This document carries out RANK3_SCOPE.md's Route D: the exact analogue, for general modulus \(q\) and reduced residue \(a\bmod q\), of UPPER_BOUND.md's Parseval identity (29) and its arc-transfer bound (30). Everything derived below is exact algebra (Parseval/orthogonality) plus explicit, non-asymptotic inequalities; no new estimate for \(\Delta(t;q,a)\)-type quantities is claimed or attempted, since RANK3_SCOPE.md §2-3 already show that estimate (Route A) is the missing input and is not supplied by (SW)/(V)/(LS). This is exactly the boundary RANK3_SCOPE.md draws: "the identity half is plausible... nothing in the text suggests an obstruction to writing it down. But converting that identity into a bound... still needs a uniform estimate."
A note on sources. This attempt's worktree does not contain RANK3_POLYRANGE.md; only its guessed weight, as quoted in this document's own assignment — \(\mu(q)^2/\phi(q)^2\), summed over \(a\), "not derived and may be wrong" — was available to check against. §5 below checks it against what is actually derived here. If RANK3_POLYRANGE.md exists elsewhere with content beyond that one guess, it was not read to produce this document, and whoever next holds both should reconcile them.
1. Setup
Recall from UPPER_BOUND.md §§3,6: for \(q\ge1\), \(a\) a reduced residue mod \(q\), and \(\beta\) with \(\|\beta\|\le\delta_q=Q/(qN)\), \[ P_{q,a}(\beta)=\frac{\mu(q)}{\phi(q)}K_N(\beta),\qquad R_{q,a}(\beta)=F_N(a/q+\beta)-P_{q,a}(\beta), \] \[ U_{(q)}=\sum_{a\bmod q}^\int_{\|\beta\|\le\delta_q}|P_{q,a}(\beta)|^2|R_{q,a}(\beta)|^2\,d\beta, \qquad Z_{(q)}=\sum_{a\bmod q}^\int_{\|\beta\|\le\delta_q}|R_{q,a}(\beta)|^4\,d\beta, \tag{D0} \] the star meaning \(a\) ranges over the \(\phi(q)\) reduced residues; both \(U_{(q)}\) and \(Z_{(q)}\) are read off \(F_N\) and \(K_N\) alone, with no prime-pair content assumed. \(F_N,K_N,\Lambda,\psi\) are as in UPPER_BOUND.md §1; \(L=\log N\).
Orthogonality decomposition. For \(n\ge1\), \(\exp1(n\cdot a/q)\) depends on \(n\) only through \(n\bmod q\), so grouping the sum defining \(F_N\) by residue class, \[ F_N(a/q+\beta)=\sum_{b\bmod q}\exp1(ab/q)\,S_b(\beta),\qquad S_b(\beta)=\sum_{\substack{n=1\\n\equiv b\,(q)}}^N\Lambda(n)\exp1(n\beta), \tag{D1} \] exactly, for every \(q,a,\beta\) — this is the identity named in the assignment, and it costs nothing beyond regrouping a finite sum.
Coprime/non-coprime split. Split the outer sum in (D1) by \(\gcd(b,q)\). For \((b,q)=1\), (SW)'s content is \(\psi(t;q,b)=t/\phi(q)+O_{B,H}(NL^{-H})\) uniformly for \(q\le L^B\); the natural model for \(S_b\) coming from that same heuristic (density \(1/\phi(q)\) among all \(n\le N\), not just those \(\equiv b\)) is \(K_N(\beta)/\phi(q)\), exactly the model UPPER_BOUND.md's own paragraph before (17) uses. Define, for \((b,q)=1\), \[ D_b(\beta)=S_b(\beta)-\frac{K_N(\beta)}{\phi(q)} =\sum_{n=1}^N\Big(\Lambda(n)\mathbf 1_{n\equiv b(q)}-\frac1{\phi(q)}\Big)\exp1(n\beta). \tag{D2}\] For \(\gcd(b,q)>1\), \(S_b\) is a residual: \(\Lambda(n)\ne0\) only at prime powers, and \(\gcd(n,q)>1\) forces \(n=p^k\) with \(p\mid q\); each such prime contributes \(\sum_{p^k\le N}\log p\le L\), and \(q\) has \(\omega(q)\le\log_2q\) prime factors, so \[ \rho_2(q):=\sum_{\substack{n\le N\\ \gcd(n,q)>1}}\Lambda(n) =\sum_{p\mid q}\ \sum_{p^k\le N}\log p \le\omega(q)L\ll L\log(2q), \tag{D3} \] the same quantity UPPER_BOUND.md's paragraph before (17) bounds ("\(O(L\log(2q))\) from prime powers whose prime divides \(q\)"). Since the sets \(\{n\le N: n\equiv b(q)\}\) for distinct non-coprime \(b\) partition \(\{n\le N: \gcd(n,q)>1\}\), \(\sum_{b:(b,q)>1}|S_b(\beta)|\le\rho_2(q)\) for every \(\beta\) (triangle inequality inside each \(S_b\), then summing the disjoint ranges).
