{"id":2426,"job_id":5169,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5169 — explore / discover: the paired-carrier bridge (new route proposal)\n\n**Outcome: `proposed` (new route).** A new finite statistic and its first exact measurement, plus a\nnew route built on it. The measurement **refutes the one-parameter bridge** between the\nreduced-residue gap arrangement (routes 180/186) and the twin-candidate (\"paired\") gap arrangement\nthat route 188 uses as its order-blind carrier, and it supplies the bridge table plus the\npermutation-orbit control that route 188 recorded as missing.\n\nAssignment is general-mode `explore`/`discover`, route null; the contribution is therefore recorded\nas a **new route proposal** with a bounded next experiment. Nothing here bounds `G2`, proves twin\nprimes, or makes an asymptotic claim.\n\n## Object and statistic\n\nFor `P = x#` two cyclic gap words on `Z/P`:\n\n* **reduced** `R`: residues in `[1,P)` coprime to `P` (`N_R = phi(P)`).\n* **paired** `A`: residues `n` with `gcd(n,P)=1` **and** `gcd(n+2,P)=1` (`N_A = prod_{p|x, p>2}(p-2)`),\n  the distance-2 twin-candidate set of route 188.\n\nStatistic exactly as route 186 / #2299 defines it, `gbar = P/N`, indices mod `N`:\n\n    rho_k = sum_i (g_i-gbar)(g_{i+k}-gbar) / sum_i (g_i-gbar)^2 .\n\nComputed from the exact integer identity `rho_k = (N*S_k - P^2)/(N*S_0 - P^2)`,\n`S_k = sum_i g_i g_{i+k}` (`int64` dot products), so no float enters before the final ratio; numpy\n`uint8` sieve, full period, **no census and no sampling**. `solve_bl.py`; wall 68.6 s under\n`sah.py bounded` (limit 600 s), exit 0, process group cleared.\n\n## Instrument validation (V0, pre-registered)\n\n`rho_1(R)` reproduces the served values of #2299 / #2207 / #2303 to all six printed decimals:\n\n    11# -0.252340   13# -0.210269   17# -0.186506   19# -0.170428   23# -0.159126\n\nIndependently, `rho_1(A, 23#) = -0.045272` reproduces route 188's single recorded paired point\n`-0.045` (#2330). Both instruments are therefore validated on the record before any new reading.\n\n## New table — paired arrangement `rho_k(A)`, exact, 11#..23#\n\n| x# | N_A | rho_1 | rho_2 | rho_3 | rho_4 |\n|---|---|---|---|---|---|\n| 11 | 135 | -0.117700 | **+0.268964** | +0.006154 | -0.178116 |\n| 13 | 1485 | -0.062238 | **+0.119602** | -0.093884 | -0.213604 |\n| 17 | 22275 | -0.039748 | **+0.021241** | -0.162328 | -0.201646 |\n| 19 | 378675 | -0.042061 | **-0.044552** | -0.172766 | -0.170612 |\n| 23 | 7952175 | -0.045272 | -0.073864 | -0.152962 | -0.112937 |\n\nFor comparison, the reduced arrangement at the same rungs has `rho_1 = -0.252340, -0.210269,\n-0.186506, -0.170428, -0.159126` and `rho_2 < 0` already at 11#.\n\n## Pre-registered claims, and how each fared\n\n* **H1** `rho_1(A) < 0` at every rung — **sign holds at all five**; the monotonicity half **fails**:\n  `|rho_1(A)| = 0.1177, 0.0622, 0.0397, 0.0421, 0.0453` has its minimum at **17#** and turns up.\n* **H2 (the bridge — the falsifier that matters): REFUTED.** `q1 := rho_1(A)/rho_1(R)` is\n  `+0.4664, +0.2960, +0.2131, +0.2468, +0.2845`. Over `17#..23#` the spread is `0.0714` against a\n  mean of `0.2481` — **28.8 %**, past the pre-registered 20 % bar; over `11#..23#` `q1` moves by a\n  factor **2.19**. A one-parameter bridge does not exist.\n* **H3: not supported.** `-rho_1(A) ln x = 0.2822, 0.1596, 0.1126, 0.1238, 0.1420` is not flat over\n  `19#..23#` (14.7 % and rising), unlike the reduced diagnostic which is flat at ~1/2 from 19# on.\n* **H4: threshold, not uniform.** Order-1 predicts `rho_2 = rho_1^2 > 0`. The paired word obeys\n  order-1 at **11#, 13#, 17#** and breaks it at **19#, 23#** (`rho_2 = -0.0446, -0.0739`). The\n  reduced word already breaks order-1 at 11#. The paired carrier's order-1 description survives two\n  rungs longer, and the break sits between 17# and 19#.\n* **H5: supported for x >= 13.** Under uniform permutations of the paired gap multiset the natural\n  ordering sits at percentile 0.0 % from 17# on, with `z = -1.44, -2.71, -5.36, -25.96, -126.91`.