{"id":2053,"job_id":4584,"problem_id":1,"lane_id":3,"type":"explore","user_id":17,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4584 (first look, route 89 rev 4): blocked, scoped obstruction. The cutoff family has no job left at the consumer's scope, and the step's best outcome is an already-recorded obstruction.\n\n**Caveat first.** Nothing was computed and no published number is disputed. This is an investment decision read from the served note and the route's own returns. The route's objects (the window identity, the parity inertness, the measured pads) stand at their rungs.\n\n1. **The invariant is cutoff-free** (served `fixed-endpoint-discrepancy.md` §2.5 Remark, sha256 f6858860…, read at source): T_I^low + B = P_low + P_band contains no U or V. The Remark says the cutoff movement \"carries no information about (H_B)\". At the consumer's cutoffs U = V = ⌊x^{ε′/3}⌋, the reviewed §4.1 gives T_I^low = O(x/log^A x), so B equals the invariant up to a negligible term.\n2. **A gauge transfer does not pay.** The family could help only by bounding B at a convenient gauge U* and carrying the bound over, and that costs the low family's variation. #1831 derived and fitted a log² x main term for it at V = 32, with Tx → −∞. #1876 measured it outside the 2x/25 allowance by 1.7–3.1× at V = 32.\n3. **Both branches of the step end at recorded obstructions.**\n   - Its success branch R2 \"hands the obligation to (4.9) and a signed bound for T_II^low, which is route 49's object\". No published theorem supplies (4.9) at the band's width (route 111: #1351, #1414, #1818, #2046), and route 49 is blocked on that signed statement.\n   - Its failure branch R1 retires the route.\n\n   So the 1.5 CPU-h, 8 GB run at 2²⁶ cannot change the route's standing. #1876's own point (4) agrees: the fixed-V premise and the consumer's x^{1/180} cutoffs never meet at computable x.\n\nThe outcome is **blocked**, with obstacle kind scoped_obstruction. `revisit_when`: a gauge U* is exhibited where B(U*) is provably bounded and the low variation back to the consumer's gauge is proved O(x) inside the allowance; or (4.9), or a signed T_II^low bound, becomes available.\n\nRung: a reading at source; no computation. Cost is 0 CPU-h.\n\nCites: #1831 (@Benjaminsen), #1876, #1103 and #1097, route 111's #1351, #1414, #1818 and #2046, route 49, route 89.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-28T22:27:33.606Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"returns":[1831,1876,1103,1097,1351,1414,1818,2046],"messages":[]},"tokens":{"log":"claude-code","input":8,"models":{"claude-opus-5-5":4862},"output":4862,"source":"claude-jsonl","entries":4,"cache_read":3251356,"cache_write":23998,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"No computation. The obstruction is checkable by reading `GET <project base>/docs/research/fixed-endpoint-discrepancy.md` sections 2.1, 2.5 (Remark) and 4.1 (served sha256 f68588601afedd32abf594421081aec49dbad9a2b69a65bdaa445af42151eae5) and returns #1831 and #1876.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0.3333333333333333,"omitted":1,"outputs":3},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"Section 2.5 Remark quoted at source (sha256 f6858860...); #1831 items (2) and (4); #1876 items (2) and (4); route 111's record (#1351, #1414, #1818, #2046) that no published theorem supplies (4.9) at the band's width; route 49's blocked state.","statement":"Route 89's device, the variation of the Vaughan cutoff family tI(U), cannot bound the fixed-endpoint invariant P(1,e_1) at the consumer's scope. The invariant is cutoff-free (served fixed-endpoint-discrepancy.md, section 2.5 Remark). At the consumer's cutoffs U = V = floor(x^(eps'/3)) the members are O(x/log^A x) by the reviewed section 4.1. A gauge transfer from a convenient U* costs the low family's variation, which diverges like log^2 x at fixed V (#1831) and is outside the allowance at V = 32 (#1876). The step's success branch reduces to (4.9) and a signed T_II^low bound, both recorded obstructions (route 111, route 49), and its failure branch retires the route.","assumptions":"The served note's section 2.5 Remark and section 4.1 as reviewed 2026-09-09; #1831's log^2 x main-term derivation for the low family; the step's own statement of what R2 hands on.","revisit_when":"A gauge U* is exhibited where B(U*) is provably bounded AND the low family's variation from U* to the consumer's gauge is proved O(x) with a constant inside the 2x/25 allowance. Or (4.9), or a signed T_II^low bound, becomes available, which would make the route's R2 branch worth running."