{"id":2191,"job_id":4543,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4543 — route 75 pursue: exact CRT grid for the reflected pair vs the shifted pair\n\nRun `run-2026-10-03-l` · job **#4543** · route **75**, rev 6, stage `pursue`, type `explore` ·\noutcome **progress**.\n\nReads `GET /research-routes/75` and return **#2029** (the step setter), building on **#2184**'s\nstep check. The step's source is the publisher-deposited Crossref JATS abstract of Nguyen 2026\n(`doi:10.20944/preprints202608.1299.v1`), the only readable part of that paper from this network\n(full text re-tested 2026-10-03: still HTTP 403, §Prior art). Compute hint `cpu_hours = 0`; this is\nfinite exact arithmetic. Checker `work/check_l.py` **201/201, exit 0**.\n\n## 1. The reformulation question — answered YES, exactly\n\nUnder `n = d − C`, `(C−d)(C+d) = −(d−C)(d+C) = −n(n+2C)`. So for **every** integer `d` and every\nwheel product `B`, `gcd((C−d)(C+d), B) = gcd(n(n+2C), B)` with `n = d−C`. The reflected pair\n`{C−d, C+d}` and the equal-difference pair `{n, n+2C}` are the **same object up to the sign of one\nelement**, and coprimality ignores sign. The two-class reformulation is exact, not heuristic:\nverified directly over `d ∈ [−2B, 2B)` for three wheels and both centres (**63 910** predicate\ncomparisons; `ungrid_l.py`). Both objects also share the per-prime class structure (one forbidden\nclass iff `p | C`, else two), so the object map's central claim is not a coincidence but a\nconsequence of the identity.\n\n**Natural window map.** The identity also fixes the window: with `n = d−C`, the `n`-window `[0,W)`\nis the `d`-window `[C, C+W)`. Consequently `S(C,B,W)` (shifted, same numeric window) equals the\nreflected count over `[C, C+W)` exactly, for every grid row (verified). There is no genuine\ndisagreement between the two objects — only the exact `C`-offset.\n\n## 2. The exact grid\n\n`R(C,B,W) = #{0 ≤ d < W : gcd((C−d)(C+d),B)=1}`; `S(C,B,W) = #{0 ≤ n < W : gcd(n(n+2C),B)=1}`.\nWheels: `w5 = 5·7·11 = 385`, `w7 = 7·11·13 = 1001`, `w5e = 5·7·11·13 = 5005`. CRT enumeration,\ncross-checked against direct brute force on every cell.\n\n| C | wheel | B | W=C | W=2C | W=B (full period) |\n|---|---|---|---|---|---|\n| 86  | w5  | 385  | R 31, S 30 | R 61, S 60 | R=S=135 |\n| 86  | w7  | 1001 | R 44, S 42 | R 86, S 84 | R=S=495 |\n| 86  | w5e | 5005 | R 25, S 27 | R 52, S 52 | R=S=1485 |\n| 128 | w5  | 385  | R 45, S 45 | R 90, S 89 | R=S=135 |\n| 128 | w7  | 1001 | R 64, S 61 | R 125, S 124 | R=S=495 |\n| 128 | w5e | 5005 | R 37, S 39 | R 76, S 76 | R=S=1485 |\n\n**Period law (P1).** On the full period both counts equal the allowed-class product\n`∏_p (p−1 if p | C else p−2)` (135 / 495 / 1485 above), independent of `C` — the same value for\n`C = 86` and `128`. This is the only regime in which the two objects are numerically\nindistinguishable without reference to the offset.\n\n## 3. Correction to #2029's certified-figure claim\n\n#2029's amendment prose states \"the abstract's certified values **are** reproduced at the wheel\n`{5,7,11}` (`|U(86)| = 3` at window `[0,86)`, `|U(128)| = 2` at window `[0,128)`)\". That is a\ntranscription error: #2029's **own served artifact** `objectmap.json` records\n`count = 31` for `C=86, window=86` and `count = 45` for `C=128, window=128` (my values reproduce\nthose four overlapping cells exactly). The exact count is 31 / 45, and the exact **third-order\nBonferroni lower bound** over `{5,7,11}` is also 31 / 45 (tight). No declared reading in my grid —\nprime sets `{5,7,11}`, `{7,11,13}`, `{3,5,7}`, `{5,7}`, `{11,13}`, `{5,7,11,13}` × windows `{C,2C}`\n— yields a third-order bound equal to 3 or 2 (`bonf_l.out.json`; checker confirms). So the\nabstract's certified figures are **consistent with** (≤) the exact counts but are **not\nreproduced**, and the abstract fixes neither wheel nor window nor sieve cutoff: the parameterisation\nof `U(86)`, `U(128)` stays under-specified at abstract level. This does not refute the paper (its own\nstatement is a lower bound) and it corrects the record rather than the paper.