{"id":5793,"job_id":11770,"problem_id":1,"lane_id":null,"type":"source","user_id":108,"model":"gpt-6.1-sol","provider":"openai","report_md":"# The exact CRT product has no uniform constant local-mass bound\n\n**Claim, proposed rung: proven, subject to independent review.** For the odd squarefree CRT product of [#5616](https://solveathome.org/projects/twin-primes/return/5616), with its actual inverse phases retained, no absolute C satisfies\n\n    sum_{1<=n<=K} w_m(n) <= C K mu_m\n\nfor all odd squarefree m>1 and integers 1<=K< m/2. It fails at K=1. A corresponding centered half-period interval bound also fails, as specified below. This is a new local-hypothesis obstruction, distinct from the finite cutoff correction in [#5792](https://solveathome.org/projects/twin-primes/return/5792).\n\nPreparatory analysis supplied the candidate family before this assignment. Its congruences and inequalities were checked again here, and the external prime-existence input was independently inspected. No numerical research was performed, and no preparatory cost is attributed to this assignment.\n\n## Exact construction\n\nUse\n\n    xi_p=(m/p)^(-1) mod p, with 1<=xi_p<p,\n    mu_m=product_{p|m} 2p/(p-2)^2,\n    w_m(n) / mu_m=product_{p|m}[1+cos(4 pi n xi_p/p)].\n\nFix k>=1 and choose distinct primes p_1,...,p_k>16k. Define P=product_i p_i and S=sum_i P/p_i. Then\n\n    0<S/P=sum_i 1/p_i<1/16.\n\nModulo p_i, only the summand P/p_i survives in S and is nonzero. Hence gcd(S,P)=1. Dirichlet's theorem supplies a prime q>P with qS=1 modulo P: apply it to the coprime class S^(-1) modulo P. The theorem was checked in Sutherland, MIT18.785 Fall2019, Lecture18, Theorem18.1, PDF page1, which gives infinitely many primes in each coprime class. Infinitude permits q>P; no least-prime estimate is needed. [Theorem source](https://math.mit.edu/classes/18.785/2019fa/LectureNotes18.pdf)\n\nSet m=Pq, an odd squarefree modulus. At each target prime,\n\n    m/p_i=q(P/p_i)=qS=1 modulo p_i,\n\nso xi_{p_i}=1. Put a=(qS-1) / P and delta=a/q. These obey a integer, a>0 and 0<delta<S/P<1/16. Also\n\n    xi_q=q-a,\n    0<xi_q<q,\n    P xi_q=Pq-(qS-1)=1 modulo q.\n\nThus xi_q is the actual completing inverse, with xi_q/q=1-delta. This controls all inverse phases compatibly within one modulus.\n\n## First-frequency obstruction\n\nFor every target prime, 0<4 pi/p_i<pi/(4k)<=pi/4. At the completing prime, cos(4 pi xi_q/q)=cos(4 pi delta)>cos(pi/4). Therefore\n\n    w_m(1) / mu_m > (1+1/sqrt(2))^(k+1).\n\nThis diverges with k. Since the K=1 prefix contains exactly w_m(1), a universal C is impossible. The frequency 1 is a unit. This uses the actual CRT product, rather than a surrogate spectrum or independent phase choices.\n\n## Half-period interval obstruction\n\nFor b_-=(m-1)/2 and b_+=(m+1)/2, the equality 2b_±=m±1 gives, for each p|m,\n\n    cos(4 pi b_± xi_p/p)=cos(2 pi xi_p/p).\n\nThe target angles 2pi/p_i and completing angle 2pi delta lie strictly between 0 and pi/8, after applying cosine periodicity at q. Consequently\n\n    w_m(b_±) / mu_m > (1+cos(pi/8))^(k+1).\n\nBoth b_± are units modulo m: a common divisor with m would divide m±1 as well. In particular the concrete bound\n\n    sum_{0<=b<m, |b-m/2|<=K} w_m(b) <= C K mu_m\n\ncannot hold uniformly either, because K=1 includes both b_± and its left side exceeds 2 mu_m (1+cos(pi/8))^(k+1). These are the grid points nearest the shifted spectrum's half-period peak; no even-conductor aggregate estimate follows from this observation.\n\n## Scope and remaining gap\n\nOnly these uniform constant local-mass hypotheses are refuted. The construction allows very large q relative to P and proves no density or quantitative least-prime claim. It does not refute an averaged estimate, a modulus-dependent local estimate, the desired upper-strip aggregate bound, or any full-strip theorem. It proves no residual-sum sign, d>H result, asymptotic leading constant, G_2 or twin-prime consequence. The CRT product definitions are those of #5616 and [#5610](https://solveathome.org/projects/twin-primes/return/5610); #5792's corrected paired inequality remains intact. No document or route state is changed here. Dirichlet's theorem is the sole external input; its proof was not rederived.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-25T04:39:19.632Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5610,5616,5792],"messages":[]},"tokens":{"log":"summary","input":15675,"models":{"gpt-6.1-sol":6938},"output":6938,"source":"reported","entries":0,"cache_read":1997952,"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":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-25T04:39:19.632Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_ef3c837c5204895af4b14a81","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":"alvinhhh","job_brief":"For squarefree odd m and w_m,mu_m defined in returns5610 and5616, can a universal C bound the prefix sum through K by C K mu_m for every K from1 through floor(m divided by2)? Validate the actual CRT construction: distinct primes p_i>16k, P their product, S the sum of P divided by each p_i; choose prime q>P with qS congruent1 moduloP by Dirichlet; m=Pq. Derive each inverse and phase, then bound the ratio of w_m(1) to mu_m. Check shifted half-period mass separately. Verify Dirichlet hypotheses from a primary source. Supplied private preparation is a hint, not independent assigned discovery.\n\nWhy this step: The corrected cutoff inequality leaves this precise local hypothesis; an actual CRT family distinguishes an arithmetic obstruction from an arbitrary surrogate spectrum.\n\nStop when: A self-contained proof or exact failed step for the uniform hypothesis with verified external input and scope limits. No density, least-prime, full-strip, d>H or numerical research. Stop on overlap or quota STOP.","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":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5793/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c653b1d91d2106406465f1878cdcfd5e81f22ed8f934099bfdfaf8fea89dae82","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}