Consistency check: (D1) reproduces \(P_{q,a}\). Summing the coprime part of (D1) with the model in place of \(S_b\), \[ \sum_{b\bmod q}^\exp1(ab/q)\frac{K_N(\beta)}{\phi(q)} =\frac{K_N(\beta)}{\phi(q)}\sum_{b\bmod q}^\exp1(ab/q) =\frac{K_N(\beta)}{\phi(q)}c_q(a) =\frac{\mu(q)}{\phi(q)}K_N(\beta), \] using \(c_q(a)=\mu(q)\) for \((a,q)=1\) (classical: write \(\mathbf1_{(b,q)=1}=\sum_{d\mid\gcd(b,q)}\mu(d)\) and sum over \(b\bmod q\); only \(d=q\) survives since \((a,q)=1\)). This is exactly \(P_{q,a}\) as already defined in UPPER_BOUND.md — the given \(P_{q,a}\) is precisely what falls out of modeling every coprime \(S_b\) by \(K_N/\phi(q)\) and summing against the character-like weight \(\exp1(ab/q)\); it is not a separate ansatz. Consequently \[ R_{q,a}(\beta)=\underbrace{\sum_{b\bmod q}^*\exp1(ab/q)D_b(\beta)}{R^{(1)}{q,a}(\beta)} +\underbrace{\sum_{\substack{b\bmod q\\(b,q)>1}}\exp1(ab/q)S_b(\beta)}{R^{(2)}{q,a}(\beta)}, \qquad|R^{(2)}_{q,a}(\beta)|\le\rho_2(q). \tag{D4} \]
The per-residue remainder. For \((b,q)=1\) and integer \(0\le t\le N\), define, exactly as named in the assignment (the dummy index is written \(b\) here, not \(a\), because \(a\) already names the arc center in \(P_{q,a}\)): \[ \Delta(t;q,b)=\psi(t;q,b)-\frac t{\phi(q)},\qquad \psi(t;q,b)=\sum_{\substack{n\le t\\n\equiv b(q)}}\Lambda(n). \] At \(q=1\) there is one residue \(b=0\), \(\phi(1)=1\), and \(\Delta(t;1,0)=\psi(t)-t=\Delta(t)\), UPPER_BOUND.md's own notation; every identity below reduces to (29)-(30) exactly at \(q=1\), checked at the end of §3.
