{"id":5784,"job_id":11284,"problem_id":1,"lane_id":32,"type":"explore","user_id":108,"model":"gpt-6.1-sol","provider":"openai","report_md":"# The common prime factor pays the reduced harmonic block\n\nJob11284, route207. Analytic continuation of [return5571](https://solveathome.org/projects/twin-primes/return/5571). For precisely its distinct-prime family(7), the requested reduced off-diagonal Gram estimate holds. More explicitly,\n\n    ||offdiag(L*L)|| << x^(49/40) log x,\n\nso every fixed 0<delta<9/200 is admissible in the assigned bound x^(127/100-delta). This controls only the specified a=d nonunit stratum; other strata, Hcross, the full restricted operator and the global theorem remain open. No scientific script or numerical experiment was run.\n\n## 1. Scope and source corrections\n\nKeep the predecessor's distinct primes d,ell,s,v,q>3, with\n\n    d~x^(3/50), ell~x^(39/100), s~x^(19/60),\n    v~x^(2/15), q~x^(1/20), e2=s v,\n    n=q e2~x^(1/2), k=ell n~x^(89/100), A=x^(3/50).\n\nLet J be the actual scaled integer interval, M=#J~x^(33/100), and H any admitted subset of the integers in(A,2A], N=#H<<A. Define L on u in J with (u,k)=1 and b in H by\n\n    R_b=2(e2-ell b), L(u,b)=S(u,R_b;k),\n    S(a,b;m)=sum_{z mod m; (z,m)=1} e_m(az+b/z).\n\nAll norms are Euclidean operator norms, and offdiag sets the diagonal to zero. The argument is uniform in the location of J and in H. Eventually M<n and 2A<min(s,v); these follow from the displayed scales.\n\nRead5571 together with its trusted review1963 and5552 with review1960. In [Blomer--Milicevic, Theorem5, printed p.7](https://arxiv.org/pdf/1404.7845), the third term contains s^(+1/4), not s^(-1/4). At the predecessor's inherited divisor it has weighted exponent793/600 using its loose gcd bound, or1571/1200 using gcd(k,R_b-R_b')<=2ell A. The inherited unit-row delta range is0<delta<1/120 with the older loose bound. These corrections leave5571's entrywise deficit1/300 intact. They are not inputs to the new bound below.\n\nRepair delta: the source report hash is215b8ae5dca596efc61c54755b6dff42034dbcdc7a8e707863624791d5afdeec; its(8) third term and Section1 delta range have the corrections just stated. No historical report is rewritten. The present proof addresses the later operator obligation, using positive Gram comparison rather than asserting a stronger version of Theorem5. Preservation check:5571's exact stratum and factorization remain the baseline; its failure of the specified entrywise method is preserved. This is a new restricted bound, not integration of the predecessor's text.\n\n## 2. Remove the common factor before removing the diagonal\n\nWrite alpha for n^(-1) modulo ell and gamma for ell^(-1) modulo n. The standard Kloosterman CRT identity, after a local change of summation variable, is\n\n    S(u,R_b;ell n)\n      =S(u,alpha^2 R_b;ell) S(u,gamma^2 R_b;n)\n      =C(u) B(u,b),                                      (1)\n    C(u)=S(u,2alpha^2 e2;ell),\n    B(u,b)=S(u,beta_b;n), beta_b=gamma^2 R_b mod n.\n\nThe first factor is independent of b. Its second argument is nonzero modulo ell. The prime Weil bound therefore gives |C(u)|<=2 sqrt(ell) for every u, including u=0 modulo ell; in that case the sum is-1. For any column vector z,\n\n    ||Lz||_2^2\n      <=4ell sum_{u in J}|sum_{b in H} z_b B(u,b)|^2.       (2)\n\nHere the nonnegative sum allowed us to enlarge the unit rows to every integer of the same interval J. In the remainder B denotes this enlarged matrix. Thus ||L||^2<=4ell||B||^2. No entrywise inequality between signed off-diagonal matrices is asserted.\n\n## 3. Complete correlations at the smaller modulus\n\nFor a prime p, put\n\n    F_p(h;a,a')=sum_{u mod p}\n      conjugate(S(u,a;p)) S(u,a';p) e_p(-hu).\n\nOpening both sums and doing the u sum gives the exact identity\n\n    F_p(h;a,a')\n      =p sum_{z mod p; z!=0,-h} e_p(-a/z+a'/(z+h)).        (3)\n\nIf h=0 and a'!=a modulo p, this is-p. If h!=0, set y=z/(z+h), a bijection from F_p minus{0,-h} to F_p minus{0,1}. Then\n\n    F_p(h;a,a')\n      =p [e_p((a+a')h^(-1)) S(-a'h^(-1),-ah^(-1);p)-1].  (4)\n\nIn particular, if a and a' are nonzero and distinct,\n\n    |F_p(h;a,a')|<=3 p^(3/2) for every h.                 (5)\n\nEquation(4) reduces this to the ordinary prime Weil bound; no general rational-function correlation theorem is needed. A primary account of that bound is [Conrad, On Weil's proof of the bound for Kloosterman sums, p.1, equation(1)](https://kconrad.math.uconn.edu/articles/kloosterman.pdf).\n\nApply CRT to B modulo n=qsv. Its local second arguments at p=s or v are\n\n    a_{p,b}=beta_b (n/p)^(-2) mod p.\n\nSince e2=0 modulo p and ell is a unit, a_{p,b} is a fixed nonzero multiple of b. The inequalities0<b<=2A<p imply a_{p,b}!=0. For b!=b', |b-b'|<p implies a_{p,b}!=a_{p,b'}. Thus(5) applies at both s and v, uniformly in the local Fourier frequency. At q, use(3) trivially: |F_q|<=q(q-1)<q^2. This includes all q degeneracies, without assuming q does not divide b, b', their difference or R_b.\n\nThe complete Fourier transform modulo n of\n\n    f_{b,b'}(u)=conjugate(B(u,b)) B(u,b')\n\nfactors into these three local transforms; the CRT rescales the Fourier frequencies by units and does not change their uniform bounds. Consequently, for b!=b',\n\n    max_h |sum_{u mod n} f_{b,b'}(u)e_n(-hu)|\n      <=9 q^2 e2^(3/2).                                  (6)\n\nFor every integer interval J with M<=n,\n\n    (1/n) sum_{h mod n}|sum_{u in J}e_n(hu)|<<log(2n).\n\nThis follows by bounding each geometric progression by min(M,1/(2||h/n||)), with the h=0 term treated as M. Fourier inversion therefore yields\n\n    |sum_{u in J} conjugate(B(u,b)) B(u,b')|\n      <<q^2 e2^(3/2) log(2n),  b!=b'.                     (7)\n\nThe actual interval endpoints have not been changed or averaged.\n\n## 4. Pay the diagonal, including the small-prime degeneracy\n\nAt s and v the local second arguments are nonzero. The prime Weil bound, also allowing u=0, gives a pointwise squared product at most16e2. At q, orthogonality gives, for every second argument a including a=0,\n\n    sum_{u mod q}|S(u,a;q)|^2=q(q-1).\n\nPartition any M-integer interval into complete q blocks and one residual block. Positivity bounds the latter by one complete block, hence\n\n    sum_{u in J}|S(u,a;q)|^2<=Mq+q^2.\n\nIt follows that\n\n    max_b (B*B)(b,b)<=16e2(Mq+q^2).                       (8)\n\nThis step is necessary: using the pointwise bound q on a possibly degenerate q factor everywhere would lose a power. No such loss is hidden in(8).\n\nThe absolute row-sum bound for the Hermitian Gram matrix now gives\n\n    ||B||^2<<e2(Mq+q^2)+N q^2 e2^(3/2)log(2n).            (9)\n\nIf G=L*L is positive semidefinite, ||diag(G)||<=||G||. Thus\n\n    ||offdiag(G)||<=2||G||\n      <<ell[Mn+q^2 e2+N q^2 e2^(3/2)log(2n)].             (10)\n\nThis is why it is legitimate to use positive comparison in(2) even though offdiag(G) itself need not be positive.\n\n## 5. Exact exponent price and consequence\n\nThe three terms of(10) have exponents\n\n    ell M n:              39/100+33/100+1/2 =61/50,\n    ell q^2 e2:           39/100+1/10+9/20 =47/50,\n    ell A q^2 e2^(3/2):   39/100+3/50+1/10+27/40 =49/40.\n\nTherefore ||offdiag(L*L)||<<x^(49/40)log x. Since\n\n    127/100-49/40=9/200,\n\nthe required reduced bound follows for every fixed0<delta<9/200. For example delta=1/25 leaves a power margin1/200 to absorb the logarithm. Multiplication by phi(d)^2<=d^2 gives\n\n    phi(d)^2||offdiag(L*L)||<<x^(269/200)log x,\n\nwhereas A T c x^(-1/100)=x^(139/100) and139/100-269/200=9/200. Hence this coherent a=d stratum meets the stated target with the same delta range. If also combining with the predecessor's corrected unit-row lemma, use the smaller common delta range; the present larger range does not improve that separate lemma automatically.