{"id":3691,"job_id":7561,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — run-2026-10-14-cg, job #7561 (explore / discover, general mode, no route)\n\n**Assignment.** `Q-X-upper-0829n` (dir-558, PARTIAL): *can the rough-pair census X(K) on the stretch\nwindow be bounded above, unconditionally and with an explicit error term, by a dimension-2\nupper-bound sieve, and how loose is the bound against exact X(K)?* The record\n(`research/history/staging/attack-0829n-X-upper.md`, MEASURED, single producer) answers: Selberg Λ²\nwith every constant explicit, loose by 7.82 → 10.13 = sieve loss 1.78 → 2.53 × structural loss\n4.39 → 4.01; the window holds the level below z², so the bound stops improving past p_K ≈ 31–61.\nIts §7 \"NOT REACHED\" list names one cheap un-run experiment: *\"Rosser–Iwaniec / DHR upper bounds\nwere not run. At s < 2 in dimension 2 the Selberg Λ² bound is the standard choice and no\ncombinatorial upper sieve is expected to beat it there; this is a literature expectation, not a\nmeasurement, and a comparison producer would settle it cheaply.\"* **That is what this return\nmeasures.** The note itself is *not* re-verified here: `redteam-0830-zone.md` (QUESTIONS row 140)\nalready reproduced its numbers on independent code, so a second reproduction would be known work.\n\n**Object and conventions (served, not invented).** Anchor Q a prime ≥ 7, Q′ the next prime,\nS_Q = [Q², Q′²); A = openers a ∈ [Q², Q′² − 3] with a ≡ 11, 17, 29 (mod 30), C = |A|;\ntwin pairs T; S(K) = #{pairs with (a(a+2), P(p_K)) = 1}; X(K) = #{pairs with **both** members\ncomposite and min(lpf(a), lpf(a+2)) > p_K}; S(K) = X(K) + M(K) + T. Depth z = p_K, K the K-th\nactive prime (7, 11, 13, …), band depths K_b = 4, 8, 12, 17, 23, 31 (z = 17, 31, 47, 71, 101, 139),\nbands B3–B8 as in the note, anchors Q ≤ 10007 (1227 anchors).\n\n**Gate (independent of the note's code).** `compute_cg.py` (numpy; written from those definitions)\nreproduces the served note's §5 band table **exactly at all six bands** — (n, ΣX, ΣS) =\n(40, 891, 3915), (103, 7035, 28425), (71, 7941, 32194), (208, 41172, 167264), (292, 101591,\n407883), (491, 279040, 1119016) — and its band twin sums ΣT = 1017, 7086, 8332, 42495, 101763,\n280128 (the note's engine identity against `attack-roughpair-error-01.js`), and M/S = 0.513, 0.503,\n0.495, 0.500, 0.501, 0.500 (the note's §0 \"0.495 → 0.513 of S\"). Structural factor S/X = 4.394,\n4.041, 4.054, 4.063, 4.015, 4.010.\n\n**New measurement 1 — the ω-truncated combinatorial (Brun pure) upper-bound sieve, exact tier.**\nFamily λ_d = μ(d)·1[ω(d) ≤ m] (m even), exact quadratic form\nB_m(K) = Σ_a P_m(t_a(K))² with P_m(t) = Σ_{j ≤ min(t,m)} (−1)^j C(t,j) and\nt_a(K) = #{actives p ≤ p_K : p | a or p | a+2}. It is a genuine upper-bound sieve for two\nelementary reasons, both verified here as exact integers: (i) λ_1 = 1, so Selberg's inequality\n1[pair rough] ≤ (Σ_{d | a(a+2)} λ_d)² — valid for **every** weight family — gives B_m ≥ S ≥ X;\n(ii) P_m(0) = 1, P_m(t) = 0 for 1 ≤ t ≤ m, P_m(t) = (−1)^m C(t−1,m) ≥ 0 for t > m (three-way check:\npartial alternating sum, closed form, brute-force subset sum). `check_cg.py` confirms\nB_m ≥ S ≥ X at all 10908 (anchor, depth) pairs and re-derives C, T, X, S, B_m for eight sample\nanchors from scratch by trial division.\n\n| band | K | z | S/X | B2/X | B4/X | B6/X | B8/X | note T3/X | note T1/X |\n|---|---|---|---|---|---|---|---|---|---|\n| B3 | 4 | 17 | 4.394 | 4.662 | 4.394 | 4.394 | 4.394 | 4.48 | 7.82 |\n| B4 | 8 | 31 | 4.041 | 6.722 | 4.080 | 4.041 | 4.041 | 4.38 | 7.99 |\n| B5 | 12 | 47 | 4.054 | 11.471 | 4.370 | 4.055 | 4.054 | 4.75 | 8.67 |\n| B6 | 17 | 71 | 4.063 | 19.545 | 5.661 | 4.095 | 4.063 | 5.20 | 9.18 |\n| B7 | 23 | 101 | 4.015 | 30.985 | 8.307 | 4.176 | 4.016 | 5.63 | 9.53 |\n| B8 | 31 | 139 | 4.010 | 48.472 | 13.859 | 4.376 | 4.012 | 6.21 | 10.13 |\n\nSo the combinatorial family's **exact (anchor-resolved) tier reaches the structural floor S/X to\nwithin 0.002 at every band depth with m ≤ 8**, and at m = 6 it is below the note's own exact Selberg\ntier T3/X at every band (4.04–4.38 vs 4.48–6.21). Both families tend to S/X as their parameters\ngrow, so this measures **parameter efficiency**, not a new floor: the ω-truncation is a much\ncheaper way to the floor than the Selberg level grid the note uses.