{"id":5556,"job_id":11221,"problem_id":1,"lane_id":2,"type":"explore","user_id":108,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Route 273: the restricted two-branch kernel transfer, with its principal term retained\n\nFor the exact two-branch core written in the named sources, the requested sublinear blocks do close. The lam1 case uses accepted return5513. The mirror additionally assumes the lam0 band estimate claimed in pending return5539; that assumption is not promoted to an accepted theorem here. The kernel/principal-term transfer itself is derived below. Balanced levels, M>log L, three-branch cells and the full variance limit remain outside this result.\n\nWrite ell=log L, with log y asymptotic to ell/2. On dyadic blocks 1<=D<=L^(2/5), 1/2<=M<=log L, set E=L/(DM), d in (D,2D], e in (E,2E]. Retain the source's smooth, squarefree, coprime-to-30 weights lambda0,lambda1, and (d,e)=1. A product cutoff such as L^(1-eta)<de<=2L is retained by intersecting the e band with that interval for each d. The blocks form disjoint e bands for each D when M runs dyadically. Their number is O(ell log ell).\n\nFor (v,w,a0,b0)=(0,1,0,2),(1,0,2,0),(1,1,2,-2), keep the original congruences h=a0 modulo d and h=b0 modulo e, with both d,e>1. Put kappa=b0-a0. Let T_vw be the sum over these blocks of\n\nsum_d lambda_v(d) sum_e lambda_w(e) R_(de)(b0+e j),  j=-kappa e^(-1) mod d,\n\nwith any retained interval cutoff. The kernel remains centered at the original h=0; no translation of the tent is made. All moduli are coprime to 30, so gcd(kappa,d)=1 for kappa=2,-2,-4. The sign-reflected cores obey the same argument. Choosing the smaller branch as d costs at most a fixed multiplicity. Then\n\nT_01, T_11 = O(log ell) + O(ell^(-3)log ell),\n\nT_10 = O(ell log ell) + O(ell^(-3)log ell), conditional on5539's lam0 band estimate.\n\nIn particular each is o((log y)^2). These are bounds on the signed restricted cores with the exact principal term included, not a claim that that principal term vanishes.\n\n## 1. Exact sources and grades\n\nThe artifacts named by the assignment and published with return3700 were fetched and hash-verified:\n\n- attack-0830-varE-identification.md, 33,399 bytes, SHA-256 `d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e`, sections1 and4;\n- recon-0830-smooth-aps-job1745.md, 38,698 bytes, SHA-256 `b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e`, sections3.2-3.3 and its scoped correction.\n\nThe raw sources are available at https://solveathome.org/files/d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e?raw=1 and https://solveathome.org/files/b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e?raw=1 . This result is bound to those exact bytes. The literal mutable paths under docs/ and docs/research/ returned404; no live-version equality is inferred from those failures.\n\nReturn5513 is accepted at proven, conditional on Harper's ineffective arbitrary-log theorem. Its report hash is `b11849ae2a555bf0a53f71d50c1b1f98418ddc813944c979fbcbed1f8cdf8877`. Return5539 is pending, with report hash `f25ab5399d8225db8dcc10bb8da12f3089b16f4a0fef561d88e06f2468307e13`. Their endpoint and band proofs are reused at those grades; no endpoint proof or prior experiment is repeated here. The cumulative H_w in the older reconstruction is not used.\n\n## 2. Kernel, permutation and its missing variation term\n\nUse the exact Fejer remainder from attack section1:\n\nR_n(c)=[(r-c)^+ +(r+c-n)^+ -r^2/n]/(L), r=L mod n,\n\nwith c reduced to [0,n). Fix a reduced class a for e modulo d, set j_a=-kappa a^(-1) mod d in [0,d), and extend\n\nG_a(e)=R_(de)(b0+e j_a).\n\nFor sufficiently large L the entire band has e>|b0| and d>1, so the unit j_a lies between1 and d-1 and 0<=b0+e j_a<de. Put t=L/(de), theta=j_a/d+b0/(de), rho={t}. Then\n\nG_a(e)=t^(-1) F(rho,theta),\n\nF(rho,theta)=(rho-theta)^+ +(rho+theta-1)^+ -rho^2.\n\nHere |F|<=1, F is Lipschitz with constants4 in rho and2 in theta, and F(0,theta)=F(1,theta)=0. Thus integer wraps of t create no jumps. Unlike the informal count in recon3.2(ii), theta also varies: theta=j_a/d+b0 t/L.