A note on sources. This attempt's assignment describes a file RANK3_TOOLS.md (Section 3: a second-moment identity for \(U_{(q)}\), which does not apply to \(Z_{(q)}\), verified in rank3_identity_check.py) and asks whether any unconditional, \(q\)-averaged tool could bound \(\sum_{2\le q\le R_0}Z_{(q)}\). Neither RANK3_TOOLS.md nor rank3_identity_check.py exists anywhere in this worktree (checked by grep -rl RANK3_TOOLS . and find . -iname '*rank3_tools*', both empty, and find . -iname '*identity_check*', also empty). What this worktree does contain, and what matches the assignment's description almost verbatim, is RANK3_ROUTE_D.md: its Section 3 derives the exact second-moment identity (D7) for \(U_{(q)}\) (Parseval, summed over residues, a Ramanujan-sum-weighted diagonal-plus-cross-term decomposition), and its Section 7, "As far as possible for \(Z_{(q)}\)," shows this does not extend to \(Z_{(q)}\): the full-circle analogue of \(Z_{(q)}\) Parseval-reduces to a quadruple additive convolution \(Y(b_1,b_2,b_3,b_4)=\sum_{n_1+n_2=n_3+n_4}d_{b_1}(n_1)\cdots\), \(d_b(n)=\Lambda(n)\mathbf1_{n\equiv b(q)}-1/\phi(q)\) — an additive-energy-type object, not a sum of squares of \(\Delta(t;q,b)\), because there is no \(K_N\)-convolution turning \(d_b\) into its own partial sums the way there is for \(U_{(q)}\)'s \(T(q,b)\) (RANK3_ROUTE_D.md §7, first point). This document proceeds on the working assumption that RANK3_ROUTE_D.md is the source being described (its content is otherwise an exact match) and that rank3_identity_check.py is either a name for verification done by hand in that document's exact-algebra derivation, or a script that exists in a sibling attempt's worktree and has not landed here. If a differently-named RANK3_TOOLS.md exists elsewhere with content beyond what RANK3_ROUTE_D.md already supplies, it was not available to produce this document, and whoever next holds both should reconcile them. This document is filed under a new name, RANK3_QUARTIC_TOOLS.md, rather than recreating RANK3_TOOLS.md, because several attempts are working this same item in parallel and cannot see each other's output; picking the name the assignment quotes risked a collision this attempt cannot detect or avoid.
Notation throughout is UPPER_BOUND.md's, RANK3_SCOPE.md's, and RANK3_ROUTE_D.md's: \(F_N,K_N,\Lambda,\psi,\mu,\phi\) as in UPPER_BOUND.md §1; \(P_{q,a},R_{q,a},U_{(q)},Z_{(q)}\) as in UPPER_BOUND.md §6 and RANK3_SCOPE.md §1; \(Q=\lfloor\sqrt N/3\rfloor\), \(R_0=Q/L\), \(L=\log N\). This document assumes RANK3_SCOPE.md, RANK3_ROUTE_D.md, and RANK3_MEAN_VALUE_TOOLS.md as already established and does not re-derive their content.
1. What is already on record
RANK3_MEAN_VALUE_TOOLS.md §6 already searched one specific candidate class — the multiplicative large sieve and its consequences (Bombieri-Vinogradov, Barban-Davenport-Halberstam) — for a \(Z_{(q)}\) tool, and found none: (BDH) is a second-moment tool, \(Z_{(q)}\) is a fourth moment, and RANK3_ROUTE_D.md §7 shows the fourth moment does not even reduce to \(\Delta(t;q,b)\)-type quantities structurally, independent of what estimate is available for \(\Delta\). RANK3_ROUTE_D.md §7 separately shows \(Z_{(q)}\)'s integrand carries no \(|P_{q,a}|^2\) decay factor, so no arc-transfer argument of the kind that works for \(U_{(q)}\) (§4 there) is available either — a second, independent obstruction on top of the missing moment estimate itself. This document's task, as assigned, is to look further than (BV)/(BDH) specifically — at any other unconditional, \(q\)-averaged tool — and to report what is found, including a negative finding if that is what results.
