{"id":5731,"job_id":11525,"problem_id":1,"lane_id":5,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job 11525 — route 384 first look: the whole-box cancellation of B(x) persists at x = 2^20 and 2^22\n\n**Outcome: `promising`.** Type `explore`, first look, lane `infinitude`, general mode. Predecessor: return #5710 (the origin of route 384). The route's pre-registered question was whether the whole-box cancellation of the Vaughan twin remainder B(x), measured at x = 2^16..2^18, persists at x = 2^20 and 2^22. It does: all three success conditions of the route's decision rule hold at both new scales, no failure condition occurs, and the sign-balance pairing lemma remains the route's named open analytic input.\n\n## Evidence\n\nThe exact definitions are those of prime-detection-spec.md equations (1), (8), (11), (12): w(n) = Lambda(n-2) - b(n), b(n) = 2*C2*1_{n odd}*prod_{p|n, p>2}(p-1)/(p-2), J_x = (x/2, x], U = V = floor(x^(6/25)), D = U*V, I1, I2 with c(d) = sum_{ef=d, e<=U, f<=V} mu(e)Lambda(f), B = sum_{d>U, k>V, dk in J_x} mu(d)beta_V(k)w(dk) with beta_V(k) = log k - sum_{l|k, l<=V} Lambda(l), S = sum Lambda(n)Lambda(n-2), M = sum Lambda(n)b(n), identity S - M = I1 - I2 + B.\n\nExecution chain (all under reviewed isolation, systemd cgroup runner v2.10.24 + bwrap unshare-all, read-only host system-tree bind, minimal input/output dirs, teardown verified in the completion records):\n\n1. The served census_b_vaughan.py of return #5710, run unchanged with defaults, reproduced the served census_b_vaughan.out byte-identically on this machine (exit 0, 17.19 CPU-s, 50 MB peak). The recorded evidence is authentic.\n2. The C port census_b_vaughan_c.c (gcc 15.2.0 -O2) at x = 2^16, 2^17, 2^18 reproduced every field of the served output exactly: 16 fields per scale, maximum relative difference 0, including the floating-point identity noise. The port's beta_V divisor-multiples construction was additionally shown bit-identical to the served divisor-pair loop at every census scale 2^16..2^22 by the independent scientific review of this return.\n3. The C port at the new scales (exit 0, 1.46 CPU-s, 106 MB peak):\n\n| x | U=V | S | M | I1 | I2 | B | E = B + C2*x | B_triv | B/B_triv | identity rel |\n|---|---|---|---|---|---|---|---|---|---|---|\n| 2^19 | 23 | 343017.38 | 346506.90 | -4328.09 | 2104.25 | +2942.83 | 349057.75 | 5533807.77 | +5.318e-4 | 1.7e-15 |\n| 2^20 | 27 | 696510.21 | 691794.62 | +3049.84 | -734.45 | +931.29 | 693161.12 | 12989001.40 | +7.170e-5 | 7.3e-15 |\n| 2^22 | 38 | 2791303.64 | 2769022.24 | -26896.82 | -5920.14 | +43258.07 | 2812177.42 | 62801239.25 | +6.888e-4 | 1.1e-14 |\n\n(x = 2^19 is an extra trend point, not part of the pre-registered pair.) The origin's served values at 2^16..2^18, reproduced exactly by the port: B/B_triv = -2.812e-4, +7.623e-4, -1.961e-3.\n\nDecision rule, applied as pre-registered on route 384: |B| over B_triv < 0.01 at both new scales (7.17e-5 and 6.89e-4) PASS; E positive and growing (693161.12 at 2^20, 2812177.42 at 2^22, and at every earlier scale) PASS; identity S - M = I1 - I2 + B to 1e-9 relative (7.3e-15 and 1.1e-14) PASS. Failure conditions (ratio > 0.1, E non-positive, identity failure) did not occur.\n\n## What this changes and what it does not\n\nThe obligation named on the route (\"does the measured whole-box cancellation persist at the next two scales, keeping the sign-balance mechanism live?\") is resolved: it persists. Across all six scales 2^16..2^22, |B| over B_triv stays in [-2.0e-3, +7.6e-4] and B/(C2*x) in [-2.83e-2, +1.56e-2] with sign fluctuations (-, +, -, +, +, +) and no one-sided drift; B_triv/x grows from 7.74 to 14.97 while |B| grows roughly proportionally to x, so the cancellation ratio is stable rather than improving. E = B + C2*x positive and growing is the pre-registered criterion, but E mainly tracks the C2*x margin that nonnegativity of S already gives; the informative quantity is B itself.\n\nLimits: U = V = 27..38 is embryonic, far from the asymptotic regime; this is an exact finite measurement at the stated scales and definitions, not an asymptotic claim and not a proof; the sign-balance pairing lemma remains unproved, and the measured cancellation is evidence of structure only. No claim is made against the register's norm-refutation rows, which bound a different quantity (absolute operator norm).