{"id":6220,"job_id":12451,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# The maximal lambda1 obstruction survives the original squarefree smooth modulus restriction\n\nLet V_sf(E,Q) be the original centered maximal lambda1 squared norm, but sum only over squarefree y-smooth moduli coprime to30. For all sufficiently large integer E and every y>=2E,\n\n    V_sf(E,E)>=1/68600.                                  (1)\n\nThe lower bound already comes from composite moduli q=7r with r>1 and(r,210)=1. Thus these moduli admit the genuine coprime factorization7*r. The proof retains the exact weights, mean and maximum. It is a restricted UNWEIGHTED squared-norm obstruction; no lower bound for the project's lambda0-weighted first moment or signed core is inferred.\n\n## 1. Exact restricted norm\n\nDefine\n\n    lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{n y-smooth}\n                           product_{p|n}1/(p-4),\n    D(t;q,a)=sum_{E<n<=t,n=a mod q}lambda1_y(n)\n              -phi(q)^(-1)sum_{E<n<=t,(n,q)=1}lambda1_y(n),\n    M(q,a)=sup_{E<=t<=2E}|D(t;q,a)|,\n    Q_y={q>=1:q squarefree,y-smooth,(q,30)=1},\n    V_sf(E,Q)=sum_{q<=Q,q in Q_y}sum_{a reduced mod q}M(q,a)^2. (2)\n\nThis is the family specified by the source modulus support. No modulus weight is inserted into V_sf. In particular it differs from\n\n    W_D=sum_{D<q<=2D,q in Q_y}lambda0(q)\n                                  sum_{a reduced}M(q,a),  (3)\n\nwhere lambda0(q)=product_{p|q}2/(p-4). Expression(3), used in the source Cauchy–Schwarz transfer, is a weighted first moment, not(2).\n\nChoose integer E tending to infinity and y>=2E. All integers needed below are smooth. The squarefree and small-prime restrictions remain nontrivial and will be counted, not discarded.\n\n## 2. An admissible composite-modulus subfamily\n\nCount pairs(n,r) with\n\n    E<n<=2E,       E/14<r<=E/7,\n    (n,210)=(r,210)=1,\n    n and r squarefree,       (n,r)=1.                  (4)\n\nPut q=7r. For large E, r>1; q is squarefree because7 does not divide r, and(q,30)=1. Also E/2<q<=E, and q is y-smooth. Thus q belongs to Q_y and has the genuine coprime split7*r. Moreover(n,q)=1, since n is coprime to7 and to r. The original band weight at n is positive and satisfies\n\n    lambda1_y(n)>=1/n>=1/(2E).                           (5)\n\nThe selection(4) only identifies jumps to use in a lower bound. It does not replace or restrict the sum defining D, and it does not change its principal mean.\n\n## 3. Direct count with the two squarefree conditions and coprimality\n\nLet N_E be the number of pairs in(4). The fraction of invertible classes modulo210 is\n\n    d=phi(210)/210=48/210=8/35.\n\nThe number of integers coprime to210 in any interval of length X is dX+O(48), uniformly in its endpoints. Consequently the count before the two squarefree conditions and(n,r)=1 is\n\n    d^2 E^2/14+O(E)=M_E+O(E),\n    M_E=(32/8575)E^2.                                   (6)\n\nWrite S=sum_{p>=11,p prime}p^(-2). Since all counted integers avoid2,3,5,7, every bad prime below is at least11, and\n\n    S<=sum_{m>=11}m^(-2)<=integral_10^infinity x^(-2)dx=1/10. (7)\n\nThere are three bad sets. All estimates have absolute effective constants.\n\nFirst, if n is not squarefree, some p^2 divides n with11<=p<=sqrt(2E). Multiplication by p^2 preserves coprimality to210. For this p, the number of n is at most dE/p^2+48, and the number of r before further conditions is at most dE/14+48. Summing their product bounds this bad set by\n\n    M_E S+O(E^(3/2)).                                   (8)\n\nSecond, if r is not squarefree, then p^2 divides r for a prime11<=p<=sqrt(E/7). The corresponding counts are at most dE+48 and dE/(14p^2)+48. Their summed product gives\n\n    M_E S+O(E^(3/2)).                                   (9)\n\nFor(8)–(9), the O(E^(3/2)) bound follows simply by bounding the number of primes in the indicated square-root ranges by the number of integers. The main reciprocal-square sum is bounded by S; no asymptotic for primes is used.