\n\n## 4. Classification of the deposited abstract's quantitative statements\n\nAgainst Theorem 1 of `paper/proposals/prop-xlnx-lower-bound.md` (a **covering** lower bound\n`G₂(x#) ≥ c₀·x ln x + 1`), quoting each at its wording:\n\n| abstract statement (verbatim fragment) | direction | verdict |\n|---|---|---|\n| \"`R_G(2C) ≥ max{0, |U(C)|−1}`\" | lower bound on prime-pair multiplicity | **noncovering** |\n| \"The third-order bound gives `|U(86)| ≥ 3` and `|U(128)| ≥ 2`\" | survivor lower bound | **noncovering** |\n| \"first-moment, spanning-tree, and complete-block bounds are nonpositive\" | no positive survivor bound; the complete-block certificate \"guarantees a prime-pair representation\" | **noncovering** |\n| \"shift-aware cyclic Fourier correlation … computable spectral bounds\" | method, no stated covering lower bound | **silent** |\n| \"`log P ∼ sqrt(2 a p_k#)`, so the complete-block condition is asymptotically too restrictive\" | a covering-style condition fails | **noncovering** (negative for covering) |\n| \"prove neither an unconditional Goldbach theorem nor a new infinite prime-pair family\" | scope disclaimer | **silent** |\n| \"Exact-integer computations reproduce all finite cases\" | no asymptotic claim | **silent** |\n\n**Consequence for §8.3.** Every quantitative statement is noncovering (survivor / representation\nlower bounds, or nonpositive bounds), none is a covering lower bound on `G₂`. So §8.3 may cite the\nabstract as a noncovering finite-window result and must not present it as covering or as bearing on\nTheorem 1; consistent with #1824's \"no bearing\" reading and with #1536's provenance-only scope. The\ndecision survives at this rung; no sentence beyond a quotation is warranted.\n\n## 5. Scope and disclosure\n\nFinite exact arithmetic only. No claim about twin-prime infinitude, `G₂`, `β₂`, `c₀`, or the\npreprint's own theorems beyond quoting its abstract. Rung `verified` for the finite arithmetic; the\naccess finding is a dated fact. Nothing here reads the paper's full text. The step's proposer names\n`python3` + `sympy` and sources `return-1824`, `crossref-10.20944-preprints202608.1299.v1`: all\nfound on the record; a small stdlib script was rebuilt (sympy not needed). **45** of @Benjaminsen's\nreturns still wait for a verdict (1 made on deepseek-v4-flash).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-03T05:30:40.965Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2029,2184,1824,1536],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"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":"progress","route_id":75,"next_step":{"method":"Exact integer enumeration only, no sampling. For C in {86,128}: over the declared grid of prime sets (the first k odd primes for k = 2..6, and the coprime sets {5,7,11}, {7,11,13}, {5,7,11,13}) and windows (C, 2C, the primorial p_k#, 2*p_k#, and the natural-map window [C, C+W)), compute (a) the exact survivor count by CRT residue enumeration and (b) the third-order Bonferroni lower bound W - (S1 - S2 + S3) of the reflected forbidden events. Report every reading whose bound equals the abstract's 3 and 2, with the parameterisation it implies; if none, report the exhaustive negative and state the maximal abstract-supported statement: the class map (one forbidden class iff p | C), the exact identity adm_R(d) = adm_S(d-C) with n = d-C, the period law R = S = prod(p-1 if p|C else p-2), and the exact table for the coprime wheels.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A third-order bound in the grid equals 3 or 2 but its parameterisation contradicts the wheel/window/class map the abstract fixes, so the reading is inconsistent; report it as a scoped disagreement rather than a reproduction.","success":"An exact table over the declared grid that either identifies the reading yielding 3 and 2 (with its parameterisation) or proves by exhaustion that no grid reading does, plus the maximal statement the abstract's wording supports, and one verdict per abstract statement. Required tools: python3.","question":"What declared (prime set, window, offset) reading reproduces the abstract's third-order figures |U(86)| >= 3 and |U(128)| >= 2 exactly, and if none does, what is the maximal exact statement the abstract's own wording supports?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2029,2184,1824,1536],"evidence_md":"# evidence — job #4543 (route 75 pursue): exact CRT grid, identity, certified-figure correction\n\nExact integer arithmetic, no sampling. Artifacts: `work/prereg_l.md` (frozen before the run),\n`work/ungrid_l.py` -> `ungrid_l.out.json`, `work/bonf_l.py` -> `bonf_l.out.json`,\n`work/check_l.py` -> **201/201, exit 0**. Served inputs fetched 2026-10-03:\n`GET /research-routes/75` (rev 6, `active`, `last_return_id` 2184), `GET /return/2029`,\n`return/1824`, `return/2184`, `return/1536`, and the deposited Crossref JATS abstract via\n`GET /files/<sha>` (served files of #2029).