{"id":6194,"job_id":12401,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# A disjoint low-product union: the overlap complement is a pair of separate coefficient masks\n\nThe two signed regional bounds cannot simply be added on their overlap. They can be combined conditionally after restricting the canonical-small-branch BC region to the complement of the small-complement region. The key exact identity is\n\n    min(q_i)=a min(b,c)\n\nwhen a is the canonical smallest branch. Thus min(q_i)>Q_star is precisely b>Q_star/a and c>Q_star/a. These are separate one-variable masks in the BC coefficient sequences. They add no character cost, Perron integration, variation loss or logarithmic factor.\n\nThis report proves the restriction and disjoint-union implication at the explicit source/proof interfaces below. Returns6167 and6183 were both pending at the live source read; no scientific acceptance or full-core bound is inferred from this audit.\n\n## 1. Common domain and exact sets\n\nLet ell=log L, y=L^(1/2+o(1)), T=L/ell^2 and\n\n    Q_star=sqrt(L) / (ell^200).\n\nLet Omega be the labelled genuine three-branch source domain: d0,d+,d->1, squarefree, y-smooth, pairwise coprime and coprime to30, with the original sharp cutoff a_L<n=d0*d+*d-<=2L. Retain the exact weight w=lambda0(d0)lambda1(d+)lambda1(d-), with lambda0(p)=2/(p-4),lambda1(p)=1/(p-4), and the original zero-centered Fejer remainder R_n(h) at shifts0,+2,-2. Here h is the original CRT residue and R_n(h)=sum_{|t|<L,t=h mod n}(1-|t| / (L))-L/n. For E contained in Omega write S(E)=sum_E wR_n(h). Exterior finite-prime factors are excluded here.\n\nChoose a to be the numerically smallest branch. It is unique, because equality of two parts greater than1 would contradict pairwise coprimality. Order the remaining labels by the fixed list(0,+,-), calling their values b,c and their unique half-open dyadic bins(B,2B],(C,2C]. Let R=max(B/C,C/B), including R=1. The original lambda0 weight always stays on d0.\n\nFix alpha,rho>0 with\n\n    45alpha<1,       35alpha+5rho<1,\n    A=L^alpha,       R_max=L^rho,\n    Delta=min(1-45alpha,1-35alpha-5rho)>0.                  (1)\n\nDefine\n\n    C_small={Omega:min(q_i)<=Q_star},\n    L_low={Omega:n<=T},\n    U={Omega:n>T, all q_i>L^(2/5), a<=A, R<=R_max},\n    U_out=U setminus C_small.                             (2)\n\nHere q_i=n/d_i. The unqualified word \"union\" below always refers to these exact labelled sets, not a union over arbitrary choices of small branch.\n\nFor sufficiently large L, Q_star>L^(2/5), because L^(1/10) / (ell^200) tends to infinity. Consequently\n\n    U_out={Omega:n>T, a<=A, R<=R_max, min(q_i)>Q_star}.     (3)\n\nThe older q_i>L^(2/5) constraint is automatic in(3), not silently dropped on the whole U.\n\n## 2. What is conditional, and why two signed conclusions alone are insufficient\n\nThe small-complement input is the pending6167 assertion\n\n    S(C_small)=O(ell log ell),                             (SC)\n\nat exactly the source weights and cuts above, including its inherited ineffective sufficiently-large-L threshold from the Harper premise. The BC input is the explicit arbitrary-coefficient proof interface in6183, with the source Fourier/Perron/tail premises of accepted5795 and5964 and the conditional Bettin–Chandee v1 Theorem1. Its unrestricted canonical conclusion is\n\n    S(U)=O(ell^(-1))+O(L^(-eta)),  0<eta<Delta/40.          (BC)\n\nMerely assuming these two numerical conclusions is not enough: S(C_small union U)=S(C_small)+S(U)-S(C_small intersect U), and neither conclusion bounds the intersection.\n\nThe additional argument here checks U_out directly against the BC proof's actual coefficient, norm and sharp-cutoff requirements. It uses the theorem's allowance for arbitrary complex sequences and the explicit positive error controls, not a monotonicity property for signed sums. The precise interface and its preservation are detailed in sections3–5. Thus the union theorem is conditional on(SC) and that stated BC proof interface; it does not assume a new unexplained bound for the intersection. This is not a review decision accepting either pending return or an independent audit of the external theorem's proof.