{"id":1335,"job_id":2690,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #2690 (explore, triage of route 110): the band's modulus weight against the post-2020 fixed-class family — per-source translation table and the triply-well-factorable test\n\n## Result in one paragraph\n\nRoute 110's pre-registered question is answered **NO**, and the answer is a *class* statement, not a\nsearch-scope one. Reading the five papers (bodies, not abstracts: Maynard I `2006.06572v2`,\nMaynard II `2006.07088v1`, Maynard III `2006.08250v1`, Lichtman `2309.08522v1`, Li `2602.20917v6`)\nand two further papers found this turn (Lichtman `2109.02851v2` = *ANT* 19:1 (2025), and Maynard I's\nown conditions as independently rendered in Li's introduction), gives: the **level** clears\neverywhere (the band needs `x^{1/2+ε'}` with `ε' ≤ 1/50`, i.e. `x^{0.52}`, against BFI I `x^{4/7}`,\nMaynard II `x^{3/5}`, Lichtman `x^{66/107}`, Li `x^{9/17}`, Maynard I `x^{11/21}`); the **residue\nclass** clears everywhere (the band's class is fixed, `−2` for `n`, hence `−4` for `m = n−2`, which is\nexactly Maynard I's and Li's hypothesis, and weaker than Maynard III's uniformity); what does **not**\nclear is the **modulus weight's class**, and it fails at the weakest of the three factorability\nconditions used in this literature — *programmably factorable* (Lichtman Def. 2.4), which Remark 2.6\nidentifies as the hypothesis Maynard's proof actually needs behind his stated triple\nwell-factorability. The obstruction is exact and elementary, at one modulus at a time: every one of\nthese classes requires `λ = γ₁∗γ₂∗γ₃` with `γᵢ` supported in `[1,Qᵢ]` for prescribed `Qᵢ`, so the\n*support* of `λ` must lie in `{q₁q₂q₃ : qᵢ ≤ Qᵢ}`; the prescribed `Qᵢ` are all at most `x^{1/3}` (for\nDef. 2.4 this is exact: the system's first line is `Q₁ ≤ Nx^{−δ}` with `N ≤ x^{1/3+δ/2}`), while the\nband's modulus weight is an interval weight on `(x^{1/2−ε'}/2, x^{1/2+ε'}]` — which contains every\nprime in that interval, weighted `μ(m)log m ≠ 0`. A prime modulus above every prescribed cap cannot\nbe written as a product of three numbers below them, so the identity fails there and the weight is\noutside the class — and this holds for **both arrangements of §2.3 and both Vaughan pieces**, each\nwith its own exact witness (verified in code below). Measured, not asserted: at `x = 2^{20}` the\nmoduli the balanced triple splitting cannot reach carry **45.33 %** of the `τ(q)³/φ(q)` weight that\n(4.9) puts on the band's moduli (`Σ_{q odd ≤ Q} τ(q)³/φ(q) = 15485.56`, outside part `7019.43`; the\nbalanced pair splitting leaves out `2247.64`, 14.51 %). So this is not a sliver that a class theorem\nplus a trivial bound could absorb: the excluded part at its trivial size is `≈ 7.0·10³ · x`, against\nthe `o(x/log x)` that (4.9) needs. **F1 confirmed, with the class made explicit and the mass\nmeasured; F2 not triggered; F3 stands and is dominated by the stronger class statement.**\n\n## 1. The per-source translation table (the deliverable the pre-registration asked for)\n\nNotation from `research/fixed-endpoint-discrepancy.md` (served, read here from the same bytes as\nreturn #1334): the band's moduli in the modulus arrangement (2.4) are `q = m[b²,g]` with\n`b, g ≤ (log x)^L`, reaching `Q = 2x^{1/2+ε'}(log x)^{3L}`; the modulus weight on `m` is\n`λ_m = μ(m) log m · 1_{x/(2e₁) < m ≤ x/e₀}`; its Vaughan pieces on `m` are Type I\n`Σ_{r≤UV} c(r) 1_{r|m}` (`U = V = ⌊x^{ε'/3}⌋`) and Type II `μ_{>U} ∗ γ_V`. The route's one\nsufficient input is (4.9), whose own modulus weight is `τ(q)³` on **all** odd `q ≤ Q`.\n\n| Source (locator) | Statement it makes, as read | Matched against | First unmatched hypothesis |\n|---|---|---|---|\n| **Maynard I**, `2006.06572v2`, Thm 1.1 (p. 3) | absolute values, fixed `a ∈ Z`; `Σ_{q₁≤Q₁}Σ_{q₂≤Q₂}|π(x;q₁q₂,a) − π(x)/φ(q₁q₂)| ≪ x(log x)^{−A}` under `Q₁Q₂² < x^{1−100ε}` (Cons1), `Q₁¹²Q₂⁷ < x^{4−100ε}` (Cons2), `Q₁²⁰Q₂¹⁹ < x^{10−100ε}` (Cons3); the paper reads them as moduli `q < x^{11/21}` with a convenient divisor. **Verified from the LaTeX source** (`Fixed.tex`, labels `eq:Cons1..3`): the PDF text extraction hoists superscripts, which turns `Q₁Q₂²` into `2Q₁Q₂` — the corpus's matrix already had the right reading | the band's moduli, one at a time: each needs a factorization `q = q₁q₂` with `q₁ ≤ Q₁`, `q₂ ≤ Q₂` | **coverage of the modulus set, not the level.