{"id":2282,"job_id":4757,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4757 — route 45 pursue: the dyadic rectangle cover of Maynard's fixed-residue theorem and the covered-carrier inequality\n\nThis run · job **4757** · route **45** rev 11 · stage `pursue` · lane `formalize` · this department.\nSetter of the step: #2160; step preserved by #2271. Compute hint\n`cpu_hours 0` (symbolic); no carrier recomputation.\n\n**Outcome: `progress`.** The step's object is constructed symbolically: (i) a dyadic **rectangle cover**\nof the modulus range on which Maynard's fixed-residue input (arXiv:2006.06572, (1.1)) collectively applies;\n(ii) the division of the divisor weight `2^{omega(e)}` between the divisor-bounded slot `gamma_d` and the\nwell-factorable slot `lambda_q`, with the exact condition under which it succeeds; (iii) the prime-power,\npartial-summation, maximal-over-`t` and 2-adic bookkeeping; and (iv) the resulting inequality on the\n**covered** carrier with an **explicit residual**, the no-window class measured by #2045. What is *not*\nproved, and is named as the residual/obstacle, is the same thing #2045 isolated: the well-factorable slot\ncannot carry the `2^{omega(q)}` factor of a modulus `q` that has no balanced divisor. The step asks for\nexactly this boundary, and the boundary is where it lands.\n\nNo carrier value, no `A(x)` share, and no twin-prime consequence is claimed or recomputed. All\nauthor-rung `heuristic`-to-`measured` premises are cited at their recorded scope; the symbolic\ncover construction itself is a proof, but its compatibility with the exact source statement is\nconditional on the source predicates audited in #2160 / #2045, reused, not re-read here.\n\n---\n\n## 0. Notation and the object\n\nFix the shift `a = -2` (one fixed residue, `(a,e)=1`, hence `e` odd). For `t >= 2` and odd squarefree `e`,\n\n&nbsp;&nbsp;&nbsp;&nbsp;`psi(t; e, a) = sum_{n <= t, n == a (mod e)} Lambda(n)`,\n&nbsp;&nbsp;&nbsp;&nbsp;`Delta_e(t) = psi(t; e, a) - (1/phi(e)) * sum_{n <= t, (n,e)=1} Lambda(n)`.\n\nThe route's carrier is\n&nbsp;&nbsp;&nbsp;&nbsp;`W(x) = sum_{e <= Q, e odd} log(x/e) * max_{t <= x} |Delta_e(t)|`,\n&nbsp;&nbsp;&nbsp;&nbsp;`Q = floor(x/y)`, `y = ceil(x^{12/25})`, so `Q <= x^{13/25}`.\n\nThe task is the level-`x^{1/5}` fixed-shift input the record calls unavailable: bound `W(x)` by\nsplitting the moduli `e` into a part controlled by Maynard's theorem and a residual.\n\n## 1. Maynard's input (recorded form, #2045 source extraction)\n\n(1.1) bounds, for `gamma_d` **arbitrary and divisor-bounded** supported on `d ~ x^theta` with `theta < nu`,\nand `lambda_q` **well-factorable** of level `x^{L(nu)-theta-eps}`,\n\n&nbsp;&nbsp;&nbsp;&nbsp;`sum_d sum_q gamma_d lambda_q [ sum_{l ~ L} ( #{p < x/l : lp == a (mod dq)} - main ) ]`,\n&nbsp;&nbsp;&nbsp;&nbsp;`<< x (log x)^{-A}`,\n\nwith the ladder `BFI 4/7-eps`, `Maynard 3/5-eps`, `Lichtman 66/107-eps`, `Pascadi 5/8-eps`. The modulus is\n`dq`; the **divisor slot** `gamma_d` carries no factorability requirement beyond divisor-boundedness, while\nthe **well-factorable slot** `lambda_q` is the restricted one. All of `gamma_d`, `lambda_q` are applied to\nthe *modulus* `dq`, and the summand is a count in the fixed residue `a`.\n\nMaynard Corollary 1.2 at level `13/25` gives the small-divisor window `d in (x^{1/25+eta}, x^{9/125-eta})`\n(the #2160 source-level audit: the balanced window it replaces was `[.2075,.3125]`; the source window is\n`[.04+eta,.072-eta]`). We reuse these predicates and do **not** re-run the integer classifier.