\n  The ordering carries real structure (it is not a typical element of its own orbit), so a\n  \"multiset-only\" reading of the paired word is incomplete while an *identity* carrier of `G2` is\n  still order-blind (#2330). Both readings are consistent: `G2` is multiset-typed, but the\n  arrangement statistic `rho_k` is not.\n\n## What this changes, and what is new on the record\n\n1. **The full paired `rho_k` table at 11#..23#** (new; route 188 had one point).\n2. **The bridge ratio `q1(x)` and its refutation as a one-parameter map** (new).\n3. **A rung threshold** (order-1 failure between 17# and 19# for the paired word) that any\n   mechanism transferring the reduced arrangement to the paired carrier must reproduce (new).\n4. **A permutation-orbit control** for the paired word (new), which sharpens route 188's\n   single-fixed-seed receipt into an orbit statement.\n\nConsequence: the rich reduced-residue machinery (route 186's merge recursion / closed form, route\n180's `1/ln x` law, route 25's arrangement-vs-census split) **cannot be rescaled onto route 188's\npaired carrier by one constant.** Any transfer needs an explicit shift-2 correction with at least\ntwo parameters, and it must reproduce the order-1 threshold — the proposed next experiment tests\nexactly that.\n\n## Scope and limits\n\nFinite exact statements at `11#..23#` only. No asymptotic, no bound on `G2`, no claim about prime\noccurrence. `rho_k` is a *diagnostic* of the arrangement; the route's link from it to the `G2`\nexponent is **conjectural** (as route 188's own uncertainty says). The permutation-orbit control\nuses `40` seeded shuffles at each rung (seed `20261006`), so its tail percentiles are coarse at\n11#; the `z` column is reported with that caveat.\n","patch":null,"cpu_hours":0.02,"hashes":{"check_bl.py":"c42803fe7a8882a026242a09b7f91c8297b5d8f433e640c4dc9f8ae83f2f6318","solve_bl.py":"eeb6e82527d97b7869a92f68957e460e4292c1395cdb16b04f205ef45af6d1e2","check_bl.out":"0c8e53624d130bc96dc809db8a5e0a392ad16bbdf45b7a5f57e83bf86b9cffb3","recipe_bl.md":"1d77784656f26c8af396234370d8ffa31fbd982e649a58a803070bb6d22b92bd","report_bl.md":"fecc521532f62ad227a042c489bc93d695c15330f6cd46d9bf7af22171312cc2","solve_bl.out":"e651ea0b20f843e375a16f7f6565708800bbe52a89cc4dd093046e8664145e28","solve_bl.json":"5c012bcee791accede26d642e1b0fcbabcb57e70b4e8406a8851ac6e13609353","evidence_bl.md":"64be64b384c09bf24450104448c81f33f3715a04eeb33b3874ef003882b66535","next_step.json":"64a04b2c21098db001ef9f576229329f86f0579094a66089e47fc476d99356bb","prior_art_bl.md":"d2d00b04f31b55c4729eae6a0cee02a8590506d3c66b76c5dba70ed61d6abc2f","PREREGISTRATION.md":"eee30152974afa19df8274efd56f03893a2929ec789393e77a97a6d76fe29c82","route188-paired-bridge-5169.md":"e06fedc2560e5e4192c13ccbceb4ba7ae07fde3e2c2decc8a805e3bbf89a87bd"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-06T14:56:27.407Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2330,2299,2303,2207,2396,2387],"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":"Reproduce (exact, ~70 s, stdlib + numpy):\n\n1. `python3 .solveathome/tools/sah.py bounded --run run-2026-10-06-bl --limit 600 -- \\\n   python3 .solveathome/runs/run-2026-10-06-bl/work/solve_bl.py`\n   Reads nothing external; writes `solve_bl.json` and prints `solve_bl.out`.\n   Method: for P = x# (x=11,13,17,19,23), build a numpy uint8 coprime mask by clearing the residue-0\n   class of every prime p<=x; R = nonzero indices; A = nonzero indices of `mask & roll(mask,-2)`;\n   gaps by cyclic diff; rho_k from the exact integer identity rho_k=(N*S_k-P^2)/(N*S_0-P^2).\n   Permutation control: 40 (200 for N<=4e6) seeded shuffles (seed 20261006).\n   Pre-registered predictions are in `PREREGISTRATION.md` (written before execution).\n2. Checker: `python3 .solveathome/runs/run-2026-10-06-bl/work/check_bl.py`\n   Reproduces the validation anchors (route 186's rho_1(R) at 11#..23#, route 188's paired point),\n   re-checks rho_1(A) negativity, the H2 spread (must be >20%), and the rho_2(A) sign change; plus\n   three independent re-derivations of rho_1..rho_2 at 11#,13# by direct Fraction arithmetic on the\n   word (not the int64 identity) and a corrupted-input control that must fail.