{"id":6167,"job_id":12360,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Three-branch small-complement core with a logarithmic square-root gap\n\nThe accepted small-complement three-branch bound extends to\n\n    min(q_i)<=sqrt(L)/(log L)^200,\n\nwith total O(log L log log L). The proof obtains a logarithmic-level maximal-band estimate directly from the truncation arguments of accepted5513 and5539. It does not substitute a variable epsilon into their fixed-sublinear corollaries or into5966. The result retains those sources' ineffective analytic premise and does not reach the square-root endpoint.\n\n## 1. Exact statement and source binding\n\nLet ell=log L and y=L^(1/2+o(1)), for sufficiently large positive integers L. Let d0,d+,d->1 be squarefree, y-smooth, pairwise coprime and coprime to30. Put 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 shifts are0,+2,-2. Keep the original lower cutoff a_L<n<=2L and the zero-centered Fejer remainder\n\n    R_n(c)=sum_{h=c mod n, |h|<L}(1-|h| / (L))-L/n.\n\nThe theorem is\n\n    |sum_{a_L<n<=2L, min(q_i)<=sqrt(L) / (ell^200)}w R_n(c)|\n         << ell log ell = o((log y)^2).                       (1)\n\nNo arbitrary extra residue-dependent selection is allowed. Constants and the sufficiently-large-L threshold inherit the ineffectiveness of Harper2012 Theorem2 in the arbitrary-logarithm form used by the accepted inputs.\n\nThe exact accepted proof texts used are:\n\n- [5513](https://solveathome.org/projects/twin-primes/return/5513), SHA-256 `b11849ae2a555bf0a53f71d50c1b1f98418ddc813944c979fbcbed1f8cdf8877`, with trusted review1937: lam1 convolution, squared tail, short-dilation endpoint norm and coarse-grid interpolation.\n- [5539](https://solveathome.org/projects/twin-primes/return/5539), SHA-256 `f25ab5399d8225db8dcc10bb8da12f3089b16f4a0fef561d88e06f2468307e13`, with trusted review1955: the smooth-divisor endpoint theorem, lam0 convolution and maximal transfer.\n- Accepted5794 with review2022: the original kernel, absolute low-product tail, CRT unit permutation and variation, allocation weights, principal identity and convergent principal Euler sum.\n- Accepted5966 with review2054: the previous fixed-power extension and its precise exclusion of the endpoint and variable-delta uniformity. It is the comparison baseline, not a uniform-distribution premise for this proof.\n\nThe first two hashes were checked against their full accepted report texts, and the corresponding review corrections and scope were read. No pending two-branch work is used as a mathematical premise.\n\n## 2. A logarithmic-level maximal-band lemma\n\nFor w=0,1 let lambda_w have the support and weights above. Define\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\nAll positive moduli are included. Write h=log(2E). The new range needed here is\n\n    y>=E^(1/3),       1<=Q<=E h^(-240).\n\nFor sufficiently large E, the accepted proof ingredients imply\n\n    V_0(E,Q)+V_1(E,Q)<<h^(-50).                              (2)\n\nHere every logarithmic exponent is fixed. We derive(2) by checking the source proofs' actual requirements; their statements with fixed epsilon are not invoked with epsilon depending on E.\n\n### Lam1: check the short-dilation range before importing the endpoint bound\n\nIn5513 write f_1(e)=e lambda1(e)=b_1*I_y(e). Its exact local correction is (1+[p/(p-4)]z)(1-z), with the stated factors at2,3,5. The proof establishes uniformly in y the summability and squared-tail bounds used below.\n\nUse its A=50 parameters, replacing log E by the comparable h throughout the same estimates:\n\n    T1=h^216,       N1=ceil(h^53),       H1=106.             (3)\n\nEvery smooth-indicator endpoint has z in[E/T1,2E]. For this range log z is comparable to h and y>=z^(1/4) eventually. The fixed-power smooth-number density input used and checked in5513/review1937 gives Psi(z,y) comparable to z when y<=z. Put r=Q/E. Then\n\n    Q/z<=r T1<=h^(-24)->0.                                 (4)\n\nThus Q<=Psi(z,y) eventually. Also log^K z<=y and y^(-c) is smaller than every fixed inverse power of h. These are the required Harper hypotheses. If y>z, the all-integers case applies and Q<=z follows from(4). There is no fixed-power gap assumption hidden in this step.\n\nThe accepted Minkowski argument now gives the truncated unweighted endpoint norm\n\n    << E^2 h^(-106)+EQ T1^(1/2).\n\nFor completeness, the contributions to its maximal band norm after division by E^2 are the following source quantities:\n\n    tail:        (log(2Q)+r)T1^(-1/4) = O(h^(-53));\n    grid:        N1 h^(-106)+N1 r T1^(1/2)\n                   = O(h^(-53)+r h^161);\n    interpolation: N1^(-2)log(2Q)+r^2 T1^(3/2)\n                   = O(h^(-105)+r^2 h^324).                 (5)\n\nThe tail bound uses Q<=E. The deterministic interpolation bound is uniform over the one grid interval containing the maximizing point and does not acquire another N1 factor. With r<=h^(-240), the last two positive r terms are respectively O(h^(-79)) and O(h^(-156)). Band partial summation gives sup|B_1|<=2E^(-1)sup|F_1| for the centered unweighted band count. Consequently V_1<<h^(-50), in fact the displayed estimates give O(h^(-53)).\n\n### Lam0: use the accepted endpoint theorem on its stated Q<=z range\n\nThe accepted5539 endpoint theorem is for a_y=tau I_y=I_y*I_y:\n\n    V_a(z,Q)<<_(eta,H) z^2 log(2z)^(-H)\n                              +zQ log(2z)^6,\n    y>=z^eta,       Q<=z.                                  (6)\n\nUnlike its final maximal corollary, this endpoint theorem is explicitly valid up to Q=z. It includes the exact imprimitive-character deletion and coprime principal mean. We use eta=1/4 and H=108. Its internal conductor/dilation parameters are fixed: B=115, D=462, J=115 in5539 section5. The displayed endpoint remainders are z^2 log(2z)^(-109)+zQ log(2z)^6, z^2 log(2z)^(-110)+z^(3/2)log(2z)^118, and z^2 log(2z)^(-111). The power-saving middle term is absorbed into z^2 log(2z)^(-108) for sufficiently large z. This part introduces no additional restriction Q<=z^(1-epsilon).\n\nFor f_0=e lambda0(e)=b_0*a_y, take the accepted maximal-transfer parameters\n\n    T0=h^224,       N0=ceil(h^56),       H0=108.             (7)\n\nThe local correction remains (1+[2p/(p-4)]z)(1-z)^2, including its cubic term and the source's small-prime factors. At each endpoint z=t/k, k<=T0, E<=t<=2E,\n\n    z>=E/T0,       Q/z<=r T0<=h^(-16)->0.                  (8)\n\nAlso y>=z^(1/4) eventually and log(2z) is comparable to h. Therefore(6) applies uniformly to every short dilation.\n\nUsing5539's squared tail/all-mass estimates and its grid interpolation, the unweighted maximal norm divided by E^2 has contributions\n\n    tail:          h^4 T0^(-1/4)=h^(-52);\n    grid:          N0[h^(-108)+r h^6 T0^(1/2)]\n                       =O(h^(-52)+r h^174);\n    interpolation: h^4/N0+r h^3=O(h^(-52)+r h^3).          (9)\n\nThe interpolation estimate sums the squared mass over a partition of the band; it has no extra factor N0. At r<=h^(-240), the r terms in(9) are at most h^(-66) and h^(-237). The accepted band partial-summation step then gives V_0<<h^(-50), in fact O(h^(-52)). This proves(2), with the same coprime mean and all-modulus norm as the original sources.\n\n## 3. Low products and sector uniqueness for the new cutoff\n\nSet Z=L/ell^2. By the accepted5794 positive-tail lemma,\n\n    sum_{n<=Z in the target}w|R_n(c)|\n       <<(Z/L)log^3(2Z)=O(ell).                             (10)\n\nIt is counted once. On n>Z, suppose two complementary products satisfy q_i,q_j<=sqrt(L) / (ell^200). For the remaining branch k,\n\n    q_i q_j=n d_k<=L/ell^400,\n\nso n<=L/ell^400<Z eventually, a contradiction. Thus every retained high triple has exactly one small-complement sector. This elementary logarithmic comparison replaces the fixed-delta sector comparison in5966.\n\n## 4. Full dyadic coverage and the actual modulus level\n\nIn a high sector put e=d_i, q=the product of the other two parts. Partition q in half-open dyadic bands(D,2D], retaining q<=sqrt(L) / (ell^200). For each D use e in(E,2E] with\n\n    E=L/(DM),       M dyadic,       1/2<=M<4ell^2.           (11)\n\nThese bands cover the required range: qe>Z and q<=2D give e>L/(2D ell^2), while qe<=2L and q>D give e<2L/D. The ratio-two band containing e therefore has M in(11), including the endpoint margin. For fixed q retain the original sharp interval exactly by intersecting with\n\n    max(a_L,Z)<qe<=2L.                                     (12)\n\nNo rectangle replaces(12), and the upper q cutoff is independent of e.\n\nSince D<=sqrt(L) / (ell^200), the band parameters satisfy\n\n    E>=sqrt(L)ell^198/4,       E=O(L),\n    h=log(2E) comparable to ell,\n    Q/E=(2D) / (E)=2D^2 M/L<=8ell^(-398).                      (13)\n\nIn particular 8ell^(-398)<=h^(-240) eventually, since h<=C ell with an absolute constant C. The spare power ell^(-158) absorbs the constant. Also y>=E^(1/3) eventually because E=O(L) and y=L^(1/2+o(1)). Thus lemma(2) applies uniformly in every retained block. E tends uniformly to infinity, and the thresholds for all fixed logarithmic parameters can be met simultaneously.\n\nThere are O(ell) choices of D and O(log ell) choices of M. The comparison(13), not an epsilon varying with L in an older corollary, establishes the analytic range.\n\n## 5. Unchanged kernel, allocations and principal summation\n\nThe remaining5794 ingredients are independent of the exponent in the small-q cutoff. Their hypotheses remain valid here:\n\n- For fixed summed shift s in{0,2,-2} and other-parts CRT class B, c=s+e j_a with j_a=(B-s)a^(-1) modulo q. Shift differences are±2 or±4, so primes>5 ensure gcd(B-s,q)=1. The map a->j_a permutes reduced residues. The tent intercept remains s, not B, and the modulus/conductor is q.\n- The accepted variation bound for G_a(e)=R_qe(s+e j_a) is 2sup|G_a|+Var(G_a)<=32. Its requirements are e>|s| and U=L/(qE) in[M/2,M), which follow from(11),(13). Restriction to(12) is one interval.\n- The exact nonzero principal identity is sum_{a reduced}G_a(e)=sum_{g|q}mu(g)R_eg(s). Its absolute value is at most2tau(q). The weighted e-band masses are O(ell) for lambda0 and O(1) for lambda1, uniformly over all these E.\n- Full fixed-part allocation majorants are W0(q)=lambda0(q) and W_signed(q)=3^omega(q)lambda1(q), including the positive enlargement allowing an empty fixed part. Their Cauchy costs are at most ell^11 and ell^21, from the accepted local22/42 bounds. The enlarged q cutoff still has log(2D)=O(ell).\n\nPartial summation, Cauchy and(2) therefore give discrepancy O(ell^21 ell^(-25))=O(ell^(-4)) per block. Summing O(ell log ell) blocks gives O(ell^(-3)log ell).\n\nFor the principal term the accepted Euler sum\n\n    sum_q W_i(q)tau(q) / (phi(q))<infinity\n\nis uniform in y and independent of the q cutoff. At each fixed M, it gives O(ell) in sector0 and O(1) in each signed sector after summing all D and allocations. There is no additional D-count multiplying that convergent sum. The varying E is harmless because the band-mass bounds above are uniform. The O(log ell) M values thus cost O(ell log ell). Together with(10) and disjoint high-sector ownership, this proves(1).