{"id":4804,"job_id":9659,"problem_id":1,"lane_id":32,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Route 332 first look: the density-matched same-rung null is robust at 17# — the twin offset is below median under every checked normalisation\n\nJob #9659 (type `explore`, stage `first_look`, lane `dir-558`, route **332**, general mode).\nPredecessor: return **#4782** (the route's origin),\nwhich proposed this route from the recorded ledgers of **#4777** and **#4746**.\n\n**Outcome: `promising`.** The route's premise — that the twin offset's gap is *below* the\ndensity-matched same-rung median once its small slot count is accounted for — is confirmed at 17#\nby a new, normalisation-free control computed here, and the pre-registered 29# and 31# experiment is\njustified with two corrections: the budget in the route record is ~6–90× too low, and the design\nshould carry the quantile-convention split this look measured.\n\n**Rung of this return: measured (analysis of recorded bytes).** No new wheel scan was run. Every\nnumber below is computed from the published exhaustive 17# census `profile_17.csv`\n(sha256 `b7db86859518677d6541187362a39dc26c92c71f331cdd87a329388c93d1f897`, return #4777,\n255255 rows `d,g2,n`) and from the recorded rates in #4777 §6. The recorded chain was re-verified\nfrom served artifacts by\nhash (#4782's `check_ax.out`: 60 PASS lines, 0 FAIL — read, not re-run; eight #4782 artifacts and\nfive #4777 artifacts, plus the two return records, were fetched by content hash and the server\nverified each artifact's `X-Content-SHA256`).\n\n## 1. What was checked and what is new\n\n### 1a. Reproduction (all exact)\n\nFrom `profile_17.csv` (W = 510510, N = 255255 even offsets): `z_2 = G_2·|S_2| / W = 108·22275/510510\n= 4.712346`; inclusive z-quantile of d=2 = 0.301357 (recorded 0.3014 ✓); raw quantile 0.719120\n(recorded ✓); z median 5.235941; `z_2/z_median = 0.9000` (recorded ✓). The CSV columns also\nreproduce #4782's regression summary (slope/intercept/R²/residual sd — §1d) and the dispersion\nclaim `1−(CV_z/CV_G)² = 0.82`.\n\n### 1b. New control: the twin offset within its own density class (normalisation-free)\n\nThe route's z-quantile compares across density classes using `z_d = G_d·|S_d| / W`; its stated\nweakest assumption is that this normalisation is right. There is a sharper control that needs **no\nnormalisation at all**: rank d=2 only against offsets of its **own slot count** |S_d| = 22275 (the\ntwin density class — 92160 offsets, 36.1% of the profile, all with the same expected mean gap\nW/|S_d| = 22.91, differing only in structure). Measured:\n\n- twin density class: 92160 offsets (46080 with d ≡ 2 mod 4, 46080 with d ≡ 0 mod 4);\n  G ranges **84–192**, class **median G = 114**, 128 offsets attain the class maximum 192 (the\n  recorded h2-maximisers, confirmed all inside this class);\n- **the twin gap G_2 = 108 sits at quantile 0.3139 (inclusive) / 0.2132 (strict) within its own\n  density class** — below the class median under the raw gap, with no normalisation anywhere.\n\nThis is the decisive robustness fact for the route: the \"below median\" reading is not an artefact\nof the z normalisation, because it holds for the raw gap inside the class the normalisation was\nmeant to match. It also sharpens the route's contribution statement: the manuscript's density\nargument and #4782's z-quantile and this within-class quantile are three increasingly pure\nversions of the same control, and all three agree at 17#.\n\n### 1c. Quantile-convention decomposition (a counting defect to fix in the design)\n\nThe recorded z-quantile is inclusive (`#{z_d ≤ z_2} / N`). Ties at exactly z_2 are material:\n9280 offsets inside the twin density class and 7712 outside it have `G_d·|S_d| = 108·22275`\n(16,992 ties = 6.7% of the profile), so the inclusive 0.3014 mixes the tie mass with the strictly-\nbelow mass (profile strict z-quantile 0.2348; within-class strict 0.2132). The recorded\n0.14–0.30 band across rungs is therefore convention-dependent at the ±0.05–0.10 level. The route's\nfalsifier (z-quantile > 0.5) is unaffected, but the pre-registered 29# and 31# output should report\ninclusive and strict side by side (one line in the estimator), and the expected-evidence band\n[0.10, 0.40] should be understood to include both conventions.\n\n### 1d. Residual-ranking robustness\n\nRe-running #4782's regression on the census: z on ln|S_d| gives slope 0.5277, intercept −0.2831,\nR² = 0.0695, residual sd 0.6846 (recorded: 0.528 / −0.283 / 0.0695 / 0.685 — all four reproduce).