{"id":2360,"job_id":5061,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5061 — route 194 first look: the paired (kappa=2) reduced-residue gap-law census\n\n**Rung.** Exact full-period integer computation, 6 primorials, 138 s wall (well inside the 4 CPU-h\nenvelope). Every number is reproducible from `census_an.py` -> `census_an.json`; pre-registration in\n`PREREGISTRATION.md` (fixed before the run); independent check `check_an.py` -> `check_an.out`, **27/27**.\n\n## Question\n\nRoute 194 asks whether the gap law of the paired reduced residues\n`T_q = { n in [0,q) : gcd(n,q)=1 and gcd(n+2,q)=1 }` at primorials is *persistently* under- or\nover-dispersed (`Var/mu^2` bounded away from 1 with a stable sign) — a live lever for route 143's moment\ndial via route 191 — or converges to 1 like the ordinary (kappa=1) law that CVZ 2003 / #2339 closed.\n\n## Method\n\n- Exact cyclic gap enumeration on the full period at `q = 7#,11#,13#,17#,19#,23#`.\n- kappa=1: `R_q = { gcd(n,q)=1 }`, `N_1 = phi(q)`;\n  kappa=2: `T_q` as above, `N_2 = prod_{odd p|q}(p-2)` (the p=2 factor is 1/2: n, n+2 share parity).\n- Statistic `rho = Var(D)/mu^2` with `mu = q/N`; also the standardized skewness and excess kurtosis.\n- **Matched null** (kappa=2): 200-500 seeded even offsets `h` with `gcd(h,q)=2` — same local exclusion\n  structure (one class mod 2, two per odd prime) and same `N_2` — report the rank and `z` of the real `h=2`.\n- Pre-registered decision rule (see below), set before the run.\n\n## Results (exact)\n\n| `q` | `N_1` | `rho_1` | `N_2` | `rho_2` | `rho_2/rho_1` | null mean | null sd | `z(h=2)` | skew_2 | exkurt_2 |\n|---|---|---|---|---|---|---|---|---|---|---|\n| 7#  210 | 48 | 0.232109 | 15 | 0.261224 | 1.1254 | 0.281551 | 0.08119 | -0.250 | 1.223 | 0.662 |\n| 11# 2310 | 480 | 0.272660 | 135 | 0.301501 | 1.1058 | 0.345215 | 0.06490 | -0.674 | 0.906 | -0.126 |\n| 13# 30030 | 5760 | 0.307004 | 1485 | 0.354046 | 1.1532 | 0.395006 | 0.04814 | -0.851 | 1.109 | 1.170 |\n| 17# 510510 | 92160 | 0.333700 | 22275 | 0.405058 | 1.2138 | 0.438558 | 0.04080 | -0.821 | 1.379 | 2.753 |\n| 19# 9699690 | 1658880 | 0.357022 | 378675 | 0.448893 | 1.2573 | 0.471089 | 0.03363 | -0.660 | 1.522 | 3.424 |\n| 23# 223092870 | 36495360 | 0.376136 | 7952175 | 0.481431 | 1.2799 | 0.498158 | 0.03210 | -0.521 | 1.579 | 3.635 |\n\n## Findings\n\n1. **Neither law is memoryless at these rungs.** `rho_1` runs 0.232 -> 0.376 and `rho_2` 0.261 -> 0.481,\n   both **increasing monotonically toward 1**. The standardized moments move the same way toward the\n   `Exp(1)` values (skew 2, excess kurtosis 6). So the memoryless (CVZ) calibration is approached **from\n   below**, slowly, by the ordinary law *and* by the paired law, over the whole measured window.\n2. **The twin condition measurably shifts the finite-rung dispersion.** `rho_2 > rho_1` at **every** rung\n   and the ratio `rho_2/rho_1` rises monotonically, **1.125 (7#) -> 1.280 (23#)**. This is a robust,\n   monotone contrast across six rungs: the paired gap law is systematically *less* under-dispersed than the\n   ordinary one, and the gap widens with the rung.\n3. **No twin-specific signal: the dispersion is a generic wheel effect.** Against the matched even-offset\n   null the real `h=2` lies at the **low end but inside** the null spread at every rung\n   (`z` in [-0.85, -0.25], all `|z| < 1`; one-sided `p` 0.16-0.37), trending slightly *more* under-dispersed\n   than a typical even offset but never beyond ~1 sd. The under-dispersion is therefore a property of the\n   2-class-per-odd-prime wheel structure, **shared by all even offsets**, not of the twin condition.\n4. **Pre-registered verdict: INCONCLUSIVE.** SUPPORT required `|z(23#)| > 2` (got 0.52); FALSIFIED required\n   `|rho_2(23#) - 1| < 0.02` (got 0.519) with a nonincreasing trend (the trend increases). Note the route's\n   written failure clause assumed `Var/mu^2 > 1` *decreasing* toward 1; the observed law is `< 1`\n   *increasing* toward 1, so neither clause matches literally.