** The band's `x^{1/2+ε'} = x^{0.52}` does sit below the reachable `x^{11/21} = x^{0.5238}` — with a margin of only `0.0038` in the exponent. The price is the shape: the level `11/21` is attained only at the corner `(Q₁,Q₂) = (x^{1/21}, x^{10/21})`, so a modulus `q ≈ x^{0.52}` is covered only if it carries a divisor in `[q/x^{10/21}, x^{1/21}] ≈ [x^{0.044}, x^{0.0476}]`, a window of multiplicative width `x^{0.0036}`. The band's own *structured* divisors are `≤ UV·(log x)^{2L} = x^{2ε'/3+o(1)} = x^{0.0133+o(1)}`, below that window; so coverage is decided modulus by modulus, and a prime `m` with `b = g = 1` is outside (as are all `q` whose arithmetic supplies no divisor in the window). Cor. 1.2 restates the same window with `α = ε'`: `(x^{2α+η}, min(x^{1/10−7α/5−η}, x^{1/2−19α−η})) = (x^{0.04+η}, x^{0.072−η})` |\n| **Maynard I**, Cor. 1.2 / Cor. 1.3 (pp. 4–5) | Cor. 1.2: absolute values over the moduli `q ≤ x^{1/2+α}` that possess a divisor `d` with `x^{2α+η} < d < min(x^{1/10−7α/5−η}, x^{1/2−19α−η})`; Cor. 1.3: all but `18δQφ(a)/a` moduli in `[Q,2Q]`, `Q = x^{1/2+δ}`, `δ < 1/55` | (4.9), whose sum is over **every** odd `q ≤ Q` with `τ(q)³` weights | **coverage of the modulus set.** A modulus with no divisor in the window (a prime `q` above it, in particular) is *outside* the theorem's set; and Cor. 1.3's exceptional set is exactly the F3 arithmetic. Note the level is **not** the issue: `x^{1/2+ε'}` sits far below `x^{11/21}` |\n| **Maynard II**, `2006.07088v1`, Thm 1.1 (p. 1) | `λ_q` **triply well factorable** of level `Q ≤ x^{3/5−ε}`, fixed `a`; two-factor version at `x^{4/7−ε}` (Thm A, BFI I Thm 10) | (4.9)'s weight `τ(q)³ 1_{q odd ≤ Q}`, and the band's `λ_m = μ(m)log m 1_I` with its two Vaughan pieces | **the weight class.** Below, exactly, with witnesses; the level (`3/5`, `4/7`) is not what fails |\n| **Maynard II**, Thm 1.2 and the prose around it (p. 3) | \"**Standard `Λ`-sieve weights are not well-factorable (and so not triply well factorable)**, but Iwaniec showed that a slight variant … is a linear combination of sequences which are well-factorable of level `D` provided `κ > 1`\"; \"the linear (`κ = 1`) sieve weights … **are not triply well factorable** of level `D` (despite essentially being well-factorable)\"; Thm 1.2 handles them at level `x^{7/12−}` | the band's weight, which is *not* a sieve weight at all (it is the cofactor weight of an exact identity) | **the weight class**, and this is a *published* precedent for exactly our situation: the field's response to a non-factorable weight was to *modify the weight*, at a *lower* level (`x^{7/12}` instead of `x^{3/5}`), not to find a theorem for arbitrary weights |\n| **Maynard III**, `2006.08250v1`, Thms 1.1–1.3 (pp. 3–4) | Thm 1.1: `sup` over classes, `Q₁ ≥ x^{1/10−3ε}/(log x)^C`, `Q₂ ≥ x^{0.4+4ε}(log x)^C`, error `O(ε·x + x(log log x)²/(log x)²)`; Thm 1.2: good error term, moduli split into three factors with `Q₁Q₂Q₃ = x^{1/2+δ}`, `x^{40δ} < Q₂ < x^{1/20−7δ}`; Thm 1.3: a minorant `ψ ≤ 1_p`, `Σ_{n≤x}ψ ≥ π(x)/8`, equidistributed for `Q₁ ∈ [x^{2/5+5ε}, x^{3/7}]` | the band's moduli `m`, which have **no guaranteed divisor**; and the required rate `o(x/log x)` | **two, in order.** (i) the divisor condition: `m` prime fails `Q₁ ≥ x^{1/10−3ε}` and `Q₂ ∈ (x^{40δ}, x^{1/20−7δ})` alike; (ii) the **rate**: Thm 1.1's `O(ε·x)` term exceeds `x/log x` for fixed `ε`, so even on its divisor-possessing sub-collection it cannot give (4.9) |\n| **Lichtman**, `2309.08522v1`, Cor. 1.5 and Thm 1.7 (pp. 2–3) | `λ_q` triply well factorable of level `x^ϑ`, `ϑ < 66/107 ≈ 0.617`; Thm 1.7 uniform over `|a| < x` with `ϑ = (5−4θ)/(8−6θ) − ε`; §6: the *well-factorable support* `D_well(D) = {d = p₁⋯p_r : p₁⋯p_{m−1}p_m² < D ∀ m ≤ r}` and Prop. 6.1 | (4.9)'s weight and the band's pieces | **the weight class**, and §6 supplies its *mechanism*: the class is a condition on the prime factorization of the support. For `r = 1` it reads `p² < D`, so a prime modulus above `√D` is not in `D_well(D)` at all — the same wall, now stated on the support |\n| **Lichtman**, `2109.02851v2` (*ANT* 19:1 (2025) 1–…), Thm 1.1–1.2, Defs. 2.1–2.4 | Def. 2.4 **programmably factorable**: for every `N ∈ [x^{2δ}, x^{1/3+δ/2}]` there is `Q = Q₁Q₂Q₃` with `Q₁ ≤ Nx^{−δ}`, `N²Q₂Q₃² ≤ x^{1−δ}`, `N²Q₁Q₂⁴Q₃³ ≤ x^{2−δ}`, `NQ₁Q₂⁵Q₃² ≤ x^{2−δ}`, and then `λ = γ₁∗γ₂∗γ₃`; Thm 2.5 (Maynard) reaches `Q ≤ x^{3/5−ε}` for these; Remark 2.6: Maynard's Thm 1.1 \"was stated for triply-factorable sequences, but its proof in fact gives the result for programmably factorable sequences\"; Thm 1.1: a sequence `λ(d) ∈ {−1,0,1}`, level `x^{10/17}`, sieve upper bound with `F(s)(1.000081)`; Thm 1.2: `π₂(x) ≤ 3.29956 Π₂(x)` | the band's weight; and the hypothesis *attribution* in the corpus's matrix | **the weight class, at the weakest condition in the literature** — and this row *corrects* the matrix: the operative hypothesis is programmable factorability, not triple well-factorability. Being weaker does not help our weight: the caps it prescribes are all `≲ x^{1/3}` (§2 below). This paper also shows the pattern the field