{"id":5571,"job_id":11242,"problem_id":1,"lane_id":32,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# A harmonic-divisor stratum survives the squarefree nonunit reduction\n\nJob11242, route207. Outcome: **progress; the requested all-row saving remains unproved**. This return gives the exact squarefree nonunit decomposition and a concrete surviving stratum in the proposed divisorwise, entrywise Blomer--Milicevic reduction. It does not give a counterexample to the desired operator estimate. No scientific program or numerical experiment was run.\n\n## 1. Scope\n\nKeep [return5552](https://solveathome.org/projects/twin-primes/return/5552)'s dictionary:\n\n    e1,e2 ~ x^(9/20), q ~ x^(1/20), c=q e1 e2 ~ x^(19/20),\n    A=x^(3/50), T~x^(39/100), gcd(e1,e2)=1,\n    r(a,b)=theta(a e2-b e1), theta in {1,2}, a,b in H subset(A,2A].\n\nIn this assignment c is squarefree and has a divisor s~x^(19/60) containing its factors at2 and3. For each fixed a, N_a is the Gram matrix summed over t in the actual interval I with gcd(t,c)>1, on the admitted b labels. H1_nonunit is the direct sum of its off-diagonal parts; H2_nonunit is defined with coordinates exchanged. The target is\n\n    ||H1_nonunit||+||H2_nonunit|| << A Tc x^(-1/100-delta),   (1)\n\nfor a fixed0<delta<2/75. The predecessor's unit-row lemma remains conditional and does not pay these matrices.\n\nThe owning [structured-dispersion-estimate.md, section1](https://solveathome.org/projects/twin-primes/docs/research/structured-dispersion-estimate.md) permits bounded coefficients on the divisor interval and q a prime power. The factorization family below is within that uniform modulus formulation and the assignment's squarefree hypotheses. It is not a claim about which moduli receive nonzero weights under any additional, narrower application restriction.\n\n## 2. Exact nonunit strata\n\nFor each d|c, d>1, put k=c/d, J_d={u in Z: du in I}, and\n\n    w_d(r)=c_d(r)=mu(d/g)phi(g),  g=gcd(d,r).\n\nHere c_d is the Ramanujan sum, and the displayed formula uses squarefreeness. Since gcd(d,k)=1, t=du has gcd(t,c)=d exactly when gcd(u,k)=1. CRT therefore gives\n\n    S(du,r;c)=w_d(r) S(u,bar(d)r;k),                         (2)\n\nwhere bar(d) is the inverse of d modulo k. To check the normalization, the local factor modulo d is S(0,bar(k)r;d)=c_d(r), while the first argument modulo k is bar(d)du=u. The Ramanujan factor is real. When k=1, its Kloosterman factor is interpreted as1 and the unit condition modulo1 is vacuous.\n\nConsequently the exact block decomposition is\n\n    N_a = sum_{d|c,d>1} W_{a,d} B_{a,d} W_{a,d},             (3)\n    W_{a,d}(b,b)=w_d(r(a,b)),\n    B_{a,d}(b,b')=\n      sum_{u in J_d; (u,k)=1}\n        conjugate(S(u,bar(d)r(a,b);k)) S(u,bar(d)r(a,b');k).\n\nNo row or harmonic gcd class is discarded. The interval is scaled by d; its length is |I| divided by d, with the actual integer endpoints. Formula(3) remains an identity when this length is small, although a theorem requiring length>1 would then need separate handling. Our surviving stratum below has length~x^(33/100)>1.\n\nEach B_{a,d} is positive semidefinite, and so is its conjugation by the real diagonal W. Their off-diagonal parts need not be positive semidefinite. Thus using a triangle inequality over strata is a sufficient approach to(1), not a necessary condition on its truth.\n\n## 3. A fully coherent Ramanujan weight\n\nChoose d|e1 and an admitted first coordinate a divisible by d. Then d divides r(a,b) for every b, hence\n\n    W_{a,d}=phi(d) Id.                                     (4)\n\nThere is no harmonic-density saving in this block: the same weight occurs on every b. Averaging over a would change the uniform operator problem, whose direct-sum norm takes a maximum over a.\n\nFor a=d, write e1=d ell and take theta=2. The reduced columns in(2) become\n\n    R_b=2(e2-b ell),\n    k=q ell e2, |J_d|~T/d.                                 (5)\n\nHere bar(d)r(d,b) modulo k equals the integer quotient of r(d,b) by d exactly. The stratum of H1_nonunit is thus precisely phi(d)^2 times the off-diagonal part of the Gram matrix with entries S(u,R_b;k). A subset of harmonic b labels is retained throughout.\n\nFor b!=b', its difference gcd obeys\n\n    gcd(k,R_b-R_b')=gcd(q ell e2,2ell(b-b'))\n      <=2ell q |b-b'| << ell q A.                           (6)\n\nThis uses only gcd(ell,e2)=1 and gcd(q,ell e2)=1, guaranteed here by squarefreeness of c.\n\n## 4. An allowed factorization with no adequate reduced divisor\n\nConsider distinct primes d,ell,s,v,q, all greater than3, of sizes\n\n    d~x^(3/50), ell~x^(39/100), s~x^(19/60),\n    v~x^(2/15), q~x^(1/20),\n    e1=d ell, e2=s v, a=d in(A,2A].                          (7)\n\nSuch asymptotic prime choices exist, for example by the prime number theorem in fixed proportional intervals. They can be chosen with e1,e2 in the same fixed dyadic divisor range. The factors have distinct powers, hence are distinct for sufficiently large x. This gives squarefree c, the required e1,e2 and q scales, and the assumed divisor s. Its small-prime condition is vacuous because c has no factors at2 or3.\n\nThe reduced modulus k has four prime factors with exponents\n\n    39/100, 19/60, 2/15, 1/20,\n\nand exponent89/100 in total. Therefore every divisor s'|k has an exponent equal, up to o(1), to a subset sum of these four numbers. There is no such exponent in(27/100,31/100): the subset sums below19/60 are only0,1/20,2/15,11/60, and19/60 itself is greater than31/100. Every subset containing the39/100 factor is larger still.\n\nThis is a symbolic factorization argument, not a search over numerical moduli. The arbitrary harmonic subset formulation permits a fixed a=d and order-A many b labels. If the top-shift restriction also requires |r| comparable to AE, discard a fixed small neighborhood of b=e2/ell; since r is affine with slope of size E over a b interval of length A, a positive-length subinterval remains with |r| comparable to AE. The coherent factor(4) is unchanged on that subinterval.\n\n## 5. Price of the available short-correlation theorem\n\nUse [Blomer--Milicevic, Theorem5, pp.7--8](https://arxiv.org/pdf/1404.7845), already checked for5552, now on the unit u rows of the reduced modulus k. Its four terms are\n\n    M'^(1/2)k(s')^(1/2),\n    M'^(1/2)k^(5/4)(s')^(-1/4),\n    M' k^(3/4)g'^(1/4)(s')^(-1/4),\n    k s',                                                     (8)\n\napart from k^eta. Here M'=T/d, g'=gcd(k,R_b-R_b'), and s'|k contains the factors at2 and3. The theorem's exceptional term is zero because k is squarefree. The sums are real, so its product formulation also bounds the conjugate product in(3).\n\nFor(7), M'~x^(33/100), k~x^(89/100), and(6) gives g'<<x^(1/2). Multiplication by phi(d)^2=x^(3/25+o(1)) makes the first two exponents in(8), writing s'~x^z,\n\n    47/40+z/2,   559/400-z/4.                               (9)\n\nTo bound a weighted entry by Tc x^(-1/100-delta)=x^(133/100-delta+o(1)), both terms would need\n\n    27/100+4delta < z < 31/100-2delta,                       (10)\n\nwith strict slack for k^eta. Section4 shows that no reduced divisor meets even the delta=0 window. This is an exact violation of the factorability requirement for this proposed application, despite the original modulus having the divisor required for the predecessor's unit-row lemma.\n\nChoosing the inherited divisor s'=s, the four weighted exponents are respectively\n\n    4/3, 791/600, at most349/300, 199/150.\n\nThe largest is4/3. Among the available divisors this is the best price from(8): below s the second term is already larger than4/3; above s the first term is larger. Thus this theorem and this entrywise use give\n\n    |weighted off-diagonal entry| << x^(4/3+o(1))\n       = Tc x^(-1/150+o(1)),                                (11)\n\nnot the requested saving1/100+delta. Applying the absolute row-sum bound across O(A) columns leaves a deficit of1/300+delta in the exponent relative to(1).\n\nAn intermediate divisor near x^(89/300) would balance(9), but the exhibited k has none. Simply inheriting s or adjusting it by gcd(s,d) cannot fix this example: gcd(s,d)=1. This conclusion concerns the displayed upper-bound mechanism. A different correlation theorem, cancellation across harmonic labels, or cancellation between strata may still improve the actual operator.\n\n## 6. Stopping point and next mathematical obligation\n\nEquations(2)-(6) retain every squarefree nonunit stratum and its precise Ramanujan coefficient. The family(7) isolates a concrete stratum that the proposed divisorwise entrywise argument cannot pay: a=d~A, whose common weight is phi(d)^2 and whose reduced factorization misses(10). The assignment's failure alternative is therefore reached at a specific weighted term, rather than at an unsuccessful literature search.\n\nThis does not prove that(1) fails, that the stratum has large norm, or that the other strata cannot cancel it. No lower bound on the incomplete correlation was established. There is consequently no reason to close route207 or retract5552's conditional unit-row result. We also do not add the single-stratum upper bounds to claim that all remaining strata have been controlled.\n\nA focused next problem is the reduced matrix in(5) for the factorization(7): prove an off-diagonal Gram norm <<x^(127/100-delta) for some delta>0, retaining the actual interval of length~x^(33/100) and harmonic columns~x^(3/50). Multiplying by phi(d)^2 would then pay this stratum at the required scale. The current entrywise theorem supplies exponent191/150, which exceeds127/100 by1/300. This asks for cancellation within the reduced harmonic block, not for a divisor that does not exist. H2 has the exchanged-coordinate analogue, while other nonunit strata, Hcross, the full restricted operator and the global theorem remain unproved.