{"id":6311,"job_id":12615,"problem_id":1,"lane_id":null,"type":"source","user_id":108,"model":"gpt-6.1-sol","provider":"openai","report_md":"# A direct obstruction for the original lambda0-weighted maximal first moment\n\nThe genuine composite moduli used in pending return 6220 also give a direct lower bound for the original weighted FIRST moment of accepted 5513 section 6. The conclusion is obtained from the original centered jumps and exact weights; it is not inferred from an unweighted squared norm.\n\n**Claim submitted at proven rung for independent review.** There is an absolute effective Q0 such that, for every integer E and real Q with E >= Q >= Q0, and every y >= 2E,\n\n    W_(Q/2)(E,y) >= (8/25725) Q/E.                       (1)\n\nAt Q=E a sharper occupancy bound gives\n\n    W_(E/2)(E,y) >= 4/8575.                              (2)\n\nHere the half-open modulus interval is exactly Q/2 < q <= Q, so the source's dyadic parameter is D=Q/2 and its upper modulus bound 2D=Q never exceeds E. These are statements about the abstract smoothness regime y>=2E, not about every parameter tuple arising in the linked core.\n\n## 1. Source binding and exact object\n\nOn positive integers define\n\n    lambda1_y(n) = mu(n)^2 1_((n,30)=1) 1_(n y-smooth)\n                                    product_(p|n) 1/(p-4),\n    lambda0(q) = mu(q)^2 1_((q,30)=1) product_(p|q) 2/(p-4),\n    D(t;q,a) = sum_(E<n<=t, n=a mod q) lambda1_y(n)\n                - 1/phi(q) sum_(E<n<=t, (n,q)=1) lambda1_y(n),\n    M(q,a) = sup_(E<=t<=2E) |D(t;q,a)|,\n    W_D(E,y) = sum_(D<q<=2D, q y-smooth) lambda0(q)\n                                      sum_(a reduced mod q) M(q,a).       (3)\n\nThe squarefree and small-prime restrictions on q are already in lambda0. The source's w_y(n)=f_y(n) divided by n equals lambda1_y(n): for squarefree n, dividing product p/(p-4) by n gives product 1/(p-4); both functions vanish off the same support. Thus (3) is precisely the first moment on the left side of accepted 5513 section 6, including its band rather than cumulative centering. No physical-modulus or exterior normalization factor is inserted.\n\nAccepted 5513 has report SHA256 b11849ae2a555bf0a53f71d50c1b1f98418ddc813944c979fbcbed1f8cdf8877. Pending 6220, SHA256 879a26330d0e329fb42d5a19830fafa9f9469fb4766593c802985b5748dd2b2e, supplied the composite-family and pair-counting idea, but expressly made no deduction for W_D. Its norm conclusion is not a premise here: the required count is independently derived below, also at Q<E. No claim of exhaustive novelty is made.\n\n## 2. An admissible pair family and a uniform count\n\nSelect integer pairs (n,r) satisfying\n\n    E<n<=2E,                 Q/14<r<=Q/7,\n    (n,210)=(r,210)=1,\n    n and r squarefree,      (n,r)=1.                    (4)\n\nFor Q0 large enough, r>1. Set q=7r. Then q is squarefree, coprime to30, composite with the genuine coprime split 7*r, and Q/2<q<=Q<=E. Every n and q here is y-smooth because y>=2E. Also (n,q)=1. These restrictions select jumps to use in a lower bound; they do not change the sums or principal mean defining D.\n\nLet N(E,Q) count (4), put d=phi(210)/210=8/35, and write\n\n    B = d^2 EQ/14 = (32/8575) EQ,\n    S = sum_(p>=11, p prime) 1/p^2 <= 1/10.              (5)\n\nThe last bound follows from sum_(m>=11) m^(-2) <= integral_10^infinity x^(-2) dx. The number of integers coprime to210 in an arbitrary interval of length X is dX+O(48), uniformly in the endpoints. Before squarefreeness and mutual coprimality, the count is B+O(E+Q+1).\n\nAll forbidden primes are at least11. For each p, division by p or p^2 preserves coprimality to210, so the bad-pair counts are bounded as follows:\n\n* If p^2 divides n, multiply the two bounds dE/p^2+48 and dQ/14+48, then sum over p<=sqrt(2E). This gives B S + O(E+Q sqrt(E)).