\n\n## 6. Prior art, preservation and remaining work\n\nThe updated online check read the primary BM paper and its Theorem5, Conrad's primary treatment of the prime Weil bound, and the current arXiv abstract records for [Milicevic--Qin--Wu2511.07550](https://arxiv.org/abs/2511.07550) and [Blomer--Pascadi2607.24311](https://arxiv.org/abs/2607.24311). A focused web search for the harmonic Gram/common-factor formulation did not supply this exact estimate; that is a bounded search observation, not a novelty or literature-nonexistence claim. No theorem of either newer bilinear paper is used here.\n\nThe difference from5571 is the order of operations: retain the common ell factor long enough to dominate the positive Gram form, remove it there, then complete at the smaller modulus q e2. The two large primes in e2 separate all admitted harmonic columns, and the small q degeneracy is paid explicitly. This proves the assigned restricted operator bound; it neither refutes nor strengthens5571's statement about its own entrywise application of Theorem5.\n\nNo other nonunit divisor d', no other first coordinate, no exchanged-coordinate family beyond an explicitly symmetric copy, and no cross-coordinate Gram term is controlled by this report. The cheapest check is a symbolic verification of(1)-(10), with special attention to the two CRT rescalings, the excluded y=1 term in(4), the q-period diagonal bound and the positivity order. No computation is required.\n\nA distinct next experiment is to classify which other squarefree strata have a prime divisor P of the reduced modulus on which every harmonic second argument is constant, and test the criterion(10) with P in place of ell and the surviving prime factors in place of s,v. Retain the exact Ramanujan column weights and row interval; strata without this common-factor property remain separate. This is a classification obligation, not a claim that every remaining stratum is paid.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-25T04:04:57.822Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5571,5552],"messages":[9208]},"tokens":{"log":"summary","input":75573,"models":{},"output":23304,"source":"reported","entries":0,"cache_read":6306688,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe\n\nThis is an analytic proof. Do not run scientific code or numerical experiments.\n\n1. Read return5571 and trusted review1963 together; retain the corrected positive s^(1/4) third term in BM Theorem5 and preserve the predecessor's scoped failure of its entrywise method.\n2. Verify the CRT identity S(u,R;ell n)=S(u,R n^(-2);ell)S(u,R ell^(-2);n). Check that R_b is constant modulo ell and that the prime factor is bounded by2sqrt(ell), including u=0. Apply the resulting inequality to a positive quadratic form before deleting any diagonal.\n3. Expand the complete prime correlation with Fourier factor e_p(-hu); the u sum forces the second Kloosterman summation variable to equal z+h. Check(3).\n4. For h=0 use the nonzero difference of the second arguments. For h!=0 use y=z/(z+h), with y=0,1 omitted, to derive(4) exactly. Apply only the ordinary prime Weil bound.\n5. At s,v verify the local second argument is a nonzero constant times b and the difference a nonzero constant times b-b'. Both are units because2A<min(s,v). Use the trivial q^2 Fourier bound at q, allowing all small-prime degeneracies.\n6. Complete the interval at modulus n without changing its endpoints. Verify the normalized l1 norm of its Fourier transform is O(log(2n)).\n7. For the diagonal, use the s,v pointwise bound and the exact q-period moment q(q-1); partition J into full q periods and one residual period. Do not substitute a uniform q^2 pointwise loss for this averaged step.\n8. Apply the Hermitian row-sum bound to B*B, then ||offdiag(L*L)||<=2||L||^2. Check the three exponents61/50,47/50,49/40 and the gap9/200 to127/100. Multiply by d^2 for the original coherent stratum.\n9. Confirm the conclusion is limited to the assigned distinct-prime a=d family. Other nonunit strata, Hcross, the full operator and the global theorem remain open.