2. The exact per-residue identity: (29), one copy per residue
Fix \((b,q)=1\). \(K_N D_b\) is the coefficient-wise convolution of \(\mathbf1_{[1,N]}\) with \(d_b(n):=\Lambda(n)\mathbf1_{n\equiv b(q)}-1/\phi(q)\) (both supported on \(1\le n\le N\)); its coefficient at \(\exp1(m\beta)\) is \(\sum_{n=\max(1,m-N)}^{\min(N,m-1)}d_b(n)\), which for \(2\le m\le N+1\) is \(\sum_{n=1}^{m-1}d_b(n)=\Delta(m-1;q,b)\), and for \(N+2\le m\le2N\) is \(\Delta(N;q,b)-\Delta(m-N-1;q,b)\) — the identical computation UPPER_BOUND.md performs in the proof of (29), with \(\Lambda(n)-1\) replaced by \(d_b(n)\). Parseval and the same completed square (an identity in the partial sums alone, so it transfers verbatim) give, for every \((b,q)=1\), \[ T(q,b):=\int_{\mathbb T}|K_N D_b|^2 =\sum_{t=1}^N\Delta(t;q,b)^2+\sum_{t=1}^{N-1}[\Delta(N;q,b)-\Delta(t;q,b)]^2 =\frac{N+1}2\Delta(N;q,b)^2+2\sum_{t=1}^{N-1}\Big[\Delta(t;q,b)-\frac{\Delta(N;q,b)}2\Big]^2. \tag{D5} \] \(T(q,b)\) is a genuine, exact per-residue copy of (29): at \(q=1\), \(T(1,0)=T_N\).
The same computation on two residues \(b\ne b'\) gives the real, exact bilinear cross term \[ X(b,b'):=\int_{\mathbb T}K_ND_b\,\overline{K_ND_{b'}} =\sum_{t=1}^N\Delta(t;q,b)\Delta(t;q,b') +\sum_{t=1}^{N-1}[\Delta(N;q,b)-\Delta(t;q,b)][\Delta(N;q,b')-\Delta(t;q,b')], \tag{D6} \] with \(X(b,b)=T(q,b)\); \(X\) is an inner product \(\langle K_ND_b,K_ND_{b'}\rangle_{L^2(\mathbb T)}\), so \((X(b,b'))_{b,b'}\) is a real symmetric positive-semidefinite Gram matrix and \(|X(b,b')|\le\sqrt{T(q,b)T(q,b')}\) by Cauchy-Schwarz. No identity of this kind exists in either source document for \(q\ge2\); (D5)-(D6) are new.
3. The exact identity for \(U_{(q)}\)'s coprime-driven part, summed over \(a\)
By (D4), \(K_NR^{(1)}{q,a}=\sum{b}^\exp1(ab/q)K_ND_b\), so \[ \int_{\mathbb T}|K_N|^2|R^{(1)}{q,a}|^2 =\sum{b,b'}^\exp1\!\big(a(b-b')/q\big)X(b,b'). \] Summing over the \(\phi(q)\) reduced residues \(a\) and using \(\sum_a^\exp1(a(b-b')/q)=c_q(b-b')\) (the Ramanujan sum already defined in UPPER_BOUND.md §2, with \(c_q(0)=\phi(q)\)): \[ \boxed{\ \sum_{a\bmod q}^\int_{\mathbb T}|K_N|^2|R^{(1)}{q,a}|^2\,d\beta =\underbrace{\phi(q)\sum{b\bmod q}^*T(q,b)}{\Sigma{\rm diag}(q)} +\underbrace{\sum_{\substack{b\ne b'\bmod q}}^*c_q(b-b')X(b,b')}{\Sigma{\rm cross}(q)}.\ } \tag{D7} \] This is exact — pure Parseval and character orthogonality, no approximation anywhere — and it is the per-\((q,a)\) analogue of (29) the task asks for. At \(q=1\) there is only \(b=b'=0\), \(\Sigma_{\rm cross}(1)=0\) (empty sum), \(\Sigma_{\rm diag}(1)=\phi(1)T(1,0)=T_N\): (D7) reduces to (29) exactly, with the single arc \(a=1\).