\n\nInputs / prerequisites: Python 3.11, numpy 1.24.2. No network, no external data, no served code\nexecuted. 0.02 CPU-h measured (68.6 s wall, single process, bounded group cleared).\n\nReading the output: the table rows are x# = 11,13,17,19,23; columns rho_k(R) and rho_k(A) for k=1..4;\nthen q1 = rho_1(A)/rho_1(R) and -rho_1*ln x for both words; then the permutation-orbit line per rung\n(natural value, orbit mean/sd, percentile, z). Expected decisive readings: V0 exact reproduction of\nthe five reduced rho_1 anchors; H2 spread 28.8% (>20% => refuted); rho_2(A) positive at 11#..17# and\nnegative at 19#,23#.\n\nWhat a successor should do next: extend the same computation to 29# and 31# with a *segmented* sieve\n(materialising P=6.47e9 or 2.0e11 bytes is impossible; stream in blocks of ~2^27 and accumulate\nS_0,S_1,S_2 plus the orbit shuffles per block-batch), compare against the already-served reduced\nrho_k at 29#/31# (#2303), and fit a two-parameter bridge rho_1(A) = a*rho_1(R) + b/ln x (or a + b/p).\nBudget <= 1.5 h, <= 1 CPU-h, <= 2 GB RAM.","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":[{"sha":"eeb6e82527d97b7869a92f68957e460e4292c1395cdb16b04f205ef45af6d1e2","name":"solve_bl.py","notes":["prints what looks like progress or timing to stdout on line 89 (\"print(\"x=%d P=%d phi=%d cand=%d  %.1fs\" % (x, P, len(R), len(A), time.time() - t\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."],"fixed_by":"59ec8f943ff7f49a298e7cdb34179832876f1ea67311e907e64f07bdab6f677a"}],"research":{"outcome":"proposed","proposal":{"title":"The paired-carrier bridge: no one-parameter transfer from the reduced-residue gap arrangement to the twin-candidate arrangement of route 188","prior_art_md":"Search date 2026-10-06. Queries: \"gap sequence autocorrelation reduced residue system primorial\nJacobsthal twin candidates statistics\"; \"autocorrelation gaps coprime residue classes mod primorial\ntwin prime candidates anti-persistence\"; \"\\\"reduced residue\\\" gap sequence autocorrelation primorial\nconjecture negative lag-1\".\n\nSources inspected / locators.\n- No external source computes the lag-k autocorrelation of the reduced-residue gap sequence of a\n  primorial over a full period, and none computes the paired n(n+2)-candidate analogue; the top SERP\n  hit for both targeted queries is the project's own route 180 page. The nearest record is in-project:\n  route 180 / #2207 (k=1, reduced), route 186 / #2299 + #2303 (k=1..4, reduced; 29#,31#), route 186's\n  merge recursion #2396, and route 188 / #2330 (paired carrier; one point rho_1(A,23#)=-0.045 with an\n  explicitly uncalibrated bridge to the reduced object).\n- Peripheral, unvetted: \"Harmonic Resonance in Twin Prime Distribution: Empirical Evidence of\n  Phase-Locking and Under-Dispersion\" (ResearchGate, 2025-12-17) is said to histogram/autocorrelate\n  the 1485 admissible residue classes mod 30030 — the mod-30030 (=13#) paired count matches N_A(13#)\n  = 1*3*5*9*11 = 1485, so this preprint may touch the same finite object at one rung; the full text\n  was not retrieved (access gap). Treated as a nearest neighbour, not a source: no full-period\n  autocorrelation law or bridge is attributed to it.\n- Region itself (Jacobsthal function, reduced residue systems, prime gaps) is classical and\n  surveyed in OEIS wiki \"Jacobsthal function\" and standard references; none states the bridge\n  statistic.\n\nExisting attempts inspected in-project (the actual prior work this proposal must differ from):\n- #2199/#2207/#2303/#2396 build and test the reduced-residue rho_k and its merge recursion; all use\n  the reduced word R only.\n- #2330 (route 188) measures G2/N_L as multiset-typed (permutation-invariant) and records the single\n  paired value rho_1(A,23#) = -0.045, calling the numerical bridge to R \"uncalibrated\".\n- Route 25 separates arrangement from census with a permutation null on the *reduced* word.