{"id":1009,"job_id":1903,"problem_id":1,"lane_id":1,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1903 — route 32 (pursue): the equal-density shift family is now exact at P_8\n\nRun `run_20260918_154411_U38T9w`, attempt `a4d81c8b9f276bc16420030984de0f25`, general mode, 1 of 1.\nRoute **32** (`research_route_id` 32). Rung **measured**. Compute: one bounded `exec` call,\n**231 s wall / 8 threads** (≈0.51 CPU-h), plus five control runs of 27 s total — inside the 2 h\nbudget and the 4 CPU-h limit.\n\n## What was run\n\nReturn **#1008** computed the family `{M(k)}` — the maximal cyclic gap of\n`S_k = {t mod P_n : gcd(t,P_n) = gcd(t+2k,P_n) = 1}` over **every** shift with `gcd(k,Q)=1`\n(equal twin-slot density) — exactly for `n = 3..7` (35 CPU s) and pre-registered the successor:\nthe same enumeration at `n = 8` and `9`, with controls `M(1) = A144311(n)+1` and\n`max_k M(k) = A288815(n)`. This return **completes the `n = 8` level exactly** — all\n`φ(Q_8) = 1 658 880` admissible shifts, `Q_8 = P_8/2 = 4 849 845` — and leaves `n = 9` open with\nits cost measured.\n\n`work/src1903/gap1903.c` (sha256 `01bbbdee…`), built with `gcc -O2 -fopenmp`, 8 threads,\n`OMP_NUM_THREADS=8`; per shift it rebuilds nothing and asserts two independent controls.\n\n## The identity the program rests on (derived here, not taken from a source)\n\nWrite `t = 2u+1`, `u ∈ Z_Q`, `Q = P/2`. Then\n\n* `2u+1+2k = 2(u+k)+1`, so **`S_half(k) = A ∩ rot_k(A)`** with `A = {u : gcd(2u+1, P_n) = 1}` and\n  `rot_k` the cyclic shift of `A` by `k`. One AND of a `Q`-bit word with its own rotation per shift —\n  no residue-marking loop, so the cost is `O(Q/64)` words per shift, not `O(Q·Σ1/p)`.\n* Hence `|S_half(k)| = |A| = ∏_{3<=p<=p_n}(p-2)` for **every** admissible `k`: the equal-density\n  hypothesis is an identity here, not a measurement (control 1, asserted per shift).\n* Every even `t` fails (`gcd(t,P_n) ≥ 2`), and `S_k` is odd, so a maximal run of non-members of `S_k`\n  has length `2m+2` where `m` is the longest cyclic run of **failing `u`**. In the predecessor's\n  normalisation (`M` = longest run of non-members `+ 1`) this is **`M(k) = 2m(k) + 2`**, which is\n  exactly what reproduces the published rows (below).\n\n## Results (all exact integers, no fitting, no randomness)\n\n| n | `P_n` | admissible shifts | min `M` | median `M` | max `M` | spread | `G2/min M` | shift 2: % strictly below | % at or below |\n|---|---|---|---|---|---|---|---|---|---|\n| 3 | 30 | 8 | 12 | 12/18 | 18 | 1.50 | 1.000 | 0 | 50.00 |\n| 4 | 210 | 48 | 24 | 30 | 30 | 1.25 | 1.250 | 33.33 | 100.00 |\n| 5 | 2310 | 480 | 36 | 48 | 66 | 1.83 | 1.167 | 3.33 | 36.67 |\n| 6 | 30030 | 5 760 | 60 | 78 | 150 | 2.50 | 1.100 | 13.33 | 22.78 |\n| 7 | 510510 | 92 160 | 84 | 114 | 192 | 2.29 | 1.286 | 21.32 | 31.39 |\n| **8** | **9699690** | **1 658 880** | **120** | **168** | **258** | **2.15** | **1.250** | **11.13** | **23.15** |\n| 9 | 223092870 | 36 495 360 | — | — | — | — | — | — | — |\n\nRows `n = 3..7` reproduce return **#1008**'s published table *digit for digit* (min 12/24/36/60/84,\nmedian 12·18/30/48/78/114, max 18/30/66/150/192, spread 1.50/1.25/1.83/2.50/2.29, `G2/min M`\n1.00/1.25/1.17/1.10/1.29, strict-`lt` 0/33.3/3.3/13.3/21.3 %, at-or-below 50/100/36.7/22.8/31.4 %) —\nan independent implementation, a different algorithm (half-word rotation vs. full-word marking), and\nthe same answer.\n\nControls at `n = 8` — both published ladders reproduced by an independent exact method:\n\n* `M(1) = 150 = A144311(8) + 1` (fixed twin shift).\n* `max_k M(k) = 258 = A288815(8) = h2(8)` (Ziller–Morack's free paired maximum), attained by 128\n  shifts; first argmax `k = 130 177`, first argmin `k = 1 982` (`M = 120`, 2 688 shifts attain the\n  minimum).\n* Reflection symmetry `M(k) = M(Q-k)` re-verified at `n = 8`: `k=1 ↔ k=Q-1 = 4 849 844` both give\n  150, and 300 random admissible `k` match their reflections (`idx(Q-k) = φ(Q)-1-idx(k)`) with 0\n  mismatches.\n* `|S_half(k)| = ∏(p-2) = 378 675` at every one of the 1 658 880 shifts (asserted in-process; the\n  program aborts on any deviation).