For \(q\ge2\), \(\Sigma_{\rm cross}(q)\) has no counterpart at \(q=1\): it is a real number of either sign (the left side is \(\ge0\), so \(\Sigma_{\rm cross}(q)\ge-\Sigma_{\rm diag}(q)\), but nothing here pins its sign further). A cancellation-free upper bound is available from the Gram property of \(X\): \(|c_q(k)|\le\phi(q)\) trivially (a sum of \(\phi(q)\) unit-modulus terms), and \[ \sum_{b\ne b'}^|X(b,b')|\le\sum_{b\ne b'}^\sqrt{T(q,b)T(q,b')} \le\Big(\sum_b^*\sqrt{T(q,b)}\Big)^2-\sum_b^*T(q,b) \le(\phi(q)-1)\sum_b^*T(q,b) \] (Cauchy-Schwarz twice), so \(|\Sigma_{\rm cross}(q)|\le\phi(q)(\phi(q)-1)\sum_b^*T(q,b)\), and \[ \Sigma_{\rm diag}(q)+\Sigma_{\rm cross}(q)\ \le\ \phi(q)^2\sum_{b\bmod q}^*T(q,b). \tag{D8} \] (D8) assumes no cancellation in \(\Sigma_{\rm cross}\) at all; whether the true value of \(\Sigma_{\rm cross}(q)\) is close to \(0\) (as the diagonal-only guess in §5 below implicitly assumes) or close to its Cauchy-Schwarz ceiling is not determined by UPPER_BOUND.md or RESULTS.md — both are consistent with the exact identity (D7).
4. Assembling \(U_{(q)}\): the \(R^{(2)}\) correction and the arc transfer
By \((x+y)^2\le2x^2+2y^2\), \(|R^{(2)}_{q,a}|\le\rho_2(q)\) pointwise, and \(\int_{\mathbb T}|K_N|^2=N\): \[ \int_{\mathbb T}|K_N|^2|R_{q,a}|^2 \le2\int_{\mathbb T}|K_N|^2|R^{(1)}_{q,a}|^2+2\rho_2(q)^2N. \] Summing over \(a\) and applying (D8): \[ \sum_{a\bmod q}^*\int_{\mathbb T}|K_N|^2|R_{q,a}|^2\,d\beta \le2\phi(q)^2\sum_{b\bmod q}^*T(q,b)+2\phi(q)\rho_2(q)^2N. \tag{D9} \] Multiplying by \((\mu(q)/\phi(q))^2=|P_{q,a}|^2/|K_N(\beta)|^2\) gives the full-circle analogue of \(T_N\) at \((q,a)\), summed over \(a\); note the \(\phi(q)^2\) in (D8)-(D9) cancels the \(1/\phi(q)^2\) exactly: \[ \sum_a^T_N(q,a):=\sum_a^\int_{\mathbb T}|P_{q,a}|^2|R_{q,a}|^2\,d\beta \le2\mu(q)^2\sum_{b\bmod q}^*T(q,b)+\frac{2\mu(q)^2\rho_2(q)^2N}{\phi(q)}. \tag{D10} \] Arc transfer, generalizing (30). On \(\|\beta\|>\delta_q=Q/(qN)\), \(|K_N(\beta)|\le(2\|\beta\|)^{-1}<qN/(2Q)\), so \(|P_{q,a}(\beta)|^2\le \mu(q)^2q^2N^2/(4Q^2\phi(q)^2)\) there, giving, exactly as in (30), \[ 0\le T_N(q,a)-\int_{\|\beta\|\le\delta_q}|P_{q,a}|^2|R_{q,a}|^2 \le\frac{\mu(q)^2q^2N^2}{4Q^2\phi(q)^2}\int_{\mathbb T}|R_{q,a}|^2\,d\beta. \] Exactly, \(\int_{\mathbb T}|R_{q,a}|^2=\sum_{n=1}^N|\Lambda(n)\exp1(na/q)-\mu(q)/\phi(q)|^2 \le\sum_n(\Lambda(n)+1)^2\le2d_N+2N\ll NL\) (Chebyshev, as UPPER_BOUND.md §2 uses for \(d_N\)), uniformly in \(a\) and \(q\). Summing over the \(\phi(q)\) values of \(a\) and combining with (D10): \[ \boxed{\ U_{(q)}\ \le\ 2\mu(q)^2\sum_{b\bmod q}^*T(q,b) \ +\ O\!\left(\frac{\mu(q)^2\rho_2(q)^2N}{\phi(q)}\right) \ +\ O\!\left(\frac{q^2N^3L}{Q^2\phi(q)}\right).\ } \tag{D11} \] This is the requested exact-identity-plus-explicit-error-term result for \(U_{(q)}\): (D7) is exact, (D8) is a one-sided but fully explicit Cauchy-Schwarz bound, (D3) and the transfer step are explicit and of the same shape as (30). At \(q=1\): \(\mu(1)^2=1\), \(\phi(1)=1\), \(\rho_2(1)=0\) (no prime divides \(1\)), and (D11) reads \(U_1\le2T_N+O(N^2L)\), matching (29)-(30) up to the factor of 2 coming from using \((x+y)^2\le2x^2+2y^2\) instead of expanding exactly — using the exact expansion at \(q=1\) (no \(R^{(2)}\) term exists there) removes that factor and reproduces (30) exactly. Two further facts fall out for free: since \(\mu(q)=0\) for non-squarefree \(q\), (D11) gives \(U_{(q)}=0\) identically for every non-squarefree \(q\) — \(U_{(q)}\) for \(q\le R_0\) is supported only on the squarefree \(q\) in that range, a simplification not stated in RANK3_SCOPE.md.
