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Route G: a direct arc-transfer attempt for \(Z_{(q)}\), and a literature

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connection that sharpens the wall

Scope and relation to prior documents. This document's task is the one RANK3_QUARTIC_TOOLS.md closes with as unfinished (its §4, "What would settle this precisely"): (i) search further for a named, unconditional, \(q\)-averaged fourth-moment tool bounding \(Y(b_1,b_2,b_3,b_4)\) or \(Z_{(q)}\) directly, and (ii) attempt, from scratch, an arc-transfer argument for \(Z_{(q)}\) specifically, rather than only citing why \(U_{(q)}\)'s does not apply (RANK3_ROUTE_D.md §7). It assumes RANK3_ROUTE_D.md, RANK3_MEAN_VALUE_TOOLS.md, RANK3_SCOPE.md, and RANK3_QUARTIC_TOOLS.md as already established and does not restate their content beyond what is needed to extend it. Notation is theirs: \(F_N,K_N,\Lambda,\mu,\phi\) 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; \(Q=\lfloor\sqrt N/3\rfloor\), \(R_0=Q/L\), \(L=\log N\); \(Y\) as defined in RANK3_ROUTE_D.md §7 and restated in the assignment.

This document contributes three things beyond what is already on record: a direct, mechanism-by-mechanism attempt to build an arc-transfer argument for \(Z_{(q)}\) (§1); a specific literature connection — Montgomery–Vaughan's mean-square theorem for Goldbach's problem — that turns "no tool found" into "no tool can exist at the full-circle level, provably" (§2); and a correction, with more data, to a numerical overclaim in RANK3_QUARTIC_TOOLS.md §2 (§3). Like that document, this one cannot run a live literature search in this environment (only grep/rg/ls/cat/ head/tail/wc/find over this worktree and the lab's Python interpreter are available; there is no network access and no bibliographic database), so §2's citation is stated as recollection, not as something independently verified against a live source here — the same limitation RANK3_QUARTIC_TOOLS.md §3 already flags for its Heath-Brown citation.

1. A direct attempt at an arc-transfer argument for \(Z_{(q)}\)

RANK3_ROUTE_D.md §7 states, correctly, that \(Z_{(q)}\)'s integrand \(|R_{q,a}|^4\) carries no factor comparable to \(U_{(q)}\)'s \(|P_{q,a}|^2\) that decays away from the arc, so the transfer step used for \(U_{(q)}\) (§4 there) has no direct analogue. That is a statement about one mechanism (bounding the arc integral by the full-circle integral via a decaying weight). It does not by itself rule out every mechanism. This section tries three others, each concretely, and shows each collapses into a tool already priced and found insufficient.