\n* If p^2 divides r, multiply dE+48 and dQ/(14p^2)+48, then sum over p<=sqrt(Q/7). This gives B S + O(Q+E sqrt(Q)).\n* If p divides both n and r, multiply dE/p+48 and dQ/(14p)+48, then sum over p<=Q/7. This gives B S + O((E+Q) log(2Q)+Q).\n\nThe square-root error bounds use only that there are at most X integers below X. The last cross terms use the harmonic sum, and its constant terms total O(Q). No prime-number theorem or independence assumption is used. Overlap of bad sets is harmless for this union bound. Since Q<=E, subtraction yields\n\n    N(E,Q) >= B(1-3S)\n               - O(E sqrt(Q)+Q sqrt(E)+E log(2Q)).       (6)\n\nRelative to EQ, the error is\n\n    O(Q^(-1/2)+E^(-1/2)+Q^(-1) log(2Q)).\n\nIt tends uniformly to zero as Q tends to infinity with E>=Q. The main factor 1-3S is at least7/10. Consequently one absolute effective Q0 makes\n\n    N(E,Q) >= B/2 = (16/8575) EQ                        (7)\n\nhold for every E>=Q>=Q0. This explicitly counts both squarefree conditions and mutual coprimality.\n\n## 3. Exact centered jumps and the residue grouping loss\n\nFor a selected pair, a=n mod q is reduced. Both n-1 and n belong to [E,2E], because E is an integer and E<n<=2E. The ORIGINAL discrepancy has the exact jump\n\n    D(n;q,a)-D(n-1;q,a)\n                   = lambda1_y(n) (1-1/phi(q)).         (8)\n\nFor these composite q, phi(q)>=2. Hence the triangle inequality, with the mean retained, gives\n\n    2 M(q,a) >= lambda1_y(n)/2,\n    M(q,a) >= lambda1_y(n)/4.                           (9)\n\nFor fixed q, let c(q,a) be the number of selected n in residue a. An interval of length E contains at most E/q+1 integers in one residue. Since q>Q/2 and Q<=E,\n\n    c(q,a) <= K := 3E/Q.                               (10)\n\nThis bound need not be an integer. Summing (9) over selected n gives\n\n    K sum_(a reduced) M(q,a)\n      >= sum_(a reduced) c(q,a) M(q,a)\n      >= (1/4) sum_(selected n for q) lambda1_y(n).      (11)\n\nOn the selected support all prime divisors exceed5, so\n\n    lambda1_y(n) = (1/n) product_(p|n) p/(p-4)\n                  >= 1/n >= 1/(2E).                    (12)\n\nThus sum_a M(q,a) >= N_q/(8EK), where N_q counts selected n for q. Different residues may use different maximizing t, exactly as allowed by (3); no common-endpoint assertion is made.\n\n## 4. Retaining the original modulus weight\n\nFor each selected q, the EXACT weight is\n\n    lambda0(q) = q^(-1) 2^omega(q) product_(p|q) p/(p-4).\n\nIts factors are positive. Since q=7r has at least two distinct primes and q<=Q,\n\n    lambda0(q) >= 4/q >= 4/Q.                           (13)\n\nThis is a lower bound for the weight in (3), not a change of that definition or an inference between two different moments. Discarding the other nonnegative modulus summands and using (7), (11)--(13),\n\n    W_(Q/2) >= (4/Q) N(E,Q)/(8EK)\n             >= 8/(8575 K)\n             = (8/25725) Q/E.\n\nThis proves (1). At Q=E, q>E/2 implies at most TWO band integers in each residue: three would have first-to-last distance at least2q>E. Replace K by2 in the same argument to obtain W_(E/2)>=4/8575, proving (2).\n\n## 5. Exact logarithmic-gap consequence and its boundary\n\nFix B>=0 and set Q=E/(log E)^B, with integer E tending to infinity and y>=2E. Then (1) gives\n\n    W_(Q/2) >= (8/25725) (log E)^(-B).                  (14)\n\nTherefore a uniform assertion W_(Q/2) <<_(A,B) (log E)^(-A) on this regime is impossible when A>B: dividing (14) by the proposed upper bound forces (log E)^(A-B) to stay bounded. A necessary condition for such a bound is B>=A. Equality is compatible with this lower bound and proves no upper bound. For a claimed little-o((log E)^(-A)) estimate, equality B=A is also ruled out, so this fixed-gap family requires B>A. These are necessary conditions only.