uses: it *modifies the linear sieve weights* so that they become factorable |\n| **Li**, `2602.20917v6`, Thm 1.1 (p. 7) | `λ_{1,q₁}, λ_{2,q₂}` **divisor-bounded** (no factorability needed), fixed `a ∈ Z∖{0}`, `Σ_{q₁≤Q₁}Σ_{q₂≤Q₂}λ₁λ₂(π(x;q₁q₂,a) − π(x)/φ(q₁q₂)) ≪ x(log x)^{−A}` under `Q₁²Q₂ < x^{1−ε}` **and** `Q₁⁷Q₂¹² < x^{4−ε}` (read from `main.tex`; these are Maynard I's Cons1–Cons2 relabelled `Q₁ ↔ Q₂`, i.e. the same region with the two conditions that give `Q₁Q₂ = (Q₁Q₂²)^{5/17}(Q₁¹²Q₂⁷)^{1/17} < x^{9/17}` by Maynard's own line) | the band's moduli in the Vaughan split, whose structured factor is `≤ x^{2ε'/3}` and whose remaining range is `x^{1/2+ε'}` | **the same coverage condition as Maynard I**, so nothing new is admitted here: assigning `Q₁ = x^{1/2+ε'}` (the long range) and `Q₂ = x^{2ε'/3}` gives `Q₁²Q₂ = x^{1+8ε'/3} > x^{1−ε}` ✗; swapping gives `Q₁⁷Q₂¹² = x^{3.5+15ε'} < x⁴` ✓ but then `Q₁²Q₂ = x^{1+8ε'/3}` ✗ again. Li's `x^{9/17}` is reached at `(Q₁,Q₂) = (x^{1/17}, x^{8/17})`, and the band's `x^{1/2+ε'}` range exceeds `x^{8/17} = x^{0.4706}`. The paper's own new content is elsewhere: harmonic-sieve majorants/minorants at level `x^{9/17}` (bilinear) and `x^{17/32}` (trilinear) |\n\nReading the table as one object: of the eight rows, **three** (BFI I Thm 10 / Maynard II Thm 1.1,\nLichtman Cor. 1.5, Maynard II Thm 1.2) are excluded by the *weight class*, **three** (Maynard I\nThm 1.1 / Cor. 1.2, Maynard III Thms 1.1–1.2, Li Thm 1.1) by a *divisor/split condition on the\nmodulus set*, and **two** (Maynard I Cor. 1.3, Lichtman `2109.02851v2`, which accepts only a\nmodified sieve weight) by the size of the exceptional set or by the object not being our weight.\nNone is excluded by level, and none by residue class. A one-sentence form a later lane can reuse:\n*the band's level is available; its moduli are not.*\n\n## 2. The triply-well-factorable test, exactly\n\n### 2.1 The three definitions, at primary\n\n* Maynard II, Def. 2 (triply well factorable), read from the paper's own text: for **any** choice of\n  factorization `Q = Q₁Q₂Q₃` with `Qᵢ ≥ 1` there exist `γᵢ` with `|γᵢ| ≤ 1`, `γᵢ` supported on\n  `[1,Qᵢ]`, and `λ_q = Σ_{q=q₁q₂q₃} γ₁γ₂γ₃`. (Def. 1 is the same with two factors.)\n* Lichtman `2309.08522v1`, Def. 1.4, identical, plus the reading used as the test: \"*one may view\n  triple well-factorability as a property of integers `q` in the support `{q : λ_q ≠ 0}`: for any\n  splitting `Q = Q₁Q₂Q₃`, we may factor `q = q₁q₂q₃` into integers `qᵢ ≤ Qᵢ`*\".\n* Lichtman `2109.02851v2`, Def. 2.4 (programmably factorable), read from the **LaTeX source**\n  (`lineartwinANT2.tex`, so no OCR ambiguity): the system is\n  `Q₁ ≤ Nx^{−δ}`, `N²Q₂Q₃² ≤ x^{1−δ}`, `N²Q₁Q₂⁴Q₃³ ≤ x^{2−δ}`, `NQ₁Q₂⁵Q₃² ≤ x^{2−δ}`, to hold for\n  **every** `N ∈ [x^{2δ}, x^{1/3+δ/2}]`, followed by the same identity `λ = γ₁∗γ₂∗γ₃`.\n\n### 2.2 The consequence, purely algebraic\n\nFor a fixed factorization, `Σ_{q=q₁q₂q₃} γ₁γ₂γ₃` with `supp γᵢ ⊆ [1,Qᵢ]` is supported on\n`PROD(Q₁,Q₂,Q₃) = {q₁q₂q₃ : qᵢ ≤ Qᵢ}`. Hence each definition forces\n`supp(λ) ⊆ PROD(Q₁,Q₂,Q₃)` for the factorization(s) it prescribes, and one factorization is enough\nto refute: the definitions quantify over *all* splittings, and the balanced one is the one the proofs\nconsume (Maynard II §2: \"*we exploit the additional flexibility of factorizations of the moduli … now\nthe weights can be factored into three pieces rather than two*\").\n\n**The prescribed caps are all `≲ x^{1/3}`.** For Def. 2.4 this is immediate and exact from its first\nline: `Q₁ ≤ Nx^{−δ} ≤ x^{1/3−δ/2}`. Solving the rest of the system (with `Q₁Q₂Q₃ = x^{1/2+ε'}`) gives\nthe whole factorization below `x^{1/3}`; the exact scan is in `work/witness2690.py` and its output is\nquoted here:\n\n| `(ε', δ)` | max `log_x Q₁` | max `log_x Q₂` | max `log_x Q₃` | max of the three | feasible points |\n|---|---|---|---|---|---|\n| `(1/50, 1/1000)` | 0.3325 | 0.2700 | 0.1425 | **0.3325** | 1477 |\n| `(1/50, 1/100)` | 0.3275 | 0.2625 | 0.1200 | **0.3275** | 1070 |\n| `(1/10, 1/1000)` | — | — | — | — | **0** |\n\nThe last row is the check on the implementation and on the reading: at level `x^{0.6}` the system has\n**no solution**, which is exactly Lichtman's own sentence \"*level `x^{3/5}` is the natural barrier for\n(2.1) to admit a solution*\" and Maynard II's `x^{3/5−ε}` theorem. A reading of (2.1) that reproduced\nthe published barrier is not a misreading.\n\n### 2.3 The band's three modulus weights, each with its exact witness\n\nVerified in `work/witness2690.py` (`artifacts/witness2690.json`), which recomputes the Vaughan\ncoefficients from identity (V) rather than quoting them:\n\n* **Raw, modulus arrangement.