(a) Triangle-inequality transfer via \(P_{q,a}\)'s own decay. The one piece of \(R_{q,a}=F_N-P_{q,a}\) that does decay off the arc is \(P_{q,a}\) itself: \(|P_{q,a}(\beta)|=(|\mu(q)|/\phi(q))|K_N(\beta)|\ll (|\mu(q)|/\phi(q))\min(N,1/\|\beta\|)\), the standard Dirichlet-kernel bound. Expanding \(|R_{q,a}|^4=|F_N-P_{q,a}|^4\) via \(|x-y|^4\le8(|x|^4+|y|^4)\) (the same inequality UPPER_BOUND.md uses at (28), and RANK3_ROUTE_D.md §7 uses for the \(R^{(2)}\) piece) gives, on any arc, \[ \int_{\rm arc}|R_{q,a}|^4\le8\int_{\rm arc}|F_N|^4+8\int_{\rm arc}|P_{q,a}|^4. \] The second term is exactly computable and small: \(|P_{q,a}|^4= (|\mu(q)|/\phi(q))^4|K_N|^4\), and \(\int_{\mathbb T}|K_N|^4\,d\beta\) is the classical, purely combinatorial fourth moment of the Dirichlet kernel — the number of solutions to \(n_1+n_2=n_3+n_4\), \(1\le n_i\le N\), which is \(\Theta(N^3)\) with an explicit elementary constant and no arithmetic input at all (this is the \(q=1\), unweighted, \(\Lambda\)-free case of the same convolution counting problem). Summed over \(a\) and \(q\le R_0\), this term costs \(O\!\big(N^3\sum_{q\le R_0}\phi(q)(\mu(q)/\phi(q))^4\big)=O(N^3\sum_q \phi(q)^{-3})=O(N^3)\) at worst (the sum over \(q\) converges), so it is never the obstruction. The whole difficulty is pushed onto the first term, \(\int_{\rm arc}|F_N|^4\): a raw, unweighted, unrestricted local fourth moment of the prime exponential sum itself on a shrinking arc around \(a/q\), with no residue-class bias subtracted at all. This matches, and sharpens, the structural observation already on record in RANK3_QUARTIC_TOOLS.md §2 (that \(Z_{(q)}\) does not vanish, and shows no saving, on non-squarefree \(q\), where \(R_{q,a}=F_N\) exactly): here the same object, \(\int_{\rm arc}|F_N|^4\), reappears as the irreducible core of \(Z_{(q)}\) even at squarefree \(q\), once the transfer is attempted. So any arc-transfer argument for \(Z_{(q)}\), not just the specific one RANK3_ROUTE_D.md §7 rules out, must in the end bound \(\int_{I_{q,a}}|F_N|^4\) directly — there is no way to route around this term by manipulating \(P_{q,a}\) alone, since \(P_{q,a}\)'s own fourth moment is cheap and the cross terms in a finer expansion (below) do not remove \(|F_N|^4\) either.

(b) Hölder/interpolation transfer. A second natural mechanism: bound the arc integral of \(|R_{q,a}|^4\) by \(\big(\sup_{\rm arc}|R_{q,a}|\big)^2 \cdot\int_{\rm arc}|R_{q,a}|^2\), i.e. an \(L^\infty\)-times-\(L^2\) interpolation rather than a full-circle comparison. This substitutes the problem for two already-named quantities: a pointwise bound on \(R_{q,a}\) on the arc, and the arc-restricted second moment (an \(U_{(q)}\)-type object, already summed in RANK3_SCOPE.md §1). The pointwise bound is exactly Vaughan's (V), and RANK3_SCOPE.md §2 ("(V) is weakest exactly here") already prices it: for \(q\lesssim N^{2/5}\) — a sub-range fully inside \(2\le q\le R_0\), since \(R_0\sim N^{1/2}/(3L)\gg N^{2/5}\) — (V) gives \(|F_N|\ll NL^{5/2}\), the trivial order, no saving. Carrying that through: \(\sup_{\rm arc}|R_{q,a}|^2\ll N^2L^5\) at worst, and \(\int_{\rm arc}|R_{q,a}|^2\le\int_{\mathbb T}|R_{q,a}|^2\ll NL\) by Parseval (coefficients are \(O(\Lambda(n))\), and \(\sum_{n\le N}\Lambda(n)^2\ll NL\)); the product is \(O(N^3L^6)\) — the same order as the trivial bound on \(Z_{(q)}\) itself, with no saving, and for exactly the same reason (V) fails to save \(U_{(q)}\) or the direct on-arc \(Z_{(q)}\) bound at small \(q\) (RANK3_SCOPE.md §2, third bullet). This mechanism is not new: it is the on-arc-via-(V) route RANK3_SCOPE.md §2 already prices and finds wanting, arrived at from a different starting point (an attempted transfer, rather than a direct arc bound), confirming it is not a route around that obstruction.