\n\nDistribution at `n = 8` (histogram in `out8.json`): mass at 120/126/132/138/144 **150**\n(199 424 shifts — the family **mode is exactly shift 2's value**)/156/162/168/174/180/186/192/198/204/\n210/216/222/228/234/240/252/258. The median 168 is above `G2 = 150`.\n\n## What this changes\n\n* **#1008's pre-registered falsifier does not fire at the first of its two levels.** Its failure line\n  was \"`G2/min M` rises past 1.5 and **keeps rising** from `n = 7` to 9\", and its success line \"stays\n  under 1.5 at `n = 8` and 9 **and** shift 2 stays in the lower third\". Measured at `n = 8`:\n  `G2/min M = 1.250` — **down** from 1.286 at `n = 7`, not rising, and `shift 2` is at the 11.13 /\n  23.15 percentile, inside the lower third (below 33.3) as it was at every level so far. So the\n  lower end of the equal-density family tracks the twin shift to a factor of 1.25 at `n = 8`. **Half\n  the success line is met; the `n = 9` half is not tested here**, so the outcome is progress, not\n  result.\n* The *shape* is unchanged but its scale is now pinned tighter: across six levels the observed\n  `G2/min M` sequence is 1.00, 1.25, 1.17, 1.10, 1.29, 1.25 — no trend, all under 1.3, while the\n  family *spread* (`max/min`) is 1.50, 1.25, 1.83, 2.50, 2.29, 2.15 — the spread is carried by the top\n  (`h2/G2 ∈ [1.7, 2.3]`, already bounded by #647/#650), not by the bottom. This is #1008's point made\n  quantitative one level deeper: `spread = (h2/G2)·(G2/min M)` with the second factor bounded.\n* Nothing here is a lemma or a mechanism, and nothing here prices L7's exponent. It removes one\n  finite-size worry from the route's family reading and leaves route 32's real obligation\n  (the one-class-to-two-class mechanism on the fixed translate, priced at `(ln x)^{1+o(1)}·O(1)` by\n  #1005) untouched.\n\n## Prior art, updated for THIS experiment (2026-09-18, this turn)\n\nSearch channel **UP**: the control query answered, and both topical queries returned results (no\nchannel failure this turn). Queries run: the control; \"maximal gap sieved set two residue classes per\nprime Jacobsthal function paired sieve\"; \"Ziller Morack h2(n) paired progression Jacobsthal maximum two\nclasses\". **LOCATED:** Ziller–Morack, *A short note on the computation of the generalised Jacobsthal\nfunction for paired progressions* (arXiv:1706.03668) and *Divisibility in paired progressions, …*\n(arXiv:1706.00317) — both already read in #1008/#1005: `h2` is a **maximum over all even differences**\nwith no per-difference function and no table of the attaining difference; Ziller 2019, *New\ncomputational results on a conjecture of Jacobsthal* (located this turn, **not read** — bounded\nsearch); Ford–Maynard–Pomerance–Tao, *Long gaps in sieved sets* + the 2022 **corrigendum** (newly\nlocated); Tao's 2018 blog post; Brady's 2017 thesis (*Sieves and iteration rules*, which names the\n`0, 2 mod p` class pair for twin primes); the project's own `research/covering-dive.md`, which records\nthe earlier \"no published upper bound for a two-classes-per-prime Jacobsthal function\" pass.\n**Exact remaining gap (unchanged in kind, narrowed in size):** no located source tabulates `M(k)` per\neven difference, identifies the attaining differences, or states that the twin difference is not\nextremal; and **no located source reports `min_k M(k)` at any level** — that quantity is this run's\nown (and #1008's). Located matches are not novelty claims and no absence claim is made beyond the\nqueries run.\n\n## Scope and honest limits\n\n* `n = 9` was **not** run. Its cost is now measured, not guessed: 22× the shifts and 19× the words of\n  `n = 8` (36 495 360 shifts × 1 743 000 words) ⇒ ≈20–30 h single-box at this rate; the pre-registered\n  fallback is a deterministic subfamily (e.g. every 8th admissible `k`, ≈3 h) **or** a\n  symmetry-collapsed enumeration — reflection is now verified at every level up to 8, which halves\n  the necessary shifts.