Proof. Let \(\xi^6=a\). Since \(F\) contains all sixth roots of unity, all roots of \(X^6-a\) lie in \(F(\xi)\), and \(\sigma\mapsto\sigma(\xi)/\xi\) embeds its Galois group into \(\mu_6\). The extension is therefore abelian. A prime outside \(6a\) is unramified; for example this follows from the discriminant of \(X^6-a\), which is supported on \(6a\). At such a prime, arithmetic Frobenius is characterized on the residue field by \(x\mapsto x^{q_{\mathfrak p}}\). Consequently its quotient on \(\xi\) reduces to \(a^{(q_{\mathfrak p}-1)/6}\). Reduction is injective on \(\mu_6\) outside \(6\), proving the displayed identity. Multiplicativity of the Artin map gives the assertion for ideals. Artin reciprocity makes this map factor through a ray group with modulus supported on the primes dividing \(6a\); these are the power-residue and ray-group assertions in [22].

For the last statement, an ideal \(A\) outside \(S\) is coprime to every \(\pi_{\mathfrak p}\) and to every unit. Its symbol therefore has no zero there, and the power of each \(\chi_A(\pi_{\mathfrak p})\in\mu_6\) depends only on \(v_{\mathfrak p}\pmod6\). There are at most \(6^{|S|+1}\) possible displayed numerators after this reduction. Apply the first assertion to each and take a common multiple of their ray moduli. Their prime supports are all contained in \(S\) because \(S\) contains the primes above \(6\). ◻

Proof. Let \(k=\mathcal O/(p)\) and let \(\psi(x)=e(x/p)\) be its nontrivial additive character. For a multiplicative character \(A\) of \(k^\times\), extended by zero, put \[\tau(A)=\sum_{x\in k}A(x)\psi(x),\qquad J(A,B)=\sum_{x\in k}A(x)B(1-x).\] If \(A\) is nonprincipal, the change of variables \(x=ty\) gives \[|\tau(A)|^2 =\sum_{t\in k^\times}A(t)\sum_{y\in k^\times}\psi((t-1)y)=P.\] The inner sum is \(P-1\) for \(t=1\) and \(-1\) otherwise, and \(\sum_{t\in k^\times}A(t)=0\). Conjugation also gives \(\tau(A)\tau(\overline A)=A(-1)P\). When \(AB\) is nonprincipal, grouping the product \(\tau(A)\tau(B)\) by \(x+y\) gives \[\tau(A)\tau(B)=J(A,B)\tau(AB).\] The group with \(x+y=0\) vanishes because \(AB\) is nonprincipal; for a nonzero sum, division by \(x+y\) gives the displayed Jacobi factor. In particular \(|J(A,B)|=\sqrt P\) if \(A,B,AB\) are all nonprincipal.

The number of solutions of \(4x(1-x)=y\) is \(1+H^3(1-y)\). Summing \(H(y)\) times this identity and using \(\sum_yH(y)=0\) gives \(J(H,H^3)=H(4)J(H,H)\). Write \(\tau_j=\tau(H^j)\). The Gauss–Jacobi identity and \(\tau_2\tau_4=P\) now give \[\frac{\tau_1\tau_3}{\tau_4} =H(4)\frac{\tau_1^2}{\tau_2},\qquad \tau_1\tau_2=\overline{H(4)}\frac{\tau_3\tau_2^3}{P}.\] Here \(H^2(-1)=1\) because \(-1\) is a cube.

It remains to fix the cubic Jacobi sum, including its unit. Put \(m=(P-1)/3\). The definition of \(H^2\) gives, in \(k\), \[J(H^2,H^2)\equiv\sum_{x\in k}x^m(1-x)^m=0\pmod p.\] Indeed the polynomial has degree \(2m<P-1\), and the sum over \(k\) of each monomial of degree less than \(P-1\) is zero in \(k\) (the constant case is \(P=0\) in \(k\)). The Jacobi sum belongs to \(\mathcal O\) and has absolute value \(\sqrt P\).

For \(x\ne0,1\), choose \(j_x\in\{0,1,2\}\) such that \(H^2(x(1-x))=\omega^{j_x}\). The products of \(x\) and of \(1-x\) over these \(x\) are both \(-1\), so \(\prod_{x\ne0,1}x(1-x)=1\). Hence \(\sum_{x\ne0,1}j_x\equiv0\pmod3\). Since \[\omega^j\equiv1+j(\omega-1)\pmod{(\omega-1)^2}, \qquad ((\omega-1)^2)=(3),\] we obtain \(J(H^2,H^2)\equiv P-2\equiv-1\pmod3\). Divisibility by \(p\) and equality of norms imply \(J(H^2,H^2)=up\) for a unit \(u\). As \(p\) is primary and the six units have distinct residues modulo \(3\), the congruence forces \(u=-1\). Finally, \(\tau_2^3=J(H^2,H^2)\tau_2\tau_4=-pP\). Dividing the two Gauss identities by the appropriate powers of \(\sqrt P\) proves the lemma. ◻

Proof. Use the Fourier transform with kernel \(e(-zy)\) and measure \(d\mu\). For \(\varepsilon>0\) apply Poisson summation on \(\mathcal O\) to \(f_\varepsilon(z)=e(z^2/c)e^{-\pi\varepsilon|z|^2}\). Direct Gaussian integration gives, with \(r_\varepsilon=1+3q_c\varepsilon^2/16\), \[\widehat f_\varepsilon(y) =\frac{|c|}{2\sqrt{r_\varepsilon}} e^{-\pi\varepsilon q_c|y|^2/(4r_\varepsilon)} e\bigl(-cy^2/(4r_\varepsilon)\bigr).\] To check the normalization, rotate \(z\) by half the argument of \(c\). The quadratic matrix of the resulting real two-variable Gaussian is \[\begin{pmatrix} \varepsilon&-4i/(\sqrt3|c|)\\ -4i/(\sqrt3|c|)&\varepsilon \end{pmatrix}, \qquad \det=\frac{16r_\varepsilon}{3q_c}.\] Its eigenvalues are conjugates with positive real part. The positive square root of the determinant, the factor \(2/\sqrt3\) in \(d\mu\), and completion of the square give the formula above.

Put \(I_\varepsilon=\int_{\mathbb C}e^{-\pi\varepsilon|z|^2}d\mu(z) =2/(\sqrt3\varepsilon)\). For a fixed ideal \((b)\), a fixed class \(v\bmod b\), and \(a>0\), Gaussian Poisson summation on \(b\mathcal O\) gives \[\frac1{I_\varepsilon}\sum_{z\in v+b\mathcal O} e^{-\pi a\varepsilon|z|^2} \longrightarrow\frac1{a q_b}\qquad(\varepsilon\downarrow0).\] The zero dual vector gives the limit and every nonzero dual vector is exponentially small. The phase \(e(z^2/c)\) is periodic modulo \(c\). Consequently the normalized left side of Poisson’s identity tends to \(\Gamma(c)/|c|\).

On the right side one may replace \(r_\varepsilon\) by \(1\). For fixed \(c\), the total error from the phase is at most \[C_c\varepsilon^2\sum_{y\in\mathcal O}|y|^2 e^{-C_c^{-1}\varepsilon|y|^2}=O_c(1),\] because the sum is \(O_c(\varepsilon^{-2})\). The amplitude error is \(O_c(\varepsilon)\), as is the damping error, by the same lattice estimate. All three are \(o(I_\varepsilon)\). The phase \(e(-cy^2/4)\) is periodic modulo \(2\mathcal O\). In the preceding Gaussian mean take \(b=2\) and \(a=q_c/4\); each of its four classes has normalized mean \(1/q_c\). The normalized right side therefore tends to \((2|c|)^{-1}\sum_{y\bmod2}e(-cy^2/4)\), which proves Equation (10).

The representatives \(0,1,\omega,\omega^2\) modulo \(2\) give the asserted four-term expression. Multiplication by an odd \(v\) permutes these classes, so the same formula gives \(\Gamma(cv^2)=\Gamma(c)\). The group of units modulo \(4\) has order \(12\). Squaring a lift modulo \(4\) depends only on its class modulo \(2\), and its three possible squares are \(1,\omega,\omega^2\). The four representatives in the statement are distinct modulo this subgroup; also \(\lambda^2=-3\equiv1\pmod4\). Evaluating the four-term expression on them gives the table. Evaluating \(\Gamma(ab)/(\Gamma(a)\Gamma(b))\) on the two generators \(-1,\lambda\) gives the displayed exponent, proving the last claims. ◻

Proof. At a good prime, counting square roots in the residue field gives \[\sum_{x\bmod p}e(x^2/p) =\sum_{y\bmod p}(1+\chi_p(y)^3)e(y/p) =\sum_{y\bmod p}\chi_p(y)^3e(y/p).\] The additive sum without the character is zero. Hence \(\Gamma(p)=\gamma_3(p)\), and replacing \(x^2/p\) by \(ux^2/p\) for a unit \(u\) multiplies the sum by \(\chi_p(u)^3\). For squarefree \(c\), representing a residue as \(\sum_{p\mid c}(c/p)x_p\) gives, in both sums, \[\Gamma(c)=\prod_{p\mid c}\chi_p(c/p)^3\Gamma(p),\qquad \gamma_3(c)=\prod_{p\mid c}\chi_p(c/p)^3\gamma_3(p).\] The cross terms in the square are integral in the additive character. Thus \(\Gamma(c)=\gamma_3(c)\) for squarefree \(c\). In particular, for distinct good primes \(p,q\), \[\frac{\Gamma(pq)}{\Gamma(p)\Gamma(q)} =\chi_p(q)^3\chi_q(p)^3.\]

We use cubic reciprocity in the following precise form: for coprime primary \(a,b\) one has \((a/b)_3=(b/a)_3\); see [8]. Thus the quotient \(\chi_b(a)/\chi_a(b)\) is a sign. At \(p,q\) its cube is \(\chi_q(p)^3\chi_p(q)^3\), so this sign equals \(\mathfrak r(p,q)\). Multiplicativity in both arguments and the bicharacter property extend the equality to every coprime primary pair. The definition by \(\mathfrak r\) on noncoprime pairs involves no division of zero symbols.

The element \(-2\) is primary and \(-1\) is a cube. Cubic reciprocity therefore gives \[\chi_c(4)=(2/c)_3=(-2/c)_3=(c/(-2))_3.\] This is a character of \(c\bmod2\). Together with the dependence of \(\Gamma\) on \(c\bmod4\), this proves that \(G\) is a function on a fixed ray group (for example the ray group modulo \(12\) suffices here). Indeed, if two primary generators represent the same ray class modulo \(12\), their quotient differs from an element congruent to one modulo \(12\) by a unit; reduction modulo \(3\) forces that unit to be one. The generators therefore have the same residue modulo \(4\). Its multiplicative refinement is the definition of \(\mathfrak r\). Since \(\Gamma(v^2)=1\) and \(\chi_v(4)\) has order dividing three, \(G(v^2)=\overline{\chi_v(4)}^{\,2}=\chi_v(4)\). The table also gives \[\Gamma(p)^2=(-1)^{(q_p-1)/2}=\chi_p(-1).\] For the last equality use \(\chi_p(-1)=(-1)^{(q_p-1)/6}\) and the equality of these parities. Now \(\Gamma(p^3)=\Gamma(p)\), and \(\overline{\chi_p(4)}^{\,3}=1\), proving \(G(p^3)=\gamma_3(p)\). Finally \(\mathcal R(p,p)=1/\Gamma(p)^2=\Gamma(p)^2\), as this square is a sign.

For a symmetric sign-valued bicharacter its diagonal is multiplicative: \[\mathcal R(vw,vw) =\mathcal R(v,v)\mathcal R(v,w)^2\mathcal R(w,w) =\mathcal R(v,v)\mathcal R(w,w).\] The prime identity consequently gives \(\mathcal R(n,n)=\chi_n(-1)\) for every primary \(n\) outside \(S\). On the square class \((-1)^e\lambda^f\), the table gives both this diagonal and \(\mathfrak r(-1,n)\) the value \((-1)^f\). This use of \(-1\) is in the residue square-class group; the ideal \((-1)\) is the identity in an ideal ray class group and is not being substituted there.

For coprime squarefree primary \(a,b\) outside \(S\), the Chinese remainder theorem gives \[\gamma_j(ab)=\chi_a(b)^j\chi_b(a)^j\gamma_j(a)\gamma_j(b).\] One obtains this by representing a residue as \(bx+ay\) with \(x\bmod a\) and \(y\bmod b\). Cubing the factor for \(j=2\) gives one. The product of the factors for \(j=1\) and \(j=2\) is \((\chi_a(b)\chi_b(a))^3=\mathcal R(a,b)\). The prime identities from Lemma 6, followed by Equation (12), therefore prove Equation (14) by induction on the number of prime factors. ◻

Proof. For an integer \(m\) with \(2m>J+d\), integration by parts gives \[(1+|\boldsymbol t|^2)^m\widehat w(\boldsymbol t) =\int_{\mathbb R^d}(1-\Delta)^m w_{\log}(\boldsymbol u) e^{-i\boldsymbol t\cdot\boldsymbol u}\,d\boldsymbol u.\] Its absolute value is bounded by the volume of \(\Omega\) times \(C_{m,d}p_{2m}(w)\). The weighted integral is finite because \(2m>J+d\); one can choose \(2m\le J+d+2\). Fourier inversion gives the monomial formula. For row vectors \(B(\boldsymbol t)\) in \(\ell^2\), the precise inequality used there is \[\left\|\frac1{(2\pi)^d}\int\widehat w(\boldsymbol t) B(\boldsymbol t)\,d\boldsymbol t\right\|_{\ell^2} \le\frac1{(2\pi)^d}\int|\widehat w(\boldsymbol t)| \|B(\boldsymbol t)\|_{\ell^2}\,d\boldsymbol t.\]

In one dimension, the fundamental theorem of calculus, averaging a base point over the enlarged interval, and Cauchy–Schwarz give \(\sup_I|F|^2\ll\int_{I+[-1,1]}(|F|^2+|F'|^2)\). Applying this successively in each coordinate proves Equation (16). It can be summed over rows before the derivative integrals, since all terms are nonnegative. A logarithmic scale range of length \(O(\log Z)\) needs \(O(\log Z)\) unit intervals; a height range \([-T_1,T_1]\) needs \(O(1+T_1)\) intervals. Derivatives with respect to a logarithmic scale insert \(yW'(y)\), together with the constant derivative of a central normalization. Derivatives of a normalized twist \(y^{it}\) insert powers of \(\log y\). These are again annular profiles with finite seminorm bounds. For example, if a fixed-parameter squared row bound is \(O((1+|t|)^H)\) and there are \(d_t\) height parameters, the covering and integration cost at most a fixed multiple of \((1+T_1)^{H+d_t}\).

On \(|\boldsymbol t|>T_1\) one has \((1+|\boldsymbol t|)^{-N}\le T_1^{-N}\), proving Equation (17). The Mellin shift is immediate from its definition. To prove Equation (18), put \(F_\sigma(u)=e^{\sigma u}W(e^u)\). For \(|T|\ge1\), integration by parts gives \[\mathcal MW(\sigma+iT)=(-iT)^{-N} \int_{\mathbb R}F_\sigma^{(N)}(u)e^{iTu}\,du, \qquad F_\sigma^{(N)}(u)=e^{\sigma u}\sum_{j=0}^N \binom Nj\sigma^{N-j}D_y^jW(e^u).\] Taking absolute values gives the bound on a compact real strip. For \(|T|<1\), the absolute integral gives it after increasing the constant. The two stated classes of \(W\) have the required endpoint decay: compact support suffices in the first case, while \(e^{\sigma u}\) at \(-\infty\) and Schwartz decay at \(+\infty\) suffice in the second.

For Equation (19), use \[1+|T|\le(1+|T+v|)(1+|v|),\qquad 1+|T|+|v|+|w|\le2(1+|T+v|)(1+|v|)(1+|w|).\] Move \((1+|T|)^N\) to the left, bound the weighted \(a(v)\) pointwise, and integrate the remaining polynomial weight against the Gaussian in \(T+v\). The remaining \(w\) integral is the one displayed. This proves the joint bound and also the asserted join estimate. For the integrated assertion, both the map \((t_1,t_2,t_3)\mapsto(t_1+t_2,t_2,t_3)\) and its inverse have fixed operator norm. A fixed polynomial weight in the original variables is therefore bounded by a fixed polynomial weight in the transformed ones. On a complement of a box one inserts the additional inverse power of that weight and integrates the Gaussian and the two Mellin factors separately. The appended block triangular map has the same property because all of its coefficients and those of its inverse are fixed.

A kernel depending on a common product, such as \(y_1y_2\), gives the same norm power on both variables in the monomial formula; this preserves an equal-product-scale difference. A nonsmooth arithmetic mask has no such derivative bound and must instead be resolved before this lemma is applied. ◻

Proof. Such a partition is obtained by normalizing the integer translates of a nonnegative compactly supported smooth function positive on \([-1/2,1/2]\). Let \(\operatorname{supp}\chi\subset[-C,C]\). Each logarithmic derivative of \(W_{\mathrm G}(e^k x)\) is a polynomial in \(k+\log x\) times the same Gaussian. The product rule therefore gives \[p_j(w_k)\le C_j(1+|k|)^j \exp\bigl(-\tfrac14((|k|-C)_+)^2\bigr).\] This is summable after multiplication by \(e^{A|k|}\) for every fixed \(A\). The first assertion follows, and the separation norms follow from Lemma 9. In the profile’s logarithmic Fourier transform, rescaling an annulus inserts only the unit phase \(e^{-ikt}\), so it does not change these norms; any real normalization of a surrounding polynomial remains explicit. On the indicated tail, the logarithm of the last bound after multiplication by \(Z^Be^{A|k|}\) is at most \(-c\kappa^2(\log Z)^2+O(\log Z)\) for some fixed \(c>0\); summing the Gaussian tail proves the stated estimate for every \(M\). Finally, set \(u=\log y\) and complete the square in its Gaussian integral to obtain \(\mathcal MW_{\mathrm G}(s)=e^{s^2}\). The same Gaussian bounds every \(e^{\sigma u}(d/du)^jW_{\mathrm G}(e^u)\) in \(L^1(\mathbb R)\), uniformly for \(\sigma\) in a compact interval. ◻

Proof. Differentiating \(\widehat f\) inserts \(x^\beta\), and multiplication by a monomial in \(\xi\) differentiates \(x^\beta f\) before Fourier transformation. The \(L^1\) bound for the Fourier transform, the product rule, and the bound of \((1+|\xi|)^A\) by a finite sum of monomials of degrees at most \(\lceil A\rceil\) give the first assertion. For a radial function, \(r\partial_r\) corresponds to \((\xi\cdot\partial_\xi)/2\). Expanding its \(j\)th power gives finitely many polynomial multiples of Fourier derivatives, so the first assertion gives the radial one. A fixed linear change of coordinates changes only its constants.

