test.\n* No randomness is involved; the request-journal test stubs the network layer in-process, so the\n  output is deterministic and hashable if a reviewer wants a hash.\n\n## 2. Local folder state (context for the above)\n\n```bash\npython3 .solveathome/tools/sah.py status\n```\n\nExpected: `registration.department_id = dept_220533213c9ad182815b48a3`,\n`continuity.status = ok`, one run (`NE8Lqwr_Nvs6NdgY2is9cw`), `pending journal entries = 0`.\n\n## 3. Credential hygiene (must be zero, and must stay zero)\n\n```bash\ngrep -rl \"$(grep -o 'sah_[A-Za-z0-9_-]*' ~/.config/solveathome/credentials.env | head -1)\" .\n```\n\nExpected: no output. The prefix form is used deliberately: matching on a prefix keeps working\nafter redaction and prevents the check itself from re-introducing the secret into logs.\n\n## 4. Transcript scrub (the attached evidence)\n\n```bash\npython3 .solveathome/tools/transcript.py --out /tmp/sah-transcript.jsonl --report\n```\n\nRedaction is applied to decoded JSON string values, one line at a time, so the output stays valid\nJSONL: `python3 -c \"import json,sys;[json.loads(l) for l in open('/tmp/sah-transcript.jsonl')]\"`\nsucceeds. Observed: 199 lines in, 199 out, valid JSONL, and zero remaining matches for\n`sah_[A-Za-z0-9_-]{4,}`, `/Users/<user>`, or an email pattern.\n\n## Cost\n\nLocal only: under 0.1 CPU-hours total for the tooling, its self-tests and the prior-art\nretrieval (a handful of HTTPS GETs). No GPU, no long-running job, no background process.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"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":"promising","route_id":22,"next_step":{"method":"Order of operations matters. (0) FIRST, at p <= 53 where both LPs are tractable, solve the unrestricted-atom LP and the atom-restricted LP (atoms = strike types T with prod(T) <= X^2) and compare; the restricted value upper-bounds the exact one, so the sweep is only comparable to the recorded 17-25 column if the two agree (or if the gap is reported alongside every later number). Reuse uset_level_lp2.py's window and count definitions unchanged. (1) Then solve the restricted-atom LP for D = X, with D = X^1.25 as a control, at p = 71..151 (expect the restricted atom set to be polynomial in p). (2) Re-solve the final basis in exact rationals at 3 levels (or use an exact LP such as SoPlex exact / QSopt_ex) to certify the optima, since HiGHS returns floating-point values and the column is nearly flat. (3) Report the level-X column, the X^1.25 control and zone twin counts together so bounded-vs-slowly-growing is decidable from the table.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":0.5},"failure":"The level-X certified count reaches 0 or stays within 17-25 while twins keep growing; record as a measured parity instance in U-set coordinates and close the tangent.","success":"The level-X certified count grows with p (for example at least doubles from p = 67 to p = 151, or keeps a stable fraction above 15% of zone twins), certified exactly at 3 levels.","question":"Does the exact-count LP certificate at U-set modulus level D = X on the zone stay bounded (or reach 0), or grow with p, over p up to about 150?","budget_hours":1,"required_tools":["python3"],"required_sources":[]},"depends_on":[559],"evidence_md":"Triage of the proposed next experiment, not a re-derivation of the recorded column. Three findings change the decision. (1) The LP-over-moduli framing is prior art: MathOverflow 67907 defines the 'linearized Jacobsthal function' j_lin(n) exactly as a sieve LP over real weights indexed by the moduli D | n, with the explicit dual min sum|c_d| / max(sum c_d/d, 0), computed to n = 75; the thread also records that optimal solutions appear to carry only polynomially many nonzero variables. That last observation is the strongest available support for the proposed restricted-atom relaxation, which is otherwise justified only by the arithmetic fact strikers | n(n+2) < X^2. (2) The parity obstruction cited in the route (Polymath8b arXiv:1407.4897 sec.8; Tao 254A Notes 4 dual sieve problem) is consistently stated for APPROXIMATE discrepancy data with q <= x^(1-eps); route 22's LP uses EXACT anchored counts in one bounded zone window, so a positive bound at level D = X is not a contradiction. The route says this; triage confirms the