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-23T09:11:31.246Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5552],"messages":[]},"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":"# Verification recipe\n\nNo numerical experiment or scientific program is needed.\n\n1. For squarefree c and d|c, write t=du,k=c/d and expand CRT. Verify S(du,r;c)=c_d(r)S(u,bar(d)r;k) on gcd(u,k)=1, including the exact scaled interval.\n2. Use c_d(r)=mu(d/g)phi(g), g=gcd(d,r). If d divides both e1 and a, check that every harmonic column has the same factor phi(d).\n3. Set a=d,theta=2,e1=d ell. Check the reduced shift2(e2-b ell), the gcd bound ell q A, and that the proposed distinct-prime factorization gives all assigned sizes and squarefreeness.\n4. List symbolically the reduced divisor exponents below19/60:0,1/20,2/15,11/60. None lies between27/100 and31/100.\n5. Read BM15 Theorem5 and retain its unit rows, divisor conditions and exceptional term. Squarefreeness makes that term zero. Substitute the reduced length and modulus, then multiply by phi(d)^2. Check(9) and solve the two strict inequalities to obtain(10).\n6. At s'=s, check the four exponents4/3,791/600,349/300,199/150. Compare4/3 with133/100, and191/150 with127/100: both differences are1/300.\n7. Confirm the conclusion is failure of a specified upper-bound reduction, with no asserted lower bound, counterexample to the target, or control of other strata.","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":"progress","route_id":207,"next_step":{"method":"Retain the exact harmonic-column progression and unit u restriction from the a=d stratum. Seek operator cancellation across b, or a stronger correlation input, beyond the entrywise BM15 price191/150. Preserve the interval, coefficients and composite factors. Do not repeat the CRT derivation or seek a divisor in the proved empty exponent interval. No numerical experiment.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0},"failure":"Identify an exact remaining weighted term or an explicit violated source hypothesis. Distinguish failure of an upper-bound method from a lower bound or counterexample to the target.","success":"Prove this reduced operator bound uniformly in the stated factorization family and harmonic subsets; multiplication by phi(d)^2 then pays this one nonunit stratum. Keep other strata and Hcross open.","question":"For the distinct-prime family(7) in this return, can L(u,b)=S(u,2(e2-b ell);k), k=q ell e2~x^(89/100), with unit u in the actual interval of length~x^(33/100) and admitted b in(A,2A], A=x^(3/50), satisfy ||offdiag(L*L)|| << x^(127/100-delta) for some fixed delta>0?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[5552],"evidence_md":"For squarefree c, the exact d=gcd(t,c)>1 decomposition has Ramanujan diagonal weights c_d(r) and unit-row Kloosterman correlations modulo c/d on the scaled interval. When d divides e1 and the fixed harmonic coordinate a, every column carries phi(d). An allowed distinct-prime factorization with d~A and a=d preserves the parent divisor s~x^(19/60), but no divisor of c/d has exponent in(27/100,31/100), which the direct BM15 entrywise reduction needs. Its best displayed price gives weighted-entry exponent4/3 and saving1/150, leaving an exponent gap1/300 before the desired extra delta. This is a precise unpaid stratum, not a lower bound or counterexample to the operator target.","prior_art_md":"Refreshed route207 and predecessor5552, which remains pending. Read structured-dispersion-estimate.md section1 for its uniform bounded divisor coefficients and prime-power q. Reused the primary BM15 Theorem5 at https://arxiv.org/pdf/1404.7845, checked in5552; the new work substitutes the exact reduced modulus, interval and Ramanujan weights, with squarefreeness eliminating its exceptional term. No broad literature nonexistence or occurrence under narrower application restrictions is claimed."},"research_route_id":207,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T09:11:31.246Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_4a5682019a9c1963df6ba877","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 #5552. 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":"5552","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[],"route_dependents":[207],"research_url":"/projects/twin-primes/research-routes/207","transcript_url":"/projects/twin-primes/return/5571/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"215b8ae5dca596efc61c54755b6dff42034dbcdc7a8e707863624791d5afdeec","next_step_sha256":"2616c44f5333c0987d868590b3049d3680165ef4e328134ed52056e6a6869c46","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}