For the Mellin assertion, differentiation under the absolutely convergent integral inserts \((-s)^j\). On the fixed strip this is bounded by a constant times \((1+|t|)^j\). Write \(y=e^u\) and integrate the Fourier transform of \(e^{-\sigma u}W(e^u)\) by parts more than \(h+j+1\) times. The product rule bounds the resulting \(L^1\) derivatives by the stated weighted logarithmic derivatives, because \(\sigma\) ranges over a fixed set. For a pure twist the Mellin integrand is \(\mathcal MW(-\sigma-it+i\omega)\). Substitute \(v=t-\omega\) and use \((1+|v+\omega|)^{h+j}\le(1+|v|)^{h+j}(1+|\omega|)^{h+j}\). Finally the product rule for a fixed annular cutoff and the chain rule for \(Ry\) give the last statement. ◻

Proof. A finite-order character has trivial infinite type, since \(\mathbb C^\times\) is connected. The primitive Hecke functional equation in this case is \[\Lambda(s,\psi)=(3Q)^{s/2}(2\pi)^{-s}\Gamma(s)L(s,\psi),\qquad \Lambda(s,\psi)=\varepsilon(\psi)\Lambda(1-s,\overline\psi), \quad |\varepsilon(\psi)|=1.\] This is the functional equation for primitive characters of trivial infinite type stated in [11]; a nonprincipal \(L(s,\psi)\) is entire. Absolute Euler convergence bounds \(L(11/10+it,\psi)\) uniformly in \(Q,t\). Applying the functional equation and Stirling’s formula on \(\Re s=-1/10\) gives \[|L(-1/10+it,\psi)|\ll Q^{3/5}(3+|t|)^{6/5}.\] The estimate for bounded \(t\) follows by continuity of the gamma quotient on that line, so the constant is uniform there as well.

For completeness, the growth hypothesis needed to use the strip principle can be obtained directly from a lattice theta integral. Let \(\mathfrak f=(f)\) be the conductor and define the periodic residue character \(\phi(a)=\psi((a))\) when \((a,\mathfrak f)=1\), with value zero otherwise. For the principal conductor \((1)\) put \(\phi(a)=1\) for all \(a\), including \(a=0\). Since the ideal class group is trivial, conductor one has no nonprincipal character. Periodicity modulo \(\mathfrak f\) follows from the ray character property; \(\phi\) is trivial on the six units because \((u)=(1)\) for a unit. Thus, for \(\Re s>1\), \[6\pi^{-s}\Gamma(s)L(s,\psi) =\int_0^\infty\bigl(\Theta_\phi(v)-\phi(0)\bigr)v^{s-1}\,dv, \qquad \Theta_\phi(v)=\sum_{a\in\mathcal O}\phi(a)e^{-\pi vq_a}.\] The factor six counts the generators of each ideal. If \(\widehat\phi(h)=Q^{-1}\sum_{a\bmod f}\phi(a)e(-ha/f)\), finite Fourier inversion and Gaussian Poisson summation give \[\Theta_\phi(v)=\frac{2}{\sqrt3v} \sum_{h\bmod f}\widehat\phi(h) \sum_{b\in\mathcal O} \exp\left(-\frac{4\pi|b-h/f|^2}{3v}\right).\] Changing the generator \(f\) only reindexes the finite sum. For each fixed conductor, the nonzero dual vectors have a positive minimum length. With \(A_\phi=2\widehat\phi(0)/\sqrt3\), this proves \[\Theta_\phi(v)=A_\phi v^{-1}+O_\phi(v^{-1}e^{-c_\phi/v})\quad(0<v\le1), \qquad \Theta_\phi(v)=\phi(0)+O_\phi(e^{-c_\phi v})\quad(v\ge1)\] for some \(c_\phi>0\). Splitting the Mellin integral at one consequently gives \[\begin{split} 6\pi^{-s}\Gamma(s)L(s,\psi) ={}&\frac{A_\phi}{s-1}-\frac{\phi(0)}s +\int_0^1\bigl(\Theta_\phi(v)-A_\phi/v\bigr)v^{s-1}\,dv\\ &+\int_1^\infty\bigl(\Theta_\phi(v)-\phi(0)\bigr)v^{s-1}\,dv. \end{split}\] Both integrals are entire and bounded in height on each bounded real strip, with constants that may depend on the fixed conductor. Stirling’s formula for \(1/\Gamma(s)\) therefore gives at most exponential height growth for \(L(s,\psi)\) on that strip. In the principal case the same conclusion holds after multiplication by \((s-1)/(s+1)\). Only this qualitative growth, for each fixed conductor, is used in the strip principle.

To see explicitly that the fixed-conductor growth constants do not enter the uniform strip bound, put \[H_Q(s)=Q^{-3/5}\frac{L(s,\psi)}{(s+2)^2}.\] It is holomorphic in the closed strip and is bounded by one absolute constant \(C\) on both vertical boundary lines, by the bounds already proved. For \(\delta>0\), apply the maximum principle on the rectangle of height \(T\) to \(H_Q(s)\exp(\delta(s-1/2)^2)\). On the vertical sides the exponential has modulus at most \(e^{9\delta/25}\), because \(|\Re s-1/2|\le3/5\). On the horizontal sides its modulus is at most \(e^{\delta(9/25-T^2)}\), which tends to zero faster than the qualitative fixed-\(Q\) exponential growth as \(T\to\infty\). First let \(T\to\infty\) for this fixed \(Q,\delta\), and then let \(\delta\downarrow0\). The result is \(|H_Q(s)|\le C\) throughout the strip, with the same constant for every \(Q\). Multiplication by \(Q^{3/5}|s+2|^2\) proves Equation (21).

For the principal function, the factor \((s-1)/(s+1)\) removes its only pole in the strip and is bounded on the boundary. Its functional equation gives the same boundary bounds with \(Q=1\). This equation also follows from the preceding Poisson formula with \(\phi=1\): after writing \(v=2t/\sqrt3\), the theta relation is \(\Theta_\phi(2t/\sqrt3)=t^{-1}\Theta_\phi(2/(\sqrt3t))\), whose split Mellin integral is invariant under \(s\mapsto1-s\). Apply the same damped-rectangle argument to \((s-1)\zeta_F(s)/((s+1)(s+2)^2)\). ◻

Proof. Write \(R_j=2-a-je\) and \(\mathcal C=2Q(3+|t|)^2\). All the disks used here lie in \(\Re s>1/2\), where the principal regularizer is holomorphic. On the zero-free disk choose a holomorphic logarithm \(g=\log\mathcal L\) whose value near the center is the Euler logarithm, together with the logarithm of the regularizing factor in the principal case. The value \(g(2+it)\) is uniformly bounded. Lemma 12 and Euler convergence to the right of \(11/10\) give \[\Re g(s)=\log|\mathcal L(s)|\ll\log\mathcal C \qquad(|s-(2+it)|\le R_3).\] The possible heights differ from \(t\) by at most \(3/2\), which is included in \(\mathcal C\). Borel–Carathéodory on radii \(R_3,R_4\), whose difference is \(e\), now gives \(|g|\ll_e\log\mathcal C\) on the disk of radius \(R_4\).

On the disk of fixed radius \(r_0=49/100\), one has \(\Re s\ge151/100>1\). The absolutely convergent Euler logarithm is uniformly bounded there. For the principal function the logarithm of \((s-1)/(s+1)\) is also bounded on this disk, using its branch in \(\Re s>1\). Since \(r_0<R_6<R_4\), Hadamard’s three-circles theorem applied to \(g\) gives \[\max_{|s-(2+it)|\le R_6}|g(s)| \ll_e(\log\mathcal C)^\theta, \qquad \theta=\frac{\log(R_6/r_0)}{\log(R_4/r_0)}<1.\] For fixed \(e\), continuity on the compact interval \(1/2\le a\le1\) makes \(\theta\) uniformly smaller than one. For every \(\epsilon>0\), \(\exp(C_e(\log\mathcal C)^\theta)\ll_{e,\epsilon}\mathcal C^\epsilon\). Applying this to both \(e^g\) and \(e^{-g}\) proves Equation (22). Cauchy’s estimate between radii \(R_6\) and \(R_8\) gives the stated bound for \(g'=\mathcal L'/\mathcal L\).

For the global assertion first suppose \(\beta_*\le b\le1\). Choose \(0<e<\min(10^{-3},v/8)\) and put \(a=b\). The disk of radius \(R_2\) is contained in \(\Re s>b+2e>\beta_*\), so it is zero-free by the definition of \(\beta_*\) and absolute Euler convergence beyond one. The disk of radius \(R_8\) covers the points \(b+v\le\Re s\le2\) at its center’s height; Euler convergence covers \(\Re s\ge2\). The preceding constants are uniform in \(b\). If \(b>1\), Euler convergence on \(\Re s\ge1+v\) suffices. Finally \[\frac1{\zeta_F(s)}=\frac{s-1}{s+1}\mathcal L(s)^{-1}, \qquad \left|\frac{s-1}{s+1}\right|\le1\quad(\Re s\ge0),\] which proves the principal reciprocal assertion. ◻

Proof. Every factor is nonzero on \(\Re s>0\). For \(\Re s\ge\sigma_0\), both its absolute value and the absolute value of its inverse are at most \((1-q_p^{-\sigma_0})^{-1}\). For sufficiently large \(q_p\), depending on \(\sigma_0,\epsilon\), \(-\log(1-q_p^{-\sigma_0})\le\epsilon\log q_p\). The finitely many smaller primes contribute a fixed constant. This proves the first estimate. For any real \(\sigma\), \[\prod_{p\mid R}(1+q_p^{-\sigma}) \le q_R^{(-\sigma)_+}2^{\#\{p:p\mid R\}} \ll_\epsilon q_R^{(-\sigma)_++\epsilon}.\] The last inequality follows by separating the finitely many primes with \(q_p<2^{1/\epsilon}\). Finally, logarithmic differentiation gives \[\left|\frac{D_R'}{D_R}(s)\right| \le\sum_{p\mid R}\frac{\log q_p}{q_p^{\sigma_0}-1} \ll_{\sigma_0}\log(2q_R).\] If \(q_R\le Z^B\), choose the exponent in the first estimate smaller than the desired exponent of \(Z\) divided by the bounded value of \(B\); the case \(B=0\) is immediate. This proves the stated interpretation for deleted factors. ◻

Proof. We first express the character masks through finitely many translated theta values. We then transform their rational cusps and compute the resulting automorphy multipliers and local Fourier factors; the final Mellin comparison gives the reflection identity and the bounds for its transformed test.

The fixed theta input. Let \(\Gamma=\mathrm{SL}_2(\mathcal O)\), \(\Gamma_1(3)=\{g\in\Gamma:g\equiv I\pmod3\}\), and \(\Gamma_2=\langle\mathrm{SL}_2(\mathbb Z),\Gamma_1(3)\rangle\). The unconditional formulas of [8] give a smooth cubic theta function \(\theta\) on \(\mathbb H^3=\mathbb C\times\mathbb R_{>0}\) with \[\theta(gw)=\kappa(g)\theta(w)\quad(g\in\Gamma_2),\qquad \kappa\!\begin{pmatrix}a&b\\c&\delta'\end{pmatrix} =\left(\frac ca\right)_3\quad \left(\begin{pmatrix}a&b\\c&\delta'\end{pmatrix}\in\Gamma_1(3),\ c\ne0\right).\] Here \(\kappa\) is one on \(\mathrm{SL}_2(\mathbb Z)\) and is one on an element of \(\Gamma_1(3)\) with lower left entry zero. The subscript \(3\) denotes the ordinary cubic residue symbol; for a good prime \(p\notin S\), one has \((x/p)_3=\chi_p(x)^2\). These automorphy and coefficient formulas have no hypothesis about zeros of \(L\)-functions.

Put \[\mathsf L(t)=\begin{pmatrix}1&0\\t&1\end{pmatrix},\qquad (t_0,t_1,t_2)=(0,\omega^2,\omega),\qquad \Theta_b(w)=\theta(\mathsf L(t_b)w).\] If \(v\in\mathbb Z+3\mathcal O\), then \(\mathsf L(v)\) is a product of an element of \(\mathrm{SL}_2(\mathbb Z)\) and a lower translation in \(\Gamma_1(3)\) with multiplier one. The analogous assertion holds for upper translations. Therefore \[\theta(\mathsf L(\lambda)w)=\Theta_1(w),\qquad \theta(\mathsf L(-\lambda)w)=\Theta_2(w),\] because \(\lambda-\omega^2=2+3\omega\) and \(-\lambda-\omega=-1-3\omega\). If \(u_0=s+t\omega\) with \(0\le s,t<3\), and \(E=\bigl(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\bigr)\), then \[\begin{pmatrix}u_0&-1\\1&0\end{pmatrix}=E\mathsf L(-u_0), \qquad -u_0-t_t\in\mathbb Z+3\mathcal O,\] and hence \(\theta\bigl(\bigl(\begin{smallmatrix}u_0&-1\\1&0\end{smallmatrix}\bigr)w\bigr) =\Theta_t(w)\). These are the only cusp representatives needed below.

For any of them, define \(d_H\) by \[ \bar\theta(H(z,v))=C_H(v)+ \sum_{\mu\ne0}d_H(\mu)vK_{1/3}(4\pi|\mu|v)\check e(\mu z). \tag{29}\] Here \(C_H(v)\) is independent of \(z\). The three functions \(\Theta_0,\Theta_1,\Theta_2\) are respectively \(F_1,F_{19},F_{10}\) in [8]. Its Appendix A, rows \(1,19,10\), expresses their coefficients in terms of \(\tau,\tau_1,\tau_2\) in its Equations (5.7), (5.13), and (5.14). In their notation the conjugate expansion uses \(\overline{d_j(-\mu)}\), as in their Equation (5.16). Those formulas give the support inclusion in Equation (24). For completeness, the two possible magnitudes from \(\tau\) at \(\mu=u\lambda^k n b^3\) are \[3^{k/6+8/3}|b|\quad\hbox{and}\quad3^{k/6+3}|b|,\] and \(\tau_1,\tau_2\) have magnitude \(9|b|\) and \(k=-4\). Here the Gauss sums use only the cubic symbol, including at primes in \(S\). To make their domain explicit, for every primary squarefree \(n\) define, as in [8], \[ \widetilde g_3(n)=q_n^{-1/2}\sum_{v\bmod n} \left(\frac vn\right)_3\check e(v/n), \qquad \widetilde g_3(1)=1. \tag{30}\] The ordinary cubic symbol is defined at every prime other than \(\lambda\) and is extended by zero on nonunits and multiplicatively in the denominator. Every primary \(n\) is prime to \(\lambda\). At each prime divisor of \(n\) the cubic character is nontrivial, and \(\check e(v/n)=e(\lambda v/n)\) is primitive because \(\lambda\) is a unit there. The finite-field Gauss identity and the Chinese remainder theorem therefore give \(|\widetilde g_3(n)|=1\) for every such squarefree \(n\). This includes the primary prime \(-2\) of norm \(4\); no sextic character with that denominator is used. Also, whenever \((t,n)=1\), the change of variable \(y=tv\) gives \[q_n^{-1/2}\sum_{v\bmod n}\left(\frac vn\right)_3\check e(tv/n) =\left(\frac tn\right)_3^{-1}\widetilde g_3(n).\] This covers the twists \(t=\lambda^2,\omega\lambda^2,\omega^2\lambda^2\) in the other source coefficient families. Thus the unnormalized Gauss sums occurring in all three cusp families have magnitude \(|n|\). Consequently each coefficient magnitude displayed above is at most \(27\,3^{k/6}|b|\), including \(k=-4\). The formulas require only that \(n\) be squarefree and that \(n,b\) be primary; they do not require \((n,b)=1\). Prime valuations of \(nb^3\) also show that its representation by such \(n,b\), when it exists, is unique. In particular the same bound is valid when the variables share primes or contain primes excluded from the primal sums.

At infinity the condition \(x=\lambda^3\mu\equiv1\pmod3\) isolates exactly \(\mu=\lambda^{-3}nb^3\) with \(n,b\) primary and \(n\) squarefree. The infinity support has \(\lambda^3\mu\in\mathcal O\); every other ramified exponent in Equation (5.7) of [8] makes \(x\) divisible by \(\lambda\), and the negative sign in the remaining pair gives \(x\equiv-1\pmod3\). At the isolated index the coefficient in \(\bar\theta\) is \[ d_I(\lambda^{-3}nb^3)=3^{5/2}|b|\widetilde g_3(n). \tag{31}\] This is the unmasked source formula, valid for every primary squarefree \(n\) and every primary \(b\), with no coprimality condition between them. Only when \((n,S)=1\) may it be expressed in the global sextic notation: substituting \(y=\lambda v\) in \(\gamma_2(n)\) gives \[ \widetilde g_3(n)=\left(\frac\lambda n\right)_3^{-1}\gamma_2(n) =\chi_n(\lambda)^{-2}\gamma_2(n)\qquad((n,S)=1). \tag{32}\] The cubic supplementary laws in [8] make \((\lambda/x)_3\) periodic on all primary \(x\) modulo \(9\). On primary elements prime to \(S\) it equals \(\chi_x(\lambda)^2\). A character in \(\mathcal X\) is periodic on primary elements prime to \(\mathfrak E\) modulo its conductor, and the zero condition \((x,\mathfrak E)\ne1\) is periodic modulo \(\mathfrak E\). Thus a sufficiently divisible \(L\) with prime support \(S\) works for every member of the fixed finite family. This verifies its choice before the moving local primes are selected.

Finite Fourier inversion with the zero extensions. For \(j\in\{0,\ldots,5\}\) define \[C_{p,j}(h)=q_p^{-1}\sum_{x\bmod p}\chi_p(x)^j e(-hx/p).\] The character pairing furnished by \(e\) is nondegenerate on \(\mathcal O/(p)\). Changing variables in a Gauss sum, and using the orthogonality of a nontrivial multiplicative character, gives \[ C_{p,j}(h)=\begin{cases} q_p^{-1/2}\tau_{p,j}^{-}\chi_p(h)^{-j},&j\ne0,\ h\ne0,\\ 0,&j\ne0,\ h=0,\\ -q_p^{-1},&j=0,\ h\ne0,\\ 1-q_p^{-1},&j=0,\ h=0. \end{cases} \tag{33}\] Here and below \(h=0\) means \(h\equiv0\pmod p\). Each nontrivial power of \(\chi_p\) is primitive modulo the prime \(p\), so the usual finite-field Gauss-sum identity gives \(|\tau_{p,m}^{\pm}|=1\).

