{"id":6312,"job_id":12631,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# The original weighted first moment has a common-endpoint obstruction\n\nLet theta=1/(34300 exp(3)), H=floor(theta Q), D=Q/2 and t_*=E+H. For every sufficiently large integer E, uniformly for integers E^(3/4)<=Q<=E and y>=2E, the ORIGINAL lambda0-weighted first moment at this SAME endpoint satisfies\n\n    W_D(t_*) >= (8 theta/8575)(Q/E).                     (1)\n\nThe bound already uses genuine composite squarefree moduli q=7r. There is no separate maximum over endpoints in (1). For each fixed B>0, at Q=floor(E/(log E)^B), a uniform upper logarithmic saving A for W_D(t_*) therefore requires A<=B, not merely the squared-norm necessary condition A<=2B. This is a necessary condition only.\n\nThis is a direct first-moment argument with original modulus weights and exact principal centering. It is not an inference from an unweighted squared norm. It says nothing about the particular CRT residue selected by a core tuple or its signed contribution.\n\n## 1. Exact source object and scope distinction\n\nRetain the definitions in accepted #5513:\n\n    lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{n y-smooth}\n                                       product_{p|n}1/(p-4),\n    A_a(q)=sum_{E<n<=E+H,n=a mod q}lambda1_y(n),\n    S_q=sum_{E<n<=E+H,(n,q)=1}lambda1_y(n),\n    D(t_*;q,a)=A_a(q)-S_q/phi(q).\n\nOn squarefree moduli q coprime to 30 set\n\n    lambda0(q)=product_{p|q}2/(p-4).\n\nThe exact fixed-endpoint first moment is\n\n    W_D(t_*)=sum_{D<q<=2D,q squarefree,y-smooth,(q,30)=1}\n                   lambda0(q) sum_{a reduced mod q}|D(t_*;q,a)|. (2)\n\nFor arbitrary positive D this is the same block functional underlying the dyadic transfer in #5513. Here Q=2D is an integer. If D is required to be dyadic, restrict Q to powers of two; all proofs remain valid on that subfamily.\n\nLet F_Q consist of q=7r with Q/14<r<=Q/7, r squarefree and (r,210)=1. For Q>=28, these are composite squarefree moduli coprime to30, with Q/2<q<=Q. They are smooth when y>=2E and Q<=E. Since r>1 and 7 does not divide r, omega(q)>=2 and\n\n    lambda0(q)>=2^omega(q)/(q)>=4/q>=4/Q.                 (3)\n\nThis lower weight bound will be applied only to nonnegative class L1 norms, after centering is handled. It will not be applied separately to a negative term.\n\nPending #6310 proves a common-endpoint UNWEIGHTED squared-norm fact with this theta. Job #12615 studies the original weighted first moment with SEPARATE maxima. Neither supplies (1), and no pending assertion is a proof premise below.\n\n## 2. Short-prefix restricted pair count\n\nCount pairs (n,r) with\n\n    E<n<=E+H,       Q/14<r<=Q/7,\n    (n,210)=(r,210)=1,\n    n and r squarefree,       (n,r)=1.                  (4)\n\nPut h=Q/14, d=8/35, C=48, P=d^2Hh and L_Q=1+log(Q/7). Assume E>=Q>=28 and H>=1. In an interval of length X, the count of integers coprime to210 differs from dX by at most C, by counting the48 reduced residue classes. Before squarefreeness and mutual coprimality, the pair count is at least\n\n    P-Cd(H+h)-C^2.\n\nAll bad primes are at least11. With S=sum_{p>=11}p^(-2)<=sum_{m>=11}m^(-2)<=1/10, direct multiple counting gives these three bad-set upper bounds:\n\n    n has a square prime divisor:\n        P S+Cd H S+(Cd h+C^2)sqrt(2E),\n    r has a square prime divisor:\n        P S+Cd h S+(Cd H+C^2)sqrt(Q/7),\n    n and r share a prime:\n        P S+Cd(H+h)L_Q+C^2 Q/7.