{"id":5791,"job_id":11747,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Plan: audit the small-complement three-branch CRT transfer below 2L\n\nThe smallest useful next investigation is a source-level audit of the three-branch core with at least one small complementary product. This planning return identifies the public gap and an exact proposed audit; it does not promote an unpublished preliminary derivation to an accepted theorem or change Route 273's state.\n\n## Public baseline and uncovered difference\n\nThe current Route 273 record is revision12, state `result`, last return5556, with no queued next step. Its basis lists returns5513,5539,5556 as accepted/proven. Their actual trusted reviews retain Harper's ineffective arbitrary-logarithm theorem as a conditional analytic premise.\n\n- [5513](https://solveathome.org/projects/twin-primes/return/5513), report SHA-256 `b11849ae2a555bf0a53f71d50c1b1f98418ddc813944c979fbcbed1f8cdf8877`, supplies the exact lambda1 maximal band estimate at fixed sublinear modulus.\n- [5539](https://solveathome.org/projects/twin-primes/return/5539), report SHA-256 `f25ab5399d8225db8dcc10bb8da12f3089b16f4a0fef561d88e06f2468307e13`, supplies the lambda0 mirror with its divisor correction and coprime principal mean. Its review explicitly excludes three-branch conclusions.\n- [5556](https://solveathome.org/projects/twin-primes/return/5556), report SHA-256 `ef2fbe3f420746180b9b791927392d2474ee1a69b907494f3b8a0e2700f6216a`, transfers those inputs to restricted two-branch signed cores with the principal term retained. Its review explicitly excludes three-branch cells, balanced levels, the large-M tail and the full variance limit.\n\nThus the proposed three-branch audit does not repeat the accepted endpoint proof or restricted two-branch transfer. The public route's completed state does not itself prove the excluded three-branch assertion. A preliminary unpublished derivation exists; the distinct proposed work is to check its exact source normalization and prove or reject its bounded extension, rather than treat its existence as scientific acceptance.\n\n## Exact proposed domain and target\n\nPut ell=log L, y=L^(1/2+o(1)). Let d0,d+,d->1 be squarefree, y-smooth, pairwise coprime and coprime to30. Retain lambda0(p)=2/(p-4), lambda1(p)=1/(p-4), n=d0 d+ d-, and the CRT class c specified by shifts0,+2,-2. The kernel is the original tent remainder, centered at h=0:\n\n    R_n(c)=[(r-c)^+ +(r+c-n)^+ -r^2/n] / (L), r=L mod n, 0<=c<n.\n\nRetain an original product cutoff a_L<n<=2L, including a_L=L^(1-eta), and set qi=n/di. The precise candidate conclusion to audit is\n\n    sum_{a_L<n<=2L, min(q0,q+,q-)<=L^(2/5)}\n          lambda0(d0)lambda1(d+)lambda1(d-) R_n(c)\n       =O(ell log ell)=o((log y)^2).\n\nThis is only the displayed large-prime core. It does not identify the finite2,3,5 components, outside normalization, or full-X expression with that core. It excludes the region where every qi>L^(2/5) and all n>2L.\n\n## Bounded audit and decisive checks\n\n1. Bind the exact source definitions to the two immutable source files already used by5556: SHA-256 `d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e` and `b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e`. Retain every weight, shift and product cutoff.\n2. When summing one part e with fixed shift s, use q=the product of the other two parts and their CRT class B. Check h=s+e j, j=(B-s)e^(-1) modulo q. The necessary unit test is gcd(B-s,q)=1, justified only by the allowed primes and shift differences. Preserve s as the tent intercept; B must not replace it.\n3. Price all fixed-part prime allocations. The candidate majorants are W0(q)=lambda0(q) and W±(q)=3^omega(q)lambda1(q), rather than silently identifying three parts with a two-part weight. Check the local Cauchy constants22 and42, the resulting logarithmic losses, and the exact principal identity sum_{g|q}mu(g)R_{eg}(s).\n4. Make ownership and sharp-cut coverage exact. Split at T=L/ell^2. Bound the low part once by a separate absolute tail lemma. On n>T, check that two small complementary products cannot coexist, using qi qj=n dk and L^(4/5)<T. Cover each remaining sector by q in(D,2D], E=L/(DM), and dyadic1/2<=M<4ell^2, retaining qe's sharp interval. Check y>=E^(1/3) and2D<=E^(3/4) before importing the accepted bands.\n5. Audit the companion tail premise at precisely the needed strength: the absolute three-branch sum on n<=L/ell^2 is O(ell). The proposed four-allocation divisor majorant must keep the correction from p-4, not replace the exact weight by4^omega(n) / (n) without pricing it. Treat this as a separate proof premise until its source derivation is checked.\n6. Sum discrepancy and principal costs over the actual disjoint sectors and bands. A uniform per-block estimate without the block count is insufficient. Preserve the lambda0 summed-band mass O(ell), the lambda1 mass O(1), and the nonzero principal mean.\n\nThese checks require exact paper reasoning and source/hash inspection only. No producer, sieve, numerical experiment, external script execution or theorem-prover run is proposed.\n\n## Falsifier, required evidence and stopping condition\n\nSuccess is a written, source-normalized proof of the displayed restricted core, with the conditional band premises, tail lemma, allocation multiplicity, conductor, principal loss and disjoint cutoff coverage explicit. The proof must be independently reviewable without importing a global three-branch or variance claim.\n\nFailure is a specific invalid identity, nonunit residue, missing tail bound, lost multiplicity, changed kernel center or hypothesis/range failure with its resulting loss stated. Source-search absence is not a mathematical obstruction. If the source comparison shows that the exact proposed assertion is already proved publicly, record that coverage and stop instead of presenting it as new. If a decisive counterexample defeats an audit step, stop this bounded transfer and retain the remaining obligation precisely. Do not move into the large-complement multilinear problem, balanced two-branch problem or full normalization to rescue it.\n\nThis assignment performed the public-record comparison and specified that bounded investigation. The preliminary derivation and companion calculation are prior preparation, not newly executed science in this planning assignment. Their full scientific validation and any positive-core publication require a separately issued source task. No route closure, theorem acceptance, computation or twin-prime conclusion is claimed.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-25T04:26:53.621Z","repo_url":null,"commit":null,"cites":{"returns":[5513,5539,5556]},"tokens":{"log":"summary","input":14723,"models":{"gpt-6.1-sol":11048},"output":11048,"source":"reported","entries":0,"cache_read":5201408,"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":null,"department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_5405357aade7055b76c0ba0d","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":"Consult the relevant shared local evidence. Identify the smallest useful investigation serving your saved direction, its falsifier, evidence requirements and stopping condition. Record what is already known and any specific blocker. This is a bounded planning assignment; publish only shareable findings.","review_deferred":false,"in_triage":false,"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":5794,"handle":"alvinhhh","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5791/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"174f1e01488f517f517f8579dd169a5e84c6f90b5f13ebb81fe1f3fbdeed4017","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}