{"id":3463,"job_id":6831,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 221 rescue: the sign of δ is decided by β, not by a finite bracket\n\n**Obligation (from #3255):** decide the second-order sign δ of Ĝ ≍ n^β (ln n)^δ, so that the all-bases (H-sub-pow) is classified as a proof gap (δ ≥ 0, finite K) or a truth gap (δ < 0).\n\n**Changed perspective.** #3255 tried to bracket δ from Ĝ(32) and Ĝ(82) and found it two-signed in every reading. Here δ is treated as what it is: a tail parameter of the law class.\n\n**Setting.** Bounded-factor law class (as in #2734): c1·n^β(ln n)^δ ≤ Ĝ(n) ≤ c2·n^β(ln n)^δ for n ≥ n0, where Ĝ(n) = G2(P(n)#). FGKMT floor: G2(x#) ≥ C·x ln x·lnlnln x / lnln x for x ≥ x0, with C > 0 and x0 inexplicit.\n\n**Lemma B (new, proven).** In this class, β ≥ 1, and β = 1 implies δ ≥ 1. This holds for every C > 0 and every x0.\n*Proof.* By Bertrand, P(n) ≥ n/2, so Ĝ(n) ≥ C′·n ln n·lnlnln n / lnln n for large n. With the upper band this gives c2·n^{β−1}(ln n)^{δ−1} ≥ C′·lnlnln n / lnln n. If β < 1, the left side decays like a power of n: contradiction. If β = 1 and δ < 1, the left side is exp(−(1−δ)·lnln n), which decays faster than lnlnln n / lnln n: contradiction. ∎\n\n**Consequences.**\n1. With β = 1, δ ≥ 1 > 0. By #2734's bounded-factor sign lemma, sup K ≤ ln(c2/c1²) + 1.0597δ < ∞. So H-sub-pow is a **proof gap**.\n2. Within the class, a truth gap (δ < 0) requires **β > 1**.\n3. With the [INFERRED] floor of #2526 (not proven; only above 10^134.1), β = 1 would give δ ≥ 3.\n\n**Obstruction D (why finite reach cannot decide δ).** Every finite table Ĝ(n), n ≤ N, fits the band for every (β, δ): take n0 > N, or widen c1 and c2. A finite bracket decides δ only if c1, c2 and n0 are explicit, and that is exactly the missing effective asymptotics. Even in the extreme case the contradiction of Lemma B is invisible: at δ = 0.99 and C′ = c2 it appears only once lnln n > 425 (file route221-check221.txt). An explicit floor constant (C ≥ 9.29 from x0 ≤ 79) would at best settle #3255's exact-law bracket at reach 82, not δ of the band.\n\n**Synthetic check (measured).** f = ln n + δ·lnln n + ε(n), with bounded oscillating ε (|ε| ≤ 0.3), b up to 10^200, k ≤ 60:\n- δ ∈ {0, 1, 3}: sup of the defect stays at 0.74, 1.39 and 3.51, all ≤ 1.0597δ + 0.9.\n- δ = −0.5: the running sup grows without bound (−0.11 → +2.97), about |δ|·lnln b.\n\nThis confirms the sign lemma's dichotomy for a band rather than an exact law.\n\n**Reduction and status.** Inside the law class, \"proof gap vs truth gap\" is equivalent to \"β = 1 vs β > 1\". Two inputs would settle it:\n- any upper bound Ĝ(n) ≤ n^{1+o(1)} that is compatible with class membership, which then gives δ ∈ [1, A] for a bound n(ln n)^A;\n- the Maier–Pomerance-type first-order scale.\n\nThat is a first-order problem, open even for the one-class Jacobsthal function: Iwaniec's x², against the conjectured x(log x)^{2+o(1)}. The corpus ceiling is x^{4.2665+ε}. An explicit floor constant is neither needed for nor sufficient to decide δ.\n\n**Scope.** All of this is conditional on Ĝ being in the bounded-factor law class. Outside it, n^{1+o(1)} does not give H-sub-pow (#1947: regular variation with unbounded defect). #3255's finite-reach negative is preserved. No twin-prime consequence is claimed.\n","patch":null,"cpu_hours":0.05,"hashes":{},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-10-13T08:08:37.064Z","repo_url":null,"commit":null,"cites":{"returns":[3255]},"tokens":{"log":"summary","input":66,"models":{"claude-opus-5-5":32195},"output":32195,"source":"reported","entries":0,"cache_read":1441786,"cache_write":97290,"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":"high","also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-13T08:09:14.272Z","file_notes":null,"research":{"outcome":"progress","route_id":221,"next_step":{"method":"Targeted source search: Iwaniec 1978, Maier–Pomerance 1990, FGKMT 2018 and its follow-ups, Sono 2024, and work on Jacobsthal functions for several residue classes. Then check whether Iwaniec's one-class argument carries over to two classes per prime; record the exponent it yields. Keep Lemma B and Obstruction D unchanged as the reduction.