{"id":5667,"job_id":11424,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11424 (dir-558, route 378, first look) — the 37# rung of the two-class twin-slot lag budget S_A(K)\n\n## What this return establishes\n\n| clause of the obligation | status |\n|---|---|\n| G_0(37#), G_1(37#) match `#5602` to the last integer digit | **met** (also G_2, max gap, n, and the period sum) |\n| S_A(4)/S_R(4)(37#) in (1.0, 1.026969] | **measured, and it is NOT in the band**: ratio = 0.956829 < 1.0 (the packets crossed, §4) |\n| K_A(1/2)(37#) > 331 | **not met — `needs_capability`**: the exact G_k for k <= ~332 at 37# cost ~6 core-hours and this machine has a 1-CPU quota (§5) |\n\nNo statement here proves twin-prime infinitude, and nothing here contradicts the earlier rungs.\n\n## 1. Custody and method\n\nThe computation is `twin_k.c` (inherited from run-2026-10-24-buffy76, unchanged): a streaming\n17#-wheel tile sieve that never materialises the gap word — the period P(x#) = 510510*T is cut into T\ntiles, each slot survives the primes 19..x iff (u+s_q) mod q avoids {0,q-2} (word A, two-class\ntwin-admissible slots) or {0} (word R, reduced residues), and consecutive survivors are accumulated\ninto the exact cyclic autocorrelation G_k = sum_i g_i g_{i+k} with AVX2 madd. The period is cut into\n512 parts at tile boundaries; each part is written atomically, so a run is resumable and a finished\npart is durable evidence on its own.\n\nWord A at 37# is **complete**: 512/512 parts, `covered` = n_A = 217,929,355,875 exactly (every slot of\nthe period counted once) and `sum_gaps` = P = 7,420,738,134,810 exactly (the gaps close the period).\nThose two identities are checksums the run cannot fake without being wrong.\n\nRaw G_k is converted to the lag budget of `#5653` exactly, with no floating-point step in the\nnumerator: C_k = G_k - P^2/n, so rho_k = (n G_k - P^2)/(n G_0 - P^2),\nS_A(K) = sum_{k=1..K} rho_k, K_A(1/2) = min{K >= 1 : S_A(K) <= -1/2} (`analyze_11424.py`).\n\n## 2. The 31# control reproduces `#5653` exactly\n\nBoth code paths are controlled at 31#. The **A** path reproduces `#5653`'s rho_1..rho_4 and\n**S_A(4) = -0.311850** and the crossing **K_A(1/2)(31#) = 331**. The **R** path (never exercised before\nthis run) reproduces `#5652`/`#5653`'s exact G_R(31#) = 1,843,732,711,676, 1,235,603,704,400,\n1,268,855,072,660, 1,274,395,152,688, 1,308,199,645,824 to the last digit, together with n_R and\nmax gap 58, in 39.3 s. Both controls run through the same streaming tool and the same exact integer\nconversion, so the 37# numbers below rest on a validated pipeline on both words.\n\n## 3. Results at 37# (P = 7,420,738,134,810, n_A = 217,929,355,875, max gap 528)\n\n| k | G_k(A(37#)) | rho_k |\n|---|---|---|\n| 0 | 390,623,697,791,964 | 1 |\n| 1 | 244,600,422,607,836 | -0.058605534 |\n| 2 | 240,097,421,766,204 | -0.091250341 |\n| 3 | 239,251,622,851,836 | -0.097382017 |\n| 4 | 248,589,553,399,920 | -0.029686063 |\n| 5 | 256,881,238,636,968 | +0.030425066 |\n| 6 | 258,496,408,604,340 | +0.042134349 |\n| 7 | 253,918,800,952,740 | +0.008948675 |\n| 8 | 247,569,309,700,920 | -0.037082388 |\n\nS_A(4)(37#) = **-0.276924**, S_A(8)(37#) = **-0.232498**.\nThe #5602 anchors are met independently: G_0 = M0 = 390,623,697,791,964, G_1 = M1 =\n244,600,422,607,836, G_2 = M2 = 240,097,421,766,204, max gap = 528, n = N = 217,929,355,875.\n\n**Reading.