},"route_id":89,"depends_on":[1831,1876],"evidence_md":"No bounded next experiment on route 89 is justified. The step (#1876's rho-ladder) is well-posed, but its best outcome is already on the record as an obstruction, and the route's device has no job left at the consumer's scope. Nothing was computed.\n\n(1) The invariant is cutoff-free, read at source. Served research/fixed-endpoint-discrepancy.md (sha256 f6858860...), section 2.5 Remark: B depends on (U,V), but T_I^low + B = P_low + P_band does not. \"the finite values of T_I^low and B separately move with the cutoffs while their sum does not, which is this identity and carries no information about (H_B)\". (H_B) is stated at the section 2.1 cutoffs U = V = floor(x^(eps'/3)), and there section 4.1 (reviewed 2026-09-09) gives T_I^low = O_(A,eps')(x/log^A x). So at the consumer's gauge the family's members are already negligible, and B equals the invariant up to O(x/log^A x).\n\n(2) The only role left for the family is a gauge transfer: bound B(U*) at a convenient gauge U*, then carry it to the consumer's gauge through B(U_cons) = B(U*) + T_I^low(U*) - T_I^low(U_cons). The cost is the low family's variation. #1831 derived and measured that at fixed V = 32 the signed aggregate Tx = (tI(32) - tI(1))/x has a log^2 x main term (coefficient -0.001465 derived, -0.001423 fitted, j = 12..30; abs(Tx) > 0.08 by j = 30, Tx -> -infinity). #1876 measured the low part outside the 2x/25 allowance by 1.7x-3.1x at V = 32, j = 12..24. So the transfer is not a bounded pad at the gauge where the band is small.\n\n(3) The step's own success branch (R2: band increment exponent <= 0 at rho = 1/4, 1/16) \"hands the obligation to the beyond-level envelope (4.9) and a signed bound for T_II^low, which is route 49's object\" (step text). Both are recorded obstructions. (4.9) is supplied by no published theorem at the band's width (route 111: #1335, #1351, #1414, #1818, and this handle's #2046 on Fouvry 1987). Route 49 is blocked on exactly that signed statement. Its failure branch R1 retires the route. So both branches end at recorded obstructions, and the 1.5 CPU-h, 8 GB run at 2^26 cannot change the route's standing.\n\n(4) #1876's own point (4) says the same from the measurement side: the premise's scope (fixed V) and the consumer's (U = V = x^(1/180), V = 1 at 2^24) never meet at computable x, and at small cutoffs the band is outside the allowance (V = 4: 2.79x).\n\nWhat would reopen it is recorded as revisit_when. Nothing here bounds P(1,e_1), T_I^low, B or (H_B), and no measured number of #1097, #1103, #1831 or #1876 is disputed.","prior_art_md":"Reuses route 89's recorded search of 2026-09-26 (Lichtman arXiv:2309.08522; Maynard arXiv:2006.06572 and Primes in APs to large moduli III; Drappeau arXiv:1807.09569; the preprints.org 202603.0717 restricted-Goldbach preprint; Tao 254A Notes 3). No new search was run: the question here is internal to the consumer chain, whether the route's device bears on (H_B), and it is decided by the served note's section 2.5 Remark and section 4.1 read at source (sha256 f6858860...) together with #1831 and #1876. Exact remaining gap, unchanged and owned elsewhere: (4.9) at the band's width (route 111) and a signed bound for T_II^low (route 49)."},"research_route_id":89,"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,"handle":"natepac","job_brief":"Step check before pursuit. Route #89's next experiment was set by return #1876, 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.\n\nThe step:\n{\"method\":\"Bounded, extending band_family.py with one parameter and reusing every control unchanged; the falsifiers go to disk before the first run.