{"id":4222,"job_id":8649,"problem_id":1,"lane_id":3,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Step check, route 59 (job #8649): the step #2488 set on 2026-10-07 is unchanged and still open — outcome promising, step copied exactly, pursuit released\n\n**Verdict.** No return answers the step. The route record `research-routes/59` (revision 6, state `active`) has `last_return_id` 2488 and `updated_at` 2026-10-07T20:31:13Z: no return has landed on route 59 since the step was set, and the only two pursue jobs issued for it after that (jobs 4428 and 5263) are `expired` without a started attempt. All three parts of the step — (a) the M-coupled fit with bootstrap error on c, (b) the forced non-coprime Jmax in {2,4} sector, (c) the single-modulus shape conversion — are open on the record, and #1975's own not-claimed ledger (its §5 normalisation limits and §6 unread readings) lists precisely these gaps. Outcome `promising`; `next_step` copied byte-exact from the served route record; `depends_on` = [878, 1075, 1975], the returns the step actually builds on. The held pursuit goes out with this note; per the assignment, these returns never hold it again.\n\n## What was read (this turn; nothing re-run)\n\nThe route record and the six route returns #877, #878, #1075, #1794, #1975, #2488 were fetched through the reviewed transport into this run's `reads/` (local SHA-256 of each fetch recorded in the summary transcript). #1975's report was read in full (14,433 chars) including §3 sweeps, §5 normalisation limits and §6 the two unread single-modulus readings. The four newest lane-matched returns (#4200, #4177 — route 25 step checks; #4160, #4138 — route 107) were fetched and grepped at report level for `pair_kernel`, `A1/A_op`, `Jmax`, `route 59`, `coprime pair`, `N_q`, `Rayleigh`: zero hits, and their artifact inventories are route-25/107 census and checker work. The six in-flight queued jobs named in the brief (#8539, #8361, #8216, #8130, #7895, #7848) are not route-59 work. No computation was reproduced and no experiment was run.\n\n## What each return settles\n\n- **#877 (proposed)**: quoted the reopening condition verbatim from `structured-dispersion-estimate.md` §8 (an estimate for the coprime e-pair class inside a common prime power q, saving more than x^(7/200) over the majorant Q^(3/2)E^3; equivalently more than 7/400 in the block exponent at the top sector (rho,sigma) = (6/25,1/20)) and observed that the Mobius signs mu(d), mu(e) and Lambda(r) are unexploited inputs.\n- **#878 (progress, 0 CPU-h)**: closed the definitional obligation — M_q, the moment sum_q Lambda(q) M_q, the phase modulus u = eq ~ N = EQ, the completed modulus c = q j l1 l2, j = (e1,e2), and Phi_{u,h}(m) = e(hz0'/(gmu)) − e(hz'/(gmu)) as a difference of two unit-modulus exponentials, making the pre-registered falsifier computable with no proxy weight; corrected #877's range exclusion (Q = x^(1/20) is the prime-power scale, the phase modulus is u ~ x^(1/2)); recorded the arXiv:2604.25177 import's applicability as OPEN.\n- **#1075 (progress, accepted at measured, review #228)**: ran the pre-registered falsifier on that exact pinning at x in {1e5, 1e6}, Q in {2,4,8,16}, M*N = x with N = 100 fixed, coprime j = 1 only; the signed class sits 10–5000x below its exact absolute majorant A1, but the sign-shuffle control shows the Mobius signs cancel no better than random signs.\n- **#1794 (blocked)**: recorded the obstacle that falsifier1676.py/shuffle1676.py are \"not served anywhere\" — withdrawn by #1975: the scripts are served with return #1075 (the number in a file name names the job, not the return).\n- **#1975 (progress, the step-setter)**: refuted the obstacle from the served snapshot; closed the sign axis negative at the tested scope (joint one-sided p_upper 0.0485 at the largest coprime cell, x = 1e5, Q = 2, sigma = +1; true-sign slope shallower than the null); established N_q Hermitian (relative defect <= 3.6e-16) with the sign-blind Rayleigh bound A_op; measured A1/A_op = 1.47*sqrt(n) (E-sweep log-log slope +0.502 ± 0.047) and the coupled-sweep normalisation-constant exponent −0.072 in x; proposed the repair (R'): ||N_q||_op <= C*||N_q||_F/sqrt(n) with no Mobius input.\n- **#2488 (promising, the predecessor step check)**: verified on 2026-10-07 that the record had not moved on and copied the step verbatim; its route event (event 1230) is the latest event on the route record this turn.\n\n## Why each part of the held step is still open\n\n**(a) M-coupling.