Fourier inversion on \(\mathcal O/(L)\) and on each \(\mathcal O/(p)\) now gives \[\phi(x)\prod_p\chi_p(x)^{j_p} =\sum_h C(h)e\!\left(x\left(h_0/L+\sum_p h_p/p\right)\right), \qquad C(h)=\widehat\phi(h_0)\prod_pC_{p,j_p}(h_p).\] This uses every additive frequency, including nonunit \(h_0\) and zero local frequencies. We do not invoke the twisted formula in [8], whose unit-Fourier-support hypothesis need not hold for these masks. The theta function with this multiplier on its infinity coefficient \(x=\lambda^3\mu\in\mathcal O\) is the finite sum \[ \mathcal F(z,v)=\sum_hC(h)\bar\theta(z+z_h,v),\qquad z_h=\lambda^2\left(h_0/L+\sum_p h_p/p\right). \tag{34}\] For an isolated infinity index \(x=nb^3\), the piecewise definition of \(\phi\) first discards every term for which \((nb,\mathfrak E)\ne1\). This is a mask on the whole completed index: \((nb^3,\mathfrak E)=1\) if and only if both \(n\) and \(b\) are \(\mathfrak E\)-units. Only on that remaining domain do we use complete multiplicativity, including \(\chi_b(\lambda)^6=1\), to write \[\phi(nb^3)\prod_p\chi_p(nb^3)^{j_p} =\chi_n(\lambda)^2\Psi(n)\Psi(b)^3 \qquad((nb,\mathfrak E)=1).\] It remains true when \(n,b\) share a moving prime, since the corresponding local powers retain their zero values, including for \(j_p=0\). For the discarded terms the completed coefficient is defined to be zero directly; no value of \(\chi_n(\lambda)\) is evaluated there. Combining Equations (31) and (32) on the retained domain now identifies the direct Mellin series with the product \(T\). The dual coefficient functions \(d_H\) keep the full source support in Equation (24); no \(S\)-mask is placed on them.

Fix \(h_0\) and declare \(p\) active precisely when \(h_p\ne0\). Equation (33) forces all primes with \(j_p\ne0\) to be active. Choose once for each \(h_0\bmod L\) a representative and a reduced expression \(\lambda^2h_0/L=a_F/c_F\). Multiplying numerator and denominator by one unit, normalize the numerator to be \(1\pmod3\) if \(\lambda\mid c_F\) and the denominator to be \(1\pmod3\) otherwise. Since every active prime is primary, this unit depends only on \(h_0\). For \(r=\prod_{p\ {\rm active}}p\), Equation (34) then gives \[ z_h=\frac ac,\qquad c=c_Fr,\qquad a=a_Fr+\lambda^2c_F\sum_{p\mid r}(r/p)h_p. \tag{35}\] At a prime dividing \(c_F\), the numerator is congruent to \(a_Fr\) and is a unit. At \(p\mid r\), it is congruent to \(\lambda^2c_F(r/p)h_p\) and is again a unit. Thus this fraction is reduced, even when \(h_0\) is a nonunit modulo \(L\). The empty product \(r=1\) is allowed. Replacing any lift by another changes \(z_h\) by an element of \(\lambda^2\mathcal O=3\mathcal O\), a period of \(\theta\).

Choice of the cusp matrix and its multiplier. The next construction determines the automorphy multiplier and reflected additive phase for each reduced cusp fraction just obtained. Set \(M=\lambda^{12}L^4\). For every active \(p\), the Chinese remainder theorem permits the chosen nonzero class \(h_p\bmod p\) to be lifted with \(h_p\equiv0\pmod{M^2}\). Restrict \(r\) to one class modulo \(M^2\). Equation (35) shows that \(a\) is then fixed modulo \(Mc_F\); in fact its moving terms are divisible by \(M^2c_F\), and \(a_Fr\) is fixed modulo \(M^2\), which is divisible by \(Mc_F\) because \(c_F\) divides \(L\) up to a unit. It also shows \(a\equiv1\pmod3\) if \(\lambda\mid c\), and \(c\equiv1\pmod3\) otherwise.

Choose \(\delta'\) by the following compatible local congruences: \[\begin{array}{ll} \delta'\equiv a^{-1}\pmod{Mc_F} &\text{at every prime dividing }c_F,\\ \delta'\equiv0\pmod M &\text{at every prime dividing }M\text{ but not }c_F,\\ \delta'\equiv a^{-1}\pmod p& (p\mid r). \end{array}\] The first line means the indicated prime-power parts of \(Mc_F\). All inverses exist because \(a/c\) is reduced. Put \(b=(a\delta'-1)/c\) and \(g=\bigl(\begin{smallmatrix}a&b\\c&\delta'\end{smallmatrix}\bigr)\). Then \(g\in\Gamma\), and division of \(a\delta'-1\) by \(c\) gives \[b\equiv0\pmod M\text{ at primes of }c_F,\qquad b\equiv-c^{-1}\pmod M\text{ at the other primes of }M.\] Consequently \(a,b,c,\delta'\) have fixed residues at the needed powers of every prime dividing \(L\), independent of the classes \(h_p\). Whenever a zero entry would occur in a residue-symbol calculation, one may first translate \(z_h\) by \(\lambda^2M^2\mathcal O\) and then add a multiple of the combined congruence modulus to \(\delta'\). These changes preserve all displayed congruences and avoid the finitely many values making an entry zero. They do not change the translated theta function.

There are three cases, distinguished by \(v_\lambda(c)\). If \(3\mid c\), put \(H=I\). Then \(G=g\in\Gamma_1(3)\), and cubic reciprocity for the primary \(a\) and the primary primes of \(r\) gives \[ \kappa(G)=\left(\frac{c_F}{a}\right)_3 \left(\frac ar\right)_3. \tag{36}\] If \(v_\lambda(c)=1\), take the unique \(u_0\in\{\lambda,-\lambda\}\) with \(u_0\equiv c\pmod3\) and put \(H=\mathsf L(u_0)\); these are the two possibilities because \(\mathcal O/(\lambda)\simeq\mathbb F_3\). The congruences above show that \(G=gH^{-1}\in\Gamma_1(3)\). Write \(A=a-u_0b\). Both \(A\) and \(a\) are primary, and they are coprime: \((a,b)=1\) and \((a,u_0)=1\). The determinant equation gives \[a(c-u_0\delta')=cA-u_0, \qquad bc\equiv-1\pmod a.\] It follows, by cubic reciprocity for \(a,A\), that \[\begin{split} \kappa(G) &=\left(\frac{-u_0}{A}\right)_3 \left(\frac aA\right)_3^{-1}\\ &=\left(\frac{-u_0}{A}\right)_3 \left(\frac{c/u_0}{a}\right)_3 =\left(\frac{-u_0}{A}\right)_3 \left(\frac{c_F/u_0}{a}\right)_3 \left(\frac ar\right)_3. \end{split}\] For the second equality, reduce \(A\) modulo \(a\) and use \((A/a)_3=(-u_0b/a)_3=(u_0/a)_3(c/a)_3^{-1}\); the cubic symbol of \(-1\) is one. Finally suppose \((\lambda,c)=1\). Choose \(u_0=s+t\omega\equiv a\pmod3\) with \(0\le s,t<3\) and put \(H=\bigl(\begin{smallmatrix}u_0&-1\\1&0\end{smallmatrix}\bigr)\). Here \(c\equiv1\), \(\delta'\equiv0\), and \(b\equiv-1\pmod3\), so \[G=gH^{-1}=\begin{pmatrix}-b&a+bu_0\\-\delta'&c+\delta'u_0\end{pmatrix} \in\Gamma_1(3).\] Both \(-b\) and \(c\) are primary. Moreover \(-bc=1-a\delta'\equiv1\pmod9\). Factor \(\delta'=u\lambda^k\delta_0\) with \(\delta_0\) primary. The supplementary laws make \((u\lambda^k/(1-a\delta'))_3=1\); cubic reciprocity makes \((\delta_0/(1-a\delta'))_3=1\) because the numerator after reciprocity is \(1\pmod{\delta_0}\). Thus \[\left(\frac{\delta'}{-bc}\right)_3=1, \qquad \kappa(G)=\left(\frac{\delta'}{-b}\right)_3 =\left(\frac{\delta'}c\right)_3^{-1} =\left(\frac ac\right)_3 =\left(\frac a{c_F}\right)_3\left(\frac ar\right)_3.\] The last inverse uses \(a\delta'\equiv1\pmod c\).

In all three cases we have proved \[ \kappa(G)=\kappa_F\prod_{p\mid r}\chi_p(a)^2, \qquad |\kappa_F|=1. \tag{37}\] The factor \(\kappa_F\) is constant as the active \(h_p\) vary in the fixed sector. If \(3\mid c\), it is \((c_F/a)_3\): the unit and \(\lambda\)-power factors are determined by \(a\bmod9\), and reciprocity determines the remaining fixed-prime factors from the residue of \(a\) at the primes of \(c_F\). In the middle case it is \((-u_0/A)_3((c_F/u_0)/a)_3\), determined in the same way by the fixed residues of \(A=a-u_0b\) and \(a\). In the last case it is \((a/c_F)_3\), which is determined directly by the fixed denominator \(c_F\) and \(a\bmod c_F\). The displayed congruences for \(a,b,c,\delta'\) therefore determine \(\kappa_F\) using only \(h_0\) and \(r\bmod M^2\). They also determine the chosen one of the three functions \(\Theta_b\).

The reflected additive phase. Write \(x=\lambda^4\mu\) and \(D_F=\lambda^3c_F\). At each \(p\mid r\), Equation (35) gives \(a\equiv\sigma_ph_p\pmod p\). Hence \(\delta'\equiv\sigma_p^{-1}h_p^{-1}\pmod p\). Additive Chinese remaindering for the coprime factors of \(D_Fr\) gives the exact identity \[ \begin{split} \check e(-\delta'\mu/c) &=e(-\delta'x/(D_Fr))\\ &=\vartheta(x)\prod_{p\mid r}e(\epsilon_ph_p^{-1}x/p), \qquad \vartheta(x)=e(-\delta'_F r^{-1}x/D_F). \end{split} \tag{38}\] Here \(\delta'_F\) is the class of \(\delta'\) modulo \(D_F\), and \(r^{-1}\) in the formula for \(\vartheta\) is taken modulo \(D_F\). For example, the local numerator at \(p\) is \(-\delta'(D_Fr/p)^{-1}=\epsilon_ph_p^{-1}\); this verifies the signs and the powers of \(\lambda\). The chosen congruences fix \(\delta'_F\), since \(D_F\) divides the corresponding local moduli, and the sector fixes \(r^{-1}\bmod D_F\). A common multiple of the finitely many \(D_F\) is therefore a fixed modulus for all the characters \(\vartheta\).

Conjugating Equation (37) contributes \(\bar\kappa_F\prod_{p\mid r}\chi_p(\sigma_p)^{-2}\chi_p(h_p)^{-2}\). For \(j\ne0\), multiplication by Equation (33) leaves the local sum \[q_p^{-1/2}\tau_{p,j}^{-} \sum_{h\ne0}\chi_p(h)^{-j-2}e(\epsilon_ph^{-1}x/p).\] Upon putting \(y=h^{-1}\), the sum is a Gauss sum of exponent \(j+2\). If \(j\ne4\), its value is \(q_p^{1/2}\tau_{p,j+2}^{+}\chi_p(\epsilon_px)^{-j-2}\), also at \(p\mid x\) because both sides then vanish. If \(j=4\), it is the Ramanujan sum \[\sum_{y\ne0}e(\epsilon_pxy/p)=-1+q_p1_{p\mid x}.\] For an active \(j=0\), the same substitution gives \[-q_p^{-1}\sum_{h\ne0}\chi_p(h)^{-2}e(\epsilon_ph^{-1}x/p) =-q_p^{-1/2}\tau_{p,2}^{+}\chi_p(\epsilon_px)^{-2}.\] These are exactly \(\omega_{p,j}B_p(x)\); an inactive zero exponent contributes \(1-q_p^{-1}\). This proves every local factor, including its zero extension. The choices of \(h_0\) and the active subset give at most \(q_L2^{\#\{p:j_p=0\}}\) terms. Splitting the \(j_p=4\) factors multiplies this by at most \(2^{\#\{p:j_p=4\}}\), proving the stated counts.

The denominator and the fixed-sector data constructed before these local sums depend on the local prime set only through \(h_0\) and the active radical \(r\); the remaining active-frequency dependence is exactly the local dependence just summed. We use the same representatives and normalized fractions for each \(h_0\), and the same fixed-sector choices of cusp function, fixed additive character, and \(\kappa_F\) whenever those data recur. An absent prime and an inactive zero-exponent prime have the same active radical; their other active \(\sigma_p,\epsilon_p\) are therefore also identical. The inactive coefficient \(1-q_p^{-1}\) is the only change. This proves the simultaneous branch compatibility in the statement.

The marked reflection in Section 3.2 will use the following dependence on pairs of active primes. Since \[\epsilon_p=-\lambda^{-5}(c/p)^{-2}\pmod p,\] there is a scalar \(\xi_{p,j}\) of absolute value one, depending on \(p,j\) but on no other active prime such that \[ \chi_p(\sigma_p)^{-2}\omega_{p,j} =\xi_{p,j}\chi_p(c/p)^{2j+2}. \tag{39}\] For \(j=4\) this uses \(2j+2\equiv-2\pmod6\); for the other exponents it follows immediately by inserting the expression for \(\epsilon_p\). Thus the contribution of another active prime to the phase at \(p\) has exponent exactly \(2j+2\) modulo six. In particular an active exponent \(j=1\) has column coupling \(\chi_p(x)^{-3}=\chi_p(x)^3\).

Mellin normalization and continuation. Let \[J(s)=\int_0^\infty \left.\partial_{\bar z}\mathcal F(z,v)\right|_{z=0} v^{2s-1}\,dv.\] The Bessel identity used in [8] is \[\int_0^\infty K_{1/3}(y)y^{w-1}\,dy =2^{w-2}\Gamma((w-1/3)/2)\Gamma((w+1/3)/2) \qquad(\Re w>1/3).\] On \(\Re s>1\) the coefficient sums converge absolutely, so we may differentiate and integrate their expansions term by term. Since \(\partial_{\bar z}\check e(\mu z)=2\pi i\bar\mu\check e(\mu z)\), the identity with \(w=2s+1\) gives \[J(s)=\frac i4(2\pi)^{-2s}\Gamma(s+1/3)\Gamma(s+2/3) \sum_{0\ne\mu\in\lambda^{-3}\mathcal O}d_I(\mu)\phi(\lambda^3\mu) \prod_p\chi_p(\lambda^3\mu)^{j_p} \bar\alpha(\mu)q_\mu^{-s}.\] First use the whole-index mask to restrict to \((nb,\mathfrak E)=1\) as above, then insert Equations (31) and (32). Using \(q_{\lambda^{-3}}=27^{-1}\) and \(\bar\alpha(\lambda^{-3})=-i\) gives \[ J(s)=\frac{3^{5/2}}4\left(\frac{27}{(2\pi)^2}\right)^s \Gamma(s+1/3)\Gamma(s+2/3)T(s,\Psi). \tag{40}\]

For one translated term \(z_h=a/c\), use the matrix \(g=GH\) constructed above. The coordinate action of \(g^{-1}\) on hyperbolic space, from [8], gives \[g^{-1}(z+a/c,v)= \left(-\frac{\delta'}c-\frac{\bar z}{c^2(v^2+|z|^2)}, \frac{v}{q_c(v^2+|z|^2)}\right).\] At \(z=0\), the derivative of its first coordinate with respect to \(\bar z\) is \(-(cv)^{-2}\); the derivatives of its conjugate coordinate and of its vertical coordinate with respect to \(\bar z\) are zero. The chain rule and \(\bar\theta(gw)=\bar\kappa(G)\bar\theta(Hw)\) therefore give \[\left.\partial_{\bar z}\bar\theta(z+a/c,v)\right|_{z=0} =-\frac{\bar\kappa(G)}{c^2v^2} \left.\partial_z\bar\theta(H(z,1/(q_cv)))\right|_{z=-\delta'/c}.\] In particular the derivative removes \(C_H(v)\) from every cusp expansion in Equation (29). Substituting that expansion and then \(v\mapsto1/(q_cv)\) in the integral gives, for \(\Re s<0\), the contribution \[ \begin{split} -\frac i4(2\pi)^{2s-2}\bar\alpha(c)^2q_c^{1-2s}\bar\kappa(G) \prod_{\pm}\Gamma(3/2-s\pm1/6)\\ {} \times\sum_{\mu\ne0}d_H(\mu)\alpha(\mu) \check e(-\delta'\mu/c)q_\mu^{-(1-s)}. \end{split} \tag{41}\] To verify the scalar directly, before applying the Bessel integral the factor is \(-2\pi i c^{-2}q_c^{2-2s}\) and the remaining integral is \(\int_0^\infty u^{2-2s}K_{1/3}(4\pi|\mu|u)\,du\). The Bessel identity with \(w=3-2s\), together with \(c^{-2}=\bar\alpha(c)^2/q_c\), gives exactly the display.

We justify both the continuation and the contour operations here. The coefficient bound in Equation (24) implies absolute convergence of the reflected series when \(\Re(1-s)>1\): the separate majorants are \[\sum_n^{\rm sf}q_n^{-(1-\Re s)},\qquad \sum_b q_b^{1/2-3(1-\Re s)},\qquad \sum_{k\ge-4}3^{k(1/6-(1-\Re s))}.\] Each converges in that region. The fixed support has a positive lower bound for \(|\mu|\). The exponential decay of \(K_{1/3}\) and of all its derivatives, the coefficient bound, and the direct expansion imply exponential decay at \(v\to\infty\) for every logarithmic derivative of each translated derivative. The chain-rule formula just obtained implies exponential decay at \(v\to0\) as well, with constants depending on the fixed translated term. Repeated integration by parts in \(\log v\) now shows that \(J\) is entire and decreases faster than every power of \(|\Im s|\) on each fixed strip.

Equation (40), with reciprocal gamma functions on its right, continues \(T\) to an entire function. Its direct series is bounded on any line to the right of \(1\). Equation (41) and Stirling’s formula give a polynomial bound on any line to the left of \(0\); the exponential parts of the two gamma products cancel. On a strip between such lines, the rapid bound for \(J\) and the reciprocal gamma factors first give \(|T(\sigma+it)|\ll (1+|t|)^C e^{\pi|t|}\) for some \(C\). Divide \(T\) by a sufficiently large power of \(B+s\), with \(B\) chosen so that \(B+s\) has no zero on the strip, and multiply by \(e^{\varepsilon s^2}\). The horizontal sides of a growing rectangle tend to zero for each \(\varepsilon>0\). The maximum principle, followed by \(\varepsilon\to0\), transfers the polynomial bounds on the two vertical sides to the strip. This proves the stated polynomial strip growth.