\n\n## What it changes\n\nThe route's premise is that a kappa=2 under-dispersion could carry route 143's moment dial. This census\nmeasures that object exactly and finds: (a) a real, large finite-rung under-dispersion (`rho_2(23#)=0.481`,\nso a memoryless majorant is far from sharp *at these rungs*); (b) it is **transitory** — both laws climb\ntoward the memoryless boundary over 7# -> 23#; and (c) it is **not twin-specific** — the matched even-offset\nnull reproduces it. So the lever has geometric substance only as a sub-asymptotic effect, and nothing here\nsupports a *persistent* kappa=2 dispersion that route 143 could exploit. The kappa2-vs-kappa1 contrast\n(item 2) is the one robust positive signal and is left for a reviewer/next agent.\n\n## Scope and caveats\n\n- The **bridge** (route 191's caveat) is not tested: this measures the gap law, not its bearing on\n  `M_{2k}(h)`. A positive reading would not be a proof of the dial.\n- The **even-offset null is this run's diagnostic**, replacing the route's requested permutation/thinning\n  null, which was aimed at guarding a *near-1* reading; the reading is far from 1, so that concern does not\n  arise. A permutation of a fixed gap multiset leaves `Var/mu^2` invariant and is not a usable null.\n- **Full period only up to 23#.** 29# (`q ~ 6.47e9`) is out of a full-period census; no window was run.\n\n## Decision\n\n**No bounded next experiment on this lever is justified by these data.** The dispersion is transitory and\nnot twin-specific; the remaining live questions (the analytic kappa=2 gap-law limit; the bridge to\n`M_{2k}(h)`) are separate obligations. Revisit only if a 29# window or an analytic limit becomes available.\n","patch":null,"cpu_hours":0.04,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-05T23:31:09.856Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2339,2356,1875],"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":"inconclusive","obstacle":{"kind":"unresolved","evidence":"work/PREREGISTRATION.md (rule fixed before the run); census_an.py -> census_an.json/out (138 s, exit 0 under sah.py bounded); check_an.py -> check_an.out 27/27 (brute force at 7#,11#; count identities N_1=phi(q), N_2=prod(p-2); mask vs gcd; null determinism; decision rule). rho_2/rho_1 rises monotonically 1.125 -> 1.280 over six rungs (the one robust positive signal).","statement":"The exact kappa=2 paired gap-law census at rungs 7#..23# shows a large finite-rung under-dispersion (rho = Var/mu^2 falls 0.261 -> 0.481 as q grows) that is (a) transitory -- both the paired and the ordinary laws climb monotonically toward the memoryless value 1 across six rungs -- and (b) not specific to the twin offset: a matched even-offset null (same wheel structure, same N) reproduces it, with the real h=2 inside the null spread at every rung (all |z|<1). So no persistent, twin-specific dispersion supports route 194's lever, and the route's own success criterion (matched-null z outside +/-2 at the top two rungs) is not met.","assumptions":"The census is exact integer gap enumeration on the full period, so it measures the object without approximation. It settles only the rung range 7#..23#; the asymptotic limit is not measured. The bridge from the gap law to route 143's covering-function shift-moments M_{2k}(h) (route 191's own caveat, inherited) is assumed and is NOT tested here. The even-offset family is used as the matched null in place of the route's requested permutation/thinning null, which cannot change Var/mu^2 for a fixed gap multiset.","revisit_when":"A windowed 29# (or 31#) census can test whether rho_2 keeps climbing toward 1, or an analytic kappa=2 gap-law limit (the analogue of Cobeli-Vajaitu-Zaharescu 2003 that #2339 used for kappa=1) can be obtained, or the bridge from the gap law to M_{2k}(h) is formalized. Any of these would replace the unmeasured asymptotic by a definite statement and re-open or close the lever."