5. What weight this identity actually carries — checking the guess
RANK3_POLYRANGE.md's guess, as quoted in this document's assignment, is that the conversion carries weight \(\mu(q)^2/\phi(q)^2\), summed over \(a\), by direct analogy with (29)-(31)'s \(1\cdot\sum_t\Delta(t)^2\to32U_1\) (i.e., \(q=1\) has \(\mu(1)^2/\phi(1)^2=1\)). Three distinct weights appear in the computation above, and they disagree with each other, so "the" weight is not a single number without specifying which quantity it multiplies:
- If \(\Sigma_{\rm cross}(q)\) is discarded assuming it is genuinely negligible (an assumption not derived here, and not decidable from UPPER_BOUND.md/RESULTS.md — see end of §3), the diagonal piece alone in (D7) carries weight \(\mu(q)^2/\phi(q)\) on \(\sum_b^*T(q,b)\) — one power of \(\phi(q)\), not two.
- Without assuming any cancellation — the only bound actually proved here, (D11) — the weight on \(\sum_b^*T(q,b)\) is \(2\mu(q)^2\), i.e. \(O(1)\), with no \(\phi(q)\) in the denominator at all: the \(\phi(q)^2\) from the Cauchy-Schwarz ceiling on \(\Sigma_{\rm cross}\) in (D8) exactly cancels the \(1/\phi(q)^2\) in \(|P_{q,a}|^2\).
- The guessed weight, \(\mu(q)^2/\phi(q)^2\), is smaller than both of these by at least one further power of \(\phi(q)\) (relative to the diagonal-only weight) or two further powers (relative to the proved, cancellation-free weight).
So the guess is not merely undetermined, it is optimistic against what this derivation actually establishes: reaching \(\mu(q)^2/\phi(q)^2\) would require \(\Sigma_{\rm cross}(q)\) to cancel the diagonal term down by a full extra factor of \(\phi(q)\) beyond assuming it is simply negligible. Nothing in UPPER_BOUND.md or RESULTS.md supplies or suggests such a cancellation; it is a new claim, not implied by the \(q=1\) analogy the guess was built on (at \(q=1\) there is no cross term to cancel anything, so the analogy is silent on this point by construction). Whether \(\Sigma_{\rm cross}(q)\) is in fact small is exactly as open as the \(\Delta(t;q,b)\)-uniformity question RANK3_SCOPE.md's Route A already names; this document does not resolve it either way, and the sources give no way to.