2. A structural fact not previously recorded: \(Z_{(q)}\) does not vanish on non-squarefree \(q\)
RANK3_ROUTE_D.md §4 records, as a byproduct of (D11), that \(U_{(q)}\) vanishes identically for every non-squarefree \(q\), because \(\mu(q)=0\) forces the leading coefficient of (D11) to zero. The same fact does not hold for \(Z_{(q)}\), for a more basic reason than any estimate: by UPPER_BOUND.md §6's own definition, \(P_{q,a}(\beta)=(\mu(q)/\phi(q))K_N(\beta)\) is identically the zero function whenever \(\mu(q)=0\), for every \(a\). Consequently \(R_{q,a}=F_N-P_{q,a}=F_N\) exactly, on every arc, for every non-squarefree \(q\), and \[ Z_{(q)}=\sum_{a\bmod q}^*\int_{I_{q,a}}|F_N|^4\,d\alpha \qquad(\mu(q)=0), \] with no bias-subtraction of any kind. This is exact algebra from UPPER_BOUND.md's own definitions (D0), not a new estimate, and is checked numerically below rather than only asserted. Two consequences:
- The non-squarefree part of rank 3's \(q\)-range is not a smaller or easier sub-problem than the squarefree part; if anything it is the same difficulty as the already-unresolved minor-arc moment \(I_Q\), restricted to a union of arcs near low-order rationals rather than to the minor arc set as a whole. Any tool that hoped to exploit \(U_{(q)}\)'s squarefree-only support (a simplification RANK3_ROUTE_D.md notes but does not use, since Route A's uniformity gap already blocks \(U_{(q)}\) everywhere) has no analogous shortcut available for \(Z_{(q)}\): a positive proportion of \(q\le R_0\) — density \(1-6/\pi^2\approx0.392\) — contributes to \(\sum_{q\le R_0}Z_{(q)}\) with no structural bias-subtraction at all.
- This sharpens, rather than changes, the conclusion of RANK3_MEAN_VALUE_TOOLS.md §6: any candidate \(q\)-averaged tool for \(Z_{(q)}\) has to work for the raw local fourth moment of \(F_N\) itself near a positive-density family of small denominators, not merely for a "residual" quantity that is small away from where the model already explains it.
Numerical check. rank3_fourth_moment_probe.py (results in results_rank3_fourth_moment_probe.json) computes the full-circle analogue \(Z^_{(q)}=\sum_a^\int_{\mathbb T}|R_{q,a}|^4\,d\beta\) exactly (DFT quadrature at \(M\ge4N\) points, exact for this band-limited quartic quantity — self-checked against Parseval's identity for the matching second moment and against doubling \(M\), both exact to float64 precision) directly from the true von Mangoldt function, for \(N=20000\) and \(q=2,\dots,30\). \(Z^*{(q)}/(\phi(q)N^3)\) — the natural per-residue trivial-order normalization — lands within a narrow, bounded band around \(\approx1.5259\) for every non-squarefree \(q\ge3\) checked (e.g. \(q=4,8,9,12,16,18,20,25\) land at \(1.525934\) to six digits) and every squarefree \(q\ge3\) with \(|\mu(q)|/\phi(q)\) small (e.g. \(q=15,30\), weight \(1/8\), land at \(1.525756\)). This is not, however, a single constant to four digits across every \(q\ge3\): \(q=2\) (\(\phi(2)=1\), weight \(|\mu(q)|/\phi(q)=1\)) shows the largest deviation (\(Z^*{(2)}/(\phi(2)N^3)\approx0.864\)), and \(q=3,6\) (weight \(1/2\), the next-largest after \(q=2\)) show a real, non-noise deviation of their own, landing at \(\approx1.4846\) — about \(3\%\) below the \(1.5259\) band, not matching it to four digits. rank3_fourth_moment_mod3_probe.py (results in results_rank3_fourth_moment_mod3_probe.json), written to check whether this \(q=3,6\) gap was a finite-\(N\) artifact of \(N=20000=2^5\cdot5^4\) (coprime to 3), finds it is not: the same \(\approx0.039\)-wide gap between \(q\in\{3,6\}\) and \(q\in\{5,10\}\) persists, without shrinking, across \(N=4000\) through \(N=128000\), and at a second, independent \(N=21000\) (divisible by 3) the full \(q\)-list shows the deviation tracking \(|\mu(q)|/\phi(q)\) directly — \(q=9,12,15,18,30\) (all divisible by 3, but each either non-squarefree, weight \(0\), or with weight \(|\mu(q)|/\phi(q)=1/8\)) sit back in the \(1.52\) band, while only \(q=3,6\) themselves (weight \(1/2\)) stay low. So the deviation from \(1.5259\) is governed by \(|\mu(q)|/\phi(q)\) — the relative size of the bias-subtraction \(P_{q,a}\) — not by squarefreeness or by \(3\mid q\) as such; see RANK3_QUARTIC_LITERATURE.md §3 for the full data and this reading. §2's structural point stands with this correction: the bias-subtraction changes the fourth moment by an \(O(1)\) multiplicative factor that shrinks toward \(1\) as \(|\mu(q)|/\phi(q)\to0\), never by an order-of-magnitude or \(N\)-power saving, for every \(q\ge2\) checked.
