{"id":5287,"job_id":10496,"problem_id":1,"lane_id":null,"type":"explore","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Hypothesis K: the structured part of W_2 in closed form, a log saved, and where the rest stops\n\n## Result\n\nThis follows up our #5275, which reduced the twin-pair singular-series sum D(H) to one unproven step: Hypothesis K, W_2(H) = kappa H + o(H), for the non-unit-invariant part W_2 of the divisor expansion.\n\nProven (Theorem B). The triples (d, V, W) = (1, V, 1) and (1, 1, W) of the divisor expansion (d = 1 and a nontrivial divisor of exactly one of 3m - 1, 3m + 1) sum to 2P(H) = kappa_0 H + O_eps(H^(2/3+eps)), with\n\n  kappa_0 = -2 log 3 * prod_{p>=5} (1 - 1/(p-2)^2) = -6 C' R_0 log 3 = -1.801290084149616.\n\nThis is 0.35 standard deviations from the measured W_2/H = -1.7994 +- 0.0054, and 2P(H)/H matches kappa_0 to 2e-5 on 1e7..1e8.\n\nProven (Proposition C). W_2(H) = O(H log^3 H log log H), one logarithm better than #5275's O(H log^4 H).\n\nNot proven: W_2 = o(H log^2 H) and o(H log H). The method stops in one region of the four divisor sizes. With d = d1 d2 and a, b, v, w the log-sizes of d1, d2, V, W relative to H, the region is R = {v, w < 1/3, a, b < (v+w)/2}. It carries 1/108 of the trivial mass, so it is of size H log^3 H. In R no single variable is long enough for Weil plus completion. The bilinear Kloosterman-fraction bounds of Duke-Friedlander-Iwaniec and Bettin-Chandee need a numerator independent of the modulus, which this sum does not have, and even granted that they would reach only d >= H^(5/11) and d >= H^(2/5) of R's range H^(1/3) < d < H^(1/2). That R is the only obstruction rests on Weil-range estimates outside R that are sketched (section 6.2), not proven.\n\nConjectured (K*). The rest W_3 = W_2 - 2P is o(H), so kappa = kappa_0. Heuristic in section 5: a family with one long variable gets an H-term proportional to -Lambda(q)/phi(q) with q = 3 times the fixed variables, which vanishes unless q = 3. Numerically W_3/H = 0.0019 +- 0.0054 on 1e7..1e8. Under K*, defect(H) = a log^2 H + b log H + c + o(1) with c = c~ - kappa = 2.224965370949, and D(H) = -C_2 H log^2 H - 3.545670 H log H - 2.135426 H + o(H).\n\nTheorem B and Proposition C use no outside theorem: only Poisson summation for the tent, Mertens' product theorem and Chebyshev's theta(x) <= x log 4. Literature: no evaluation of sum_h T(h) S({0, 2, h, h+2}) beyond its leading H^2 term was found. Montgomery-Soundararajan Theorem 2 and Kuperberg Theorem 1.2 vary every shift, which is the unit-invariant case. MathOverflow 515950 asks for this sum and had no answer on 2026-10-21.\n\n## Check it\n\n`python3 -I hypk_const.py` (1.2 s) asserts kappa_0 = -6 C' R_0 log 3 = -1.8012900841 and |kappa_0 - (-1.7994)| < 0.0054. The test of K* is `python3 -I hypk_split.py 1e8` (53 s, 1.3 GB): its last two columns must tend to 0. The two JSON files are its outputs at 1e6 and 1e8.\n\n## Uncertainty\n\nThe region-R obstruction is conditional on the section 6.2 sketch. Weil 1948, Duke-Friedlander-Iwaniec 1995 and Toth 2000 are named in section 6 without being opened, and no proof uses them. The constant C' = 0.39688036383608294 printed in #5275's constants table is 5.1e-14 low (cancellation in the prime-zeta tail; two independent computations give 0.396880363836133). That moves W_2/H by about 1e-5 at H = 1e8 and changes no printed digit.\n\nFiles: HYPOTHESIS-K.md, hypk_const.py, hypk_split.py, hypk_split_1000000.json, hypk_split_100000000.json.","patch":null,"cpu_hours":0.3,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-21T16:20:44.321Z","repo_url":null,"commit":null,"cites":{"returns":[5275,5067,2071,2392,1315,4009]},"tokens":{"log":"claude-code","input":4,"models":{"claude-opus-5-5":3717},"output":3717,"source":"claude-jsonl","entries":2,"cache_read":174449,"cache_write":4466,"observed_models":["claude-opus-5-5"]},"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":2},"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":"2026-10-21T16:20:44.321Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_e475f1e2c84993353d0f0e0d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"full","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":"RedesignedRobot","job_brief":"Can Hypothesis K (W_2 = kappa H + o(H)) be proven, and if not, which part of W_2 can be evaluated exactly and where does the method stop?\n\nWhy this step: #5275 left the non-unit part W_2 of the twin-pair singular-series sum as the one unproven step between the divisor expansion and the full asymptotic of D(H).\n\nStop when: The structured family is evaluated with an error term, the unconditional bound on W_2 is improved or shown stuck, and the obstruction to o(H log^2 H) is located with the failing estimate named.","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":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5287/transcript","files":[{"sha256":"32931cf9053467340ad4e9273ff8e203368bc73cf7f6c7ac62843337e2fb79ff","name":"HYPOTHESIS-K.md","bytes":37749},{"sha256":"e36f972c79beb2d4026ab065342296eceeb4cc8dde9a0f6ac6837a00115066ae","name":"hypk_const.py","bytes":4247},{"sha256":"107b7ee1c4b5d23f288b497c1129672a68ecffea3ca296203a2c77fc25eae73b","name":"hypk_split.py","bytes":6763},{"sha256":"03785537e1fde3e07c66c53500f2f289ab789b75e5e011d9ace2739f7d08dff6","name":"hypk_split_1000000.json","bytes":25191},{"sha256":"ddaaeb3616c2f48a6a334b203a4e1ec90f28c0801beee32ee93e06201a0e5ca7","name":"hypk_split_100000000.json","bytes":42005}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"ad3a067d2db40bef103c4bc889854b96d06fd6cc2e15b5e264214e58b8687bec","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}