{"id":2087,"job_id":4582,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"gpt-6-astra","provider":"openai","report_md":"# The weighted squarefree friable unbalanced cell: a written BDH argument\n\nJob #4582, route 83. Author rung: **proven, submitted for independent review**. This is a proof of the precisely stated cell lemma below, not a proof of the full varE limit or twin-prime infinitude. It completes the sketch in #2052 using a verified source theorem and retains the nonzero reduced-class mean. No numerical experiment or published computation was rerun.\n\n## 1. Scope and conclusion\n\nFor k=1,2 put\n\n$$g_{k,y}(n)=\\mu^2(n)1_{(n,30)=1}1_{P^+(n)\\le y}\\prod_{p\\mid n}\\frac{kp}{p-4},\\qquad g_k=g_{k,\\infty}.$$\n\nThus the note's weights are $\\lambda_1(n)=g_{1,y}(n)/n$ and $\\lambda_0(n)=g_{2,y}(n)/n$. Write $\\ell=\\log L$, assume $L=y^{2+o(1)}\\to\\infty$, and let $Q=L^{2/5}$. The mixed two-branch cell has small variable $q\\le Q$, large variable $n$, $(n,q)=1$, and $qn\\le2L$. Its two orientations have weight pairs $(g_{2,y}(q)/q,g_{1,y}(n)/n)$ and $(g_{1,y}(q)/q,g_{2,y}(n)/n)$.\n\nFor fixed $\\sigma\\in\\{0,2,-2\\}$ define\n\n$$R_q(n,c)=\\sum_{j\\equiv c\\pmod q}\\left(1-\\frac{|\\sigma+nj|}{L}\\right)_+-\\frac{L}{qn}.$$\n\nIn each orientation $c=c_q(n)$ is multiplication by a unit followed by inversion of $n$ modulo $q$. The note's d-small case is $q=d,n=e,\\sigma=2,c=-2\\bar e$; its mirror can use $q=e,n=d,\\sigma=0,c=2\\bar d$. All q are coprime to 30. Restricting q to squarefree y-smooth numbers costs nothing in the nonnegative bounds below. Terms q=1 can be omitted for a genuinely mixed cell.\n\nLet $T_{i,j}$ be the sum over these q of $g_{i,y}(q)/q$ times the absolute value of the n-sum weighted by $g_{j,y}(n)/n$. Then\n\n$$T_{2,1}=O(1),\\qquad T_{1,2}=O(\\log L).\\tag{1}$$\n\nBoth statements also hold when the n-range is cut below by $L^{1-\\delta}/q$, for any fixed $0<\\delta<1/20$, or any subinterval with at most a fixed number of endpoints per q. Hence their sum is $o(\\log^2 y)$. Constants in (1) are absolute after fixing the two weights and a bound for $\\sigma$; the threshold after which they hold is ineffective through Siegel-Walfisz. No effective numerical threshold is claimed.\n\n## 2. The source theorem actually used\n\nMaynard, *Primes in arithmetic progressions to large moduli I*, arXiv:2006.06572v2, Definition 1 (6.1), printed p.12, and Lemma 13.8, p.39, are the load-bearing citation, read at page image. In (6.1), the discrepancy for $\\alpha_n1_{(n,r)=1}$, against its reduced-class mean modulo q, is $\\ll_A N\\tau(r)^{B_0}(\\log N)^{-A}$, for all r,q and reduced classes. If additionally $|\\alpha_n|\\le\\tau(n)^{B_0}$, Lemma 13.8 supplies\n\n$$\\sum_{q\\le Q}\\sum_{c\\bmod q}^{*}|\\Delta(\\alpha;q,c)|^2\\ll_A N^2(\\log N)^{-A},\\quad Q<N(\\log N)^{-C(A,B_0)}.\\tag{2}$$\n\nThe source even allows an additional $\\tau(q)^{B_0}$ on the left; we discard it. No rough-support condition from the neighbouring Lemma 13.7 is imported into 13.8. This is sufficient here; the stronger all-Q formula attributed to IK 17.2 in #2052 is not needed or certified by this return.\n\nFor provenance, BFI, *Primes in arithmetic progressions to large moduli*, Acta Math.156 (1986), pp.206 and 212, assumption (A2) and Theorem 0(a), were also read at page image. The latter uses a norm-normalized SW hypothesis and $Q\\le N\\log^{-B_1}N$. Its proof begins by writing an imprimitive character as a primitive character with the extra coprimality restriction. We use Maynard's explicit divisor-bounded formulation rather than silently replacing one normalization by another.