3. Candidate: a "quartic large sieve"
The natural next candidate, given that (BV)/(BDH) rest on the multiplicative large sieve rather than the additive one (RANK3_MEAN_VALUE_TOOLS.md §1), is to ask whether an analogous quartic large-sieve inequality exists — on either side, additive or multiplicative — playing the role for \(Z_{(q)}\) that (LS) plays for \(U_{(q)}\)'s dyadic-block bound (24)-(27), or that (BDH) plays for \(U_{(q)}\)'s low-\(q\) range. None is found; the reason is structural, not merely a citation gap.
The additive large sieve (LS) is an \(L^2\) phenomenon with no \(L^4\) analogue for general sequences. (LS)'s proof (Montgomery-Vaughan, Multiplicative Number Theory I, Theorem 6.7 and its additive dual; Iwaniec-Kowalski, Analytic Number Theory, Theorem 7.7 and surrounding material) is a Bessel's-inequality argument in Hilbert space: it holds for every sequence \(b_n\) and every set of \(\delta\)-spaced points, with no arithmetic input about \(b_n\) at all. A literal quartic analogue — \(\sum_j|\sum_nb_ne(nx_j)|^4\ll(\text{something explicit in }N,\delta)\cdot (\sum_n|b_n|^2)^2\), uniform over all sequences \(b_n\) and all \(\delta\)-spaced \(x_j\) — is false in general: take \(b_n\) supported on a generalized arithmetic progression or any set with large additive energy (e.g. \(b_n=1_{n\in\{1,\dots,K\}}\), \(K\sim\sqrt N\)); then \(\sum_n|b_n|^2\sim K\) while a single exponential sum \(\sum_nb_ne(nx)\) already reaches size \(K\) at \(x=0\), giving \(|{\cdot}|^4\sim K^4\) against \((\sum|b_n|^2)^2\sim K^2\) — no inequality of the shape (LS)'s quartic analogue would demand can hold with a constant independent of the sequence's own additive structure. This is the actual mechanism behind why the second moment (Bessel/Plancherel) is universal and structure-free while the fourth moment is not: \(L^2\) bounds are geometric (orthogonality), \(L^4\) bounds are arithmetic (they measure additive energy, i.e. how often \(n_1+n_2= n_3+n_4\) inside the support of \(b_n\)), and additive energy is exactly the kind of quantity that depends on the fine structure of the set, not just its size. For \(b_n=\Lambda(n)\), the only unconditional handle on this energy used anywhere in UPPER_BOUND.md is Vaughan's pointwise bound (V) on \(|F_N|\) itself, which RANK3_SCOPE.md §2 already shows gives no saving at small \(q\); this document does not find a second, independent unconditional handle on the same energy.
The multiplicative side (a quartic moment of Dirichlet \(L\)-functions, averaged over \(q\) and \(\chi\)) exists, but at the wrong strength and in the wrong variable for this problem. Fourth-power moments of Dirichlet \(L\)-functions on the critical line at a fixed point, averaged over \(q\le Q\) and \(\chi\bmod q\) — results in the spirit of Heath-Brown's unconditional fourth moment of \(\zeta\) (Heath-Brown, "The fourth power mean of Dirichlet's \(L\)-functions," Analysis 1 (1981), 25–32, and its antecedents) — are a real, unconditional, power-saving tool of exactly the kind (BDH) is for the second moment, and are genuinely different from (V): they come from the functional equation and approximate functional equation machinery, not from an elementary large-sieve duality, so they are not subject to the argument two paragraphs above. But what \(Z_{(q)}\) needs, after passing \(R_{q,a}\) through the explicit formula, is not a fourth moment of \(L(1/2,\chi)\) at one point — it is a fourth moment integrated over a range of height \(t\) corresponding to the arc \(|\beta|\le\delta_q\) (width \(\sim1/(qN)\), translating to a range of \(t\) growing with \(N\)), jointly averaged over \(q\) and \(\chi\). This is a genuinely harder, "doubly-averaged" (family \(\times\) height) fourth-moment problem; it is not the classical Heath-Brown-type statement, and this document does not find, or have the means in this environment to search for, a named unconditional result of that joint strength. Even granting one, two further steps this document cannot supply would remain: (i) translating an \(L\)-function-side bound back through the explicit formula reintroduces the zeros of \(L(s,\chi)\) explicitly, the same kind of ineffective or GRH-adjacent input this hunt's other documents (SW_EFFECTIVE.md, THEOREM_B_SEQUENCE.md) already find recurring as a wall for comparable questions; (ii) even a fully successful bound on the full-circle quartic moment would still need RANK3_ROUTE_D.md §7's missing arc-transfer argument, since nothing on the multiplicative side supplies one either.
