This document answers one question, precisely scoped: RANK3_ROUTE_D.md §6 shows that the direct, per-arc transfer bound used in its equation (D11) — the generalization of UPPER_BOUND.md's (30) from \(q=1\) to general \(q\) — costs \(O(N^3)\) once summed over \(2\le q\le R_0\), as large as the unconditional baseline (1), regardless of any estimate for \(\Delta(t;q,b)\). The question posed is whether UPPER_BOUND.md's dyadic large-sieve method (24)-(27), which solves an analogous problem for \(q>R_0\), can be adapted to bring that transfer cost down, ideally to \(O(N^{2+\epsilon})\).
The answer is not an adaptation. The transfer step does not need a better bound; it needs to be deleted. The quantity RANK3_ROUTE_D.md's §4 spends an \(O(N^3)\)-costing inequality bounding is already bounded, for free and exactly, by monotonicity of integration — a fact that holds independently of \(\Delta(t;q,b)\), of the large sieve, and of \(q\). Once this is used instead, \(\sum_{q=2}^{R_0}U_{(q)}\)'s only remaining unproved dependence is exactly \(\sum_{q=2}^{R_0}\mu(q)^2\sum_b^*T(q,b)\) — RANK3_SCOPE.md's Route A, untouched here — plus an explicit term this document bounds by \(O(NL^6)\), negligible against any target of size \(N^{2+\epsilon}\) or larger. §5 below explains why the dyadic large-sieve mechanism specifically does not fit this quantity even when asked to, which is a separate fact from its being unnecessary.
No estimate for \(\Delta(t;q,b)\) is attempted or assumed anywhere below, matching RANK3_ROUTE_D.md's own scope and the assignment's instruction.
1. The quantity at issue, restated
Notation is RANK3_ROUTE_D.md's throughout: \(q\ge2\), \(a\) a reduced residue mod \(q\), \(P_{q,a}(\alpha)=(\mu(q)/\phi(q))K_N(\alpha-a/q)\), \(R_{q,a}=F_N-P_{q,a}\), both defined as finite exponential sums on all of \(\mathbb T=\mathbb R/\mathbb Z\), and \[ I_{q,a}=\{\alpha\in\mathbb T:\|\alpha-a/q\|\le\delta_q\},\quad\delta_q=Q/(qN), \] the arc from UPPER_BOUND.md (7). RANK3_ROUTE_D.md's equation (D10)-adjacent text defines, exactly as the assignment names it, \[ T_N(q,a):=\int_{\mathbb T}|P_{q,a}|^2|R_{q,a}|^2\,d\beta, \] and UPPER_BOUND.md's \(U_{(q)}\) (the quantity that actually enters the sufficient bound (23) via \(U_Q=\sum_{q\le Q}U_{(q)}\)) is \[ U_{(q)}=\sum_{a\bmod q}^\int_{I_{q,a}}|P_{q,a}|^2|R_{q,a}|^2\,d\alpha. \] The transfer quantity the assignment names is \(\sum_{a}^\big[T_N(q,a)-\int_{I_{q,a}}|P_{q,a}|^2|R_{q,a}|^2\big]\), summed over \(2\le q\le R_0\). RANK3_ROUTE_D.md §4 bounds this quantity, per \((q,a)\), by \(\dfrac{\mu(q)^2q^2N^2}{4Q^2\phi(q)^2}\int_{\mathbb T}|R_{q,a}|^2\), which after summing over \(a\) and \(q\le R_0\) is the \(O(N^3)\) term §6 identifies.
2. Monotonicity: the trivial bound that makes the transfer step unnecessary
Lemma. For every \(q\ge1\) and every reduced \(a\bmod q\), \[ 0\ \le\ \int_{I_{q,a}}|P_{q,a}|^2|R_{q,a}|^2\,d\alpha\ \le\ T_N(q,a). \tag{T1} \]
Proof. \(I_{q,a}\subset\mathbb T\), and the integrand \(|P_{q,a}|^2|R_{q,a}|^2\) is real-valued and nonnegative everywhere on \(\mathbb T\) (it is a product of two squared moduli of finite exponential sums, defined at every point, not only on the arc). Integrating a nonnegative function over a subset of its domain cannot exceed integrating it over the whole domain. \(\square\)
This is not asymptotic, not conditional on anything about \(\Delta(t;q,b)\), and does not use any property of \(P_{q,a}\) or \(R_{q,a}\) beyond nonnegativity of the integrand and \(I_{q,a}\subseteq\mathbb T\). It holds at every \(N,q,a\), including \(q=1\). §6 of this document checks it numerically against direct evaluation of \(F_N,K_N\) at several small \((N,q,a)\); it holds in every case, as it must.