Shift the integral on the left of Equation (23) from \(\Re s=a\) to \(\Re s=1/2-\sigma<0\) with \(\sigma>1/2\). The entire continuation, polynomial strip bound, and rapid Mellin decay justify the shift: Equation (18) applied to the weighted logarithmic derivatives of \(V\) makes the horizontal joins tend to zero. Inserting Equation (41) divided by Equation (40), and putting \(t=1/2-s\), gives the gamma quotient \(R(t)\) and the factor \[-\frac i{81}\bar\alpha(c)^2q_\mu^{-1/2} \left(\frac{Kq_\mu X}{q_c^2}\right)^{-t}.\] Indeed \((2\pi)^{4s-2}27^{-s}3^{-5/2} =3^{-4}K^{-t}\) and \(q_c^{1-2s}q_\mu^{s-1} =q_\mu^{-1/2}(q_\mu/q_c^2)^{-t}\). The reflected coefficient series is absolutely convergent on this line, so it may be interchanged with the integral. Summing its finite local factors using Equation (38) and the computed Gauss sums proves Equation (23) and Equation (25). Since \(|\phi|\le1\), we have \(|\widehat\phi(h_0)|\le1\); every other factor in the scalar has absolute value at most one. This proves \(|\zeta|\le1/81\).

Uniform estimates for the transformed test. The first negative pole of the numerator of \(R(t)\) is \(-5/6\). The reciprocal denominator gamma functions are entire. Hence the integrand defining \(V^\sharp\) is holomorphic for \(\Re t>-5/6\). On \(-1/4\le\sigma\le A\), Stirling’s formula gives \[|R(\sigma+iu)|\ll_A(1+|u|)^{4\sigma}.\] The estimate is uniform also for bounded \(u\), since this closed strip has no numerator pole. Equation (18) makes \(\widehat V\) rapidly decreasing pointwise on every vertical strip. Rectangular contour shifts are therefore valid within \(-1/4\le\Re t\le A\), including between any two nonnegative lines. Applying \((x\partial_x)^j\) contributes \((-t)^j\). The three lines \(-1/4,0,A\) bound the resulting integral respectively by constant multiples of \[x^{1/4}\mathfrak M_{A,\lceil4A\rceil+j+2}(V),\quad \mathfrak M_{A,\lceil4A\rceil+j+2}(V),\quad x^{-A}\mathfrak M_{A,\lceil4A\rceil+j+2}(V).\] Combining them proves Equation (26). The rapid large-\(x\) bound, with \(A\) arbitrarily large, also makes every dual sum in Equation (23) absolutely convergent when \(V^\sharp\) is represented on the zero line. Thus the shift of the kernel contour does not require a termwise shift of a conditionally convergent Dirichlet series.

For Equation (27), set \(f_Y(v)=\chi(e^v)V^\sharp(Ye^v)\). This is supported on a fixed compact interval. Integrating its Fourier transform by parts \(J+2\) times and using the trivial bound for \(|u|\le1\) gives \[\int_{\mathbb R}(1+|u|)^J|\widehat f_Y(u)|\,du \ll_J\sum_{j=0}^{J+2}\|f_Y^{(j)}\|_{L^1(\mathbb R)}.\] Leibniz’s rule and Equation (26) bound each norm by the right side of Equation (27), because \(e^v\) stays in a fixed compact subinterval of \((0,\infty)\). Finally, for \(|u|\ge1\), \(B+2\) integrations by parts give \[|\widehat V(\eta+iu)| \le |\eta+iu|^{-B-2} \int_0^\infty|(x\partial_x)^{B+2}V(x)|x^\eta\frac{dx}{x}.\] For \(|u|\le1\) use the corresponding undifferentiated integral. On \(-A\le\eta\le1/4\), \(x^\eta\le1+x^{-A}+x^{1/4}\), proving Equation (28). The identity \(\widehat{x^{iu_0}V}(\eta+iu)=\widehat V(\eta+i(u+u_0))\) and \(1+|u|\le(1+|u+u_0|)(1+|u_0|)\) give the asserted norm-twist bound. In particular the Gaussian test \(V(x)=(2\sqrt\pi)^{-1}\exp(- (\log x)^2/4)\), whose Mellin transform is \(e^{t^2}\), satisfies the hypotheses of this Proposition directly. ◻

Proof. If \((A,m_{\rm good})=1\), multiplicativity and the symmetric reciprocity factor in Lemma 8 give Equation (42). If a good prime is shared, both sides are zero: the left side is the zero extension of \(\chi_A(m)\), and its local factor on the right is zero even for a six-divisible exponent. This proves the identity on the entire stated domain.

On this domain \(\chi_A(u m_S)\) depends only on \(u\) and the \(S\)-valuations modulo six, because every numerator factor is a unit modulo \(A\). For each of the finitely many representatives \(d=u\prod_{\mathfrak l\in S}\pi_{\mathfrak l}^{e_{\mathfrak l}}\), \(0\le e_{\mathfrak l}<6\), Lemma 5 identifies \(A\mapsto\chi_A(d)\) with a finite-ray character whose conductor is supported at primes over \(6d\), all in \(S\). Once the good ray class is fixed, \(A\mapsto\mathcal R(A,m_{\rm good})\) is one of a fixed finite family of characters supported at the primes over \(2,3\). Their products with the finite family of base characters give the asserted fixed family. Choosing a common multiple of its conductors and the \(S\)-mask gives the common \(\mathfrak E\) and then \(L\). Additional good local residue-symbol factors in a completed coefficient are placed in \(\mathcal P\) and combined at a shared prime with the same zero convention; they do not alter this fixed modulus. The argument requires \(m\ne0\) and makes no local-prime assertion for the row \(m=0\). ◻

Proof. On the support of \(W\), the monomial \(h\) is bounded above and below by positive constants depending only on \(\Omega\) and the exponents. Each operator \(y_i\partial_{y_i}\) acting on \(V^\sharp(Yh(\boldsymbol y))\) is \(a_i\) times the Euler derivative of \(V^\sharp\) at \(Yh(\boldsymbol y)\). It produces no additional power of \(Y\). Leibniz’s rule and Equation (26) prove Equation (44). Applying Lemma 9 with \(j=J+d+2\) gives Equation (45). ◻

Proof. The shell \(2^jU<a q_\mu\le2^{j+1}U\) contains \(O(1+2^jU/a)\) points of either fixed lattice. Its contribution is at most \(C_A(1+2^jU/a)(2^jU)^{-A}\). Summing the two geometric series gives \(O_A(C_A(U^{-A}+a^{-1}U^{1-A}))\), which implies the display because \(U\ge1\). The last assertion follows from Equation (26). ◻

Proof. The factorization \(b=gt^2\) permits \(g\) and \(t\) to share primes. Since \(nb=c^2mht^2\), the zero convention gives the exact identity \[ \chi_k(nb)^3=\chi_k(mh)^3\,1_{(k,ct)=1}. \tag{51}\] Indeed a square has sixth power under the displayed quadratic character; this is one on units and zero at a collision. The identity therefore also holds when \(n\) or \(g\) shares a prime with \(t\).

For the chosen ranges there are \(O(T)\) possible \(t\). Hilbert-space Cauchy gives \[\left\|\sum_{q_t\asymp T}F_t\right\|_2^2 \ll T\sum_{q_t\asymp T}\|F_t\|_2^2.\] Fix \(t\) in the positive sum. The condition \((k,t)=1\) is now a fixed row restriction. For the remaining condition use the full identity \[1_{(k,c)=1}=\sum_{r\mid(k,c)}\mu(r).\] For each \(k\) the number of possible \(r\) is at most \(d_{\mathcal O}(k)\). Rowwise Cauchy followed by summation over \(k\) costs \((KNB)^\epsilon\) and gives a positive sum over fixed squarefree good \(r\). Write \(k=rk'\) and \(c=rc'\). Squarefreeness gives the fixed row restriction \((k',r)=1\) and the coefficient restriction \((c',r)=1\); terms with \((r,t)>1\) vanish, and otherwise \((k',t)=1\) is another fixed row restriction. There is no remaining \((k',c')\) condition in an individual \(r\)-summand: the full collision mask was already expanded. A fixed ray sector for \(k\) becomes a fixed sector for \(k'\) once \(r\) is fixed.

The factor \(\chi_r(mh)^3\) is a bounded column factor. Split the squarefree parts of \(m,h\) supported on \(S\) from their good parts: \(m=m_Sm_{\rm g}\) and \(h=h_Sh_{\rm g}\). There are only finitely many choices of \(m_S,h_S\). Their symbols are bounded row factors, while \(j=m_{\rm g}h_{\rm g}\) is squarefree and good, of norm \(O(NG/C^2)\). For fixed \(r,t,m_S,h_S\), group the remaining columns by \(j\). If \(C_j\) is the resulting coefficient, the quadratic large sieve gives \[\sum_{k'}\left|\sum_j C_j\chi_{k'}(j)^3\right|^2 \ll_\epsilon (KNB)^\epsilon \left(\frac K{q_r}+\frac{NG}{C^2}\right)\sum_j|C_j|^2.\] The outer sum may retain all its fixed restrictions. On a nonempty range, both quotient lengths in the last display are bounded below by a fixed positive constant; replacing them by their maxima with one changes only the comparison constant.

For each \(j\), the number of factorizations into \(m_{\rm g},h_{\rm g}\) is divisor-bounded. There are \(O(C/q_r)\) possible \(c'\) in its range. Cauchy over these two choices therefore gives \[\sum_j|C_j|^2 \ll_\epsilon (KNB)^\epsilon\frac C{q_r} \sum_{(c',m,h)\in\mathcal D_r(t)} |\beta(rc'm,rc'ht^2)|^2.\] The bounded column factor from \(\chi_r\) has been discarded only in this positive sum. The fixed bad-part symbols were row factors of unit modulus and were removed from the outer modulus before applying the sieve. The ideals \(m,h\) here are the original ideals, reconstructed from their fixed \(S\)-parts; the \(S\)-part of \(c'\) is untouched. Combining the displays proves Equation (49).

Suppose now \(|\beta|\le1\). Ideal counting gives \[\#\mathcal D_r(t)\ll \frac C{q_r}\frac NC\frac GC.\] For \(q_r\asymp r_0\), its product with the preceding factor \(C/q_r\) is \(O(NG/r_0^2)\). There are \(O(r_0)\) possible \(r\) in this range and \(O(T)\) possible \(t\). Since \(B\asymp GT^2\), their contribution to Equation (49) is at most \[(KNB)^\epsilon\frac{NB}{r_0} \left(\frac K{r_0}+\frac{NG}{C^2}\right) \ll (KNB)^\epsilon NB(K+NB).\] Here \(r_0,C,T\) are bounded below on a nonempty range and \(G\ll B\). The number of dyadic choices for \(C,G,T,r_0\) is logarithmic in the norm ranges. Hilbert-space triangle over the \(C,G,T\) blocks, followed by a rescaling of \(\epsilon\), proves Equation (50). ◻

Proof. For the fixed \(h_0\) and the fixed class of \(R\) modulo \(M_{\rm ref}^2\), the class of \(F_{\rm act}R\) is fixed. Proposition 15 therefore gives one common \(d\) and one common additive character \(\vartheta\) for the whole row set. Write \(x_F=u\lambda^{k+4}N_FB_F^3\), so that \(\lambda^4\mu=x_Fnb^3\). The residual local factors are exactly \[\prod_{p\mid R}B_p(\lambda^4\mu) =\chi_R(x_F)^3\chi_R(nb^3)^3 =\chi_R(x_F)^3\chi_R(nb)^3.\] Both equalities include zeros; no unit symbol has been divided. In particular, any collision with \(x_F\) is a restriction depending only on \(R\). Every remaining local factor is fixed and depends only on the dual index. The scalar \(\zeta_R\) of the reflection depends on \(R\) but not on \(\mu\).

Define on the specified dual support \[\delta(n,b)= \frac{d(\mu)\alpha(\mu)\vartheta(\lambda^4\mu)} {27\,3^{k/6}|B_Fb|}.\] The coefficient bound gives \(|\delta(n,b)|\le1\). Its denominator is nonzero, and the definition remains valid when the numerator is zero. Using \(q_\mu=3^kq_{N_Fn}q_{B_Fb}^3\) gives the exact factorization \[\frac{d(\mu)\alpha(\mu)\vartheta(\lambda^4\mu)}{\sqrt{q_\mu}} =27\delta(n,b)q_{\lambda^k}^{-1/3} q_{N_Fn}^{-1/2}q_{B_Fb}^{-1}.\] A small Ramanujan factor or an active zero-mask factor contributes \(q_p^{-1/2}\). A divisibility factor contributes \(q_p^{1/2}\); for a prime assigned to \(N_F\) it cancels \(q_p^{-1/2}\) from the squarefree denominator, and for a prime assigned to \(B_F\) it leaves \(q_p^{-1/2}\) after extraction. When a prime assigned to \(N_F\) also divides \(b\), its entire occurrence in \(b\) remains in \(q_b^{-1}\) and in \(\delta(n,b)\). These facts give exactly the common power \(Z^{-\nu_0}y_n^{-1/2}y_b^{-1}\).

Absorb \(\delta\), the signs of small Ramanujan terms, the bounded fixed local characters, and the fixed dual indicators into \(\beta(n,b)\). This function is bounded by one and is independent of \(R\). Put \(\rho(R)=81\zeta_R\chi_R(x_F)^3\); it is bounded by one because \(|\zeta_R|\le1/81\). The factor \(27/81=1/3\) now proves Equation (54). Multiplicativity of the norm proves Equation (55). It does not use any coprimality between \(N_F\) and \(b\).

By the definitions of the log-norms, \[Y=q_{c_F}^{-2}Z^{v+3\ell_b+e_\lambda-T_0}.\] The finite set of possible \(c_F\) depends only on \(L\). Consequently \(\mathfrak m_A(Y)\ll_L Z^{-(T_0-v-3\ell_b-e_\lambda)_+/4}\). Apply Lemma 17 in dimension three and Equation (46) to the structural identity. The single density is common to all \(R\) because \(W\) uses the row and both dual norms as coordinates. At each Fourier mode the norm powers have absolute value one and can be absorbed into the bounded row scalar and the common dual coefficient. Only now enlarge the positive row norm from \(\mathcal I_H\) to squarefree good \(R\) with \(q_R\ll Z^H\), retaining any fixed restrictions. No kernel estimate is used on the added rows.

Equation (50), with \(K=Z^H\), \(N_{\rm col}=Z^v\), and \(B_{\rm col}=Z^{\ell_b}\), bounds the separated squared norm by \[Z^{\epsilon+\max(H,v+\ell_b)+v+\ell_b}.\] The squared common coefficient in the structural identity has exponent \(-v-2\ell_b-2e_\lambda/3-S_0-B_0\). The square of the extracted profile scalar contributes \(-(T_0-v-3\ell_b-e_\lambda)_+/2\). Combining these exponents and the dimension-three separation norm proves Equation (56). ◻

Proof. Fix one character \(\nu\). Lemma 16, with \(\Psi_{\rm base}=\nu\), supplies a fixed finite family of literal characters for the reflection of \(\nu(A)\chi_A(m)\). Choose their common \(\mathfrak E,L\) before any good row prime varies. Partition the rows into the finitely many unit, \(S\)-valuation residue, and good-ray sectors from that lemma. This keeps every good local prime, including those whose row valuation is divisible by six.

For a row in the prescribed \(O\)-dyad, split its valuation-one primes into \(m_{\rm sup}\) supported on \(S\) and the squarefree good product \(R\). Thus \((m)=m_{\rm pow}m_{\rm sup}R\) on disjoint prime supports. Freeze the unit, the actual ideals \(m_{\rm pow},m_{\rm sup}\), and then partition \(R\) into actual smooth annuli \(q_R\asymp Z^H\), with \(H\ge0\) and the unit in the bounded zero dyad. Keep this row cutoff in each reflected vector. Equation (57) gives \[ H\le M-O+O_{\rm fixed}(1/\log Z). \tag{59}\] The fixed row restrictions, including coprimality with the frozen factors, are independent of the dual variables.

After the row-sector conversion, take \(\mathcal F\) in Definition 20 to be the good primes of \(m_{\rm pow}\), with their valuations reduced modulo six. The primes of \(R\) have exponent one. Fix \(h_0\), a class of \(R\) modulo \(M_{\rm ref}^2\), and the local active and Ramanujan choices at the frozen primes. Every active frozen good prime divides \(m_{\rm pow}\) to exponent at least two. Consequently \[ q_{F_{\rm act}}^2\le q_{m_{\rm pow}},\qquad 2A_0\le O+O_{\rm fixed}(1/\log Z). \tag{60}\] This uses the actual powerful dyad’s upper comparison. The possible \(S\)-prime factors of \(m_{\rm pow}\) only increase its norm.

Apply Equation (23) at \(X=Z^N\) to the Mellin representation in Equation (53). Its dual sum is absolutely convergent. Insert fixed smooth dyadic partitions in the residual squarefree and cube norms, and fix the unit and ramified exponent of the dual index. These operations give the blocks of Definition 20 with uniformly bounded seminorms for \(W\). No partition of the primal test \(V\) is needed: the compact profile here is the product of the dual and row cutoffs with the normalized inverse roots, and \(V^\sharp\) remains the kernel.

Fix a small \(\tau_{\rm ref}>0\). The row lengths \(H,O\), the active length \(A_0\), and \(N\) lie in fixed bounded ranges. The same is true of \(S_0,N_0,B_0\), since their prime products divide \(F_{\rm act}\). The tail argument following Lemma 21 therefore removes all unretained whole dual blocks with any prescribed power saving, with a finite seminorm of \(V\). Its polynomial cost includes the row and local counts below but not the discarded dual indices. Choose its kernel order first, and increase \(A,J\) in the statement so that \(\mathfrak M_{A,J}(V)\) dominates this tail seminorm and the block seminorm. This is possible because increasing \(A\) widens the supremum defining \(\mathfrak M_{A,J}\) and increasing \(J\) increases its weight.

Write \(e=e_\lambda\). The source restriction \(k\ge-4\) gives \(e\ge-4\log 3/\log Z\). For a retained block, \[v+3\ell_b+e\le T_0+\tau_{\rm ref}.\] Since \(v,\ell_b\ge0\) and \(T_0\) is bounded, this also bounds every retained dual length. There are only logarithmically many retained choices for the dual annuli and for \(k\). Choose a nonnegative \(\epsilon_Z=O_{\rm fixed}(1/\log Z)\) large enough to cover the support offsets in Equations (59) and (60), and to ensure \(e\ge-\epsilon_Z\).