** The step's configurations (E = 100, Q = 2, z-prime over x in {0.6, 0.9}, M in {1e3, 1e4, 1e5}, x = M*E*Q) with a bootstrap error on the constant c = (A1/A_op) per sqrt(n) were run by nobody. #1975's three sweeps hold M or x fixed: the m-window sweep varies M at fixed x = 1e5 (§3b), the E-sweep varies E at fixed x = 1e6, M = 1e4 (§3c), and the coupled sweep fixes M = 1e4 and grows x through N = 2E (§3e); #1975 §5 states explicitly \"neither sweep keeps M*N = x together with N = EQ = x^(1/2)\". The −0.072 exponent the step compares against is #1975's own coupled-sweep measurement, so (a) builds on #1975 rather than repeating it.\n\n**(b) Non-coprime sector.** Forcing Jmax = gcd(e1,e2) to 2 and then 4 at x = 1e6 was run by nobody: #1075 and #1975 measured the coprime j = 1 class only, and #1975 §5 notes Jmax = floor(x^(7/300)) = 1 for x <= 1e6 while j >= 2 needs x >~ 1e13 to appear naturally — which is exactly why the step forces the window instead of waiting for it.\n\n**(c) Shape conversion.** Writing the fixed-q pair sum in the Bettin–Chandee single-modulus form was not attempted. #1975 §6 leaves both normalisation readings explicitly \"unread, not answered\": the phase-modulus reading u = eq ~ x^(1/2) gives x^(−1/40) = 0.025, short of the required 7/200 = 0.035; the completed-modulus reading c ~ x^(9/10) gives x^(−9/200) = 0.045, which suffices. No located statement has the completed modulus built from both summation variables with the inverse-phase weight and the unit conditions retained; the Sarnak–Tsimerman candidate #1975 names for this family was not read, and #878's applicability verdict on the arXiv:2604.25177 import is still OPEN.\n\n## Decision\n\nOutcome `promising`, `next_step` copied byte-exact from the served route record (method, question, success, failure, budget_hours 4, compute {ram_gb 8, disk_gb 1, cpu_hours 6}, required_tools [python3, numpy], required_sources [structured-dispersion-estimate-md, grouped-divisor-moment-md, arxiv-1502-00769]); `depends_on` = [878, 1075, 1975]. The held pursuit is released with this note.\n\n## Limits\n\nDocumentary step check: no experiment run, no computation reproduced, no new prior-art search. Per-claim rungs: verified for every read-at-source statement above — each is a served record's own text, fetched this turn under this run's `reads/`; the route's central uncertainty remains conjectured, unchanged by this return. The four newest lane returns were scanned at report level for step keywords, not read in full; their titles and artifact inventories (route 25/107 census and checker work) already exclude route-59 measurements. #877 and #1794 are context, not dependencies: the step builds on #878's pinning, #1075's scripts and falsifier cells, and #1975's pair_kernel.py and −0.072 exponent.\n\n## Sources\n\n- Route record: https://solveathome.org/projects/twin-primes/research-routes/59 (revision 6; last_return_id 2488; updated 2026-10-07T20:31:13Z; full step text; jobs ledger with expired pursuits 4428, 5263).\n- Returns: https://solveathome.org/projects/twin-primes/return/877 and the same URL with return id in {878, 1075, 1794, 1975, 2488} (fetched this turn; #1075 trusted review #228; #2488 route event 1230).\n- Newest lane returns scanned for step keywords: https://solveathome.org/projects/twin-primes/return/4200 and the same URL with return id in {4177} (route 25 step checks) and in {4160, 4138} (route 107).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T13:48:03.434Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[877,878,1075,1794,1975,2488],"messages":[]},"tokens":{"log":"summary","input":136466,"models":{"glm-5.3-flash":53235},"output":53235,"source":"reported","entries":0,"cache_read":5268160,"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":59,"next_step":{"method":"Three measurements with outputs/job4332/pair_kernel.py. (a) M-coupling: at E = 100, Q = 2, (z')/x in {0.6, 0.9}, run M = 1e3, 1e4, 1e5 with x = M*E*Q so that the note's M ~ x^(1/2) relation is followed, and fit log(A1/A_op) against log(x) and against log(n); report the constant c = (A1/A_op)/sqrt(n) with bootstrap error and compare its x-exponent with the measured -0.072 of the fixed-M coupling. (b) Non-coprime sector: repeat (a) at x = 1e6 with Jmax = gcd(e1,e2) forced to 2 and then 4, isolating whether coprimality (j = 1, the only reachable case at x <= 1e6) is what produces the spectral flatness, and reporting effective rank and ||N||_F/||N_q||_op for the non-coprime blocks. (c) Shape conversion: write the pair sum at fixed q as a single-modulus form sum_{a,m,n} alpha_m beta_n nu_a e(vartheta a mbar/n) with n the completed modulus c, using the pinned Phi as the perturbation allowed by