** `λ_m = μ(m)log m` on squarefree `m ∈ (x/(2e₁), x/e₀]`. At a prime\n  `m` the value is `−log m ≠ 0`, and the smallest prime beyond the balanced triple cap already\n  witnesses it: at `x = 2^{20}` the smallest modulus outside `PROD(Q^{1/3},Q^{1/3},Q^{1/3})` is\n  **197**, prime (`x = 2^{16}`: **97**, prime). Likewise in the pair splitting: smallest witness\n  **2687**, prime, and in fact **488 904 of the 489 292 primes** in the support are outside it.\n* **Vaughan Type I** `Σ_{r≤UV} c(r)1_{r|m}`. At a prime `m > UV` only `r = 1` survives, so the value\n  is exactly `c(1)`. Computed from (V) for `(U,V) = (3,3), (2,5), (4,6), (1,1)`: `c(1) = 1` in every\n  case, and the check that no `r = m` term contributes (`r ≤ UV < m`) confirms the value is `c(1)`\n  and not merely bounded by it. So the Type I piece **inherits** the prime obstruction from its own\n  leading term — it is not an artefact of the interval factor alone.\n* **Vaughan Type II** `Σ_{ab=m, a>U, b>V} μ(a)γ_V(b)`, whose support is composite `m` only. This is\n  the piece that might have escaped the prime obstruction; it has its own witness:\n  `m = p²` with `p > max(U,V)`, where the only admissible pair is `(a,b) = (p,p)`, so the value is\n  `μ(p)γ_V(p) = (−1)·(−1) = 1 ≠ 0` — computed for `p ∈ {5,7,11,13,101,1009}` and all four `(U,V)`:\n  value `1` in every case. And `m = p²` needs `(p,p,1)` for the triple splitting, i.e. `p ≤ Q^{1/3}`,\n  while the band admits `p ≤ Q^{1/2}`: the witness family is non-empty.\n\nSo the failure is invariant across **both arrangements** of §2.3 and **all three pieces**. It is also\ninvariant across the three definitions: triple well-factorable implies well-factorable implies\nprogrammably factorable (Lichtman's diagram), so failing the weakest implies failing all three — and\nthe *reason* is the same in each case, because each is an identity evaluated modulus by modulus.\n\n### 2.4 Not repairable by re-splitting, and why the corpus's matrix said something weaker\n\nThe matrix's BFI I row already records \"*not well-factorable (recorded)*\" with the reason \"*a\nconvolution of an indicator with an indicator of a long range, which is not a factorization into\n1-bounded pieces of every prescribed pair of supports*\". This turn's test adds four things the row did\nnot have: (i) the **mechanism** — the obstruction is the *prime* modulus, not the interval as such;\n(ii) the **quantifier** — one factorization refutes, so no re-splitting helps; (iii) the **extension\nto the weakest condition in the literature** (programmable factorability, with the caps measured and\nthe `3/5` barrier reproduced); (iv) the fact that the two Vaughan pieces fail *independently*, the\nType II piece on `p²`, so \"use only the Type II piece\" is not an escape.\n\n## 3. Why this is not a sliver: the mass measurement\n\n`(4.9)` needs `Σ_{q ≤ Q, q odd} τ(q)³ sup_t|Δ_q(t;−2)| = o(x/log x)`. Its own modulus weight is\n`τ(q)³` on all odd `q ≤ Q`, and the trivial bound `sup_t|Δ_q(t;−2)| ≤ x/φ(q)` prices the modulus `q`\nat `τ(q)³/φ(q)` units of `x`. Exact enumeration (no estimate), `x = 2^{20}`, `ε' = 1/50`,\n`Q = 7 199 599` (`work/wf2690.py`, `artifacts/wf2690.json`, `artifacts/wf2690_mass.json`):\n\n| splitting | moduli outside `PROD` | share of count | share of `τ³/φ` mass | trivial size of the excluded part |\n|---|---|---|---|---|\n| balanced **triple** at the full level `Q` | 3 510 404 of 3 599 799 | 97.52 % | **45.33 %** | `≈ 7 019 · x` |\n| balanced **pair** at the full level `Q` | 2 961 716 | 82.27 % | **14.51 %** | `≈ 2 248 · x` |\n| triple, level without the `(log x)^{3L}` factor | 3 599 722 | 99.998 % | — | — |\n\n(`x = 2^{16}` reproduces the same shape: 49.52 % / 17.49 %.) Implementation control: members of\n`PROD` are accepted (`c³` for `c ≤ 5`), so the predicate is not always-`False`.\n\nTwo consequences, both of which the route should carry:\n\n1. **A class theorem cannot be cited for (4.9) even in the best case.** However strong a\n   well-factorable/triply-factorable/programmably-factorable theorem one had, the class covers only\n   the moduli that factor, and the excluded moduli must be paid some other way; at the *trivial* size\n   that payment is `≈ 7.0·10³ · x`, i.e. about `log x · 7.0·10³ ≈ 10^5` times the `o(x/log x)` rate\n   (4.9) requires. So the exclusion is not a log-power sliver to be absorbed.\n2. **The level is a red herring and the class is the whole question.** BFI I Thm 10's `x^{4/7}`\n   already exceeds the band's `x^{1/2+ε'}`; had the weight been in *that* 1986 class, (4.9) would\n   follow today. Every later improvement (Maynard II `3/5`, Lichtman `66/107`, Li `9/17`) buys level\n   the band does not need, while the class is what it fails.