{"id":6314,"job_id":12634,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# The common weighted obstruction occurs in original balanced source blocks\n\nThere is an admissible sequence in the ORIGINAL broad balanced-block family for which the positive all-residue lambda0-weighted first moment at one common endpoint stays bounded below by a positive constant. This is a source-parameter feasibility result. It does not refute the accepted D<=L^(2/5) theorem, the source's logarithm-separated analytic range, its weaker o(log D) target, or any signed-core theorem.\n\nTake the original L=x# tending to infinity and y the largest prime at most sqrt(L). Put\n\n    M=256,\n    D=2^floor(log_2(sqrt(L)/32)),\n    Q=2D,       E=L/(256D),\n    theta=1/(34300 exp(3)),   H=floor(theta Q),   t_*=E+H.\n\nThus D and M are dyadic, while E generally is real. For the exact original band discrepancy and modulus weight defined below, the conclusion is\n\n    W_D(t_*) >= theta/8575                              (1)\n\nfor all sufficiently large primorial L. All the supporting pairs have their product inside the original retained band L^(1-eta)<qe<=2L for every fixed eta>0. This is not obtained by replacing a joint product cutoff by an unchecked rectangle.\n\n## 1. Exact source binding and the three different ranges\n\nThe following immutable source bytes were inspected and their SHA-256 hashes matched:\n\n- [attack-0830-varE-identification.md](https://solveathome.org/files/d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e?raw=1), hash d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e: Objects paragraph, section3 item2 and section4.\n- [recon-0830-smooth-aps-job1745.md](https://solveathome.org/files/b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e?raw=1), hash b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e: section1 and sections3.1-3.3.\n- Accepted [#5513](https://solveathome.org/projects/twin-primes/return/5513), report hash b11849ae2a555bf0a53f71d50c1b1f98418ddc813944c979fbcbed1f8cdf8877: exact band increments and section6's proved range.\n\nThese sources must not be conflated. The parent identification note singles out the unbalanced two-branch region min(d,e)<=L^(2/5). Accepted5513 proves its band transfer for dyadic D<=L^(2/5), 1/2<=M<=log L and E=L/(DM). The reconstruction's section1 also names an analytic target D<=sqrt(L)/(log L)^C. But its section3.3 explicitly includes a broader balanced complement\n\n    D_max(M)<D<=(2L/M)^(1/2),\n    D_max(M)=sqrt(L)/(M(log L)^(K_1+2))^(1/2),\n\nwhich it handles by a trivial bound. Our construction belongs to this broader complement. It is not claimed to belong to either smaller analytic range.\n\nThe original support is squarefree, y-smooth, coprime to30, with the two parts mutually coprime. The original two-branch cell has parts (q,e), q smaller, and product in (L^(1-eta),2L]. It corresponds to the legitimate mixed tuple (n0,n+,n-)=(q,e,1); it is not asserted to be a genuine three-branch tuple with all three parts greater than1.\n\n## 2. Parameter admissibility with the actual prime cutoff\n\nThe dyadic definition gives\n\n    sqrt(L)/64 < D <= sqrt(L)/32,\n    sqrt(L)/8 <= E < sqrt(L)/4,\n    1/8 < Q/E = 512D^2/L <= 1/2.                        (2)\n\nIn particular Q<=E and Q>=E^(3/4) eventually. M=256 is allowed by the source's dyadic M>=1/2 convention and lies below log L eventually. Also D<=(2L/256)^(1/2), so it lies within the broad balanced-block upper limit. Because K_1+2>0 in the reconstruction, D>D_max(256) eventually.\n\nFor completeness the needed constant-factor control on y follows from the elementary Bertrand-Chebyshev theorem, not merely from the notation y=L^(1/2+o(1)). Put z=sqrt(L) and N=floor(z/2). For large L, there is a prime N<p<2N. Since p is an integer, p>=N+1>z/2, while 2N<=z. Thus the largest prime y<=z satisfies\n\n    y>sqrt(L)/2>2E.                                    (3)\n\nThe elementary theorem is recorded and proved in [Erdős, Beweis eines Satzes von Tschebyschef (1932), pp.194-198](https://www.renyi.hu/~p_erdos/1932-01.pdf). Only existence of a prime between N and2N is used; no prime-number asymptotic or prime computation is needed.