Corollary. Summing (T1) over the \(\phi(q)\) reduced residues \(a\), \[ U_{(q)}\ \le\ \sum_{a\bmod q}^*T_N(q,a). \tag{T2} \]
The right side of (T2) is exactly the quantity RANK3_ROUTE_D.md's equation (D10) already bounds — and (D10) is derived (via (D7), the one-sided Cauchy-Schwarz bound (D8), and the \(\rho_2(q)\) correction (D3)) before RANK3_ROUTE_D.md's §4 introduces the arc-transfer step at all. Nothing in the derivation of (D10) uses the transfer inequality; (D10) is a self-contained bound on \(\sum_a^*T_N(q,a)\), reproduced here: \[ \sum_{a\bmod q}^*T_N(q,a)\ \le\ 2\mu(q)^2\sum_{b\bmod q}^*T(q,b) +\frac{2\mu(q)^2\rho_2(q)^2N}{\phi(q)}. \tag{D10} \] Combining (T2) with (D10) directly: \[ \boxed{\ U_{(q)}\ \le\ 2\mu(q)^2\sum_{b\bmod q}^*T(q,b) +\frac{2\mu(q)^2\rho_2(q)^2N}{\phi(q)}.\ } \tag{T3} \] (T3) is exactly RANK3_ROUTE_D.md's boxed (D11) with its third term — the transfer term \(O(q^2N^3L/(Q^2\phi(q)))\) — removed, and it is a strictly stronger (or, in the degenerate case, equal) bound: (D11) is obtained from (T3) by adding a nonnegative quantity to its right side, which can only weaken an upper bound, never repair one. There is consequently nothing to adapt: the transfer step contributes an unnecessary and, once summed over \(q\), dominant slack term to an inequality that is already true without it.
3. What this removes from the O(N^3) in RANK3_ROUTE_D.md §6
§6's computation is a bound on \(\sum_{q=2}^{R_0}\) of (D11)'s third term alone; it does not touch (D11)'s first two terms (those are the diagonal Parseval quantity and the \(\rho_2\) correction, both present in (T3) unchanged). Since (T3) has no third term, that entire computation — \(\sum_{q=2}^{R_0}O(q^2N^3L/(Q^2\phi(q)))\asymp N^3\) — is not a cost of bounding \(\sum_{q=2}^{R_0}U_{(q)}\) via (T3). Summing (T3) over \(2\le q\le R_0\): \[ \sum_{q=2}^{R_0}U_{(q)}\ \le\ 2\sum_{q=2}^{R_0}\mu(q)^2\sum_{b\bmod q}^*T(q,b) +2N\sum_{q=2}^{R_0}\frac{\mu(q)^2\rho_2(q)^2}{\phi(q)}. \tag{T4} \]
4. The surviving explicit term is \(O(NL^6)\), not \(O(N^3)\)
The second sum in (T4) is bounded using only tools already present in UPPER_BOUND.md and RANK3_ROUTE_D.md. By (D3), \(\rho_2(q)\ll L\log(2q)\), so \(\rho_2(q)^2\ll L^2\log^2(2q)\). By UPPER_BOUND.md (15), \(q/\phi(q)\le\zeta(2)(1+\log q)\), hence \(1/\phi(q)\le\zeta(2)(1+\log q)/q\). Since \(\mu(q)^2\le1\), \[ \sum_{q=2}^{R_0}\frac{\mu(q)^2\rho_2(q)^2}{\phi(q)} \ll L^2\sum_{q=2}^{R_0}\frac{\log^2(2q)(1+\log q)}{q} \ll L^2\sum_{q=2}^{R_0}\frac{\log^3(2q)}{q}. \] By the standard comparison \(\sum_{2\le q\le x}\log^3(2q)/q\ll\log^4(2x)\) (partial summation against \(\int_2^x(\log2t)^3\,dt/t=[(\log2t)^4/4]_2^x\)), and \(R_0=Q/L\), \(Q=\lfloor\sqrt N/3\rfloor\), so \(\log(2R_0)\ll L\): \[ \sum_{q=2}^{R_0}\frac{\mu(q)^2\rho_2(q)^2}{\phi(q)}\ll L^2\cdot L^4=L^6. \] Therefore \[ \boxed{\ 2N\sum_{q=2}^{R_0}\frac{\mu(q)^2\rho_2(q)^2}{\phi(q)}\ \ll\ NL^6.\ } \tag{T5} \] This is smaller than \(N^{1+\epsilon}\) for every fixed \(\epsilon>0\) and sufficiently large \(N\); it is negligible against the target \(N^{2+\epsilon}\), let alone against the \(O(N^3)\) this document removes.