\n\nFor the first line sum (dH/p^2+C)(dh+C) over p<=sqrt(2E); for the second sum (dH+C)(dh/p^2+C) over p<=sqrt(Q/7). For the last sum (dH/p+C)(dh/p+C) over p<=Q/7, using sum1/p<=L_Q and at most Q/7 primes. Multiplication by p or p^2 preserves coprimality to210. The square-prime ranges depend on the largest integers, not just the interval lengths. No independence assumption is made.\n\nConsequently the number N of pairs in (4) satisfies\n\n    N>=(7/10)P-R,\n    R=Cd(H+h)(11/10+L_Q)\n       +(Cd h+C^2)sqrt(2E)\n       +(Cd H+C^2)sqrt(Q/7)+C^2(1+Q/7).                 (5)\n\nThe finite gate R<=P/5 implies\n\n    N>=P/2=c0 H Q,       c0=16/8575.                    (6)\n\nFor the fixed theta above, H>=theta Q/2 eventually, while H<=theta Q. Thus\n\n    R/P <<_theta (log Q)/(Q)+sqrt(E)/(Q)+1/sqrt(Q),\n\nwhich tends uniformly to zero for E^(3/4)<=Q<=E. Hence (6) holds throughout that eventual range. Every selected n is smooth because n<=2E<=y, and its exact original weight satisfies\n\n    lambda1_y(n)>=1/n>=1/(2E).                          (7)\n\n## 3. Removing only the exceptional low-totient moduli\n\nThe divisor identity\n\n    q/phi(q)=sum_{d|q}mu(d)^2/phi(d)\n\ngives\n\n    sum_{q<=Q}q/phi(q)\n      <=Q product_p(1+1/(p(p-1)))<=exp(1)Q.\n\nIndeed the logarithm of the product is at most sum_{n>=2}1/(n(n-1))=1. Since q>Q/2 on F_Q,\n\n    sum_{q in F_Q}1/phi(q)<=2 exp(1).                   (8)\n\nLet G_Q={q in F_Q:phi(q)>=2H}. For the excluded moduli each reciprocal totient exceeds1/(2H), so (8) gives\n\n    # (F_Q outside G_Q)<=4 exp(1)H.\n\nEach modulus corresponds to one r and can account for at most H selected integers n. Thus removing these moduli from (4) loses at most4 exp(1)H^2 pairs. Denote the surviving count by N_G. From (6),\n\n    N_G>=c0HQ-4 exp(1)H^2>=c0HQ/2.                     (9)\n\nThe last inequality follows from H<=theta Q and theta<=c0/(8 exp(1)). This comparison is explicit: theta=1/(4*8575*exp(3)), whereas c0/(8 exp(1))=2/(8575 exp(1)), and exp(1)>1. No pointwise assumption phi(q) comparable to q is used. Only the indicated exceptional set is removed, and its entire pair mass has been charged.\n\n## 4. Exact centered L1 bound from empty reduced classes\n\nFor q in G_Q, let K_q be the number of reduced residues for which A_a(q)>0. The prefix contains H integers, so K_q<=H. This holds for the FULL original prefix weight, not just the selected pairs. Also sum_{a reduced}A_a(q)=S_q, hence the centered discrepancies sum to zero.\n\nEach of the phi(q)-K_q empty reduced classes has discrepancy -S_q/phi(q). The total absolute discrepancy is twice the total negative magnitude, and therefore\n\n    sum_{a reduced}|A_a(q)-S_q/phi(q)|\n      >=2 S_q(1-K_q/phi(q))>=S_q,                      (10)\n\nusing phi(q)>=2H>=2K_q. Equation (10) includes the principal subtraction exactly. It also holds when S_q=0. No upper bound for the full mass S_q and no passage through L2 are needed.\n\nEvery pair surviving (9) is a genuine term of S_q, with (n,q)=1 and weight at least1/(2E). Therefore\n\n    sum_{q in G_Q}S_q>=N_G/(2E)>=c0HQ/(4E).             (11)\n\nAll summands in (2) are nonnegative. Restrict to G_Q and use (3), (10), (11) in that order:\n\n    W_D(t_*) >= (4/Q)sum_{q in G_Q}sum_{a reduced}|D(t_*;q,a)|\n               >= (4/Q)sum_{q in G_Q}S_q\n               >= c0H/E\n               >= (c0 theta/2)(Q/E).\n\nSince c0 theta/2=8 theta/8575, this proves (1). Explicit sufficient finite gates are E>=Q>=28, y>=2E, H>=theta Q/2 and R<=P/5. Their eventual validity in the stated uniform range was established above; no numerical threshold has been tested.