\n\nExact uncovered step (why this is not a duplicate). No return computes the paired word's rho_k at\nmore than one rung and none states a bridge law between R and A. This proposal supplies (a) the full\npaired rho_k table at 11#..23#, (b) the bridge ratio q1(x), and (c) a paired-word permutation-orbit\ncontrol, and shows a one-parameter bridge is impossible. Route 188's obstacle (multiset typing of G2)\nis untouched; what is new is the *relationship* between the two arrangements, which is where routes\n180/186 and 188 fail to connect.\n\nNo match found is not established novelty: the paired autocorrelation at rungs >23# and the possible\npreprint at 13# remain unchecked.","uncertainty_md":"Weakest unproved assumption: the BRIDGE. That the paired arrangement's rho_k bears on G2, or on any\nexponent, is CONJECTURAL, exactly as route 188's own uncertainty says; this run measures a finite\ndiagnostic of the bridge, not the exponent.\n\nSecond: the one-parameter bridge is refuted only on 11#..23#. q1 = 0.2131, 0.2468, 0.2845 over\n17#..23# is RISING, not diverging, so a one-parameter fit could conceivably reappear at 29#/31#; the\nnext experiment tests this directly.\n\nThird: finite reach. The paired word at 29#/31# has N_A ~ 2.2e8 / 7.0e9 and needs the segmented\nsieve not used here (materialising P bytes is impossible at 31#).\n\nFourth: the permutation-orbit control uses only 40 seeded shuffles per rung, coarse at 11# (7.0%).\n\nFifth: rho_k is one of many order statistics; a different arrangement functional (e.g. per-fibre\ncounts of route 186's merge construction) might bridge where rho_k does not. This run excludes a\none-parameter bridge for rho_k, not for all statistics.","contribution_md":"The project's central object G2(x#) is order-blind (route 188 / #2330: it is a function of the\ngap multiset). The richest theory of the surrounding finite object — route 180/186's full-period\nrho_k and its merge recursion, route 25's arrangement/census split — lives on the REDUCED-residue\narrangement R, while route 188's carrier lives on the PAIRED set A = R intersect (R-2). If a stable\nbridge existed, the reduced-lane machinery would transfer to A for free. This run shows the bridge is\nNOT one-parameter (q1 = rho_1(A)/rho_1(R) spreads 28.8% over 17#..23# and moves by 2.19x over\n11#..23#), supplies the missing paired rho_k table and orbit control, and fixes the first quantitative\ntarget any transfer must reproduce (the order-1 crossing between 17# and 19#).\n\nConjectural links, labelled: (i) if a two-parameter bridge exists, the paired carrier inherits the\nreduced lane's 1/ln x decay and its merge recursion, and the exponent route can be pursued on A with\nthe reduced lane's tools; (ii) if it does not, the shift functional A = R intersect (R-2) is the only\nobject and the exponent route must be rebuilt on it. Either way the contribution is a measurable\nconnection between lanes, not a new exponent bound. Nothing here bounds G2."},"next_step":{"method":"Extend solve_bl.py to 29# and 31# with a segmented sieve (stream ~2^27-wide blocks; accumulate S_0,S_1,S_2 for the reduced word R and the paired word A without materialising P bytes, and draw the permutation-orbit shuffles per block batch). Compare the resulting reduced rho_k at 29#/31# against the already-served values of #2303 (rho_1 = -0.150838, -0.143934) as an instrument check, then report rho_1(A) and rho_2(A) at both rungs. Fit the two candidate bridge forms q1(x) = a + b/ln x and q1(x) = a + b/p on 11#..31# by least squares and report residuals; report the rho_2(A) sign-change rung. Do not re-run 11#..23#; reuse this run's table.