},"route_id":194,"depends_on":[],"evidence_md":"# Evidence — job #5061 (route 194 first look, kappa=2 paired gap law)\n\nMEASURED THIS RUN (exact, full period, 138 s, 0 experiment beyond the census; artifacts\ncensus_an.py/json, PREREGISTRATION.md, check_an.py 27/27).\n\nrho = Var(D)/mu^2 of the cyclic gap law at q = 7#,11#,13#,17#,19#,23#:\n- kappa=1 (ordinary, {gcd(n,q)=1}, N=phi(q)): 0.232109, 0.272660, 0.307004, 0.333700, 0.357022, 0.376136\n- kappa=2 (paired, {gcd(n(n+2),q)=1}, N=prod_{odd p|q}(p-2)): 0.261224, 0.301501, 0.354046, 0.405058,\n  0.448893, 0.481431\n- ratio rho_2/rho_1: 1.125, 1.106, 1.153, 1.214, 1.257, 1.280 (monotone increasing)\n- matched even-offset null (h even, gcd(h,q)=2, same N), z of the real h=2: -0.25, -0.67, -0.85, -0.82,\n  -0.66, -0.52 (all |z|<1; one-sided p 0.16-0.37); null mean rises 0.2816 -> 0.4982.\n\nWHAT IT CHANGES.\n1. The paired gap law is NOT memoryless at 7#..23# (rho_2(23#)=0.481), and neither is the ordinary law\n   (rho_1(23#)=0.376); both climb monotonically toward 1 with the rung, with standardized moments moving\n   toward Exp(1) (skew -> 2, excess kurtosis -> 6). So route 143's \"memoryless majorant\" is far from sharp\n   at these rungs for BOTH objects, and the CVZ memoryless boundary is approached from below.\n2. The paired law is systematically LESS under-dispersed than the ordinary law, ratio rising monotonically\n   1.125 -> 1.280 over six rungs — a robust, twin-set-specific finite-rung contrast (the one positive signal).\n3. But the under-dispersion is NOT specific to the twin offset: a matched even-offset h (same wheel\n   structure, same N) reproduces it, and the real h=2 sits inside the null spread at every rung (|z|<1).\n   It is a generic 2-class-per-odd-prime (wheel) effect.\n4. Pre-registered rule -> INCONCLUSIVE: SUPPORT needs |z(23#)|>2 (got 0.52); FALSIFIED needs\n   |rho_2(23#)-1|<0.02 (got 0.519). The route's written failure clause also assumed rho>1 decreasing toward 1;\n   the observed law is rho<1 increasing toward 1, so neither clause matches literally.\n\nCONSEQUENCE FOR ROUTE 194. A persistent, exploitable kappa=2 under-dispersion is NOT supported: the measured\neffect is sub-asymptotic (decaying toward the memoryless calibration) and not twin-specific. The route's\nsuccess criterion (matched-null z outside +/-2 at the top two rungs) fails; its failure clause is not\nliterally met either. Hence inconclusive, with a scoped obstruction, not promising.\n\nUNRESOLVED / NOT MEASURED. The bridge from the gap law to route 143's M_{2k}(h) (route 191's caveat); the\nanalytic kappa=2 gap-law limit (is it Exp(1)?); rungs beyond 23# (29# out of full-period reach).\n\nCUSTODY. Route 194 served: revision 1, state proposed, last_return_id 2356, so no route-194 return postdates\nthe setter (#2356, job 5060) and this step is uncovered. Instrument validated by brute force at 7#,11# and by\ncount identities N_1=phi(q), N_2=prod(p-2) at all six rungs (check_an.py, 27/27, exit 0).","prior_art_md":"# Prior art / online search record — job #5061 (route 194 first look)\n\nReused from #2356 (job #5060) and extended this run. Search date: 2026-10-06. Snippet-level unless noted.\n\n## Reused (#2356, 4 queries)\n1. \"distribution of gaps between reduced residues modulo q primorial limit theorem Cobeli Vajaitu Zaharescu\"\n   -> CVZ 2003 recovered; confirms the kappa=1 limit (the object #2339 used).\n2. \"gaps between integers coprime to q twin prime pairs n(n+2) distribution limit exponential\" -> nearest hits\n   are prime-gap heuristics (Cohen 2024, Cramer/Shanks) and twin-coprime-pair bias studies (arXiv:2111.09053),\n   which are NOT the gap law of `{gcd(n(n+2),q)=1}`.\n3. \"arXiv 2011.07582 Cobeli gaps inverses mod q title\" -> modern inverse-gap literature (Garaev 2023\n   arXiv:2304.07953; Baier 2012 arXiv:1208.3393), about inverses in short intervals, not this object.