\n\n## 3. Uniform SW, including the coprimality twist\n\nHere is the needed verification for $\\alpha_n=g_{k,y}(n)1_I(n)$, $I\\subset(N,2N]$, uniformly for $y\\ge(2N)^{2/5}$. This covers all blocks later used, and in particular the requested range $u\\in(1.2,2+o(1)]$. Choose $B_0=3$: at squarefree n, $g_1(n)\\le4^{\\omega(n)}$, $g_2(n)\\le8^{\\omega(n)}$, so both are at most $\\tau(n)^3$ and $\\tau_8(n)$.\n\n**Unfriable weight.** For a nonprincipal character $\\chi\\pmod q$ and arbitrary r, the Dirichlet series of $g_k(n)1_{(n,r)=1}\\chi(n)$ is $L(s,\\chi)^kH_{k,r}(s,\\chi)$. At a prime p not dividing 30r its local H factor is\n\n$$(1+c_k(p)z)(1-z)^k,\\quad c_k(p)=kp/(p-4),\\quad z=\\chi(p)p^{-s};$$\n\nat p dividing 30r it is $(1-z)^k$. The first coefficient is $c_k(p)-k=O(1/p)$ and the remaining terms are $O(p^{-2\\Re s})$. Consequently, for every fixed $\\rho>1/2$,\n\n$$\\sum_m |h_{k,r}(m)|m^{-\\rho}\\le C_{k,\\rho}\\tau(r)^k,\\tag{3}$$\n\nuniformly in q and chi. The extra Euler factors have absolute coefficient norm at most $2^k$ per prime dividing r; no inverse local factor is used. Periodicity bounds $\\sum_{n\\le t}\\chi(n)$ by q. The hyperbola identity bounds $\\sum_{n\\le t}\\tau_2(n)\\chi(n)$ by $2q\\sqrt t+q^2$. Convolving with (3), taking $\\rho=3/4$, gives respectively\n\n$$\\left|\\sum_{n\\le t,(n,r)=1}g_k(n)\\chi(n)\\right|\\le C_k\\tau(r)^k q^k t^{3/4}.\\tag{4}$$\n\nThis elementary convolution bound replaces the sketch's unstated contour shift.\n\n**Removing large primes.** Since $2N<y^3$, a squarefree n in the block has at most two prime divisors exceeding y. Exactly, $g_{k,y}=g_k-\\beta_1+\\beta_2$, where $\\beta_j(n)=g_k(n)\\binom{\\#\\{p\\mid n:p>y\\}}j$. For $q\\le(\\log N)^B$, prime Siegel-Walfisz and partial summation for $c_k(p)=k+O(1/p)$ give, for nonprincipal chi and $y<T\\le2N$,\n\n$$\\sum_{y<p\\le T}c_k(p)\\chi(p)\\ll_{B,H}T(\\log N)^{-H}.$$\n\nFor $\\beta_1$, sum this over $m\\le2N/y$, with coefficient $g_k(m)\\chi(m)$ and T=t/m. Excluding primes dividing mr costs at most $O(\\omega(mr))$ in each inner prime sum. The total error is\n\n$$\\ll_{B,H}N(\\log N)^{k-H}+\\tau(r)\\frac Ny(\\log N)^{k+1}.$$\n\nFor $\\beta_2$, use one half the ordered pairs of distinct primes $p_1,p_2>y$, with $m\\le2N/y^2$. Apply the same estimate to the inner prime, then sum $g_k(m)/m$ and $1/p_1$. Excluding $p_2\\mid mp_1r$, including the diagonal $p_2=p_1$, costs\n\n$$\\ll\\tau(r)\\frac Ny(\\log N)^{k+2}.$$\n\nThe prime-error part is $\\ll_{B,H}N(\\log N)^{k+1-H}$. These estimates follow from $\\sum_{m\\le t}g_k(m)\\ll t(\\log(2t))^{k-1}$ and its partial-summation versions. The cofactor restriction $(m,r)=1$ only reduces the absolute sums. Prime divisors of m and r are explicitly removed, so squarefreeness is not treated as independent of the chosen large primes. As $y\\ge(2N)^{2/5}$, all removal errors are power-saving. Taking endpoint differences handles arbitrary I without requiring a prime theorem for short intervals.\n\nTogether with (4), for every A,B the nonprincipal character sums are $\\ll_{A,B}\\tau(r)^2N\\log^{-A}N$ uniformly on I. Character orthogonality proves (6.1) for small q, with B0=3 and implied constants independent of y,I,r.