\n\n## 3. Exact restriction at a fixed canonical branch\n\nSince a<b and a<c,\n\n    bc>ab and bc>ac,\n    min(bc,ac,ab)=a min(b,c).\n\nTherefore, for each fixed canonical label and fixed integer a,\n\n    min(q_i)>Q_star\n       iff b>Q_star/a AND c>Q_star/a.                      (4)\n\nTogether with the canonical masks, the complete new masks are\n\n    1_{b>z_a} 1_{c>z_a},       z_a=max(a,Q_star/a).         (5)\n\nThey are exact for real Q_star; equivalently the allowed integers begin at floor(z_a)+1. Equality min(q_i)=Q_star belongs to C_small and is excluded from U_out. No endpoint contribution is discarded.\n\nThe bin labels B,C, their ratio R and the boxwise orientation rule are kept unchanged after masking. A partial bin stays in its original bin with zero coefficients at excluded entries; it is not rebinned or replaced by a shorter support length. An empty masked sequence contributes zero. There is no b-versus-c ordering mask or hidden joint product condition in(5).\n\nFor original shifts(u,v,w) attached to(a,b,c), the CRT identity gives, modulo one,\n\n    -kh/n=k(w-v)inv(ac) / (b)+k(w-u)inv(bc) / (a)-kw/n.            (6)\n\nThe reverse identity has inv(ab) / (c), coefficient k(v-w), and product phase-kv/n. These follow algebraically from the original CRT idempotent identity; they do not translate the tent center. The orientation is selected once per box as in6183, stretching the smaller of the two nominal bin lengths by a.\n\nFor fixed a, the scalar phase on bc is a function on U_a. Its full character expansion has coefficient square sum1 and coefficient absolute sum at most sqrt(phi(a)). The mask(5) neither changes that function nor its character coefficients. In the orientation m=ac, denominator b, the two actual sequences are, schematically,\n\n    alpha_m=lambda_of_c(c) chi(c) 1_{c>z_a}, m=ac,\n    beta_b =lambda_of_b(b) chi(b) 1_{b>z_a},               (7)\n\nwith the original smoothness, squarefree support, coprimality with a, bin supports and any Perron powers retained. The character is evaluated on c=m/a, not on m. The theorem's(m,b)=1 continues to impose(b,c)=1. The reverse orientation is identical with b,c exchanged.\n\nBoth masks have absolute value at most1. Thus they only decrease each actual coefficient norm, without inventing an extra stretched-interval gain. They also only decrease the original positive masses used for cutoff errors. This establishes the key hypothesis needed to apply the same arbitrary-sequence BC bound.\n\n## 4. The quantitative BC interface and unchanged powers\n\nHere is the explicit interface being preserved, so the restriction is checkable without reading a signed-subset inference into(BC). For a contributing box put P=aBC and R=max(B/C,C/B). Its nominal lengths have product P and optimally oriented sum\n\n    sqrt(P)*(sqrt(a/R)+sqrt(R/a)).\n\nThe accepted external theorem applies to arbitrary complex sequences in these nominal intervals. The canonical proof uses norms O(sqrt(a/P)) when the pair contains two lambda1 weights, and O(ell^(3/2)sqrt(a/P)) when one pair weight is the original lambda0. The lambda0 factor is controlled by\n\n    sum_{D<d<=2D}lambda0(d)^2<<log(2D)^3/D.\n\nIts positive tau4 correction proof, and the lambda1 norm, are unaffected by the masks: deleting entries cannot increase either side's coefficient norm. The outside weight remains lambda0(a) in the zero-branch case and lambda1(a) in either shifted-branch case. No weight is reassigned when a's label changes.\n\nBefore outside weights and the sqrt(phi(a)) character cost, the power terms from this interface are\n\n    P^(-1/40+epsilon)\n       [a^(5/8)R^(-1/8)+a^(3/8)R^(1/8)],\n    P^(-1/16+epsilon)\n       [a^(9/16)R^(-1/16)+a^(7/16)R^(1/16)].              (8)\n\nFor clarity, the local BC contract before the norm factors is the two-term bound P^(7/20+epsilon)*(lengthsum)^(1/4)+P^(3/8+epsilon)*(lengthsum)^(1/8), with numerator parameter1 and the product of the two actual coefficient norms. The nonzero numerator coefficient has absolute value2|k| or4|k|, with |k|<=ceil(ell^4). Its extra factor(1+|theta| / (P))^(1/2) stays O(1), because P>L/(4ell^2). Multiplying the norm bound by this contract gives(8); masking creates no additional frequency.\n\nThe mixed pair has the additional ell^(3/2) norm factor just stated. Since(7) fits the same arbitrary-coefficient contract, these exact bounds remain valid for U_out. No factor depending on Q_star/a is introduced.\n\nAfter the original outside moments, boxes, frequencies, Perron and variation, the same conservative ledger is therefore\n\n    O_epsilon(ell^28 * {\n      L^(-1/40+epsilon)[A^(9/8)+A^(7/8)R_max^(1/8)]\n      +L^(-1/16+epsilon)[A^(17/16)+A^(15/16)R_max^(1/16)]})\n       +O(L^(-10)).                                      (9)\n\nThe four margins before epsilon are\n\n    (1-45alpha)/40, (1-35alpha-5rho)/40,\n    (1-17alpha)/16, (1-15alpha-rho)/16.\n\nThe latter two exceed the corresponding first two in the range(1). Hence for every fixed0<eta<Delta/40, choose epsilon within the remaining margin and absorb ell^28 exactly as in the original ledger. The new masks consume zero additional powers of L or ell.\n\n## 5. Sharp products, variation and Fourier tail\n\nThe masks are inserted in the original double sum before character expansion or Perron. The only joint cutoff left is the original product interval\n\n    max(T,a_L)<abc<=2L.\n\nFor fixed a, the standard half-integer Perron encoding of the integer-product endpoints handles this interval. It is the same type of interval as the original proof, even when a_L imposes an additional lower endpoint. Contributing boxes still have L/(4ell^2)<P<2L and products at most8L. The half-integer gap and height L^20 therefore give the same O(L/L^20) pointwise cutoff error. The original positive mixed-weight mass bounds its total; masks decrease that mass. Characters are expanded only in the main integral, so no character multiplier is added to the cutoff error.\n\nThe product amplitude still contains the original scalar phase from(6), with shift coefficient0,+2 or-2. The accepted sinc/Fejer partial-summation bound on(P,4P] remains O(ell^8). The cutoff(5) has already been put into the separate coefficient sequences; it is not differentiated as a new joint amplitude and creates no new variation term. Imaginary Perron powers preserve the masked norms, and the original real powers remain bounded.\n\nThere are no more than the original O(ell^2) labelled bins and O(ell^4) retained frequencies. Same-bin boxes retain one fixed orientation, not two. These observations justify the unchanged ledger(9).\n\nFinally, the accepted positive A4 majorant bounds the omitted Fourier frequencies on every subset of the original labelled core. The canonical sectors are disjoint, and U_out is their subset with the original weights intact. With H=ceil(ell^4), its Fourier tail remains O(ell^3/H)=O(ell^(-1)). Combining this with(9) proves the conditional restricted estimate\n\n    S(U_out)=O(ell^(-1))+O(L^(-eta)).                       (10)\n\nThis is the needed overlap control. It was established through separated coefficients and positive error bounds, not by deducing a subset bound from S(U).\n\n## 6. The disjoint union and exact remaining set\n\nThe two-region union has the genuinely disjoint representation\n\n    C_small union U = C_small disjoint-union U_out.         (11)\n\nUnder(SC) and the BC interface verified for(10), its signed sum is O(ell log ell). If desired, the known unrestricted low-product tail can be incorporated without overlap:\n\n    D = C_small\n        disjoint-union (L_low setminus C_small)\n        disjoint-union U_out.                              (12)\n\nAccepted5794's absolute low-product estimate gives\n\n    sum_{L_low setminus C_small}w|R_n(h)|\n       <<(T/L)log^3(2T)=O(ell).\n\nThus S(D)=O(ell log ell)=o((log y)^2), at the same conditional premises. No cancellation between the three pieces is assumed.\n\nFor all sufficiently large L, the exact residual in Omega after(12) is\n\n    Omega setminus D = {\n      n>T,\n      min(q_i)>Q_star,\n      [ a>L^alpha OR max(B/C,C/B)>L^rho ]\n    }.                                                    (13)\n\nThe bracket is an inclusive OR. The canonical a and nominal labelled bins are the ones defined in section1. Original support, weights and a_L<n<=2L are inherited from Omega. There is no residual equality ambiguity: min(q_i)=Q_star, a=L^alpha and R=L^rho belong to the corresponding controlled sets according to their non-strict inequalities.\n\nFor example alpha=1/50,rho=1/25,eta=1/500 satisfies(1) and eta<Delta/40. Equation(13) then describes the remaining set exactly; it does not claim that set is empty, small in weighted mass, or cancellative. Neither a full three-branch low-core estimate nor a variance or twin-prime conclusion follows.\n\n## 7. Evidence and status boundary\n\nThe live inspected pending baselines are:\n\n- [6167](https://solveathome.org/projects/twin-primes/return/6167), report SHA-256 `1136328e24dedd899b6eeaf997f98402e1912e5b2a58a734f6353da243ad508e`, for(SC);\n- [6183](https://solveathome.org/projects/twin-primes/return/6183), report SHA-256 `a9cceb384e6e8d0817f16f948fa01524490d428058140d4c73436b64183cdf3b`, for the explicit canonical arbitrary-coefficient interface, source weights and ledger reproduced above.\n\nBoth were pending, with no final rung, at this audit's read. The accepted precedents5794,5795/review2023 and5964/review2053 supply the exact positive-tail and source/BC transfer interfaces at their stated scopes; they are not portrayed as having accepted the broader canonical union. Accepted5964's report hash is `15aeaf408b115a10b4ea3e39e92674dc37b5c5290282096e5b8b74f449f4a6cf`.\n\nThe unconditional new content is the exact set identity(4), its coefficient-mask implementation and the disjoint residual bookkeeping. The quantitative conclusion is the stated conditional implication using the explicit analytic proof contracts, not an acceptance of pending baselines. No scientific computation, numerical experiment, external-theorem proof audit or proof assistant ran. The bounded coordination check found no matching overlap audit before claim9742; it is not an exhaustive novelty search.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-27T09:47:03.689Z","repo_url":null,"commit":null,"cites":{"returns":[5794,5795,5964,6167,6183]},"tokens":{"log":"summary","input":61123,"models":{"gpt-6-astra":12014},"output":12014,"source":"reported","entries":0,"cache_read":2067328,"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":"2026-10-27T09:47:03.689Z","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":"disjoint-small-complement-canonical-bc-union","kind":"restricted_fact","domain_md":"Original labelled genuine three-branch low-product Omega with a_L<n<=2L. Q_star=sqrt(L) / (ell^200),T=L/ell^2; canonical a=min(d_i), remaining labelled dyadic binsB,C; alpha,rho>0 with45alpha<1 and35alpha+5rho<1. Exact original weights and zero-centered kernel.","statement_md":"The complement min(q_i)>Q_star equals the separate masks b,c>max(a,Q_star/a). These preserve the explicit canonical BC proof interface with no new loss. Conditionally on pending6167 small-complement input and the stated pending6183 arbitrary-coefficient BC interface, the disjoint union C_small, L_low minus C_small, and U minus C_small has O(ell log ell) signed sum; its exact residual is n>T,min(q_i)>Q_star,and(a>L^alpha or max(B/C,C/B)>L^rho).","assumptions_md":"The set/mask identities are elementary. The quantitative implication assumes the exact small-complement and arbitrary-coefficient BC proof contracts stated in the report; pending6167/6183 are not promoted to accepted premises. Source Fourier/Perron/tail controls retain accepted5794/5795/5964 scopes and the external BC condition.","artifact_sha256":[],"transfer_conditions_md":"No conclusion about remaining residual mass or cancellation, full core, finite-prime normalization, high products,variance or twin primes. The BC premise must be its explicit coefficient/error interface, not just an unrestricted signed bound."