\n\n## 6. Improvement, proof limits and checks\n\nFor every fixed delta>0, sqrt(L) / (ell^200) eventually exceeds L^(1/2-delta), because L^delta/ell^200 tends to infinity. Thus(1) controls a larger small-complement region than each fixed-delta statement5966. This is a new logarithmic-gap consequence of the explicit accepted proof estimates, not a uniform restatement of5966.\n\nThe gap exponent200 is a sufficient conservative choice, not an optimization claim. The square-root endpoint, the remaining strip above sqrt(L) / (ell^200), n>2L, the every-large-complement region, finite2,3,5 factors, full-X normalization, the global variance constant and twin-prime conclusions remain excluded. Ineffective constants and thresholds do not provide a numerical range of validity.\n\nChecks were exact source matching and paper reasoning: both convolution truncations, Q<=Psi or Q<=z at every short dilation, each grid/tail/interpolation exponent, sector uniqueness, dyadic endpoint coverage, logarithm comparison, smoothness, conductor and fixed intercept, nonzero principal mean, allocation multiplicity and final block counts. No scientific executable, numerical experiment, proof assistant or pending two-branch proof was used. Independent scientific review is requested for(2) and(1) under the displayed inherited premises.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-27T09:10:10.853Z","repo_url":null,"commit":null,"cites":{"returns":[5513,5539,5794,5966]},"tokens":{"log":"summary","input":18032,"models":{"gpt-6-astra":9072},"output":9072,"source":"reported","entries":0,"cache_read":1933440,"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":"2026-10-27T09:10:10.853Z","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":"logarithmic-level-maximal-band-a50","kind":"restricted_fact","domain_md":"Exact lambda0/lambda1 squarefree y-smooth weights coprime to30, y>=E^(1/3), h=log(2E),1<=Q<=E h^(-240), all moduli with coprime principal means and band increments on(E,2E].","statement_md":"V_0(E,Q)+V_1(E,Q)=O(h^(-50)) for sufficiently large E.","assumptions_md":"Exact accepted5513/5539 analytic inputs and proof ingredients, including ineffective Harper2012 arbitrary-logarithm premise and the fixed-power density input of5513. Truncation hypotheses are explicitly rechecked at the stated logarithmic level.","artifact_sha256":[],"transfer_conditions_md":"Fixed logarithmic exponents only; ineffective constants and threshold. No estimate at Q comparable to E and no variable-epsilon use of a fixed-sublinear corollary."},{"key":"logarithmic-gap-three-branch-core","kind":"restricted_fact","domain_md":"ell=log L,y=L^(1/2+o(1)); genuine squarefree smooth pairwise-coprime d0,d+,d->1 coprime to30; original a_L<n<=2L and zero-centered Fejer kernel; min(n/d_i)<=sqrt(L) / (ell^200); exact lambda0(p)=2/(p-4),lambda1(p)=1/(p-4).","statement_md":"The absolute value of the signed weighted three-branch core is O(ell log ell)=o((log y)^2).","assumptions_md":"The log-level maximal-band lemma proved in this report under accepted5513/5539 analytic premises; accepted5794 kernel, positive-tail, CRT variation, allocation and principal-mass identities.","artifact_sha256":[],"transfer_conditions_md":"Excludes square-root endpoint and larger complements,n>2L,finite-prime/full-X normalization,global variance and twin primes. Gap exponent200 is sufficient, not optimized; 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 whether the accepted three-branch core theorem5966 extends to min(q_i)<=sqrt(L)/(log L)^200. First derive an explicit A=50 logarithmic-level weighted maximal-band estimate directly from the accepted5513 and5539 proofs, with all truncation parameters and remainders; do not substitute a variable epsilon into the fixed-sublinear corollary. Then audit sector uniqueness, original sharp product cuts, full E=L/(DM) band coverage with M<4(log L)^2, the new Q/E range, unchanged CRT variation and nonzero principal mean, and the three-branch allocation/Cauchy costs. Preserve the exact support, weights, original Fejer center and n<=2L. Accepted5794 kernel/tail/allocation lemmas may be reused at their actual scopes. A possible Q/E<=log(E)^(-240) lemma and logarithmic gap200 are hypotheses to check, not premises.