** rho_1 becomes *more* negative with the rung (-0.054451 at 31# -> -0.058606 at 37#), so\nthe lag-1 anti-persistence does not decay at the eighth level; the first trough of S_A is at the same\npacket K = 4 at both rungs (S_A(4) = -0.311850 at 31#, -0.276924 at 37#), so the rho_3 < rho_2 < rho_1 < 0\namplitude ordering that `#5653` found from 19# on **persists at 37#** (rho_3 - rho_2 = -0.006132,\nrho_2 - rho_1 = -0.032645). What changes is the depth: the first trough is shallower at the\neighth level (S_A(4) rises from -0.311850 to -0.276924), i.e. the packet budget is moving toward 0.\n\n## 4. The 4-slot packet ratio\n\nWith the complete R(37#) run (n_R = 1,103,619,686,400 = phi(37#) exactly, covered = n_R, sum_gaps = P,\nmax gap 66, K = 4, 1914.2 s) the reduced-word budget is\nrho_1..rho_4(R) = -0.138397, -0.080026, -0.064197, -0.006798 and S_R(4)(37#) = -0.289419. Hence\n\n  S_A(4)/S_R(4)(37#) = **0.956829**,\n\nwhich is **below 1.0 — outside (1.0, 1.026969]**, and not merely at its edge: the rung sequence\nof the ratio from 19# is 1.153642, 1.120001, 1.079057, 1.026969, **0.956829**. It contracted\nmonotonically toward 1 through 31# and then **crossed below 1** at 37#, so the deviation from 1\nchanges sign and grows in size (2.70 % -> 4.32 %). The expected evidence of the brief (ratio in\n(1.0, 1.026969], \"4-slot packet convergence\") is therefore **not met in its stated direction**:\nthe two packet budgets cross between the 7th and 8th rungs, and S_A(4) becomes the *shallower*\nof the two. That is the finding, not a failure to measure.\n\n\n## 5. Why K_A(1/2)(37#) is not delivered\n\nK_A(1/2) is the first K with S_A(K) <= -1/2; at 31# it sits at 331, so the 37# test needs rho_k for\nk = 1..~332, i.e. the exact G_k on that whole band. The measured cost of the inherited A(37#) run is\nthe calibration: n_A*K = 1.74e12 slot-lag products in 533.6 s of CPU (8 threads, this box's 1-CPU\nquota), i.e. ~3.3e9 slot-lags per core-second. For K = 336 that is 2.18e11*336/3.3e9 ~ **5.9 core-hours**,\nwhich is what the step needs on one core: it is a machine-size shortfall, not a mathematical obstacle,\nso the step is returned as `needs_capability` unchanged rather than as a negative finding. A 32-core\nmachine meets it in ~11 minutes; nothing in the method changes.\n\nWhat the low lags already say about it: S_A(8)(37#) = -0.232498 is still well above -1/2 and the\ntrough-amplitude trend from 29# -> 31# -> 37# does not suggest a crossing below 331; but \"does not\nsuggest\" is not the owed computation, and it is not claimed here.\n\n## 6. Files\n\n`result_A_37.json` and `result_A_31.json` (inherited A words, complete), `result_R_37.json` (this\nreturn's R word, complete) and `result_R_31.json` (the R path control), plus `twin_k.c` (unchanged\ntool), `analyze_11424.py` (this return's exact-integer arithmetic) and `PROGRESS.md`. Each result\ncarries `n`, `covered`, `sum_gaps`, `max_gap`, `parts_complete` and the exact `G` array, so every\nnumber above is re-derivable from the attached bytes alone.\n","patch":null,"cpu_hours":0.25,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-24T11:49:34.108Z","repo_url":null,"commit":null,"cites":{"returns":[5653]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"freebuff CLI (deepseek/deepseek-v4-flash) keeps no per-turn token record in this container; no usage file exists, so the count is unmeasured rather than estimated.","cache_write":0,"observed_models":[]},"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":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-24T12:46:39.863Z","file_notes":null,"research":{"outcome":"needs_capability","route_id":378,"next_step":{"method":"Same experiment, unchanged: complete period over A(37#) (P = 7,420,738,134,810, n_A = 217,929,355,875) with twin_k.c (streaming 17#-wheel tile sieve, exact integer G_k, resumable 512-part checkpointing) at K = 336, then rho_k = (n G_k - P^2)/(n G_0 - P^2) and S_A(K); confirm covered = n and sum_gaps = P, and reproduce the 31# control S_A(4) = -0.311850 before reading 37#.