(c) A finer binomial expansion, isolating cross terms. Expanding \(|R_{q,a}|^4=|F_N-P_{q,a}|^4\) exactly (not via the crude \(8(x^4+y^4)\) bound of (a)) produces cross terms of the shape \(|F_N|^3|P_{q,a}|\), \(|F_N|^2|P_{q,a}|^2\), \(|F_N||P_{q,a}|^3\), each controllable by Cauchy–Schwarz in terms of moments of \(F_N\) up to order 3 and moments of \(P_{q,a}\) (which, being a scaled \(K_N\), has every moment computable in closed form as above). This does not remove the \(|F_N|^4\) term found in (a) — that term has coefficient 1 in the exact expansion and no cross term cancels it, since cross terms only ever pair \(F_N\) with the small, decaying \(P_{q,a}\), never eliminate a pure \(F_N\) power. So a finer expansion changes the bookkeeping around the edges (potentially sharpening the constant, or converting some \(|F_N|^3|P_{q,a}|\)-type terms into \(U_{(q)}\)-adjacent quantities via Cauchy–Schwarz) but does not change the conclusion of (a): the exact \(|F_N|^4\) term is unavoidable and irreducible, appears with full weight, and is not itself reducible to \(\Delta(t;q,b)\)-type quantities (that is RANK3_ROUTE_D.md §7's first point, restated here for the isolated term rather than for the full \(R_{q,a}^4\)).

Conclusion of this section. Three natural transfer mechanisms — direct decay comparison, sup-times-\(L^2\) interpolation, and a finer cross-term expansion — all reduce to the same two facts already on record: \(\int_{\rm arc}|F_N|^4\) is the irreducible core (not removable by any splitting against \(P_{q,a}\), since \(P_{q,a}\)'s own moments are cheap), and every tool available to attack that core directly — (V) via interpolation, or a full-circle-to-arc comparison — is priced at trivial order in exactly this \(q\)-range by RANK3_SCOPE.md §2. No fourth mechanism suggests itself from the algebra of \(|x-y|^4\) beyond these three (binomial expansion has exactly these term-shapes, and Hölder/ interpolation admits no other split of \(4=p+q\) with \(p,q\ge0\) integers that avoids reproducing either (a) or (b)). This is offered as a genuine, if negative, answer to "can a fresh arc-transfer argument be constructed at all": not merely that RANK3_ROUTE_D.md's specific mechanism fails, but that the natural alternatives fail for the same underlying reason, and identify the same missing ingredient — a non-trivial bound on \(\int_{\rm arc}|F_N|^4\) itself, at \(q\lesssim N^{2/5}\), which no tool named in UPPER_BOUND.md or found in §2 below supplies.

2. Literature: Montgomery–Vaughan's mean-square theorem, and why it

sharpens rather than fills the wall

RANK3_QUARTIC_TOOLS.md §3 already covers two candidate directions and finds neither adapts: a structure-free "quartic large sieve" cannot exist (the \(L^4\) inequality is sensitive to additive energy, unlike the structure-free \(L^2\) large sieve), and the multiplicative-side quartic moment of Dirichlet \(L\)-functions (Heath-Brown 1981 and antecedents) is a fixed-height statement, the wrong variable for \(Z_{(q)}\)'s height-integrated need. This section adds a third, more classical direction, specific to \(\Lambda\)-weighted quadruple convolutions rather than to \(L\)-function moments or large sieves: the second-moment (in \(n\)) theory of Goldbach representations.

The connection. Write \(r(n)=\sum_{n_1+n_2=n}\Lambda(n_1)\Lambda(n_2)\) for \(2\le n\le2N\) (summing over \(1\le n_1,n_2\le N\)); these are exactly the Fourier coefficients of \(F_N(\alpha)^2\). By Parseval, \[ \sum_{n=2}^{2N}r(n)^2=\int_{\mathbb T}|F_N(\alpha)|^4\,d\alpha, \] i.e. the full-circle, \(q=1\), no residue restriction at all case of exactly the raw quantity Section 1(a) isolates as \(Z_{(q)}\)'s irreducible core (restricted there to a single arc rather than the whole circle). The classical circle-method literature on Goldbach's problem's error term — Montgomery and Vaughan, "The exceptional set in Goldbach's problem," Acta Arith. 27 (1975), 353–370, building on the Hardy–Littlewood/Vinogradov-era major-minor arc apparatus for this exact convolution — establishes, as recollected here (not verified against a live source in this environment, per this document's preamble):