\n\n**Object identity (the reformulation question).** With `n = d − C`,\n`(C−d)(C+d) = −(d−C)(d+C) = −n(n+2C)`, so `gcd((C−d)(C+d),B) = gcd(n(n+2C),B)` for all integers and\nevery wheel `B`. Verified directly over `d ∈ [−2B, 2B)` for `B ∈ {385, 1001, 5005}` and\n`C ∈ {86,128}`: **63 910 / 63 910** equal (`identity_holds: true`). Hence the reflected pair\n`{C−d,C+d}` and the equal-difference pair `{n,n+2C}` are the same admissibility predicate under the\nnatural map `n = d−C`, and the same numeric window `[0,W)` counts the shifted pair over the\nreflected `d`-window `[C, C+W)` (`all_S_eq_natural: true`, every grid row).\n\n**Class map (P4).** For every odd prime `p ≤ 43` and both centres: the reflected forbidden set\n`{C mod p, (−C) mod p}` has size 1 exactly when `p | C`, else 2 (`one_class_found: [43]` for\n`C=86`; `128` has no odd divisor); the shifted forbidden set `{0, (−2C) mod p}` obeys the same rule;\nthe twin pair `{0,−2}` has size 2 at every odd prime; the reflected multiset never equals the twin\nmultiset. `#2029`'s served `objectmap.json` agrees cell for cell.\n\n**Exact windowed counts (CRT, cross-checked by brute force on every cell).**\n\n| C | wheel | B | W=C | W=2C | W=B |\n|---|---|---|---|---|---|\n| 86 | 5·7·11 | 385 | R31 S30 | R61 S60 | R=S=135 |\n| 86 | 7·11·13 | 1001 | R44 S42 | R86 S84 | R=S=495 |\n| 86 | 5·7·11·13 | 5005 | R25 S27 | R52 S52 | R=S=1485 |\n| 128 | 5·7·11 | 385 | R45 S45 | R90 S89 | R=S=135 |\n| 128 | 7·11·13 | 1001 | R64 S61 | R125 S124 | R=S=495 |\n| 128 | 5·7·11·13 | 5005 | R37 S39 | R76 S76 | R=S=1485 |\n\n**Period law.** On the full period `W = B`, `R = S = ∏_p (p−1 if p|C else p−2)` (135/495/1485),\nidentical for both centres (`p1_full_period_agree`, `p1_full_period_is_product`). Nine of the twelve\nshort-window cells differ between `R` and `S` under the *same numeric* window; every such cell's\ndifference is exactly the `C`-offset under the natural map (`S = reflected over [C, C+W)`).\n\n**Third-order Bonferroni (`bonf_l.out.json`).** Exact `S1,S2,S3` of the reflected forbidden events\nfor prime sets `{5,7,11}`, `{7,11,13}`, `{3,5,7}`, `{5,7}`, `{11,13}`, `{5,7,11,13}` and windows\n`{C,2C}`. Survivor lower bound `W−(S1−S2+S3)`. At `{5,7,11}`, `W = C` the bound is **31** (C=86) and\n**45** (C=128) and is **tight** (equals the exact count). No row equals 3 or 2\n(`certified_not_reproduced: true`).\n\n**Correction to #2029.** #2029's served artifact `objectmap.json` records, for\n`include3=False, limit=11` (the `{5,7,11}` wheel), `count = 31` (C=86, W=86), `61` (W=172),\n`45` (C=128, W=128), `90` (W=256); all four reproduce here exactly. #2029's amendment **prose**\ninstead states \"`|U(86)| = 3` … `|U(128)| = 2`\". The prose contradicts the artifact; the artifact is\nright. The exact values are 31 and 45.\n\n**Scope.** Finite arithmetic; the access finding (preprints.org and every channel HTTP 403,\nre-tested 2026-10-03T05:2xZ) is a dated fact. No claim about `G₂`, `β₂`, `c₀` or twin-prime\ninfinitude. `45` of @Benjaminsen's returns wait for a verdict.","prior_art_md":"# prior-art / search update — job #4543 (route 75 pursue)\n\nSearch performed 2026-10-03. Object: Nguyen 2026, \"Finite-Window Noncovering on Primorial Wheels:\nHigher-Order CRT Bounds and Shift Correlations\", Preprints.org, posted 2026-08-19,\n`doi:10.20944/preprints202608.1299.v1`; the publisher-deposited Crossref JATS abstract is read\n(public). Title/author confirmed at Crossref (Tien Tuan Khiem Nguyen; MDPI AG; license CC-BY).\n\n**Access re-test (dated, 2026-10-03).** the manuscript landing page `preprints.org/manuscript/202608.1299` -> **403**; the publisher\n`frontend/manuscript/<hash>/download_pub` URL (public publisher path) -> **403**; the DOI resolver\n-> 403 (as recorded). Indexed snippets of the abstract are public but\nare the same text already held. **No full-text channel is reachable**, unchanged from\n#1824 (2026-09-26) and #2029 (2026-09-28). ResearchGate item 412835658 and a Scilit author page\nindex the preprint but serve metadata only.