The exponent in Equation (56) satisfies \[E_0\le\max\{H,T_0-S_0-B_0\}+\frac53\epsilon_Z+\tau_{\rm ref}.\] To see this, use \(\max(H,v+\ell_b)-\ell_b=\max(H-\ell_b,v)\) and discard the nonpositive kernel term. The first branch is at most \(H+2\epsilon_Z/3\), since \(\ell_b,S_0,B_0\ge0\). In the second branch, retention gives \(v\le T_0-3\ell_b-e+\tau_{\rm ref}\), so it is at most \(T_0-S_0-B_0+5\epsilon_Z/3+\tau_{\rm ref}\). Moreover, \[\begin{split} T_0-S_0-B_0 &=2H+2A_0-N-N_0-S_0-4B_0\\ &\le 2M-O-N+O(\epsilon_Z), \end{split}\] by Equations (59) and (60). Thus every retained block, with its frozen row parts fixed, has exponent at most \[\max\{M-O,2M-O-N\}+\tau_{\rm ref}+O(\epsilon_Z).\]

It remains to count the frozen parts and the finite expansions. Every powerful ideal is uniquely \(x^2y^3\) with \(y\) squarefree: at a prime, the exponent of \(y\) is the parity of the powerful exponent, and an odd positive exponent is at least three. Ideal counting and \(\sum_yq_y^{-3/2}<\infty\) therefore give \[\#\{m_{\rm pow}:q_{m_{\rm pow}}\ll Z^O\}\ll Z^{O/2}.\] The squarefree \(m_{\rm sup}\) is a divisor of the fixed radical of \(S\) and has only finitely many choices. The local branches at the good primes of \(m_{\rm pow}\) have divisor-bounded multiplicity, as do their Ramanujan assignments. Rowwise divisor Cauchy and then summation of the local choices cost an arbitrarily small power of \(Z\). The choices of \(h_0\), fixed ray sectors, and units are finite, and the actual row and retained dual annuli cost only powers of \(\log Z\). This counts the powerful contribution \(O/2\) exactly once.

Choose \(\tau_{\rm ref}\) and the local small-power losses within the prescribed \(\epsilon\) allowance. Then increase the fixed-data threshold for \(Z\) so that the \(O(\epsilon_Z)\) terms lie within that allowance. The block bound, the preceding counts, and the already chosen tail saving prove Equation (58). The twist assertion follows from the norm-twist bound for \(\mathfrak M_{A,J}\) in Proposition 15. ◻

Proof. The assertion is immediate when \(J=0\). The characters \(e(mz_j)\) are well defined on \(\mathbb C/\mathcal O\) by self-duality. Choose representatives in \(\mathbb C\) for the other cases. We use the Fourier transforms \[\widehat F(y)=\int_{\mathbb C}F(x)e(-xy)\,d\mu(x),\qquad \check\phi(x)=\int_{\mathbb C}\phi(y)e(xy)\,d\mu(y).\] Choose a nonnegative \(\phi\in C_c^\infty(\mathbb C)\) of integral one, supported in a sufficiently small disk about zero. Since \(|e(xy)-1|\le (4\pi/\sqrt3)|x||y|\), its support can be chosen so that \[|\check\phi(x)-1|\le\frac12\qquad(|x|\le1).\] Thus \(F(x)=4|\check\phi(x)|^2\) is a nonnegative Schwartz function and \(F(x)\ge1\) on the unit disk. If \(\widetilde\phi(y)=\overline{\phi(-y)}\), Fourier inversion and the product formula, with convolution taken with respect to \(d\mu\), give \[\widehat F=4\phi*\widetilde\phi.\] In particular, \(\widehat F\) is bounded and supported in a disk of some fixed radius \(L\). This is the bandlimited majorant we need.

Put \(R=\sqrt Q\). Positivity of \(F\) first gives \[\sum_{q_m\le Q}\left|\sum_j a_je(mz_j)\right|^2 \le \sum_{m\in\mathcal O}F(m/R) \left|\sum_j a_je(mz_j)\right|^2.\] All sums on the right converge absolutely. For any \(z\in\mathbb C\), the Fourier transform of \(x\mapsto F(x/R)e(xz)\) at \(y\) is \(R^2\widehat F(R(y-z))\). The factor \(R^2\) is the Jacobian of the real two-dimensional dilation; no lattice-volume factor occurs because \(d\mu\) has covolume one on \(\mathcal O\). Poisson summation therefore expands the right side as \[R^2\sum_{j,k}a_j\overline{a_k} \sum_{n\in\mathcal O} \widehat F\bigl(R(n-z_j+z_k)\bigr).\]

For each fixed \(j\), a term in the inner sum can be nonzero only if the lift \(z_k+n\) lies within distance \(L/R\) of \(z_j\). The set of all lifts \(\{z_k+n:1\le k\le J,\ n\in\mathcal O\}\) is \(\delta\)-separated in \(\mathbb C\). Distinct classes have this property by hypothesis, and two distinct lifts of one class differ by a nonzero Eisenstein integer, whose absolute value is at least one and hence at least \(\delta\). The disks of radius \(\delta/3\) about lifts in a disk of radius \(L/R\) are disjoint and lie in the concentric disk of radius \(L/R+\delta/3\). Comparing their Euclidean areas bounds the number of these lifts by \[\left(1+\frac{3L}{R\delta}\right)^2 \ll 1+(R\delta)^{-2}.\] The same bound holds with \(j\) and \(k\) interchanged. Taking absolute values in the Poisson expansion, using the fixed bound for \(\widehat F\), and then using \(2|a_ja_k|\le |a_j|^2+|a_k|^2\), we obtain \[\sum_{q_m\le Q}\left|\sum_j a_je(mz_j)\right|^2 \ll R^2\bigl(1+(R\delta)^{-2}\bigr)\sum_j|a_j|^2.\] Since \(R^2=Q\), this is the asserted estimate. Replacing \(Q\) by a fixed multiple proves the final statement. ◻

Proof. For a primary \(s\) outside \(S\), put \(r_s=q_s/Y\) and \[c_\sigma(s)=1_{s\in\sigma}\frac{\chi_s(b_*)}{\tau\xi(s)}.\] The inverse is evaluated only on these elements prime to \(S\). Here \(|\tau|=1\), \(|\xi(s)|=1\), and \(|\chi_s(b_*)|=1\), because \(b_*\) is supported on \(S\). Thus \(|c_\sigma(s)|\le1\). The definition of \(A\) and the Gauss expansion give the exact identity \[A_{m,\sigma,v}(Y) =Y^{-3/2}\sum_s c_\sigma(s)W_1(r_s)r_s^{-1+iv} \sum_{d\bmod s}\chi_s(d)e(-md/s).\] The sum is over primary ideals outside \(S\). For every such \(s\), the character \(\chi_s(d)\) is zero exactly when \((d,s)>1\), and has absolute value one otherwise. This includes nonsquarefree \(s\): a local exponent divisible by six is the indicator of the units at that prime, not the constant function one. For \(s=1\) the single residue class is included and \(\chi_1=1\).

We verify that the fractions \(d/s\) for the contributing pairs are distinct modulo \(\mathcal O\). Suppose that \((d,s)=(d',s')=1\) and \(d/s-d'/s'\in\mathcal O\). Then \(ds'-d's\) is divisible by \(ss'\). Reducing this divisibility modulo \(s\) and using the inverse of \(d\) modulo \(s\) gives \(s\mid s'\). The symmetric argument gives \(s'\mid s\). The primary generator of an ideal outside \(S\) is unique, so \(s=s'\); the original congruence then gives \(d=d'\pmod s\). The same conclusion includes the unit modulus. Changing a residue representative only translates its fraction by an element of \(\mathcal O\).

Let \(0<r_-<r_+\) be fixed with \(\operatorname{supp}W_1\subset[r_-,r_+]\). For two distinct contributing fractions and every \(n\in\mathcal O\), the Eisenstein integer \(ds'-d's-nss'\) is nonzero. Hence \[\left|\frac d s-\frac{d'}{s'}-n\right| =\frac{|ds'-d's-nss'|}{|ss'|} \ge\frac1{|ss'|}\ge\frac1{r_+Y}.\] The last inequality uses \(q_s,q_{s'}\le r_+Y\). Thus these fractions are \(\delta_Y\)-separated in \(\mathbb C/\mathcal O\), where \(\delta_Y=\min(1,r_+^{-1})/Y\le1\).

Writing the expanded polynomial as \(\sum_{s,d}a_{s,d}e(-md/s)\), with only the unit residue classes retained, its coefficient square mass is exactly \[\begin{split} \sum_{s,d}|a_{s,d}|^2 &=Y^{-3}\sum_s |c_\sigma(s)W_1(r_s)|^2r_s^{-2} \varphi_{\mathcal O}(s),\\ \varphi_{\mathcal O}(s) &:=\#(\mathcal O/s\mathcal O)^\times,\qquad \varphi_{\mathcal O}(1):=1. \end{split}\] There is no dependence on \(v\) in this expression. Since \(\varphi_{\mathcal O}(s)\le q_s\le r_+Y\), \(r_s\ge r_-\) on the support, and there are \(O(Y)\) ideals with \(q_s\le r_+Y\), this mass is \(O(Y^{-1})\). Lemma 23, applied to the points \(-d/s\) with separation \(\delta_Y\), now proves the first bound. Finally \(q_{b_*}\ge1\), so \(Q=q_{b_*}Y^2\) implies \(Q+Y^2\le2Q\), proving the second. ◻

Proof. Set \(X=Y=Z^{1/2}\) and \(Q=q_{b_*}XY=q_{b_*}Z\). The function \(\Omega\) in Equation (63) is bounded and supported in a fixed compact subinterval of \((0,\infty)\). Thus all rows in that identity lie in \(0<q_m\le C_\Omega Q\) for a fixed constant \(C_\Omega\).

The specialization \(M=N=1\) of Lemma 22, followed by the finite ray decomposition defining \(B_{m,\sigma}\), gives for every \(\epsilon_1>0\) \[\sum_{0<q_m\le C_\Omega Q}|B_{m,\sigma}(Z)|^2 \ll_{\mathcal A,\epsilon_1} Z^{1+\epsilon_1}.\] Indeed, \(q_{b_*}\) is fixed, so the row ball is a fixed multiple of the \(M=1\) ball. The completed scale is \(Z\), which is \(N=1\) in that lemma. The lemma applies directly to the Gaussian profile in the completed integrals defining \(B\); the fixed finite Fourier sum over ray characters costs only a fixed factor by the triangle inequality in the row Hilbert space. In particular this application retains the original row zero masks.

Apply Cauchy–Schwarz to the row sum in Equation (63). The factor \((q_m/Q)^{-iv}\) has absolute value one, \(|\xi(m)|\le1\) with its zero extension, and \(\Omega\) is bounded. Lemma 24 is uniform in \(v\), while \(B\) is independent of \(v\). Since \(W_0\) is smooth and annular, \(\int_{\mathbb R}|\widehat W_0(iv)|\,dv<\infty\). The finite sum over \(\sigma\in T\) therefore gives \[\begin{split} |I_\eta(X,Y,Z)| &\ll_{\mathcal A,\epsilon_1} Q^{-1/2}\left(\frac QY\right)^{1/2}Z^{1/2+\epsilon_1/2}\\ &=Z^{1/4+\epsilon_1/2}. \end{split}\] Taking \(\epsilon_1=2\epsilon\) proves the proposition. ◻

Proof. Each maximum exists: the collection is finite, the zeros of each nonprincipal primitive \(L\)-function are discrete in compact rectangles, and absolute Euler convergence excludes zeros with real part greater than one. The sequence \(M_j(u)\) is nondecreasing and lies in \([51/100,1]\). If every one of its \(I-1\) increments exceeded \(e\), then \[M_I(u)-M_1(u)>(I-1)e>1,\] which is impossible. Choose an index with \(M_{i+1}(u)-M_i(u)\le e\), and round \(M_i(u)\) down on the stated grid. This gives the two inequalities. When \(a>51/100\), the maximum \(M_i(u)\) is attained by an actual zero, giving \(\rho\). This reasoning allows a zero on the line one.

For \(|t|\le(3i+2)T_1\), the closed disk centered at \(2+it\) with radius \(2-a-2e\) has real part at least \(a+2e\). Its radius is less than \(3/2<T_1\), so every point in it has imaginary part of absolute value less than \(3(i+1)T_1\). The bound \(M_{i+1}(u)<a+2e\) therefore excludes every zero of every primitive function in this disk. Lemma 13 controls the concentric disk of radius \(2-a-6e\). Apply it with center \(2+i\Im s\), and use absolute Euler convergence farther right. Since \(Q_\psi\ll_{\mathcal A}U\) and \((3+|t|)^2\ll_e U^{1/50}\), the arbitrarily small power in that lemma can be chosen so that its contribution is \(U^{\epsilon_1/2}\). The estimates for polynomial-size deleted Euler products following that lemma supply the other \(U^{\epsilon_1/2}\).

For the last assertion, the primitive functional equation recalled in the proof of Lemma 12 [11] relates the value at \(1-a-6e+it\) to the conjugate primitive function at \(a+6e-it\). The conjugate presentation is in \(\mathcal X_u\). The conductor and gamma quotient contribute at most \[Q_\psi^{a-1/2+6e}(3+|t|)^C.\] The reflected primitive value has an arbitrarily small \(U\)-power by the preceding disk bound. Finally, \(1-a-6e\ge-6e\), so the upper bound for the deleted product is \(U^{6e+\epsilon_1}\), after reducing the preliminary losses. This proves the reflected estimate. The ranges of \(i\) and \(a\) give \(O_e(1)\) pairs. ◻

Proof. We spell out the height restriction because a full infinite contour shift would not be justified by the bin. For a fixed annular \(W\), a bounded real number \(\sigma\), and a pure twist \(y^{i\omega}\), Mellin inversion on a line \(c+\sigma>1\) gives \[\begin{split} &U^{-r/2}\sum_n\mu(n)\psi(n) W(q_n/U^r)(q_n/U^r)^{-\sigma+i\omega}\\ &\quad=\frac1{2\pi i}\int_{(c)} \mathcal MW(s)\, U^{r(s+\sigma-i\omega-1/2)} L_{\rm orig}(s+\sigma-i\omega,\psi)^{-1}\,ds. \end{split}\] Here \(\mathcal MW\) is the Mellin transform defined in Lemma 9. The twist occurs in the \(L\)-argument, not in the transform whose tails are estimated. Shift only the portion with added frequency at most the assigned fraction of \(T_1\) to \(\Re(s+\sigma)=a+6e\). The horizontal joins and the shifted portion remain in the zero-free rectangle of Lemma 27. Its bound gives \(U^{(a-1/2+6e)r+\epsilon_1}\) for the central portion.

Leave the remaining tails on the absolute line, which we may take to be \(\Re(s+\sigma)=2\). There the reciprocal Euler product is absolutely bounded. On the inverse joins the reciprocal remains inside the buffered rectangle. Since the real join interval and the length range are bounded, the remaining arithmetic and scale factors on a join are \(O(U^B(1+|\Im s|)^J)\) for fixed \(B,J\) chosen before the external tail order. The pointwise Mellin bound in Equation (18), applied on that compact real interval with derivative order \(N+\lceil J\rceil+1\), bounds each horizontal join by \(O(U^B T_1^{-N})\). The separate vertical tails have the same bound by Equation (17). Both estimates apply to the untwisted profile, uniformly in the stated family. Because \(T_1=Z^\tau\), \(\tau>0\), and all lengths lie in a fixed bounded interval, a sufficiently large fixed \(N\) makes these errors smaller than the claimed bound. Taking \(e\) and \(\epsilon_1\) small in the prescribed \(\epsilon\) proves Equation (81).

For the plain polynomial the same calculation uses \(L_{\rm orig}\) instead of its reciprocal. Shifting to \(\Re(s+\sigma)=a+6e\) gives \[|S_m|\ll U^{(a-1/2+6e)m+\epsilon_1} +O(U^B T_1^{-N}).\] Alternatively shift its central portion to \(\Re(s+\sigma)=1-a-6e\). The function is entire because the row is nonprincipal. The reflected estimate of Lemma 27 gives \[|S_m|\ll U^{a-1/2+12e+\epsilon_1} U^{(1/2-a-6e)m}(3+T_1)^C +O(U^B T_1^{-N}).\] The joins here need only the global upper strip bound and the negative-strip bound for the deleted product, since no reciprocal is present. These give fixed \(B,J\) as above, so the pointwise Mellin estimate bounds the joins and the integrated Fourier estimate bounds the absolute-line tails. Taking the better central bound, choosing \(N\) to dominate also the possibly negative exponent when \(m>1\), and using the displayed height hypothesis proves Equation (82). A primitive conductor smaller than \(U\) reduces the reflected conductor factor. The original deleted Euler factors have already been included in Lemma 27.

There are only finitely many separated variables in any one \(L\)-argument. Assign each a fixed fraction of the single \(T_1/2\) allowance; do not assign that allowance anew at successive shifts. The profile windows and the finite height orders used in the row estimates are fixed before \(N\). Lemma 9 differentiates only the untwisted separating profiles when increasing \(N\), which proves the last assertion. ◻

Proof. The truncated-inverse and Gamma-integral construction is a form of the classical zero detector; compare [21] and [14]. The buffered row-dependent rectangle and the simultaneous lower bounds needed here are established below. Choose the zero \(\rho\) supplied by the bin and set \[D_*=U^t,\qquad Y_*=U^{20}.\] Let \(V_{\le}\) be a fixed smooth function equal to one on \([0,1]\) and zero on \([2,\infty)\). Define \[C_\psi(s)=\sum_l\mu(l)\psi(l)V_{\le}(q_l/D_*)q_l^{-s}, \qquad J=\frac1{2\pi i}\int_{(2)} Y_*^z\Gamma(z)L_{\rm orig}(\rho+z,\psi) C_\psi(\rho+z)\,dz.\] The row character is nonprincipal, so \(L_{\rm orig}\) is entire. Moving the line to \(\Re z=-1/4\) crosses only the pole of \(\Gamma(z)\) at zero, and its residue vanishes because \(L_{\rm orig}(\rho,\psi)=0\). The global strip bound in Equation (21), together with the deleted Euler factors, bounds the new line by \[U^{-5}U^2D_*^{\,1-\sigma+1/4+o(1)}T_1^2.\] Indeed \(\Re(\rho-1/4)\ge26/100>0\), so the deleted product has an arbitrarily small \(U\)-power; the deliberately weaker \(U^2\) includes the primitive conductor bound. The finite polynomial is bounded by the displayed power of \(D_*\) using the ideal count. Finally, the fixed polynomial in the added height is integrable against \(\Gamma(-1/4+iv)\), since \((1+|\gamma+v|)^2\le(1+|\gamma|)^2(1+|v|)^2\). Uniformly for \(1\le t\le3/2\), \[-3+t(5/4-\sigma) \le-3+\frac32\left(\frac54-\frac{51}{100}\right) =-\frac{189}{100}.\] Thus \(J=o(1)\). This Gamma integration uses only global upper bounds and consumes no reciprocal height allowance.

