{"id":6732,"job_id":13405,"problem_id":1,"lane_id":32,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job #13405 — route 408 step check: the (g,e) step's success clause is unsatisfiable as written — outcome `progress`, step replaced\n\nRead-only comparison. No experiment was run and no published computation was reproduced; cpu_hours 0\n(served reads and one stdlib record-integrity checker `check_13405.py`: **9/9 gates PASS exit 0**;\n8/8 planted-fault modes each exit 1 with the named FAIL — `check_13405.out`, `check_13405_faults.out`).\nEvery quoted fact below was verified verbatim against the served bytes in `reads/`.\n\n## Custody\n\nServed route-408 `next_step` canonical sha256 `e917d26c5dc7e2748f7d19d70d72fc3c9859882ac40ef35c8d2c84705d4b6a2c`\n= #6280's served `next_step_sha256`, and the brief's step object is field-identical. Route 408 active,\nrevision 4, `last_return_id 6280`, `updated_at 2026-10-27T16:26:47.372Z`; no route-408 return postdates\n#6280 (route jobs: 12469 → #6252, 12502 → #6264, 12514 → #6280, 12536 expired, 13405 this check).\n\nThe step's success clause reads, verbatim: *\"Some dyadic block's (g,e) ratio is at most 0.5 while the top four\nblocks (m > H/8) still carry >= 0.9 of the (g,e) mass: the (g,e) split separates where the m-split does not,\nso Step 1 can be posed block-locally on that block with the aligned closed form available inside it.\"*\n\n1. **The direction of the (g,e)-to-m comparison is on the record three times.** #6264 (accepted, measured):\n   the m-aggregate `sum_m |U_nu(m)|` is \"a lower bound for it (triangle inequality over the (g,e) split of\n   each m)\". #6252 (accepted, measured): \"`sum_m |U| <= sum_{ge} u_0 u_2 |D^un|`\". #6280 (accepted, measured):\n   \"a lower bound by the triangle inequality over the `(g,e)` split of each `m` (this is #6264's stated\n   limit and it is unchanged)\". The step itself makes the same direction a requirement: the m-aggregate\n   \"must not exceed the second sum\".\n2. **Per block and branch the ratios therefore order as (g,e)-ratio >= m-ratio.** The step requires the\n   aligned (g,e) sum to reproduce the block AL exactly, so both ratios share the denominator; the triangle\n   inequality gives the (g,e)-object numerator >= the m-aggregate numerator, block by block, branch by branch.\n3. **#6280's served block table puts every block m-ratio above 0.5 at H = 3e8:** branch-0\n   `sum|U_0| / AL_0` = 0.644446 … 0.586438 across the twelve blocks; branch-1 `sum|U_1| / AL_1` = 0.722832 …\n   0.654860 (block 0 has no branch-1 term). Minimum over all 23 recorded ratios: 0.586438.\n4. **Corollary (two lines, from recorded facts only):** at H = 3e8 the success clause cannot fire in any\n   block of #6280's recorded table — (g,e)-ratio >= m-ratio >= 0.586438 > 0.5. #6280's F2 also records the\n   combined block ratios at H = 1e8 (0.636–0.694), likewise above 0.5. At H = 1e5 and 1e6 (the step's first\n   scales) per-block branch ratios are not tabulated, but the step's own control (\"whose per-branch ratios\n   the new run must bracket\", i.e. #6264's H = 1e5 row plus #6280's 0.586–0.723 block profile) makes a\n   <= 0.5 block a violation of its own bracketing requirement rather than a clean success. The success\n   clause is dead as written, and the step's rule also has a hole: a (g,e)-ratio between 0.5 and 1.1x its\n   m-ratio fires neither clause.\n5. **Minor defect:** the success clause labels the top four blocks \"m > H/8\"; block j = (H/2^(j+1), H/2^j],\n   so blocks 0–3 span m > H/16.\n\nThis is not a new theorem: it is the conjunction of the triangle direction stated in #6252/#6264/#6280\nwith #6280's served block table, offered for review. No number was recomputed.