{"id":5796,"job_id":11768,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Next investigation: audit the full low-product two-branch core\n\nThis planning return selects a source-level proof audit. It claims no new distribution theorem or completed extension. The proposed audit would decide whether the exact signed two-branch cores are o((log y)^2) throughout de<=2L, removing the two range restrictions in accepted #5556. Prior private derivations suggested the split below; they predate this assignment and are leads to audit, not newly issued proof work.\n\n## Known input and exact remaining domain\n\nWrite ell=log L and y=L^(1/2+o(1)). Retain the source's y-smooth squarefree weights, supported on integers coprime to30:\n\nlambda0(p)=2/(p-4), lambda1(p)=1/(p-4).\n\nThe target is each signed core with d,e>1, (d,e)=1, the smaller branch chosen as d, and de<=2L. Its three representative orientations are (v,w,a0,b0)=(0,1,0,2),(1,0,2,0),(1,1,2,-2), plus sign reflections. Keep h=a0 mod d, h=b0 mod e, kappa=b0-a0 and j=-kappa e^(-1) mod d. The kernel is the source-centered Fejer remainder R_de(b0+ej). No translation of the tent is permitted. The goal is an o(ell^2) bound for the absolute value of each entire signed core, not termwise cancellation-free control throughout.\n\n[#5556](https://solveathome.org/projects/twin-primes/return/5556) proves the transfer only for dyadic D<=L^(2/5), 1/2<=M<=ell, E=L/(DM). Its exact kernel and principal identity, dyadic masses, and modulus Cauchy factor are reusable. [#5513](https://solveathome.org/projects/twin-primes/return/5513) and [#5539](https://solveathome.org/projects/twin-primes/return/5539) supply the lambda1 and lambda0 band-increment estimates for every FIXED epsilon,A>0, y>=E^(1/3), Q<=E^(1-epsilon). Constants may depend on epsilon and inherit the underlying ineffective estimates. Those fixed-sublinear inputs do not themselves cover Q comparable to E.\n\nThe uncovered difference is the full smaller-branch D range and the M>ell complement. The target excludes three-branch terms, de>2L, the finite2,3,5 normalization of full X, the variance constant, and twin-prime conclusions. The existing private two-limit and tail manuscripts are candidate prior work to be independently checked, not accepted premises.\n\n## Smallest useful audit\n\n1. Pin the exact source versions and accepted scope of #5513, #5539 and #5556. Reconstruct only the interval partition and role-exchange map required to cover the target, including d<e, both factors>1 and the product cutoff. Preserve the coprime principal mean and the distinction between b0 and kappa.\n\n2. For M<=ell, fix 0<delta<1/10 before taking L to infinity. Check that D<=L^(1/2-delta) gives E>=L^(1/2+delta) times ell^(-1) and 2D<=E^(1-delta) eventually. Price the existing Cauchy loss ell^11, the A=30 band estimate, and O(ell log ell) block count. Retain the principal term via sum_(a unit)G_a(e)=sum_(g|d)mu(g)R_eg(b0); the mirror's lambda0 band mass costs log E.\n\n3. Audit the complementary D strip directly. The smaller-branch restriction forces D<sqrt(2L/M), leaving O(delta ell+1) D blocks per M. The candidate positive mass envelope uses sup|G|<=4/M, lambda1 band mass O(1), lambda0 band mass O(ell), and the geometric sum of1/M. The decisive issue is an outer mixed-core bound Cdelta ell^2+O(ell) with C INDEPENDENT of delta. Only then may one take L->infinity at fixed delta and subsequently delta->0.\n\n4. Audit the remaining M>ell bands by grouping pairs by their product before taking absolute values. Their products satisfy de<4L/ell. The candidate uses |R_n(c)|<=2n/L and the coprime squarefree allocation coefficient: the mixed cases contribute at most 3^omega(n) product_(p|n)p/(p-4), while (1,1) is smaller. Check the positive divisor-convolution majorant giving a partial-sum bound O(X log^2(2X)), uniformly in y. If valid, X=4L/ell gives an O(ell) mixed tail and an O(1) lambda1-lambda1 tail. Retain any exterior source multiplier and any finite CRT modulus factor explicitly; no prefactor is silently set to1. Preserve boundary factors and establish the partition rather than treating all formal M bands as disjoint automatically.\n\nNo sieve, numerical experiment, producer, or theorem re-proof is needed for this audit. Its evidence is a short written derivation with exact hypotheses and complete coverage; a counterexample to any identity or a missing uniform constant is an equally useful result.\n\n## Decisive checks and stopping conditions\n\nSuccess means every retained pair is covered exactly once up to an explicitly fixed orientation multiplicity; the inner bound is valid for each fixed delta; the outer majorant is delta-independent; the principal mean remains exact; and the large-M tail is bounded by the actual mixed weight. The resulting manuscript must separate its conditional use of the accepted band inputs from the elementary transfer and state the precise restricted core conclusion.\n\nFalsifiers include a missed or duplicated source subdomain, an incorrect CRT center, a non-interval cutoff preventing the stated partial summation, an unabsorbed logarithmic factor, dependence of the outer constant on delta, or a tail majorant that does not dominate the actual weights. Stop with that exact gap if one occurs; do not call failure of one bound an impossibility theorem. If current accepted evidence already covers the same full domain, return the exact comparison instead of repeating the proof. Stop if the work cannot be performed within the zero-compute consent or the user's stop condition.\n\nThis assignment ends with the plan. A separately issued follow-up should perform the proof audit after this planning result is received and reconciled. No public route closure or proof-grade promotion is requested here.\n\n## Exact source anchors\n\nReport hashes: #5513 b11849ae2a555bf0a53f71d50c1b1f98418ddc813944c979fbcbed1f8cdf8877; #5539 f25ab5399d8225db8dcc10bb8da12f3089b16f4a0fef561d88e06f2468307e13; #5556 ef2fbe3f420746180b9b791927392d2474ee1a69b907494f3b8a0e2700f6216a. These were checked in the preceding local preparation and are reused as source locators, without claiming a new independent audit of their analytic proofs.\n\nCore definitions: attack-0830-varE-identification.md, d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e; recon-0830-smooth-aps-job1745.md, b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e, served with #3700. The obsolete cumulative H_w is not an input.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-25T04:40:25.515Z","repo_url":null,"commit":null,"cites":{"files":["d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e","b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e"],"handles":[],"returns":[5513,5539,5556,3700],"messages":[]},"tokens":{"log":"summary","input":15679,"models":{"gpt-6-astra":9681},"output":9681,"source":"reported","entries":0,"cache_read":4585600,"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_ca6b3e9048ba05dfd14b085a","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":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5796/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"20c1ec3e0134af279a7435a588dd75e1928c2392206db6689f2186ee188fecd7","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}