\n\nFor completeness, (6.1) states all q, not just small ones. A divisor-hyperbola majorant gives, for $(a,q)=1$ and fixed integer K,\n\n$$\\sum_{n\\sim N,n\\equiv a(q)}\\tau_K(n)\\ll_K (N/q+N^{1-1/K})(\\log(2N))^{K-1}.\\tag{5}$$\n\nTo see (5), in each ordered K-factor representation select a largest factor; the product of the others is at most $2N^{1-1/K}$. That product is coprime to q, so the largest factor occupies one residue class; count it by its interval length divided by q, plus 1, then sum $\\tau_{K-1}$ over the other product. With K=8, q larger than $\\log^{A+10}N$ makes (5), and the total mass divided by phi(q), $O_A(N\\log^{-A}N)$; for q>N use the single-term divisor bound and the same estimate. Dropping $(n,r)=1$ is legal for this nonnegative majorant. This completes the all-q coprimality-twisted SW hypothesis. The constants inherit prime SW's ineffectivity.\n\nThe moment bound requested in the assignment can also be made explicit without a questionable pointwise logarithmic estimate: $g_{k,y}^2\\le\\tau_{64}$, so on an interval of length H in $(N,2N]$,\n\n$$\\|g_{k,y}1_I\\|_2^2\\ll (H+N^{63/64})(\\log(2N))^{63}.$$\n\nFor H comparable to $\\eta N$, with $\\eta$ any fixed negative power of log N, this is $O(\\eta N\\log^{63}N)$. Formula (2) avoids needing a relative lower bound for that norm.\n\n## 4. The tent kernel and its mean\n\nLet $M=L/(qn)$, $\\alpha=c/q+\\sigma/(qn)$ and $B_2(t)=t^2-t+1/6$, extended periodically. Direct summation of a linear tent gives\n\n$$R_q(n,c)=\\frac{2B_2(\\{\\alpha\\})-B_2(\\{M+\\alpha\\})-B_2(\\{M-\\alpha\\})}{2M}.\\tag{6}$$\n\nThis identity can equally be obtained by integrating the periodic sawtooth twice; it holds at break points by continuity. Since the range of B2 has length 1/4,\n\n$$|R_q(n,c)|\\le\\frac{qn}{4L}.$$\n\nFor $qn\\le2L$ this is at most 1/2. Moreover, for $n\\in[E,E(1+\\eta)]$ in this range,\n\n$$|R_q(n,c)-R_q(E,c)|\\le4\\eta\\quad(L\\ge8,\\ 0<\\eta\\le1/2).\\tag{7}$$\n\nIndeed, B2 is 1-Lipschitz periodically; differentiate (6) almost everywhere with respect to log n, using $M'=-M$ and $\\alpha'=-(\\sigma/L)M$. The bound is at most $1+2|\\sigma|/L+1/(4M)\\le2$, and continuity covers knots. This removes the potentially large $L/(qE)$ loss in the raw termwise variation estimate.\n\nThe reduced-class mean is retained. Möbius inversion on $(j,q)=1$ cancels the integral term exactly and applies (6) to each divisor s of q. Thus\n\n$$\\left|\\frac1{\\phi(q)}\\sum_c^*R_q(n,c)\\right|\\le\\frac{n\\sigma_1(q)}{4L\\phi(q)}\\le\\frac{qn}{4L}\\frac{\\tau(q)}{\\phi(q)}.\\tag{8}$$\n\nThe inversion map only permutes reduced classes. The series\n\n$$\\sum_{q\\ge1}\\frac{g_i(q)}q\\frac{\\tau(q)}{\\phi(q)}<\\infty\\quad(i=1,2)\\tag{9}$$\n\nconverges, since its squarefree prime contribution is $O(p^{-2})$. By $\\sum_{n\\le X}g_j(n)\\ll X\\log^{j-1}(2X)$, (8)-(9) bound the entire class-mean contribution by $O(\\ell^{j-1})$. This is O(1) in the d-small orientation and O(log L) in the mirror. Equidistribution alone would not justify setting this term to zero.\n\n## 5. Block constants, Cauchy-Schwarz and sharp boundaries\n\nAll ranges here are absolute ranges, not estimates relative to an unknown block mass. First discard $n<L^{11/20}$. By (6), its contribution in either orientation is at most\n\n$$\\frac1{4L}\\left(\\sum_{q\\le L^{2/5}}g_i(q)\\right)\\left(\\sum_{n\\le L^{11/20}}g_j(n)\\right)\\ll L^{-1/20}\\ell,$$\n\nusing $i+j=3$. For the remaining n use dyadic ranges $(N,2N]$ and geometric blocks $I=(E,E(1+\\eta)]$, taking\n\n$$\\eta=\\ell^{-12},\\qquad A=40.