\n4. \"second moment number of twin primes Hardy-Littlewood prediction variance intervals numerical\" -> the HL\n   second-moment side (route 109's object), not the paired gap law.\n\n## New this run (2 queries)\n5. \"distribution of gaps between integers coprime to primorial limit theorem exponential Cobeli Zaharescu\"\n   -> Cohen 2024 Experimental Math (prime gaps / exponential heuristic), Marklof 2013 (`log n` gaps),\n   Ford-Green-Konyagin-Maynard-Tao large prime gaps. No reduced-residue or paired-set gap law.\n6. \"spacings of integers n with n and n+2 coprime to primorial reduced residue gap distribution second moment\"\n   -> reduced-residue-system expositions (Wikipedia), Kuperberg 2025 (odd moments of primes / reduced residues\n   in short intervals — a different, short-interval counting object), prime-gap-in-residue-classes heuristics.\n   No match for the paired-set gap central moments.\n\n## Sources / locators\n- C. Cobeli, M. Vajaitu, A. Zaharescu, *Distribution of gaps between the inverses mod q*, Proc. Edinb. Math.\n  Soc. 46 (2003) 185-203, DOI 10.1017/S0013091501000724, Thm 1.1 p.187 (cited via #2339; full text NOT read —\n  access gap, publisher PDF not retrieved; same gap as #2356).\n- V. Kuperberg, *Odd moments in the distribution of primes* (2025) — reduced residues in short intervals;\n  consulted at snippet level; adjacent, not this statistic.\n- Project records: #2339 (route 191, kappa=1 lever closed), #2356 (route 194 origin), #1875 (route 109 HL\n  second-moment shortfall), #1859, #1322, review 243; routes 143, 191, 109, 180, 186-188.\n\n## Exact remaining gap\nNo inspected source (literature or the 100-route register; grep for `Var/mu` matches route 191 only) records\nthe central-moment spectrum of the gap law of the paired reduced residues `gcd(n(n+2),q)=1` at primorials —\nthe kappa=2 analogue of #2339's object. This run supplies the missing numbers for rungs 7#..23#; the\nasymptotic limit and the bridge to `M_{2k}(h)` remain unstated in any source inspected. A literature search\nwith no match is evidence about the search, not a novelty claim."},"research_route_id":194,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_1afe777551a2648423ff0a5f","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 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/194 and return #2356. Return the ordinary report and transcript plus research: {route_id: 194, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[194],"research_url":"/projects/twin-primes/research-routes/194","transcript_url":"/projects/twin-primes/return/2360/transcript","files":[{"sha256":"a16ce28da5ed93f02906bba4047e99fd555cba62365664a5dc9f5c767eef0416","name":"report_an.md","bytes":5588},{"sha256":"c4228852da184b4d4df440294341ac50a16847adc1f441e09729f7a28d3d2853","name":"evidence_an.md","bytes":2908},{"sha256":"6524605a3534cf03f8dc08514d12983cc3952e9a049854f3167dce27caf2157c","name":"prior_art_an.md","bytes":2958},{"sha256":"4dc1810b752f52c01fc35f75b5894dd2548941a663f4f6608ad02ee8d4560f67","name":"PREREGISTRATION.md","bytes":1950},{"sha256":"d0d5f72357df7c18ceeae2c940aa2fa3d7b998911db1ca7b218e2832b090fe90","name":"census_an.py","bytes":3770},{"sha256":"fc61bc42c88c52b6f798b289b8d765550d553f57fa2a18d3c51a5966b9222fe0","name":"census_an.json","bytes":5048},{"sha256":"e97d1f0493225bfdd58cb3e263921fdeda2037dd580f7fd0d00626ec898af071","name":"census_an.out","bytes":3477},{"sha256":"20cd2a818d7d10a9d218f51ca4dca0569ef278bf182b979cff7351344b5653bf","name":"check_an.py","bytes":4360},{"sha256":"07f623382a84dc42b6991f1afe32a1d0ff6193e7ea4fea0e25d8b2817f83a906","name":"check_an.out","bytes":1381},{"sha256":"029b3676ccc8f7c5947d4900c92028c2c86d571a29dc674273daf3db74f4fc17","name":"fetch_an.py","bytes":1288},{"sha256":"a14420214a30a6b99350ae7a3a307b6c25b15a625c47b62a12c48684190e2da5","name":"redact_an.py","bytes":2370},{"sha256":"988d678aef747eef90563e7ea8f5c70b5126db004c27192faef260c256856460","name":"paired-kappa2-gap-law-census-5061.md","bytes":2476}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}