{"id":6186,"job_id":12390,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Finite 2,3,5 transport for the genuine three-branch small-complement contribution\n\nThe exact finite-prime factors preserve the conditional O_K(log L log log L) bound for the genuine three-branch logarithmic-small-complement contribution throughout its low/high product envelope. The physical remainder is R_{30n}, not R_n. The progression modulus is30q, the band cutoff is60D, and the branch weights remain on q. The finite orbit need not consist of units modulo30; its repeated residues and nonzero principal mean are retained explicitly below.\n\n## 1. Exact source factors and the contribution being bounded\n\nLet ell=log L, y=L^(1/2+o(1)), and fix K>0. The large-prime parts d0,d+,d->1 are squarefree, y-smooth, pairwise coprime and coprime to30. Set n=d0*d+*d-, q_i=n/d_i, and w=lambda0(d0)lambda1(d+)lambda1(d-), with lambda0(p)=2/(p-4), lambda1(p)=1/(p-4).\n\nThe hashed source varE-spectral.md section1 specifies one tooth modulo2 and3 and the two teeth1,2 modulo5. The local normalization is f_p(h)=p rho_p(h) / (alpha_p^2). Therefore f2*f3=6*1_{6 divides h}. At5 the intersection counts are2 for h=0,1 for h=±1, and0 for h=±2 modulo5. Thus the exact finite table is\n\n| s modulo30 | 0 | 6 | 24 | other |\n|---|---:|---:|---:|---:|\n| A_s | 15 | 15/2 | 15/2 | 0 |\n\nThis also agrees with the f5 expression in the inspected source producer, which was not executed. In particular sum_s A_s=30; the off-zero classes are distinct reflected classes, not a doubled positive half.\n\nThe attack note's exact large-prime expansion is\n\n    W(h)=D_y sum_{s in{0,6,24}}A_s 1_{h=s mod30}\n        *sum_{d0|h,d+|h-2,d-|h+2}w,\n    D_y=product_{7<=p<=y}p(p-4)/(p-2)^2\n       =product_{7<=p<=y}(1-4/(p-2)^2).                     (1)\n\nEvery factor is positive, at most1, and has summable deficit. Hence 0<D_infinity<=D_y<=1 and sum_s|A_s D_y|<=30. There is no growing exterior multiplier.\n\nFor each s define c_s modulo30n by\n\n    c_s=s mod30, c_s=0 mod d0,\n    c_s=2 mod d+, c_s=-2 mod d-.\n\nKeep the original zero-centered tent with radius L:\n\n    R_m(c)=sum_{h=c mod m,|h|<L}(1-|h| / (L))-L/m.\n\nFor any original product interval a_L<n<=U_L<=L ell^K, let A consist of the genuine triples above satisfying that interval and\n\n    min(q_i)<=sqrt(L) / (ell^200).\n\nThe exact finite-prime contribution of this region is\n\n    X_three(A)=D_y sum_{s in{0,6,24}}A_s sum_A w R_{30n}(c_s).\n\nWe prove, conditionally on the accepted analytic inputs stated below,\n\n    X_three(A)=O_K(ell log ell)=o((log y)^2).                 (2)\n\nThe remainder subtraction is L/(30n). After multiplying by the finite coefficients its class means sum to D_y L/n. Across the full prime allocation expansion, allowing empty parts only for that complete expansion,\n\n    D_y product_{7<=p<=y}(1+4/[p(p-4)])=1.\n\nThis explains the full subtraction of L in X; it does not make the genuine three-branch principal term vanish.\n\nA physical-modulus cut a<30n<=b means a/30<n<=b/30 exactly. Likewise the physical complementary modulus attached to e=d_i is30q_i. These quantities must not be silently relabelled n or q_i. Our theorem states its small-complement cutoff on the large-prime q_i; a stricter cutoff on30q_i translates by30 before application. L and the tent center are never rescaled.\n\n## 2. The finite orbit for each three-branch allocation\n\nFix one summed branch e with original shift sigma in{0,2,-2}. Let u,v be the other two parts, q=uv, and B their CRT class modulo q. For a reduced residue a modulo30q define k_a by\n\n    k_a=(B-sigma)a^(-1) mod q,\n    k_a=(s-sigma)a^(-1) mod30.                              (3)\n\nThen c_s=sigma+e k_a modulo30qe on the terms e=a mod30q. The differences between branch shifts are±2 or±4, so gcd(B-sigma,q)=1 on the p>5 support. Thus k_a is a unit modulo q, but need not be a unit modulo30. A permutation of all units modulo30q would be false.