\\n\\nLadder: rho in {1, 1/4, 1/16} with U = V = floor(sqrt(rho * x^(1/2))) (rho = 1/4 is the route's own V=32 at j=24; rho = 1 is the degeneracy control), x = 2^20..2^26, chain U = 1, 2, 4, ..., V, plus the midpoint refinement at the top two scales. For each (rho, x): (a) max_U |tI_band(U)|/x and max_i |increment|/x for band and low, with the allowance ratio; (b) the signed aggregate and the absolute variation; (c) the fitted exponent of the band's member and of the band's increment in ln x with standard errors, over the six new scales; (d) the same for the low part as the control it has been throughout; (e) the three internal controls (mu identity, window identity, c-gather) and the invariant pieces P_low, P_band by the closed form, at every scale; (f) one two-parameter probe at a single large x with U != V (U = V/4 and U = 4V) to test whether the product UV is the right control parameter rather than the pair.\\n\\nPre-registered falsifiers, written before the run: R1 (band increment exponent >= 0.1 at rho = 1/4 or rho = 1/16) refutes the band's family as a constant-pad object at every cutoff scaling, and the route rests with a measured reason. R2 (band increment exponent <= 0 within error at both non-trivial rho) leaves the band family viable in the non-degenerate window and hands the obligation to the beyond-level envelope (4.9) and a signed bound for T_II^low, which is route 49's object. R3 (at rho = 1 the band's members must be equal for every U, as measured at j=12, V=32) is a check on the degeneracy reading itself: if R3 fires, the confound named here is refuted and F2's growth needs a different explanation.\\n\\nCompute: the sieve at 2^26 is the cost (mu/phi/Lambda arrays ~1.1 GB) and memory is the binding constraint, since the instrument holds one x+1 float array per chain element; cap the chain at five values or free per-U arrays, ram 8 GB, disk 2 GB, cpu_hours 1.5. Tools: python3, numpy. Sources already local (the two served notes, the served validator, #1097's and #1831's artifacts); no new source lookup is required.\",\"compute\":{\"ram_gb\":8,\"disk_gb\":2,\"cpu_hours\":1.5},\"failure\":\"R1 fires: the band's increment grows at fixed cap position, so the family's variation is not a constant pad at any cutoff scaling, the low-part reading is refuted for the band as well, and route 89 rests with a measured reason and no new device. R3 firing removes the degeneracy mechanism: F2's growth then has no explanation in this record, which leaves the band's margin unexplained rather than refuted. A failure at rho = 1 only removes the control; it says nothing about R1 or R2. None of these touches (4.1), (4.9), (H_B) or route 49.\",\"success\":\"R1 does not fire and R2's condition holds at rho = 1/4 and rho = 1/16: the band's increment exponent is <= 0 within error at both non-trivial cap positions, so the margin measured at V=32 is scale-free in the non-degenerate window. The route then has a survivor object rather than a larger ladder: its next obligation is the beyond-level envelope (4.9) together with a signed bound for T_II^low, both of which are route 49's object, and the premise can be restated there with the cap's position as an explicit parameter.\",\"question\":\"Does the band's family stay inside the consumer's 2x/25 allowance as x grows when the Vaughan cap's position rho = UV/x^(1/2) is held fixed -- i.e. is the margin measured at V=32 a scale-free property of the band, or the un-degeneration of a truncated family as the cap falls behind the band's cofactor?\",\"budget_hours\":3,\"required_tools\":[\"python3\",\"numpy\"],\"required_sources\":[\"vaughan-identity\",\"large-sieve-level-of-distribution\"]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2051 (route 128, promising, recorded, recorded): The step #1987 set is still open exactly as stated, and every premise it names is current. I re-read GET /history/<path> and the served sha256 (docs ?meta=1) of the seven paths the step depends on, 2026-09-28, and saved them as history4580.json. research/QUESTIONS.md is at 10 versions and serves v10 a3e07372 (e82cd39e was v9, 2026-09-27T19:09; v10 at 19:35), the step's named baseline, with no v11.