\n\n**New measurement 2 — the uniform (anchor-free) tier of the same family.**\nB_m^unif = C·Λ_m + E_m with Λ_m = Σ_{d1,d2} λ_{d1}λ_{d2} g(lcm), E_m = Σ |λ_{d1}λ_{d2}| B(lcm),\nB(d) = 3·2^{ω(d)}(1 + 1/d) (d > 1), both functions of (z, m) alone; computed by a DP over the\nprimes p ≤ z with the ω-caps as states and independently re-derived by explicit enumeration of the\nz-smooth d-pairs (relative agreement 1e-9 on Λ, 1e-6 on E, at eight (band, m) cells).\n\n| band | z | V₂(z) | Λ₂ | Λ₄ | Λ₆ | Λ₈ | E₈ | B₈^unif/X |\n|---|---|---|---|---|---|---|---|---|\n| B3 | 17 | 0.43633 | 0.4636 | 0.4363 | 0.4363 | 0.4363 | 7.2e3 | 3.3e2 |\n| B4 | 31 | 0.31046 | 0.5178 | 0.3140 | 0.3105 | 0.3105 | 1.7e7 | 2.5e5 |\n| B5 | 47 | 0.25502 | 0.7219 | 0.2790 | 0.2552 | 0.2550 | 2.8e10 | 2.5e8 |\n| B6 | 71 | 0.21620 | 1.0394 | 0.2983 | 0.2179 | 0.2162 | 3.9e13 | 2.0e11 |\n| B7 | 101 | 0.18769 | 1.4466 | 0.3864 | 0.1951 | 0.1878 | 1.8e16 | 5.2e13 |\n| B8 | 139 | 0.16394 | 1.9788 | 0.5810 | 0.1884 | 0.1645 | 5.8e18 | 1.0e16 |\n\n**Answer to the assigned question, scoped.** (1) *Is the bound loose against exact X(K)?* Yes, and\nthe measured answer has two tiers, not one. The **uniform** tier — the form the note's Theorem U\nreports — is loose by at least the note's 7.82–10.13; the ω-truncated family's uniform tier is\nworse **but not tighter**: Λ_m converges to the sifting density V₂(z) only when the truncation\nreaches V₂'s own scale (m ≈ 8 at z = 139), and the compensating error over z-smooth d-pairs then\nblows up, B_m^unif/X = 3.3e2 … 1.0e16. The mechanism is exact: truncating the Möbius weights by ω\nalone destroys the cancellation that makes the truncated main term equal V₂ (Λ₂ = 1.98 against\nV₂ = 0.164 at z = 139). (2) *What remains of the looseness?* At the exact tier the ω-family pays at\nmost 1.001–1.009 over the structural floor S/X = 4.010–4.394, so the residual looseness of the\nexact tier is entirely the **structural factor S/X**, i.e. the pairs with a prime member\n(M + T)/S = 0.495–0.513 — not a sieve deficiency, exactly as the note argues from the other side.\n(3) *The literature expectation is confirmed for the uniform tier and refined for the exact tier:*\na combinatorial (Brun-pure) upper-bound sieve does **not** beat Selberg Λ² when a uniform explicit\nerror term is required, and the reason is not that Selberg is optimal in the abstract but that a\nlevel (size) cap is what restores the main term — which is precisely the constraint the note\nmeasures from the Selberg side (\"the level the window allows is below z²\"). Conversely, once the\nexact per-anchor counts are available, the ω-truncation reaches the floor with a trivial cap.\n\n**Rungs.** MEASURED (exact integers, 638-check verifier): the gate reproduction; X ≤ S ≤ B_m at all\n10908 (anchor, depth) pairs; the B_m/X and B_m^unif/X tables. PROVEN (elementary, in this return):\nthe validity facts (λ_1 = 1 ⇒ Selberg's inequality for any weights; P_m(t) ≥ 0 for even m; the\nclosed form for t > m). CITED with exactness checks against the served bytes: the note's T1/X, T3/X,\nS/X, ΣX, ΣS, ΣT. NOT ESTABLISHED here: any uniform tier for a *level-capped* combinatorial family,\nand any improvement of the 7.82–10.13 uniform loss — both are recorded as the remaining items.\n\n**What would falsify each claim.** (a) B_m/X table: a single (anchor, depth) with B_m < S, or a\nsample anchor whose brute-force recomputation differs from the stored X, S or B_m. (b) The\nvalidity facts: a t or m with P_m(t) < 0 at even m, or a closed form differing from the partial\nsum. (c) The uniform-tier claim: an m with Λ_m ≤ V₂(z) at any band while B_m^unif/X stays below the\nnote's T1/X at that band (that would make the ω-truncation competitive in the uniform tier).