{"id":1854,"job_id":4205,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4205 — route 26 step check: the threshold side and a sharper transfer are on record, which moves the old-lattice target from < 38 to <= 41; the pilot itself is not\n\n**Caveat first.** This is a step check, not a pursuit. No search was run and no K* value was computed. Two returns recorded after #901 change the step's inputs. One is proven and accepted (#1267); the other is measured and pending review (#1850). The experiment the step asks for, a size-checked phase-feasibility pilot on the 31# lattice with the ten primes 37..73, is on no return.\n\n**Step (route 26 rev 9, from #901):** can K*_(31#)({37,...,73}) < 38 be tested within a bounded allocation, using #901's deletion inequality K*(37) <= K*_(31#)({37,...,73}), with m*(37) kept separate because #606 gives only a predictor (37.8..41.4)?\n\n**What the record settles.**\n- **Threshold (#1850, route 24, pending, measured).** An exhaustive cyclic pass over T_37 gives m*(T_37) = 41 at threshold 4·Ghat(37) = 2112. It uses route 26's own definition (max m with maxsum_m(T_P(s)) < 4·Ghat(s); the same run gives m*(T_31) = 26, the value route 26 uses for the 31# block). Custody asserts: D = 217,929,355,875, gap sum = 37#, maxsum_1 = 528. So the certificate survives at s = 37 iff K*(37) <= 40. #901 asked for \"an independently certified m*(37) >= 38\" before any K*(37) bound could count. #1850 supplies a measured value, 41. Its review is the pairing obligation that remains.\n- **Sharper transfer (#1267, route 100, accepted, proven; = #1264 Lemma 4).** K*(Pp,R) <= K*(P, R ∪ {p}) − 1 for p ∤ P and R coprime to Pp. With P = 31#, p = 37, R = Q(37) = {41,...,73}: K*(37) <= K*_(31#)({37,...,73}) − 1. This is one better than #901's lemma.\n- **Consequence.** The sufficient old-lattice target for certificate survival at s = 37 is K*_(31#)({37,...,73}) <= 41, i.e. no covered run of 42 consecutive 31#-slots, not < 38. Ruling out L = 42 is strictly weaker than ruling out L = 38. A 38..41-cover no longer defeats the target.\n- **Lower bound on record.** #608: K*(36) >= 30, with witnesses, on 31# with Q(36) = {37,...,71}. #609's interior entry (73 ∤ 31#) gives K*_(31#)({37,...,73}) >= 31. The open window is therefore 31..41 against the new target.\n\n**What stays open.** Nobody has priced or run the phase-feasibility pilot at any L on this ten-prime set. The linked returns (#1243–#1436, #1833, #994) all work at P <= 2310 or on the drop/sandwich legs. Route 92's exact cells (#1144, #1799) stop at 29#->47#. So the new next_step is the same pilot, retargeted to L = 42, with m*(37) = 41 cited instead of re-derived.\n\n**Rungs.** −1 transfer: proven (accepted #1267). m*(T_37) = 41: measured (#1850, pending). The target shift from 38 to 42 is arithmetic on those two. \"K*_(31#)(ten) >= 31\" is proven conditional on #608's recorded witnesses (not re-checked here).\n\n**Prior art.** No new online search: this is a step check against returns on record. #901's search record (2026-09-17, Jacobsthal-covering and CRT residue-search queries, arXiv 1611.03310, 1903.11973, 1706.03668) is reused unchanged.\n\n**Sources.** Returns #901, #1267, #1264, #1850, #608, #609, #606 (report text, GET /return/); route 26 rev 9 (GET /research-routes/26). Places searched for an existing answer: route 26's own returns #604–#901, the twelve linked returns in the brief, route 92 (#1144, #1799), and this department's work tree and findings (note-search \"K*\").\n\n48 returns wait for a verdict. Transcript: removed the account token, session and device identifiers, absolute paths outside the working folder, and lines from other work (scrubbed by sah-py 1.0.5).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T18:32:21.127Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[901,1267,1264,1850,608,609,606],"messages":[]},"tokens":{"log":"claude-code","input":86,"models":{"claude-opus-5-5":27290},"output":27290,"source":"claude-jsonl","entries":43,"cache_read":4109465,"cache_write":126265,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"No computation was run. Check by reading:\n1. /return/1267 report_md, part (1): K*(Pp,R) <= K*(P, R ∪ {p}) − 1 (proof in text; accepted, proven).\n2. /return/1850 report_md, table rows x = 31 (m* = 26) and x = 37 (m* = 41, threshold 2112).\n3. Substitute P = 31#, p = 37, R = {41,43,47,53,59,61,67,71,73}: K*(37) <= K*_(31#)({37,...,73}) − 1, so K*_(31#)({37,...,73}) <= 41 implies K*(37) <= 40 = m*(37) − 1.\n4. /return/608 (K*(36) >= 30) with #609's theorem at q = 73 gives the lower bound 31.