What this settles. The first fact is a proved lower bound, not just an upper one, on \(\int_{\mathbb T}|F_N|^4\,d\alpha\) — the unrestricted quantity is \(\Theta(N^3)\), unconditionally. This means no tool of any kind (quartic large sieve, \(L\)-function moment, additive-combinatorial energy bound, or anything else) can ever produce a bound on \(\int_{\mathbb T}|F_N|^4\) better than \(O(N^3)\), on pain of contradicting a proved theorem. This is a strictly stronger statement than RANK3_QUARTIC_TOOLS.md §4's "no candidate tool is found" — it says no candidate tool could exist for the full-circle version of the quantity Section 1 shows is \(Z_{(q)}\)'s irreducible core. Combined with Section 1's reduction, the entire hope for a useful bound on \(\sum_{2\le q\le R_0} Z_{(q)}\) has to come from the arc-restriction — the fact that \(I_{q,a}\) has width \(\sim1/(qN)\ll1\), a small fraction of the full circle — buying a saving that the full-circle quantity provably does not have on its own; it cannot come from the quartic moment itself somehow being smaller in size than \(\Theta(N^3)\), because it is not. Numerically, this matches RANK3_QUARTIC_TOOLS.md §2's finding exactly (and Section 3 below): \(Z^*_{(q)}/(\phi(q)N^3)\) is bounded above and below by positive constants for every \(q\ge2\) checked, exactly the signature of a \(\Theta(N^3)\) quantity, not a quantity with room for a hidden \(N\)-power saving waiting to be found by a cleverer tool.

What this does not settle. The second (mean-square exceptional set) fact is a genuine, unconditional saving — but relative to a different bias subtraction (pointwise in \(n\), via \(\mathfrak S(n)\)) than the one \(R_{q,a}=F_N-P_{q,a}\) uses (indexed by \((q,a)\) via the arc decomposition), and it is a statement about the whole circle, not an individual arc restricted to width \(\sim1/(qN)\) around one rational. It does not, as it stands, supply the arc-restricted bound Section 1 shows is needed, and this document does not find a way to convert it into one; it is recorded here because it is the closest named, unconditional, fourth-moment (in the sense of a quadruple \(\Lambda\)-convolution) result this document locates, and because it upgrades the wall from a search gap to a proved obstruction at the full-circle level. Whether the log-power saving in the mean-square-exceptional-set result has any analogue at the arc-restricted, \((q,a)\)-indexed level is a question this document does not answer and does not attempt to guess at; it is a possible next step, not a result here.

Vinogradov-type mean value theorems do not apply either, for a different reason than (BV)/(BDH). A natural fourth candidate, given the shape of \(Y\) (a quadruple additive convolution \(n_1+n_2=n_3+n_4\)), is Vinogradov's mean value theorem and its modern strengthenings (Wooley's efficient congruencing; Bourgain–Demeter–Guth decoupling). These bound the number of solutions to systems of equations \(\sum u_i^j=\sum v_i^j\) for \(j=1,\dots,k\) with a power-saving over the trivial count, for degree \(k\ge2\) monomial curves. The equation defining \(Y\), \(n_1+n_2=n_3+n_4\), is the degree-1 (single, linear) case — the additive energy of the interval \([1,N]\) itself — which is elementary and exact (\(\sum_k r_K(k)^2=\Theta(N^3)\), the same computation behind \(\int_{\mathbb T}|K_N|^4\) in Section 1(a)), not a case where Vinogradov's machinery or decoupling supplies anything beyond what direct combinatorics already gives. What is missing is not a sharper count of solutions to \(n_1+n_2=n_3+n_4\) — that count is known exactly — but a saving in the \(\Lambda\)-weighted version, i.e. arithmetic information about which quadruples the primes occupy relative to the uniform measure, which is additive-energy-of-a-specific-set information (RANK3_QUARTIC_TOOLS.md §3's point, restated: this is not a structure-free geometric statement, and Vinogradov/decoupling machinery is exactly such a structure-free geometric tool, aimed at a different, higher-degree problem).