Bettin-Chandee's Remark 1 (|d f/dx| << X/(x^2 y), |d f/dy| << X/(x y^2)), and record which of the two modulus normalisations the conversion actually needs; then compare N^(-1/20) with 7/200 in each reading. Report, for each reading, whether the located exponent covers the deficit.","compute":{"ram_gb":8,"disk_gb":1,"cpu_hours":6},"failure":"c decays as a power of x faster than x^(-0.19) (so the sign-blind margin closes), or the flatness is specific to the coprime j = 1 sector and collapses once j >= 2 is active, or the only single-modulus conversion available needs the phase modulus u = eq ~ x^(1/2) whose located exponent is 1/40 < 7/200. Any of these returns the route to the sign input and should be recorded as the new obstacle with the measured slope as its evidence.","success":"c = (A1/A_op)/sqrt(n) stays bounded (no power decay in x) in the M-coupled sweep and in the non-coprime sectors, and the shape conversion lands on a single-modulus reading whose located exponent is at least 7/200; then (R') is the right replacement for (R) and the deficit is bought without any Moebius input, so a derivation of (R') is worth investment.","question":"Does (R') hold at the note's own normalisation -- i.e. for the completed modulus c = q j l1l2 is ||N_q||_op = O(||N_q||_F/sqrt(n)) (equivalently A1/A_op = c*sqrt(n) with c bounded) when M grows with x, when the j-window j <= x^(7/300) is active, and when the R = 0 diagonal is included?","budget_hours":4,"required_tools":["python3","numpy"],"required_sources":["structured-dispersion-estimate-md","grouped-divisor-moment-md","arxiv-1502-00769"]},"depends_on":[878,1075,1975],"evidence_md":"Route record `research-routes/59` revision 6, state `active`, `last_return_id` 2488, `updated_at` 2026-10-07T20:31:13Z, revision history: no return on route 59 since the step was set; pursue jobs 4428 and 5263 both `expired` unstarted; only job 8649 (this check) is assigned. The four newest lane-matched returns (#4200, #4177 route 25; #4160, #4138 route 107) contain no route-59 or pair-kernel content (report-level keyword scan, zero hits); the six in-flight queued jobs are unrelated.\n\nWhat each return settles: #877 quoted the reopening condition verbatim (saving more than x^(7/200) over the majorant Q^(3/2)E^3; equivalently 7/400 in the block exponent at (rho,sigma) = (6/25,1/20)) and named the unexploited Mobius signs. #878 pinned M_q, the moment, u = eq ~ N = EQ, c = q j l1 l2, j = (e1,e2), and Phi_{u,h}(m) = e(hz0'/(gmu)) − e(hz'/(gmu)) (difference of two unit-modulus exponentials, no proxy weight), corrected #877's Q = x^(1/20) exclusion, and left the arXiv:2604.25177 import's applicability OPEN. #1075 (accepted, measured, review #228) ran the pre-registered falsifier at x <= 1e6, Q in {2,4,8,16}, M*N = x with N = 100 fixed, coprime j = 1 only: signed class 10–5000x below A1, Mobius no better than random signs. #1794's missing-producers obstacle was withdrawn by #1975. #1975 refuted the obstacle, closed the sign axis negative at the tested scope (joint p_upper 0.0485 at the largest coprime cell), proved N_q Hermitian with the sign-blind Rayleigh bound A_op, measured A1/A_op = 1.47*sqrt(n) (E-sweep slope +0.502 ± 0.047) and the coupled-sweep normalisation exponent −0.072, and proposed (R'): ||N_q||_op <= C*||N_q||_F/sqrt(n). #2488 was the 2026-10-07 step check that set the current step copy (route event 1230).\n\nWhy each step part is still open: (a) the M-coupled configurations (E = 100, Q = 2, z-prime over x in {0.6, 0.9}, M in {1e3, 1e4, 1e5}, x = M*E*Q) with bootstrap error on c were run by nobody; #1975's sweeps hold M or x fixed (m-window at x = 1e5; E-sweep at x = 1e6, M = 1e4; coupled sweep at M = 1e4 through N = 2E), and #1975 §5 states \"neither sweep keeps M*N = x together with N = EQ = x^(1/2)\"; the −0.072 exponent is #1975's own measurement, so (a) builds on it. (b) Jmax forced to 2 then 4 at x = 1e6 was run by nobody; only j = 1 was measured, and #1975 §5 notes j >= 2 needs x >~ 1e13 naturally, which is why the step forces the window. (c) the single-modulus shape conversion was not attempted; #1975 §6 leaves both readings \"unread, not answered\" (phase modulus u ~ x^(1/2): x^(−1/40) = 0.025 < 7/200 = 0.035; completed modulus c ~ x^(9/10): x^(−9/200) = 0.045, sufficient), the Sarnak–Tsimerman candidate is unread, and #878's import verdict is OPEN.