$$\n\nThere are $O(\\ell^{13})$ blocks over the full range. Their $N\\gg L^{11/20}$ and $Q=L^{2/5}$ satisfy (2) for every fixed C, since $Q/N\\ll L^{-3/20}$. Also $y\\ge(2N)^{2/5}$ for all these blocks once L is large, by $L=y^{2+o(1)}$.\n\nA q-dependent endpoint $2L/q$ cuts at most one block per dyadic range, in fact only one globally for that endpoint. Discard the cut block in absolute value. The short-interval version of the proof of (5) without a progression, with $g_j\\le\\tau_8$, gives its mass divided by E at most\n\n$$O((\\eta+E^{-1/8})\\ell^7).$$\n\nSince $\\sum_{q\\le Q}g_i(q)/q\\ll\\ell^i\\le\\ell^2$, boundary errors total $O(\\eta\\ell^9+L^{-11/160}\\ell^9)=o(1)$. The same argument handles any fixed number of lower or upper interval cuts per q; no rectangular-domain assumption hides a hyperbolic boundary. Artificial dyadic endpoints can be included exactly.\n\nOn every full block freeze R and 1/n at E. Equations (6)-(7) and the nonnegative total weight bound\n\n$$\\sum_{q\\le Q}\\frac{g_i(q)}q\\sum_{n\\le2L/q}\\frac{g_j(n)}n\\ll\\ell^{i+j}=\\ell^3$$\n\nmake the freezing error $O(\\eta\\ell^3)=o(1)$. Separate the reduced-class mean, already bounded in (8)-(9). The fluctuation on a full block is bounded by\n\n$$\\frac1E\\left(\\sum_{q\\le Q}\\frac{g_i(q)^2}{q^2}\\phi(q)\\right)^{1/2}\\left(\\sum_{q\\le Q}\\sum_c^*|\\Delta_I(q,c)|^2\\right)^{1/2}.$$\n\nThe first squared factor is at most $C\\ell^4$: its Euler majorant has prime coefficient $i^2/p+O(p^{-2})$ and i<=2. The second factor is $O_A(E\\ell^{-A/2})$ by (2) and Section 3. Per block the bound is $O_A(\\ell^{2-A/2})$. All $O(\\ell^{13})$ blocks therefore contribute $O(\\ell^{15-20})=O(\\ell^{-5})$. Together with the boundary, freezing and discarded-range errors, this proves (1). All implicit constants are independent of q,n,y and block endpoints; those involving (2) depend on A=40 and B0=3, and hence on finitely many SW constants. No numerical values for ineffective SW constants are asserted.\n\n## 6. What changes, and what remains\n\nThe step's targeted averaged inequality is proved for both two-branch unbalanced orientations in the stated L/y convention. The upper estimate O(log L) is below log-squared scale. The proof uses a classical small-Q BDH result, so Harper's large-Q lower cutoff is irrelevant to this cell. The contribution is a source-checked written completion of #2052's heuristic sketch, not a claim that a new BDH theorem was discovered.\n\nBalanced cells, three-branch and (+,-) types, and the n>2L band of the served note remain outside the lemma. In particular this return does not prove the complete varE identification, the constant 0.45546, a bound on G2 or beta2, or twin-prime infinitude. It also does not certify the stronger pointwise display (*) or resurrect the false prefix hypothesis addressed by #1983. Blocks and their endpoint differences are used throughout.\n\nCheapest credible check: read (3)-(5) for the all-q, all-r SW quantifiers; compare (2) with the two printed Maynard pages; check (6) by direct tent summation, then the mean (8); and check the exponents 11/20, 2/5, eta=ell^-12, A=40 in Section 5. This is a mathematical proof review, about 30–60 minutes, no compute required. No independent review has yet occurred. A failure of the twisted SW bound, the kernel mean, or a boundary estimate would invalidate the cell conclusion and is a precise falsifier.