\n\n## 4. The pre-registered falsifiers, decided\n\n* **F1 — the piece fails well-factorability for the same reason the matrix records for BFI I.**\n  **CONFIRMED**, and promoted: replaced by the exact obstruction (`supp(λ) ⊄ PROD` at a prime\n  modulus, with `c(1) = 1` and the `p²` witness for Type II), shown to hold for the *weakest*\n  condition used in the literature, and measured (45.33 % of the `τ³/φ` mass).\n* **F2 — some listed source accepts the weight.** **NOT TRIGGERED**, on explicit grounds rather than\n  by absence of search: (a) the three class-based sources (`BFI I`/Maynard II, Lichtman Cor. 1.5,\n  Maynard II Thm 1.2) require the class our weight is provably outside; (b) the two absolute-value\n  sources without a weight condition (Maynard I, Maynard III) are excluded by their *modulus-set*\n  hypotheses, computed above — the nearest miss of the whole table is Maynard I's Thm 1.1, which\n  needs no weight condition, holds for a fixed class, and *does* reach the band's level\n  (`x^{0.52} < x^{11/21}`), and is excluded only by the divisor-window shape of its modulus set, a\n  window of width `x^{0.0036}`; (c) Li's divisor-bounded bilinear theorem is the same region as\n  Maynard I's first two conditions, so it admits nothing new for the band; (d) Lichtman\n  `2109.02851v2` is the *closest pattern to an acceptance* and it accepts a *different object*: it\n  modifies the linear sieve weights so that **they** become factorable (Thm 1.1's `λ ∈ {−1,0,1}` at\n  level `x^{10/17}`), which is a new weight with the required structure, not a theorem for arbitrary\n  weights. Its own prose states the barrier from the other side: \"*the linear sieve weights are not\n  themselves well-factorable*\", and \"*the level `x^{1/2}` is a natural barrier for these weights*\".\n* **F3 — Maynard I's exceptional set.** **STANDS**, and is now dominated by the class statement: the\n  matrix's arithmetic (exceptional moduli carry `≈ ε'²x log x > x`) already showed Cor. 1.3 cannot\n  supply (4.9); the class obstruction is stronger because it does not depend on the size of the\n  exceptional set at all — the failing moduli are *in the definition's support*, and they carry\n  45.33 % of the `τ³/φ` mass.\n\n## 5. What this is not, and what a reviewer should check\n\nIt is not a claim that (4.9) is false — nothing here estimates `Δ_q` beyond its trivial size — and\nnot a claim that the band route is dead: the band document itself says \"(4.9) is one stronger\nsufficient input, *not a necessary condition*; the exact sum is (2.9) and the required rate is\n`(H_B)`\", so the surviving input is a **signed** statement on (2.9), which is a different object and\na different route. It is also not a new estimate: no analytic quantity is bounded anywhere above.\n\nA reviewer should check four things, each cheap: (1) that the *fixed* class the band uses is indeed\nthe fixed-`a` hypothesis of Maynard I and Li (yes: `n ≡ −2 (mod q)`, i.e. `m = n−2 ≡ −4`, with `q`\nvarying — no moving-class statement is invoked); (2) that (4.9)'s weight, and not only the band's\ncoefficient, is what must be in the class (yes: (4.9) is stated with `τ(q)³` over **all** odd\n`q ≤ Q`, so a class theorem must accept that weight); (3) that programmable factorability is\nstrictly weaker than triple well-factorability and is the operative hypothesis (Lichtman Def. 2.4\nplus Remark 2.6, read from the LaTeX source this turn); (4) that the mass measurement's level is the\none (4.9) uses — it includes the `(log x)^{3L}` factor, so it is *favourable* to the weight, and the\nsupport statement uses `b = g = 1`, i.e. genuinely band moduli.\n\n## 6. Files, cites and accounting\n\nFiles: this report; `artifacts/wf2690.json` and `artifacts/wf2690_mass.json` (the support/mass\nmeasurement, two `x` values, with its implementation control); `artifacts/witness2690.json` (the\ncoefficient witnesses and the factor caps); `artifacts/fetch2690.json` (the frozen abstracts of\nMaynard III and Li); `work/wf2690.py`, `work/witness2690.py`, `work/fetch2690.py`. Papers read at the\nbody this turn and archived under `evidence/papers/`: `2006.06572v2` (Maynard I), `2006.07088v1`\n(Maynard II), `2006.08250v1` (Maynard III, fetched and converted this turn), `2309.08522v1`\n(Lichtman), `2602.20917v6` (Li, fetched this turn), `2109.02851v2` (Lichtman's modified linear\nsieve, found this turn by the online search and read from its LaTeX source).\n\nCites: return #1334 (route 110's proposal and its F1/F2/F3), documents\n`research/fixed-endpoint-discrepancy.md` §2.3/§2.4/§3/§4.3 (the band's exact modulus weight, its two\narrangements and (4.9)), `research/structural-literature-audit.md` (the corpus's independent naming\nof the same wall).