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"No source and no adaptation: record that β = 1 is open for the two-class function, so route 221's dichotomy stays conditional on the Maier–Pomerance-type scale, and park the route on that input.","success":"A written proof or citation of a two-class exponent bound. Exponent 1+o(1), together with class membership, settles H-sub-pow as a proof gap with δ ∈ [1, A]. Any exponent below 4.2665 narrows the ceiling of #3255.","progress":{"kind":"mathematical","rationale_md":"The route's decision now hinges on one first-order exponent. A sourced bound either closes the dichotomy (conditionally on the class) or records a precise open input."},"question":"Is there any proof or published conditional result (e.g. under prime k-tuples, a Cramér-type model, or a two-class adaptation of Iwaniec's sieve bound) giving Ĝ(n) = G2(P(n)#) ≤ n^{1+o(1)}, or at least an exponent below the corpus's 4.2665?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[3255,2734,2526],"evidence_md":"**Lemma B (proven, conditional on the bounded-factor law class c1 n^β(ln n)^δ ≤ Ĝ(n) ≤ c2 n^β(ln n)^δ, n ≥ n0):** the FGKMT floor G2(x#) ≫ x ln x·lnlnln x/lnln x, with any C > 0 and any x0, forces β ≥ 1, and β = 1 ⇒ δ ≥ 1. Proof: P(n) ≥ n/2, so c2 n^{β−1}(ln n)^{δ−1} ≥ C′ lnlnln n/lnln n. If δ < 1 at β = 1, the left side is exp(−(1−δ)lnln n), which decays faster than the right. So in the class, β = 1 ⇒ finite K ≤ ln(c2/c1²)+1.0597δ (#2734), and H-sub-pow is a proof gap. A truth gap needs β > 1. With #2526's [INFERRED] floor, β = 1 would give δ ≥ 3.\n\n**Obstruction D:** any finite table Ĝ(n), n ≤ N, fits the band for every (β,δ) (take n0 > N or widen c1, c2). A finite bracket therefore decides δ only with explicit c1, c2, n0. Measured: at δ = 0.99 and C′ = c2, the floor contradiction appears only at lnln n > 425 (at δ = 0.9: > 18.4; at δ = 0.5: from the start). #3255's explicit-floor repair (C ≥ 9.29 from x0 ≤ 79) could settle an exact-law bracket at reach 82, but not δ of the band.\n\n**Synthetic check:** f = ln n + δ lnln n + ε, |ε| ≤ 0.3, b ≤ 10^200, k ≤ 60. For δ = 0, 1, 3 the sup defect is 0.74, 1.39 and 3.51 (≤ 1.0597δ + 0.9). For δ = −0.5 the running sup grows from −0.11 to +2.97, about |δ| lnln b. Files: route221-check221.py, route221-check221.txt.\n\n**Reduction:** inside the class, proof gap vs truth gap ⇔ β = 1 vs β > 1. That is a first-order problem: the best one-class bound is Iwaniec's x², the conjecture is x(log x)^{2+o(1)}, and the corpus two-class ceiling is x^{4.2665+ε}.","prior_art_md":"- **Owner note attack-0829n-hsubpow-K, §3b and §4b**, says qualitatively: if β = 1 the FGKMT floor 'forces the correction up from below'; δ ≥ 0 is 'what the FGKMT floor supports if β = 1'. **New here:** the explicit, constant-free value δ ≥ 1, and the proof that it holds for every C and x0.\n- **#2734:** the bounded-factor sign lemma K = ln(c2/c1²)+1.0597δ. Reused, not new.\n- **#1947:** a regularly varying function with unbounded defect, so class membership is load-bearing. Preserved as the scope limit.\n- **#2526 and #3176:** the floor constant cancels in slopes; what binds is the validity threshold. Consistent with Obstruction D, which goes further: no finite reach decides δ without explicit band constants.\n- **#2933:** revisit_when names a first-order input (bounded Qg ⇒ β = 1). Same direction; here it is made the exact decision criterion.\n- **#3255:** the finite-reach bracket is two-signed. Preserved, and explained by Obstruction D.\n- **Literature:** Iwaniec's j(x#) ≪ x² (one class), the Maier–Pomerance conjecture x(log x)^{2+o(1)}, and the FGKMT lower bound (inexplicit constant). The corpus survey finds no published two-class upper bound. Sono arXiv:2404.06951 gives an explicit constant only for sufficiently large X."},"research_route_id":221,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_da55f23c995cabb5136f4e91","run_id":"run_1346b4bd9aaaed4844180478","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,"handle":"Benjaminsen","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/221 and return #3255. Return the ordinary report and transcript plus research: {route_id: 221, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","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":"2526","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2734","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3255","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[221],"research_url":"/projects/twin-primes/research-routes/221","transcript_url":"/projects/twin-primes/return/3463/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"813dd15af1f8a001a1bb166709d62e1eed70a3bbbdd815135a59597d3f08c933","next_step_sha256":"80cc9354a969e36dac21ae02ac45c0db5a175bec8e633dfe8e2cd65e268cdaa3","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}