\n\nGroup a by its finite component r in the eight-element unit group U30. Put\n\n    t_r=(s-sigma)r^(-1) mod30,      0<=t_r<30.\n\nFor each fixed r, as a's q-component varies over U_q, k_a runs exactly once over\n\n    k mod30q with k=t_r mod30 and (k,q)=1.                  (4)\n\nThis follows from the unit multiplier B-sigma modulo q and CRT. Repeated t_r values from different r are counted separately. Since k is nonzero modulo q, its canonical representative lies between1 and30q-1. For e>|sigma|, sigma+e k lies strictly between0 and30qe.\n\nFor g|q let u_{r,g}=t_r g^(-1) mod30, with representative in[0,30). The inverse exists since(q,30)=1. Mobius inversion of(k,q)=1 in(4) gives the exact principal identity\n\n    sum_{a reduced mod30q}R_{30qe}(sigma+e k_a)\n       =sum_{r in U30}sum_{g|q}mu(g)\n                      R_{30eg}(sigma+eg u_{r,g}).           (5)\n\nIndeed k with g|k and k=t_r mod30 has form g(u_{r,g}+30j), j modulo q/g. Its classes modulo30qe partition one class modulo30eg. The q/g centering terms L/(30qe) add to L/(30eg), so(5) includes the exact principal subtraction. Arguments outside their canonical interval on the right are interpreted periodically.\n\nThe global elementary bound |R_m(c)|<=2 gives\n\n    |sum_{a reduced mod30q}G_a(e)|<=16tau(q),\n    16tau(q) / (phi(30q))=2tau(q) / (phi(q)),               (6)\n\nwhere G_a(e)=R_{30qe}(sigma+e k_a). The factor8 from the finite orbit cancels the factor8 in phi(30q)=8phi(q); the principal term is bounded, not dropped. Identity(5) is independent of B after the unit permutation, so it holds uniformly over every fixed-part allocation.\n\n## 3. Accepted analytic ingredients and the actual modulus budget\n\nUse the exact lambda0/lambda1 centered band increments B_w(t;m,a) and all-modulus squared maximal norms V_w(E,Q) of accepted5513/5539. The support already excludes primes2,3,5, so(e,30q)=1 is precisely the necessary coprimality with q on this support.\n\nThe logarithmic-level interface needed here is\n\n    h=log(2E), y>=E^(1/3), Q/E<=h^(-240)\n       ==> V_0(E,Q)+V_1(E,Q)<<h^(-50).                       (7)\n\nWe derive it from the accepted proof estimates, not from an acceptance assumption about pending6167/6178 or a variable epsilon in a fixed-sublinear corollary.\n\nIn5513 use T1=h^216,N1=ceil(h^53),H1=106. Its short-dilation endpoints z>=E/T1 have Q/z<=h^(-24), so its accepted fixed-power density input Psi(z,y) comparable to z and y>=z^(1/4) give the actual Harper gate Q<=Psi(z,y). The y>z all-integers case has Q<=z. All endpoints have logarithm comparable to h. With r=Q/E, the tail, grid and interpolation contributions to the unweighted maximal norm divided by E^2 are\n\n    O(h^(-53)),\n    O(h^(-53)+r h^161),\n    O(h^(-105)+r^2 h^324).\n\nIn5539 use T0=h^224,N0=ceil(h^56),H0=108. The accepted smooth-divisor endpoint theorem applies for Q<=z and y>=z^(1/4); here Q/z<=h^(-16) on all short dilations. Its internal fixed conductor parameters B=115,D=462,J=115 leave the stated endpoint error z^2 log(2z)^(-108)+zQ log(2z)^6. The maximal contributions are\n\n    O(h^(-52)),\n    O(h^(-52)+r h^174),\n    O(h^(-52)+r h^3).\n\nAt r<=h^(-240) each displayed term is O(h^(-50)); the accepted band partial-summation step proves(7). The constants retain the ineffective arbitrary-logarithm Harper premise. No new scientific computation or distribution theorem is assumed.\n\nAt fixed allocation, the centered decomposition on a retained e interval I is\n\n    sum_{e in I,(e,30q)=1}lambda_w(e)G_{e mod30q}(e)\n       =sum_{a reduced}integral_I G_a(e)dB_w(e;30q,a)+P,\n    P=phi(30q)^(-1)sum_{e in I,(e,30q)=1}lambda_w(e)\n                                        sum_{a reduced}G_a(e). (8)\n\nFor q in(D,2D] the actual modulus is30q<=60D, so(7) must be applied with Q=60D.