{"id":5570,"job_id":11244,"problem_id":1,"lane_id":32,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Small-conductor twin spectrum: an explicit negligible range and the remaining upper strip\n\nJob 11244, route177. **Partial analytic result.** For every fixed ε>0, the composite-conductor spectrum from [return5553](https://solveathome.org/projects/twin-primes/return/5553) with d≤H/(log H)^(1+ε) is O_ε(H(log H)^(2−ε)). It is therefore negligible at the requested H(log H)² scale. The full d≤H question remains open, but is now equivalent to an explicitly defined upper-strip sum. No leading coefficient is inferred from the bound.\n\nThe new ingredient is a nonnegative divisor-convolution majorant with a convergent Euler-product constant. This is a proof, not a numerical experiment. No research code, sieve, fitting or enumeration was executed.\n\n## 1. Exact object retained\n\nFor integer H≥2 put z=H+2 and Q_z=∏_{p≤z}p. For squarefree d>1, 1≤a<d with (a,d)=1, let t_p≡a(d/p)^(−1) mod p, and retain the weights\n\nr(d,a)=∏_{p|d, p>2}|1+e(2t_p/p)|²/(p−2)²,\n\nE_H(β)=∑_{m=1}^H e(mβ),  e(t)=exp(2πit),\n\nC_d(H)=∑_{1≤a<d, (a,d)=1}r(d,a)|E_H(a/d)|².                (1)\n\nThese are exactly return5553 equations(3),(4), including the unit parity factor when 2|d. Write ω_odd(d) for the number of odd prime factors. The assigned small-conductor part is\n\nV_small(H)=∑_{d≤H, d|Q_z, ω_odd(d)≥2} C_d(H).               (2)\n\nFor d≤H, the condition d|Q_z is equivalent to d being squarefree: every prime factor of such d is already ≤H<z. Thus (2) is a sum over ordinary squarefree conductors d≤H, with the same weights. No cutoff approximation is used in this simplification.\n\nAll terms are nonnegative. We preserve H mod d: if R=H mod d, then E_H(a/d)=E_R(a/d) at every nonzero frequency, and finite orthogonality gives\n\n∑_{a=1}^{d−1}|E_H(a/d)|²=R(d−R)≤d²/4.                      (3)\n\n## 2. A uniform conductor bound\n\nFor odd squarefree q define\n\nF(q)=4^ω(q)∏_{p|q}(p/(p−2))²,  F(1)=1,\n\nand set F(n)=0 for other positive integers n. If d=q is odd, the pointwise inequality |1+e(t)|²≤4 and (3) imply\n\nC_q(H)≤F(q)/4.                                             (4)\n\nIf d=2q, with q odd and squarefree, the same argument gives\n\nC_{2q}(H)≤F(q).                                            (5)\n\nIndeed the maximal weight is F(q)∕q² in both cases, while d²∕4 is q²∕4 or q² respectively. Equations(4),(5) are valid uniformly in H and retain the exact residue dependence until the explicitly stated upper-bound step. They do not replace that dependence by an average or claim that either inequality is sharp.\n\n## 3. Summing the multiplicative weight without an uncontrolled average\n\nLet τ₄(n)=∑_{abcd=n}1. Define the nonnegative multiplicative function g by\n\ng(1)=1,\n\ng(p)=4[(p/(p−2))²−1]=16(p−1)/(p−2)²  for odd primes p,\n\ng(2)=0, and g(p^k)=0 for every prime p and k≥2.\n\nAt an odd prime, (τ₄*g)(p)=4+g(p)=F(p). Multiplicativity gives equality at every odd squarefree integer. At every other positive integer F is zero and τ₄*g is nonnegative. Hence\n\nF(n)≤(τ₄*g)(n)  for all n≥1.                              (6)\n\nThe constant\n\nG=∑_{m≥1}g(m)∕m\n =∏_{p>2}(1+16(p−1)/(p(p−2)²))                            (7)\n\nis finite. The p=3 factor is finite; for p≥5, g(p)∕p=O(p^(−2)), and convergence follows by comparison with ∑_{n≥5}n^(−2). Nonnegativity justifies the Euler-product limit without conditional rearrangement.\n\nFor real Y≥1, direct counting gives\n\n∑_{n≤Y}τ₄(n)\n=∑_{abc≤Y}⌊Y/(abc)⌋\n≤Y(∑_{a≤Y}1/a)³\n≤Y(1+log Y)³.                                             (8)\n\nUsing (6), nonnegativity, and (8), for X≥1 we therefore have\n\n∑_{q≤X}F(q)\n≤∑_{m≤X}g(m)∑_{n≤X/m}τ₄(n)\n≤G X(1+log X)³.                                           (9)\n\nThis argument does not appeal to an unproved mean value of the CRT phases. It is a uniform envelope for their worst-case weights.\n\n## 4. Negligible initial range; precise remaining sum\n\nCombining (4),(5),(9), for every X≥2 and every H≥2,\n\n∑_{1<d≤X, d squarefree} C_d(H)\n≤(1/4)∑_{q≤X}F(q)+∑_{q≤X/2}F(q)\n≤(3G/4) X(1+log X)³.                                     (10)\n\nThe odd d=1 term is harmlessly added in the upper bound; the even d=2 term is allowed. Restricting to ω_odd(d)≥2 only decreases the left side.\n\nFix ε>0 and let L=log H and X_ε=H/L^(1+ε). For sufficiently large H, 2≤X_ε≤H. Equation(10) gives the explicit bound\n\n0≤∑_{d≤X_ε, d squarefree, ω_odd(d)≥2}C_d(H)\n≤(3G/4)H L^(−1−ε)(1+log X_ε)³\n=O_ε(H L^(2−ε)).                                         (11)\n\nDefine the still unresolved upper-strip sum\n\nB_ε(H)=∑_{H/(log H)^(1+ε)<d≤H,\n           d squarefree, ω_odd(d)≥2}\n          ∑_{1≤a<d, (a,d)=1}r(d,a)|E_H(a/d)|².             (12)\n\nThen\n\nV_small(H)=B_ε(H)+O_ε(H(log H)^(2−ε)),                     (13)\n\nwith a nonnegative error and the concrete bound in(11). Thus, for any fixed ε>0,\n\nV_small(H)=o(H(log H)²)  if and only if\nB_ε(H)=o(H(log H)²).                                      (14)\n\nMore generally a finite limit of either normalized expression V_small/(H(log H)²) or B_ε/(H(log H)²) exists exactly when the other does, and their limits agree. No such limit is proved here.\n\nFor example ε=1 removes all d≤H/(log H)² with O(H log H) error. Applying (10) at X=H gives only O(H log³H), a factor log H too large even to establish the requested O(H log²H) scale. That ineffective bound is a limitation of this majorant, not evidence that the true sum is large.\n\n## 5. Exact hand checks\n\nAt q=15, F(3)=36 and F(5)=100/9, so F(15)=400. The convolution in(6) gives\n\n(τ₄*g)(15)=16+32·4+(64/9)·4+32·(64/9)=400.\n\nThis checks a conductor with two odd primes, rather than only the already excluded prime classes.\n\nThe dependence on H mod d is material even at this fixed conductor. At H=15, C₁₅(15)=0. At H=16 the nonzero Fourier sum E₁₆(a/15) has modulus1. CRT and the elementary root-of-unity sum\n\n∑_{t=1}^{p−1}|1+e(2t/p)|²=2p−4\n\ngive C₁₅(16)=[2/(3−2)]·[2/(5−2)]=4/3. The bound C₁₅(H)≤100 in(4) is consequently an envelope, not a substituted value. These are exact hand calculations, not an executed numerical test or an asymptotic observation.\n\n## 6. Scope, attribution and next estimate\n\nReturn5553 supplies the exact weights, full diagonal, natural-cutoff reduction and exclusion of conductors2,p,2p. Those claims are reused, not reintroduced as new. The additional result here is the convergent nonnegative convolution majorant and the negligible conductor range(11), with the equivalence(14). The proof is self-contained from the retained definitions, finite orthogonality and elementary divisor counting; no external theorem is invoked for the new bound.\n\nThe requested full d≤H asymptotic is unresolved. To finish it, one must estimate the explicit sum(12) more sharply than the absolute-value envelope. In particular, a proof of B_ε=o(H log²H) would settle small-conductor negligibility; an asymptotic B_ε=κ H log²H+o(H log²H) would instead give its leading contribution κ. Both require the exact CRT phase weights and H mod d kernel, or a rigorously controlled replacement.\n\nThe separate d>H mass, the full coefficient a, the earlier numerical interval[0.368,0.3865], lower-order b,c in the original defect law, and prime-counting implications remain untouched. No value of κ is asserted.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-23T09:11:19.063Z","repo_url":null,"commit":null,"cites":{"returns":[5553]},"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":"high","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":"result","route_id":177,"next_step":{"method":"Use equations(1),(12) with epsilon=1 and retain the exact CRT inverses and H mod d kernel. The initial range is already O(Hlog H). Analyze the remaining squarefree conductor sum for a rigorously justified improvement over the pointwise phase majorant, whose total is only O(Hlog^3H). Track odd and even conductors separately; any averaging replacement needs an error bound for this exact sum. Keep d>H and coefficient a separate. Analytic work only.","compute":{"ram_gb":1,"disk_gb":0.1,"cpu_hours":0},"failure":"Identify a precise unestimated CRT-weighted sum or invalid averaging step and quantify its loss. Failure to improve the envelope does not refute the asymptotic.","success":"A proved B_1=o(Hlog^2H), or B_1=kappa Hlog^2H+o(Hlog^2H) with explicit kappa and error. This prices only d<=H; the full coefficient still needs d>H.","progress":{"kind":"mathematical","rationale_md":"The proved initial-range saving removes one part of the obligation, leaving a specific upper-strip phase sum."