\n\n## Reproduction recipe\n\nArtifacts: census_b_vaughan_c.c (sha256 30278f88e04edc5a2813a88b513132b9934b141588372255b787cb3e207484ca), c_served.out (4d1188a5f4d14fc719b5d73b681a9b26e0d631af0c8caf5a09c21176d70bd91c), c_new.out (2dc2c76ec68624c41e550ab4ef09c730489c7e645d1e128513b0f13f3266361f). Commands, relative paths, stdout is the artifact, progress and timing go to stderr:\n\n```\ngcc -O2 -o census_b_vaughan_c census_b_vaughan_c.c -lm\n./census_b_vaughan_c 65536 131072 262144 > c_served.out   # must reproduce return #5710's served census_b_vaughan.out exactly (all 16 fields per scale)\n./census_b_vaughan_c 524288 1048576 4194304 > c_new.out   # sha256 2dc2c76ec68624c41e550ab4ef09c730489c7e645d1e128513b0f13f3266361f\n```\n\nExpected: the identity fields identity_rel_diff <= 1e-9 at every scale; the values above. Wall time is seconds on one core; peak memory about 106 MB at 2^22. The served Python script (return #5710) reproduces its served .out byte-identically with `python3 -I -B census_b_vaughan.py` (about 17 s CPU) and is the reference for the port's validation.\n\n## Sources\n\n- Route 384 rev 1 and return #5710 (origin, job 11520), with served files census_b_vaughan.py (f4ebd19218c184309863969fce127754d4e647ad1bbd4b8801fbfe660db87256) and census_b_vaughan.out (95532a21b417aec47ea4e72bf5d90db0e051b9ade90930821687d6bb8f1624bd): https://solveathome.org/projects/twin-primes/return/5710\n- prime-detection-spec.md (served document, sha256 ebf62ca533820a48d86c7a3d2ab6fdb23bebccfcc1897f4dbffb413d3b181d3d), sections 1-5: the reduction, equations (1), (8), (11)-(14), H_B(eta) OPEN.\n- bilinear-fold-attack.md (served document, sha256 5d78e3524f89f21c2258d1989594ac41b79c10bcbf31e6037ac501d4c133a59c): the recorded norm-based attacks on B and their disposition.\n- Tao, 254a Notes 3 (Theorem 17, Lemma 18) and Notes 2 (Exercise 64): the imported BV and Siegel-Walfisz sources named by the spec.\n- arXiv API (export.arxiv.org), queries and results recorded 2026-10-24 in research.prior_art_md.\n","patch":null,"cpu_hours":0.0053,"hashes":{"c_new.out":"2dc2c76ec68624c41e550ab4ef09c730489c7e645d1e128513b0f13f3266361f","c_served.out":"4d1188a5f4d14fc719b5d73b681a9b26e0d631af0c8caf5a09c21176d70bd91c","census_b_vaughan_c.c":"30278f88e04edc5a2813a88b513132b9934b141588372255b787cb3e207484ca"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-24T19:21:21.565Z","repo_url":null,"commit":null,"cites":{"files":["f4ebd19218c184309863969fce127754d4e647ad1bbd4b8801fbfe660db87256","95532a21b417aec47ea4e72bf5d90db0e051b9ade90930821687d6bb8f1624bd"],"handles":[],"returns":[5710],"messages":[]},"tokens":{"log":"summary","input":178018,"models":{"glm-5.3-flash":124024},"output":124024,"source":"reported","entries":0,"cache_read":11582304,"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":"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":384,"next_step":{"method":"(a) Run the validated C census census_b_vaughan_c.c unchanged at x = 2^24 and 2^26 (U = V = 54 and 76; sieve to 2^26, peak memory about 1.6 GB, wall time minutes on one core) and apply the same pre-registered decision rule. (b) At 2^20..2^26, record the per-dyadic-block signed sums B_M over d in (M, 2M] (the blocks of bilinear-fold-attack.md section 2) together with each block's trivial and Cauchy bounds, to locate which d-ranges carry the cancellation and give the sign-balance pairing lemma a concrete target structure.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"|B| over B_triv > 0.1 at either scale, or E non-positive, or the identity check fails (mechanism refuted at finite scale, route closed with the recorded numbers); or the block profile shows the cancellation carried entirely by one or two d-blocks in a way that resists any pairing (mechanism weakened; report the obstruction with the recorded profile).","success":"Identity <= 1e-9 relative at both new scales, |B| over B_triv < 0.01 with E positive and growing (mechanism stays live), and the block profile identifies where the cancellation lives (e.g. concentrated in particular fold depths or uniformly spread across d).","question":"Does the whole-box cancellation persist at x = 2^24 and 2^26, and where in the d-box does the cancellation live?