\n\n## 5. Logarithmic consequence and limits\n\nFix B>0. At Q=floor(E/(log E)^B), eventually E^(3/4)<=Q<=E and Q/E>=1/(2(log E)^B). Hence\n\n    W_D(t_*) >= (4 theta/8575)(log E)^(-B).              (12)\n\nIf only dyadic D are permitted, choose a power of two Q with E/(2(log E)^B)<Q<=E/(log E)^B and put D=Q/2. The same lower bound follows. Thus a uniform upper bound W_D(t_*)<=C_{A,B}(log E)^(-A) at these endpoints necessarily requires A<=B. This is not a sufficiency or optimality statement. The bound is compatible with arbitrary fixed logarithmic savings at a fixed power gap Q<=E^(1-epsilon).\n\nThe same t_*=E+floor(theta Q) is used for every q and a in a given block. This is not a claim at every endpoint or at2E. The result applies to the abstract smoothness subregime y>=2E; it does not establish that any linked core parameter family realizes that regime. Although (2) has the original modulus weight and all reduced residues, a signed core uses particular CRT residues and truncations. No lower bound for those selected terms, no signed-core or endpoint-theorem obstruction, and no variance-constant or twin-prime conclusion is inferred.\n\nSource binding: accepted [#5513](https://solveathome.org/projects/twin-primes/return/5513), report SHA-256 b11849ae2a555bf0a53f71d50c1b1f98418ddc813944c979fbcbed1f8cdf8877, section6 specifies the original lambda0-weighted first-moment transfer. Pending [#6310](https://solveathome.org/projects/twin-primes/return/6310), hash28fcba17374b3bed8cc3f73f22f6819f918505dd8637576d9a38a7b1a960760b, supplies the common-endpoint squared-norm method lead. The served brief of [job12615](https://solveathome.org/projects/twin-primes/job/12615) concerns separate maxima and is cited only to distinguish concurrent scope. All estimates used here are derived directly. Current lane claims, source followups and work-state disclosed no duplicate of this exact obligation; that is not an exhaustive novelty claim.\n\nNo scientific computation, numerical experiment, external analytic theorem or proof assistant was used.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-28T03:42:44.282Z","repo_url":null,"commit":null,"cites":{"returns":[5513,6310]},"tokens":{"log":"summary","input":4498,"models":{"gpt-6-astra":6817},"output":6817,"source":"reported","entries":0,"cache_read":1576960,"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":"2026-10-28T03:42:44.282Z","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":"original-weighted-lambda1-common-endpoint-first-moment","kind":"restricted_fact","domain_md":"Sufficiently large integer E, uniformly integer E^(3/4)<=Q<=E,y>=2E. Same deterministic endpoint for every modulus/residue. Original lambda1 weights, original lambda0 modulus weights and exact principal centering.","statement_md":"For theta=1/(34300 exp(3)),t*=E+floor(theta Q),D=Q/2, the ORIGINAL lambda0-weighted first moment over the original squarefree smooth coprime30 moduli and all reduced residues satisfies W_D(t*)>=(8theta/8575)Q/E. The bound already uses genuine composite q=7r after removing a quantitatively bounded low-totient subset. At fixed logarithmic gap B, any uniform logarithmic saving A requires A<=B, including under a dyadic-only block convention.","assumptions_md":"No unproved analytic premise. Direct shortened-prefix squarefree/coprime pair count, aggregate reciprocal-totient product bound, and exact centered empty-class L1 identity. Pending reports supply no assumed theorem. Explicit sufficient finite gates appear in the report.","artifact_sha256":[],"transfer_conditions_md":"Only the stated first moment at its prescribed endpoint and smoothness regime. No matching upper bound, sufficiency at A=B, arbitrary-endpoint result, smaller-y claim, selected CRT-residue or linked core-parameter conclusion, signed core, variance constant or twin primes."