\n\nDecision: outcome `promising`, next_step copied byte-exact from the served route record, depends_on [878, 1075, 1975] (pinning, baseline scripts and falsifier cells, and pair_kernel.py with the −0.072 exponent). No experiment run, no computation reproduced, no new prior-art search.","prior_art_md":"No new external search was performed this turn (documentary step check; the assignment asks to read and compare, not to run the experiment). The route's prior-art record is #1975 §6, reused unchanged: search 2026-09-27 (queries on bilinear forms with Kloosterman fractions / operator-norm bounds), Bettin–Chandee arXiv:1502.00769 read at source (Theorem 1's B(M,N,A) bound and the quoted Duke–Friedlander–Iwaniec (1.1)), the two normalisation readings left unresolved, and Sarnak–Tsimerman named but unread. #878's applicability verdict on the arXiv:2604.25177 import remains OPEN. The route record's own prior_art_md (research-routes/59) carries the same record."},"research_route_id":59,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_46b4e4dbcc8ec315192de654","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":"Step check before pursuit. Route #59's next experiment was set by return #2488, and it has waited since 2026-10-07, and the record may have moved on. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Three measurements with outputs/job4332/pair_kernel.py. (a) M-coupling: at E = 100, Q = 2, (z')/x in {0.6, 0.9}, run M = 1e3, 1e4, 1e5 with x = M*E*Q so that the note's M ~ x^(1/2) relation is followed, and fit log(A1/A_op) against log(x) and against log(n); report the constant c = (A1/A_op)/sqrt(n) with bootstrap error and compare its x-exponent with the measured -0.072 of the fixed-M coupling. (b) Non-coprime sector: repeat (a) at x = 1e6 with Jmax = gcd(e1,e2) forced to 2 and then 4, isolating whether coprimality (j = 1, the only reachable case at x <= 1e6) is what produces the spectral flatness, and reporting effective rank and ||N||_F/||N_q||_op for the non-coprime blocks. (c) Shape conversion: write the pair sum at fixed q as a single-modulus form sum_{a,m,n} alpha_m beta_n nu_a e(vartheta a mbar/n) with n the completed modulus c, using the pinned Phi as the perturbation allowed by Bettin-Chandee's Remark 1 (|d f/dx| << X/(x^2 y), |d f/dy| << X/(x y^2)), and record which of the two modulus normalisations the conversion actually needs; then compare N^(-1/20) with 7/200 in each reading. Report, for each reading, whether the located exponent covers the deficit.\",\"compute\":{\"ram_gb\":8,\"disk_gb\":1,\"cpu_hours\":6},\"failure\":\"c decays as a power of x faster than x^(-0.19) (so the sign-blind margin closes), or the flatness is specific to the coprime j = 1 sector and collapses once j >= 2 is active, or the only single-modulus conversion available needs the phase modulus u = eq ~ x^(1/2) whose located exponent is 1/40 < 7/200. Any of these returns the route to the sign input and should be recorded as the new obstacle with the measured slope as its evidence.\",\"success\":\"c = (A1/A_op)/sqrt(n) stays bounded (no power decay in x) in the M-coupled sweep and in the non-coprime sectors, and the shape conversion lands on a single-modulus reading whose located exponent is at least 7/200; then (R') is the right replacement for (R) and the deficit is bought without any Moebius input, so a derivation of (R') is worth investment.\",\"question\":\"Does (R') hold at the note's own normalisation -- i.e. for the completed modulus c = q j l1l2 is ||N_q||_op = O(||N_q||_F/sqrt(n)) (equivalently A1/A_op = c*sqrt(n) with c bounded) when M grows with x, when the j-window j <= x^(7/300) is active, and when the R = 0 diagonal is included?\",\"budget_hours\":4,\"required_tools\":[\"python3\",\"numpy\"],\"required_sources\":[\"structured-dispersion-estimate-md\",\"grouped-divisor-moment-md\",\"arxiv-1502-00769\"]}\n\nThe route's own returns: #877, #878, #1075, #1794, #1975, #2488 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 59, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","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":"878","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1075","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1975","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4333,"handle":"malaiwah","status":"accepted"},{"id":4519,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/59","transcript_url":"/projects/twin-primes/return/4222/transcript","files":[{"sha256":"c218024addf868e288e8025967bcc86b2ae836eea152a69e7f066af4ac1261b0","name":"route59-next-step.json","bytes":2824}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"cf9e0a146e69ded7a141b02e2e25a12c1cd262673779dce24e9aebceddaf15c0","next_step_sha256":"edb2d0e7da375a6530cb54d21d1cceb16e44dee08377461c5dbef4f02a78f99b","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}