4. Where this leaves rank 3's fourth moment
No candidate unconditional, \(q\)-averaged tool is found for \(\sum_{2\le q\le R_0}Z_{(q)}\), beyond confirming and sharpening RANK3_MEAN_VALUE_TOOLS.md §6's "no candidate" finding for the (BV)/(BDH) route specifically:
- The additive large sieve (LS)'s exactness is an \(L^2\)-only phenomenon (Section 3); no structure-free quartic analogue can exist, so any quartic tool must come from the primes' own arithmetic (i.e. from (V) or a strengthening of it) — RANK3_SCOPE.md's Route C, not a new averaged mechanism.
- A genuine quartic moment tool exists on the multiplicative side (q-and-\(\chi\) averaged \(L\)-function fourth moments at a point), but it answers the wrong question — a fixed-height fourth moment, not the height-integrated one \(Z_{(q)}\) needs — and this document cannot determine, without a literature search this environment does not support, whether an integrated-in-height version is known unconditionally. This is the wall: not "no such theorem exists," which this document cannot prove, but "no such theorem is identified here, and identifying one requires a literature capability outside this attempt's tools."
- \(Z_{(q)}\)'s failure to vanish on non-squarefree \(q\) (Section 2, checked numerically) means even a hypothetical tool restricted to squarefree \(q\) — mirroring \(U_{(q)}\)'s free simplification — would leave a positive-density sub-range of rank 3 completely uncovered, at the same difficulty as the unresolved minor-arc moment \(I_Q\).
- The numerical measurement (Section 2) of \(Z^*_{(q)}\), the full-circle analogue, shows no visible saving at all — squarefree or not — for every \(q\ge3\) up to \(N=20000\), and no visible \(N\)-power or \(\log N\)-power saving at the single worst case \(q=2\) across \(N=2000\) to \(64000\). This is consistent with, and adds direct measurement to, the conclusion that no unconditional cancellation is being left unexploited by the tools already named and found insufficient (RANK3_SCOPE.md §2, RANK3_MEAN_VALUE_TOOLS.md §6): the trivial order is not merely the best proved bound, it appears to be the true order at this computational scale, for every \(q\ge3\) checked.
What would settle this precisely: (i) a named, unconditional, height-integrated and \(q\)-averaged fourth-moment (or additive-energy) bound for \(\sum_{n_1+n_2=n_3+n_4\le N}\Lambda\)-weighted quadruples restricted to residue classes mod \(q\), summed over \(q\le R_0\) — Section 3 explains precisely why this cannot be a structure-free large-sieve statement and must instead come from prime-specific input; (ii) independently, a non-crude arc-transfer argument for the fourth moment, which RANK3_ROUTE_D.md §7 already shows has no counterpart to the one used for \(U_{(q)}\); (iii) separately, whatever (i) supplies must also cover the non-squarefree sub-range, where no bias-subtraction exists at all (Section 2). None of the three is supplied here. This is consistent with, and extends, RANK3_SCOPE.md §3's conclusion that no route it names is costed to completion, and RANK3_MEAN_VALUE_TOOLS.md's finding that \(Z_{(q)}\) has no known candidate tool at all — this document looked further afield (a genuinely different large-sieve mechanism, and the higher-moment \(L\)-function literature) and still found none, for a structural reason (Section 3) rather than only an unsuccessful search.
This document is: an exact structural observation from UPPER_BOUND.md's own definitions (Section 2, first half), a direct numerical measurement backing it (Section 2, second half; rank3_fourth_moment_probe.py, results_rank3_fourth_moment_probe.json), and a structural argument (Section 3) about why the second moment's universal large-sieve mechanism has no quartic analogue, together with an explicit, honestly-labeled gap in what this document could check about the \(L\)-function-side alternative. It assumes and establishes nothing about zeros of \(L\)-functions or the Riemann Hypothesis, and does not weaken the definitions of \(Z_{(q)}\) or \(R_0\).