3. Numerical data: the deviation from \(\Theta(N^3)\) tracks

\(|\mu(q)|/\phi(q)\), not squarefreeness

RANK3_QUARTIC_TOOLS.md §2 measures \(Z^*_{(q)}/(\phi(q)N^3)\) at \(N=20000\) and reports it as "the same constant... for every \(q\ge3\) checked." Re-running rank3_fourth_moment_probe.py (unchanged; results reproduced exactly in results_rank3_fourth_moment_probe.json) and cross- checking with a new script, rank3_fourth_moment_mod3_probe.py (results in results_rank3_fourth_moment_mod3_probe.json), shows this is not quite right as stated: at \(N=20000\), \(q=3\) gives \(1.484566\) and \(q=6\) gives \(1.484700\) — about \(3\%\) below the \(\approx1.5259\) band that \(q=5,7,10,15,30\) and every non-squarefree \(q\) checked land in, not matching it "to at least four digits."

The new script checks two things: whether this \(q\in\{3,6\}\) gap is a finite-\(N\) artifact tied to \(N=20000=2^5\cdot5^4\) (coprime to 3), and what actually governs it.

So the deviation of \(Z^*{(q)}/(\phi(q)N^3)\) from the common \(\approx1.5259\) band is governed by \(|\mu(q)|/\phi(q)\) — the relative weight of the bias-subtraction term \(P{q,a}\) itself — not by squarefreeness or by \(3\mid q\) as such: \(q=2\) (weight \(1\)) deviates most (\(\approx0.864\)); \(q=3,6\) (weight \(1/2\)) deviate next most (\(\approx1.48\)); \(q=5,7,10\) (weight \(1/4,1/6,1/4\)) and every \(q\ge9\) checked (weight \(\le1/6\)) sit within a percent or so of \(1.5259\). This is consistent with, and refines, the reading in Section 2 above: \(Z^*_{(q)}\) is \(\Theta(\phi(q)N^3)\) with an \(O(1)\) constant depending mildly on \(|\mu(q)|/\phi(q)\) (largest at \(q=2\), shrinking toward a common limit as \(|\mu(q)|/\phi(q)\to0\)), never an \(N\)-power or even a clear \(q\)-power saving. RANK3_QUARTIC_TOOLS.md §2 has been corrected in place to state this precisely rather than "the same constant to four digits," with a pointer to this document for the full data; its core conclusion (no order-of-magnitude saving, for any \(q\ge2\) checked) is unaffected by the correction.

4. Where this leaves rank 3's fourth moment

The wall, stated precisely: what would still be needed is a genuinely new, unconditional bound on \(\int_{I_{q,a}}|F_N|^4\,d\alpha\) — the raw, unrestricted local fourth moment of the prime exponential sum on a single arc of width \(\sim1/(qN)\) — that beats the trivial order at \(q\lesssim N^{2/5}\), summed over \(q\le R_0\). This document does not find such a bound, and Section 2 gives an unconditional reason (not merely an unsuccessful search) why the natural places to look for one — the \(L^4\) large sieve, \(L\)-function fourth moments, Vinogradov-type mean value theorems, and the classical Goldbach-representation second-moment literature — either cannot supply a structure-free version of it or answer a differently-shaped question. This is consistent with, and extends, RANK3_SCOPE.md §3's conclusion that no route it names is costed to completion, and RANK3_QUARTIC_TOOLS.md §4's finding that \(Z_{(q)}\) has no known candidate tool: this document looked at the arc-transfer question directly (rather than only citing why one specific mechanism fails) and at one further literature direction (the classical Goldbach second-moment theory, rather than only large sieves and \(L\)-function moments), and still finds no route past this range, for reasons this document can name precisely rather than only report as absent.

This document assumes and establishes nothing about zeros of \(L\)-functions or the Riemann Hypothesis, does not weaken the definitions of \(Z_{(q)}\), \(R_0\), or the reduced-residue/coprimality conditions, and is: a direct three-mechanism arc-transfer attempt (§1), a literature connection with an explicit, appropriately-hedged citation (§2), and a numerical correction backed by a new script and its output (§3, rank3_fourth_moment_mod3_probe.py, results_rank3_fourth_moment_mod3_probe.json).