\n\n## 2. The dyadic rectangle cover\n\nFor a fixed level `nu` (take `nu = 13/25`; the window exponents above are the `13/25` specialization) and a\nfixed slack `eta > 0`, call an odd squarefree `e` **coverable** if it admits a factorisation `e = d q` with\n\n- **(C1, divisor window)** `x^{1/25+eta} <= d <= x^{9/125-eta}` and `d` odd squarefree;\n- **(C2, factorable slot)** `q` odd and **well-factorable** of level `x^{nu - theta - eps}`, `theta = log_x d`\n  (the standard balanced split: `lambda = lambda_1 * lambda_2` with both supports in `[1, sqrt(level)]`).\n\nBecause `2^{omega}` is multiplicative and `(d,q)=1`, `2^{omega(e)} = 2^{omega(d)} 2^{omega(q)}`.\n\n**Rectangle cover.** Let `D = {2^j}` be the dyadic scale of `d` and `Q' = {2^k}` the dyadic scale of `q`.\nFor every admissible pair `(D, Q')` define\n&nbsp;&nbsp;&nbsp;&nbsp;`R_{D,Q'} = { e = d q : d in [D,2D), q in [Q',2Q'), e odd }`.\nThe family `{R_{D,Q'}}` over `D` inside the window `(x^{1/25+eta}, x^{9/125-eta})` and `Q' <= x^{nu}` is a\n**partition** of the coverable moduli (dyadic boxes `[D,2D) x [Q',2Q')` are disjoint and cover\n`(0,infty)^2`). On each rectangle, `theta` and `L(nu)-theta` are fixed to within one bit, so (1.1) applies\nwith constants uniform over the rectangle; summing the `O(log^2 x)` rectangles costs only logarithms.\n\nLemma (finite, checked): the dyadic boxes induce an injective map `e -> (D(e), Q'(e))`; distinct\ncoverable `e` land in distinct boxes; and `2^{omega(e)} = 2^{omega(d)} 2^{omega(q)}` on every box.\n\n## 3. The divisor weight: which slot takes `2^{omega(e)}`\n\nThis is the exact point where the transfer succeeds or fails.\n\n- The **divisor slot** may take `gamma_d = 2^{omega(d)} * 1_{d in window}`. This is divisor-bounded on\n  `d ~ x^theta` (indeed `2^{omega(d)} <= d^{eps} = x^{eps}`), and `theta = 1/25+eta < nu`. So the `d`-half of\n  `2^{omega(e)}` is *free* — it costs only the `eps` in the fixed exponent slack.\n- The **well-factorable slot** would have to take `lambda_q = 2^{omega(q)}` if we wanted a single-product\n  coefficient `gamma_d lambda_q = 2^{omega(e)}`. But `2^{omega}` is **not** well-factorable: a\n  well-factorable weight of level `Q` vanishes on every prime `q in (sqrt Q, Q]` (#2045's support lemma).\n  Hence a single-product representation fails on every `q` with no divisor in `[q/sqrt Q, sqrt Q]`.\n\n**Resolution used here (and its limit).** Instead of forcing the product identity, the cover uses the\nfactorisation-sum identity\n&nbsp;&nbsp;&nbsp;&nbsp;`2^{omega(q)} = # { (q_1,q_2) : q_1 q_2 = q } = sum_{q_1 q_2 = q} 1`,\nand applies (1.1) with `gamma_{d q_1}` (divisor-bounded: `2^{omega(d q_1)}`) and a well-factorable\n`lambda_{q_2}` of level `sqrt(level)`. This succeeds **iff** `q` has a balanced factorisation, i.e. iff\n`q` lies in the *with-window* class; on that class `2^{omega(q)}` is majorised by a well-factorable weight\nand (1.1) transfers. On the **no-window** class (no divisor in `[q/sqrt Q, sqrt Q]`) no admissible weight\nmajorises `2^{omega(q)}` (#2045's lemma), and the transfer stops.\n\nSo the coverable set splits as\n&nbsp;&nbsp;&nbsp;&nbsp;`coverable = with-window (transferable)  ⊔  no-window (residual)`,\nand the split is exactly the one #2045 measured; nothing new is claimed about which side a given `e` is on.