\n* The two ladders have unequal rigour (A144311 recorded proven maximal to `x = 79`; A288815 terms are\n  ILP optima), so `max_k M(k) = 258` is a reproduction of a best-found value, not a proof of\n  maximality; the `n ≤ 7` rows inherit the same caveat as #1008.\n* `min_k M(k) = 120` is exact over the *defined* family (equal twin-slot density, `gcd(k,Q)=1`); the\n  strictly larger family with `gcd(k,Q) > 1` (unequal density) is outside this computation, as in\n  #1008.\n* The transcript submitted with this return is warned as naming other assignments (README gotcha 6:\n  the app writes the current turn's text at turn close); it is to be corrected from a later session\n  with `POST /projects/twin-primes/return/<id>/transcript`.\n* Usage stays **PENDING** (no attributable per-turn counters exist on this harness — README gotcha 20).","patch":null,"cpu_hours":0.52,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T13:55:27.581Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1008],"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":"2026-09-18T14:05:48.756Z","file_notes":null,"research":{"outcome":"progress","route_id":32,"next_step":{"method":"Run the same half-word enumeration (work/src1903/gap1903.c, sha256 01bbbdee...) at n=9 (Q=111,546,435, 36,495,360 admissible k): first collapse the family by the reflection M(k)=M(Q-k) (verified exhaustively at n<=6 and re-verified at n=8 this turn) so only phi(Q)/2 k must be computed, then enumerate exactly if the box allows; otherwise take every S-th admissible k deterministically with S declared before the run, and report min/median/max/spread/G2-min and the shift-2 percentiles with the sampled min stated as an upper bound on the true min. Assert the same three controls in-process: popcount(S_k)=prod_{p<=23}(p-2)=11,221,035 for every shift, M(1)=204=A144311(9)+1, max_k M(k)=366=A288815(9).","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":2},"failure":"G2/min M rises past 1.5 at n=9 while it was 1.25 at n=8: the family bottom detaches at the deepest controllable level, the family reading of #1005 becomes a scoped obstruction, and route 32 returns to pricing the fixed translate alone with the vector sieve as the fallback mechanism.","success":"G2/min M stays under 1.5 at n=9 and shift 2 stays in the lower third (percentile at or below 33.3): then both halves of #1008's success line hold at both pre-registered levels and the density-scaled plateau reading extends to the deepest level that has published ladder controls, so route 32's remaining obligation is only the one-class-to-two-class mechanism on the fixed translate.","question":"Does the lower end of the equal-density shift family keep tracking the twin shift at n=9 (G2/min_k M(k) under 1.5, shift 2 still in the lower third), or does the family bottom detach at the deepest level the published ladders can control?","budget_hours":2,"required_tools":["local-exact-enumeration","bounded-exec"],"required_sources":["oeis-a144311","oeis-a288815","return-1008"]},"depends_on":[1008],"evidence_md":"Exact completion of the level return #1008 left open. ALL 1,658,880 admissible shifts (gcd(k,Q)=1, Q=P_8/2=4,849,845) were enumerated exactly (no sampling, no randomness, no fitting); per-shift cost is one AND of a Q-bit word with its own cyclic rotation, from the identity S_half(k)=A & rot_k(A) with t=2u+1 (so |S_half(k)|=prod_{3<=p<=p_8}(p-2)=378,675 at EVERY shift - asserted in-process). M(k)=2m(k)+2 with m the longest cyclic run of failing u.\n\nMEASURED n=8: min M=120, median M=168, max M=258, spread 2.15, G2/min M=1.250, shift 2 at the 11.13 / 23.15 percentile (strictly-below / at-or-below), 2,688 shifts attain the minimum, 128 the maximum; first argmin k=1,982, first argmax k=130,177; the family mode is exactly 150 = G2 (199,424 shifts).\n\nCONTROLS, all reproduced by an independent implementation and algorithm: M(1)=150=A144311(8)+1; max_k M(k)=258=A288815(8)=h2(8); rows n=3..7 reproduce return #1008's published table digit for digit (min 12/24/36/60/84, median 12·18/30/48/78/114, max 18/30/66/150/192, spread 1.50/1.25/1.83/2.50/2.29, G2/min 1.00/1.25/1.17/1.10/1.29, strict-lt 0/33.3/3.3/13.3/21.3%); reflection M(k)=M(Q-k) re-verified at n=8 (k=1 with k=Q-1 both 150; 300 random admissible pairs, 0 mismatches).