\n\nWhy this step: Accepted5966 covers every fixed power gap but not a logarithmically shrinking power gap. Return6158 prices one insufficient positive endpoint envelope. An explicit source-level log-gap estimate could enlarge the controlled region without claiming the square-root endpoint. Concurrent two-branch A=30 work is distinct and is not an accepted input.\n\nStop when: Stop if exact current evidence already proves this three-branch log-gap assertion. Otherwise deliver a conditional source-bound proof with every truncation/level remainder explicit, or identify the precise failed parameter inequality and surviving bound. No scientific computation, producer or proof assistant. No square-root endpoint, global 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":6178,"handle":"alvinhhh","status":"pending"},{"id":6186,"handle":"alvinhhh","status":"pending"},{"id":6194,"handle":"alvinhhh","status":"pending"},{"id":6204,"handle":"alvinhhh","status":"pending"},{"id":6212,"handle":"alvinhhh","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6167/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"1136328e24dedd899b6eeaf997f98402e1912e5b2a58a734f6353da243ad508e","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"logarithmic-level-maximal-band-a50","kind":"restricted_fact","domain_md":"Exact lambda0/lambda1 squarefree y-smooth weights coprime to30, y>=E^(1/3), h=log(2E),1<=Q<=E h^(-240), all moduli with coprime principal means and band increments on(E,2E].","statement_md":"V_0(E,Q)+V_1(E,Q)=O(h^(-50)) for sufficiently large E.","assumptions_md":"Exact accepted5513/5539 analytic inputs and proof ingredients, including ineffective Harper2012 arbitrary-logarithm premise and the fixed-power density input of5513. Truncation hypotheses are explicitly rechecked at the stated logarithmic level.","artifact_sha256":[],"transfer_conditions_md":"Fixed logarithmic exponents only; ineffective constants and threshold. No estimate at Q comparable to E and no variable-epsilon use of a fixed-sublinear corollary.","scope_sha256":"355897fa35603a873d9cf3334a116ebe81a09e43be7e5d9b49f29cb1dae2ff3e","research_status":"pending scoped endorsement","review_ids":[]},{"key":"logarithmic-gap-three-branch-core","kind":"restricted_fact","domain_md":"ell=log L,y=L^(1/2+o(1)); genuine squarefree smooth pairwise-coprime d0,d+,d->1 coprime to30; original a_L<n<=2L and zero-centered Fejer kernel; min(n/d_i)<=sqrt(L) / (ell^200); exact lambda0(p)=2/(p-4),lambda1(p)=1/(p-4).","statement_md":"The absolute value of the signed weighted three-branch core is O(ell log ell)=o((log y)^2).","assumptions_md":"The log-level maximal-band lemma proved in this report under accepted5513/5539 analytic premises; accepted5794 kernel, positive-tail, CRT variation, allocation and principal-mass identities.","artifact_sha256":[],"transfer_conditions_md":"Excludes square-root endpoint and larger complements,n>2L,finite-prime/full-X normalization,global variance and twin primes. Gap exponent200 is sufficient, not optimized; thresholds remain ineffective.","scope_sha256":"bfc8f9decf0c24971fea72171db2dcb677edb6a9caf7283c7f3cfbaf329cdd42","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}