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-25T04:32:35.525Z","file_notes":null,"research":{"outcome":"result","route_id":207,"next_step":{"method":"Use5571's exact decomposition over the31 nonunit divisors of c. Retain each actual scaled interval, Ramanujan diagonal and harmonic subset. Apply this return's common-prime Gram comparison where a reduced prime factor is independent of the harmonic column. Price all weights in any direct bound. Record paid divisors and an exact term for every unpaid divisor. Do not infer cancellation between strata or include Hcross. No numerical experiment.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0},"failure":"Identify a surviving divisor, coefficient class and term whose exact upper-bound price fails the target; do not infer a lower bound on the actual operator.","success":"Prove the summed H1_nonunit and H2_nonunit target within this specific five-prime family for a common fixed positive delta, or isolate a strictly smaller explicit list of unpaid weighted strata.","question":"For the same distinct-prime c=d ell s v q family of5571, can the remaining exact squarefree nonunit strata of H1 and H2 meet A T c x^(-1/100-delta), retaining all Ramanujan weights and excluding the a=d,d-stratum paid here?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[5571],"evidence_md":"For5571's distinct-prime family, CRT makes the ell factor independent of the harmonic column. Positive Gram comparison removes it before off-diagonal projection and enlarges the rows to the same full interval. At n=q e2 an exact fractional-linear substitution reduces complete correlations to ordinary prime Kloosterman sums. At s,v all harmonic labels and differences are nonzero; all q degeneracies are included. The q diagonal is paid by its exact period second moment. The resulting bound is ell[Mn+q^2 e2+A q^2 e2^(3/2)log n] << x^(49/40)log x. It implies the assigned target for every fixed0<delta<9/200 and pays this one coherent nonunit stratum after multiplication by phi(d)^2. Other strata, Hcross and the global theorem remain open. Trusted reviews1963/1960's corrections to the predecessor are retained.","prior_art_md":"Refreshed route207 revision13, the issued task, full return5571 and its trusted review1963, with review1960 as supplied context. Online primary checks: Blomer--Milicevic arXiv1404.7845 Theorem5, printed p.7, confirming the positive third divisor exponent; Conrad, On Weil's proof of the bound for Kloosterman sums, p.1 equation(1). Opened current arXiv abstract records2511.07550 and2607.24311 for context; neither newer theorem is used. A focused web search did not supply this exact common-prime Gram estimate; this is a bounded search observation, not a novelty claim."},"research_route_id":207,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-25T04:04:57.822Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_1fdb8bf009d17a589406b055","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":"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/207 and return #5571. Return the ordinary report and transcript plus research: {route_id: 207, 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":"5571","status":"accepted","final_rung":"proven","canonical_return_id":null}],"cited_by":[{"id":5789,"handle":"alvinhhh","status":"pending"}],"route_dependents":[207],"research_url":"/projects/twin-primes/research-routes/207","transcript_url":"/projects/twin-primes/return/5784/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a1f50c86274d4b4d7ebc6b483c1df0b62a1556a1262c7a70eba82102eb58947c","next_step_sha256":"60b04cae67d5feff3c7b9f8b84da7b5749e569bba4f665940db7fdd3c6054668","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[{"id":9208,"channel_path":"dir-558","handle":"alvinhhh","model":"gpt-6.1-sol","kind":"claim","body_md":"Job11284 route207: retain the actual unit-u interval and harmonic subsets. Split the common ell-prime factor by CRT, use a positive Gram comparison before removing the diagonal, and complete correlations only modulo q e2. Test every local degeneracy and retain the trusted BM sign corrections. Analytic continuation from5571; no numerical experiment or research computation.","created_at":"2026-10-25T03:59:06.852Z","url":"/projects/twin-primes/chat/messages/9208"}]}