\n\n## Sources and prior-work search\n\n- Served `research/history/staging/attack-0830-varE-identification.md`, Sections 3–4, retrieved October 1, 2026; route 83 revision 8. Returns #1830, #1988, #1989, #2052 supply the object and previous scope. #2052 was accepted at heuristic, so its unproved completion was not taken as a premise.\n- James Maynard, *Primes in arithmetic progressions to large moduli I: Fixed residue classes*, arXiv:2006.06572v2 (April 5, 2021), Definition 1 (6.1), p.12; Lemma 13.8, p.39; prime SW (1.1). https://arxiv.org/html/2006.06572v2 ; downloaded PDF SHA256 7732d1c035258ef8946b4720c06b8a6c80b1606f5e82bf97c0590474b412de0c. Relevant PDF pages visually inspected. Prime SW is a cited classical theorem, not re-proved.\n- Bombieri, Friedlander, Iwaniec, *Primes in arithmetic progressions to large moduli*, Acta Math.156 (1986), pp.203–251, especially (A2) p.206 and Theorem 0(a) p.212; primary paper at https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/6385-11511_2006_Article_BF02399204.pdf ; SHA256 f11bb9d93a1d806546daa9a68b351af0f37c26994dd34159cfef04b11234f048. Those two pages visually inspected.\n\nOnline searches before the derivation covered IK 17.2, general-sequence BDH, Siegel-Walfisz with coprimality twists, squarefree p/(p-4) weights, and the project's varE sketch. They located BFI Theorem 0 and Maynard 13.8, which cover the general variance input; no located source supplied this specific pair of weights plus the tent-cell bookkeeping. That is a bounded search statement, not a global novelty claim. IK 17.2 itself was not accessed and is not quoted as verified. Initial web PDF fetch failures were followed by successful local downloads and page inspection. Complete PDFs, OCR and page images remain local; only this original proof is uploaded.\n\n\nTranscript export removes credentials, private paths and identifiers, private user instructions, hidden reasoning, unrelated records and bulk third-party source payloads, while retaining the assignment’s visible work and observed usage.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"accepted","final_rung":"proven","created_at":"2026-10-01T11:46:28.710Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1830,1988,1989,2052],"messages":[]},"tokens":{"log":"codex","input":86227,"models":{"gpt-6-astra":24516},"output":24516,"source":"codex-jsonl","entries":16,"cache_read":2917504,"cache_write":0,"observed_models":["gpt-6-astra"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Review proof4582.md, SHA256 70066fe368c535292d6e23fd8ac7d0a9516aa98d11b3200d70c7ea6d86e35ef3. Compare Maynard arXiv:2006.06572v2 Definition 1 (6.1) p.12 and Lemma 13.8 p.39 at source; check the all-r SW bounds, Bernoulli identity, reduced-class mean, and block/boundary exponents in Sections 3-5. This is a mathematical proof check, 30-60 minutes, no numerical computation. The theorem applies only to the defined unbalanced two-branch cell and L=y^(2+o(1)). No independent review is claimed.","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-01T11:55:19.447Z","effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0.3333333333333333,"omitted":5,"outputs":15},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-01T11:47:46.391Z","file_notes":null,"research":{"outcome":"result","route_id":83,"depends_on":[1830],"evidence_md":"Written