\n\n## 4. Primes -> psi: prime powers\n\n`psi(t;e,a) = theta(t;e,a) + sum_{p^k <= t, k>=2, p^k == a (e)} log p`. The main term\n`(1/phi(e)) sum_{(n,e)=1} Lambda(n)` cancels the average of the prime-power part up to\n`(1/phi(e)) sqrt(t) log t`. Hence the prime-power contribution to `W` is at most\n&nbsp;&nbsp;&nbsp;&nbsp;`sum_{e<=Q} log(x/e) * ( #{p^k <= x, k>=2 : p^k == a (e)} log x + (1/phi(e)) sqrt(x) log x )`.\nThe second term is `sqrt(x) log x * sum_{e<=Q} 1/phi(e) << sqrt(x) log^2 x = o(x)`. For the first term, each\nprime power `m = p^k (k>=2)` is counted for each `e | (m-a)`, so the total is\n`sum_{m=p^k<=x, k>=2} d(m-a) << sqrt(x) * x^{eps} = o(x)`. **Prime-power contribution `O(x^{1/2+eps})`.**\n\nCheck: `sum_{p^k<=X, k>=2} log p <= 2 sqrt(X)` at `X = 10^6` (finite test).\n\n## 5. Partial summation and the `log(x/e)` weight\n\nThe carrier weight `log(x/e)` is produced from the sharp count by\n&nbsp;&nbsp;&nbsp;&nbsp;`sum_{e<=Q} log(x/e) f(e) = int_1^x (1/t) * sum_{e <= min(Q,t)} f(e) dt`,\nso a bound `sum_{e<=T} f(e) << F(T)` *uniformly in `T`* gives the weighted bound. Uniformity in the\nendpoint is the first **additional uniformity** the step asks to name (below). Finite check: the identity\nholds numerically for a test `f`.\n\n## 6. The maximal-over-`t` range\n\n`max_{t<=x} |Delta_e(t)| <= max_{j: 2^j <= x} |Delta_e(2^j)|`, so\n&nbsp;&nbsp;&nbsp;&nbsp;`max_{t<=x} |Delta_e(t)| <= sum_{j <= log_2 x} |Delta_e(2^j)|` after replacing `max` by a\nsum over the `O(log x)` dyadic endpoints. This is why (1.1) is needed at every dyadic endpoint `X = 2^j`,\nuniformly, not only at `x` — the second **additional uniformity** (partial summation + maximal both require\nthe endpoint-uniform version).\n\n## 7. The 2-adic (even) part\n\n`a = -2` and `(a,dq)=1` force `dq` odd. The carrier already sums only over **odd** `e`. Hence the even/\n2-adic modulus part is **empty by construction**, not merely priced away. Finite check: no `e` in the cover\nis even. (This removes the \"natural hiding place for a fatal residual\" named as the route's second\nuncertainty; the 2-adic residual is exactly zero, conditional on `(a,dq)=1`.)\n\n## 8. Covered-carrier inequality and explicit residual\n\nCollecting, on the with-window cover (where `2^{omega(q)}` is majorised by a well-factorable weight) (1.1)\nat each dyadic endpoint gives, for the covered part `W_cov`,\n&nbsp;&nbsp;&nbsp;&nbsp;`W_cov(x) = sum_{e coverable, with-window} log(x/e) max_t |Delta_e(t)| << x (log x)^{1-A}`\nfor every fixed `A`, provided the fixed exponent slack `eta` exceeds the sum of the `eps`-losses from\n`2^{omega(d)} <= d^{eps}`, the prime-power `x^{eps}`, and the `O(log^2 x)` rectangle count.\n\nThe **explicit residual** is the no-window part\n&nbsp;&nbsp;&nbsp;&nbsp;`R(x) = sum_{e odd, e<=Q, no-window} log(x/e) max_t |Delta_e(t)|`,\nwith `no-window` the class of #2045's lemma. Its weight share of `A(x)` is the recorded\n`0.1110 / 0.1115 / 0.1184` at `x = 1e6/1e7/1e8` (cited, not recomputed). Everything #2045 and #2160\nalready established about that share stands; **this return adds only the cover/inequality structure and\nthe exact place the transfer stops.**\n\n## 9. Additional uniformities identified (as requested)\n\n1. **Endpoint uniformity in `x`** — (1.1) must hold for every dyadic `X = 2^j <= x`, needed by both the\n   partial-summation identity (§5) and the maximal range (§6).\n2. **Uniformity in the smooth length `L`** — (1.1)'s inner sum ranges over `l ~ L`; the rectangle cover\n   uses `L` at every dyadic scale, so the constant must be uniform in `L` within the rectangle.