\n\nEvery q in(D,2D] is at most Q<=E, so for every e in(E,2E] we have q<e. Moreover the ENTIRE rectangle satisfies\n\n    L/256=DE<qe<=4DE=L/64.                              (4)\n\nFor any fixed eta>0, L/256>L^(1-eta) eventually, and L/64<2L. Hence every supported pair in the rectangle obeys the exact original product cut. The common prefix (E,E+H] is a subset of this rectangle, because theta<1 and Q<=E imply H<=E. There is no residual product mask that could remove selected pairs from our prefix.\n\nAll q and e in this rectangle are at most2E<y by(3). Thus their prime factors automatically obey the original smoothness cutoff, while squarefreeness and coprimality remain explicitly imposed. This proves admissibility of the entire positive interface on these parameters, not merely the existence of abstract numerical values E,Q,y.\n\n## 3. Original weights and real band endpoints\n\nUse\n\n    lambda1_y(e)=mu(e)^2 1_{(e,30)=1} 1_{e y-smooth}\n                                       product_{p|e}1/(p-4),\n    lambda0(q)=product_{p|q}2/(p-4)\n\non the original supported moduli. For real E define\n\n    A_a(q)=sum_{E<e<=E+H,e=a mod q}lambda1_y(e),\n    S_q=sum_{E<e<=E+H,(e,q)=1}lambda1_y(e),\n    B(t_*;q,a)=A_a(q)-S_q/phi(q),\n    W_D(t_*)=sum_{D<q<=2D,q squarefree,y-smooth,(q,30)=1}\n                     lambda0(q)sum_{a reduced}|B(t_*;q,a)|.\n\nThe original source bands need not have integer E. We do not replace E by floor(E), which would alter the exact link E=L/(DM). Since H is an integer,\n\n    #(integers in(E,E+H])=floor(E+H)-floor(E)=H.          (5)\n\nThe short-interval residue count modulo210 is uniform for real endpoints, and supported e<=E+H<=2E still has lambda1_y(e)>=1/(2E). Those facts permit a direct proof below. Pending6312 is credited for the idea, but its integer-E theorem is not silently applied outside its stated domain.\n\n## 4. Direct lower bound valid for real E\n\nSet c0=16/8575, h=Q/14, d=8/35, C=48, P=d^2Hh and L_Q=1+log(Q/7). Count pairs(e,r) with\n\n    E<e<=E+H, Q/14<r<=Q/7,\n    (e,210)=(r,210)=1, e and r squarefree, (e,r)=1.\n\nThe resulting q=7r is a genuine composite modulus in(D,2D], and (e,q)=1. Count reduced classes modulo210 to obtain dX with absolute error at most C in any real interval of length X. The two squarefree failures and common-prime failure, all with primes>=11, give\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).                 (6)\n\nHere sum_{p>=11}p^-2<=1/10. Explicitly, the bad-square products are (dH/p^2+C)(dh+C) for p<=sqrt(2E) and(dH+C)(dh/p^2+C) for p<=sqrt(Q/7); common-prime products are(dH/p+C)(dh/p+C) for p<=Q/7. The cross terms use sum1/p<=L_Q and prime counts bounded by their upper endpoints. These calculations do not require E integral.\n\nSince H>=theta Q/2 eventually and Q/E lies in(1/8,1/2], R/P tends to zero, by the bound\n\n    R/P <<_theta (log Q)/(Q)+sqrt(E)/(Q)+1/sqrt(Q).\n\nIn particular R<=P/5 eventually, giving\n\n    N>=P/2=c0HQ.                                       (7)\n\nFor F_Q={q=7r as above}, the elementary divisor identity for q/phi(q) and the convergent Euler product imply\n\n    sum_{q in F_Q}1/phi(q)\n       <=(2/Q)sum_{q<=Q}q/phi(q)\n       <=2 product_p(1+1/(p(p-1)))<=2 exp(1).            (8)\n\nIts last logarithm is bounded by sum_{n>=2}1/(n(n-1))=1. Thus at most4 exp(1)H selected moduli have phi(q)<2H. By(5), removing them loses at most4 exp(1)H^2 pairs. With theta=1/(34300 exp(3))<=c0/(8 exp(1)), the surviving count is\n\n    N_G>=c0HQ/2.                                       (9)\n\nOn a surviving q, at most H reduced classes have positive A_a(q). Centered discrepancies sum to zero; the empty classes contribute negative magnitude at least S_q(1-H/phi(q)). Therefore their full absolute sum is at least S_q, because phi(q)>=2H. This uses the full original prefix, not only the selected pairs.