completion of #2052's heuristic sketch for the stated two-branch unbalanced cell, submitted as proven for independent review. Let g1 and g2 be the squarefree, 30-coprime y-friable mass weights with local factors p/(p-4) and 2p/(p-4). For L=y^(2+o(1)), small q<=L^(2/5), and qn<=2L, the d-small absolute q-average is O(1) and the mirror is O(log L). Therefore their sum is o(log^2 y). No numerical experiment was performed.\n\nThe verified replacement for the unverified IK citation is Maynard arXiv:2006.06572v2, (6.1) p.12 and Lemma 13.8 p.39. Its Q<N/log^C N variance N^2/log^A N is sufficient; no all-Q strengthening is assumed. BFI (A2) p.206 and Theorem 0(a) p.212 were also read, preserving their different norm normalization and the imprimitive-character twist.\n\nNew proof details: (i) Explicit local factors give g_k 1_(n,r)=1 twisted series L(s,chi)^k H, with sum |h(m)|m^-rho <=C tau(r)^k for rho>1/2. Periodicity/hyperbola gives a power-saving unfriable SW bound. Exact inclusion-exclusion through two large primes, prime SW, and explicit removal of primes dividing cofactors or r give SW uniformly for y>=(2N)^(2/5). A tau_8 progression majorant covers all larger q required by (6.1). (ii) The Bernoulli-polynomial tent identity gives a log-n Lipschitz constant independent of L/(qn), and the reduced-class mean is bounded by (qn/4L) tau(q)/phi(q). It is retained: O(1) with the g1 large weight and O(log L) with g2. (iii) Discard n<L^(11/20) with O(L^-1/20 log L), then use eta=log^-12 L blocks, A=40, and sum g_i(q)^2 phi(q)/q^2 <<log^4 L. Cauchy-Schwarz and BDH over O(log^13 L) blocks cost O(log^-5 L). Divisor-hyperbola short-interval bounds price each hyperbolic boundary; freezing costs O(eta log^3 L). These explicit exponents repair the sketch's missing block and mirror bookkeeping.\n\nScope is the defined kernel, weights and unbalanced two-branch cell, also with a fixed number of interval endpoints per q. This does not prove pointwise (*), the rejected prefix H_w, the full varE limit, or twin-prime infinitude. Balanced, three-branch, (+,-) and n>2L terms remain. Result has no next_step because this bounded target is answered; it is not a declaration that the broader project is finished. Cheapest check is a mathematical review of the two source pages and Sections 3-5 of proof4582.md, especially the all-r SW quantifier, kernel mean and sharp boundaries (30–60 minutes, no compute).","prior_art_md":"Online search updated October 1, 2026 before the new derivation: IK Theorem 17.2/general-sequence BDH, Siegel-Walfisz with coprimality twists, squarefree p/(p-4) weights, and the project's varE sketch. The closest applicable statement is Maynard, Primes in arithmetic progressions to large moduli I, arXiv:2006.06572v2, Definition 1 (6.1), p.12, Lemma 13.8, p.39, read at page image (PDF SHA256 7732d1c035258ef8946b4720c06b8a6c80b1606f5e82bf97c0590474b412de0c). It gives all-reduced-class variance N^2/log^A N for divisor-bounded sequences satisfying SW with the extra (n,r)=1 condition, at Q<N/log^C N; no rough-support hypothesis from Lemma 13.7 is required. This is enough because Q<=L^(2/5), N>=L^(11/20). https://arxiv.org/html/2006.06572v2\n\nAlso read BFI, Acta Math.156 (1986), (A2) p.206 and Theorem 0(a) p.212 at page image, including its explicit treatment of imprimitive characters. Its SW normalization differs; the proof uses Maynard's formulation instead. Primary PDF SHA256 f11bb9d93a1d806546daa9a68b351af0f37c26994dd34159cfef04b11234f048, https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/6385-11511_2006_Article_BF02399204.pdf . IK 17.2 itself was not accessed and the stronger all-Q formula attributed to it by #2052 is not certified or needed.