Combining (T4) and (T5): \[ \boxed{\ \sum_{q=2}^{R_0}U_{(q)}\ \le\ 2\sum_{q=2}^{R_0}\mu(q)^2\sum_{b\bmod q}^*T(q,b)\ +\ O(NL^6).\ } \tag{T6} \]
The only unproved dependence left in (T6) is \(\sum_{q=2}^{R_0}\mu(q)^2\sum_b^*T(q,b)\), which is exactly a \(\Delta(t;q,b)\)-uniformity statement summed over \(q\le R_0\) — RANK3_SCOPE.md's Route A, not attempted here per the assignment's own scope. What (T6) establishes is that the transfer step contributes no separate obstruction on top of Route A: RANK3_ROUTE_D.md §6's claim that Route D "surfaces a second, independent requirement" beyond Route A's uniform estimate does not survive replacing its transfer step with (T1)-(T3). The requirement was an artifact of that specific (avoidable) step, not a property of \(U_{(q)}\) or of the identity (D7) itself.
5. Why the dyadic large-sieve method does not adapt to this quantity — and does not need to
The assignment asks, in the alternative, for a precise reason if dyadic large sieve does not adapt to the transfer quantity. It does not, and the reason is a mismatch between what (24)-(27) is built to bound and what the transfer quantity is.
(24)-(27) bounds, for a dyadic block \(R<q\le\min(2R,Q)\), the sum \(\sum_{q,a\text{ in block}}\int_{I_{q,a}}|P_{q,a}|^2|R_{q,a}|^2\) directly on the arcs, by: (i) enlarging every arc in the block to the single common interval \(|\beta|\le Q/(RN)\) (valid because every \(q\) in the block has comparable size, so \(\delta_q=Q/(qN)\) is comparable across the whole block); (ii) applying the additive large sieve (LS) at a fixed \(\beta\), to the set of points \(\{a/q:R<q\le2R\}\), which are genuinely \((4R^2)^{-1}\)-spaced and therefore admissible LS inputs; and (iii) integrating the resulting \(\beta\)-uniform bound over the (now common, small) enlarged interval. The mechanism is fundamentally about a sum over many centers at one shared \(\beta\), bounded by how well-spaced those centers are.
The transfer quantity, by contrast, is \(\sum_a^\big[T_N(q,a)-\int_{I_{q,a}}|P_{q,a}|^2|R_{q,a}|^2\big] =\sum_a^\int_{\mathbb T\setminus I_{q,a}}|P_{q,a}|^2|R_{q,a}|^2\,d\beta\): for a single modulus \(q\), an integral over \(\beta\) of the complement of one arc. There is no second index to space out and no shared \(\beta\) to fix — LS has nothing to act on here, because LS bounds a sum of squared values of a fixed exponential polynomial evaluated at well-separated points, and this quantity is not of that shape at all: it is one center, integrated over the part of the circle its own arc excludes. Grouping \(q\) into dyadic blocks does not change this, because the object being bounded does not involve other moduli in the block; it is defined per \(q\) before any sum over a block is taken. Handing this quantity to the (24)-(27) machine has no well-defined entry point, independent of whether the resulting bound would be good enough.
This is a different fact from the one in §§2-4: those show the transfer quantity need not be bounded at all (for the purpose \(U_{(q)}\) needs, upper bounding by monotonicity dominates), which is why the type-mismatch above is not an obstruction to anything — it only rules out the one route the assignment named as a first guess, in favor of the route that already works.