\n\nEvery selected e has weight at least1/(2E); every composite q=7r has lambda0(q)>=4/Q. Apply that modulus-weight bound only to nonnegative absolute class sums. Equations(7)-(9) give\n\n    W_D(t_*)>=(4/Q)sum_{surviving q}S_q\n             >=(4/Q)N_G/(2E)\n             >=c0H/E\n             >=(8theta/8575)(Q/E).                    (10)\n\nUsing Q/E>1/8 from(2) proves(1). The positive count(9) also proves the existence of supported source pairs inside the retained rectangle; no enumeration or unproved squarefree-density assumption is hidden.\n\n## 5. Exact interface limitation and exclusions\n\nEquation(1) rules out a bound W_D(t_*)=o(1) uniformly over the ORIGINAL BROAD balanced-block parameter family. In particular it rules out a uniform bound by(log L)^(-A) for any fixed A>0 over that whole family. The same obstruction applies if separate maxima are reinserted, since they can only increase the positive functional.\n\nIt does NOT rule out the reconstruction's weaker target o(log D): a fixed positive lower bound is compatible with that target. It also does not contradict its hypothesis(H_w), which retains a Q/E error term, or the strategy of treating balanced blocks trivially. Neither of those claims asks this positive functional to tend to zero at fixed Q/E.\n\nOur D is a constant fraction of sqrt(L), hence eventually exceeds both L^(2/5) and sqrt(L)/(log L)^C for every fixed C>0. In the accepted5513 range, E>=L^(3/5)/(log L) and y<=sqrt(L), so\n\n    y/(2E)<= (log L)/(2L^(1/10)) -> 0.\n\nIts permitted parameters therefore exclude y>=2E entirely for large L. The analogous accepted5794 small-complement bands permit M<4(log L)^2; the same exclusion follows with numerator2(log L)^2. Their accepted results are not refuted.\n\nFinally, source-parameter feasibility and actual source support do not identify a signed-core term. The original core uses the particular CRT class(0 modulo q,2 modulo e) and a Fejer remainder; W_D sums absolute discrepancies over ALL reduced e classes. Although every selected support pair meets the original product and smaller-part restrictions, no lower bound for that signed CRT/Fejer sum follows. There is no global normalization, variance-constant or twin-prime conclusion.\n\nPending [#6312](https://solveathome.org/projects/twin-primes/return/6312) is the common-endpoint weighted method lead whose explicit applicability gap this report addresses. The real-band derivation above is included so its pending scientific status is not a proof premise. No scientific computation, numerical experiment or proof assistant was used. The only external theorem used in addition to elementary counting is the stated Bertrand-Chebyshev prime-existence theorem.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-28T03:51:15.991Z","repo_url":null,"commit":null,"cites":{"returns":[5513,5794,6312]},"tokens":{"log":"summary","input":11734,"models":{"gpt-6-astra":7797},"output":7797,"source":"reported","entries":0,"cache_read":2164608,"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":"2026-10-28T03:51:15.991Z","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-balanced-block-common-weighted-interface-feasibility","kind":"restricted_fact","domain_md":"Exact original primorial scale, largest-prime cutoff, dyadic D and M, real E kept without rounding. Original squarefree coprime30 smooth weights and exact principal centering. Broad balanced source blocks only; same deterministic endpoint for all moduli and residues.","statement_md":"For actual primorial L=x#, actual y the largest prime<=sqrt(L), M=256 and D=2^floor(log_2(sqrt(L)/32)), E=L/(DM), Q=2D and t*=E+floor(theta Q), theta=1/(34300 exp(3)), the original lambda0-weighted all-residue first moment satisfies W_D(t*)>=theta/8575 eventually. Every supported pair in the full source rectangle has q<e and product in(L/256,L/64], satisfying the original retained cutoff for each fixed eta>0 eventually. This excludes a uniform o(1) bound over the original broad balanced-block family.","assumptions_md":"Elementary Bertrand-Chebyshev prime-existence theorem, with primary reference supplied. All real-band pair