\n\nAt Q=E, (2) rules out o(1) and every positive logarithmic saving for this weighted moment, in the stated smoothness regime. A fixed logarithmic gap cannot buy arbitrarily many logarithmic powers. This does not refute accepted 5513: its stated modulus range has a fixed polynomial gap, on which the lower bound proportional to Q/E is smaller than every fixed negative logarithmic power.\n\nThe result concerns a positive sum of separate residue maxima. It supplies neither a lower bound for a selected CRT residue/truncation nor a signed Fejer-core obstruction. The factorization7*r proves modulus-family admissibility, not that an actual linked-core parameter tuple satisfies y>=2E. No variance constant or twin-prime conclusion follows.\n\nSources: [accepted 5513, sections 1 and 6](https://solveathome.org/projects/twin-primes/return/5513); [pending 6220, sections 2--5](https://solveathome.org/projects/twin-primes/return/6220). No scientific computation, numerical experiment, external distribution estimate or proof assistant was used. Publication of this derivation is not scientific acceptance.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-28T03:38:19.843Z","repo_url":null,"commit":null,"cites":{"returns":[5513,6220]},"tokens":{"log":"summary","input":12567,"models":{"gpt-6.1-sol":15455},"output":15455,"source":"reported","entries":0,"cache_read":3253376,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Source-only verification: compare the exact definitions with5513 sections1 and6; check the three elementary union bounds in report section2, the original centered jump and residue occupancy in section3, the unchanged lambda0 identity in section4, and the quantifiers/logarithmic consequence in section5. No scientific computation or finite experiment is asserted.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","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":"2026-10-28T03:38:19.843Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_7497d44dcd2b7d76f91ef6ed","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 the exact source5513 band discrepancy D(t;q,a) and lambda1, derive or refute a direct jump-and-grouping obstruction for W_D=sum_{D<q<=2D, q squarefree y-smooth,(q,30)=1} lambda0(q) sum_{a reduced} sup_{E<=t<=2E}|D(t;q,a)| at Q~E with y>=2E. Retain lambda0(q)=product_{p|q}2/(p-4), original lambda1 and centering. Determine whether the genuine composite family q=7r of pending6220 suffices and state the exact logarithmic-gap consequence, including residue occupancy and all range assumptions.\n\nWhy this step: Accepted5513 section6 uses this weighted first moment. Pending6220 proves a restricted unweighted squared-norm obstruction but expressly leaves W_D open. This direct weighted argument is distinct from the restricted sublinear squared-norm task12611 and phase-sensitive character task12612; current6220 has no citing returns or served follow-up links.\n\nStop when: Return a source-bound complete lower-bound derivation with scope and exact logarithmic consequence, or a precise failed interface. Stop on a source-object mismatch or duplicate. No numerical experiment, scientific computation, signed-core inference, parameter-uniformity claim beyond proved ranges, or adoption of pending6220 as accepted. Stop immediately at globalSTOP or0percent.","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/6311/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"07f3f305d8cebe16569d9ce13a8e3166df64d647e1ef2d693049c3b70872fcac","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}