This checks one specific question: can the Siegel-Walfisz major-arc argument of UPPER_BOUND.md Section 5, equations (17)-(19) — proved uniformly for denominators \(q\le Q'=\lfloor L^B\rfloor\), any fixed \(B\), \(L=\log N\) — be spliced onto Section 6's mixed moment \(U_Q\) and fourth moment \(Z_Q\) (defined just before its (23), for \(Q=\lfloor\sqrt N/3\rfloor\)) so as to bound \(U_{2\le q\le Q'}\) and \(Z_{2\le q\le Q'}\) separately, at the strength (19) gives their combined quantity \(M_{Q'}\). All equation numbers in parentheses refer to UPPER_BOUND.md unless marked "here."
Answer: no. A first attempt at this splice looks like it works, but it silently substitutes one arc system for another. Once that is corrected, the splice fails outright for every \(q\) in the claimed range, including the smallest one, \(q=2\) — not because of anything to do with the size of \(q\), but because of the width of the arc that \(U_Q\), \(Z_Q\) actually integrate over. Section 3 below gives the exact reason.
1. The two arc systems are not the same, for the same q
By (7), the arc dissection is built from one integer parameter \(Q\), chosen once: \(\delta_q=Q/(qN)\), \(I_{q,a}=\{\alpha:\|\alpha-a/q\|\le\delta_q\}\). The width of the arc attached to a given modulus \(q\) depends on which \(Q\) was chosen for the whole dissection, not on \(q\) alone.
- Section 6's \(U_Q\), \(Z_Q\) — the objects the task asks about — are built with \(Q=\lfloor\sqrt N/3\rfloor\) (Section 6's opening line: "Now set \(Q=\lfloor\sqrt N/3\rfloor\)"). For a fixed modulus \(q\), their arc \(I_{q,a}\) has half-width \(\delta_q=Q/(qN)\asymp1/(q\sqrt N)\).
- Section 5's (17) is proved for a different dissection, with its own \(Q'=\lfloor L^B\rfloor\) in place of \(Q\) (Section 5's opening line: "First take \(Q=\lfloor L^B\rfloor\), for fixed \(B\), rather than a square-root arc denominator"). For the same modulus \(q\), that arc has half-width \(\delta'_q=Q'/(qN)\asymp L^B/(qN)\).
For any fixed \(q\) (in particular for every \(q\le Q'\)), the ratio of widths is \[ \delta_q/\delta'_q=Q/Q'\asymp\sqrt N/L^B\to\infty. \] Section 6's arc for modulus \(q\) is therefore always far wider than Section 5's arc for the same \(q\) — and (17) is a statement about Section 5's narrower arc only: it reads "\(q\le Q\), \(|\beta|\le Q/(qN)\)" with \(Q\) there meaning Section 5's own \(Q'\), by construction. It says nothing about \(F_N\) on the wider annulus of Section 6's arc that lies outside Section 5's arc, for any \(q\).
2. The gap is not just unstated — the proof of (17) genuinely fails there
This is not only a labeling mismatch. The paragraph justifying (17) (immediately after its statement) needs the partial-summation cost against \(\exp1(t\beta)\), which is \(O(1+N|\beta|)\), to be a fixed power of \(L\), so that Siegel-Walfisz's arbitrary log-power saving \(NL^{-H}\) absorbs it after raising \(H\). Concretely:
- On Section 5's own arc, \(|\beta|\le Q'/(qN)\), so \(N|\beta|\le Q'/q\le Q'=\lfloor L^B\rfloor\), a fixed power of \(L\). This is exactly what the proof needs, and is why (17) holds there for every fixed \(B,H\).
- On Section 6's arc, \(|\beta|\le Q/(qN)\) with \(Q=\lfloor\sqrt N/3\rfloor\), so \(N|\beta|\le Q/q\asymp\sqrt N/(3q)\). For any fixed \(q\) (in particular for every \(q\) in the claimed range \(2\le q\le Q'\), since \(Q'=\lfloor L^B\rfloor=o(\sqrt N)\)), this is a growing power of \(N\), not a fixed power of \(L\). Multiplying it against the Siegel-Walfisz error \(NL^{-H}\) gives \(N^{3/2}L^{-H}/q\), which exceeds the trivial size \(O(N)\) of \(F_N\) once \(N^{1/2}\gg qL^H\) — eventually true for every fixed \(q\) and every fixed \(H\). So the argument does not just fail to reach Section 6's arc; run there, it produces no saving over the trivial bound at all.
So the obstruction is driven by arc width, which is set by the global dissection parameter \(Q\), not by the size of \(q\). This is the opposite of what a first reading suggests: one might expect small \(q\) to be the "easy" end, since (17)'s own \(q\)-range is \(q\le L^B\) and \(2\le q\le L^B\) sits entirely inside it. But that \(q\)-range restriction in (17) is paired with a specific, narrow \(\beta\)-range tied to Section 5's own \(Q'\); once \(q\) is fixed and the arc is instead Section 6's wider one, the \(\beta\)-range attached to that same small \(q\) already exceeds where the Siegel-Walfisz saving can survive. Small \(q\) on Section 6's dissection is not a favorable case for (17) — it is, if anything, the case with the widest arc of all (since \(\delta_q=Q/(qN)\) is largest at small \(q\)).