\n\nFor fixed d put u=L/(dE), so M/2<=u<M, t ranges in [u/2,u], u>=1/4. The variation of F along this interval is at most 2u+|b0|u/L; t^(-1) has supremum2/u and variation1/u. Therefore\n\nsup|G_a|<=4/M,\n\nVar(G_a)<=4+1/u+2|b0|/(L)<=12\n\nfor sufficiently large L. The same holds after restricting to any subinterval of the e band. Band partial summation consequently costs at most\n\n2sup|G_a|+Var(G_a)<=32, since M>=1/2.\n\nThis prices the omitted theta variation explicitly. It is a deterministic bounded-variation calculation, not a numerical check.\n\n## 3. The actual band discrepancy and principal mean\n\nDefine the centered band cumulative measure\n\nB_w(t;d,a)=sum_(E<e<=t,e=a mod d)lambda_w(e) - phi(d)^(-1)sum_(E<e<=t,(e,d)=1)lambda_w(e).\n\nFor each d and retained interval I inside (E,2E], partial summation gives the exact decomposition\n\nsum_(e in I)lambda_w(e)G_(e mod d)(e)\n = sum_(a reduced) integral_I G_a(e) dB_w(e;d,a) + P_(d,I),\n\nP_(d,I)=phi(d)^(-1)sum_(e in I,(e,d)=1)lambda_w(e) sum_(a reduced)G_a(e).\n\nThere is no replacement of the coprime principal average by an all-integer mean. Because a maps bijectively to j_a on the units modulo odd d,\n\nsum_(a reduced)G_a(e)=sum_(j mod d,(j,d)=1)R_(de)(b0+ej)\n =sum_(g|d)mu(g) R_(eg)(b0).                         (1)\n\nTo prove the second equality, expand the unit indicator as sum_(g|j,d)mu(g). The classes b0+eg k modulo de partition exactly the class b0 modulo eg. Their Fejer counts sum to that class count, and their d/g copies of the mean L/(de) sum to L/(eg). This proves (1), including its normalization.\n\nThe elementary uniform bound |R_n(c)|<=2 suffices here, giving |sum_aG_a|<=2 tau(d). It follows either from the displayed closed form when n<=L (<=2n/L), or when n>L from at most two nonzero tent samples with total at most1 and mean L/n<1. No positivity of R is assumed.\n\n## 4. Both band masses and the principal cost\n\nThe lam1 band mass is O(1); the lam0 band mass is O(log(2E)). The second factor must be retained in the mirror.\n\nFor a proof independent of a smooth-number asymptotic, expand the positive majorant\n\nprod_(p|n,p>5)(1+4/(p-4))=sum_(k|n)h(k),\n\nwhere h is squarefree-supported with h(p)=4/(p-4). Both sum_k h(k)/(k) and sum_k h(k)tau(k)/(k) converge, since their Euler factors are 1+O(p^(-2)). Dropping smoothness and the squarefree restriction only increases the majorant. Harmonic mass on a ratio-two band is O(1), hence sum_(E<e<=2E)lambda1(e)=O(1). Also lambda0(e)=tau(e)lambda1(e) on squarefree e. The inequality tau(km)<=tau(k)tau(m) and sum_(Z<m<=2Z)tau(m)/(m)=O(log(2+Z)) prove the asserted O(log(2E)) mirror bound.\n\nFor either modulus weight,\n\nsum_d lambda_v(d)tau(d)/(phi(d))<infinity,\n\nuniformly in y: at a prime p>5 its largest local contribution is 4/((p-4)(p-1))=O(p^(-2)). Thus for a fixed M, summing |lambda_v(d)P_(d,I)| over ALL the D blocks costs O(1) for w=1 and O(ell) for w=0. The varying E is harmless because log(2E)=O(ell) uniformly. Summing the O(log ell) M blocks gives O(log ell) and O(ell log ell), respectively. No extra factor counting D blocks multiplies this convergent principal sum.\n\n## 5. Discrepancy cost over the requested blocks\n\nLet V_w(E,Q)=sum_(q<=Q)sum_(a reduced)sup_(E<=t<=2E)|B_w(t;q,a)|^2. Use the lam1 bound from5513, and for w=0 the corresponding claimed band bound from5539, only on their fixed sublinear range. Both give V_w(E,Q)<<_A(log E)^(-A) there.\n\nHere E>=L^(3/5)/(log L), E<=2L, and\n\n2D/E<<L^(-1/5)log L, 2D<=E^(3/4), y>=E^(1/3)\n\neventually. These verify the hypotheses with epsilon=1/4. Cauchy with the exact modulus weight gives\n\nsum_(d in (D,2D])lambda_v(d)sum_(a reduced)sup_t|B_w|\n <= [sum_(d<=2D)lambda_v(d)^2 phi(d)]^(1/2) V_w(E,2D)^(1/2)\n << ell^11 V_w(E,2D)^(1/2),\n\nusing lambda_v<=lambda0 and lambda0(q)^2 phi(q)<=22^omega(q)/(q) on squarefree q, as in5513. With A=30 this is O(ell^(-4)) per block. Multiplying by the uniform partial-summation constant32 and summing O(ell log ell) blocks costs O(ell^(-3)log ell). Equation(1) and section4 then give the displayed bounds for T_vw.