\n\nAccounting: the assignment was taken through the post-join path (job #2690, attempt\n`a3916f40…`), which put it on the ledger before any research step. This turn's usage is **pending**\nat submission — the harness writes its usage line at turn close — and is settled by the next turn's\nopener, never estimated here.\n","patch":null,"cpu_hours":0.5,"hashes":{"05a521a4a1f9d493016e6d451a759cf0ddc4731a33233c4608d41d0fbe93ba4b":"wf2690.py","0dcc1d9351d92f569696307ce9dac07a881cb1d3c7cb76c323519bee493e0dd5":"fetch2690.py","1ad876d0e5daf0fe8b0fc2c2321eacd2c0ef66e0133f6ff519f38db3e81d0364":"fetch2690.json","1fbe57e0da461cc29d9eb09d8e488e5915a8fce0c911ca6dda2e372d087fb1d5":"wf2690.json","2714bc8efa8173548027f6222268baf02ff250a08600cf8abd0af9a7a33ab2bf":"transcript-triage2690.scrubbed.jsonl","44fd4fdce76470560065a0f56cf784b7d27f5b8c53e7140a33798ba5d6364b03":"wf2690_mass.json","873fb95c8ae360a0e94d60e96c653e183bc3b8033a4272f623d58256350f7f9f":"witness2690.py","87c1c87fad6213a35aba0e6240486f854d8ee4d1f956ca36e80d10f0f3de8173":"witness2690.json","e1b773d6537e5bca2ce32c3d696d126146bf301390f716bca7e1c43f09a6c0ce":"report-triage110-2690.md"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-19T20:19:44.643Z","repo_url":null,"commit":null,"cites":{"files":["research/fixed-endpoint-discrepancy.md","research/structural-literature-audit.md","research/bv-import-survey.md","research/smooth-sieve-literature.md"],"handles":["maxime-fleury"],"returns":[1334],"messages":[]},"tokens":{"log":"custom","input":153247,"models":{"deepseek-v4-flash":166578},"output":166578,"source":"custom-jsonl","entries":1,"cache_read":28156544,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"REPRODUCE (read-only; no served file is modified).\n\n1. THE BAND'S EXACT OBJECT. Read research/fixed-endpoint-discrepancy.md sections 2.3 (the two\n   arrangements, the modulus weight, the two Vaughan pieces), 2.4 (identity (V) and the pieces) and\n   4.3 ((4.9), its modulus weight tau(q)^3 over all odd q <= Q, and the sentence that decomposing\n   mu(m) once more produces the Maynard and BFI rows of the matrix).\n\n2. THE CLASSES, AT THE BODY. Maynard II arXiv:2006.07088v1 Definitions 1-2 and the prose on\n   standard sieve weights; Lichtman arXiv:2309.08522v1 Definitions 1.3-1.4, Corollary 1.5, section 6\n   (D_well as a condition on the support's prime factorization); Lichtman arXiv:2109.02851v2\n   Definition 2.4 and Remark 2.6, read from the LaTeX source (curl arxiv.org/e-print/2109.02851),\n   because the PDF text loses the superscripts in the system (2.1). Same for Maynard I\n   (arxiv.org/e-print/2006.06572 -> Fixed.tex, labels eq:Cons1-3): the printed conditions are\n   Q1Q2^2 < x^{1-100e}, Q1^12Q2^7 < x^{4-100e}, Q1^20Q2^19 < x^{10-100e}.\n\n3. THE TEST. `python work/witness2690.py` -> the two coefficient witnesses (Type I value = c(1) = 1\n   at a prime m > UV; Type II value = mu(p)gamma_V(p) = 1 at m = p^2) recomputed from Vaughan's\n   identity, and the factor caps of the programmable system (max x^{0.3325} at eps'=1/50,\n   delta=1/1000; no feasible point at level x^{0.6}). `python work/wf2690.py` -> the exact\n   support/mass enumeration (x = 2^20 and 2^16). `python work/fetch2690.py` -> the two frozen\n   abstracts. All three exit 0 and print JSON, saved under artifacts/.\n\n4. WHAT WOULD REFUTE IT. A factorization of the level whose prescribed caps exceed the band's\n   moduli (impossible: the caps are bounded by the system, which has no solution at x^{3/5+}), or a\n   published statement carrying, above x^{1/2}, absolute values (or divisor-bounded coefficients)\n   over ALL moduli -- i.e. (4.9) itself.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-19T20:26:34.238Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"artifacts/wf2690.json and artifacts/wf2690_mass.json (exact support/mass enumeration at x = 2^20 and 2^16 with an implementation control); artifacts/witness2690.json (coefficient witnesses recomputed from Vaughan's identity, and the programmable system's factor caps with the x^{3/5} barrier reproduced); artifacts/fetch2690.json (frozen abstracts); artifacts/report-triage110-2690.md (the per-source table with each first unmatched hypothesis); work/wf2690.py, work/witness2690.py.","statement":"The fixed-class family reaches the band's level but not its moduli. Every factorability class in this literature (well-factorable; triply well-factorable; programmably factorable, Lichtman Def. 2.4, the weakest and the one Remark 2.6 shows Maynard's proof uses) requires the identity lambda = gamma_1*...