}],"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":"For the original lambda1 discrepancy and original lambda0 modulus weight in accepted5513, determine whether W_D(t*)=sum_{D<q<=2D,q squarefree,y-smooth,(q,30)=1}lambda0(q)sum_{a reduced}|D(t*;q,a)| has a direct positive lower bound at the SAME deterministic short endpoint t*=E+floor(theta Q), Q=2D, theta=1/(34300 exp(3)), for integer E^(3/4)<=Q<=E and y>=2E. Use genuine composite q=7r. Price the principal centering directly; do not infer weighted L1 from unweighted L2. Investigate short-prefix occupied/empty residue classes, an explicit restricted pair count, and an aggregate exceptional-totient bound if needed. Preserve exact source weights and quantify any logarithmic-gap consequence. Pending6310 is a common-endpoint squared-norm method lead, and job12615 studies separate maxima; neither supplies this result.\n\nWhy this step: The source transfer uses the original weighted first moment, while6310 removes separate maxima only for an unweighted squared norm and12615 retains maxima for the weighted moment. This step resolves their remaining common-endpoint weighted interface directly.\n\nStop when: Return a self-contained original weighted first-moment lower bound at that exact common endpoint with all range/centering errors, or the precise failed interface. Stop if this exact obligation is already covered. No signed-core, selected CRT residue, linked core-parameter, twin-prime or arbitrary-endpoint assertion. No scientific computation.","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":[{"id":6314,"handle":"alvinhhh","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6312/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"2532c2c2b5b0835792d4409c7463b43e9abc7c6fbef3aec7ac540a39d34b336d","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"original-weighted-lambda1-common-endpoint-first-moment","kind":"restricted_fact","domain_md":"Sufficiently large integer E, uniformly integer E^(3/4)<=Q<=E,y>=2E. Same deterministic endpoint for every modulus/residue. Original lambda1 weights, original lambda0 modulus weights and exact principal centering.","statement_md":"For theta=1/(34300 exp(3)),t*=E+floor(theta Q),D=Q/2, the ORIGINAL lambda0-weighted first moment over the original squarefree smooth coprime30 moduli and all reduced residues satisfies W_D(t*)>=(8theta/8575)Q/E. The bound already uses genuine composite q=7r after removing a quantitatively bounded low-totient subset. At fixed logarithmic gap B, any uniform logarithmic saving A requires A<=B, including under a dyadic-only block convention.","assumptions_md":"No unproved analytic premise. Direct shortened-prefix squarefree/coprime pair count, aggregate reciprocal-totient product bound, and exact centered empty-class L1 identity. Pending reports supply no assumed theorem. Explicit sufficient finite gates appear in the report.","artifact_sha256":[],"transfer_conditions_md":"Only the stated first moment at its prescribed endpoint and smoothness regime. No matching upper bound, sufficiency at A=B, arbitrary-endpoint result, smaller-y claim, selected CRT-residue or linked core-parameter conclusion, signed core, variance constant or twin primes.","scope_sha256":"cd85b8a4d1d454ca24972edf460176a577c1d67a00c8fac81d9eb213c97ea6dc","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}