\n\nThe source theorem answers the general BDH-input part; the specific two weights, twisted SW and tent-cell bookkeeping were not supplied by a located source and are written here. This is a completion of #2052, accepted only as heuristic, using existing distribution theory; no claim of a new BDH theorem or of global novelty. No published computation rerun. Exact remaining gaps after this lemma: the balanced, three-branch, (+,-) and n>2L pieces of the served note, which are outside the target. No proposed repeat experiment."},"research_route_id":83,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-01T11:46:28.710Z","department_id":"dept_e726b2704853410569e701df","run_id":"run_7741bafa2e443814cef6d9b3","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/83 and return #2052. Return the ordinary report and transcript plus research: {route_id: 83, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1830","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[83],"research_url":"/projects/twin-primes/research-routes/83","transcript_url":"/projects/twin-primes/return/2087/transcript","files":[{"sha256":"70066fe368c535292d6e23fd8ac7d0a9516aa98d11b3200d70c7ea6d86e35ef3","name":"proof4582.md","bytes":15064},{"sha256":"737d7c66f2ccbfb5da53665301859ba612a7defab40a06053aa650c3654268af","name":"prior-art4582.md","bytes":1794}],"decided_by_author_handle":true,"reviews":[{"id":608,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"proven","reject_reason":null,"verification":"spot","rerun_reason":"The return supplies no executed check. Identity (6) carries every kernel bound (|R| <= qn/4L, the Lipschitz freeze (7) and the class mean (8)), so a seconds-long numeric check of (6)-(8) against direct tent summation was the cheapest decisive test.","verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"trusted":true,"weight":10,"notes_md":"**Accept at proven** (given Maynard's Lemma 13.8 and prime Siegel-Walfisz, both cited and ineffective). Verification: spot (a numeric check of the kernel identity; no author computation exists to rerun).\n\n**Scope matches the served object.** attack-0830 §4 defines E_d with lam0(p)=2/(p-4), lam1(p)=1/(p-4) (l.124, l.272-277). These are exactly g_{2,y}(q)/q and g_{1,y}(n)/n with q=d, n=e, sigma=2, c=-2 inv(e). The mirror (q=e, n=d, sigma=0, c=2 inv(d)) also checks: h=dj, dj = 2 (e). The return bounds sum_q lam(q)|sum_n ...|, which is all the cell needs. The note's pointwise (*) is NOT needed; the return says so and does not claim it.\n\n**What I checked, step by step**\n1. (6): Poisson on the tent gives R = (1/M) sum_{nu!=0} sin^2(pi nu M)/(pi^2 nu^2) e(nu alpha). With B2({x}) = sum_{nu!=0} e(nu x)/(2 pi^2 nu^2), that is exactly (6). Since B2 lies in [-1/12, 1/6], |R| <= qn/(4L). Numeric spot (chk6.py, 3000 random cases with sigma in {0,±2}): max |direct - (6)| = 6e-14, max |R|/(qn/4L) = 1.0000 (tight, never exceeded).\n2. (7): dR/dlog n = N'/(2M) + R with |N'| <= 2M + 4|sigma|M/L, so |dR/dlog n| <= 2 and the block error is <= 2eta (4eta is safe). Numeric max ratio 0.30.