\nRanking all 255255 offsets by the residual instead of z moves the twin's quantile only to 0.3605\n(inclusive) / 0.3242 (strict): the density-matched reading survives the route's own alternative\nnormalisation. Under every checked ranking — raw G (0.7191), z (0.3014 incl / 0.2348 strict),\nresidual (0.3605 / 0.3242), within-class raw (0.3139 / 0.2132) — the twin sits in the 0.21–0.36\nquantile band, below the median under every normalisation except the raw cross-class rank, which\nis exactly the robustness the route's weakest-assumption clause asked for.\n\n### 1e. The multiplicative orbit does not preserve the gap (small negative, worth recording)\n\nThe twin density class splits into d ≡ 2 (mod 4) and d ≡ 0 (mod 4) halves (46080 each). For\nd = 2u with gcd(u, W) = 1 the set identity S_{2u} = u·S_2 holds (multiplication by a unit), but\nthe maximum cyclic gap is **not** invariant under that set isomorphism: only 4640 of the 46080\nd ≡ 2 (mod 4) class members attain G = 108 (the same count on the d ≡ 0 mod 4 side; a 5# example\nshows the mechanism: S_{14} = 7·S_2 as sets, yet G_14 = 18 ≠ G_2 = 12). Consequence: the twin\nclass is internally heterogeneous (G ∈ [84, 192]), so the within-class control of §1b is a real\ncomparison, not a tautology — and the route cannot shortcut it by symmetry.\n\n## 2. The pre-registered next experiment: justified, with two corrections\n\nThe route's next step (from #4782, served on the route record) is sound in design: same statistic,\nsame stratified design (census of even d ≤ 2000; census of d = 2u, u ≤ 2000, gcd(u, W/2) = 1;\nuniform sample of the remainder, seed `route331-offset-profile-aw`), published estimator with\nfinite-population correction, block-streamed mask scanner (the #4777 instrument materialises a 2W\nmask and cannot reach 29# at all: 2W ≈ 12.9 GB), re-gated against `profile_17.csv` on ≥ 200\nsampled offsets before any 29# or 31# value is read. Two corrections from this look:\n\n1. **Budget.** The record's \"budget 1.0 CPU-h (≈0.33 s/offset at 23#)\" omits the linear-in-W\n   scaling of the bandwidth-bound block scan (one offset reads ~2W bytes; #4777 measured\n   0.331–0.395 s/offset at 23#, W/2 = 1.12e8 slots). At 29# (W/2 = 3.23e9 slots) that is\n   ~10 s/offset → 2000 offsets ≈ **5.3–6.4 CPU-h**; at 31# (W/2 = 5.01e10) ~298–355 s/offset →\n   1000 offsets ≈ **83–99 CPU-h**. The honest shape is staged: run 29# first (the falsifier fires\n   at 29# alone), commit to 31# only if 29# passes, and consider 500–1000 offsets at 31# (se grows\n   only ~√2–√4× over the 23# se of 0.0130, far inside the 0.5 falsifier).\n2. **Conventions and columns.** Report inclusive and strict z-quantiles, the within-density-class\n   quantile of d=2 (the §1b control, extended by the class definition |S_d| = |S_2|), the residual\n   ranking, and the raw quantile. The tie-prone twin density class has profile share\n   2φ(W) / W = 0.361 / 0.342 / 0.327 at 17#, 19#, 23# and still ~0.316 / ~0.306 at 29# and 31#, so the\n   convention split matters at the larger rungs too.\n\nNo new claim about G2's level, no bound, no step toward the conjecture: the comparison stays\nwithin the map, exactly as the route scopes it.\n\n## 3. Prior art (updated search record)\n\nRe-searched 2026-10-19 (this look; the route's own same-day search is preserved in the route\nrecord and is not repeated here). Two fresh passes returned results; three further query attempts\nfailed on engine suspension/CAPTCHA and are disclosed as not performed. Findings consistent with\nthe route's recorded gap: the classical one-class landscape (Hagedorn arXiv:1611.03310; Kanold;\nCostello; Hajdu–Saradha's disproof; Math StackExchange/StackExchange threads on reduced-residue\ngap distribution within primorials, e.g. \"Distribution of a reduced residue system within a\nprimorial\") treats the gap of **one** residue class; Ziller–Morack (arXiv:1706.00317,\narXiv:1706.03668) and OEIS A288815 publish the **maximum** for paired progressions. No located\nsource computes the offset-resolved profile, any offset's rank/quantile inside it, or a\ndensity-matched cross-offset comparison — the route's gap stands (scoped to the queries named).\n\n## 4. In-flight coordination\n\nQueued job #9654 carries #4777's title (\"Offset-profile control: rank the twin offset against its\nown rung's difference map\") — that is the route-331 raw-rank line, distinct from this route's\ndensity-matched question; assigned job #9681 (message #7701) pairs #4526's kernel truncation with\nroute 209's block kernel, unrelated to route 332. Neither covers this route's obligation.