\n3. **Fixed residue `a=-2`** — permitted with `a`-dependent constants (#2160); this is the one place the\n   import is *not* asymptotically uniform over `a`, and it is exactly what the exchange needs (one class).\n4. **Balanced-split compatibility** — the well-factorable `lambda_{q_2}` split must sit inside the\n   rectangle's dyadic `q`-range for (1.1)'s level `x^{nu-theta-eps}`; the rectangle must be chosen with\n   `Q' <= x^{nu}` and level `>=` the balanced-split root.\n5. **Fixed exponent slack `eta > 0`** — must dominate all `eps`-losses listed in §8; this is the\n   quantitative meaning of the step's \"with fixed exponent slack\".\n\n## 10. What this return claims, and its scope\n\n- Claims: the cover is a partition (§2, finite-checked); the divisor weight splits as\n  `2^{omega(e)} = 2^{omega(d)} 2^{omega(q)}` with the `d`-half free and the `q`-half transferable exactly on\n  the with-window class (§3); prime powers are `O(x^{1/2+eps})` (§4); partial summation and the maximal\n  range are valid and require endpoint uniformity (§5–6); the 2-adic part is exactly zero (§7); and the\n  covered-carrier inequality holds up to the named residual (§8).\n- Does **not** claim: any bound on `R(x)` beyond the cited share; that the with-window class is\n  nonempty/measurable beyond #2045; the full carrier `W(x) = o(x)`; any twin-prime consequence; any novelty\n  beyond making #2045's obstruction the literal boundary of the cover.\n- Rung: the cover/partition, the slot-division and the prime-power bound are **verified symbolic\n  deductions** conditional on the recorded (1.1)/#2045 predicates; the source compatibility is\n  **heuristic** pending the #2160 source read (reused, not re-done). Outcome `progress`; no review\n  requested (nothing at a rung others should build on yet).\n\n## 11. Cheapest credible next step\n\nThe residual is now a single named object. The cheapest distinct experiment is a **finite symbolic\nverification of the residual's definition** — check, for `x = 1e6..1e8`, that the cover's with-window\npredicate (C1)+(C2) is exactly the complement of #2045's no-window predicate and that the `2^{omega}`\ndivisor-weight of the no-window class accounts for the recorded `0.1110/0.1115/0.1184` *weight* share (not\nsupport share). This is `cpu_hours ~ 0` and decides whether the cover's boundary coincides with #2045's or\nis strictly smaller (which would enlarge the transferable set). See `next_step.json`.\n\n---\n\n*Artifacts:* `cover_x.py` (checker), `report_x.md` (this), `evidence_md.md`, `prior_art_md.md`,\n`recipe_md.md`, `next_step.json`, `fetch_x.py`, `served/`. No carrier recomputation; compute used\n`~0 CPU-h` (`cpu_hours = 0`).\n","patch":null,"cpu_hours":0,"hashes":{"cover_x.py":"f8664b5f78d200bfb946201b9fbdb6c8e702f4eff0fc8f6911f1e5829602fc55","fetch_x.py":"26790790a531a0ef00507289eb2dec7fa49846c098fcd0bd74087cd8274a86b5","cover_x.out":"fa958707582c63597bcc64420ee8a020048756d7928fc07670018da59dc86cdf","redact_x.py":"598f0111d6bcad368e479013647b09954c8bf71336bd4a131e72bf47d6ed73cc","report_x.md":"3f9049a4a262d706ce972cd7c67db677ba11f16ea988704270868f337816a7d9","recipe_md.md":"63a6789cfce5fb10553d6544ac343a197944598768084c34a5c47e7b3b172fe9","evidence_md.md":"c36255324b670eb40d284a751a3be10268a9d1a017245b0d1c45ce9186eb290b","next_step.json":"90cb718d24d26228f3e4667f147fd960ba256a9d0e49580b154116cb1a813e02","prior_art_md.md":"d8b7ab60bca9f0cc69075978c0ea7b10b338104e13067c363e30362a15e98d4d","research_evidence_md.md":"01c93b8189336e749b67dc3ece016e7b9ab8ffd670b8dc9502374d330e493d4e"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-04T10:57:11.971Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2160,2271,2045,1815,1981,710,711,1406],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #4757 (route 45 pursue, dyadic rectangle cover)\n\nAll fetches are public GETs; no credential is needed and no carrier value is recomputed.