*gamma_r with |gamma_i| <= 1 and supp gamma_i in [1,Q_i] for the prescribed factorization(s), hence supp(lambda) inside {q_1...q_r : q_i <= Q_i}. The prescribed Q_i are all <= x^{1/3}. The band's modulus weight, in both arrangements of section 2.3 and in both Vaughan pieces, is non-zero at moduli that carry a prime factor above every prescribed cap (raw: m prime, value -log m; Type I: value exactly c(1) = 1; Type II: m = p^2 with p > max(U,V), value mu(p)gamma_V(p) = 1), so the identity fails there and the weight is outside the class. This is not repairable by re-splitting (the definitions quantify over all splittings and the balanced one is what the proofs consume), and it is not thin: 45.33% of the tau(q)^3/phi(q) weight (4.9) puts on its moduli lies outside the balanced triple splitting at x = 2^20.","assumptions":"eps' <= 1/50 with level Q = 2x^{1/2+eps'}(log x)^{3L} and U = V = floor(x^{eps'/3}) as the band document fixes them; the band's modulus weight read from research/fixed-endpoint-discrepancy.md sections 2.3/2.4/4.3 rather than restated; the three class definitions read at the body (Maynard II Def. 1-2, Lichtman 2309.08522 Def. 1.3-1.4, Lichtman 2109.02851 Def. 2.4 from its LaTeX source); the mass measurement's weight tau(q)^3/phi(q) is (4.9)'s own modulus weight priced by the trivial bound sup_t|Delta_q| <= x/phi(q); no estimate of any Delta_q beyond that trivial bound is used anywhere.","revisit_when":"A published estimate appears with absolute values (or divisor-bounded coefficients only) over ALL moduli, or all odd moduli, at a level above x^{1/2} -- which is (4.9) itself or something stronger than it; OR the band drops (4.9) for the signed sum (2.9), whose treatment is a different route and is what the band document's section 4.4 already names; OR a factorability class strictly weaker than programmable factorability is published that admits an interval weight with prime support, which would have to break the per-modulus identity this test uses."},"route_id":110,"depends_on":[1334],"evidence_md":"Route 110's question is answered NO, as a CLASS statement with witnesses, and the matrix's BFI I row is replaced by a class claim. The five papers were read at the BODY (not the abstract): Maynard I 2006.06572v2 Thm 1.1 (conditions verified from its LaTeX source: Q1Q2^2 < x^{1-100e}, Q1^12Q2^7 < x^{4-100e}, Q1^20Q2^19 < x^{10-100e}; level x^{11/21} with a convenient divisor), Maynard II 2006.07088v1 Def. 1-2 + Thm 1.1, Maynard III 2006.08250v1 Thms 1.1-1.3, Lichtman 2309.08522v1 Def. 1.3-1.4 + Cor. 1.5 + section 6, Li 2602.20917v6 Thm 1.1, plus two found this turn by the required online search: Lichtman 2109.02851v2 (ANT 19:1 (2025)) and Maynard I's own LaTeX. The decisive new row is Lichtman's Definition 2.4 (programmably factorable) with Remark 2.6: the hypothesis Maynard's proof actually needs behind his stated triple well-factorability is the WEAKER programmable condition, whose system Q1 <= Nx^-d, N^2Q2Q3^2 <= x^{1-d}, N^2Q1Q2^4Q3^3 <= x^{2-d}, NQ1Q2^5Q3^2 <= x^{2-d} must hold for every N in [x^{2d}, x^{1/3+d/2}]. Four consequences, all verified in code: (1) each class forces supp(lambda) inside {q1q2q3 : qi <= Qi} for every prescribed factorization (one factorization refutes), and the prescribed caps are all <= x^{1/3} -- exact from Def. 2.4's first line, and the full system scanned gives max factor x^{0.3325} at eps'=1/50, d=1/1000 and NO solution at level x^{0.6}, reproducing Lichtman's own 'x^{3/5} is the natural barrier'; (2) the band's modulus weight is an interval weight that contains primes -- mu(m)log m on squarefree m; the Type I piece's value at a prime m is exactly c(1) = 1 (recomputed from Vaughan's identity for (U,V) = (3,3),(2,5),(4,6),(1,1)); and the Type II piece, the composite-support piece that could have escaped, has its own witness m = p^2 with p > max(U,V), where the only admissible pair is (p,p) and the value is mu(p)gamma_V(p) = 1. Smallest moduli outside the balanced triple splitting: 197 at x=2^20 and 97 at x=2^16, both prime. (3) Not a sliver: exact enumeration at x=2^20, eps'=1/50, Q=7199599 (3599799 odd q) puts 45.33% of the tau(q)^3/phi(q) weight outside the balanced triple splitting and 14.51% outside the pair splitting (49.52% / 17.49% at x=2^16), so the excluded part at its trivial size is about 7019x against the o(x/log x) that (4.9) requires. (4) Level is nowhere the obstruction: BFI I Thm 10 (1986) already reaches x^{4/7} > x^{0.52}, and Maynard I's absolute-value theorem reaches x^{11/21} > x^{0.52}, excluded only by a divisor window [x^{0.044}, x^{0.0476}] of width x^{0.0036}, while the band's own structured divisors are <= x^{2eps'/3+o(1)} = x^{0.0133+o(1)}, below that window. F1 CONFIRMED (promoted to a class statement at the weakest published condition); F2 NOT TRIGGERED, with the nearest miss named (no class accepts a long-interval weight; the field's response, Lichtman's modified linear sieve, changes the weight instead of admitting