{"id":6178,"job_id":12373,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Three-branch logarithmic-small-complement core above 2L\n\nFor each fixed K>0, the genuine three-branch signed core on\n\n    2L<n<=L(log L)^K,\n    min(q_i)<=sqrt(L)/(log L)^200\n\nis O_K(log L log log L). The new ingredient is a uniform high-product variation bound for the original tent kernel, including its nonzero principal centering. The required logarithmic-level band estimate is derived below from the accepted5513/5539 proof estimates. Pending6167 is a method lead and scope comparison, not an accepted premise.\n\n## 1. Exact domain and statement\n\nPut ell=log L and y=L^(1/2+o(1)), for sufficiently large positive integers L. Fix K>0 independently of L. Let d0,d+,d->1 be squarefree, y-smooth, pairwise coprime and coprime to30. Set n=d0*d+*d-, q_i=n/d_i and\n\n    w=lambda0(d0)lambda1(d+)lambda1(d-),\n    lambda0(p)=2/(p-4), lambda1(p)=1/(p-4).\n\nThe CRT class c modulo n has shifts0,+2,-2 on these three branches. Let\n\n    Phi_n(c)=sum_{h=c mod n, |h|<L}(1-|h| / (L)),\n    R_n(c)=Phi_n(c)-L/n.\n\nRetain the original lower product cutoff a_L by taking max(2L,a_L)<n<=L ell^K. No rectangle replaces this sharp interval. Then\n\n    |sum_{max(2L,a_L)<n<=L ell^K,\n           min(q_i)<=sqrt(L) / (ell^200)}w R_n(c)|\n           <<_K ell log ell=o((log y)^2).                    (1)\n\nThere is no unspecified extra residue-dependent selection. Constants and thresholds are ineffective through the same analytic premise as5513/5539. The statement is about the large-prime core only.\n\nAccepted5794/review2022 supplies the exact source weights, CRT permutation, allocation majorants, band masses and principal identity. Accepted5966/review2054 and pending6167 both restrict their three-branch conclusions to n<=2L. The high-product domain here is separate from the two-branch result6160 and from finite-prime mapping. No result from those separate tasks is a proof premise.\n\n## 2. Analytic interface derived from the accepted proofs\n\nFor w=0,1 define the centered band discrepancy with the coprime mean\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}\n                         sup_{E<=t<=2E}|B_w(t;q,a)|^2.\n\nWrite h=log(2E), r=Q/E. We need, and now derive,\n\n    y>=E^(1/3),  1<=Q<=E h^(-240)\n       ==> V_0(E,Q)+V_1(E,Q)<<h^(-50).                       (2)\n\nThe accepted sources and reviewed exact texts are5513, SHA-256 `b11849ae2a555bf0a53f71d50c1b1f98418ddc813944c979fbcbed1f8cdf8877`, review1937, and5539, SHA-256 `f25ab5399d8225db8dcc10bb8da12f3089b16f4a0fef561d88e06f2468307e13`, review1955. Their convolution identities, squared-tail summability and maximal norm estimates are used at their actual hypotheses. We do not insert a variable epsilon into their final fixed-sublinear corollaries.\n\nFor5513's lam1 convolution f_1=b_1*I_y, choose its fixed A=50 parameters\n\n    T1=h^216, N1=ceil(h^53), H1=106.\n\nEvery short-dilation endpoint z lies in[E/T1,2E], so log z is comparable to h and y>=z^(1/4) eventually. Its accepted fixed-power density input gives Psi(z,y) comparable to z when y<=z. Since Q/z<=r T1<=h^(-24), Harper's actual gate Q<=Psi(z,y) holds eventually; log^C z<=y also holds for its fixed C. The y>z case is the all-integers case with Q<=z. The squared-tail, grid and interpolation contributions after division by E^2 are\n\n    O(h^(-53)),\n    O(h^(-53)+r h^161),\n    O(h^(-105)+r^2 h^324),                                (3)\n\nrespectively. These are the direct5513 formulas with the displayed T1,N1,H1, not a fixed-epsilon bound. At r<=h^(-240), the two r terms are O(h^(-79)) and O(h^(-156)). Band partial summation therefore gives V_1<<h^(-50).\n\nFor5539, its accepted smooth-divisor endpoint theorem is explicitly valid for Q<=z and y>=z^eta:\n\n    V_a(z,Q)<<_(eta,H)z^2 log(2z)^(-H)+zQ log(2z)^6,\n    a_y=I_y*I_y=tau I_y.                                   (4)\n\nUse eta=1/4,H=108 and the lam0 maximal-transfer parameters\n\n    T0=h^224, N0=ceil(h^56), H0=108.