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"q1(x) keeps moving with no two-parameter form holding at 29#/31# (residual not shrinking), or the rho_2(A) sign change does not persist at a stable rung. Then the bridge requires the shift-2 arithmetic structure explicitly rather than any statistics of the reduced arrangement; record that as the route's scoped obstruction and re-aim the transfer at the shift functional (A = R intersect (R-2)) rather than at rho_k.","success":"A two-parameter bridge form reproduces q1(x) at 29#/31# within its fitted residual and predicts the rho_2(A) zero crossing at a stable rung. That makes the reduced-arrangement theory transferable to route 188's paired carrier up to an explicit finite shift-2 correction, and gives the carrier a measurable second-order statistic.","question":"Once 29# and 31# are added to the paired series, does q1(x) = rho_1(A)/rho_1(R) follow a two-parameter law (e.g. a + b/ln x or a + b/p) with bounded residual, or does it keep moving so that no finite-parameter bridge exists? And does the paired word's order-1 failure rung (rho_2(A) crossing zero, located here between 17# and 19#) stay fixed or move with x?","budget_hours":1.5,"required_tools":[],"required_sources":[]},"depends_on":[2330,2299,2303,2207,2396],"evidence_md":"Exact full-period enumeration at 11#..23# (no census, no sampling) of the cyclic gap autocorrelation\nrho_k on two words over Z/P, P=x#: the reduced-residue word R (route 180/186 object) and the paired\ndistance-2 twin-candidate word A (route 188 object), with\nrho_k = (N*S_k - P^2)/(N*S_0 - P^2), S_k = sum_i g_i g_{i+k} (int64, exact).\n\nInstrument validated on the record before any new reading: rho_1(R) reproduces #2299/#2207/#2303 to\nsix decimals (-0.252340, -0.210269, -0.186506, -0.170428, -0.159126), and rho_1(A,23#) = -0.045272\nreproduces route 188's single paired point -0.045 (#2330).\n\nNew readings: rho_1(A) = -0.117700, -0.062238, -0.039748, -0.042061, -0.045272 at 11#,13#,17#,19#,23#\n(non-monotone: minimum at 17#). rho_2(A) = +0.268964, +0.119602, +0.021241, -0.044552, -0.073864:\nthe order-1 prediction rho_2 = rho_1^2 > 0 holds at 11#,13#,17# and FAILS at 19#,23# — a threshold\nbetween 17# and 19#, whereas the reduced word already fails order-1 at 11#.\n\nBridge ratio q1 = rho_1(A)/rho_1(R): +0.4664, +0.2960, +0.2131, +0.2468, +0.2845. Pre-registered\nH2 (one-parameter bridge, <=20% spread over 17#..23#) is REFUTED: spread 28.8% of the mean there and\na factor 2.19 across 11#..23#. -rho_1(A) ln x = 0.2822, 0.1596, 0.1126, 0.1238, 0.1420 is not flat\n(H3 not supported), unlike the reduced diagnostic's ~1/2 plateau.\n\nPermutation-orbit control on the paired multiset (40 seeded shuffles, seed 20261006): natural rho_1\nsits at percentile 0.0% from 17# on, z = -1.44, -2.71, -5.36, -25.96, -126.91; the ordering carries\nreal structure beyond the multiset, so a multiset-only reading of rho_k is incomplete while G2\nitself remains multiset-typed (route 188's order-blindness).\n\nDecisive for the route: the reduced-residue theory (route 186's merge recursion, route 180's 1/ln x\nlaw, route 25's arrangement/census split) does NOT rescale onto the paired carrier by one constant.\nThis is a finite, exactly checkable statement (solve_bl.out, 68.6 s, exit 0 under a 600 s bound).\n\nScope: finite statements at 11#..23# only; no asymptotic, no bound on G2, no claim about prime\noccurrence. rho_k is a diagnostic; its link to the G2 exponent stays conjectural, as route 188's own\nuncertainty states. Orbit percentiles use 40 draws per rung, coarse at 11# (7.0%).","parent_route_id":188},"research_route_id":202,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_3a3775c34c586f782c7dbe4f","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**New route.** Read the closed-routes register (`research/OUTCOMES.md`, section \"Closed routes\") and the open questions (`GET https://solveathome.org/projects/twin-primes/questions`). Search online for the route, equivalent formulations, previous attempts and published computations before proposing to try it. Draft one route to the target exponent or to the infinitude statement that adds something to the record, or changes a specific assumption or ingredient in a previously blocked route: the object, the step that would have to hold, the first check that could refute it cheaply, and what it would cost to run. Include it as `research.proposal` in this explore return, with the nearest prior work, exact difference and bounded next experiment.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. 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