Distinct from Route B, which is a genuine, still-open obstruction. A different question — applying (24)-(27)'s method directly to bound \(U_{(q)}\) itself for \(q\le R_0\) (bypassing the Parseval identity (D7) entirely, rather than using it and discarding the transfer step) — is exactly RANK3_SCOPE.md §3's "Route B," and RANK3_SCOPE.md §2 already shows it fails: the per-block estimate (25) does not save anything at small block size, and the single block \(R=1\) (i.e. \(q=2\)) alone costs \(O(w(Q)^2N^3L)=O(N^3L^3)\) by (25), exceeding target by a full power of \(N\). That failure is genuine and is not resolved by anything in this document — it is a separate question from the transfer step addressed here, which concerns going from the already-derived identity-based bound (D10) to the arc-restricted quantity, not replacing the identity with an LS argument from scratch.
6. Numeric check
rank3_arc_transfer_probe.py evaluates \(F_N,K_N,P_{q,a},R_{q,a}\) as finite exponential sums (no asymptotics, no model assumptions) at \(N=120\) for six \((q,a)\) pairs, computing the full-circle integral \(T_N(q,a)\) exactly (a uniform grid with more than twice the maximum frequency present recovers the DC Fourier coefficient exactly, by orthogonality) and the arc integral by numerical quadrature over the actual interval \(I_{q,a}\) at \(Q=\lfloor\sqrt N/3\rfloor\). The Lemma (T1) holds in all six cases (e.g. \(q=2,a=1\): arc \(\approx9452.6\le\) full circle \(\approx10385.5\)); results are in results_rank3_arc_transfer_probe.json. The same script independently cross-checks RANK3_ROUTE_D.md's exact identity (D7) — summing \(\int_{\mathbb T}|K_N|^2|R^{(1)}{q,a}|^2\) over \(a\) against \(\Sigma{\rm diag}(q)+\Sigma_{\rm cross}(q)\) computed from \(T(q,b)\), \(X(b,b')\) via (D5)-(D6) — at four \((N,q)\) pairs, matching to numerical precision in every case. This confirms the definitions used above agree with RANK3_ROUTE_D.md's, which is what makes reusing (D10) directly (§2) valid rather than merely plausible.
7. Where this leaves rank 3
RANK3_ROUTE_D.md §8 recorded two requirements for closing rank 3 by Route D: a uniform \(\Delta(t;q,b)\) estimate (Route A, relocated) and "a non-crude, large-sieve-style transfer argument in place of the direct (30)-style bound." This document removes the second requirement outright, at every \(q\), by showing the direct (30)-style bound should not have been used to go from (D10) to \(U_{(q)}\) in the first place — monotonicity already supplies that step for free, and (T6) shows the resulting total cost over \(2\le q\le R_0\) beyond Route A is \(O(NL^6)\), far under any target of size \(N^{2+\epsilon}\) or larger. What remains open for rank 3's \(U\)-side is exactly \(\sum_{q=2}^{R_0}\mu(q)^2\sum_b^*T(q,b)\) — Route A's uniform estimate, undiminished but also not multiplied by any further obstruction from the identity-to-arc step. This document says nothing about \(Z_{(q)}\) (RANK3_ROUTE_D.md §7 already shows the arc-transfer idea does not apply there for an unrelated, structural reason: \(Z\)'s integrand carries no \(|P_{q,a}|^2\) factor and so has no reason to decay away from the arc, so even the trivial monotonicity direction used here, while still true for \(Z\), does not by itself make \(Z_{(q)}\)'s full-circle version a useful proxy — the full-circle route to \(Z\) is not attempted here or in RANK3_ROUTE_D.md), about rank 1, or about the RESULTS.md doors table's overall ranking, which RANK3_SCOPE.md §4 already shows rank 3's closure cannot move on its own regardless of what happens here.
This document is algebraic bookkeeping (monotonicity of integration of a nonnegative function, plus the elementary sum estimate in §4) around RANK3_ROUTE_D.md's and UPPER_BOUND.md's existing unconditional construction; it assumes and establishes nothing about zeros of \(L\)-functions or the Riemann Hypothesis, and does not attempt any part of the \(\Delta(t;q,b)\) uniformity question.