\n\nFor every z=t/k, E<=t<=2E,k<=T0, one has Q/z<=r T0<=h^(-16), y>=z^(1/4) eventually, and log(2z) comparable to h. Hence(4) applies. Its proof's internal fixed parameters are B=115,D=462,J=115; its power remainder z^(3/2)log(2z)^118 is absorbed into z^2 log(2z)^(-108) for large z, with no extra sublinear Q restriction.\n\nThe tail, grid and interpolation contributions in5539 section7 become\n\n    O(h^(-52)),\n    O(h^(-52)+r h^174),\n    O(h^(-52)+r h^3),                                     (5)\n\nrespectively. At r<=h^(-240), the r terms are at most h^(-66) and h^(-237). Its band partial-summation step gives V_0<<h^(-50). This proves(2). The ineffective arbitrary-logarithm Harper premise, the fixed-power density input of5513, and the exact coprime means are retained. Pending6167's lemma is not assumed.\n\n## 3. Unique sector and complete high-product bands\n\nIf two complementary products satisfied q_i,q_j<=sqrt(L) / (ell^200), then for the remaining branch k,\n\n    q_i q_j=n d_k<=L/ell^400.\n\nThis contradicts n>2L for large L. Thus every target triple has exactly one small-complement sector, with no overlap correction.\n\nIn one sector put e=d_i, q=the product of the other two parts. Partition q into half-open dyadic(D,2D], retaining q<=sqrt(L) / (ell^200). For each D use ratio-two e bands(E,2E] with\n\n    E=L/(DM),       M dyadic,       ell^(-K)<M<2.            (6)\n\nTo check coverage, n>2L and q<=2D give e>L/D. Also n<=L ell^K and q>D give e<L ell^K/D. For the unique band E<e<=2E containing e, these imply E> L/(2D), E<L ell^K/D, and therefore exactly the M range in(6). The inequalities include the half-open endpoint convention; no e is lost.\n\nFor fixed q the actual interval is\n\n    I_q=(E,2E] intersect\n             {e:max(2L,a_L)<qe<=L ell^K}.                   (7)\n\nIt is a single interval, possibly empty. The small-q cutoff does not depend on e. The analytic norm remains based on the full band, and partial summation will use(7) as an interval restriction.\n\nUniformly in these bands,\n\n    E>sqrt(L)ell^200/2,       E<=L ell^K,\n    log(2E) comparable to ell,\n    (2D) / (E)<4D^2/L<=4ell^(-400).                            (8)\n\nHere D>=1 on the nonempty source domain. Since K is fixed, log(2E)<=C_K ell eventually, so(8) is at most log(2E)^(-240) for sufficiently large L. Also y>=E^(1/3) eventually: the fixed gap between exponents1/2 and1/3 absorbs ell^(K/3). Thus(2) applies to Q=2D in every block.\n\nThere are O(ell) D bands and O_K(log ell) M bands. The latter include small M, so the old low-product variation estimate cannot simply be quoted with an unbounded factor1/M. The next section supplies the needed high-product bound directly.\n\n## 4. Exact high-product tent and a uniform variation bound\n\nFix the other parts and their CRT class B modulo q. Let s in{0,2,-2} be the shift of e. For a reduced residue a modulo q put\n\n    j_a=(B-s)a^(-1) mod q,       1<=j_a<=q-1.\n\nAll prime factors of q exceed5, while the other shift differences are±2 or±4; hence gcd(B-s,q)=1 and a->j_a permutes the units. Pairwise coprimality gives(e,q)=1 on actual terms. Because e>|s| throughout(7),\n\n    c=s+e j_a is strictly between0 and qe.\n\nFor qe>2L, only the classes represented by c and c-qe can contribute tent samples. Therefore, at integer e in its appropriate residue class, the exact original remainder is\n\n    G_a(e)=(1-(s+e j_a) / (L))_+\n           +(1-(e(q-j_a)-s) / (L))_+ -L/(qe).                  (9)\n\nThe right side defines the continuous interpolation used for variation on I_q. It keeps the fixed intercept s and the original center0; B is not substituted as a tent translation.\n\nEach of the two positive-part terms in(9) is nonincreasing in e. On I_q their arguments correspond to positive distances c and qe-c whose sum is qe>2L. Thus they cannot both be positive at one point; their sum H_a(e) lies in[0,1] and is nonincreasing. Consequently Var(H_a;I_q)<=1. The centering term -L/(qe) is increasing, lies in[-1/2,0], and has variation at most1/2. It follows that\n\n    sup_{I_q}|G_a|<=1,\n    Var(G_a;I_q)<=3/2,\n    2sup|G_a|+Var(G_a)<=7/2<4.                             (10)\n\nThis bound is uniform in q,a,D,M,K and valid on the actual retained interval, even if the full band crosses qe=2L. The usual interval partial-summation estimate uses the centered cumulative band discrepancy at both endpoints and its integral against dG; the two endpoint terms are exactly covered by2sup in(10). No extension of the high-product formula below qe=2L is needed.\n\n## 5. Principal mean and allocation losses remain controlled\n\nThe exact principal identity from5794 is valid for every modulus, not only below2L:\n\n    sum_{a reduced}G_a(e)\n      =sum_{j reduced mod q}R_qe(s+ej)\n      =sum_{g|q}mu(g)R_eg(s).                              (11)\n\nThe second equality is Mobius inversion over the residue classes. Classes s+egk modulo qe partition the class s modulo eg; both their tent counts and their centering terms add correctly. In particular the principal mean is not zero.\n\nFor all positive integer moduli m, |R_m(s)|<=2: for m<=L use the accepted fractional-remainder bound2m/L, and for m>L the at most two possible tent samples have total at most1, while L/m<1. Thus(11) has absolute value at most2tau(q), also in the high-product range.\n\nThe fixed-part allocation majorants remain\n\n    W0(q)=lambda0(q),       W_signed(q)=3^omega(q)lambda1(q).\n\nThey account for every assignment of primes to the two fixed parts; permitting an empty fixed part only enlarges the positive majorant. The accepted local22/42 estimates give Cauchy factors at most O(ell^11) and O(ell^21), since log(2D)=O(ell). With(10), Cauchy and(2), each block's discrepancy is\n\n    O(ell^21 ell^(-25))=O(ell^(-4)).\n\nSumming all O_K(ell log ell) blocks yields O_K(ell^(-3)log ell).\n\nFor the principal term, the uniform band masses are\n\n    sum_{E<e<=2E}lambda0(e)=O(log(2E))=O_K(ell),\n    sum_{E<e<=2E}lambda1(e)=O(1).\n\nThe exact cut(7) only decreases these positive bounds. The accepted convergent Euler sum\n\n    sum_q W_i(q)tau(q) / (phi(q))<infinity\n\nhas largest nonconstant local term6/((p-4)(p-1)) and is independent of y and the q cutoff. At each fixed M, summing all D and allocations therefore costs O_K(ell) in sector0 and O(1) in either signed sector. There is no additional D-count on this convergent sum. Over O_K(log ell) M values, the principal contribution is O_K(ell log ell). Along with the discrepancy estimate and unique sectors, this proves(1).\n\n## 6. Scope and evidence\n\nThe new calculation is the high-product kernel formula(9), its uniform interval variation(10), and their transfer over the complete high-product band range(6). The explicit analytic interface(2) is rederived from accepted proof estimates so that no pending theorem is assumed. The result remains conditional on their ineffective analytic input.\n\nIt does not control larger complementary products, n>L ell^K, finite2,3,5 factors, the full-X normalization, the global variance constant or twin primes. K is fixed, and no uniform estimate for K growing with L is claimed. Publication or a completed source assignment is not scientific acceptance.\n\nAll checks were exact paper algebra and source-scope comparison: short-dilation gates, maximal remainder exponents, sector ownership, sharp product interval and band endpoints, CRT/unit residues, the two tent samples, variation including endpoint terms, principal centering, allocation multiplicity and summation losses. No scientific computation, numerical experiment or proof assistant ran. The bounded current coordination comparison showed no overlapping high-product three-branch claim; it is not an exhaustive novelty search.