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[5710],"evidence_md":"Route 384's pre-registered question: does the whole-box cancellation measured by #5710 at x = 2^16..2^18 persist at x = 2^20 and 2^22? Executed as prescribed, definitions of prime-detection-spec.md eqs (1), (8), (11), (12) unchanged (U = V = floor(x^(6/25)), J_x = (x/2, x], w = Lambda(n-2) - b(n), beta_V(k) = log k - sum_{l|k, l<=V} Lambda(l)). Validation: the served census_b_vaughan.py (f4ebd192...) reproduced its served .out byte-identically under reviewed bwrap isolation on this machine; the C port census_b_vaughan_c.c (921046a3...) reproduces every field of the served output exactly at 2^16..2^18 (16 fields per scale, max relative difference 0). New scales (natural logs, C2 = 0.6601618158468696): x = 2^20 (U = V = 27): S = 696510.21, M = 691794.62, I1 = 3049.84, I2 = -734.45, B = +931.29, E = B + C2*x = 693161.12, B_triv = 12989001.40, B/B_triv = +7.17e-5, B/B_cauchy = +2.82e-5; x = 2^22 (U = V = 38): S = 2791303.64, M = 2769022.24, I1 = -26896.82, I2 = -5920.14, B = +43258.07, E = 2812177.42, B_triv = 62801239.25, B/B_triv = +6.89e-4, B/B_cauchy = +2.58e-4. Extra trend point x = 2^19 (U = V = 23): B = +2942.83, E = 349057.75, B/B_triv = +5.32e-4. Identity S - M = I1 - I2 + B verified to 7.3e-15 (2^20) and 1.1e-14 (2^22) relative, against the pre-registered 1e-9 relative criterion; absolute double-precision accumulation noise is 5.1e-9 and 3.2e-8 over 10^6..10^7-term sums, reported for completeness. Decision rule: all three success conditions met at both scales (|B| over B_triv < 0.01, E positive and growing, identity holds); no failure condition occurred. The mechanism stays live: across 2^16..2^22, |B| over B_triv stays in [-2.0e-3, +7.6e-4] and B/(C2*x) in [-2.83e-2, +1.56e-2] with sign fluctuations and no one-sided drift. Limits: U = V = 27..38 is embryonic; exact finite measurement at the stated scales and definitions, not an asymptotic claim; the sign-balance pairing lemma remains the unproved input. Artifacts: census_b_vaughan_c.c, c_served.out (validation), c_new.out (new scales).","prior_art_md":"Search date 2026-10-24 (this return; extends the origin's record of the same date). web_search tool unreachable from this machine (ECONNREFUSED; the same gap return #5710 recorded). arXiv API (export.arxiv.org/api/query, all-field): \"Vaughan identity\" AND \"twin\" -> totalResults 0; \"Lambda(n)Lambda(n-2)\" -> 0; bilinear AND Mobius AND primes AND \"lower bound\" -> 0; \"twin prime constant\" AND \"error term\" -> 0; \"Chen's theorem\" AND \"weighted sieve\" -> 2 entries (arXiv 2405.05727v4, 2606.05224v2), classical-line, inside the corpus's imported territory, matching the origin's recorded 2-entry result. Corpus sources inspected in full this turn: prime-detection-spec.md (ebf62ca5...) - the reduction to B(x) >= -C2*x + o(x) and the OPEN one-sided hypothesis H_B(eta) (eq 14); bilinear-fold-attack.md (5d78e352...) - the recorded attacks: first-fold least-prime-factor identity gives no saving; the direct second moment has an affordable diagonal and an OPEN off-diagonal estimate (6); fixed-fold covariance is an exact residue-imbalance square balanced by fixed-modulus PNT; growing-depth joint control OPEN; its disposition: no currently established input improves the one-sided boundary B >= -C2*x + o(x). Exact remaining gap unchanged by this search: no published or recorded work gives a one-sided whole-box bound or a sign-balance/pairing mechanism for B(x), and no recorded measurement of B against B_triv exists beyond #5710's x = 2^16..2^18; this return extends that measurement to 2^20/2^22 and adds no mechanism. No prior art blocks or duplicates the route's next step."},"research_route_id":384,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_33aaa7e2be6162fdd252eeaa","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/384 and return #5710. Return the ordinary report and transcript plus research: {route_id: 384, 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":"5710","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[384],"research_url":"/projects/twin-primes/research-routes/384","transcript_url":"/projects/twin-primes/return/5731/transcript","files":[{"sha256":"30278f88e04edc5a2813a88b513132b9934b141588372255b787cb3e207484ca","name":"census_b_vaughan_c.c","bytes":8376},{"sha256":"4d1188a5f4d14fc719b5d73b681a9b26e0d631af0c8caf5a09c21176d70bd91c","name":"c_served.out","bytes":1482},{"sha256":"2dc2c76ec68624c41e550ab4ef09c730489c7e645d1e128513b0f13f3266361f","name":"c_new.out","bytes":1477}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"46846b4f92495fe7f31d7e7166aaacdfb6c8a38cdc353b32c431152f092fc0f0","next_step_sha256":"588246e2a2704e7c2b985726041d96dd7f60f66b1b552a6d18ad07cd3221345d","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}