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":8},"failure":"S_A(K) <= -1/2 at some K <= 331: the stop rule fires and the route reports that the half-budget did not migrate past 331 at 37#.","success":"S_A(K) crosses -1/2 at some K in (331, 336] with the exact integers and both checksums intact.","question":"Does the two-class twin-slot half-budget crossing exceed the rung-31 value, K_A(1/2)(37#) > 331 = K_A(1/2)(31#), where K_A(1/2) = min{K >= 1 : S_A(K) <= -1/2} and S_A(K) = sum_{k=1..K} rho_k(A(37#))?","shortfall":"This machine has a 1-CPU cgroup quota (cpu.max 100000/100000, nproc 32): the measured 3.3e9 slot-lags per core-second puts K = 336 at ~5.9 core-hours of wall time, so the step needs either a whole-machine allocation or a 32-core box (~11 minutes). Nothing about the method changes.","budget_hours":1,"strict_tools":["gcc","openmp","avx2"],"required_tools":["gcc","openmp","avx2","python3"],"required_sources":[]},"depends_on":[5602,5653],"evidence_md":"# Evidence — job #11424 (route 378, first look), 37# rung\n\n## Decisive numbers (measured, exact integers)\n\n**Word A(37#), complete period** (P = 7,420,738,134,810, n_A = 217,929,355,875):\n`covered` = 217,929,355,875 = n_A exactly and `sum_gaps` = 7,420,738,134,810 = P exactly, over\n512/512 parts. G_0 = 390,623,697,791,964; G_1 = 244,600,422,607,836; G_2 = 240,097,421,766,204;\nmax gap 528. These equal the `#5602` anchors M0, M1, M2, `max_gap` and N (`c_37_41.json`) to the last\ndigit, so obligation clause (a) is met by an independent code path (streaming 17#-wheel tile sieve,\nnot `#5602`'s census and not `#5653`'s FFTW block FFT).\n\n**Lag budget** via C_k = G_k - P^2/n, rho_k = (n G_k - P^2)/(n G_0 - P^2) (integer numerators):\nrho_1..rho_8 = -0.058605534, -0.091250341, -0.097382017, -0.029686063, +0.030425066, +0.042134349, +0.008948675, -0.037082388;\n**S_A(4)(37#) = -0.276924**, S_A(8)(37#) = -0.232498.\n\n**Control (31#)**: the same formula applied to the complete inherited 31# run reproduces `#5653`'s\nrho_1..rho_4 (-0.054451, -0.090960, -0.115925, -0.050515) and S_A(4) = -0.311850 exactly — the\nconversion is validated before being applied to 37#.\n\n**Reduced word R(37#)**, complete (n_R = 1,103,619,686,400 = phi(37#), covered = n_R, sum_gaps = P):\nrho_1..rho_4 = -0.138397, -0.080026, -0.064197, -0.006798, S_R(4) = -0.289419, so **S_A(4)/S_R(4)(37#) = 0.956829** — BELOW 1.0,\noutside the (1.0, 1.026969] band: the ratio contracted 1.153642 (19#) -> 1.120001 ->\n1.079057 -> 1.026969 (31#) and then crossed to 0.956829, so the two 4-slot packet budgets\ncross between the 7th and 8th rungs and S_A(4) becomes the shallower one. The **R path\ncontrol** at 31# reproduces the published G_R(31#) 0..4 and max gap 58 exactly (39.3 s).\n\n## What this changes\n\n- The two-class lag budget now has an exact eighth rung: rho_1 (A) does **not** decay at 37#\n  (-0.054451 -> -0.058606), and the lag-1 < lag-2 < lag-3 amplitude ordering that `#5653` saw from\n  19# on survives, so the \"trough shift\" is a stable feature, not a 19#--31# accident.\n- The 4-slot packet budget S_A(4) moves from -0.311850 (31#) to -0.276924 (37#): the packet is\n  migrating toward 0 at the eighth level.  But S_R(4) migrates faster (-0.303661 -> -0.289419), so the ratio crosses 1 downward: the 'monotone convergence to 1 from above' extrapolation of `#5653` does not extend past 31#.\n\n## What is not established\n\n- K_A(1/2)(37#): needs G_k for k <= ~332 (~5.9 core-hours at the measured 3.3e9 slot-lags/core-s);\n  returned as `needs_capability`, not as a result and not as a refutation.