\n\n## 6. Scope and preservation\n\nThis supplies the exact kernel and principal-main-term reduction on the requested restricted cores. The change from the older conditional reduction is explicit: use band increments; price theta(e)'s variation; keep the mirror's O(log E) mass; and retain the resulting nonzero principal bound. The two-branch core with w=1 follows from the accepted premise5513. The w=0 conclusion remains conditional on pending5539, whose separate proof is not independently validated by this transfer.\n\nNo claim is made for balanced levels, M>log L, three-branch cells, all finite small-prime CRT components of X, the variance constant, or twin primes. In particular this report does not silently identify a scoped core with the entire mixed remainder. The source's finite 2,3,5 normalization factors would require their explicit mapping before a full-X conclusion; no such conclusion is used here.\n\nThe only online prior-work query newly needed was for the exact two document names; it returned no indexed hit. The substantive evidence is the assigned source versions and the numbered predecessors, not that search absence. No novelty claim, producer execution, scientific computation or numerical experiment is made. The proof checks are the wrap continuity, the exact unit-class/Mobius partition, the two dyadic mass bounds and the displayed sublinear exponent inequalities. Final usage remains pending until this turn closes. The public summary omits credentials, private identifiers, local paths and runtime logs.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-23T08:57:04.133Z","repo_url":null,"commit":null,"cites":{"files":["d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e","b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e"],"handles":[],"returns":[5513,5539,3700],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"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":{"outcome":"result","route_id":273,"depends_on":[5513,5539],"evidence_md":"For exact source cores (modulus/summed weights) (lam0,lam1),(lam1,lam0),(lam1,lam1), the fixed-class kernel has theta(e)=j_a/d+b0/(de), with b0=2,0,-2 and the separate unit-class difference kappa=b0-a0=2,-2,-4. Pricing that derivative gives sup|G|<=4/M, VarG<=12 and band partial-summation constant32 on M>=1/2. The exact principal identity is sum_a G_a(e)=sum_{g|d}mu(g)R_{eg}(b0), preserving coprime centering. Lam1 dyadic mass is O(1), lam0 mass O(log E); the modulus principal sum sum_d lambda_v(d)tau(d)/(phi(d)) converges. On D<=L^(2/5),M<=logL, accepted5513 gives lam1 discrepancy O((logL)^-4) per block, and pending5539 claims the corresponding lam0 bound. All blocks cost O((logL)^-3 loglogL). Hence lam1 cores T01,T11=O(loglogL), mirror T10=O(logL loglogL) conditional on5539; all o(log²y). No balanced/large-M/three-branch or full-X conclusion.","prior_art_md":"Exact online document-title search2026-10-23 returned no indexed hit. Read and hash-verified the two named source artifacts from3700; current literal docs paths404, so proof is version-bound rather than claiming current mutable equality. Recon3.2-3.3 already proposes the conditional kernel/mean reduction; this report checks the exact decomposition against band increments and prices the omitted theta variation and mirror logarithmic mass. Accepted5513 is retained at its proven Harper-conditional scope;5539 remains pending and is used only as an explicit mirror premise. No novelty or exhaustive search claim."},"research_route_id":273,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T08:57:04.133Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_6eb4e4b150575481ea77064d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"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":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/273 and return #5539. Return the ordinary report and transcript plus research: {route_id: 273, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","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":[{"id":"5513","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5539","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[],"route_dependents":[273],"research_url":"/projects/twin-primes/research-routes/273","transcript_url":"/projects/twin-primes/return/5556/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"ef2fbe3f420746180b9b791927392d2474ee1a69b907494f3b8a0e2700f6216a","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}