\n\n## What the record answers and what it does not\n\n- **Answered:** \"the (g,e) split separates where the m-split does not\" is impossible in the sense the\n  success clause defines. Block-local posing of Step 1 cannot gain at any block of #6280's recorded table\n  (H = 3e8; all 23 recorded per-branch ratios above 0.5, and the F2 combined block ratios at H = 1e8\n  likewise). The route's placement question, as posed by #6280's success clause, is closed negatively\n  without running anything.\n- **Not answered:** the actual value of Step 1's own object. No return measures\n  `sum_{ge<=H} u_0(g)u_2(e)|D^un|` or its block profile; the m-aggregate (`sum|U|` = 0.463325 / 0.468242 /\n  0.472515 of H log H at 3e7 / 1e8 / 3e8; `sum|U| / AL` = 0.669856 / 0.679241 / 0.687656) is only its triangle\n  lower bound. Whether the (g,e)-object sits within 10% of the m-aggregate in every block (neutrality\n  structural at the pair level) or above it (per-pair concentration the m-split cannot see) is open, and it\n  decides what the re-derivation of Step 1 must attack.\n\n## Census of returns after #6280 (route + linked)\n\n- Route 408: none (last_return_id 6280). In-flight lane jobs #13378, #13371, #13367, #13319, #13175,\n  #13174 are on other routes (176/107, the paired gap word, the centered prime-Möbius discrepancy, and\n  three staging-document audits) — none on route 408.\n- Lane dir-558 returns after #6280: #6284, #6501, #6582, #6634, #6663, #6668, #6693, #6703, #6704, #6708,\n  #6718, #6722, #6727. Not one cites a route-408 return (6229/6252/6264/6280) or depends on one, and none\n  contains the step's execution vocabulary (`u_0`, `D^un`, `(g,e)`). The only shared-premise link is\n  #6708 (route 177) via #6156: it reduces route 177's B_1 to the odd branch of #6156's residual machinery\n  and records that \"the unaligned residual of the middle factor-count band\" is unbounded on record —\n  consistent with route 408's Step 1 being open, and bearing nothing on the (g,e) block-placement question.\n  The two \"dyadic\" hits outside route 408 are different objects: #6284/#6501's TT/MRT scale-average witness\n  and #6727's corner-measurement at finite dyadic scales.\n\n## The replaced step\n\n`research.next_step` replaces #6280's step with the same instrument, counted work, scales and controls:\nthe (g,e) decomposition carried through the class recursion, accumulated inside the twelve dyadic m-blocks,\nper block and branch reporting the aligned (g,e) sum (must reproduce #6280's block AL exactly), the\nm-aggregate (must equal the block |U| and must not exceed the (g,e)-object — the triangle direction, kept\nas a validation gate), and Step 1's own (g,e)-weighted object. The dead success clause is removed; the\npre-registered rule is now the live dichotomy — neutrality (every block's per-branch (g,e)-ratio within 10%\nof its m-aggregate ratio) versus concentration (some block more than 10% above, with 2x marking strong\nconcentration) — which partitions the outcomes and measures the never-measured Step-1 constant either way.\n\n## Scope and limits\n\nComparison only. The corollary is a derived consequence of recorded facts at their recorded rungs\n(#6252/#6264/#6280 accepted, measured; the triangle direction quoted verbatim); no bound is proved here,\nno route-408 number was recomputed, and nothing bears on twin-prime infinitude. The corrected step keeps\n#6280's compute envelope (0.5 CPU-h, 2 GB RAM, 2 GB disk, `cc`) and its H = 1e5 → 1e6 scale plan. If the\nreviewer finds the triangle reading wrong, the old step stands and this return should be rejected.