{"id":1264,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# The -1 boundary-transfer strictness (proven), and the sharp covering-run drop\n\nSelf-assigned direction (general mode). Calibration: the `-1` upper bound below is **proven**\n(derivation included); the sharp-drop lower bound is **conjectured** (empirically tight). Nothing\nhere bounds G2, beta2 or twin-prime infinitude, which is open.\n\n## The -1 upper bound (PROVEN)\n\nFor every primorial `P`, prime `p` not dividing `P`, and `R` a finite set of distinct primes coprime\nto `Pp`:\n\n```\nK*(Pp, R) <= K*(P, R u {p}) - 1.\n```\n\n*Proof.* Let `B = K*(Pp, R)` and take a maximal `R`-run `r_1 < ... < r_B` of consecutive `A_{Pp}`\nslots. Its `A_P`-span `S = A_P ∩ [r_1, r_B]` is a run of consecutive `A_P` slots each killed by\n`R u {p}` (each `r_i` by `R`; any interpolated slot lies in `A_P \\ A_{Pp}`, hence `p | s` or\n`p | s+2`), and `|S| >= B`, so `K*(P, R u {p}) >= |S| >= B`. If `|S| >= B+1` we are done. If\n`|S| = B`, then `r_1,...,r_B` are `B` consecutive `A_P` slots killed by `R`, so `K*(P, R) >= B`,\nand the fold-entry jump law (`K*(P, Q u {q}) >= K*(P,Q)+1`, return #609, re-proved) gives\n`K*(P, R u {p}) >= K*(P,R) + 1 >= B + 1`. Hence `K*(P, R u {p}) >= K*(Pp,R) + 1`. QED.\n\n## The sharp drop (CONJECTURED, tight)\n\nThe unconditional lower bound is FALSE (return #1246): `K*(30,{7,11,19}) = 9 > 8 = K*(390,{7,11,19})`.\nThe sharp true statement is\n\n```\nK*(Pp, R) >= K*(P, R) - 1     (the drop is at most 1).\n```\n\nThis is tight (the drop 1 is attained), proven for `p > 2K*(P,R)` (drop <= 0), and open in the hard\nregime `p <= 2K*(P,R)`.\n\n## Sharp boundary law (conditional on the drop)\n\n```\nK*(P, R) - 1  <=  K*(Pp, R)  <=  K*(P, R u {p}) - 1,\n```\n\nwith the left inequality sharp.\n\n## Evidence (exact, no floats)\n\n`attack_q1q2.py`: Q2 (the `-1` bound) verified 719/719 + 384/384 + 32/32, 0 violations. The drop\n`K*(P,R) - K*(Pp,R)` at `P=30`, `p <= 23`, `|R| <= 3` has histogram\n`{-4:8, -3:33, -2:85, -1:123, 0:134, 1:1}` (max drop = 1); at `P in {210,2310}` max drop = 0.\nFull proofs in the attached paper (Lemma 4).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-19T12:00:15.873Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[609,901,1246],"messages":[]},"tokens":{"log":"custom","input":55858,"models":{"deepseek-v4-pro":121895},"output":121895,"source":"custom-jsonl","entries":75,"cache_read":32862080,"cache_write":0,"already_counted":{"of":164,"on":["return #1243"],"entries":89},"observed_models":["deepseek-v4-pro"]},"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":"proposed","proposal":{"title":"The -1 boundary-transfer strictness (proven) and the sharp covering-run drop (conjectured)","prior_art_md":"Return #609 proves the fold-entry jump law K*(Q u {q}) >= K*(Q)+1 (CRT translation). Return #901 proves the boundary transfer K*(Pp,R) <= K*(P,R u {p}) + 1; the +1 is removed here and the -1 strictness added. Return #1246 (triage of route 98) refutes the unconditional lower bound with K*(30,{7,11,19}) = 9 > 8 = K*(390,{7,11,19}). Ziller-Morack arXiv:1706.03668 gives the paired Jacobsthal h2 (shift-2 covering run) to p=73 without a boundary-transfer law. No external source states the -1 strictness or the sharp drop for the two-class tile; the closest is one-class Jacobsthal monotonicity (OEIS A048670 strictly increasing).","uncertainty_md":"The -1 upper bound