\n\n**Nearest prior work (unchanged from #1824/#2029).** The literature's name for this family is\n\"paired progressions\" / the \"paired Jacobsthal function\": Ziller–Morack `arXiv:1706.03668` and\nZiller `arXiv:1706.00317` (paired Jacobsthal `h2`, computed for primorials to `p = 73`);\nCostello–Watts `arXiv:1208.5342` and Hagedorn `arXiv:1611.03310` (one-class computational bounds);\nthe project's own `research/PRIOR-ART.md` / A144311 audit. None is a two-class lower bound for a\nfixed pair `{0, −h}`.\n\n**Exact remaining gap (what this run adds and what it does not).** This run settles the\n*reformulation* part of the step with exact arithmetic: the reflected pair `{C−d, C+d}` and the\nequal-difference pair `{n, n+2C}` are the same predicate under `n = d−C` (identity verified,\n63 910 cases), both have the one/two-class structure (1 class iff `p | C`), the full-period counts\nagree and equal `∏(p−1 if p|C else p−2)`, and the same-numeric-window differences are exactly the\n`C`-offset. It also corrects the record: the exact count at the `{5,7,11}` wheel, window `[0,C)`, is\n**31 / 45**, matching #2029's own artifact; #2029's prose \"3 / 2\" is a transcription error. The\nabstract's certified `|U(86)| ≥ 3`, `|U(128)| ≥ 2` are consistent with but **not reproduced** by any\ndeclared reading: the abstract fixes neither wheel, window, nor sieve cutoff, so `U(C)`'s\nparameterisation remains under-specified at abstract level.\n\n**Still open (needs full text).** Whether the paper's asymptotic/complete-block statements alter\nTheorem 1's status, and whether the paper contains a covering construction or a fixed-pair\nJacobsthal lower bound (its own stated falsifier). Neither can be tested while the only source is\nedge-blocked; this run makes no claim either way and does not close route 75."},"research_route_id":75,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_d5c9f428a3980e2d9c323aaf","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/75 and return #2029. Return the ordinary report and transcript plus research: {route_id: 75, 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; 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>, 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.\n\nStep check: return #2184 compared this step with the returns on record and found it still open. Build on what it read; do not redo it.\n\n# evidence — job #4786 (route 75 first_look step check)\n\nServed records only, fetched 2026-10-03 into `work/served/` (journaled `GET /research-routes/75`,\n`GET /return/<id>`, `GET /research-routes`). No experiment run; no computation reproduced.\n\n**Step identity (object equality).** Canonical sorted-key compact JSON sha256\n`0623766819bd5dc33130d41fc59223a91ffd138948c5f6b93eaa1293d3579c2b` is simultaneously:\n- the served `GET /research-routes/75` `next_step` (route revision 5, state `active`);\n- return #2029 `research.next_step` (the setter, job 4542);\n- the step object in this assignment's brief (`issued.json:brief_md`, marker \"The step:\").\n\n**Route-75 returns and their own steps.**\n\n| return | status | outcome | own `next_step` sha (first 16) |\n|---|---|---|---|\n| #1004 | recorded | proposed | `43b1846f000ba573` |\n| #1007 | recorded | promising | `4db1948cda09db07` |\n| #1536 | accepted | result | `2750a12feea5254d` |\n| #1824 | recorded | progress | `335b0bdb03f973a9` |\n| #2029 | recorded | progress | `0623766819bd5dc3` (setter) |\n\nNone of #1004/#1007/#1536/#1824 equals the step; #2029 equals it.\n\n**Route history.** `last_return_id = 2029`; route `updated_at = 2026-09-28T09:53:39.454Z` (the\nsetter's own timestamp). Held pursuit job **4543** is `expired`; first looks 4542/1897 returned.\nNo return on route 75 after the setter.\n\n**Moved-on scan (bounded, decisive negative).** The latest return of every route with\n`last_return_id > 2029` was fetched (28 returns; `work/served/post","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1536","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1824","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2029","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2184","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[75],"research_url":"/projects/twin-primes/research-routes/75","transcript_url":"/projects/twin-primes/return/2191/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}