On the original line, absolute convergence and the Mellin formula for the exponential give \[J=\sum_{l,k}\mu(l)\psi(lk)V_{\le}(q_l/D_*) (q_lq_k)^{-\rho}e^{-q_lq_k/Y_*}.\] For \(1<q_n\le D_*\), the coefficient of \(\psi(n)q_n^{-\rho}\) is \(\sum_{l\mid n}\mu(l)=0\). For \(n=1\) it is one. When \(q_n>D_*\), \(V_{\le}(2q_n/D_*)=0\). Consequently insertion of \(1-V_{\le}(2q_lq_k/D_*)\) removes exactly the unit contribution \(e^{-1/Y_*}=1+o(1)\). The resulting tail has absolute value \(1+o(1)\). Multiplying it by a fixed smooth terminal cutoff equal to one for \(q_lq_k\le U^{21}\) and zero for \(q_lq_k\ge2U^{21}\) changes it by \(o(1)\); this follows from \(Y_*=U^{20}\), exponential decay, and the ideal count.

Partition \(q_l\asymp D\), \(q_k\asymp N\) by a fixed smooth dyadic partition. There are \(O((\log U)^2)\) relevant pairs, and their support satisfies \[D\ll D_*,\qquad DN\gg D_*,\qquad DN\ll U^{21}.\] With \(x=q_l/D\), \(y=q_k/N\), the profile for a pair is \[W_1(x)V_{\le}(Dx/D_*)\,W_2(y)\,H_{D,N}(xy),\] where \(\Omega\) is a fixed annular cutoff equal to one on the product of the supports of \(W_1,W_2\), and \[H_{D,N}(s)=\Omega(s) [1-V_{\le}(2DNs/D_*)]e^{-DNs/Y_*} V_{\rm term}(DNs/U^{21}).\] Every fixed logarithmic derivative is bounded uniformly in \(D,N,U\). On a cutoff transition its argument is bounded, and a logarithmic derivative of the exponential is a polynomial in its argument times that exponential. Logarithmic Fourier inversion therefore gives \[\begin{split} H_{D,N}(xy)&=\frac1{2\pi}\int_{\mathbb R} \widehat H_{D,N}(\nu)x^{i\nu}y^{i\nu}\,d\nu,\\ \int_{|\nu|>cT_1}|\widehat H_{D,N}(\nu)| (1+|\nu|)^J\,d\nu&\ll_{J,N_0,c}T_1^{-N_0} \end{split}\] for every fixed \(J,N_0\). The coefficient \(L^1\)-norm on the full line is also uniformly bounded. Trivial bounds for the two finite polynomials have a fixed \(U\)-power because their lengths are bounded by \(U^{21+o(1)}\). After \(\tau>0\) is fixed, choose \(N_0\) so that the discarded Fourier tail is \(o(1)\).

The absolute value of the remaining sum of integrals is bounded below by a positive constant. The number of dyadic pairs and the uniform \(L^1\)-norm show that, for one pair and one \(|\nu|\le cT_1\), the product of the two unnormalized blocks has absolute value \(\gg(\log U)^{-2}\). Both profiles have the same height \(\gamma-\nu\): \[\begin{split} W_M(x)&=W_1(x)V_{\le}(Dx/D_*)x^{-\sigma-i(\gamma-\nu)},\\ W_S(y)&=W_2(y)y^{-\sigma-i(\gamma-\nu)}. \end{split}\] The first extra cutoff remains part of the inverse annular profile and creates no second frequency. Removing the factors \(D^{1/2-\sigma}\) and \(N^{1/2-\sigma}\) by central normalization gives \[|M_rS_m|^2\gg(\log U)^{-4}(DN)^{2\sigma-1} \ge U^{\delta(r+m)-\epsilon}\] for sufficiently large \(U\). The support inequalities give the two length bounds in Equation (83).

Apply Lemma 28 to these profiles, choosing its loss smaller than the present \(\epsilon\). Comparison of the product lower bound with the two upper bounds gives \[\delta\{m-\min(m,1-m)\} =2\delta(m-1/2)_+\ll\epsilon.\] Since \(\delta\ge1/50\), this implies \(m\le1/2+O(\epsilon)\) with an absolute constant. The support inequality then gives \(r\ge t-1/2-O(\epsilon)\). Dividing the product lower bound in turn by each individual upper bound gives the two individual lower bounds in Equation (84). Only the dyadic support errors are \(O(1/\log U)\); all other losses are arbitrarily small fixed powers. Lemma 9 permits the stated rowwise choices with a fixed polynomial height factor. ◻

Proof. We follow the higher-order large-sieve recursion of Blomer, Goldmakher, and Louvel [4], giving the local verification for the present primary normalization. In the finite Fourier calculation the individual residue characters act on elements; their unit-trivial quotient is the character on ideals to which we apply Poisson summation.

For good ideals \(a,b\), represented by their primary generators, put \[F_b(a)=\chi_a(b),\qquad J=\{1,2,4\}.\] This uses the original zero-extended symbol, also when \(a\) is not squarefree. It is multiplicative in both \(a\) and \(b\). In every power below, including a multiple of six, a nonunit still has value zero. For \(M>0\) let \[\mathcal I_M=\{a\text{ good}:M<q_a\le2M\},\qquad \mathcal A_M=\{a\in\mathcal I_M:a\text{ squarefree}\}.\] For a sequence supported on \(\mathcal A_N\) write \(\|\lambda\|_2^2=\sum_{b\in\mathcal A_N}|\lambda_b|^2\), and define \[\begin{align*} D_j(M,N)&=\sup_{\|\lambda\|_2=1} \sum_{a\in\mathcal A_M} \left|\sum_{b\in\mathcal A_N}\lambda_bF_b(a)^j\right|^2,\\ E_j(M,N)&=\sup_{\|\lambda\|_2=1} \sum_{a\in\mathcal I_M} \left|\sum_{b\in\mathcal A_N}\lambda_bF_b(a)^j\right|^2. \end{align*}\] An empty support gives norm zero. These norms include all finite ray classes. We will prove the symmetric bound for \(D_1\); the matrix in the lemma is its transpose with the two lengths exchanged.

The squarefree norm \(D_j\) is our target. We also need \(E_j\), since Poisson summation produces arbitrary evaluation ideals. Opening the square in its defining sum and pairing the column characters will replace a row length \(M\) by a dual length of order \(N^2/M\). A power decomposition then returns the unrestricted row sum to squarefree norms. It selects either the first-power or the second-power factor, so the exponents \(j\in\{1,2,4\}\) are kept together: this set is closed under \(j\mapsto2j\pmod6\). We will use this cycle to improve a provisional exponent in the bound for \(D_j\).

The paired summation formula. Use the fixed finite group \[\mathcal T=\ker\bigl((\mathcal O/36\mathcal O)^\times \longrightarrow(\mathcal O/3\mathcal O)^\times\bigr), \qquad t(b)=b\bmod36\mathcal O.\] The primary normalization identifies these with the ray classes modulo \(36\mathcal O\): each unit orbit has a unique representative equal to one modulo \(3\). In particular \(t\) is multiplicative. This subdivision is fixed, since the primes above \(2\) and \(3\) belong to \(S\).

For a global unit \(u\) and a good prime \(p\), reduction modulo \(p\) gives \(\chi_p(u)=u^{(q_p-1)/6}\). Since \(q_p\equiv1\pmod6\), multiplication over the prime factors of \(b\) gives \[ \chi_b(u)=u^{(q_b-1)/6}. \tag{86}\] The exponent is read modulo six: expanding a product of integers \(1+6h\) proves the equality of exponents. Thus equal \(t\)-classes have the same restriction to the six global units. They also have the same residue modulo \(4\), which controls the reciprocity factor \(\mathcal R\) of Lemma 8.

Let \(b,c\) be coprime squarefree good ideals with \(t(b)=t(c)\) and let \(j\in J\). The raw character on elements \[\Theta_{b,c}^{(j)}(z)=\chi_b(z)^j\overline{\chi_c(z)^j} \quad (z\bmod bc)\] is trivial on global units by Equation (86). It therefore defines a finite-order ray character on ideals by \[\Psi_{b,c}^{(j)}((z))=\Theta_{b,c}^{(j)}(z).\] For fractional ideals prime to \(bc\), the residue symbols are evaluated on local unit fractions; the same unit cancellation makes this definition independent of the generator. At each good prime the sextic residue character has exact order six, since the residue multiplicative group is cyclic. Hence its powers \(j\) and \(-j\) are nontrivial. CRT shows that \(\Theta_{b,c}^{(j)}\) has raw conductor \(bc\), and the ray character has the same conductor: omitting a prime would contradict this nontriviality by varying only its residue. It is extended by zero on ideals meeting \(bc\). For the pair \(b=c=1\) define \(\Psi_{1,1}^{(j)}\) to be principal.

Sextic reciprocity, with its factor fixed by \(t(b)=t(c)\), gives for every good ideal \(a\) \[ \Psi_{b,c}^{(j)}(a)=F_b(a)^j\overline{F_c(a)^j}. \tag{87}\] For coprime inputs the two reciprocity factors cancel; on the other inputs both sides are zero. With the Gauss sums already defined in the arithmetic preliminaries, CRT using \(z=cv+bw\) gives \[ \begin{split} \Gamma_{b,c}^{(j)} &:=q_{bc}^{-1/2}\sum_{z\bmod bc}\Theta_{b,c}^{(j)}(z)e(z/(bc))\\ &=\chi_b(c)^j\overline{\chi_c(b)^j}\gamma_j(b)\gamma_{-j}(c) =\eta_{t(b),j}\gamma_j(b)\gamma_{-j}(c), \end{split} \tag{88}\] where \(\eta_{t(b),j}=\mathcal R(b,c)^j\) depends only on the common class. Every \(\gamma_{\pm j}\) here has modulus one by prime Gauss orthogonality and CRT. Formula (88) retains \(\gamma_{-j}\) itself, including its factor at \(-1\) under conjugation.

Choose once a nonnegative \(W\in C_c^\infty((0,\infty))\) with \(W\ge1\) on \([1,2]\). For the Fourier transform and self-dual measure in the arithmetic preliminaries, put \(w(z)=W(|z|^2)\) and define \(\mathscr W\) by \[\widehat w(y)=\mathscr W(|y|^2).\] The transform is radial, smooth at zero, and rapidly decreasing. Thus \[ \mathscr W(x)\ll_A(1+x)^{-A},\qquad \int_{\mathbb R}|\mathcal M\mathscr W(\sigma+it)|\,dt<\infty \quad(\sigma>0). \tag{89}\] Here \(\mathcal M\) denotes the Mellin transform. The second assertion follows by integration by parts in the Mellin integral; all logarithmic derivatives are bounded at zero and rapidly decreasing at infinity. Scaling the Fourier transform gives \(M\mathscr W(M|y|^2)\) for \(z\mapsto W(|z|^2/M)\).

For \(bc\ne1\), finite Fourier inversion for the primitive raw character is \[\sum_{v\bmod bc}\Theta_{b,c}^{(j)}(v)e(vy/(bc)) =\overline{\Theta_{b,c}^{(j)}(y)}\sqrt{q_{bc}}\, \Gamma_{b,c}^{(j)}.\] For a nonunit \(y\) the left side vanishes because one nontrivial local character is summed against the trivial additive character. For the stated \(e\) and \(d\mu\), the trace pairing on the basis \((1,\omega)\) has matrix \(\left(\begin{smallmatrix}0&1\\1&-1\end{smallmatrix}\right)\), of determinant \(-1\), and the \(d\mu\)-covolume of \(\mathcal O\) is one. Thus the dual of \(bc\mathcal O\) is \((bc)^{-1}\mathcal O\), while the covolume of \(bc\mathcal O\) is \(q_{bc}\). Coset Poisson summation, followed by division by the six generators of each ideal, therefore gives \[ \sum_{\mathfrak a\ne0}W(q_{\mathfrak a}/M)\Psi_{b,c}^{(j)}(\mathfrak a) =\frac{M\Gamma_{b,c}^{(j)}}{\sqrt{q_{bc}}} \sum_{\mathfrak a\ne0}\mathscr W(Mq_{\mathfrak a}/q_{bc}) \overline{\Psi_{b,c}^{(j)}(\mathfrak a)}. \tag{90}\] Both sums here are over all nonzero integral ideals. The division by six uses unit-triviality of \(\Theta_{b,c}^{(j)}\) and radiality of the weight. The zero frequency vanishes since this character is nontrivial.

Poisson transformation of the squared norm. For \(\lambda\) supported on one \(t\)-class in \(\mathcal A_N\), set \[Q_j(M,N;\lambda)= \sum_{\substack{b,c\in\mathcal A_N\\(b,c)=1}} \lambda_b\overline{\lambda_c} \sum_{\mathfrak a\ne0}W(q_{\mathfrak a}/M) \Psi_{b,c}^{(j)}(\mathfrak a).\] We claim, for every \(\varepsilon>0\), \[ \begin{split} |Q_j(M,N;\lambda)| &\ll_{S,\varepsilon}(2+MN)^\varepsilon\|\lambda\|_2^2\\ &\quad\times \left\{M+\frac MN \max_{\substack{1\le H\\ H\ll_{S,\varepsilon}(2+MN)^\varepsilon N^2/M}} E_j(H,N)\right\}. \end{split} \tag{91}\] The maximum runs over dyadic shells and is zero if empty. If \(N\le1\), only the unit good ideal can occur and the annular ideal count proves the \(M\) term. Also \(Q_j=0\) when \(M\) is below a fixed positive constant, because \(W\) has a fixed upper support endpoint and nonzero ideals have norm at least one.

Suppose \(N>1\) and normalize \(\|\lambda\|_2=1\). Then every pair in \(Q_j\) is nontrivial and \(q_{bc}\asymp N^2\). Apply Equation (90) and take the triangle inequality over the dual ideals. For a small \(\delta>0\) put \(Z=(2+MN)^\delta\). Cauchy gives \((\sum_b|\lambda_b|)^2\ll N\). By Equation (89), the part with \(q_{\mathfrak a}>ZN^2/M\) is, for any \(A>1\), at most \[M\sum_{q_{\mathfrak a}>ZN^2/M}(Mq_{\mathfrak a}/N^2)^{-A} \ll N^2Z^{1-A}.\] The same bound holds if the cutoff is below one. Taking \(A\) large makes this an arbitrary negative power error.

In the remaining sum write \(\mathfrak a=a_0\mathfrak s\), with \(a_0\) good and \(\mathfrak s\) supported on \(S\), and choose one generator \(s_0\) for each fixed \(\mathfrak s\). Equations (87) and (88) give the separated factors \[\overline{\Psi_{b,c}^{(j)}(a_0\mathfrak s)} =\overline{F_b(a_0)^j}F_c(a_0)^j \overline{\chi_b(s_0)^j}\chi_c(s_0)^j.\] All \(\chi_b(s_0)\) have modulus one. Split \(a_0\) into shells \(a_0\in\mathcal I_H\), where \(H\ll ZN^2/(Mq_{\mathfrak s})\), and insert Mellin inversion for \(\mathscr W\) on \(\Re w=\delta\). Apart from a bounded factor depending on \(M,q_{a_0},q_{\mathfrak s}\), the two coefficient sequences are \[A_b=\lambda_b\gamma_j(b)\overline{\chi_b(s_0)^j} q_b^{-1/2+\delta+it},\qquad B_b=\overline{\lambda_b}\gamma_{-j}(b)\chi_b(s_0)^j q_b^{-1/2+\delta+it}.\] They are fixed across the \(a_0\) sum and have absolute values \(\ll N^{-1/2+\delta}|\lambda_b|\). Detecting \((b,c)=1\) by Möbius inversion and applying Cauchy in \(a_0\) gives \[\begin{align*} &\sum_{a_0\in\mathcal I_H} \left|\sum_{\substack{b,c\in\mathcal A_N\\(b,c)=1}} A_bB_c\overline{F_b(a_0)^j}F_c(a_0)^j\right|\\ &\quad\le E_j(H,N)\sum_{\mathfrak d} \left(\sum_{\mathfrak d\mid b}|A_b|^2\right)^{1/2} \left(\sum_{\mathfrak d\mid b}|B_b|^2\right)^{1/2} \ll N^{-1+3\delta}E_j(H,N). \end{align*}\] Here the divisor bound absorbs the sum over \(\mathfrak d\). Conjugating the entire first polynomial uses the same \(E_j\) norm. The factors with \(H<1\) have just the unit evaluation ideal and satisfy \(E_j(H,N)\ll N\). There are only a fixed power of \(\log(2+ZN^2)\) choices of \(\mathfrak s\) and of the dyadic shell, because \(S\) is fixed and a nonzero \(Q_j\) has \(M\) bounded below. Equation (89) integrates the \(t\) variable. Choosing \(\delta\) sufficiently small in terms of \(\varepsilon\) proves Equation (91), including the allowed global loss on its \(M\) term.

We now apply this paired estimate to \(E_j\). Partition its inner sum by \(t\), insert \(W\), and open the square. Write the two squarefree indices as \(db,dc\), where \(d\) is their gcd, \((b,c)=1\), and \((d,bc)=1\). Their residual classes agree because \(t\) is multiplicative. Index multiplicativity and Equation (87) give on every good \(a\) \[F_{db}(a)^j\overline{F_{dc}(a)^j} =1_{(a,d)=1}\Psi_{b,c}^{(j)}(a).\] Let \(\mathfrak s_S=\prod_{\mathfrak p\in S}\mathfrak p\). Möbius inversion for \((a,\mathfrak s_Sd)=1\) writes \(a=\mathfrak r\mathfrak a\) with \(\mathfrak r\mid\mathfrak s_Sd\). Choose one generator \(r_0\) of each fixed \(\mathfrak r\). The value of \(\Psi\) at \(\mathfrak r\) separates as \(\chi_b(r_0)^j\overline{\chi_c(r_0)^j}\), so the common residual sequence is \[\lambda'_b=\lambda_{db}1_{(b,d)=1}\chi_b(r_0)^j, \qquad |\lambda'_b|\le|\lambda_{db}|.\] It is supported on \(\mathcal A_{N/q_d}\) in one class. Apply Equation (91) with lengths \(M/q_{\mathfrak r}\) and \(N/q_d\). The summed coefficient mass is at most \[\sum_d\sum_{\mathfrak r\mid\mathfrak s_Sd}\sum_b|\lambda_{db}|^2 \le d_{\mathcal O}(\mathfrak s_S) \sum_m d_{\mathcal O}(m)^2|\lambda_m|^2 \ll_{S,\delta}(2+N)^\delta\|\lambda\|_2^2.\] Writing \(\Delta_{\mathrm r}=q_{\mathfrak r}\) and \(\Delta_{\mathrm d}=q_d\), we have \(1\le\Delta_{\mathrm d}\le2N\) and \(1\le\Delta_{\mathrm r}\le q_{\mathfrak s_S}\Delta_{\mathrm d}\). We obtain the recursive interface \[ \begin{split} E_j(M,N) &\ll_{S,\varepsilon}(2+MN)^\varepsilon \max_{\substack{1\le\Delta_{\mathrm d}\le2N\\ 1\le\Delta_{\mathrm r}\le q_{\mathfrak s_S}\Delta_{\mathrm d}}} \left\{\frac M{\Delta_{\mathrm r}}\right.\\ &\qquad\left.+ \frac{M\Delta_{\mathrm d}}{N\Delta_{\mathrm r}} \max_{\substack{1\le H\\ H\ll_{S,\varepsilon}(2+MN)^\varepsilon N^2\Delta_{\mathrm r}/(M\Delta_{\mathrm d}^2)}} E_j(H,N/\Delta_{\mathrm d})\right\}. \end{split} \tag{92}\] The maxima may be restricted to nonempty ranges. This is the combination of the gcd and Poisson steps in [4]; every sequence used here is common within its norm sum, and \(S\) remains fixed.