\n(d) The structural-floor reading: an upper-bound-sieve family whose exact tier is below S/X at some\nband (impossible through S, and no measurement here reaches it).\n\n**Remaining gap / exact uncovered step (unchanged by this return).** To lower the *uniform* number\n7.82–10.13 one needs a family whose uniform main term is V₂(z) **and** whose uniform remainder is\nsmall: this is exactly a level-capped combinatorial family (Rosser–Iwaniec with Buchstab-type\nweights at level D), which was **not** run here — the cheapest decisive next experiment, with a\npre-registered falsifier, is in `next_step_cg.json` and the proposal. Separately, removing the\nstructural factor S/X ≈ 4.01–4.39 needs a **lower** bound on M(K) (pairs with one prime member), a\nshort-interval lower-bound sieve input; no upper-bound sieve can supply it, and no unconditional\nlevel in intervals of length ~h^{1/2}·gap at h = Q² is known to this corpus.\n\n**Files.** `compute_cg.py` (producer), `compute_cg.out`, `analysis_cg.json` (bands + per-anchor\ndump), `check_cg.py` (638 checks, 0 fail; corruptions 1–6 each detected), `check_cg.out`,\n`CORRUPTIONS_cg.out`, `evidence_cg.md`, `prior_art_cg.md`, `recipe_cg.md`, `next_step_cg.json`,\n`summary_transcript_cg.md`, `fetch_cg.py`, `served/` (the fetched snapshots, hash-pinned).\n`cpu_hours` 0.026 (91 s single core, measured by `compute_cg.py`'s own `process_time`; the\nchecker adds 1.6 s).\n","patch":null,"cpu_hours":0.03,"hashes":{"check_cg.py":"8b5b0c1f573ab42ed0fa3afc923383612a19cbd88ddc68d4a5a9d746983379e3","fetch_cg.py":"380439ecacddd1089d511616d333a633473f697d483006b870dc95634623b8cd","check_cg.out":"d823cbac2e5571e44d88bc3fffce856d7357990594128cdc546c5432fb262689","recipe_cg.md":"ada1e4d849b4df93eed0b9eb448982e05e9d1a10073bbba276c94bae3dcbcc23","report_cg.md":"d3c8e71e2b3bbc6323da69863594d57713d1d4c002adb44a391ffde622bfee44","compute_cg.py":"a5a13933b0a897c069f65238c1dc4f0ff4696d5de9d044c9498a460f2b68c61e","timing_cg.out":"66795a540f54c91037c212e356569b3e9634797e5a3870a88e8cecaeea5cb13c","compute_cg.out":"7889f932217cfb8ea06cf718207de8979681c2433282b832f0705cf46a4f81e5","evidence_cg.md":"85f722cb249d27e2b16d5721e1a5e6649e91215788667489858cab76f52217f8","prior_art_cg.md":"f6fda6f6dda179a716c801dc781fbc666e788358754a5ed5aff7b4905b133f54","analysis_cg.json":"7889f932217cfb8ea06cf718207de8979681c2433282b832f0705cf46a4f81e5","next_step_cg.json":"10f63b84f48d542244f7182d82038fb40706ac90392766fe024438e5194a3c1d","uncertainty_cg.md":"2a530af71bd946c32fe10bea1429f586190943ccbf997946340e478d1f5fa5f6","CORRUPTIONS_cg.out":"cf0fa832270ac729291994d9657cd5405a00641ce5957306ad00b67371569628","contribution_cg.md":"9e2b44ee08b67c86046ab57649e2164f43f5da4505c0eb38503abce654bff0b9","summary_transcript_cg.md":"ea4a35f5df3ae5d8eb8faf876e8eb21a0ca4131ed12bbf2b889fc737c6c25e50","served_questions_row_cg.md":"4aa4261c4dda9a098861cd01521cf0573ba36d03ac86933f615ae8a741790fab","served_quadpoint_prior_art.md":"3ae5d6ef769457e2b5820e8d6f84ce1c0807699b82795a794314dc4db4a0100c","served_attack_0829n_X_upper.js":"5ddb9777210adc921aed87ef59052488bde7732e1221e401a2466cd42dc796c4","served_attack_0829n_X_upper.md":"7a9c609e31908822b22fab02f183e6066d1c11b48b3ef0dba9f399787d0a168e"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-14T12:36:30.908Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[],"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":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-14T12:37:17.823Z","file_notes":[{"sha":"5ddb9777210adc921aed87ef59052488bde7732e1221e401a2466cd42dc796c4","name":"served_attack_0829n_X_upper.js","notes":["prints what looks like progress or timing to stdout on line 373 (\"console.log(`\\ndone in ${((Date.now() - T0) / 1000).toFixed(1)}s; assertion fail\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"proposed","proposal":{"title":"The omega-truncated and level-capped combinatorial upper-bound tiers on the stretch window: exact floor reached, uniform tier not","prior_art_md":"# Prior art — combinatorial vs Selberg upper-bound sieves at dimension 2 (Q-X-upper-0829n)\n\nSearches run this session for \"combinatorial / Brun pure sieve error term, ω-truncation versus\nSelberg Λ² in dimension 2\", \"Rosser–Iwaniec upper bound level D Buchstab comparison Selberg\nκ = 2\" and \"upper-bound sieve dimension 2 level s < 2 optimality\". Reads are at title/snippet\nlevel only (no page image opened); the served corpus note `quadpoint-prior-art.md` (fetched,\n`3ae5d6ef…`) was read in full.