6. A cost the identity carries that RANK3_SCOPE.md did not have: the transfer step, summed over \(q\)
RANK3_SCOPE.md's Route D discussion (§3) says building the identity would only "relocate" Route A's missing uniform estimate, not add a new cost. That undercounts one item, visible only once the identity is written down: even granting a hypothetical uniform bound making every \(\sum_b^*T(q,b)\) in (D11) as small as the target requires, the transfer term \(O(q^2N^3L/(Q^2\phi(q)))\) in (D11), summed over the actual rank-3 range \(2\le q\le R_0\) with \(Q=\lfloor\sqrt N/3\rfloor\), \(R_0=Q/L\), already costs more than the whole target budget on its own. By (15), \(q^2/\phi(q)=q\cdot(q/\phi(q))\le\zeta(2)q(1+\log q)\), so \[ \sum_{q=2}^{R_0}\frac{q^2}{\phi(q)}\ll\sum_{q=2}^{R_0}q(1+\log q)\ll R_0^2\log(2R_0). \] With \(R_0=Q/L\), \(Q\asymp\sqrt N\): \(R_0^2\log(2R_0)\asymp (N/L^2)\cdot(L/2)=N/(2L)\), so \[ \sum_{q=2}^{R_0}O\!\left(\frac{q^2N^3L}{Q^2\phi(q)}\right) \ll\frac{N^3L}{Q^2}\cdot\frac N{L} =\frac{N^4L}{Q^2L}\cdot\frac1{1} \asymp N^3, \] using \(Q^2\asymp N\). That is, the crude sup-bound transfer step used here (the direct generalization of (30)'s method) costs \(O(N^3)\) once summed over \(2\le q\le R_0\) — no saving over the trivial bound at all, the same order as the unconditional baseline (1) already proves by a completely different route. This holds independent of anything assumed about \(\Delta(t;q,b)\): the obstruction is the transfer step itself, not the main term. The reason (30)'s version of this step was harmless at \(q=1\) is that it was never summed over \(q\); (24)-(27)'s dyadic large-sieve argument is exactly the kind of non-crude transfer that avoids this blow-up for \(q>R_0\), by averaging across a whole dyadic block rather than bounding each arc's tail by its own supremum. Route D, carried out with the naive per-arc transfer, would need an analogous refinement for \(q\le R_0\) — this is additional to, not a restatement of, Route A's \(\Delta(t;q,b)\)-uniformity requirement, and neither document names it.
7. As far as possible for \(Z_{(q)}\)
\(Z_{(q)}=\sum_a^*\int_{\rm arc}|R_{q,a}|^4\) carries no \(|P_{q,a}|^2\) weight (UPPER_BOUND.md §6, and the assignment's own definition), unlike \(U_{(q)}\). This changes the picture in two independent ways.
No K_N-convolution, so no telescoping. The reduction in §2 that turned \(D_b\)'s coefficients into partial sums \(\Delta(t;q,b)\) came entirely from convolving with \(K_N\) (whose coefficients are the constant \(1\) on \([1,N]\), turning a coefficient sequence into its partial sums under convolution). \(Z_{(q)}\) has no such factor: its full-circle analogue is a 4th moment of \(D_b\) itself, not of \(K_ND_b\). Repeating (D4) and Parseval on \(R^{(1)}_{q,a}=\sum_b^\exp1(ab/q)D_b\), \[ \int_{\mathbb T}|R^{(1)}{q,a}|^4 =\sum{b_1,b_2,b_3,b_4}^\exp1\!\big(a(b_1{+}b_2{-}b_3{-}b_4)/q\big)\,Y(b_1,b_2,b_3,b_4), \] \[ Y(b_1,b_2,b_3,b_4)=\sum_{\substack{n_1,n_2,n_3,n_4=1\\n_1+n_2=n_3+n_4}}^N d_{b_1}(n_1)d_{b_2}(n_2)d_{b_3}(n_3)d_{b_4}(n_4), \] exact by the same "\(\int_{\mathbb T}\exp1(k\beta)=\mathbf1_{k=0}\)" argument. Summing over \(a\) produces a 4-index Ramanujan-sum weight \(c_q(b_1+b_2-b_3-b_4)\) exactly as \(c_q(b-b')\) appeared in (D7). This is the exact per-\((q,a)\) generalization the task asks for, as far as it goes: it reduces \(Z_{(q)}\)'s coprime part to a Ramanujan-sum-weighted sum of the quadruple additive-convolutions \(Y\) of the sequences \(d_b\). Unlike (D5), \(Y\) does not reduce to a sum of squares of \(\Delta(t;q,b)\): the constraint \(n_1+n_2=n_3+n_4\) is a genuine additive condition on four independent indices, not a diagonal telescoping identity, because there is no kernel factor turning \(d_b\) into its own partial sums. This is a structural fact about the definition of \(Z\), not a gap in this derivation: at \(q=1\), \(Y(0,0,0,0)=\int_{\mathbb T}|F_N-K_N|^4\) is exactly \(Z_{(1)}\)'s full-circle analogue, and RANK3_SCOPE.md §1 already records that no identity for \(Z_{(1)}\) exists in either source document. This document does not supply one either, for \(q=1\) or any \(q\ge2\); it exhibits the same difficulty, parametrized over \(q\) and \(b\), rather than resolving it.