The initial and power bounds. Row inclusion gives \(D_j\le E_j\). Hilbert space duality and sextic reciprocity, splitting one matrix variable by its finite \(t\)-class, give \[ D_j(M,N)\ll_S D_j(N,M). \tag{93}\] The phases are unit-valued and the noncoprime entries are zero in both orientations. We also have the initial estimate \[ D_j(M,N)\ll_S M^2+N\qquad(M,N\ge1). \tag{94}\] Indeed, for the squarefree row \(a\), finite Fourier inversion gives on every \(b\), including nonunits, \[\chi_a(b)^j=\frac1{\sqrt{q_a}\gamma_{-j}(a)} \sum_{x\bmod a}\overline{\chi_a(x)^j}e(xb/a).\] Cauchy costs at most \(\#(\mathcal O/(a))^\times/q_a\le1\). The reduced fractions \(x/a\) from \(a\in\mathcal A_M\) are distinct modulo \(\mathcal O\) and separated by at least \(1/(2M)\): a nonzero numerator of \(x/a-y/c-h\) has absolute value at least one, while \(|ac|\le2M\); equality modulo \(\mathcal O\) forces the same primary denominator and residue. The dual of Lemma 23, with the coefficient generators in the ball \(q_b\le2N\), proves Equation (94).

We need the following near-monotonicity, also used in the large-sieve recursions of [13] and [4]: \[ D_j(M_1,N)\ll_S D_j(M_2,N) \quad\text{if }M_2\ge C M_1\log(2M_1N),\quad M_1,M_2,N\ge1. \tag{95}\] Here is a direct verification. The assertion is immediate for a zero norm. Otherwise choose a maximizing unit coefficient vector for \(D_j(M_1,N)\), and split the row shell at \(3M_1/2\); one part carries at least half its mass. Put \(L=M_2/M_1\). For the first part use good prime ideals of norms in \((L,4L/3]\), and for the second use norms in \((2L/3,L]\). Then \(ap\in\mathcal A_{M_2}\) when \(p\nmid a\). The fixed-field prime ideal theorem supplies \(P\gg_S L/\log L\) such primes. An ideal of norm at most \(2x\) contains at most \(d_x=\log(2x)/\log(2L/3)\) of them. For \(S_a(v)=\sum_bv_bF_b(a)^j\) and \(p\nmid a\), multiplicativity gives \[S_a(\lambda)=S_{ap}\bigl((\lambda_b\overline{F_b(p)^j})_b\bigr) +S_a\bigl((\lambda_b1_{p\mid b})_b\bigr).\] The first coefficient is zero at \(p\mid b\) and cancels the factor at \(p\) otherwise. Squaring and summing over the chosen rows and primes gives \[\frac{P-d_{M_1}}2D_j(M_1,N) \le2P D_j(M_2,N)+2d_ND_j(M_1,N).\] For the stated threshold with \(C\) sufficiently large, \(d_{M_1}+4d_N\le P/2\), proving Equation (95).

We finish with the power decomposition and recursion of [4]. Suppose, simultaneously for \(j\in J\), that for every positive loss \[ D_j(M,N)\ll_{S,\varepsilon}(MN)^\varepsilon \{M^\alpha+N+(MN)^{2/3}\},\qquad \alpha>4/3, \tag{96}\] for \(M,N\ge1\); the unit shells satisfy the same bound directly. The initial estimate gives this with \(\alpha=2\). Write a good evaluation ideal uniquely as \[a=a_1a_2^2a_3^3a_4^4a_5^5a_6^6,\] where \(a_1,\ldots,a_5\) are pairwise coprime and squarefree, and \(a_6\) is arbitrary. Insert \(a_i\in\mathcal I_{X_i}\) and put \(X=\prod_iX_i\); here \(X_i\ge1/2\) and \(\prod_iX_i^i\asymp M\). Choose \(\ell\in\{1,2\}\) with \(X_\ell=\max(X_1,X_2)\). After the other factors are fixed, the coefficient \[\lambda'_b=\lambda_b\prod_{i\ne\ell}F_b(a_i)^{ij}\] has absolute value at most \(|\lambda_b|\), and the remaining kernel is exactly that of \(D_{j\ell\bmod6}(X_\ell,N)\). Enlarging its fixed positive row restriction gives this norm bound. This includes the zero of the sixth power factor. The set \(J\) is closed under \(j\mapsto j\ell\bmod6\). Counting the fixed ideals and forgetting their coprimalities in positive sums gives \[ E_j(M,N)\ll_{S,\varepsilon}(MN)^\varepsilon \max_{\boldsymbol X} \min\left\{M^{1/3},(M/X_\ell)^{2/3}\right\} D_{j\ell\bmod6}(X_\ell,N). \tag{97}\] To verify the factor, the number of fixed choices is \(O(X/X_\ell)\). For \(\ell=1\) it is at most a constant times both \((M/X_1)^{1/2}\ll(M/X_1)^{2/3}\) and \((MX_2/X_1)^{1/3}\le M^{1/3}\); for \(\ell=2\) it is at most a constant times \((MX_1^2/X_2^2)^{1/3}\le M^{1/3}\ll(M/X_2)^{2/3}\). The number of boxes is a fixed power of \(\log(2+M)\).

Substituting Equation (96) into Equation (97) yields \[E_j(M,N)\ll_{S,\varepsilon}(MN)^\varepsilon (M^\alpha+M^{1/3}N).\] The three terms before simplification are bounded by \(M^\alpha\), \(M^{1/3}N\), and \((MN)^{2/3}\); the last is bounded by one of the first two according as \(N\le M\) or \(N\ge M\). Use this estimate in Equation (92). All auxiliary nonempty scales are bounded by fixed powers of \(2+MN\): here \(N/\Delta_{\mathrm d}\ge1/2\) and \(H(N/\Delta_{\mathrm d})\ll_{S,\varepsilon}(2+MN)^{3+\varepsilon}\). The unit column shell satisfies the same estimate directly. Thus preliminary losses can be chosen smaller to give any specified final loss. The two scale factors are \[\Delta_{\mathrm r}^{\alpha-1}\Delta_{\mathrm d}^{1-2\alpha} \ll_S\Delta_{\mathrm d}^{-\alpha}\le1, \qquad (\Delta_{\mathrm r}\Delta_{\mathrm d})^{-2/3}\le1.\] Consequently, for \(M,N\ge1\), \[ D_j(M,N)\le E_j(M,N)\ll_{S,\varepsilon}(MN)^\varepsilon \{M+N^{2\alpha-1}M^{1-\alpha}+(MN)^{2/3}\}. \tag{98}\]

Put \(h=2-3/(3\alpha-1)=(2\alpha-5/3)/(\alpha-1/3)\) and \(\beta=2-2/(3\alpha-1)\). If \(M\ge N^h\), the middle term in Equation (98) is at most \((MN)^{2/3}\). If \(M<N^h\), apply Equation (95) with a fixed multiple of \(N^h\log(2MN)\) in place of \(M\). Equation (98) there gives \((MN)^\varepsilon N^\beta\), since \(2(h+1)/3=\beta\) and \(h<\beta\). Thus in both cases \[D_j(M,N)\ll_{S,\varepsilon}(MN)^\varepsilon \{M+N^\beta+(MN)^{2/3}\}.\] Symmetry gives Equation (96) with \(\beta\) in place of \(\alpha\), simultaneously for every \(j\in J\). Starting from \(2\), the map \(\alpha\mapsto2-2/(3\alpha-1)\) decreases to \(4/3\). For a requested loss take finitely many iterations until the remaining exponent gap is smaller than a fixed fraction of that loss, and absorb it into \((MN)^\varepsilon\). Combining the bound and its symmetric form, using \(M^{4/3}\le(MN)^{2/3}\) when \(M\le N\) and the analogous inequality when \(N\le M\), proves \[ D_j(M,N)\ll_{S,\varepsilon}(MN)^\varepsilon \{M+N+(MN)^{2/3}\}\qquad(j\in J). \tag{99}\]

Dyadic subdivision of both norm balls, with Cauchy across the column shells, converts Equation (99) to the same bound for the ball matrix with entries \(F_b(a)=\chi_a(b)\). The logarithmic factors are absorbed by a smaller preliminary loss, and the unit shells are bounded directly. The matrix in Equation (85) for \(\varsigma=1\) is the transpose of this matrix at lengths \((D,K)\). A complex matrix and its transpose have the same operator norm, by the adjoint identity followed by whole-vector conjugation. The bound is symmetric in the two lengths, so it gives precisely \(K+D+(KD)^{2/3}\). Conjugating the entire inner sum gives \(\varsigma=-1\), with unchanged zeros. Finally, a fixed row restriction removes nonnegative terms, and a fixed coefficient restriction is imposed by zero extension. This proves all assertions of the lemma. ◻

Proof. Fix a small preliminary loss \(\epsilon_0>0\). Apply Proposition 29 with \(t=1\) to every row in \(\mathcal B\). Subdivide by its presentation labels in \(\psi_{u,\nu,\varsigma}(n)=\nu(n)\chi_n(u)^{\varsigma}\) and its dyadic pair of polynomial lengths. There are a fixed finite number of presentation labels and \(O_{\mathcal A}((\log U)^2)\) pairs. Within one such subdivision, the inverse length \(D=U^r\) and the pair \((\nu,\varsigma)\) are fixed. In this subdivision write \[M_u(D;W)=D^{-1/2}\sum_n\mu(n)\nu(n)\chi_n(u)^{\varsigma}W(q_n/D)\] for the inverse polynomial of that presentation. The witness gives \[ \frac12-O(\epsilon_0)\le r\le1+O(1/\log U), \qquad |M_u(D;W_u)|^2\gg_{\mathcal A,\epsilon_0} U^{\delta r-\epsilon_0}. \tag{101}\] The notation \(W_u\) retains the permitted rowwise norm-profile parameters. The polynomial here is the original one for the displayed presentation, not the polynomial of its primitive inducing character. We first bound it when those parameters are fixed, and then handle their rowwise choice.

For each sixth-power-free row, factor its ideal uniquely as \[(u)=\mathfrak s\mathfrak w\mathfrak k, \quad \mathfrak s=\prod_{p\in S}p^{v_p(u)},\quad \mathfrak w=\prod_{\substack{p\notin S\\2\le v_p(u)\le5}}p^{v_p(u)}, \quad \mathfrak k=\prod_{\substack{p\notin S\\v_p(u)=1}}p.\] Thus \(\mathfrak k\) is squarefree, \((\mathfrak k,\mathfrak w)=1\), and \(\mathfrak w\) is powerful, meaning that each of its positive prime valuations is at least two. There are only finitely many possibilities for \(\mathfrak s\), because its valuations lie in \(\{0,\ldots,5\}\) on the fixed set \(S\). Choose one generator \(s_{\mathfrak s}\) for each of these ideals. Writing \(w,k\) for the primary generators of \(\mathfrak w,\mathfrak k\), there is a unique unit \(\upsilon\) such that \[u=\upsilon s_{\mathfrak s}wk.\] We fix \(\mathfrak s\) and \(\upsilon\) whenever a row sum is taken.

Insert dyadic ranges \(q_{\mathfrak w}\asymp V\), including the unit ideal in the bounded range \(V=1\). Only \(1\le V\ll_{\mathcal A}U\) can occur, and there are \(O_{\mathcal A}(\log U)\) such ranges. The original annulus \(q_u\asymp U\) gives the actual residual annulus \[ q_{\mathfrak k}=\frac{q_u}{q_{\mathfrak s}q_{\mathfrak w}} \asymp_{\mathcal A}\frac UV. \tag{102}\] We keep this restriction when forming the sum. It will be enlarged to the ball \(q_k\ll_{\mathcal A}U/V\) only when applying a positive row bound. Let \(\mathcal U_V\) denote the rows of the current witness subdivision in this powerful-part dyad, with all original physical restrictions retained.

The number of powerful ideals of norm at most \(V\) is \(O(V^{1/2})\). Indeed, every powerful ideal has a unique expression \(\mathfrak v^2\mathfrak y^3\) with \(\mathfrak y\) squarefree; the two factors need not be coprime. The ideal count from the arithmetic preliminaries gives \[\#\{\mathfrak w:q_{\mathfrak w}\le V,\ \mathfrak w\ {\rm powerful}\} \ll V^{1/2}\sum_{\mathfrak y}q_{\mathfrak y}^{-3/2} \ll V^{1/2}.\] The last series converges by the same ideal count. For each fixed \(\mathfrak w\) in its dyad, the ball containing Equation (102) has \(O_{\mathcal A}(U/V)\) ideals. A nonempty annulus has \(U/V\) bounded below by a fixed positive constant, which also covers the unit residual ideal. Since the factorization of \((u)\) and its unit are unique, the number of rows in this powerful-part dyad satisfies \[ \#\mathcal U_V\ll_{\mathcal A}U V^{-1/2}. \tag{103}\] Discarding any of the restrictions defining \(\mathcal U_V\) in this upper count can only add rows.

For fixed \(\mathfrak w,\mathfrak s,\upsilon\), put \(v=\upsilon s_{\mathfrak s}w\). Multiplicativity in the numerator gives the exact identity \[ \chi_n(u)^{\varsigma} =\chi_n(v)^{\varsigma}\chi_n(k)^{\varsigma}. \tag{104}\] It includes the zero extensions: if \(n\) meets either factor, both sides are zero, and otherwise it is ordinary multiplicativity. In particular, the inverse polynomial for a fixed profile \(W\) is the row sum in Lemma 30 with coefficients \[c_n=D^{-1/2}\mu(n)\nu(n)\chi_n(v)^{\varsigma}W(q_n/D).\] The factor \(\mu(n)\) keeps the columns squarefree. The factor \(\chi_n(v)^{\varsigma}\) retains every original zero at \((n,v)>1\) and is fixed across the residual row \(k\). The only remaining zero depending on both \(n\) and \(k\) is the zero of the displayed sieve kernel itself. The restriction \((k,w)=1\), and any restriction inherited from the physical row set for fixed external parameters, is a fixed row restriction in this application. We do not enlarge \(S\) to contain the primes of \(w\). Consequently the constant in the sieve is independent of \(w\), even though its fixed coefficient mask may have large norm.

The common annular support and ideal counting give, uniformly for the fixed profile parameters, \[\sum_n|c_n|^2 \le D^{-1}\sum_{q_n\asymp D}|W(q_n/D)|^2\ll_{\mathcal A}1.\] The same statement holds for each of the finitely many profile derivatives used below, with the corresponding fixed seminorm. Apply Lemma 30 after enlarging the positive residual row sum from its actual annulus to \(q_k\ll_{\mathcal A}U/V\). A fixed enlargement makes the row bound at least one on every nonempty quotient range and includes the annular column support. Summing the result over the \(O(V^{1/2})\) choices of \(\mathfrak w\) and the finitely many choices of \(\mathfrak s,\upsilon\) yields \[ \sum_{u\in\mathcal U_V}|M_u(D;W)|^2 \ll_{\mathcal A,\epsilon_1}(UD)^{\epsilon_1}V^{1/2} \left\{\frac UV+D+\left(\frac{UD}{V}\right)^{2/3}\right\}. \tag{105}\] All coefficient sequences in this derivation are fixed within the row sum to which the sieve is applied. They may differ for different frozen values of \(\mathfrak w\), which are summed only after those positive bounds.

We now restore the rowwise choice of \(W_u\) in Equation (101). Here the detector was used with \(t=1\), so \(D_*=U\). After the presentation and dyadic pair are fixed, its inverse profiles have the form \[W_{\rm dyad}(x)V_{\le}(Dx/U)x^{-\sigma-i\omega}, \qquad \omega=\gamma-\nu_F,\] where \(W_{\rm dyad}\) is the fixed smooth dyadic cutoff and \(\nu_F\) is the Fourier frequency from the detector. The factor \(V_{\le}(Dx/U)\) is common to the row sum; the varying parameters are \(\sigma\in[51/100,1]\) and \(|\omega|\le(3i+1)T_1\). Differentiating in either parameter inserts a power of \(\log x\), which is bounded on the fixed annulus. Cover these two parameter ranges by unit boxes. There are at most a fixed power of \(1+T_1\) such boxes. The parameter Sobolev inequality in Lemma 9 bounds the supremum on a box by a finite sum of integrals of squared profile derivatives. Sum over the rows before these integrals. At each fixed parameter value, differentiation changes only the fixed profile coefficient, inserting the allowed logarithmic weights or profile derivatives. Equation (105) therefore applies to every such integral. This proves \[ \sum_{u\in\mathcal U_V}|M_u(D;W_u)|^2 \ll_{\mathcal A,\epsilon_1}(UD)^{\epsilon_1}(1+T_1)^{A_{\mathcal A}} V^{1/2}\left\{\frac UV+D+ \left(\frac{UD}{V}\right)^{2/3}\right\}. \tag{106}\] The derivative order and hence \(A_{\mathcal A}\) are fixed before choosing \(\tau\). The height intervals and the total allowance for additional frequencies are exactly those in the shared detector estimates; the Sobolev step does not assign a new frequency allowance or change a contour. It does not permit a coefficient to be selected separately for each \(k\).

It remains to combine this moment with the independent count Equation (103). Write \[V=U^o,\qquad D=U^r,\qquad e_o(r)=\frac o2+\max\left\{1-o,r,\frac{2(1-o+r)}3\right\}.\] These are the parameters of the actual dyads. A nonempty dyad has \(0\le o\le1+O_{\mathcal A}(1/\log U)\); the unit dyad has \(o=0\). Since the witness lengths stay bounded, the arbitrarily small power of \(UD\) in Equation (106) can be written as an arbitrarily small power of \(U\). Dividing that equation by the spike in Equation (101), and also using Equation (103), gives for this subdivision \[ \#\mathcal U_V \ll U^{\min\{1-o/2,e_o(r)-\delta r\}+O(\epsilon_0)+\epsilon_1} (1+T_1)^{A_{\mathcal A}}, \tag{107}\] after decreasing the sieve loss denoted by \(\epsilon_1\) if necessary.