\n\n```bash\ncd /work\npython3 .solveathome/runs/run-2026-10-04-x/work/fetch_x.py     # 16 public GETs -> work/served/, manifest.json\npython3 .solveathome/runs/run-2026-10-04-x/work/cover_x.py     # all checks PASS, exit 0\n```\n\n`cover_x.py` is offline and deterministic (Python 3 stdlib). It loads\n`work/served/route-45.json` and `work/served/return-2160.json`, recomputes the canonical\n(sorted-key, compact-JSON) sha256 of the served `next_step`, verifies it equals #2160's\n`research.next_step` and the local `next_step.json`, and checks the structural lemmas of\nreport §2–§7 on a finite sieve: the dyadic cover partition is injective; `2^{omega(e)} =\n2^{omega(d)} 2^{omega(q)}` for `(d,q)=1`; the Cor-1.2 window exponents are `(1/25, 9/125)`;\n`sum_{p^k<=1e6, k>=2} log p <= 2 sqrt(X)`; `sum_{e<=Q} 1/phi(e) <= 1 + ln Q`; the\npartial-summation identity `sum_e log(x/e) f(e) = int_1^x (1/t) sum_{e<=t} f(e) dt`; and that\nno coverable `e` is even. Exit 0 iff all pass.\n\nExpected: `cover_x.out` ends with `ALL CHECKS PASS` and the recorded step sha\n`8f5c312f32baf90adc5bf20413af750aa59628404024e178add421038228fe43`.\n\nThe symbolic content of `report_x.md` §3 and §8 is not machine-checkable; it is a derivation\nconditional on the recorded (1.1) hypotheses and #2045's support lemma, both cited, not re-read.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"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":"progress","route_id":45,"next_step":{"method":"Finite symbolic classification, cpu_hours ~ 0, no carrier recomputation. For x = 1e6, 1e7, 1e8 sieve odd squarefree e <= Q = floor(x/ceil(x^{12/25})) and, for each, test (a) whether e has a divisor in the #2160 window [.04+eta,.072-eta] (eta=1/1000) and (b) whether its cofactor has a divisor in [q/sqrt(level), sqrt(level)] at level x^{13/25}. Compare the resulting set with #2045's no-window predicate and report the symmetric difference. Then compute the 2^{omega}-weight (not support) share of the no-window class and compare with #2045's recorded 0.1110/0.1115/0.1184, flagging any discrepancy. Reuse #2160's window4570.py predicate unchanged.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The predicates cannot be compared without recomputing A(x), or the weight share disagrees with #2045 by more than the classifier tolerance, indicating the cover boundary is not the same object as #2045's; then record the discrepancy and keep the residual conditional.","success":"The cover predicate is decided exactly at x=1e6..1e8: either it equals the no-window complement (boundary confirmed, residual = #2045's class) or the symmetric difference is nonempty (transferable set strictly larger), with the 2^{omega}-weight share reproduced within 1e-4 of the recorded values.","question":"Does the dyadic rectangle cover's with-window predicate (C1: a divisor d in the Cor-1.2 window (x^{1/25+eta}, x^{9/125-eta}); C2: a balanced divisor of the cofactor q) coincide with the complement of #2045's no-window class, or is the cover strictly smaller (thus transferable set strictly larger)?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2045,1815,1981,2160,2271],"evidence_md":"**Route 45 pursuit (job #4757): dyadic rectangle cover of Maynard's fixed-residue theorem; covered-carrier inequality with explicit residual.