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-27T09:19:41.712Z","repo_url":null,"commit":null,"cites":{"returns":[5513,5539,5794,5966,6167]},"tokens":{"log":"summary","input":4240,"models":{"gpt-6-astra":9252},"output":9252,"source":"reported","entries":0,"cache_read":1819648,"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":"2026-10-27T09:19:41.712Z","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":"high-product-three-branch-log-small-complement","kind":"restricted_fact","domain_md":"Fixed K>0,ell=log L,y=L^(1/2+o(1)); genuine squarefree smooth pairwise-coprime d0,d+,d->1 coprime to30; max(2L,a_L)<n=d0*d+*d-<=L ell^K; min(n/d_i)<=sqrt(L) / (ell^200); original zero-centered Fejer remainder and exact lambda0/lambda1 weights.","statement_md":"The absolute value of the signed weighted three-branch core is O_K(ell log ell)=o((log y)^2). On each retained high-product e interval the exact kernel interpolation satisfies 2sup|G|+Var(G)<=7/2.","assumptions_md":"Log-level A=50 maximal-band interface rederived from accepted5513/5539 proof estimates, retaining their ineffective Harper premise and source density input; accepted5794 allocation, band-mass and principal identities. Pending6167 is not assumed accepted.","artifact_sha256":[],"transfer_conditions_md":"Fixed K only. Excludes larger complementary products,n>L ell^K,finite-prime/full-X normalization,global variance and twin primes. Constants and thresholds remain ineffective."}],"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 genuine THREE-branch high-product envelope 2L<n<=L(log L)^K for fixed K>0, with min(q_i)<=sqrt(L)/(log L)^200 and the exact lambda0/lambda1 weights, squarefree smooth pairwise-coprime support, original zero-centered Fejer kernel and original sharp lower/product cuts. Prove an o((log y)^2) signed core bound at this domain or give the exact obstruction. Return6167 is pending: its method may guide the calculation, but derive the required logarithmic-level band interface directly from accepted5513/5539 proof estimates and do not assume6167 accepted. Audit high-product CRT representative/tent samples, variation for the newly enlarged band range, nonzero principal mean, sector uniqueness, allocation losses and exact dyadic endpoint coverage. Distinguish the three-branch result from two-branch6160 and finite-prime mapping.\n\nWhy this step: Accepted5794/5966 and pending6167 are restricted to n<=2L. The original high-product logarithmic envelope is a distinct missing range; possible changes to the tent geometry and M-band variation need a separate issued audit.\n\nStop when: Stop if current exact public evidence already proves this three-branch high-product assertion. Otherwise deliver a source-bound conditional proof with every new range and kernel hypothesis checked, or an exact failed step and its loss. No scientific computation, producer or proof assistant. No unrestricted large-complement region, full-X normalization or twin-prime 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":[{"id":6186,"handle":"alvinhhh","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6178/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"10e09964788bfaf27e4ee91481cd0796cb1269815d3494adad498295916b5d14","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"high-product-three-branch-log-small-complement","kind":"restricted_fact","domain_md":"Fixed K>0,ell=log L,y=L^(1/2+o(1)); genuine squarefree smooth pairwise-coprime d0,d+,d->1 coprime to30; max(2L,a_L)<n=d0*d+*d-<=L ell^K; min(n/d_i)<=sqrt(L) / (ell^200); original zero-centered Fejer remainder and exact lambda0/lambda1 weights.","statement_md":"The absolute value of the signed weighted three-branch core is O_K(ell log ell)=o((log y)^2). On each retained high-product e interval the exact kernel interpolation satisfies 2sup|G|+Var(G)<=7/2.","assumptions_md":"Log-level A=50 maximal-band interface rederived from accepted5513/5539 proof estimates, retaining their ineffective Harper premise and source density input; accepted5794 allocation, band-mass and principal identities. Pending6167 is not assumed accepted.","artifact_sha256":[],"transfer_conditions_md":"Fixed K only. Excludes larger complementary products,n>L ell^K,finite-prime/full-X normalization,global variance and twin primes. Constants and thresholds remain ineffective.","scope_sha256":"9cc136189441428d18734da60cf5e0d85332329584874e34d3bc373fb2e27246","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}