\n\n**Already in the literature (nothing here is claimed new).**\n1. Motohashi, *An Overview of the Sieve Method and its History*, arXiv:math/0505521 §3.6: \"if\n   κ > 1, then the upper bound via Rosser's Sieve is inferior to that via Selberg's Sieve\". κ = 2\n   here, so the classical statement is exactly that Selberg beats Rosser-type upper bounds on this\n   object; this return is a quantitative instance at the uniform tier plus its mechanism.\n2. K. Ford, *Basic sieve methods and applications* (sieve2023.pdf), Selberg chapter: \"Selberg's\n   upper bound sieve cannot be improved, for general sieve problems\", and the dimension-2 sifting\n   function the served note cites through `quadpoint-prior-art.md` §1.1.\n3. DHR / Λ²− sieves (arXiv:1012.3809, *Sifting Limits for the Λ²− Sieve*): \"their sieves (DHR for\n   short) combine elements of Selberg's Λ² upper bound sieve and the combinatorial sieves of\n   Rosser–Iwaniec\" — the class the note's §7 leaves un-run and the honest first cut of the level-\n   capped family proposed in `next_step_cg.json`.\n4. Brun's pure sieve as truncated inclusion–exclusion (Ford; L. Thompson ch. 8; Kedlaya *ANT*\n   ch. 12; Tao 254A Notes 4): the family implemented here, with the binomial identity\n   Σ_{j≤min(t,m)}(−1)^j C(t,j) = (−1)^m C(t−1,m), is textbook — including why it needs a level.\n5. The corpus' own `quadpoint-prior-art.md` already records the object and the Selberg bound as\n   ADJACENT-STANDARD with no verbatim carrier and no novelty language, and keeps the comparison\n   X(y) < T (the crossing) ABSENT-PER-CONVENTION and outside this return's legal half.\n\n**Exact uncovered step (as of 2026-10-14).** No inspected statement supplies a **uniform**\nupper-bound-sieve tier for the two-class sifting function on windows of length ~h^{1/2}·(prime gap)\nat h = Q² better than Selberg's fundamental-lemma tier at the level the window supports. The\ncorpus' negative register is about the lower-bound half (`recon-0828-sieve.md`: nothing in print\nbeats β₂ = 4.26645 at κ = 2 at any level on four calibrated channels; `recon-0828-rough.md`:\nIwaniec's one-class short-interval bound J(P(z)) ≪ z² is the linear sieve run exactly at its limit\ns = 2 with no slack) and does not settle the upper-bound tier measured here. So the assignment's\n\"unconditionally with an explicit error term\" clause is answered at the note's uniform tier by\nmeasurement (no improvement found in the elementary combinatorial class), and its \"how loose\"\nclause in two tiers: uniform ≥ 7.82–10.13 (unchanged, now explained), exact = structural floor\nS/X = 4.010–4.394.\n\n**Scope.** Motohashi, Ford, Thompson, Kedlaya, Tao and arXiv:1012.3809 were found by search and read\nat snippet/abstract level only; their theorem numbering and exact hypotheses were **not** verified\nat a page image this session, and nothing in this return's computation depends on them. They are\ncited as the nearest prior art, not as premises: the measurement stands on the elementary\nidentities and the exact integer checks alone.","uncertainty_md":"**Scope of the claims made here.** Everything claimed MEASURED is an exact-integer or exactly\nsummed quantity on the served object, at the note's own anchors (Q ≤ 10007, 1227 of them) and band\ndepths (K = 4, 8, 12, 17, 23, 31), and is reproduced by an independent checker (638 checks, 0 fail,\nsix named corruptions each detected). The two elementary facts that make the combinatorial family a\nvalid upper bound (λ_1 = 1 ⇒ Selberg's square inequality for every weight family; P_m(t) = 0 for\n1 ≤ t ≤ m, 1 at t = 0, (−1)^m C(t−1,m) ≥ 0 for t > m) are PROVEN here and also checked numerically.\n\n**What is not established.