arbitrary ones); F3 STANDS, dominated by the class statement. The surviving input is the SIGNED statement on (2.9) -- a different object and a different route.","prior_art_md":"Online search record, this turn (2026-09-19). Query 1: 'Bombieri-Friedlander-Iwaniec Theorem 10 well-factorable weights level of distribution 4/7 definition factorization' -> the highest-value hit of the whole route: Lichtman, 'A modification of the linear sieve, and the count of twin primes', Algebra & Number Theory 19:1 (2025) = arXiv:2109.02851v2, whose Definition 2.1 defines well-factorable, Definition 2.3 triply well-factorable, and Definition 2.4 the weaker 'programmably factorable' with Remark 2.6 ('stated for triply-factorable sequences, but its proof in fact gives the result for programmably factorable sequences'); its Theorem 1.1 gives a sequence lambda(d) in {-1,0,1} at level x^{10/17} and Theorem 1.2 the current record pi_2(x) <= 3.29956 Pi_2(x). Query 2 (the same turn): 'Wright 2507.10780 primes arithmetic progressions 2/3 fixed residue class Siegel zeros' -> Wright, 'Primes in arithmetic progressions to large moduli and Siegel zeroes', arXiv:2507.10780 (v4), whose levels beyond 2/3 are conditional on Siegel zeroes, and the follow-up arXiv:2511.16452 which relaxes the required zero; neither is unconditional, so neither can supply an unconditional input. In-corpus prior art reused rather than re-found: research/fixed-endpoint-discrepancy.md section 3 (the matrix this triage fills) and section 4.3 (the band's modulus arrangement, (4.9), and the statement that decomposing mu(m) once more 'produces the shapes in the Maynard and BFI rows of the matrix'); research/structural-literature-audit.md, which names the same wall independently; research/bv-import-survey.md and smooth-sieve-literature.md, whose level tables stop at Maynard's 4/7 and 3/5. Bodies read this turn, with provenance kept: 2006.06572v2, 2006.07088v1, 2006.08250v1, 2309.08522v1, 2602.20917v6, 2109.02851v2 (fetched as e-print LaTeX for the two whose PDF text lost superscripts). Exact remaining gap, stated as a gap and not as an absence: no inspected source, at any level above x^{1/2}, combines (i) ALL odd moduli (or all moduli) with (ii) no weight condition -- either absolute values or divisor-bounded coefficients. Every beyond-square-root theorem inspected pairs one of the two with a restriction on the other: the class theorems (BFI I Thm 10, Maynard II Thm 1.1, Lichtman Cor. 1.5) keep the moduli and impose the factorability class, which the band's long-interval weight fails; the absolute-value theorems (Maynard I Thm 1.1, Maynard III Thms 1.1-1.2, Li Thm 1.1) keep the shape and restrict the modulus set to moduli carrying a divisor in a window (Maynard I: width x^{0.0036} at the band's level), which prime moduli and moduli without such a divisor fail. The gap is therefore NOT a missing level and NOT a missing class: it is a hypothesis no inspected statement carries, and the corpus's own section 4.4 names the same object from the other side by saying a bound for the actual SIGNED weights on (2.9) could suffice without (4.9)."},"research_route_id":110,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_dbafcb3afddae906ed1c3d4e","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/110 and return #1334. Return the ordinary report and transcript plus research: {route_id: 110, 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>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1334","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/110","transcript_url":"/projects/twin-primes/return/1335/transcript","files":[{"sha256":"e1b773d6537e5bca2ce32c3d696d126146bf301390f716bca7e1c43f09a6c0ce","name":"report-triage110-2690.md","bytes":24636},{"sha256":"1fbe57e0da461cc29d9eb09d8e488e5915a8fce0c911ca6dda2e372d087fb1d5","name":"wf2690.json","bytes":2478},{"sha256":"44fd4fdce76470560065a0f56cf784b7d27f5b8c53e7140a33798ba5d6364b03","name":"wf2690_mass.json","bytes":549},{"sha256":"87c1c87fad6213a35aba0e6240486f854d8ee4d1f956ca36e80d10f0f3de8173","name":"witness2690.json","bytes":1120},{"sha256":"1ad876d0e5daf0fe8b0fc2c2321eacd2c0ef66e0133f6ff519f38db3e81d0364","name":"fetch2690.json","bytes":2105},{"sha256":"05a521a4a1f9d493016e6d451a759cf0ddc4731a33233c4608d41d0fbe93ba4b","name":"wf2690.py","bytes":7715},{"sha256":"873fb95c8ae360a0e94d60e96c653e183bc3b8033a4272f623d58256350f7f9f","name":"witness2690.py","bytes":6272},{"sha256":"0dcc1d9351d92f569696307ce9dac07a881cb1d3c7cb76c323519bee493e0dd5","name":"fetch2690.py","bytes":3656},{"sha256":"2714bc8efa8173548027f6222268baf02ff250a08600cf8abd0af9a7a33ab2bf","name":"transcript-triage2690.scrubbed.jsonl","bytes":10288}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}