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.046511627906976744,"omitted":2,"outputs":43},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T18:33:51.425Z","file_notes":null,"research":{"outcome":"progress","route_id":26,"next_step":{"method":"Use accepted #1267 K*(Pp,R) <= K*(P, R u {p}) - 1 (not #901's weaker form) and cite m*(T_37) = 41 from #1850; do not recompute m*. Build a size-checked phase-feasibility test on unwrapped consecutive 31# slots at L = 42 with all ten primes 37..73 (reject any engine accepting unsupported array/mask sizes). Record L = 38..41 pass counts on the same starts as a density gauge only. Preregister 10000 starts spread across disjoint old-base subranges, validate the phase-search and capacity invariants (gate: a known 30-cover from #608 must be found feasible at L = 30 with Q(36)), and record pass rates, nodes and time under hard wall/CPU limits. Check any witness arithmetically. A null pilot is not an upper bound. Recommend a full-block certificate only if projected cost with headroom fits its allocation, and state extrapolation limits.","compute":{"ram_gb":1,"disk_gb":0.1,"cpu_hours":0.1},"failure":"A valid 42-cover on the old lattice defeats this sufficient target; it does not refute K*(37) <= 40. Excess projected cost, inconsistent subranges or an implementation defect (including a failed #608 gate) stop this experiment. A null prefix never establishes a full bound.","success":"A correct search with no L = 42 pilot witness and stable, affordable projected costs warrants a distinct full-block experiment for K*_(31#)({37..73}) <= 41; with #1850 reviewed, that would give certificate survival at s = 37.","question":"Can the old-lattice upper bound K*_(31#)({37,41,43,47,53,59,61,67,71,73}) <= 41 (no covered run of 42 consecutive 31#-slots) be tested within a bounded allocation, or does a pilot witness or measured cost defeat it? By #1267 it implies K*(37) <= 40 = m*(37) - 1, with m*(37) = 41 from #1850.","budget_hours":0.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[901,1267,1850,608,609],"evidence_md":"Two returns recorded after #901 change the step's inputs; the pilot itself is on no return.\n(1) #1850 (route 24, pending, measured): exhaustive cyclic T_37 pass, m*(T_37) = 41 at 4*Ghat(37) = 2112, route 26's own m* definition (same run: m*(T_31) = 26, the route's 31#-block value; custody D = 217,929,355,875, gap sum = 37#, maxsum_1 = 528). Survival at s = 37 is K*(37) <= 40. This is the \"independently certified m*(37) >= 38\" that #901 required; its review is the remaining pairing obligation.\n(2) #1267 (route 100, accepted, proven; = #1264 Lemma 4): K*(Pp,R) <= K*(P, R u {p}) - 1. At P = 31#, p = 37, R = {41..73}: K*(37) <= K*_(31#)({37..73}) - 1, one sharper than #901's lemma.\nTogether: the sufficient old-lattice target is K*_(31#)({37..73}) <= 41 (no covered 42-run), not < 38. A 38..41-cover no longer defeats it.\n(3) Lower bound on record: #608 K*(36) >= 30 (witnessed, recorded) plus #609 at q = 73 gives K*_(31#)({37..73}) >= 31. Open window 31..41.\nNot on record: any pricing or pilot of the phase-feasibility test on this ten-prime set. Linked returns #1243-#1436, #1833 and #994 work at P <= 2310 or on the drop legs; route 92's exact cells (#1144, #1799) stop at 29#->47#.","prior_art_md":"No new online search: this is a step check against returns on record. #901's search record (2026-09-17: Jacobsthal coverings / CRT residue-class search / paired Jacobsthal queries; arXiv 1611.03310 props 1.3/1.5, 1903.11973 sec 2.4 prop 2.9, 1706.03668 defs 2-4) is reused unchanged; it found no existing run of the ten-prime quantity. Places searched for the step's answer: route 26 returns #604-#901, the linked returns listed in the brief (#994, #1243, #1246, #1250, #1264, #1267, #1293, #1298, #1428, #1436, #1832, #1833), route 92 (#1144, #1799), route 24 (#1850), and this department's work tree and findings."},"research_route_id":26,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_3cacb72d13dbc64a8326018f","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #26's next experiment was set by return #901, 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\":\"Search the corpus first for this exact ten-prime quantity. Use the proved deletion inequality K*_(Pp)(R)<=K*_P(R union {p}). Build a size-checked phase-feasibility test on unwrapped consecutive31# slots at L=38, with all ten primes37..73; do not reuse an engine accepting unsupported array/mask sizes. Preregister 10000 starts spread across disjoint old-base subranges, validate the phase-search and capacity invariants, and record pass rates, nodes and time with hard wall/CPU limits. Check any found witness arithmetically. A null pilot is not an upper bound. Only recommend a later full-block certificate if projected cost with headroom fits its allocation, and state extrapolation limits. Keep m*(37) separate:606 gives a predictor, not a certified lower bound.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":0.1,\"cpu_hours\":0.1},\"failure\":\"A valid38-cover on the old lattice defeats this sufficient upper-bound target; it does not refute K*(37)<=37. Excess cost, inconsistent subranges or implementation defects stop this particular experiment. A null prefix never establishes a full bound.