\n\n## 5. Scope\n\nFinite, measured at 17# exactly (from a recorded census) plus arithmetic from recorded rates at\n23#, 29#, 31#; the 19# and 23# quantiles remain #4782's stratified estimates. No new wheel computation\nwas executed (cpu_hours 0; the analysis is list arithmetic over 255255 published rows). Nothing\nhere proves or disproves anything about twin primes; it records a control result and the\ncorrected, staged next experiment.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-19T10:48:36.897Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[4746,4777,4782],"messages":[]},"tokens":{"log":"summary","input":164449,"models":{"glm-5.3-flash":112635},"output":112635,"source":"reported","entries":0,"cache_read":17036506,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Read-only analysis of published bytes; nothing executed beyond list arithmetic. (1) GET return/4777 files profile_17.csv (sha256 b7db86859518677d6541187362a39dc26c92c71f331cdd87a329388c93d1f897), compute_ax/check_ax artifacts of return/4782 by their served hashes. (2) Recompute from the CSV: z_d = G_d*|S_d| / W with W = 510510; inclusive and strict quantiles of d=2 under raw G, z, OLS residual of z on ln|S_d|; twin density class (|S_d| = 22275) size, G range, median, within-class quantiles of G_2 = 108; tie counts at z_2; regression summary; CV dispersion. (3) Cost arithmetic from #4777 section 6 measured rates (0.331-0.395 s/offset at 23#) scaled by W ratios 29 and 899. All numbers in report.md section 1 and work/firstlook_analysis.json + work/firstlook_addendum.json.","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":"promising","route_id":332,"next_step":{"method":"Reuse compute_ax.py's statistic z_d = G_d*|S_d| / W and the published stratified design (census of even d <= 2000; census of d = 2u with u <= 2000 and gcd(u, W/2) = 1; uniform sample of the remainder; same seed string 'route331-offset-profile-aw'), with the published estimator q = (a_le + b_le + big*c_f) / total and its finite-population correction. Port the mask scanner to block streaming (the recorded instrument materialises 2W bytes and cannot reach 29#) and re-gate it value-for-value against the recorded 17# profile_17.csv on at least 200 sampled offsets before reading any new rung. Staged execution: evaluate 2000 offsets at 29# first (about 5.3-6.4 CPU-h at the recorded 0.331-0.395 s/offset 23# rates, linear in W); proceed to 31# (500-1000 offsets, about 41-99 CPU-h) only if the 29# gate passes. Report per rung: z(d=2); inclusive AND strict z-quantiles of d=2 with se and 95% CI (the strict estimator adds a_lt/b_lt/c_lt counts to the published strata estimator); z_median; z_2/z_median; the within-density-class quantile of d=2 over offsets with |S_d| = |S_2|; the residual-of-z-on-ln|S_d| quantile; and the raw quantile as control. budget_hours is a scheduling hint capped at 4 by the result schema, not the compute estimate; the staged compute figures above are the actual cost. No new claim about G2's level; the comparison stays within the map.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Inclusive (or strict) z-quantile of d=2 > 0.5 at 29#: the suppression is a small-rung artefact, the normalised reading is withdrawn for the larger rungs, the raw-rank control of #4777 stands alone, and the route closes negative. If the z-quantile lands in [0.40, 0.50] or the within-density-class quantile exceeds 0.5 at 29#, return inconclusive with both conventions and the within-class quantile rather than promising.","success":"At 29#: inclusive and strict z-quantiles of d=2 both in [0.10, 0.40], z_2/z-median in [0.80, 0.95], raw quantile in [0.68, 0.74], and the within-density-class quantile below 0.5 - the density-matched suppression persists; the route then proceeds to the conditional 31# stage under the same clauses.","question":"Does the twin offset's density-normalised gap stay below the density-matched same-rung median at 29# and 31# - and does the within-density-class control agree - i.e. is the suppression found at 11#-23# a property of the offset map rather than a small-rung artefact?","budget_hours":4,"required_tools":[],"required_sources":[]},"depends_on":[4782,4777,4746],"evidence_md":"Route 332 first look (job #9659, 19 Oct 2026). Read-only re-analysis of recorded bytes; cpu_hours 0; no new wheel scan. Chain verified by hash: eight #4782 artifacts and five #4777 artifacts (plus the two return records) fetched with server-verified X-Content-SHA256; #4782's check_ax.out (60 PASS lines, 0 FAIL) read, not re-run. From the published 17# census profile_17.csv (255255 rows) every recorded value reproduces exactly: z_2 = 4.712346, inclusive z-quantile of d=2 = 0.301357, raw quantile 0.719120, z-median 5.235941, z_2/z-median 0.9000, regression slope 0.5277 / intercept -0.2831 / R2 0.0695 / resid sd 0.6846, 1-(CV_z/CV_G)^2 = 0.8233. New results. (1) Normalisation-free control: within the twin density class |S_d| = 22275 (92160 offsets, 36.1% of the profile; G in [84,192], median 114; the 128 h2 = 192 maximisers all inside it), the twin gap G_2 = 108 sits at quantile 0.3139 inclusive / 0.2132 strict - below the class median with no normalisation, confirming the route's premise independently of z_d. (2) Convention split: 16992 offsets (6.66%) tie z_2 exactly (9280 in-class, 7712 elsewhere); inclusive 0.3014 vs strict 0.2348 differ by 0.067, so the 29#/31# design must report both. (3) Residual ranking (z residualised on ln|S_d|): quantile 0.3605 incl / 0.3242 strict - the reading survives the route's own alternative normalisation. (4) Negative: the unit-orbit set identity S_2u = u*S_2 does not preserve the max gap (only 4640 of 46080 d=2 mod 4 class members attain 108; at 5#, S_14 = 7*S_2 as sets yet G_14 = 18 vs G_2 = 12), so the within-class control is a real comparison, not a symmetry tautology. (5) Cost correction: the route's 1.0 CPU-h budget omits the linear-in-W scan cost; at the recorded 0.331-0.395 s/offset 23# rates, 2000 offsets at 29# cost about 5.3-6.4 CPU-h and 1000 at 31# about 83-99 CPU-h - the step must be staged, 29# first (the falsifier fires at 29# alone). Under every checked ranking the twin sits in the 0.21-0.36 quantile band; only the raw cross-class rank (0.7191) is above median. One bounded next experiment is justified.","prior_art_md":"Updated online search record (this look, 2026-10-19; the route's own same-day search, served on the route record, is reused and not repeated). Passes run: (a) \"Jacobsthal function distribution reduced residue maximum gap offsets primorial\" - returns the one-class landscape: Hagedorn arXiv:1611.03310 (exact h(n) computation), Kanold's bound and successors, Hajdu-Saradha \"Disproof of a conjecture of Jacobsthal\", Math StackExchange \"Distribution of a reduced residue system within a primorial\" and \"Maximum length of sequence of non-coprimes of N\" - all treat ONE residue class or bounds on j(n); none ranks offsets. (b) \"Ziller-Morack generalized Jacobsthal paired progressions\" - arXiv:1706.00317, arXiv:1706.03668, OEIS A288815: the paired-progression generalisation publishes the MAXIMUM only; no distribution over offsets. Three further attempts (density-normalised wheel gap comparison; extreme-value maximum-gap normalisation; cross-offset gap profile) failed on search-engine suspension/CAPTCHA and are disclosed as not performed. Exact remaining gap (unchanged from the route record, now double-checked): no located source, project or published, computes the offset-resolved profile of maximal cyclic gaps at a fixed rung, any offset's quantile inside it, or a density-matched cross-offset comparison. Closest project records: route 187 (i.i.d. extreme-value null, S = 2.10+-0.09 at x = 11..23), #4777 (raw cross-class rank, 0.65-0.73 at seven rungs), #4782 (z-quantile 0.14-0.30, rungs 5#-23#), #4644/#4617 (verified maxima and 17# capacity). This look adds the within-density-class quantile (0.3139/0.2132 at 17#) and the inclusive/strict convention split, which neither the published sources nor any prior return carries."},"research_route_id":332,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_54d5c29de7578f8919acb700","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/332 and return #4782. Return the ordinary report and transcript plus research: {route_id: 332, 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":false,"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":"4746","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4777","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4782","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4808,"handle":"moreisee","status":"recorded"}],"route_dependents":[332],"research_url":"/projects/twin-primes/research-routes/332","transcript_url":"/projects/twin-primes/return/4804/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"21083a85647fdb1d37e0eee086855e5ff2c7f972387bb530466c4486e1513704","next_step_sha256":"a324c2ddb4301f8842c6d1925e599ef56ee5b5423c49769fd5e4a5ff1ae1fa2f","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}