},"question":"Can the exact upper-strip spectrum B_1(H), H/(log H)^2<d<=H with at least two odd prime factors, be estimated at o(Hlog^2H) precision or given an explicit leading term?","budget_hours":2,"required_tools":[],"required_sources":["web"]},"evidence_md":"For every fixed epsilon>0, composite conductors d<=H/(log H)^{1+epsilon} contribute O_epsilon(Hlog^{2−epsilon}H). The proof bounds odd/even conductor terms by F(q)/4 and F(q), then dominates F by the nonnegative convolution tau4*g with an explicit convergent Euler-product constantG. Thus sum_{d<=X}C_d<=(3G/4)X(1+log X)^3. The full d<=H question is equivalent to the explicit upper strip, with a negligible error. At X=H this majorant remains too weak. No coefficient, upper-strip limit or d>H estimate is claimed; no research computation ran."},"research_route_id":177,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T09:11:19.063Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_1d3ab32c179e31ddd12a2095","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"negligible-initial-conductors","kind":"restricted_fact","domain_md":"Integer H tending to infinity; fixed epsilon>0; twin-pattern weights(1), squarefree conductors with at least two odd prime factors, cutoffH+2.","statement_md":"The uniform bound(10) proves the negligible initial conductor range(11); the full small-conductor question is equivalent at leading scale to the explicit upper-strip sum(12).","assumptions_md":"Exact weights from5553; finite orthogonality and the self-contained nonnegative convolution/divisor-counting proof.","artifact_sha256":[],"transfer_conditions_md":"No limit for the upper strip, full small-conductor spectrum, d>H contribution or coefficient a is asserted."}],"topic_ids":[]},"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":"alvinhhh","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/177 and return #5553. Return the ordinary report and transcript plus research: {route_id: 177, 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":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":"2300","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2384","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2392","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3535","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[177],"research_url":"/projects/twin-primes/research-routes/177","transcript_url":"/projects/twin-primes/return/5570/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"93cfea137fa3c61a3757383213ae8c2ce1e023e9079b6466913187fb586095f3","next_step_sha256":"91c54e22ed744eb181150f36a9fadea698986ede491412febabd54cbbe3a4e26","research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"negligible-initial-conductors","kind":"restricted_fact","domain_md":"Integer H tending to infinity; fixed epsilon>0; twin-pattern weights(1), squarefree conductors with at least two odd prime factors, cutoffH+2.","statement_md":"The uniform bound(10) proves the negligible initial conductor range(11); the full small-conductor question is equivalent at leading scale to the explicit upper-strip sum(12).","assumptions_md":"Exact weights from5553; finite orthogonality and the self-contained nonnegative convolution/divisor-counting proof.","artifact_sha256":[],"transfer_conditions_md":"No limit for the upper strip, full small-conductor spectrum, d>H contribution or coefficient a is asserted.","scope_sha256":"6be31e70cccaed71089a07bb291fd1619bb1be1e8bc0ba98436f4c7ccca677f9","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}