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-31T15:27:52.863Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6156,6229,6252,6264,6280,6708],"messages":[]},"tokens":{"log":"summary","input":145964,"models":{"glm-5.3-flash":91303},"output":91303,"source":"reported","entries":0,"cache_read":17861724,"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":"progress","route_id":408,"next_step":{"method":"The same exact-Phi instrument with the (g,e) decomposition carried through the class recursion and accumulated inside the twelve dyadic m-blocks, exactly as #6280's step posed it: the aligned terms are already enumerated per divisor g|m (A(g)J(m/g) psi_2(H/g) and A(g)J(m/g) T(H/(2g))), so the block AL is reproduced by summing them per block; the new work is the per-pair unaligned term, which needs each CRT class's weight projected onto the g-part and the e-part of the class (the class generator itself is reused; the counted work is sum over odd squarefree m <= H of 6^omega(m) = 3^omega(m) classes times 2^omega(m) divisors). Per block and branch report: the aligned (g,e) sum (must reproduce this return's block AL exactly), the m-aggregate sum_m|U_nu(m)| (must equal this return's block |U| and must not exceed the (g,e) object - the triangle direction, kept as a validation gate), and the (g,e)-weighted unaligned object sum_{(g,e)} u_0(g)u_2(e)|D^un(g,e)| (Step 1's own object, never measured), plus each block's (g,e)-object share of the total and the top four blocks' share (blocks 0-3 span m > H/16). The old success clause is removed: it required a block (g,e)-ratio <= 0.5, which the triangle direction makes unreachable wherever the m-ratio exceeds 0.5 (all 23 recorded table ratios at H = 3e8, min 0.586438). Pre-registered on the per-block per-branch ratios of the (g,e)-object to AL: neutrality = every block within 10% of its m-aggregate ratio (recorded ranges: branch 0 0.586438..0.644446, branch 1 0.654860..0.722832 at H = 3e8); concentration = some block more than 10% above its m-ratio, with the 2x level marking strong concentration. Scales: H = 1e5 first, where the counted work is ~1e9-1e10 class-divisor operations (controls on the record: #6264's row sum|U| 501360.8, AL 816725.6, n0 40527, and #6280's block profile at 3e7/1e8/3e8, whose per-branch ratios the new run must bracket); then H = 1e6 if the cost model is confirmed, since 6^omega grows like (log H)^5.","compute":{"ram_gb":2,"disk_gb":2,"cpu_hours":0.5},"failure":"Every block's per-branch (g,e)-ratio is within 10% of its m-aggregate ratio (branch 0 0.586438..0.644446, branch 1 0.654860..0.722832 at H = 3e8): neutrality is structural at the (g,e) level too, the dyadic split cannot absorb Step 1's object at either level, and Step 1 has to be re-derived as a global constant-level bound on the signed per-m aggregate (measured UN/(H log H) = -0.320 flat across three decades, sum|U| / AL = 0.670..0.688) rather than posed block-locally; the route's two-zone decoupling is then measured dead at both levels.","success":"Some dyadic block's per-branch (g,e)-ratio exceeds that block's m-aggregate ratio by more than 10% (per-pair concentration the m-split cannot see): the measured (g,e)-object is then Step 1's own constant, larger than the m-aggregate's 0.46-0.47 H log H, and the concentration pattern (which (g,e) pairs and blocks carry it) is the input the re-derivation of Step 1 needs; at the 2x level the divisor-measure bound is strained by that factor and the within-m cancellation structure becomes the object to bound.","question":"Step 1's own object sum_{ge<=H} u_0(g)u_2(e)|D^un| has never been measured: the m-aggregate is its triangle lower bound, and the old success clause ((g,e)-ratio <= 0.5) is unsatisfiable because the (g,e)-ratio is >= the m-ratio per block and every block m-ratio of the recorded table (H = 3e8) exceeds 0.586. Does the (g,e)-weighted unaligned object sit within 10% of the m-aggregate in every dyadic block (pair-level neutrality), or does per-pair mass concentrate where the m-split cannot see it?","budget_hours":0.5,"required_tools":["cc"],"required_sources":[]},"depends_on":[6156,6252,6264,6280],"evidence_md":"Step check of route 408's held step (set by #6280), read-only, cpu_hours 0. Custody: served route-408 next_step canonical sha256 e917d26c5dc7e2748f7d19d70d72fc3c9859882ac40ef35c8d2c84705d4b6a2c = #6280's served next_step_sha256; route active, revision 4, last_return_id 6280; no route-408 return postdates it. DECISIVE: the step's success clause ('some dyadic block's (g,e) ratio at most 0.5 ...') is unsatisfiable as written. The record fixes the direction: #6264 states the m-aggregate sum_m|U_nu(m)| is 'a lower bound for it (triangle inequality over the (g,e) split of each m)'; #6252 states sum_m|U| <= sum_{ge} u_0 u_2 |D^un|; #6280 repeats it verbatim; the step itself requires the m-aggregate 'must not exceed the second sum'. Both ratios share the denominator (the step requires the aligned (g,e) sum to reproduce block AL exactly), so per block and branch (g,e)-ratio >= m-ratio. #6280's served block table: branch-0 sum|U_0| / AL_0 = 0.644446..0.586438 over twelve blocks, branch-1 sum|U_1| / AL_1 = 0.722832..0.654860 over eleven (block 0 has no branch-1 term); minimum 0.586438 > 0.5. Corollary: at H = 3e8 no block can satisfy the success clause; at H = 1e5/1e6 a <= 0.5 block would violate the step's own bracketing control (#6264's 1e5 row, #6280's 0.586-0.723 block profile). The rule also has a hole: a (g,e)-ratio between 0.5 and 1.1x its m-ratio fires neither clause. Minor: the success clause labels the top four blocks 'm > H/8'; blocks 0-3 span m > H/16. NOT answered by the record: the value of Step 1's own object sum_{ge<=H} u_0(g)u_2(e)|D^un| - never measured; the m-aggregate (sum|U| / (H log H) = 0.463325/0.468242/0.472515 at 3e7/1e8/3e8, sum|U| / AL = 0.669856/0.679241/0.687656) is only its triangle lower bound. Census of returns after #6280: route 408 none; lane dir-558 #6284 #6501 #6582 #6634 #6663 #6668 #6693 #6703 #6704 #6708 #6718 #6722 #6727 - none cites or depends on a route-408 return (6229/6252/6264/6280) and none contains the step's execution vocabulary (u_0, D^un, (g,e)); the only shared-premise link is #6708 (route 177) via #6156, which records that the unaligned residual of #6156's middle factor-count band is unbounded - consistent with Step 1 being open, bearing nothing on the (g,e) placement question. #6284/#6501's dyadic content is the TT/MRT scale-average witness; #6727's is corner-measurement at finite dyadic scales. OUTCOME progress: the step is replaced by the same instrument, counted work, scales and controls with the dead success clause removed and the live dichotomy pre-registered (neutrality: every block's per-branch (g,e)-ratio within 10% of its m-aggregate ratio; concentration: some block more than 10% above, 2x = strong). Checker check_13405.py: 9/9 gates PASS exit 0; 8/8 planted-fault modes each exit 1 with the named FAIL."},"research_route_id":408,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-31T15:27:52.863Z","department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_ac9fcc2daf49e1ff6475259a","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 #408's next experiment was set by return #6280, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. 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\":\"The same exact-Phi instrument with the (g,e) decomposition carried through the class recursion and accumulated inside the twelve dyadic m-blocks. The aligned terms are already enumerated per divisor g|m (A(g)J(m/g) psi_2(H/g) and A(g)J(m/g) T(H/(2g))), so the block AL of this return is reproduced by summing them per block; the new work is the per-pair unaligned term, which needs each CRT class's weight projected onto the g-part and the e-part of the class (the class generator itself is reused; the counted work is sum over odd squarefree m <= H of 6^omega(m) = 3^omega(m) classes times 2^omega(m) divisors). Report per block: sum over (g,e) of A(g)J(e)(psi_2 or T) (must reproduce this return's block AL exactly), sum over (g,e) of u_0(g)u_2(e)|D^un(g,e)| (Step 1's object), and sum_m|U_nu(m)| (must equal this return's block |U| and must not exceed the second sum), plus each block's ratio of the second to the first and to AL. Scales: H = 1e5 first, where the counted work is ~1e9-1e10 class-divisor operations; then H = 1e6 if the cost model is confirmed, since 6^omega grows like (log H)^5. Controls on the record at H = 1e5: #6264's row (sum|U| 501360.8, AL 816725.6, n0 40527) and this return's block profile at the three larger scales, whose per-branch ratios the new run must bracket.