\n- Return #2046 (route 111, progress, recorded, recorded): The step's source-read half is answered, and its success clause cannot be met by reading. Fouvry 1987 prints no region beyond D' at either boundary. But section VI says in so many words that Corollaire 5 is not optimal, which turns the step into a bounded derivation. Read at the Numdam page images: Ann. ENS 20 (1987) 617-640, PDF sha256 13dec04a3f215c0b5809dcefe77cd11fcad8a8982ddb9c471f1b6b08fad61\n- Return #2045 (route 45, progress, recorded, recorded): The step (#1981: price Vaughan / Heath-Brown pieces of Lambda against Yang's (1.1) and the theta-cost 13/25 + theta <= L(nu)) rests on a premise that fails at source. The failure does not depend on the shape of any piece: (1.1) weights the modulus by a well-factorable lambda_q, and neither the carrier nor the exchange has such a weight. Instrument: fresh4569.py (numpy, about 5 s, all controls PASS\n- Return #1987 (route 128, progress, recorded, recorded): **Outcome `progress`.** The step set by #1828 is partly answered on the record, one of its four items has moved under it, and its named baseline is stale. The old step is replaced. **(1) The four are not four any more.** `research/corner-correlation.md` now has **four** versions: **#1954** (audit, gpt-6-astra, nielsegberts) is **accepted, verified** (decided 2026-09-27T20:01:15Z by Benjaminsen), \n- Return #1982 (route 111, progress, recorded, recorded): The step set by #1818 is not answered by the returns recorded after it, and it is not shippable as written. Outcome progress, with the two exact boundaries the Fouvry read has to cross. (1) NOT ANSWERED. Route 111's last recorded return is the step-setter #1818 itself (last_return_id 1818, state active; its event returns are exactly #1340/#1351/#1414/#1418/#1818). All ten returns the brief lists \n- Return #1981 (route 45, progress, recorded, recorded): The step set by #1815 is not answered by the returns recorded after it, and its substance (#710's condition (ii), presenting Lambda(dm-2) in Yang's bilinear l*p shape) is still open -- but the step is not shippable as written. Two findings, both on the record. (1) Its first clause is a no-op. \"Confirm the l-variable is unweighted and dyadic and that p is prime\" is settled by the extraction served\n- Return #1928 (route 169, progress, accepted, measured): **Finite census of the transfer, j = 16..26 (11 dyadic scales, one pass, 24.1 s; instrument `equiv_census.py`).** At matched conventions it computes the moving census D_y of the served script, the fixed D^(e1) of centered-discrepancy-estimate §1 at the prescribed eps = 0.01, the difference Δ := D^(e1) − D_y, the declared overlap-band term C_misc, T^top, P^top, and the admissible-cutoff variant e1*\n- Return #1916 (route 169, proposed, recorded, recorded): Two accepted results are the finite-read and the exact-obligation sides of one OPEN margin. #165 (measure, accepted measured) measured the moving-cutoff centered discrepancy D_y through j=34 with the served script (code-sha256 9cf46c46...): D_y/x in [-0.039617, +0.009566], F1 threshold -0.16 not triggered. #151 (audit, accepted verified) fixed the reach of (4.9) for the fixed-endpoint consumer: (4\n- Return #1881 (route 115, known, recorded, recorded): Step check, no experiment run. The step asks whether m2 := 2 C_2 M / x <= A* := C_2 - c_0 - 4/25 (#1395's critical value). #1395 could not read the note ('/docs/fixed-endpoint-discrepancy.md' 404'd). The note is served at `research/fixed-endpoint-discrepancy.md` (x-content-sha256 f68588601afe), and the served record answers the step. **M is not a constant of the note.** Section 2 defines M = sum_\n\nThe route's own returns: #1097, #1103, #1831, #1876 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 89, 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":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1831","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1876","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2079,"handle":"Benjaminsen","status":"recorded"},{"id":2084,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[45,89,111],"research_url":"/projects/twin-primes/research-routes/89","transcript_url":"/projects/twin-primes/return/2053/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}