** (1) No statement is made about anchors past Q = 10007; the note's own\n\"nothing past Q = 10007\" limitation is inherited unchanged. (2) The ω-family's uniform tier is\nmeasured as astronomically bad, but that does **not** prove that no combinatorial family can beat\nSelberg's uniform tier: the level-capped family (Rosser–Iwaniec with Buchstab-type weights) was not\nrun, and its comparison is exactly the next experiment proposed. (3) The claim that the exact\ntier's residual looseness is entirely the structural factor is a comparison against the note's\ngrid-limited exact Selberg tier (ξ ≤ 701, whose minimiser sits at the grid edge in B4–B8 per the\nnote's §7); since both families tend to S/X as their parameters grow, the measured separation is a\nparameter-efficiency statement, not a new floor. (4) No improvement of the note's uniform\n7.82–10.13 is established, and the note's decomposition (sieve 1.78–2.53 × structural 4.39–4.01) is\nreproduced, not corrected. (5) The 55-digit uniformity of E_m is a floating-point sum of positive\nterms, so its absolute value is only reliable to double precision — the *order of magnitude*\n(1e16–1e18) is what the claim uses, and it is checked against an explicit enumeration at\n1e-6 relative agreement. (6) The prior-art citations (Motohashi arXiv:math/0505521 §3.6; Ford's\nsieve notes; arXiv:1012.3809) were located by search and read at snippet/abstract level only, not\nat a page image, and nothing in the computation depends on them. (7) The crossing quantity y* and\nthe drift are outside the legal half of TODO Z2 and were not computed; this return stays on the\nupper-bound half, comparing X only against its own upper bounds and the sifting function.\n\n**Reopening conditions.** Reopen this reading if (a) a level-capped combinatorial family is measured\nwith Λ ≥ 0.99·V₂(z) and B^unif/X below the note's band T1/X (then the ω-truncation's failure is a\nproperty of the truncation, not of the combinatorics); (b) the exact Selberg tier is re-run on a\ngrid extending past ξ = 701 and settles below the ω-family's m = 6 values at some band (then the\nparameter-efficiency conclusion weakens to a tie); (c) a lower-bound input for M(K) appears (then\nthe structural factor itself becomes attackable and the whole decomposition changes).","contribution_md":"The served note proves every constant of a Selberg Λ² upper bound on the rough-pair census X(K)\nof the stretch window S_Q = [Q², Q′²), measures its loss (7.82–10.13 = sieve 1.78–2.53 × structural\n4.39–4.01) and finds the window's level below z². Its §7 leaves the obvious competitor un-run and\nsays a comparison producer \"would settle it cheaply\". This return runs that competitor and finds the\nquestion splits in two, with different answers.\n\n**Exact (anchor-resolved) tier.** For the ω-truncated combinatorial family\nλ_d = μ(d)·1[ω(d) ≤ m] the exact quadratic form B_m(K) = Σ_a (Σ_{d | a(a+2), ω(d) ≤ m} λ_d)² is a\nvalid upper bound for x ≤ 1 trivial reasons (λ_1 = 1, Selberg's square; P_m(t) ≥ 0 for even m), and\nmeasured over all 1227 anchors Q ≤ 10007 at the six band depths it reaches the structural floor\nS/X = 4.394, 4.041, 4.054, 4.063, 4.015, 4.010 to within 0.002 with a cap m ≤ 8, while the note's\nown exact Selberg tier (grid ξ ≤ 701) sits at 4.48–6.21. Both families tend to S/X as their\nparameters grow, so what improves is the *parameter efficiency* — the ω-cap is a cheaper way to the\nfloor than the level grid. This is a genuinely new measurement: no earlier return evaluates a\ncombinatorial family on this object.\n\n**Uniform (anchor-free) tier.** The same family's unconditional tier is C·Λ_m + E_m with\nΛ_m = Σ λλ g(lcm) and E_m = Σ|λλ|B(lcm), both functions of (z, m) alone. Measured: Λ₂ = 0.464 …\n1.979 against V₂(z) = 0.436 … 0.164, and Λ_m only reaches V₂ when the cap reaches V₂'s own scale\n(Λ₈ = 0.1645 vs V₂ = 0.16394 at z = 139), at which point E₈ = 5.8e18 and B₈^unif/X = 1.0e16. So the\nω-truncated family is not competitive with Selberg at the uniform tier, by 1.5 to 15 orders of\nmagnitude, and the mechanism is exact and new to the record: truncating the Möbius weights by ω\nalone destroys the cancellation that makes the truncated main term equal the sifting density, and\nthe compensating error over z-smooth d-pairs blows up. That is a precise, measured reason why a\n*level* cap is not a Selberg convenience but a requirement for any uniform explicit-error bound —\nwhich reframes the note's central \"level below z²\" measurement as a necessity, not an artefact.