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":2,\"cpu_hours\":0.5},\"failure\":\"Every block's (g,e) ratio is within 10% of that block's m-aggregate ratio (branch 0: 0.586-0.644; branch 1: 0.655-0.723 at H = 3e8): the neutrality this return measured is then structural at the (g,e) level too, the dyadic split cannot absorb Step 1's object, and Step 1 has to be re-derived rather than posed block-locally.\",\"success\":\"Some dyadic block's (g,e) ratio is at most 0.5 while the top four blocks (m > H/8) still carry >= 0.9 of the (g,e) mass: the (g,e) split separates where the m-split does not, so Step 1 can be posed block-locally on that block with the aligned closed form available inside it.\",\"progress\":{\"kind\":\"mathematical\",\"rationale_md\":\"Moves the route's placement question one level finer, to the (g,e) split that Step 1 actually needs, now that the m-level split has been measured to be value-nondegenerate but constant-neutral.\"},\"question\":\"The dyadic m-split is non-degenerate in value but neutral in constant: at H = 3e8 the branch-resolved ratios sum|U_0|/AL_0 = 0.586438..0.644446 and sum|U_1|/AL_1 = 0.654860..0.722832 across the twelve blocks, against block totals 0.642072 and 0.719248, so the split separates |U| from AL by at most a factor 1.10 and block-locality cannot pay the divisor measure. Step 1's own object is not this m-aggregate but the (g,e)-weighted sum sum_{ge<=H} u_0(g)u_2(e)|D^un|, of which the m-aggregate is a triangle-inequality lower bound (#6264's stated limit). Does the (g,e) split separate inside a dyadic block where the m-split does not?\",\"budget_hours\":0.5,\"required_tools\":[\"cc\"],\"required_sources\":[]}\n\nThe route's own returns: #6229, #6252, #6264, #6280 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #6708 (route 177, progress, recorded, recorded): # Evidence — job #13372 (route 177 step check, outcome `progress`) Read-only comparison; no experiment run, no published computation reproduced; cpu_hours 0. Checker `check_13372.py`: 14/14 gates PASS exit 0; 10/10 planted-fault modes exit 1 with the named FAIL (`check_13372.out`, `check_13372.corrupt.out`); every quoted fact verified verbatim against the served bytes in `reads/` (hash-recorded p\n\nReturn the ordinary report and transcript plus research: {route_id: 408, 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":true,"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":"6156","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"6252","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"6264","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"6280","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[408],"research_url":"/projects/twin-primes/research-routes/408","transcript_url":"/projects/twin-primes/return/6732/transcript","files":[{"sha256":"77c2993e76f7749c9e93a0de63a5746e95f2377e3e7d188b28b2f21305338b84","name":"check_13405.py","bytes":8331},{"sha256":"021a5ab96d057f4cab53c180ced060e0e2f03b15008eb6b28159520626175819","name":"check_13405.out","bytes":277},{"sha256":"ac949d9343648758c50c19251db0f18ab9fafa06cdf838c533d932271e81aaa4","name":"check_13405_faults.out","bytes":362}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"cfea403337c99fed5f539744bdddb5f6157efa30b846d6b683afc40981161edd","next_step_sha256":"823b9e6ac2ac0771e5f6b03535852bc564797e07b4cf6ed2285ab2650ee9169d","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}