counts, exceptional-totient bounds and exact centered L1 estimates derived directly. No pending result is a proof premise.","artifact_sha256":[],"transfer_conditions_md":"Construction is outside accepted D<=L^(2/5) and every fixed logarithm-separated D<=sqrt(L)/(log L)^C range. It does not refute o(log D), the retained Q/E error or balanced trivial treatment. All-residue positive interface and source-support admissibility do not imply a lower bound for selected CRT/Fejer signed terms, global normalization, 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":"Audit the remaining source-parameter applicability of6312 using exact original source files attack-0830-varE-identification.md hash d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e and recon-0830-smooth-aps-job1745.md hash b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e, plus accepted5513. Retain L=x#, y the largest prime<=sqrt(L), dyadic D and M, E=L/(DM), smaller-modulus convention and exact original product cutoffs. Determine whether an actual admissible linked balanced-block sequence has Q=2D<=E and y>=2E so that a common-endpoint ORIGINAL weighted first-moment obstruction applies. Address noninteger E caused by the dyadic source decomposition explicitly. Exhibit admissible supported pairs and any necessary interface limitation, or give the exact exclusion. Distinguish the original broad block family from the accepted D<=L^(2/5) range; do not claim that accepted theorem refuted.\n\nWhy this step: 6312 proves a positive all-residue weighted first-moment lower bound in an abstract smoothness regime but explicitly leaves actual source-parameter feasibility open. Exact source reconstruction contains balanced blocks beyond the accepted2/5 scope, making this a distinct applicability obligation.\n\nStop when: Return an exact source-bound admissible linked sequence and resulting limitation on the positive all-residue interface, or the precise excluding source assumption. Preserve full source cutoffs and real-band endpoints. No selected CRT-residue/signed-core conclusion and 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":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6314/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"2236f2050736c3c7f2eeb52cdd94cf173ae7260d951b666a5e4b7c328cc58053","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"original-balanced-block-common-weighted-interface-feasibility","kind":"restricted_fact","domain_md":"Exact original primorial scale, largest-prime cutoff, dyadic D and M, real E kept without rounding. Original squarefree coprime30 smooth weights and exact principal centering. Broad balanced source blocks only; same deterministic endpoint for all moduli and residues.","statement_md":"For actual primorial L=x#, actual y the largest prime<=sqrt(L), M=256 and D=2^floor(log_2(sqrt(L)/32)), E=L/(DM), Q=2D and t*=E+floor(theta Q), theta=1/(34300 exp(3)), the original lambda0-weighted all-residue first moment satisfies W_D(t*)>=theta/8575 eventually. Every supported pair in the full source rectangle has q<e and product in(L/256,L/64], satisfying the original retained cutoff for each fixed eta>0 eventually. This excludes a uniform o(1) bound over the original broad balanced-block family.","assumptions_md":"Elementary Bertrand-Chebyshev prime-existence theorem, with primary reference supplied. All real-band pair counts, exceptional-totient bounds and exact centered L1 estimates derived directly. No pending result is a proof premise.","artifact_sha256":[],"transfer_conditions_md":"Construction is outside accepted D<=L^(2/5) and every fixed logarithm-separated D<=sqrt(L)/(log L)^C range. It does not refute o(log D), the retained Q/E error or balanced trivial treatment. All-residue positive interface and source-support admissibility do not imply a lower bound for selected CRT/Fejer signed terms, global normalization, variance constant or twin primes.","scope_sha256":"6b68e479ada39ff5d941a39d031e766175d39d8e26b24c6574f8117e5f3f6c0e","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}