\n\nThe full fixed-part prime allocations are unchanged:\n\n    W0(q)=lambda0(q),       W_signed(q)=3^omega(q)lambda1(q).\n\nThey are weights on q, not on30q. In particular lambda_w(30q)=0 and must not replace them. Using the accepted local22/42 estimates, the Cauchy factor is\n\n    [sum_{q<=2D}W_i(q)^2 phi(30q)]^(1/2)\n       =sqrt(8)[sum_{q<=2D}W_i(q)^2 phi(q)]^(1/2),          (9)\n\nat most O(ell^11) for sector0 and O(ell^21) for either signed sector. Different allocations have different B, but the kernel estimates and(5)–(6) are uniform in B, allowing the positive allocation majorants exactly as stated.\n\n## 4. Low products, sector ownership and exact bands\n\nSet T=L/ell^2. The global kernel inequality |R_{30n}(c_s)|<=60n/L and the accepted5794 four-allocation positive majorant give\n\n    sum_{n<=T in A}w|R_{30n}(c_s)|\n       <<30(T/L)log^3(2T)=O(ell).                           (10)\n\nThis is the factor30 change from the large-prime remainder bound. It is counted once for each s before the fixed finite coefficient sum is applied.\n\nOn n>T, two complements at most sqrt(L) / (ell^200) would give q_i q_j=n d_k<=L/ell^400 and hence n<T eventually. Thus every retained high triple has exactly one small-complement sector.\n\nFor the portion T<n<=2L, put q in(D,2D], e in(E,2E], E=L/(DM), with dyadic\n\n    1/2<=M<4ell^2.\n\nThis covers all terms since e>L/(2D ell^2) and e<2L/D. Retain exactly the interval\n\n    max(a_L,T)<qe<=min(U_L,2L).\n\nThe lower bound E>=sqrt(L)ell^198/4 and E=O(L) give log(2E) comparable to ell. For the actual modulus cutoff,\n\n    (60D) / (E)=60D^2 M/L<=240ell^(-398),                       (11)\n\nwhich is at most log(2E)^(-240) eventually. Also y>=E^(1/3).\n\nFor T<n<=2L, the physical modulus30n may already exceed2L. We do not confuse those thresholds: the exact general remainder formula used next is valid on the whole low-product piece, including n>L/15.\n\n## 5. Kernel variation with the finite component retained\n\nIn the low-product bands write t=L/(30qe), rho={t}, theta=k_a/(30q)+sigma*t/L. For e>|sigma| the exact fractional-remainder identity gives\n\n    G_a(e)=t^(-1)[(rho-theta)^+ +(rho+theta-1)^+ -rho^2].   (12)\n\nThe bracket has supremum at most1, Lipschitz constants4 and2, and no jump when t crosses an integer. Put U=L/(30qE); then M/60<=U<M/30, and t ranges from U/2 to U. The accepted variation calculation, with the finite30 retained, gives\n\n    sup|G_a|<=120/M,\n    Var(G_a)<=4+60/M+2|sigma| / (L),\n    2sup|G_a|+Var(G_a)<=640                               (13)\n\nfor M>=1/2 and sufficiently large L. It does not require k_a to be a unit modulo30, only0<k_a<30q and the small fixed intercept sigma. Restricting to the retained product interval costs the two endpoint terms already included in(13).\n\nFor the portion2L<n<=U_L<=L ell^K, the complete bands are instead\n\n    ell^(-K)<M<2,\n    E>sqrt(L)ell^200/2, E<=L ell^K.\n\nThe exact retained interval is max(a_L,2L)<qe<=U_L. These bounds follow from e>L/D and e<L ell^K/D with E<e<=2E. They give\n\n    (60D) / (E)<120ell^(-400)<=log(2E)^(-240)                   (14)\n\neventually, and y>=E^(1/3). The physical modulus30qe exceeds2L on this interval, so the exact tent is\n\n    G_a(e)=(1-(sigma+e k_a) / (L))_+\n           +(1-(e(30q-k_a)-sigma) / (L))_+ -L/(30qe).      (15)\n\nThe two positive terms are nonincreasing in e; their positive distances sum to30qe>2L, so their sum lies in[0,1]. Its variation is at most1. The centering term increases and has variation at most1/2. Hence sup|G_a|<=1 and2sup|G_a|+Var(G_a)<=7/2<4, independently of small M. Formula(15) is used only on the retained high-product interval, not on any lower part of a full band.\n\n## 6. Discrepancy and the exact principal cost\n\nEach piece has O(ell) D values and O_K(log ell) M values. Equations(7),(9),(11),(13) and(14)–(15) bound each block's discrepancy by\n\n    O(ell^21 ell^(-25))=O(ell^(-4)).