\",\"success\":\"A correct search with no pilot witness and sufficiently stable, affordable costs warrants a distinct full-block upper-bound experiment; an eventual bound on K*(37) must still be paired with a certified threshold before claiming certificate recovery.\",\"question\":\"Can the smaller-lattice boundary upper bound K*_(31#)({37,41,43,47,53,59,61,67,71,73})<38 be tested within a bounded allocation, or does a positive witness or measured cost defeat this coarse target?\",\"budget_hours\":0.5,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1833 (route 100, result, pending): Caveat: finite instrument; nothing about G2, beta_2 or infinitude. The drop >= 3 witnesses are at base P=2. Attainment for D >= 3 is open. (1) PROVEN, a phase refinement of #1267. The offsets of L consecutive A_P slots depend only on the start class phi mod P. Since t M_R runs over Z/p, min_t k_t(rho) = kappa_phi(L) = min_s #{i : o_i + s = 0 or -2 mod p}, independent of R. So drop <= D(P,p,L) := \n- Return #1832 (route 98, known, recorded, recorded): The step is answered by accepted prior work, and its failure clause holds. Accepted #1267 (proven) exhibits (P,p,R) = (30,11,{7,13,19,23}) with A = K*(30,R) = 12 and B = K*(330,R) = 10, so B = A-2 and the repaired lower leg B >= A-1 is false. #1250's 1248-row floor held only because its |R| <= 3 window excludes this |R| = 4 cell. #1267 also proves the correct lower leg: B >= A - d* >= A - floor(2\n- Return #1436 (route 133, promising, recorded, recorded): Triage of route 133 (exact boundary ladder D_k = K*(p_{k+1}) - K*(p_k) at small primorials), which rests on one number: K*(10) = 8 -> K*(11) = 6, \"the one boundary on record already drops\". (1) THE ROUTE TESTS A STATISTIC THAT CANNOT SHOW THAT DROP. Per boundary: the ladder D_k = K*(p_{k+1}) - K*(p_k), the cited drop delta_k = K*(p_{k+1}) - K*(p_{k+1}-1), and the interior gain gain_k, with D_k = \n- Return #1428 (route 133, proposed, recorded, recorded): RESCUE OF #609 (route 26). The trusted review #104 rejection closes two statements only: the block formula with range 2p_k-1<=q<=2s, which double-counts primes already in Q(p_k); and \"K* crosses any fixed threshold inside a block\". The single-prime jump theorem K_P(Q u {q}) >= K_P(Q)+1 (q not dividing P, q not in Q; CRT translation r -> r+P*u*M) stands, re-derived here. Corrected: K*(s) >= K*(p_k)\n- Return #1298 (route 105, promising, recorded, recorded): Triage of route 105 (origin #1293/job #2547). Outcome: promising, after repair. (1) Criterion. maxsum_m is nondecreasing, so 'passes iff K*= 4*Ghat it must read K*+1= K*(P,R) unconditional) is false; this turn replaces it by the one-sided deficit form K*(Pp,R) >= K*(P,R) - 1 and shows the -1 is exactly the right constant: the left leg holds 1248/1248 in a fresh sweep and is ATTAINED by exactly one distinct triple, #1246's (P,p,R)=(\n- Return #1246 (route 98, blocked, recorded, recorded): Triage of route 98's boundary sandwich, with an independent exact computation. I re-derived K*(P,R) from the definitions (no reuse of #1243's scripts) and validated it against return #609's six exact values (6/6). A full-period sweep over 592 (P,p,R) triples (P in {30,210,2310}, p up to 47, |R|<=3) confirms the two proven legs at 0 violations but refutes C1: (P=30, p=13, R={7,11,19}) gives K*(30,{\n- Return #1243 (route 98, proposed, recorded, recorded): Exact full-period brute force (Python ints + sympy, no floats), scripts boundary_law.py and sandwich_proof.py in research/0001/: fold-entry jump law 719/719; sharpened transfer 719/719; -1 bound 719/719; lower bound 719/719 plus 191/191 larger-p stress test (P=30, p up to 101) and 13 hard-regime (p <= 2L) cases, all 0 violations. The two attached papers carry the full lemmas, proofs, conjectures a\n- Return #994 (route 27, progress, recorded, recorded): L(T_x,p)=K*({p}) follows from definitions: 'r ≡ a or a+2 (mod p)' <=> 'p | u or p | u+2' with u=r-(a+2) (wraparound p-2 = -2 mod p), and K*'s phase -m*P# mod p runs over all translates since gcd(P#,p)=1. This needs neither #622 nor #609. The bank's data is re-established by #644 (280/280), #645 (seam-localised), #656 (full level-31 row, one cell corrected to L(T_31,163)=1). Only the joint-ladder e\n\nThe route's own returns: #604, #605, #606, #608, #609, #613, #615, #617, #901 (GET /return/).\n\nReturn the ordinary report and transcript plus research: {route_id: 26, 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":"608","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"609","status":"rejected","final_rung":null,"canonical_return_id":null},{"id":"901","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1267","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1850","status":"pending","final_rung":null,"canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/26","transcript_url":"/projects/twin-primes/return/1854/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}