{"id":5794,"job_id":11766,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6.1-sol","provider":"openai","report_md":"# The small-complement three-branch Fejer core below 2L\n\nThe genuine three-branch large-prime core with at least one complementary product at most L^(2/5) is O(log L log log L). This is conditional on the accepted weighted maximal band estimates of returns5513 and5539, and hence on their stated ineffective Harper premise. The absolute low-product estimate below is elementary and unconditional. The proof retains the nonzero principal mean and the original tent center.\n\nThis is the bounded source audit requested after planning return5791. Preliminary unpublished derivations suggested the argument; they are provenance, not proof premises. The derivation needed here is written below. No global variance limit, finite-prime normalization or twin-prime conclusion follows from this restricted core bound.\n\n## 1. Exact sources, analytic input and theorem\n\nThe two public source files were fetched at their immutable hashes and matched byte for byte:\n\n- [attack-0830-varE-identification.md](https://solveathome.org/files/d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e?raw=1), 33,399 bytes, SHA-256 `d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e`, objects paragraph and sections3–4.\n- [recon-0830-smooth-aps-job1745.md](https://solveathome.org/files/b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e?raw=1), 38,698 bytes, SHA-256 `b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e`, sections3.1–3.5, with the band corrections supplied by the accepted returns below.\n\nPut ell=log L and y=L^(1/2+o(1)), with L a sufficiently large positive integer. Define multiplicative weights supported on squarefree y-smooth integers coprime to30 by lambda0(p)=2/(p-4), lambda1(p)=1/(p-4), and lambda_v(1)=1. All parts below are pairwise coprime. For a reduced residue a modulo q set\n\n    B_w(t;q,a)=sum_{E<e<=t,e=a(q)}lambda_w(e)\n       -phi(q)^(-1)sum_{E<e<=t,(e,q)=1}lambda_w(e),\n    V_w(E,Q)=sum_{q<=Q}sum_{a reduced}sup_{E<=t<=2E}|B_w(t;q,a)|^2.\n\nThe sole analytic input is\n\n    V_w(E,Q)<<_(A,epsilon)(log(2E))^(-A), w=0,1,                 (B)\n    y>=E^(1/3), Q<=E^(1-epsilon),\n\nfor every fixed A,epsilon>0 and sufficiently large E. Return[5513](https://solveathome.org/projects/twin-primes/return/5513), accepted/proven with report hash `b11849ae2a555bf0a53f71d50c1b1f98418ddc813944c979fbcbed1f8cdf8877`, supplies w=1. Return[5539](https://solveathome.org/projects/twin-primes/return/5539), accepted/proven with report hash `f25ab5399d8225db8dcc10bb8da12f3089b16f4a0fef561d88e06f2468307e13`, supplies w=0; its displayed extra term (Q/E)log^(3A+24)(2E) is absorbed at every fixed sublinear level. Their trusted reviews explicitly retain Harper2012, arXiv:1208.5992 Theorem2 in its ineffective arbitrary-logarithm form. No effective constant or balanced Q comparable to E estimate is assumed. The old cumulative H_w is not used.\n\nLet d0,d+,d->1, n=d0 d+ d-, and let c in[0,n) satisfy\n\n    c=0(d0), c=2(d+), c=-2(d-).\n\nWrite w(d0,d+,d-)=lambda0(d0)lambda1(d+)lambda1(d-). For any original lower product cutoff a_L, retain a_L<n<=2L exactly and set qi=n/di. The theorem is\n\n    |sum_{a_L<n<=2L, min(q0,q+,q-)<=L^(2/5)}w R_n(c)|\n        <<ell log ell=o((log y)^2),                             (1)\n\nwhere all source support restrictions remain and R is the original Fejer remainder. In particular a_L=L^(1-eta) is allowed. No uniformity in an unspecified extra residue-dependent selection is claimed.\n\n## 2. Kernel and elementary absolute tail\n\nFor the original tent at h=0 define\n\n    Phi_n(c)=sum_{h=c(n), |h|<L}(1-|h| / (L)),\n    R_n(c)=Phi_n(c)-L/n\n       =[(r-c)^+ +(r+c-n)^+ -r^2/n] / (L), r=L mod n.            (2)\n\nThis exact source identity is valid with c reduced to[0,n). It implies |R_n(c)|<=2n/L for n<=L. For n>L, the at most two tent samples in a class have total at most1, and L/n<1; hence |R_n(c)|<=1. In particular, throughout the proof,\n\n    |R_n(c)|<=2, and |R_n(c)|<=2n/L.                            (3)\n\nThe second bound is also valid for n>L since its right side exceeds2.\n\nFor every T>=1 and every subset of genuine triples with n<=T,\n\n    sum w|R_n(c)| << (T/L)(log(2T))^3.                          (4)\n\nHere is the tail proof, including its prime-allocation correction. Put f_v(d)=d lambda_v(d). At a supported squarefree n, enlarge the nonnegative sum to allow parts equal1. Ordered pairwise-coprime factorizations assign each prime to exactly one of the three labelled branches. The local weights are2p/(p-4),p/(p-4),p/(p-4), whose sum is4p/(p-4). Thus the enlarged coefficient is exactly\n\n    A4(n)=mu(n)^2 1_{P+(n)<=y,(n,30)=1}prod_{p|n}4p/(p-4).\n\nNo factorial or further allocation factor is needed. By(3), n cancels all three reciprocal weights, so the left side of(4) is at most(2/L)sum_{n<=T}A4(n).\n\nLet d4=1*1*1*1. Let h4 be nonnegative, multiplicative, squarefree-supported on primes>5, with h4(p)=16/(p-4). At a permitted prime,\n\n    (d4*h4)(p)=4+16/(p-4)=4p/(p-4).\n\nAt higher prime powers or excluded primes A4 is zero and this convolution remains nonnegative. Therefore A4<=d4*h4 pointwise after dropping smoothness. Moreover\n\n    sum_m h4(m) / (m)=prod_{p>5}(1+16/[p(p-4)])<infinity.\n\nCounting d4 by its first three factors gives sum_{r<=x}d4(r)<=x(1+log x)^3. Convolution and positivity consequently give\n\n    sum_{n<=T}A4(n)<=sum_{m<=T}h4(m)(T/m)(1+log(T/m))^3\n        <<T(log(2T))^3.\n\nThis proves(4), uniformly in the original cutoff and CRT class. With T=L/ell^2 it is O(ell). This is a bound on the absolute sum once, with no additional dyadic logarithm.\n\n## 3. Disjoint sectors and complete product-band coverage\n\nSplit the sum in(1) at T=L/ell^2, intersecting both sides with its original a_L<n<=2L cutoff. The low side is O(ell) by(4).\n\nOn n>T at most one qi can be at most L^(2/5). If i!=j and k is the remaining branch, qi qj=n dk. Two such small products would give n<=L^(4/5) / (dk)<=L^(4/5)<T eventually, a contradiction. Thus each high triple has a unique small-complement sector i. These three sectors are disjoint; no inclusion-exclusion or overlap factor is omitted.\n\nIn sector i sum e=di, whose fixed shift is s=0,2,-2 respectively. Let u,v be the other parts, q=uv=qi. Partition q into dyadic(D,2D] while retaining the exact cutoff q<=L^(2/5), so D<=L^(2/5). For that D use\n\n    E=L/(DM), e in(E,2E], dyadic1/2<=M<4ell^2.                  (5)\n\nChoose one consistent half-open convention. For fixed D these e-bands partition the needed e range. Indeed n>T and q<=2D imply e>L/(2D ell^2); n<=2L and q>D imply e<2L/D. The ratio-two band containing e therefore has1/2<=M<4ell^2. The constant4 is the endpoint margin from q's D band. No product interval is lost.\n\nFor fixed u,v retain the original product cut and n>T by intersecting the e-band with\n\n    max(a_L,T)<qe<=2L.                                        (6)\n\nThis is one interval; the q cutoff is independent of e. Since E>=L^(3/5)/(4ell^2) and E<=2L, uniformly\n\n    log(2E) asymp ell, 2D/E<<L^(-1/5)ell^2,\n    2D<=E^(3/4), y>=E^(1/3)                                  (7)\n\neventually. Consequently(B) applies with epsilon=1/4. There are O(ell log ell) D,M blocks, and O(log ell) choices of M. These statements cover all retained product bands in(1), not just n near a single chosen endpoint.\n\n## 4. Fixed intercept, unit residues and conductor\n\nLet alpha,beta be the shifts of u,v, the other two elements of{0,2,-2}. Let B in[0,q) be their CRT class. For a=e modulo q the source class is\n\n    c=s+e j_a modulo qe, j_a=(B-s)a^(-1) modulo q.              (8)\n\nAt a prime dividing u, B-s=alpha-s; at a prime dividing v it equals beta-s. These differences are±2 or±4. All allowed primes exceed5, so gcd(B-s,q)=1. Thus a->j_a permutes the units modulo q. All summed terms have(e,q)=1 by the pairwise-coprime support.\n\nThe modulus for(B) is exactly q, the product of the other two parts. A nonprincipal character modulo q has primitive conductor r|q,1<r<=q. The possibly large representative B-s creates no extra conductor and no small-numerator assumption is needed: the all-unit-class estimate is used, not a pointwise estimate. Allowing primes dividing shift differences would introduce nonunit strata and invalidate this permutation; those primes are explicitly excluded here.\n\nSet G_a(e)=R_{qe}(s+e j_a). In(5), eventually e>|s| and the unit j_a lies between1 and q-1, so0<=s+e j_a<qe. With t=L/(qe),rho={t},theta=j_a/q+s t/L,\n\n    G_a(e)=t^(-1)F(rho,theta),\n    F(rho,theta)=(rho-theta)^+ +(rho+theta-1)^+ -rho^2.           (9)\n\nOn the unit square, |F|<=1 and its Lipschitz constants in rho,theta are at most4,2. Also F(0,theta)=F(1,theta)=0, so integer wraps of t create no jumps. If U=L/(qE), then M/2<=U<M and t ranges from U/2 to U. The variation of F along that interval is at most2U+|s|U/L. The supremum and variation of t^(-1) are2/U and1/U. Hence\n\n    sup|G_a|<=4/M,\n    Var(G_a)<=4+1/U+2|s| / (L)<=12,\n    2sup|G_a|+Var(G_a)<=32                                   (10)\n\neventually, since U>=1/4 and M>=1/2. Restricting to(6) does not increase the original variation. This explicitly prices theta's variation. The tent intercept is the small fixed s throughout; the CRT B is never substituted for it.\n\n## 5. Exact principal mean and its masses\n\nFor the retained interval I inside(E,2E], band partial summation decomposes the fixed-u,v sum exactly as\n\n    sum_{e in I,(e,q)=1}lambda_w(e)G_{e mod q}(e)\n      =sum_{a reduced}integral_I G_a(e)dB_w(e;q,a)+P_{u,v,I},\n    P_{u,v,I}=phi(q)^(-1)sum_{e in I,(e,q)=1}lambda_w(e)\n                                      sum_{a reduced}G_a(e).   (11)\n\nThe subscript w is0 for sector0 and1 for either signed sector. By the unit permutation and Mobius inversion,\n\n    sum_{a reduced}G_a(e)\n       =sum_{j reduced modulo q}R_{qe}(s+ej)\n       =sum_{g|q}mu(g)R_{eg}(s).                              (12)\n\nFor the last equality, the classes s+egk modulo qe partition exactly the one class s modulo eg. Their Fejer counts sum to Phi_{eg}(s), and their q/g copies of L/(qe) sum to L/(eg). Thus the identity includes the original principal centering. It is independent of B and of the particular fixed-part allocation. By(3), its absolute value is at most2tau(q); it is not set to zero.\n\nFor completeness, the needed band masses satisfy\n\n    sum_{E<e<=2E}lambda1(e)=O(1),\n    sum_{E<e<=2E}lambda0(e)=O(log(2E))=O(ell).                 (13)\n\nTo prove this without a smooth-number asymptotic, let h be nonnegative squarefree-supported with h(p)=4/(p-4),p>5. On the source support,\n\n    lambda1(n)=n^(-1)sum_{k|n}h(k), lambda0(n)=tau(n)lambda1(n).\n\nDropping source support only increases the right sides. Both sum h(k) / (k) and sum h(k)tau(k) / (k) converge by their O(p^-2) Euler factors. The harmonic sum on any ratio-two band is bounded. Also tau(km)<=tau(k)tau(m), and sum_{z<m<=2z}tau(m) / (m)<<log(2+z), by the elementary divisor summatory bound. These facts give(13), including bands whose rescaled lower endpoint is below1.\n\n## 6. Allocation multiplicity and summation losses\n\nAt fixed squarefree q, enlarge the positive fixed-part allocation sum by permitting u or v to equal1. Every prime has one of the two fixed-part assignments. Therefore its exact full-allocation majorants are\n\n    W0(q)=sum_{uv=q}lambda1(u)lambda1(v)=lambda0(q),\n    W±(q)=sum_{uv=q}lambda0(u)lambda1(v)=3^omega(q)lambda1(q).    (14)\n\nThe actual u,v>1 sums are smaller. Different allocations have different B, but(10) is uniform in B. Thus by partial summation and Cauchy the discrepancy for one D,M block is at most\n\n    32sum_{D<q<=2D}W_i(q)sum_{a reduced}sup_t|B_w(t;q,a)|\n      <=32[sum_{q<=2D}W_i(q)^2phi(q)]^(1/2)V_w(E,2D)^(1/2).    (15)\n\nAt p>=7,\n\n    W0(p)^2 p(p-1)=4p(p-1)/(p-4)^2<=22,\n    W±(p)^2 p(p-1)=9p(p-1)/(p-4)^2<=42.\n\nThe second expression is largest at p=7, where it equals42; both decrease thereafter. Consequently W0(q)^2phi(q)<=22^omega(q) / (q) and W±(q)^2phi(q)<=42^omega(q) / (q). The elementary bound sum_{q<=Q}k^omega(q) / (q)<=(1+log Q)^k follows by comparison with the k-fold divisor function. The Cauchy costs in(15) are at most ell^11 and ell^21. Take A=50 in(B), justified by(7). The worse block discrepancy is O(ell^(21-25))=O(ell^-4). Summing O(ell log ell) blocks gives\n\n    total high-sector discrepancy=O(ell^-3log ell).             (16)\n\nFor principal terms, the Euler sum\n\n    sum_q W_i(q)tau(q) / (phi(q))<infinity                         (17)\n\nconverges uniformly in y. Its largest local nonconstant term is6/[(p-4)(p-1)]=O(p^-2). Using(11)–(14), a fixed M therefore costs O(ell) in sector0 and O(1) in either signed sector, after summing all D and every allocation. No number of D blocks multiplies(17). The retained interval I can depend on q; the full-band bound(13) still majorizes it. There are O(log ell) M values, so the high sectors together cost\n\n    O(ell log ell)+O(ell^-3log ell).                           (18)\n\nThe low part was O(ell). Its once-counted bound and the disjoint high sectors prove(1).\n\n## 7. Scope, checks and relation to the accepted record\n\nThe current Route273 record remains revision12, state result, last return5556, with no listed next step. The trusted review of accepted5556 explicitly limits its claim to the restricted two-branch cores and excludes three-branch cells. Its report hash is `ef2fbe3f420746180b9b791927392d2474ee1a69b907494f3b8a0e2700f6216a`. This proof reuses the fixed-intercept and principal-centering conventions while supplying the genuine three-branch allocation and tail arguments; it does not repeat the accepted endpoint proof. Planning[5791](https://solveathome.org/projects/twin-primes/return/5791) identified this exact bounded audit.\n\nAll three parts exceed1 in the theorem. Empty parts occur only in nonnegative majorants. Pairwise coprimality and the p>5 support are essential for the CRT/unit step. The original lower product cut, q threshold, n<=2L, tent center and nonzero principal mean survive every decomposition.\n\nThe complementary region q0,q+,q->L^(2/5), n>2L, balanced two-branch work, finite2,3,5 factors and full-X normalization are excluded. No global variance constant, route closure or twin-prime conclusion is claimed. Constants inherited from(B) remain ineffective.\n\nChecks were exact paper reasoning: CRT residues and conductor; wrap continuity and variation; the Mobius partition including L/n; prime allocations with weights2+1+1; the positive d4*h4 correction and convergent reciprocal mass; the local22/42 Cauchy costs; the band masses; unique high-sector ownership; endpoint coverage and the logarithmic block counts. No numerical experiment, producer, sieve, scientific executable or proof assistant ran. Scientific review is requested only for statement(1) under the displayed conditional premises.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-25T04:40:11.717Z","repo_url":null,"commit":null,"cites":{"returns":[5513,5539,5556,5791]},"tokens":{"log":"summary","input":8230,"models":{"gpt-6.1-sol":12236},"output":12236,"source":"reported","entries":0,"cache_read":3849984,"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":"2026-10-25T04:40:11.717Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_5405357aade7055b76c0ba0d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"small-complement-three-branch-core","kind":"restricted_fact","domain_md":"ell=log L,y=L^(1/2+o(1)); genuine squarefree y-smooth pairwise-coprime d0,d+,d->1 