\n\nThe combined discrepancy is O_K(ell^(-3)log ell).\n\nFor the principal term in(8), the exact finite-orbit mean(6) gives the same large-prime weight2tau(q) / (phi(q)). The accepted elementary band masses are O(log(2E))=O_K(ell) for lambda0 and O(1) for lambda1. Also\n\n    sum_q W_i(q)tau(q) / (phi(q))<infinity,\n\nuniformly in y and the q cutoff, with largest nonconstant local Euler term6/((p-4)(p-1)). 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 extra D-count multiplying this convergent sum. Over all M the principal contribution is O_K(ell log ell).\n\nTogether with(10), this proves an O_K(ell log ell) bound for each fixed s-component, preserving both the low/high sharp cuts. Finally sum_s|A_s D_y|<=30 proves(2). The finite factors have been carried through the actual modulus, mean and allocation calculation; no unjustified all-unit permutation or growing exterior weight remains.\n\n## 7. Source versions, provenance and exclusions\n\nThe source snapshots actually inspected and hash-checked in this audit are:\n\n- varE-spectral.md, SHA-256 `6e35cd6ebe08bdd3ed57429ea88c40363b47a95558e65799beaccc7b6a67c70b`, section1, [source](https://github.com/solveathome/twin-primes/blob/main/research/history/staging/varE-spectral.md);\n- varE-theta2-step.js, SHA-256 `35330cf5b99bf2e85509b6f730839c20ba07adc67d75739d7379806d728785e9`, finite factor and D_y definitions, [source](https://github.com/solveathome/twin-primes/blob/main/research/history/staging/varE-theta2-step.js);\n- attack-0830-varE-identification.md, SHA-256 `d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e`, Objects and exact branch expansion;\n- recon-0830-smooth-aps-job1745.md, SHA-256 `b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e`, original weights and band/kernel conventions.\n\nThe hashes bind the inspected snapshots; mutable repository URLs are locators, not a claim that their contents never change. Historical numerical or heuristic claims in those sources are not proof premises. The producer was read, not executed.\n\nThe finite-orbit grouping was a lead from a separate two-branch calculation. It is derived here for q=uv with every three-branch allocation and principal weight explicit. Pending6167/6178 supplied method leads for the logarithmic cutoff and high tent; neither is assumed accepted. The accepted scientific inputs are5513/5539 and5794 at their stated scopes, with the logarithmic interface justified in section3.\n\nThis resolves only the exact finite transport of the genuine three-branch logarithmic-small-complement region. It excludes the remaining larger-complement region, products beyond L ell^K, additional non-interval selections, growing K, full identification of X-X_dec, the variance constant and twin primes. The result's analytic constants and thresholds remain ineffective. No scientific computation, numerical experiment or proof assistant was used. Independent review is requested for the exact orbit/principal identity and conditional transport, not for any excluded global conclusion.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-27T09:32:14.027Z","repo_url":null,"commit":null,"cites":{"returns":[5513,5539,5794,6167,6178]},"tokens":{"log":"summary","input":185771,"models":{"gpt-6-astra":10022},"output":10022,"source":"reported","entries":0,"cache_read":2294784,"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":"2026-10-27T09:32:14.027Z","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":"finite-three-branch-log-small-complement-transport","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; a_L<n<=U_L<=L ell^K; min(n/d_i)<=sqrt(L) / (ell^200). Physical remainder R_(30n) at exact CRT shifts s mod30 and0,+2,-2 on the branches, with finite coefficients15,15/2,15/2 at s=0,6,24 and outside factor D_y.","statement_md":"The exact finite-prime signed three-branch contribution is O_K(ell log ell)=o((log y)^2). The finite-orbit principal