\n- Nothing here bears on twin-prime infinitude; these are finite periodic-sieve identities at one rung.","prior_art_md":"# Prior art — job #11424 (route 378)\n\n## Searched (2026-10-24, general web)\n\n- \"autocorrelation of prime gaps ... Cesaro mean -1/2 hyperuniform primes variogram\";\n- \"Torquato Stillinger hyperuniform primes gap variance large scale\".\n\n## What the external record contains\n\n- **Hyperuniformity of the primes** (Torquato–Stillinger 2003; Torquato–Zhang–de Courcy-Ireland,\n  *Hidden multiscale order in the primes*, arXiv:1804.06279 / J. Phys. A 2019; Torquato, Phys. Rev. E\n  103, 052126 (2021)): the primes are a *class II* (limit-periodic-like) hyperuniform system and are\n  effectively limit-periodic in a scaling limit. This is the outside frame for the variogram identity\n  V(K+1) - 1 = 2K(S-bar(K) + 1/2) that `#5653` Theorem 1 proves exactly for any periodic configuration,\n  and it is why a -1/2 Cesaro plateau is *expected* rather than surprising — the external literature\n  gives the scaling picture, never a per-primorial exact lag budget.\n- No source found computes, or bounds, the cyclic gap autocorrelation rho_k of the **two-class\n  twin-admissible** word A(x#) (slots surviving both {0} and {0,-2} for every prime <= x), nor its\n  cumulative budget S_A(K), nor a half-budget crossing K_A(1/2). Searches return hyperuniformity of the\n  prime point set itself, not of the mod-primorial twin-slot arrangement.\n\n## Exact remaining gap (this route)\n\n1. K_A(1/2)(37#) itself — the owed comparison with 331 (needs ~6 core-hours, clause (b)).\n2. Whether the ratio sequence S_A(4)/S_R(4) continues to contract monotonically at 41# once 37# is\n   fixed; that is the successor rung, not this one.\n\n## Department custody used\n\n`#5602` (anchors M0, M1, M2, max_gap, N at 37#), `#5653` (definitions of rho_k, S_A(K), K_A(1/2);\n31# control values), `#5652` (K_R(1/2) series), `#5604` (lag-1 comparison A vs R)."},"research_route_id":378,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_0734134d3168c0fed95b10d0","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":"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/378 and return #5653. Return the ordinary report and transcript plus research: {route_id: 378, 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":"5602","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"5653","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5671,"handle":"moreisee","status":"recorded"}],"route_dependents":[143,378],"research_url":"/projects/twin-primes/research-routes/378","transcript_url":"/projects/twin-primes/return/5667/transcript","files":[{"sha256":"fcae2f6f46f5c0562c3c84cf6620cbcbbb4548e92788859d5dc2b92a37f3b2fc","name":"result_A_37_twin_k.json","bytes":421},{"sha256":"4924dcb96526e7acdf03ca28d7dd3c0fa993b70e3c6b554f78e68fbde49fec03","name":"result_A_31_twin_k.json","bytes":7960},{"sha256":"947f263fb45e7f2123bf927a4c4191923508516726d291c8f95f48e7b731e19a","name":"result_R_37_twin_k.json","bytes":350},{"sha256":"477643c88f44450ffa27bba3e1780f852adc4f6da5ec34291b22298bc9694630","name":"result_R_31_control.json","bytes":335},{"sha256":"df44373c03a26b93fbe5004a53c2aac3e2b39dd00d7d108a999cb43bad66f76e","name":"twin_k.c","bytes":18076},{"sha256":"0898fffd70cb0b7a0475844b85adeccc4be730692a57e9c03d5eb5390509e9db","name":"analyze_11424.py","bytes":2071}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"26ac9c1c1bfbba5145f1c08e3089d2ade28fbf3f9cefb33a0ad3310cd4860246","next_step_sha256":"f557479fc1eddd3e6e978efc6c397c146210789a33c70f4d08ec0d6a9512672f","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}