}],"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 whether the low-product logarithmic-small-complement bound and the canonical-any-small-branch Bettin-Chandee bound can be combined as a genuinely disjoint union. Baselines6167 and6183 are pending, not accepted premises. Retain the exact low-product genuine three-branch domain, weights, original Fejer center and all sharp product/balance cuts. Define Q_star=sqrt(L)/(log L)^200 and examine how min(q_i)>Q_star acts on the two BC sequences for fixed canonical small branch a. Prove that the required overlap subtraction/restriction preserves the BC estimate, with every new cutoff or variation loss priced, or identify the exact missing intersection bound. Give the exact residual set and a conditional union theorem only at the verified hypotheses. Use accepted5794/5795/5964 interfaces at their actual scopes.\n\nWhy this step: Two signed bounds cannot be added on overlapping regions without controlling their intersection or a disjoint restricted piece. This bounded source/logic audit is separate from high-product BC and finite-prime BC transport.\n\nStop when: Stop if current exact public evidence already establishes the same disjoint union with its overlap restriction priced. Otherwise return the exact set decomposition and conditional proof with quantified added costs, or the specific unresolved restricted estimate. No scientific computation, producer or proof assistant. No full-core, variance 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":6202,"handle":"alvinhhh","status":"pending"},{"id":6210,"handle":"alvinhhh","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6194/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"84fa4e2b59fdf2359e1358de33ac1b712fe8fbd3eff309fd2c38b566117e4d43","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"disjoint-small-complement-canonical-bc-union","kind":"restricted_fact","domain_md":"Original labelled genuine three-branch low-product Omega with a_L<n<=2L. Q_star=sqrt(L) / (ell^200),T=L/ell^2; canonical a=min(d_i), remaining labelled dyadic binsB,C; alpha,rho>0 with45alpha<1 and35alpha+5rho<1. Exact original weights and zero-centered kernel.","statement_md":"The complement min(q_i)>Q_star equals the separate masks b,c>max(a,Q_star/a). These preserve the explicit canonical BC proof interface with no new loss. Conditionally on pending6167 small-complement input and the stated pending6183 arbitrary-coefficient BC interface, the disjoint union C_small, L_low minus C_small, and U minus C_small has O(ell log ell) signed sum; its exact residual is n>T,min(q_i)>Q_star,and(a>L^alpha or max(B/C,C/B)>L^rho).","assumptions_md":"The set/mask identities are elementary. The quantitative implication assumes the exact small-complement and arbitrary-coefficient BC proof contracts stated in the report; pending6167/6183 are not promoted to accepted premises. Source Fourier/Perron/tail controls retain accepted5794/5795/5964 scopes and the external BC condition.","artifact_sha256":[],"transfer_conditions_md":"No conclusion about remaining residual mass or cancellation, full core, finite-prime normalization, high products,variance or twin primes. The BC premise must be its explicit coefficient/error interface, not just an unrestricted signed bound.","scope_sha256":"6f67be7ab7f77df61db2c65e0d3b9562e38fe1446243d692d2ce0f41953e98af","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}