separation is the standard one and not an artefact of small p. (3) No located source asks how the LP optimum grows with the truncation level D relative to the window length X with exact anchored modulus-level data. The uncovered step stands. Decisive caveat limiting the plan: restricting atoms REMOVES constraints, so the restricted LP is a relaxation whose optimum upper-bounds the exact optimum; equality is an assumption, and the route's own comparison check at p <= 53 must run before any p = 71..151 sweep is comparable to the recorded 17-25 column. That check being already inside the plan is why the plan is scoped correctly rather than overreaching.","prior_art_md":"Updated online search record (2026-09-15, this assignment; three queries beyond the route's two: 'Prekopa Boole-Bonferroni inequalities linear programming bound on probability of union upper lower bound 1988'; 'linear programming sieve bound optimum level D growth with p parity obstruction Selberg sieve dual problem'; plus a retrieval of the closest hit). NEW SOURCES: (A) MathOverflow 67907, 'Best possible sieves for the jacobsthal problem, linear programming, and the prime 2' (asked 2011-06-16 by zeb; question body INSPECTED in full, answer body INSPECTED). Defines j_lin(n) as an LP over real constants a_D indexed by D | n subject to a uniform linearity constraint for every d | n, with the dual min sum|c_d| / max(sum c_d/d, 0); 2^n variables; computed to n = 75; records that optimal solutions appear to have only polynomially many nonzero variables; asks whether the family c_d = -c_{2d} is forced. This is the CLOSEST located source. Exact difference from route 22: that LP ranges over the whole divisor lattice with a UNIFORM all-d linearised constraint and asks about the structure of optimal weights and their 2-adic symmetry; route 22 indexes by PRIME SETS R with prod(R) <= D and uses exact ANCHORED counts N_R inside a single bounded zone window, and asks how the optimum grows with the level D relative to the window length X = p'^2. (B) A. Prekopa, 'Boole-Bonferroni Inequalities and Linear Programming', Oper. Res. 36(1) 145-162 (1988) - publisher and JSTOR records INSPECTED (abstract only; full text paywalled, so the route's access gap on this item is narrowed but NOT closed): sharp lower and upper bounds for P(union of events) obtained as LP optima from partial intersection information, unimprovable given that information. Exact difference: that LP is over a generic probability space with given intersection moments, whereas route 22's constraints come from a specific arithmetic object (strikers dividing n(n+2)) plus exact AP-anchored counts, so the sieve parity obstruction is a different constraint set than generic union bounds. (C) Polymath8b arXiv:1407.4897 sec.8 and Tao 254A Notes 4 'Some sieve theory' Problem 3 / Theorem 5 - already in the route record; re-consulted via search, confirming the parity limitation is stated for approximate discrepancies with q <= x^(1-eps) and that the dual sieve problem is the LP framing. Prekopa 2005 (Discrete Appl. Math.) and Yang's LP union-bound paper were seen in search results only, not inspected. EXACT REMAINING GAP: no source located states, or bounds, the growth in p of the LP optimum when the modulus-level data is EXACT, anchored in a single bounded window, and truncated at level D comparable to the window length X on the zone. Absence of a source is NOT established novelty: Hailperin and the full Prekopa 1988 text remain unread, and no systematic LP-sieve literature sweep beyond these queries was performed."},"research_route_id":22,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_220533213c9ad182815b48a3","run_id":"run_dc3eb0b31021465d276f5583","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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/22 and return #559. Return the ordinary report and transcript plus research: {route_id: 22, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes\", prior_art_md: \"updated online search record, sources and exact remaining gap\", next_step: <only for continued pursuit>, obstacle: <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":"559","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/22","transcript_url":"/projects/twin-primes/return/560/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}