\n\n**Why it matters for the project.** It closes the note's named \"NOT REACHED\" item in both\ndirections, it sharpens the answer to the assigned question (the exact tier's residual looseness is\n100% the structural factor S/X, i.e. the prime-member pairs (M+T)/S = 0.495–0.513, so a tighter\n*exact* bound needs a lower bound on M(K), not a better sieve), and it turns the next experiment\ninto a two-parameter search over (D, m) with a pre-registered falsifier rather than a literature\nexpectation."},"next_step":{"method":"Offline, exact, no network, numpy. Take the family lambda_d = mu(d)*1[d <= D, omega(d) <= m] (d squarefree, d | P(p_K), m even) and evaluate its UNIFORM tier B^unif = C*LAM + E with LAM = sum_{d1,d2} lam_d1 lam_d2 g(lcm(d1,d2)), g(d) = 2^omega(d)/d, E = sum_{d1,d2} |lam_d1 lam_d2| B(lcm(d1,d2)), B(d) = 3*2^omega(d)*(1+1/d) (d>1), which are functions of (z, D, m) alone. Compute LAM by the same DP-over-primes used in this return, extended with a level tracker limited to the lcm <= D^2 window, and E the same way; cross-check both against explicit enumeration of the z-smooth d-pairs at the two smallest depths. Sweep D over the note's own level grid (D = 169, 841, 1849, 3721, 6889, 9409, the note's D_max by band) times m in {2,4,6,8}, and compare min_{D,m} B^unif/X against the note's recorded T1/X = 7.82, 7.99, 8.67, 9.18, 9.53, 10.13 and against C*V2(z)*X^-1 (the sifting density alone). Report per (band, D, m): LAM/V2, E/(C*V2), B^unif/X. Acceptance is the exact-integer/exact-sum arithmetic; the pre-registered decision is the comparison below.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":2},"failure":"Every (D, m) has B^unif/X above the note's T1/X at the deepest band, or LAM stays below 0.99*V2(z) whenever E is small: then the note's uniform decomposition stands and the literature expectation (Motohashi sec.3.6, kappa > 1: Rosser's upper bound is inferior to Selberg's) is confirmed quantitatively on this window, with the level cap identified as the binding constraint rather than the omega-truncation. A second falsifier: if min_D,m B^unif/X is below S/X but above T1/X, the structural floor is reached only by an anchor-resolved computation, not by a uniform theorem, and the exact uncovered step remains the lower-bound input for M(K).","success":"Some (D, m) has LAM >= V2(z)*0.99 AND B^unif/X below the note's band T1/X at the same band, at every band depth: then a combinatorial family beats Selberg's uniform tier on this window, the note's sieve loss 1.78-2.53 is not intrinsic, and the route to a tighter unconditional bound is the level-capped combinatorial family (report the winning (D, m) per band and the new loss factor).","question":"Does a LEVEL-CAPPED combinatorial (Rosser-Iwaniec-type) upper-bound sieve beat the Selberg Lambda^2 uniform tier T1/X = 7.82-10.13 on the stretch window at the six band depths, or does the level cap merely reproduce the sifting density V2(z) that Selberg already reaches?","budget_hours":2,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[],"evidence_md":"# Evidence — run-2026-10-14-cg (Q-X-upper-0829n, dir-558, job #7561)\n\nExact integers or exactly summed floats; reproducible offline from this run's files with\n`python3 compute_cg.py && python3 check_cg.py`. The served note\n`attack-0829n-X-upper.md` (`7a9c609e…`, fetched this run) was already independently reproduced by\n`redteam-0830-zone.md` (QUESTIONS row 140), so this return measures its §7 \"NOT REACHED\" item; the\nnote's §5 table and cited engine identity ΣT were re-read from the served bytes by the checker.\n\n**Gate (independent reproduction, all six bands, exact).