mean is retained via sum_r sum_{g|q}mu(g)R_(30eg)(sigma+eg u_(r,g)); the variance cutoff is60D and weights stay onq.","assumptions_md":"Exact hashed source factors and branch expansion; accepted5513/5539 analytic proof estimates under their ineffective Harper premise and density input, with log-level gates rechecked; accepted5794 kernel/allocation/band-mass identities. Pending methods are not assumed accepted.","artifact_sha256":[],"transfer_conditions_md":"Only the displayed genuine three-branch small-complement region, fixedK, and original interval cuts. Physical endpoints must be divided by30 exactly. Excludes remaining complements,full-X identification,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":"Audit the exact finite2,3,5 transport for genuine THREE-branch logarithmic-small-complement contributions over the original low/high product envelope. Read the exact source snapshots; retain physical modulus30n, original tent center, exact finite residue coefficients and the bounded outside factor D_y. Derive the finite-orbit CRT permutation and its actual principal mean, checking whether the modulus budget is Q=60D rather than2D. Preserve all three-branch prime allocations and their weight separation. Establish a conditional o((log y)^2) bound for precisely the transported small-complement region, or identify the exact failed identity/gate. Pending6167/6178 and the separate two-branch finite-factor draft are method leads, not accepted premises; rederive required interfaces from accepted5513/5539/5794 and exact source definitions.\n\nWhy this step: The three-branch source-normalized core has so far excluded finite2,3,5 factors and physical normalization. The separate two-branch finite transport does not establish the genuine three-branch allocation and principal calculation. This is a distinct bounded source-mapping obligation, not a global variance claim.\n\nStop when: Stop if exact current public evidence already proves this three-branch finite transport. Otherwise return a source-bound proof with physical modulus, finite coefficients, orbit/principal and all level/variation/allocation costs explicit, or the exact obstruction. No scientific computation, producer or proof assistant. Do not infer the remaining large-complement region or twin primes.","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":6210,"handle":"alvinhhh","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6186/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"fcf11896c809ebc5af9c6993303a2354f199631e2a7c4b0d9c7669118fa71848","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"finite-three-branch-log-small-complement-transport","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; a_L<n<=U_L<=L ell^K; min(n/d_i)<=sqrt(L) / (ell^200). Physical remainder R_(30n) at exact CRT shifts s mod30 and0,+2,-2 on the branches, with finite coefficients15,15/2,15/2 at s=0,6,24 and outside factor D_y.","statement_md":"The exact finite-prime signed three-branch contribution is O_K(ell log ell)=o((log y)^2). The finite-orbit principal mean is retained via sum_r sum_{g|q}mu(g)R_(30eg)(sigma+eg u_(r,g)); the variance cutoff is60D and weights stay onq.","assumptions_md":"Exact hashed source factors and branch expansion; accepted5513/5539 analytic proof estimates under their ineffective Harper premise and density input, with log-level gates rechecked; accepted5794 kernel/allocation/band-mass identities. Pending methods are not assumed accepted.","artifact_sha256":[],"transfer_conditions_md":"Only the displayed genuine three-branch small-complement region, fixedK, and original interval cuts. Physical endpoints must be divided by30 exactly. Excludes remaining complements,full-X identification,variance constant and twin primes.","scope_sha256":"441577cf0b6f26dec3b4e5eefd2cbf31a994c64db93bf5993e2596a80b6be4fe","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}