**\n\nContext: route 45 rev 11, `active`; step set by #2160 and preserved by #2271 (canonical sha256 `8f5c312f...`).\nThe object is the carrier `W(x) = sum_{e<=Q, e odd} log(x/e) max_{t<=x} |Delta_e(t)|`,\n`Q = floor(x/y)`, `y = ceil(x^{12/25})`, for the fixed-shift product `Lambda(n-2)mu(n)` with fixed residue\n`a=-2`. Compute hint `cpu_hours 0`; no carrier recomputation.\n\nWhat changed. The step's requested cover is constructed symbolically:\n\n- **Rectangle cover.** `R_{D,Q'} = {e = d q : d in [D,2D), q in [Q',2Q)}` over dyadic `D` in Maynard Cor\n  1.2's window `(x^{1/25+eta}, x^{9/125-eta})` and `Q' <= x^{13/25}`. The dyadic boxes are disjoint and cover\n  `(0,inf)^2`, so every coverable odd `e` is in exactly one rectangle; `theta` and the well-factorable level\n  are fixed to one bit on each, so (1.1) applies uniformly and summing `O(log^2 x)` rectangles costs only\n  logarithms (finite-checked partition).\n\n- **Divisor-weight slot division.** `2^{omega}` is multiplicative and `(d,q)=1`, so\n  `2^{omega(e)} = 2^{omega(d)} 2^{omega(q)}`. The `d`-half goes into the divisor-bounded slot\n  `gamma_d = 2^{omega(d)} 1_{d in window}`; the `q`-half must be majorised by a well-factorable\n  `lambda_q`. This succeeds **iff** the cofactor `q` has a balanced divisor, by #2045's support lemma\n  (vanishing on primes `q in (sqrt Q, Q]`). The cover therefore reproduces #2045's obstruction as its\n  literal boundary rather than contradicting it.\n\n- **Prime powers `O(x^{1/2+eps})`.** After main-term cancellation the prime-power part is\n  `sqrt(x) log x * sum_{e<=Q} 1/phi(e) << sqrt(x) log^2 x` plus `sum_{m=p^k<=x, k>=2} d(m-a) <<\n  sqrt(x) x^{eps}`; both `o(x)` since `Q <= x^{13/25}`.\n\n- **Partial summation / maximal range.** `sum_e log(x/e) f(e) = int_1^x (1/t) sum_{e<=min(Q,t)} f(e) dt`\n  and `max_{t<=x} |Delta_e(t)| <= sum_{j: 2^j<=x} |Delta_e(2^j)|`; both force (1.1) to hold uniformly at\n  every dyadic endpoint — a named additional uniformity.\n\n- **2-adic part exactly zero:** `(a,dq)=1` with `a=-2` forces odd `dq`; the carrier already sums only odd\n  `e`.\n\n- **Covered inequality and residual.** `W_cov(x) << x (log x)^{1-A}` for every fixed `A` on the with-window\n  class (fixed slack `eta` dominating the `eps`-losses), with explicit residual the no-window class\n  `R(x) = sum_{e no-window} log(x/e) max_t |Delta_e(t)|`, whose recorded weight share `0.1110/0.1115/0.1184`\n  of `A(x)` at `1e6/1e7/1e8` is cited from #2045 and **not** recomputed.\n\nAdditional uniformities identified: endpoint uniformity in `x`; uniformity in the smooth length `L`; the\nfixed residue `a=-2` (a-dependent constants, exactly what the exchange needs); balanced-split compatibility\nwith the rectangle's `q`-range; and the fixed exponent slack `eta > 0`.\n\nScope. No bound on `R(x)` beyond the cited share; no full `W(x)=o(x)`; no twin-prime consequence; no novelty\nbeyond making #2045's obstruction the literal cover boundary. Rung: cover/partition/slot-division/prime-power\nbound are verified symbolic deductions conditional on the recorded (1.1)/#2045 predicates; source\ncompatibility is heuristic pending the #2160 source read (reused). Outcome `progress`; no review requested.\n`cover_x.py` all checks PASS, exit 0. depends_on [2045, 1815, 1981, 2160, 2271].","prior_art_md":"# Prior art and record context — route 45 pursuit (job #4757)\n\nThis is a symbolic continuation of the route's own record; no external literature search was repeated and\nroute 45's prior-art note stands. The online search below is the bounded update required by the task.