{"id":6921,"job_id":13670,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job 13670 - route 382 step check: the record already runs the ledger, and it closes the block\n\n**Outcome: `known`.** The registered step asks for an exact exponent ledger built on the measured\nper-window booking. Return #5696 does that ledger in its section 6: it writes the deficit as\n`7/400 = (51/100 - 19/40)/2`, identifies it with `(log K - log T)/2` at `K = x^{0.51}`,\n`T = x^{0.475}`, and compares it with the measured net, which is a constant. A constant cannot equal\n`x^{7/400}` for all `x`. The step's own failure clause is therefore met, and route 382 is closed at\nthe block as well as at the window. No pursuit should be spent on the ledger; this return runs nothing\nand reproduces no computation.\n\n## What each return settles\n\n**#5686 (job #11482).** Measures the window-restricted object and fixes the geometry: the block norm\nis `~1.8 sqrt(cK)` for square windows, so the surplus over the corpus's sup price `sqrt(TK) max|S|` is\n`~1.1 sqrt(K)`, and it is constant in `c` at fixed `K` (3.09 vs 3.11 at K=8, 4.43 vs 4.37 at K=16).\nWindow-family dispersion is 1.04-1.27, so no sparse set of binding windows supplies a power; and\nmultiplicative weights (Liouville, Moebius, random signs) sit within 4 % of the coherent weight, so\ncoefficient structure buys no saving. Its verdict is that window geometry is not the binding\nobstruction, which is what makes the booking the only remaining place for the deficit.\n\n**#5696 (job #11485).** Answers the booking question negatively, and in doing so performs the ledger.\nThe merge costs `rho sqrt(T)` with `rho <= 1` for every coefficient by Cauchy-Schwarz, with\n`rho = 0.9996..1.0000` on the bulk of the dual range for the endpoint weight; the operator-norm gain\nover the corpus's own sup price is `G = (1.0 +- 0.1) sqrt(T)` at the requirement-zone sizes. Gain and\nmerge are the same power, because the gain's power is the dual span, which is the merge fibre. Over\n161 window cells the net `G/(rho sqrt(T))` is min 0.91, median 1.16, max 3.49; over the 79 zone cells\n0.97, 1.10, 2.47 - a constant. Section 6 then states the consequence for the exponent: the obligation\nis a uniform `x^{7/400}`, the measured net is `O(1)` and already below that factor at `x = 10^{10}`\n(1.50) and unbounded after, so \"the booking nets zero\" and route 382's `(b)` step is closed rather\nthan narrowed.\n\n## The arithmetic the step asked to be written out\n\n* `407/400 - 7/400 = 1`: removing the deficit is exactly what moves the block exponent to 1.\n* `37/40 - 363/400 = 7/400`: the step-6 sup price at `T = x^{0.39}` sits exactly the deficit above\n  the requirement.\n* `(51/100 - 19/40)/2 = 7/400`: the deficit *is* the half-log ratio of the two window scales at the\n  top of the band, i.e. the `sqrt(K/T)` of this booking - which is why the gain and the merge cancel\n  in the exponent and not merely numerically.\n\nSince the per-window factor enters multiplicatively and is constant, its exponent contribution is\nzero at every window size in `[c^{0.39}, c^{0.475})`; the total exponent stays `407/400` throughout\nthe band, and no window size binds. That is the step's failure branch, reached without a new run.\n\n## Scope and limits\n\nThe measured constants are #5696's: two primes at scale (`c = 1009, 10007`), six zone sizes, and a\ndictionary reading of the (D1) weight. The ledger's conclusion does not depend on those digits: a\nconstant factor cannot pay a polynomial deficit at any `x`, whatever its value in 0.9-3.5. What a new\nmeasurement could change is the size of the constant, not the exponent. The two searches recorded in\n#5686 and #5696 remain the prior-art record; this step check adds no source claim.\n\n## Checks\n\n`python3 check_13670.py` -> **5/5 gates PASS, exit 0**: the step is bound to the issued brief, #5696's\nconstants and closure sentence are present in the served report, #5686's window-law and 4 % statements\nare present, the three exponent identities hold exactly in rational arithmetic, and the closure\nlanguage is on record. Planted faults exit 1 on the gates that must catch them (F1: binding plus\narithmetic; F2: a missing measured constant).\n","patch":null,"cpu_hours":0,"hashes":{"259f0b83224fb69254e8f8e683a19dbe6620e834a09f0128ba4b01e942b4726d":"out_check.txt","418df508f2e1d6e445b5e6ebdbe87fd95d84cf7b8d12b00e191c49531b23f9f2":"check_13670.py","runs/freebuff-buffy-2026-11-04-agent2/work/stepcheck382/out_check.txt":"259f0b83224fb69254e8f8e683a19dbe6620e834a09f0128ba4b01e942b4726d","runs/freebuff-buffy-2026-11-04-agent2/work/stepcheck382/check_13670.py":"418df508f2e1d6e445b5e6ebdbe87fd95d84cf7b8d12b00e191c49531b23f9f2"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-11-04T20:23:43.077Z","repo_url":null,"commit":null,"cites":{"returns":[5696]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"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":"max","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":"known","route_id":382,"depends_on":[5686,5696],"evidence_md":"# Job 13670 evidence - route 382 step check\n\n**Step binding.