No arc-transfer analogue either. The transfer step in §4 worked because \(U_{(q)}\)'s integrand carries the factor \(|P_{q,a}|^2=|K_N|^2\cdot (\mu(q)/\phi(q))^2\), which decays like \(q^2N^2/Q^2\) away from the arc and so makes the full-circle-minus-arc discrepancy small once \(Q\) is large enough. \(Z_{(q)}\)'s integrand, \(|R_{q,a}|^4\), carries no such factor: \(R_{q,a}=F_N-P_{q,a}\) has no reason to be small away from the arc (indeed \(F_N\) itself need not be small there), so bounding \(\int_{\mathbb T}|R_{q,a}|^4\) does not bound \(\int_{\rm arc}|R_{q,a}|^4\) the way it did for \(U\). This matches UPPER_BOUND.md's own practice: (28)'s bound on \(Z_{q>R_0}\) never goes through a full-circle identity, it stays on the actual arcs and uses (V) directly. So even setting aside the non-reduction of \(Y\), a full-circle route to \(Z_{(q)}\) would need a second, unrelated argument this document does not supply, and neither source names one.
The \(R^{(2)}\) correction for \(Z\) is straightforward by the same method as §4: \((x+y)^4\le8(x^4+y^4)\) (the same constant UPPER_BOUND.md uses at (28)) and \(|R^{(2)}_{q,a}|\le\rho_2(q)\) give \(\int_{\mathbb T}|R^{(2)}_{q,a}|^4\le\rho_2(q)^4\), hence \(\sum_a^\int_{\mathbb T}|R_{q,a}|^4\le8\sum_a^\int_{\mathbb T}|R^{(1)}_{q,a}|^4+8\phi(q)\rho_2(q)^4\); this is the one piece of \(Z_{(q)}\)'s treatment that is fully closed here, and it is the smaller of the two obstructions above.
8. Where this leaves rank 3
The identity half of Route D, which RANK3_SCOPE.md §3 said was plausible but unwritten, is now written down exactly for \(U_{(q)}\) — (D7) — and as far as the definition of \(Z\) permits for \(Z_{(q)}\) (the quadruple-sum reduction above, which does not close). RANK3_SCOPE.md §3's conclusion that Route D "does not lower the cost of Route A; it only relocates where the missing input would be used" needs one addition: it also surfaces a second, independent requirement (§6) — a non-crude, large-sieve-style transfer argument in place of the direct \((30)\)-style bound, since the latter costs \(O(N^3)\) once summed over \(2\le q\le R_0\) regardless of how well \(\Delta(t;q,b)\) is controlled. Neither requirement is met here or costed to completion by either source document. RANK3_SCOPE.md §4's conclusion — that closing rank 3 alone cannot move the bound reached, because rank 1's \(O_H(N^3L^{-2H})\) is the current bottleneck in (23) regardless of what happens to ranks 2 or 3 — is unaffected by anything in this document: no identity or bound derived here touches rank 1, \(M_Q\), \(I_Q\), or the unconditional total (1).
This document is algebraic bookkeeping (Parseval, orthogonality, Cauchy-Schwarz) around the existing unconditional construction in UPPER_BOUND.md; it assumes and establishes nothing about zeros of \(L\)-functions or the Riemann Hypothesis.