3. Why a "use the narrower arc instead" fix does not repair the splice
One might try to define an auxiliary quantity using Section 5's narrower arc in place of \(I_{q,a}\) for these small \(q\), where (17) is genuinely valid, and call that the desired bound. This does not bound \(U_{(q)}\), \(Z_{(q)}\) — the actual \(q\)-summands of Section 6's \(U_Q\), \(Z_Q\) that feed equation (23) — for two reasons:
- \(U_{(q)}\), \(Z_{(q)}\) are defined (Section 6, before (23)) as integrals over Section 6's actual arc \(I_{q,a}\), of half-width \(Q/(qN)\). An integral over a strictly smaller sub-arc is a different quantity; it omits the annulus between the two arc widths, where (17) supplies no information about \(R_{q,a}=F_N-P_{q,a}\) at all, and where, per Section 2 above, \(F_N\) need not be close to \(P_{q,a}\) — the trivial bounds \(F_N=O(N)\) and \(P_{q,a}=O(N/\phi(q))\) are all that is available there absent a separate argument. There is no basis here for asserting the omitted annulus contributes negligibly.
- The disjointness of the arcs (Section 3's argument, "the circular distance between distinct reduced fractions is at least \(1/(qq')\); their radii sum to \(Q(q+q')/(qq'N)<1/(qq')\)") is proved for a single, fixed \(Q\) used for every modulus in the dissection at once. \(U_Q\), \(Z_Q\), and the bound (23) built from them, are structured around that one dissection throughout Sections 6-7. Substituting a narrower arc for some moduli and the original wide arc for others is not a dissection of this kind, and nothing in Sections 3, 6, or 7 supports combining pieces built on different arc systems into a bound on \(U_Q\) or \(Z_Q\) as those quantities are actually defined.
4. The role of (18), for completeness
(18), \((|F_N|^2-|P_{q,a}|^2)^2\le8|P_{q,a}|^2|R_{q,a}|^2+2|R_{q,a}|^4\), is not where this splice fails. Given a pointwise bound on \(R_{q,a}\) over an arc, pulling that bound out of the corresponding integral and multiplying by \(\int_{I_{q,a}}|P_{q,a}|^2\) (for the \(U\)-type term) or by the arc's measure (for the \(Z\)-type term) is mechanically sound, with no missing cross terms or double-counting, and (18) itself is needed only to reconstitute a bound on the combined quantity \(M_{Q'}\) from separate bounds on \(U\) and \(Z\), not to bound \(U\) and \(Z\) individually. Indeed, restricted honestly to Section 5's own dissection (arcs of width \(Q'/(qN)\), the arcs (17) is actually proved on), this mechanism does produce matching separate bounds \(\ll_{B,H}N^3L^{-2H}\log(2Q')\) and \(\ll_{B,H}N^3Q'^2L^{-4H}\) for the two pieces of what would be \(M_{Q'}\) on that dissection, reproducing (19) exactly via (18). That derivation is correct as a fact about Section 5's own construction. The error in the original attempt at this splice was treating that fact as if it also bounded Section 6's \(U_{(q)}\), \(Z_{(q)}\) for \(q\le Q'\), which — per Sections 1-3 above — it does not, because those live on a different, much wider arc for the same \(q\).
5. Relation to RANK3_SCOPE.md
RANK3_SCOPE.md Section 2 already states that "(17)/(SW) does not reach this range at all," arguing from a mismatch of \(q\)-ranges: (17)'s uniformity is fixed at \(q\le L^B\) for fixed \(B\), while rank 3 needs \(2\le q\le R_0\) with \(R_0\) growing like a power of \(N\), so for any fixed \(B\), (17) covers only a vanishing initial segment of rank 3's range by \(q\)-size. That argument leaves open, in principle, whether the vanishing initial segment it does cover — \(2\le q\le L^B\) — could still be closed by transplanting (17) into Section 6's actual construction there. This document answers that residual question: no. The obstruction found here is different in kind from RANK3_SCOPE's: it is not that \(q\) is too large for (17), but that Section 6's arc for any fixed \(q\) is too wide for (17), because arc width there is set by the global parameter \(Q=\lfloor\sqrt N/3\rfloor\), not by \(q\). So even the sub-range where (17) nominally applies to a modulus \(q\) in isolation does not transfer into Section 6's \(U_Q\), \(Z_Q\) at that same \(q\), and rank 3's range \(2\le q\le R_0\) has no sub-piece, however small, that this route closes.
6. Summary
Splicing (17)-(19)'s device onto Section 6's \(U_Q\), \(Z_Q\), so as to bound \(U_{2\le q\le\lfloor L^B\rfloor}\) and \(Z_{2\le q\le\lfloor L^B\rfloor}\) separately, does not work. The Siegel-Walfisz pointwise estimate (17) is proved on the arcs of Section 5's own, narrower dissection (parameter \(Q'=\lfloor L^B\rfloor\)); Section 6's \(U_Q\), \(Z_Q\) integrate over the arcs of a different, much wider dissection (parameter \(Q=\lfloor\sqrt N/3\rfloor\)), and for every fixed \(q\) — including the smallest, \(q=2\) — that wider arc extends far past where (17)'s own proof (the partial-summation bound needing to be a fixed power of \(L\), not of \(N\)) survives. This holds uniformly over the whole claimed range \(2\le q\le\lfloor L^B\rfloor\), not just at its upper end, and there is no way to patch it by substituting a narrower arc, since \(U_Q\), \(Z_Q\) and the disjointness that lets them decompose by \(q\)-block are all fixed to the single dissection parameter \(Q=\lfloor\sqrt N/3\rfloor\) throughout Sections 6-7. Nothing here contradicts (1) or (19), both of which are proved entirely within Section 5's own, narrower dissection and never call on Section 6's \(U_Q\) or \(Z_Q\); this document only rules out one specific way of connecting the two constructions, matching and sharpening Section 6's own remark that "Formula (17) does not apply for all \(q\le Q\)" by identifying arc width, not the size of \(q\), as the actual obstruction.