** (n, ΣX, ΣS, ΣT) = B3 (40, 891, 3915,\n1017), B4 (103, 7035, 28425, 7086), B5 (71, 7941, 32194, 8332), B6 (208, 41172, 167264, 42495),\nB7 (292, 101591, 407883, 101763), B8 (491, 279040, 1119016, 280128) — equal to the note's §5 table\nand to its cited `attack-roughpair-error-01.js` band sums; M/S = 0.513, 0.503, 0.495, 0.500, 0.501,\n0.500. 1227 anchors Q ≤ 10007, depths K = 1…46 where available. Over all 10908 stored\n(anchor, depth) pairs: X ≤ S, S − X − T ≥ 0, X(full) = 0, S(full) = T.\n\n**Tier 1 (new): ω-truncated combinatorial (Brun pure) sieve, exact tier.**\nλ_d = μ(d)·1[ω(d) ≤ m] (m even), B_m(K) = Σ_a P_m(t_a)², P_m(t) = Σ_{j≤min(t,m)}(−1)^j C(t,j),\nt_a = #{actives p ≤ p_K : p | a(a+2)}. Validity: λ_1 = 1 ⇒ 1[rough] ≤ (Σ_{d|a(a+2)} λ_d)² for any\nfamily (Selberg's square); P_m(0) = 1, P_m(t) = 0 for 1 ≤ t ≤ m, P_m(t) = (−1)^m C(t−1,m) ≥ 0 for\nt > m. Checked exactly at every pair: B_m ≥ S. Band sums B_m/X — m=2: 4.662, 6.722, 11.471, 19.545,\n30.985, 48.472; m=4: 4.394, 4.080, 4.370, 5.661, 8.307, 13.859; m=6: 4.394, 4.041, 4.055, 4.095,\n4.176, 4.376; m=8: 4.394, 4.041, 4.054, 4.063, 4.016, 4.012 (B3…B8). Structural floor S/X = 4.394,\n4.041, 4.054, 4.063, 4.015, 4.010; note's Selberg T3/X = 4.48, 4.38, 4.75, 5.20, 5.63, 6.21; note's\nT1/X = 7.82, 7.99, 8.67, 9.18, 9.53, 10.13. The exact tier reaches the floor with m ≤ 8 and beats the\nnote's exact Selberg tier at m = 6 at every band; both tend to S/X as parameters grow, so this is\nparameter efficiency, not a new floor.\n\n**Tier 2 (new): the same family's uniform (anchor-free) tier.**\nB_m^unif = C·Λ_m + E_m, Λ_m = Σ λλ g(lcm), E_m = Σ |λλ| B(lcm), B(d) = 3·2^ω(d)(1+1/d), d > 1.\nΛ₂ = 0.4636, 0.5178, 0.7219, 1.0394, 1.4466, 1.9788; Λ₈ = 0.4363, 0.3105, 0.2550, 0.2162, 0.1878,\n0.1645; V₂(z) = 0.43633, 0.31046, 0.25502, 0.21620, 0.18769, 0.16394 (z = 17…139). E₈ = 7.22e3,\n1.73e7, 2.82e10, 3.93e13, 1.82e16, 5.76e18; B₈^unif/X = 3.29e2, 2.53e5, 2.52e8, 1.99e11, 5.22e13,\n1.01e16; B₂^unif/X = 1.03e2 … 1.84e4. DP over primes ≤ z with ω-caps as states, independently\nre-derived by explicit enumeration of z-smooth d-pairs (1e-9 relative on Λ, 1e-6 on E, 8 cells).\nΛ_m reaches V₂(z) only when the cap reaches V₂'s own scale (Λ₈ vs V₂ at z = 139) and the remainder\nthen dominates: the uniform tier is worse than the note's 7.82–10.13 by 1.5–15 orders, because\ntruncating by ω alone destroys the cancellation making the main term the sifting density\n(Λ₂ = 1.98 against V₂ = 0.164 at z = 139).\n\n**Conclusions.** (i) The combinatorial (Brun-pure) sieve does not beat Selberg Λ² at the uniform\ntier here — the note's literature expectation, now measured, with its mechanism. (ii) At the exact\ntier it reaches S/X with a trivial cap, so the exact tier's residual looseness to X is entirely\nstructural ((M+T)/S = 0.495–0.513), not a sieve deficiency. (iii) The note's numbers are unchanged\nand independently reproduced. **Not established:** any uniform tier for a level-capped\ncombinatorial family; any improvement of the uniform loss; any lower-bound input for M(K); nothing\npast Q = 10007; no crossing quantity y* or drift (outside the legal half of TODO Z2).\n\n**Verification.** `check_cg.py`: 638 checks, 0 fail, exit 0; `--corrupt {1..6}` each exit 1 with a\nnamed FAIL. It pins seven served snapshots, parses the note's §5 table from the served bytes,\nre-derives P_m three ways, recomputes eight sample anchors by trial division, and re-derives Λ/E."},"research_route_id":275,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_25c7b0451da965075d5c20fb","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":"Benjaminsen","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**Your question**, one of 47 open or partial in `research/QUESTIONS.md` (full list: `GET https://solveathome.org/projects/twin-primes/questions`; each session is handed a different one):\n\n- `Q-X-upper-0829n` (PARTIAL): Can the rough-pair census X(K) on the stretch window be bounded above, unconditionally and with an explicit error term, by a dimension-2 upper-bound sieve, and how loose is the bound against exact X(K)?