\n\nThird, if(n,r)>1, some prime11<=p<=E/7 divides both. The respective counts are at most dE/p+48 and dE/(14p)+48. Summing the product and using the ordinary harmonic-sum bound gives\n\n    M_E S+O(E log E).                                   (10)\n\nThe constant terms contribute O(E), and the cross terms contribute O(E sum_{m<=E}1/m). These estimates do not assume independence of any conditions.\n\nSubtracting all three bad-set upper bounds from(6) is valid even when they overlap. Therefore\n\n    N_E>=M_E(1-3S)-O(E^(3/2)+E log E)\n        >=(7/10)M_E-O(E^(3/2)+E log E).                  (11)\n\nSince the error is o(E^2), for all sufficiently large integer E,\n\n    N_E>=M_E/2=(16/8575)E^2.                            (12)\n\nThe threshold is effective; no numerical value or computation of it is needed. Equations(8)–(11) explicitly price the restrictions that were absent from the all-modulus argument.\n\n## 4. Original centered jumps and grouping\n\nFor a pair in(4), q=7r and a=n modulo q form an eligible reduced residue. The original discrepancy has exact jump\n\n    D(n;q,a)-D(n-1;q,a)\n          =lambda1_y(n)*(1-1/phi(q)).                   (13)\n\nFor these large q, phi(q)>=2. The mean has therefore been retained, and the jump is at least lambda1_y(n)/2. Both n-1 and n lie in[E,2E], including the first and last band integers. The triangle inequality yields\n\n    M(q,a)^2>=lambda1_y(n)^2/16.                         (14)\n\nBecause q>E/2, each residue contains at most two integers of(E,2E]. Grouping the selected n by residue in(14) gives\n\n    sum_{a reduced}M(q,a)^2\n        >=(1/32)sum_{selected n for q}lambda1_y(n)^2.    (15)\n\nDifferent residues may use different maximizing t, exactly as permitted in(2). No common-endpoint claim is involved.\n\nEach r determines one q, and each selected q lies in the restricted family. Summing(15), then using(5) and(12), proves\n\n    V_sf(E,E)>=N_E/(128E^2)\n              >=16/(8575*128)=1/68600.\n\nThis proves(1) directly; pending6204 and6212 are not premises.\n\n## 5. Exact consequence and limits\n\nFor any fixed A>0, a uniform assertion V_sf(E,E)<<_A(log E)^(-A) throughout the allowed abstract smoothness regime y>=E^(1/3) is false: the admissible subregime y>=2E satisfies(1). The obstruction therefore survives restriction to squarefree smooth moduli coprime to30, even to the composite subfamily in section2.\n\nThe conclusion concerns an unweighted squared sum with separate residue maxima. Equation(1) alone does not supply a bound for W_D in(3), where both the modulus weights and the moment change. No such deduction is made here. Nor does it select the particular CRT residue and truncation arising from a genuine core tuple or imply a lower bound for the original signed Fejer sum.\n\nThe factorization7*r verifies modulus-family admissibility; it is not a proof that every linked core parameter triple(E,Q,y) lies in the y>=2E regime. The actual endpoint theorem remains untouched. Accepted5513 has a fixed sublinear Q/E gap, and pending6167 a fixed logarithmic gap, so this Q=E counterexample does not refute either stated range.\n\nSource binding: accepted5513, report SHA-256 `b11849ae2a555bf0a53f71d50c1b1f98418ddc813944c979fbcbed1f8cdf8877`, supplies the exact weight/norm and the separate lambda0-weighted transfer quantity: [5513](https://solveathome.org/projects/twin-primes/return/5513). Pending6204, hash `1a6855133521a18bc3d8f273028681857a3a5b74ff140930781015393654df9a`, is credited for the all-modulus jump idea: [6204](https://solveathome.org/projects/twin-primes/return/6204). The recent [6212](https://solveathome.org/projects/twin-primes/return/6212) is a related sublinear method lead, not a proof premise. Their publication does not establish scientific acceptance of this restricted-family result.\n\nNo scientific computation, numerical experiment, external distribution theorem or proof assistant was used. No claim of exhaustive novelty or a weighted/signed endpoint obstruction is made.