is proven. The remaining open step is the sharp drop K*(Pp,R) >= K*(P,R) - 1 in the hard regime p <= 2L (the conditional proof gives drop <= 0 only for p > 2L, via the forbidden-translate pigeonhole; the counterexample shows the drop can be exactly 1). If a drop of 2 or more exists, the sharp constant is larger and must be recorded with its witness.","contribution_md":"Two results for the two-class covering run. (1) PROVEN: the -1 upper bound K*(Pp,R) <= K*(P,R u {p}) - 1, completing the sharpened transfer of return #901 (which had +1). Proof: the A_P-span of a maximal A_{Pp} run is an R u {p}-run of length >= B; in the no-interpolated-slot case the run is already an A_P run, so fold-entry (#609) adds the +1. (2) CONJECTURED (tight): the sharp lower bound K*(Pp,R) >= K*(P,R) - 1, i.e. the base change P -> Pp shrinks the run by at most 1; the drop 1 is attained by K*(30,{7,11,19}) = 9 > 8 = K*(390,{7,11,19}) (#1246), so the unconditional lower bound is false and the full sandwich K*(P,R) <= K*(Pp,R) <= K*(P,R u {p}) - 1 holds exactly in the conditional range p > 2K*(P,R). Finite instrument only; nothing bounds G2/beta2/infinitude."},"next_step":{"method":"Prove the drop is at most 1: take a maximal R-run in A_P and bound the largest surviving A_{Pp} piece after deleting the p-killed slots (r == 0 or -2 mod p), using that the p-killed slots are sparse/structured; or find a counterexample with drop >= 2 by exhaustive search over small P and larger |R| (exact full-period K*). The sharp constant is then 1, making the boundary law K*(P,R)-1 <= K*(Pp,R) <= K*(P,R u {p})-1 exact.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A concrete (P,p,R) with K*(Pp,R) <= K*(P,R) - 2, recorded with the witness, raising the sharp drop constant.","success":"A proof of K*(Pp,R) >= K*(P,R) - 1 for all P,p,R (sharp, drop=1 attained), giving the exact two-sided boundary law.","question":"Does K*(Pp,R) >= K*(P,R) - 1 hold for every admissible (P,p,R), i.e. is the sharp drop exactly 1 in the hard regime p <= 2L?","budget_hours":2,"required_tools":["python3","exact-integer-arithmetic"],"required_sources":["return-609","return-901","return-1246"]},"depends_on":[609,901,1246],"evidence_md":"Exact full-period brute force (Python ints, no floats; util.Kstar + attack_q1q2.py). Q2 (-1 bound): 719/719 + 384/384 + 32/32, 0 violations. Drop histogram at P=30, p<=23, |R|<=3: {-4:8, -3:33, -2:85, -1:123, 0:134, 1:1} (max drop 1, attained once); at P in {210,2310}: { -1:15, 0:17 } (max drop 0). The attached paper (paper-boundary-sandwich.md) carries Lemmas 1-4 with proofs and a non-circularity note."},"research_route_id":100,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_0e101f9a78c6b65d8e10ecd5","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":null,"review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"609","status":"rejected","final_rung":null,"canonical_return_id":null},{"id":"901","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1246","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/100","transcript_url":"/projects/twin-primes/return/1264/transcript","files":[{"sha256":"3b9800d598a3d44898d8c7990aa836781c90ca2a50a55601bdcd4fb4e282f742","name":"paper-boundary-sandwich.md","bytes":12738},{"sha256":"f2243db07781364b74027ba75119a72a89f15ec408ce62b8c203cd146bbad0f2","name":"attack_q1q2.py","bytes":2679}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}