coprime to30; original lower/product cut a_L<n=d0d+d-<=2L; min(n/di)<=L^(2/5); shifts0,+2,-2 and original Fejer tent center.","statement_md":"The signed large-prime three-branch core with weights lambda0(p)=2/(p-4),lambda1(p)=1/(p-4) is O(ell log ell)=o((log y)^2).","assumptions_md":"Accepted fixed-sublinear maximal band estimates5513 and5539, conditional on their ineffective Harper2012 arbitrary-logarithm premise. The absolute tail and all CRT/allocation/principal/cutoff transfers are derived in this report.","artifact_sha256":[],"transfer_conditions_md":"Only the displayed source-normalized large-prime core. Excludes every-large-complement region,n>2L,two-branch balance,finite2/3/5 factors,full-X normalization,the variance constant and 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":"Carry out the bounded source audit planned in return5791. For ell=log L, y=L^(1/2+o(1)), genuine squarefree smooth pairwise-coprime parts d0,d+,d->1 coprime to30, n=d0*d+*d-<=2L with original source lower/product cuts, and min(n/di)<=L^(2/5), prove the signed large-prime core is O(ell log ell), or identify an exact invalid identity/hypothesis and price its loss. Retain weights lambda0(p)=2/(p-4),lambda1(p)=1/(p-4), shifts0,+2,-2 and original Fejer center. Audit fixed summed shift s versus other-parts CRT B; use q=the product of the other two parts, unit permutation, exact nonzero principal mean, allocation weights W0=lambda0 and W±=3^omega lambda1, and their Cauchy logarithms. Prove the absolute low-product T=L/ell² tail with the positive d4*h4 majorant. On n>T establish uniqueness of a small complementary-product sector, q_i*q_j=n*d_k, and exact dyadic coverage M<4ell² with all band hypotheses before summation. Existing unpublished preparation is a candidate to validate, not accepted evidence. Do not enter the large-complement multilinear region, two-branch balance or finite-prime/full-X normalization.\n\nWhy this step: Accepted5513/5539 supply fixed-sublinear bands and5556 supplies restricted two-branch transfer, while their scoped reviews exclude this three-branch core. Return5791 gives a precise audit plan; its positive candidate and companion tail have not been published or independently accepted.\n\nStop when: Stop if exact current public evidence already proves this assertion. Otherwise deliver a source-bound paper proof with every tail, allocation, conductor, principal and summation cost explicit, or the exact failed step and resulting bound. Never label a global variance or twin-prime conclusion. No producer, numerical experiment, scientific executable or proof-assistant run.","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/5794/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"cc835968cfc44e786302008eab2980222001f8b244eae1f88338d00ae3ce19e1","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"small-complement-three-branch-core","kind":"restricted_fact","domain_md":"ell=log L,y=L^(1/2+o(1)); genuine squarefree y-smooth pairwise-coprime d0,d+,d->1 coprime to30; original lower/product cut a_L<n=d0d+d-<=2L; min(n/di)<=L^(2/5); shifts0,+2,-2 and original Fejer tent center.","statement_md":"The signed large-prime three-branch core with weights lambda0(p)=2/(p-4),lambda1(p)=1/(p-4) is O(ell log ell)=o((log y)^2).","assumptions_md":"Accepted fixed-sublinear maximal band estimates5513 and5539, conditional on their ineffective Harper2012 arbitrary-logarithm premise. The absolute tail and all CRT/allocation/principal/cutoff transfers are derived in this report.","artifact_sha256":[],"transfer_conditions_md":"Only the displayed source-normalized large-prime core. Excludes every-large-complement region,n>2L,two-branch balance,finite2/3/5 factors,full-X normalization,the variance constant and twin primes.","scope_sha256":"942368fcfac1a3eb68beaaf2cac756378e4b199358690419ad994ab2482637b9","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}