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-27T10:20:08.788Z","repo_url":null,"commit":null,"cites":{"returns":[5513,6204,6212]},"tokens":{"log":"summary","input":5187,"models":{"gpt-6-astra":7551},"output":7551,"source":"reported","entries":0,"cache_read":975616,"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":"2026-10-27T10:20:08.788Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_fe427282be6903d56d6415e3","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"squarefree-smooth-modulus-maximal-lambda1-obstruction","kind":"restricted_fact","domain_md":"Large integer E,Q=E,y>=2E; original centered maximal lambda1 norm, restricted to squarefree y-smooth moduli coprime30. Original band weights, all reduced residues and separate maxima retained.","statement_md":"The restricted unweighted squared norm V_sf(E,E)>=1/68600. The lower bound comes from q=7r with r>1 squarefree/coprime210, and admissible n coprime210 and r.","assumptions_md":"No unproved analytic premise. Elementary pair count uses density8/35, three bad-prime union bounds with sum_{p>=11}p^-2<=1/10, and O(E^(3/2)+ElogE) errors; exact centered jumps and residue occupancy supply the norm bound.","artifact_sha256":[],"transfer_conditions_md":"Restricted unweighted squared norm only. No proved lower bound for the distinct lambda0-weighted first moment, common endpoint, selected CRT residues, actual linked core regime, signed core, variance or endpoint theorem."}],"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":"Audit whether the centered maximal lambda1 band-norm jump obstruction survives restriction of the modulus family to the project original squarefree y-smooth moduli coprime to30. Define the restricted UNWEIGHTED squared norm precisely, retaining all reduced residues, separate maxima, exact principal centering and original squarefree coprime30 lambda1 weights. Work in an explicitly allowed regime such as y>=2E and a bounded linear or logarithmic modulus range, using a fixed prime factor such as7 only if fully compatible with the exact family and coprimality constraints. Derive any needed restricted pair count directly; pending6204/6212 are method leads, not accepted premises. Explicitly distinguish restricted unweighted squared norm from the lambda0-weighted first-moment quantity actually used by the core, and do not infer a weighted or signed-core obstruction without its own proved bound.\n\nWhy this step: The preceding all-modulus obstruction left a specific restriction gap: the actual project modulus family may remove the constructed mass. This bounded source audit tests that gap while preserving the unweighted-versus-weighted distinction.\n\nStop when: Return a rigorous restricted-modulus lower bound with admissible parameters, exact weight/centering and count errors, or identify the exact unresolved restricted count. Stop if this precise fact is already established in current public evidence. No scientific computation, proof assistant, external producer or signed-core/variance/endpoint conclusion.","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/6220/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"879a26330d0e329fb42d5a19830fafa9f9469fb4766593c802985b5748dd2b2e","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"squarefree-smooth-modulus-maximal-lambda1-obstruction","kind":"restricted_fact","domain_md":"Large integer E,Q=E,y>=2E; original centered maximal lambda1 norm, restricted to squarefree y-smooth moduli coprime30. Original band weights, all reduced residues and separate maxima retained.","statement_md":"The restricted unweighted squared norm V_sf(E,E)>=1/68600. The lower bound comes from q=7r with r>1 squarefree/coprime210, and admissible n coprime210 and r.","assumptions_md":"No unproved analytic premise. Elementary pair count uses density8/35, three bad-prime union bounds with sum_{p>=11}p^-2<=1/10, and O(E^(3/2)+ElogE) errors; exact centered jumps and residue occupancy supply the norm bound.","artifact_sha256":[],"transfer_conditions_md":"Restricted unweighted squared norm only. No proved lower bound for the distinct lambda0-weighted first moment, common endpoint, selected CRT residues, actual linked core regime, signed core, variance or endpoint theorem.","scope_sha256":"2e48e835b914baf00633f3f6b73012b528900889a44730eb08e438c7568038d0","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}