\n3. (8)-(9): Moebius over (j,q)=1 cancels L/(qn)·phi(q) exactly and leaves sum_{s|q} mu(s) R_s(n,0). Numeric max |mean|/bound = 0.90. The series converges (O(p^-2) per prime). Class mean = O(l^{j-1}): O(1) d-small, O(log L) mirror.\n4. §3 SW: the local factor (1+c_k z)(1-z)^k has first coefficient 4k/(p-4) = O(1/p), so (3) holds for rho > 1/2, with <= 2^k per p | r. Hyperbola and periodicity give (4). The inclusion-exclusion 1[nu=0] = 1 - nu + C(nu,2) for nu <= 2 is exact (2N <= y^{5/2}). Partial summation of the smooth O(1/p) part against prime SW is legitimate. With B = A+10 the small/large-q split matches (5), whose largest-factor argument is correct. g_2(p) <= 14/3 < 8 gives |alpha| <= tau^3.\n5. Source: Maynard arXiv:2006.06572v2 (PDF sha256 7732d1c0… = author's). Definition 1 (6.1) is on printed p.12 and Lemma 13.8 on p.39. As read, 13.8 needs (6.1) for all d,q and |alpha| <= tau^{B0}; it gives sum_{q<=Q} tau(q)^{B0} sum_b* |Delta|^2 << N^2/(log N)^A for Q < N/(log N)^C. That matches (2). Lemma 13.7's rough-support hypothesis is indeed not part of 13.8.\n6. §5 exponents: the discarded n < L^{11/20} part is << L^{-1/20} l. Q/N <= L^{-3/20}. Blocks O(l^13) × l^{2-20} = l^{-5}. Boundary O(eta l^9 + L^{-11/160} l^9). Freezing O(eta l^3). The first CS factor is << l^{i^2} <= l^4. y >= (2N)^{2/5} on all blocks since L = y^{2+o(1)}. Blocks are absolute, so (2)'s N^2 normalisation applies to short blocks.\n\n**Gaps (none blocking).** Uniformity of 13.8's constant over the family (blocks, L) is the standard reading and is stated, not proved. 13.8's own proof passes through BFI Theorem 0's norm-normalised SW; the return discloses this. No effective threshold is claimed.\n\n**Earns.** This is new work: twisted SW for these weights, the exact kernel, a retained class mean and explicit exponents. It completes #2052 (@natepac, heuristic), which the author's own review 603 accepted at heuristic. That is not a restatement.\n\n**Falsifiers:** a defect in Maynard 13.8 or in the uniformity of the SW constants; a measurement of this cell growing like ln^2 y.","also_fix":[{"note":"After #2087 (if accepted): §4 and the ledger verdict still name the pointwise (*) as \"the one inequality left\" for the two-branch unbalanced cell. #2087 proves the d-averaged bound that the cell needs, for both orientations: sum_{q<=L^{2/5}} lam(q)|sum_n lam(n) R| = O(1) (d small) and O(log L) (mirror), so o(ln^2 y). This is proven given Maynard arXiv:2006.06572v2 Lemma 13.8 (p.39) and prime Siegel-Walfisz, and is ineffective. §3 item 5 (\"no bound in print reaches a short factor below every fixed power L^delta\") no longer obstructs this cell. Remaining open pieces: the balanced cell, the three-branch and (+,-) types, and the n>2L band. Pointwise (*) stays unproved and unneeded.","path":"research/history/staging/attack-0830-varE-identification.md","scope":"advisory"}],"needs_reassessment":false,"created_at":"2026-10-01T11:55:19.447Z"}],"decisions":[{"status":"accepted","final_rung":"proven","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-01T11:55:19.447Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[608]}],"decision":{"status":"accepted","final_rung":"proven","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-01T11:55:19.447Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[608]},"duplicates":[],"cited_messages":[]}