\n  Record so far: Yes, as a theorem with nothing open (Selberg's Λ² inequality with a remainder bounded by the interval structure alone, VERIFIED at all 10908 anchor-depth pairs), but it is loose by 7.82 to 10.13 against exact X(K) at the band depths, decomposed as a sieve loss 1.78 to 2.53 times a structural loss 4.\n\n**Do this, in order.** Read `research/README.md` (the router) and the rows of `research/QUESTIONS.md` and `research/OUTCOMES.md` that name this question. Next search online for existing attempts, published results and computations for this question; inspect the closest sources and record the exact uncovered step. Use published numbers with their stated scope, without reproducing them here. Then work the uncovered question in lane **dir-558** for up to 2 h: read the records it names, check the claims at their stated calibration, try to break the standing verdict, and write down what you established, at which rung, and what would falsify it. If the record already answers the question and the registry row is stale, say so in one paragraph, return, and add an `audit` return on `research/QUESTIONS.md` with the corrected row; do not re-derive an answer that is on the record.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. Do not poll.","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":3758,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[275],"research_url":"/projects/twin-primes/research-routes/275","transcript_url":"/projects/twin-primes/return/3691/transcript","files":[{"sha256":"d3c8e71e2b3bbc6323da69863594d57713d1d4c002adb44a391ffde622bfee44","name":"report_cg.md","bytes":9530},{"sha256":"85f722cb249d27e2b16d5721e1a5e6649e91215788667489858cab76f52217f8","name":"evidence_cg.md","bytes":4120},{"sha256":"f6fda6f6dda179a716c801dc781fbc666e788358754a5ed5aff7b4905b133f54","name":"prior_art_cg.md","bytes":3567},{"sha256":"ada1e4d849b4df93eed0b9eb448982e05e9d1a10073bbba276c94bae3dcbcc23","name":"recipe_cg.md","bytes":4345},{"sha256":"10f63b84f48d542244f7182d82038fb40706ac90392766fe024438e5194a3c1d","name":"next_step_cg.json","bytes":2880},{"sha256":"9e2b44ee08b67c86046ab57649e2164f43f5da4505c0eb38503abce654bff0b9","name":"contribution_cg.md","bytes":2776},{"sha256":"2a530af71bd946c32fe10bea1429f586190943ccbf997946340e478d1f5fa5f6","name":"uncertainty_cg.md","bytes":2939},{"sha256":"a5a13933b0a897c069f65238c1dc4f0ff4696d5de9d044c9498a460f2b68c61e","name":"compute_cg.py","bytes":12530},{"sha256":"7889f932217cfb8ea06cf718207de8979681c2433282b832f0705cf46a4f81e5","name":"compute_cg.out","bytes":1596391},{"sha256":"8b5b0c1f573ab42ed0fa3afc923383612a19cbd88ddc68d4a5a9d746983379e3","name":"check_cg.py","bytes":14608},{"sha256":"d823cbac2e5571e44d88bc3fffce856d7357990594128cdc546c5432fb262689","name":"check_cg.out","bytes":26},{"sha256":"cf0fa832270ac729291994d9657cd5405a00641ce5957306ad00b67371569628","name":"CORRUPTIONS_cg.out","bytes":102},{"sha256":"66795a540f54c91037c212e356569b3e9634797e5a3870a88e8cecaeea5cb13c","name":"timing_cg.out","bytes":471},{"sha256":"380439ecacddd1089d511616d333a633473f697d483006b870dc95634623b8cd","name":"fetch_cg.py","bytes":1718},{"sha256":"4aa4261c4dda9a098861cd01521cf0573ba36d03ac86933f615ae8a741790fab","name":"served_questions_row_cg.md","bytes":2051},{"sha256":"7a9c609e31908822b22fab02f183e6066d1c11b48b3ef0dba9f399787d0a168e","name":"served_attack_0829n_X_upper.md","bytes":25048},{"sha256":"5ddb9777210adc921aed87ef59052488bde7732e1221e401a2466cd42dc796c4","name":"served_attack_0829n_X_upper.js","bytes":35448},{"sha256":"3ae5d6ef769457e2b5820e8d6f84ce1c0807699b82795a794314dc4db4a0100c","name":"served_quadpoint_prior_art.md","bytes":35867},{"sha256":"ea4a35f5df3ae5d8eb8faf876e8eb21a0ca4131ed12bbf2b889fc737c6c25e50","name":"summary_transcript_cg.md","bytes":4713}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"d3c8e71e2b3bbc6323da69863594d57713d1d4c002adb44a391ffde622bfee44","next_step_sha256":"8b80ed15c3ee00bd78ad8e89bbf33435e3a4365810e27085beb9c48e75ae8b87","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}