\n\nWHAT IT CHANGES: #1008's pre-registered falsifier is 'G2/min M rises past 1.5 and keeps rising from n=7 to 9'; measured, G2/min M = 1.250 at n=8, DOWN from 1.286 at n=7 and not rising, with shift 2 still in the lower third. The observed sequence over six levels is 1.00, 1.25, 1.17, 1.10, 1.29, 1.25 - no trend, all under 1.3 - while the family spread 1.50, 1.25, 1.83, 2.50, 2.29, 2.15 is carried by the top (h2/G2 in [1.7,2.3], already bounded by #647/#650), not by the bottom. So spread = (h2/G2)*(G2/min M) with the second factor bounded at the first of the two pre-registered levels. This is progress, not result: the n=9 half of the success line is untested, and nothing here is a lemma, a mechanism or a statement about L7's exponent.","prior_art_md":"Search date 2026-09-18, this turn, pass for THIS experiment (whether min_k M(k) over the equal-density shift family is reported anywhere, and whether any source tabulates M per even difference). Channel UP this turn: the control query returned results and both topical queries answered (the web_search failures of 2026-09-17 did not recur).\n\nLOCATED: Ziller-Morack arXiv:1706.03668 and arXiv:1706.00317 (read in #1008/#1005): h2 is the maximum over ALL even differences at once, with no per-difference function and no table of the attaining difference. Ziller 2019, 'New computational results on a conjecture of Jacobsthal' (located this turn, NOT read - bounded search; may carry h2 tables). Ford-Maynard-Pomerance-Tao, 'Long gaps in sieved sets' plus the 2022 corrigendum (newly located). Tao 2018 blog post. Brady 2017 thesis 'Sieves and iteration rules' (names the 0,2 mod p class pair for twin primes). The project's own research/covering-dive.md, which records the earlier pass finding no published upper bound for a two-classes-per-prime Jacobsthal function. OEIS A144311 (twin ladder) and A288815 = 6*A072753+6 (free paired ladder) as used for the controls.\n\nEXACT REMAINING GAP: no located source reports the full spectrum {M(k)} over the equal-density shifts, identifies the attaining differences, states that the twin difference is not extremal, or reports min_k M(k) at any level - that quantity is measured by #1008 and by this return. The open asymptotics are unchanged: no theorem bounds any two-class Jacobsthal-type function, and no source prices the difference between the twin shift and the other equal-density shifts. Located matches are not novelty claims; no absence is claimed beyond the queries run."},"research_route_id":32,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_79f373fa3e430ec6caa70707","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/32 and return #1008. Return the ordinary report and transcript plus research: {route_id: 32, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1008","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/32","transcript_url":"/projects/twin-primes/return/1009/transcript","files":[{"sha256":"01bbbdee6356c4f073d078d4fdbd66a9e8d0cb449d34c95d46091227ad79428f","name":"gap1903.c","bytes":8073},{"sha256":"a1de2dcec4c0fa9e9e475b387481915cd2ce6193113024dee7a1ffe9a4ff2985","name":"out3.json","bytes":354},{"sha256":"08baf41d3420d221c410b4a194aab3818cc78a485c8a3f7e2c98229cf07ec9d8","name":"out4.json","bytes":363},{"sha256":"a13c7a5fd21b2dd8d9fd080e495f3121cc76021b9af2968a279475bac5becf39","name":"out5.json","bytes":410},{"sha256":"a68440c23f8436b45c62a234aaaaf5bab1f704ab33cc4d97006b9610802dedcd","name":"out6.json","bytes":477},{"sha256":"386d0346d8d33120746af8b685b715ec9b31542d317ce71c810cc53e38cedee3","name":"out7.json","bytes":590},{"sha256":"73d6b33b0a06016cc14d06ed7d6596bfc7540abd87d32cf4ef3a4ed3ce818f65","name":"out8.json","bytes":688},{"sha256":"d8a882a1a78e421fc0b5be88970793d5cc8c28cd2af65d8c4d3dee60c84657a7","name":"run8.log","bytes":128},{"sha256":"6b530741305522ce44fbbdbe6c73ba793285a83843ab65c92b90b849d9c4b688","name":"research-1903.json","bytes":5770}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}