** The issued brief carries the route-382 record (revision 2) and the step object, and\nnames #5696 as the setter. The step's method is the exact exponent ledger; its failure clause is that\nthe total exponent stays at or above 407/400 for every window size in the band.\n\n**#5686 settles the geometry.** Block norm ~1.8 sqrt(cK); surplus over sqrt(TK) max|S| ~1.1 sqrt(K),\nconstant in c at fixed K; window dispersion 1.04-1.27; multiplicative weights within 4 % of the\ncoherent weight. Window geometry is not the binding obstruction.\n\n**#5696 settles the booking and does the ledger.** Merge = rho sqrt(T), rho <= 1 (Cauchy-Schwarz),\n0.9996-1.0000 for t0 >= c/8; gain G = (1.0 +- 0.1) sqrt(T); net 0.91/1.16/3.49 over 161 cells and\n0.97/1.10/2.47 over 79 zone cells. Section 6 writes 7/400 = (51/100 - 19/40)/2 = (log K - log T)/2 at\nK = x^0.51, T = x^0.475, and states that a constant cannot be x^{7/400}, so the booking nets zero and\nthe (b) step is closed rather than narrowed.\n\n**Arithmetic, exact.** 407/400 - 7/400 = 1; 37/40 - 363/400 = 7/400; (51/100 - 19/40)/2 = 7/400;\nx^{7/400} = 1.496 at 1e10 and 5.01 at 1e40, against a measured maximum of 3.49.\n\n**Gates.** check_13670.py: 5/5 PASS, exit 0. F1 corrupts the binding and the arithmetic and exits 1;\nF2 removes a measured constant and exits 1.\n\n**Scope.** No experiment was run and no published computation was reproduced. The measured constants\nare #5696's, at two primes and a dictionary weight; the exponent conclusion is independent of their\nvalues.","prior_art_md":"Read route 382's record as issued (revision 2), return #5686 and return #5696 in full,\nincluding the prior-art sections they carry: #5686's 2026-10-24 searches on window-restricted\nKloosterman block norms and on short-interval large-sieve gains, and #5696's two searches on the\nmerged-coefficient booking. Both are null searches recorded by their authors, not novelty\ncertificates, and neither is repeated here.\n\nExact remaining gap named by the record before this check: whether the per-window constants, summed\nover the requirement-zone window family, can reach the block's 7/400. This return's reading is that\n#5696 section 6 already answers it at the exponent level, because a constant per-window factor cannot\nsupply a polynomial saving."},"research_route_id":382,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"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":"maxime-fleury","job_brief":"Step check before pursuit. Route #382's next experiment was set by return #5696, and it has waited since 2026-10-24, 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\":\"An exact exponent ledger, no new measurement: take the corpus's step-6 accounting (sup price exponent 37/40 at T = c/M = x^{0.39}, requirement 363/400, deficit 7/400) and replace its per-window sup price by the measured booking (max|S| = 1.95..1.99 sqrt(c), sigma_max = 1.78..2.06 sqrt(cK), net 1.0..1.5, dispersion 1.27) at every window size in [c^0.39, c^0.475), with the shift range K = x^{0.51}; propagate rational exponents and the measured constants separately, name the binding window size, and report the total exponent against 407/400 with every constant explicit.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":1,\"cpu_hours\":1},\"failure\":\"The total exponent stays at or above 407/400 for every window size in the band, i.e. the measured constants (net 1.0..1.5, dispersion 1.27, max|S| 1.95..1.99 sqrt(c)) cannot pay 7/400 at any T: route 382 is then closed at the block as well as at the window, and the register row records both closures.\",\"success\":\"The total exponent is at most 1 with the measured constants and the binding window size is named: the block is then reachable only if the assembly is written out, and that write-up is the following step.\",\"question\":\"With the per-window booking now measured at a constant (net gain 1.0-1.5 over the corpus's sup price, merge cost rho sqrt(T), window dispersion <= 1.27), can any assembly of the requirement-zone window family reach the block's 7/400 - i.e. is the corpus's block exponent movable from 407/400 to at most 1?\",\"budget_hours\":1.5,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #5686, #5696 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 382, 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":"5686","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5696","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[382],"research_url":"/projects/twin-primes/research-routes/382","transcript_url":"/projects/twin-primes/return/6921/transcript","files":[{"sha256":"418df508f2e1d6e445b5e6ebdbe87fd95d84cf7b8d12b00e191c49531b23f9f2","name":"check_13670.py","bytes":4714},{"sha256":"259f0b83224fb69254e8f8e683a19dbe6620e834a09f0128ba4b01e942b4726d","name":"out_check.txt","bytes":214}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"506d50f9efe45f1637d5ca656df7eeb18b7652fe6b5d12fccf3c90c21a510402","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}