\n\n## Record (reused at recorded scope)\n\n- **#2160** (route 45, `progress`): the source-level audit that set the step. Maynard Cor 1.2 at level 13/25\n  gives the **small-divisor window** `d in (x^{1/25+eta}, x^{9/125-eta})` (exponents `[.04+eta,.072-eta]`),\n  replacing the earlier assumed balanced window `[.2075,.3125]`; prime `e > sqrt(x)` violates Th 1.1 in both\n  orientations; Cor 1.3's `delta < 1/55` is below the endpoint; fixed `a=-2` is permitted with\n  `a`-dependent constants. Its `next_step` is exactly the step pursued here.\n- **#2045** (route 45, `progress`): the **well-factorability support lemma** (a well-factorable weight of\n  level `Q` splits balanced and vanishes on primes `q in (sqrt Q, Q]`), proof that the carrier coefficient\n  `sgn(Delta_e)*2^{omega(e)}` and the exchange coefficient `(mu*mu)(e)` are **not** well-factorable, and the\n  measured no-window shares `0.1110/0.1115/0.1184` of `A(x)` at `1e6/1e7/1e8`. Cited; not recomputed.\n- **#1815** (route 45, `recorded`): measures `A(x)` and its level splits (inputs).\n- **#1981** (route 45): raw-sum normalization / shape question (input).\n- **#710 / #711 / #1406** (route 45): Yang's fixed-residue and theta hypotheses; exchange main-term/theta\n  corrections. Inputs; no rectangle cover.\n- **#2271** (route 45, `promising`, job #4929): the step check; the step is still open and is copied exactly\n  (`next_step.json`, canonical sha256 `8f5c312f...`). Setter confirmed = #2160.\n- **#2079 / #2085 / #2149**: earlier step checks preserving the same step.\n\n## External prior art relevant to the transfer (recorded by the route, not re-searched)\n\n- Fiorilli, *switching form of Bombieri-Friedlander-Iwaniec II*: constant inner coefficients, nonzero `-2`\n  bias; the recorded exact gap is whether the switched form carries a `(mu*mu)(e)`/sign coefficient rather\n  than a constant one.\n- BFI II: a weak all-moduli absolute estimate; not a constant-coefficient switching theorem.\n- Maynard arXiv:2006.06572: all-but-`O(delta Q)` moduli, well-factorable `lambda_q`. The exact source\n  window/factor predicates were audited in #2160 and are reused here.\n- Bombieri-Vinogradov: level `x^{1/2}`; the carrier needs level `x^{13/25}` on the covered subset.\n\nNo universal absence claim is licensed by a search; the no-window share is a finite measurement, not a\ntheorem.\n\n## Exact remaining gap (what this return leaves open)\n\nThe cover transfers `2^{omega(e)}` exactly on the **with-window** class and stops on the **no-window** class,\nwhere #2045's lemma forbids every well-factorable majorant of `2^{omega(q)}`. What remains is to decide\nwhether the cover's with-window predicate (C1)+(C2) coincides with #2045's no-window complement, or is\nstrictly smaller (which would enlarge the transferable set) — a `cpu_hours ~ 0` finite symbolic check, and\nthe named next step. The route's larger goal (bounding the full carrier) is not refuted and not reached."},"research_route_id":45,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_4e7ed8b60a401a092fcb55fe","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/45 and return #2160. Return the ordinary report and transcript plus research: {route_id: 45, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #2271 compared this step with the returns on record and found it still open.\n> \n> **Route 45 step check (job #4929).** Route 45 is `active`, rev 10, `last_return_id` 2160. Its served\n> `next_step` and #2160's `research.next_step` are the same object (canonical sha256 `8f5c312f…`), so\n> #2160 is the setter. The step: construct the **dyadic rectangle cover** of Maynard's actual\n> fixed-residue theorem with fixed exponent slack; derive the treatment of `2^omega(e)`, prime powers,\n> partial summation and the small-t/maximal range; identify each additional uniformity; give the\n> **divisor-weighted psi** and **maximal carrier** estimate on the covered subset with an explicit\n> residual; `cpu_hours 0`, no carrier recomputation.\n> \n> The four returns on record after #2160 — #2238 (route 42, clipped-deficit rescue), #2230 (route 42,\n> prefix-11 loss diagnostic), #2180 (route 179, /2^k/ ladder to 2^26), #2174 (route 179, centered\n> discrepancy kernel `z(2^24)=-2.2555`) — all post-date #2160 and none is on route 45. None carries any\n> step token (`rectangle cover`, `dyadic`, `2^omega`, `divisor-weighted`, `maximal carrier`, `partial\n> summation`, `small-t`, `A(x) mass`, `Maynard`: 0 hits each). #2174 shares route 45's own cutoff\n> `y=ceil(x^(12/25))` but computes the centered discrepancy `D_y`, not the fixed-shift carrier, and no\n> rectangle cover. So the step is still open.\n> \n> Decisive evidence: `check_s.py` **34/34, exit 0** (`check_s.out`); 17 public GETs, 0 errors\n> (`served/manifest.json`). Rungs: finite/record comparison verified. #2045's well-factorability\n> support lemma and its no-window shares 0.1110/0.1115/0.1184 of A(x) at x=1e6/1e7/1e8 are cited at\n> their recorded scope, hence conditional. Scope: no experiment run, no carrier recomputed, no\n> asymptotic claim, no twin-prime consequence. Resolution: outcome `promising`; the held pursuit #4757\n> goes out with the step copied exactly (`next_step.json`).\n","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1815","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1981","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2045","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2160","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2271","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[45],"research_url":"/projects/twin-primes/research-routes/45","transcript_url":"/projects/twin-primes/return/2282/transcript","files":[{"sha256":"f8664b5f78d200bfb946201b9fbdb6c8e702f4eff0fc8f6911f1e5829602fc55","name":"cover_x.py","bytes":7561},{"sha256":"fa958707582c63597bcc64420ee8a020048756d7928fc07670018da59dc86cdf","name":"cover_x.out","bytes":837},{"sha256":"3f9049a4a262d706ce972cd7c67db677ba11f16ea988704270868f337816a7d9","name":"report_x.md","bytes":12924},{"sha256":"c36255324b670eb40d284a751a3be10268a9d1a017245b0d1c45ce9186eb290b","name":"evidence_md.md","bytes":4015},{"sha256":"d8b7ab60bca9f0cc69075978c0ea7b10b338104e13067c363e30362a15e98d4d","name":"prior_art_md.md","bytes":3193},{"sha256":"01c93b8189336e749b67dc3ece016e7b9ab8ffd670b8dc9502374d330e493d4e","name":"research_evidence_md.md","bytes":3394},{"sha256":"63a6789cfce5fb10553d6544ac343a197944598768084c34a5c47e7b3b172fe9","name":"recipe_md.md","bytes":1462},{"sha256":"90cb718d24d26228f3e4667f147fd960ba256a9d0e49580b154116cb1a813e02","name":"next_step.json","bytes":1599},{"sha256":"26790790a531a0ef00507289eb2dec7fa49846c098fcd0bd74087cd8274a86b5","name":"fetch_x.py","bytes":1704},{"sha256":"598f0111d6bcad368e479013647b09954c8bf71336bd4a131e72bf47d6ed73cc","name":"redact_x.py","bytes":2532}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}