We compute the largest exponent in this expression first on \(0\le o\le1\) and \(1/2\le r\le1\). Put \(x=1-o\) and \(A=(1+x)/2\). Then \[e_o(r)-\delta r =\max\left\{ A-\delta r,\quad \frac{1-x}{2}+(1-\delta)r,\quad \frac12+\frac x6+\left(\frac23-\delta\right)r \right\}.\] The minimum of \(A\) with a maximum of three real numbers is the maximum of its minima with those numbers. The first such minimum is \(A-\delta r\le1-\delta/2\). For fixed \(r\), the second is the minimum of an increasing and a decreasing affine function of \(x\). They cross at \(x=(1-\delta)r\in[0,1]\), so its maximum over \(x\) is \[\frac{1+(1-\delta)r}{2}\le1-\frac\delta2.\] Both functions in the third minimum increase with \(x\). Its maximum over \(x\) is therefore \[\min\left\{1,\frac23+\left(\frac23-\delta\right)r\right\}.\] The affine expression in \(r\) takes the values \(1-\delta/2\) and \(4/3-\delta\) at the two endpoints. It follows, and equality is attained at one of those endpoints, that \[ \max_{\substack{0\le o\le1\\1/2\le r\le1}} \min\{1-o/2,e_o(r)-\delta r\} =\min\left\{1,\max\left(1-\frac\delta2,\frac43-\delta\right)\right\}. \tag{108}\] This computation is uniform for \(0\le\delta\le1\).

The functions just used are maxima and minima of affine functions whose slopes are uniformly bounded on a fixed neighborhood of these compact ranges. Replacing the actual \(o\) by its nearest point in \([0,1]\), and the actual witness \(r\) by its nearest point in \([1/2,1]\), changes the exponent by at most \(O_{\mathcal A}(\epsilon_0+1/\log U)\), by Equation (101) and the actual dyad bounds. Thus this replacement is made only in the final optimization, not in the row sum or its sieve application. Choose \(\epsilon_0\) and the preliminary sieve loss small enough in terms of the prescribed \(\epsilon\). The \(O((\log U)^2)\) witness pairs and \(O(\log U)\) powerful-part dyads cost only another arbitrarily small power. Summing Equation (107) proves Equation (100).

Finally, the bin \(a=51/100\) need not contain an actual zero, so no use of Proposition 29 is made there. Counting all elements in its physical annulus gives \(O_{\mathcal A}(U)\) rows. Since \(1/50<1/3\), the displayed formula for \(R\) gives \(R(1/50)=1\), as claimed. ◻

Proof. Multiply the kernel bound by Equation (114). Integrating \((1+|\boldsymbol t|)^{J-L}\) outside the box proves Equation (115) when \(L>N+J+m\). On \(t_j=T\), integration in the other \(m-1\) coordinates gives \[\int_{\mathbb R^{m-1}} (1+|T|+|\widehat{\boldsymbol t}_j|)^{J-L} d\widehat{\boldsymbol t}_j \ll(1+|T|)^{J-L+m-1}.\] Taking \(L>N+J+m-1\) proves Equation (116). Restricting to \(\mathcal D\) decreases these positive integrals, and a bounded real interval contributes only its length. ◻

Proof. Section 6 gives \(a\le\beta_*\), including the floor bin. Isolate the finite sum over \(\mathcal B\) on the absolute lines in Equation (113). Move \(z\) from \(2\) to \(z_0\), keep \(w=3\), and move \(s\) from \(3\) to \(\beta_*+20e\). These moves stay in \(\mathcal D_1(\epsilon_0)\) for a fixed positive \(\epsilon_0\), the two numerator arguments remain to the right of their poles, and the reciprocal stays in \(\Re s>\beta_*\). Its global bound follows from Lemmas 13 and 14, using the conductor and deletion radical bounds in Section 6. The correction obeys Equation (112). Lemma 33 with \(T\to\infty\) therefore justifies these moves.

Now move \(w\) from \(3\) to \(1-a-6e\) while \(s\) stays on that global line. Throughout the move, \[\Re(s+w)\ge1+(\beta_*-a)+14e\ge1+14e.\] The numerator character of every retained row is nonprincipal, so its \(L\)-function is entire. Also \(\Re w\ge-6e>-1/100\) and \(\Re z=z_0\). Thus the correction remains holomorphic in \(\mathcal D_1(\epsilon_0)\), for example with \(\epsilon_0=10e\), and no pole is crossed. On a \(w\)-join the reciprocal is still global. The global upper strip bound for the numerator, the deleted-factor bound and Equation (112) give a majorant of the form Equation (114) on the axes integrated there. The trace estimate makes the joins tend to zero.

Before moving \(s\) farther, restrict the three original Mellin heights to the stated box. On the discarded part keep \(s\) on \(\Re s=\beta_*+20e\). There the global reciprocal bound, the global numerator bound, the deleted factors and the all-height correction bound give \(Z^{B_\eta}\) times a fixed polynomial in the heights: the row range is bounded by \(q_u\ll Z^{d_{\max}}\), and the number of rows is \(O(U)\). The integrated tail in Lemma 33 gives \(O_{\eta,N}(Z^{B_\eta}T_1^{-N})\).

Move only the retained \(s\) segment to \(a+16e\). On this rectangle, \[\Re(s+w)\ge1+10e,\] and the other inequalities defining \(\mathcal D_1(10e)\) still hold. The real part of the reciprocal argument is at least \(a+16e>a+6e\). The stated height box places both scalar arguments strictly inside \(|\Im v|\le(3i+2)T_1\) for sufficiently large \(Z\). Lemma 27 therefore excludes its zeros throughout the retained rectangle. Since \(6z_0>1\), the scalar zeta factor has no pole on this move. Thus no pole is crossed.

For an \(s\)-join, its imaginary coordinate is fixed at a constant multiple of \(T_1\). The scalar denominator in Equation (113) depends only on \(s\), so its buffered estimate remains valid when the \(\Im z\) and \(\Im w\) integrations are extended to their whole axes. On those extended axes use the global numerator estimate and Equation (112). The trace estimate then gives \(O_{\eta,N}(Z^{B_\eta}T_1^{-N})\). No \(w\)- or \(z\)-join in this argument extends an \(s\)-axis that has been moved into a merely buffered region. ◻

Proof. Apply Lemma 34 to the original triple integral, using for its three box constants the corresponding \(c_k\) in Equation (122). These constants are positive and at most \(1/2\). The full correction and the fixed bin therefore give the retained integral and the asserted errors before any pointwise pieces are introduced. Any auxiliary integrations used for the central bound are subject to the additional hypotheses in the statement.

Now use the central decomposition and form the pointwise sets \(\mathcal B_{\lambda,\nu}\). At each retained point apply the triangle inequality to the original row sum, and then apply Equation (121) to the sets present at that point. There are at most a fixed multiple of \((1+\log Z)^D\) such sets pointwise. Bound each of their exponents by the supremum over \(\mathcal P_Z\) and integrate only the original holomorphic row sum. Thus neither measurability of an individual pointwise subdivision nor one common label set across contour points is needed. On the retained contours, Lemma 27 bounds the reciprocal by \(O_{\eta,e,\varepsilon_d}(U^{\varepsilon_d})\). The tests have a uniformly bounded joint \(L^1\) norm, since the transformation \((\Im s,\Im z,\Im w)\mapsto(\Im(s+z),\Im z,\Im w)\) is invertible. The scalar factor \(\zeta_F^S(6z_0)\) is absolutely bounded. It remains to compute the real power of \(Z\).

The outside factor, including \(q_u^{-z_0}\) and the real displacements in Equation (119), contributes \[l_x(1/2-z_0)+a+z_0-1-a l_y-dz_0+(16-6l_y)e.\] Equation (121) adds \(d(R+\delta/2+\varepsilon_c)+\ell(z_0-1/2)+g+\varepsilon_c\). The reciprocal adds \(d\varepsilon_d\). Subtracting \(C(\sigma_0)=\sigma_0+l_x/2-1+h/6\) and using \(h=1-l_x+\ell\) gives Equation (123) and the remaining terms in Equation (124). The pointwise logarithmic multiplicity is bounded by \(Z^{\varepsilon_p}\) for sufficiently large \(Z\). This proves the claim. ◻

Proof. For \(u=1\), the scalar factor in the high identity is \[\frac{\zeta_F^S(6z)\zeta_F^S(w)}{L_F^S(s,\eta)}.\] Isolate this term on the absolute contours. Move to \(\Re s=\beta_*+e\), \(\Re w=1+e\) and \(\Re z=1/6+e\). The reciprocal stays in \(\Re s>\beta_*\), and both zeta arguments stay to the right of one. The paths lie in \(\mathcal D_2\), so the correction is holomorphic. The all-height majorant and Lemma 33 justify the horizontal limits.

Move \(w\) to \(19/20\), crossing its simple pole at \(w=1\). In that residue move \(z\) to \(33/200=1/6-1/600\), crossing its simple pole at \(z=1/6\). The unresidued \(w\) integral keeps \(\Re z=1/6+e\). All these paths lie in \(\mathcal D_2\), and \(M(z)\) is holomorphic for \(\Re z>0\). Thus the two stated scalar poles are the only poles crossed. The reciprocal remains on its global line throughout, even when \(\eta\) is principal. Every extended vertical \(w\)- or \(z\)-axis used to bound a tail lies strictly on one side of its scalar pole; \(w=1\) and \(z=1/6\) occur only as residues, and the horizontal joins have large nonzero height. On those axes the zeta functions have fixed polynomial bounds. Consequently the integrated and trace estimates of Lemma 33 apply without an unremoved pole in their majorants. After the \(w\) residue, its Mellin factor has become the constant \(\widehat W_1(1)\). The remaining heights use the two-dimensional kernel in Equation (117), with \((y_1,y_2)=(\Im(s+z),\Im z)\); its trace estimate justifies the subsequent \(z\)-join.

On the principal contours, Equation (125) and the outside powers give the raw exponent \[\frac{l_x}{2}+\Re s-1+h\Re z+l_y(\Re w-1).\] For the unresidued \(w\) integral this is \(C(\beta_*)+(1+h)e-l_y/20\); for the leftover \(z\) integral in the \(w\) residue it is \(C(\beta_*)+e-h/600\). The reciprocal on \(\Re s=\beta_*+e\) has an arbitrarily small power of its fixed target conductor and a fixed polynomial in height by Lemmas 13 and 14. The tests integrate those height powers. Requesting the small powers in the correction and in \(A_\eta^{-1}\) to be sufficiently small gives the first two terms of Equation (127).

The product of the scalar residues is \(c_S\): the residue of \(\zeta_F^S(6z)\) at \(1/6\) is one sixth of the residue of \(\zeta_F^S(v)\) at one. Positivity of \(c_S\) was proved when the normalization was defined.

At the double residue the outside power is \[X^{1/3}Z^{s-5/6}Z^{\ell/6}=Z^{C(s)}, \qquad \Phi(s+1/6-1)=e^{(s-5/6)^2}.\] Equation (126) therefore makes the normalized double residue \[\frac1{2\pi i}\int_{(\beta_*+e)} Z^{C(s)}e^{(s-5/6)^2}\frac{H_\eta(s)}{L_F^S(s,\eta)} (1+\mathcal R_{\eta,Z}(s))\,ds.\] When present, the error factor is estimated on this line; the global reciprocal bound and Gaussian give \(O_{\eta,e,\epsilon}(Z^{C(\beta_*)+e-\mu+\epsilon})\). No continuation of that error factor is required.

Move only the main integral right to \(\Re s=2\). The reciprocal is holomorphic for \(\Re s>\beta_*\) by the definition of \(\beta_*\) and absolute Euler convergence beyond one. At a principal pole its reciprocal has a zero, so that case adds no residue. The function \(H_\eta\) is holomorphic there by Lemma 26, and is bounded there by Equation (110). The global reciprocal estimate is polynomial in height uniformly on the fixed real strip, while the Gaussian is \(O(e^{-(\Im s)^2})\). Rectangular contours therefore have vanishing horizontal sides, and the shifted integral is exactly \(f_\eta(Z)\). This also covers \(\beta_*=1\). The three remainders give the asserted bound. ◻

Proof. Every row under consideration has nonprincipal numerator by the classification preceding Lemma 36. The denominator may be principal for a bounded unit row; its reciprocal is nevertheless holomorphic and bounded on \(\Re s=\beta_*+e\) by the principal specialization of Lemma 13. For all rows, the conductor and deletion radical bounds in Section 6, together with Lemma 14, give an arbitrarily small power of \(q_u\) and a fixed height polynomial for that reciprocal.

For a small dyad move its finite sum to \((\Re s,\Re w,\Re z)=(\beta_*+e,1/2,z_0)\). The reciprocal stays global. The numerator is entire, and the scalar zeta argument stays to the right of one. These paths may be taken in \(\mathcal D_1(3/8)\): at the final point \(\Re(s+w)=\beta_*+e+1/2>1+3/8\), and the other inequalities are immediate. The all-height correction bound and Lemma 33 justify the moves. On the final lines, Equation (21) and the deletion bound give \(U^{3/5+\epsilon}\) times a fixed height polynomial for the nonprincipal numerator. There are \(O(U)\) element rows in the dyad, including unit factors, and \(q_u^{-z_0}\ll U^{-z_0}\). The correction is bounded by Equation (128). The tests integrate all fixed height powers. The resulting exponent outside the row power, relative to \(C(\beta_*)\), is \[h(z_0-1/6)-l_y/2+e,\] and the row power is \(1+3/5-z_0=63/50\). Requesting the component small powers to sum to the displayed \(\epsilon\) proves Equation (129). The row exponent is positive, so dyadic summation up to \(Z^{d_{\min}}\) gives Equation (130), after decreasing the preliminary losses.

For each fixed \(Z\) and large dyad, move its finite row sum to the absolute lines \((2,2,v)\). The path from \((3,3,2)\) remains in \(\Re s,\Re w\ge2\) and \(\Re z\ge2\), so the scalar factors are holomorphic there. For this fixed dyad choose \(D\) large enough to contain its rows in \(q_u\le Z^D\); Equation (112) and the external trace bound make its horizontal limits vanish. Constants used only to justify this equality may depend on the fixed dyad. The estimate on the final lines is uniform in the dyad by Equation (131). The scalar \(L\)-factors are absolutely bounded there, as is \(\zeta_F^S(6z)\), and Equation (131) applies. The outside power, including the correction but not the row count, is \[l_x(1/2-v)+1+v+l_y+\ell v=B_0+hv.\] The \(O(U)\) rows and \(q_u^{-v}\) give \(U^{1-v}\); the tests integrate the fixed height polynomial. This proves Equation (132). Because \(1+\epsilon-v<0\), summing its geometric dyadic tail gives Equation (133). For fixed \(\zeta>0\), the coefficient of \(v\) in that exponent is \(-\zeta\), proving the last assertion. The original row series is absolutely convergent on its starting lines, and the displayed bounds give an absolutely summable final tail, so the individual dyadic contour identities may be summed. The number \(v\) and the finite test orders it requires are fixed before \(Z\) tends to infinity. ◻

Proof. For the fixed target choose \[0<\tau_\eta\le \min\left\{\frac{d_{\min}}{100},\tau_{0,\eta}, \frac{m}{4(A_\eta+1)}\right\}, \qquad T_1=Z^{\tau_\eta}.\] Then \((1+T_1)^{A_\eta}\ll_\eta Z^{m/4}\), so the first term in Equation (134) is \(O_\eta(Z^{C(\beta_*)-3m/4})\). Next choose a fixed integer \(N\) such that \[B_\eta-N\tau_\eta<C(\beta_*)-m/2.\] This is possible because \(\tau_\eta>0\) and \(B_\eta\) is finite. The second term is then bounded by the required power. Increase the lower threshold for \(Z\) after these choices. Both \(m\) and \(\omega\) were fixed before the target, whereas \(\tau_\eta,N\) and the threshold may depend on it. Since \(J_\eta\) and \(f_\eta\) do not contain \(T_1\), this proves the asserted family-wide exponent without changing either function. ◻

Proof. Use \(l_x=l_y=h=1/2\), \(\ell=0\) and \(C=C_{\mathrm I}\). The exact identity and local correction were verified at the start of Section 8; Equation (120) estimates its intermediate rows, and the principal and outer specializations there give the margins listed above. It remains to choose their losses.

Proof. Passing from a primitive Hecke character to one that it induces changes only finitely many factors \(1-\eta(\mathfrak p)N\mathfrak p^{-s}\). They are nonzero for \(\Re s>0\), so the Hecke assertion extends to all finite-order characters. The same observation for factors \(1-\chi(p)p^{-s}\) reduces the Dirichlet assertion to a primitive Dirichlet character \(\chi\) of conductor \(q\).

Let \(\chi_{-3}\) be the quadratic character of conductor three, and let \(\eta\) be the finite-order Hecke character given by \(\mathfrak a\mapsto\chi(N\mathfrak a)\) on ideals coprime to \(3q\). It is a ray character: if \(\alpha\equiv1\pmod{q\mathcal O}\), then \(N\alpha\equiv1\pmod q\), so the norm character is trivial on the corresponding principal ray subgroup. Let \(S_{\mathbb Q}\) be the rational primes dividing \(3q\), and let \(\mathcal S_{\mathbb Q}\) be the primes of \(F\) above them. Superscripts by these sets denote deletion of those Euler factors.

For \(p\notin S_{\mathbb Q}\), the local factors agree as follows. If \(p\) splits in \(F\), then \(\chi_{-3}(p)=1\), the two prime ideals have norm \(p\), and the Hecke factor is \((1-\chi(p)p^{-s})^{-2}\). This is the product of the Dirichlet factors for \(\chi\) and \(\chi\chi_{-3}\). If \(p\) is inert, then \(\chi_{-3}(p)=-1\), the unique prime ideal has norm \(p^2\), and its factor is \[(1-\chi(p)^2p^{-2s})^{-1} =(1-\chi(p)p^{-s})^{-1}(1+\chi(p)p^{-s})^{-1}.\] These exhaust the primes outside \(S_{\mathbb Q}\). Absolute Euler convergence for \(\Re s>1\), followed by uniqueness of meromorphic continuation, gives \[ L_F^{\mathcal S_{\mathbb Q}}(s,\eta) =L^{S_{\mathbb Q}}(s,\chi) L^{S_{\mathbb Q}}(s,\chi\chi_{-3}). \tag{135}\] The product character on the right may be imprimitive; deletion of \(S_{\mathbb Q}\) makes the identity independent of that choice. Every deleted factor is nonzero for \(\Re s>0\).

Both Dirichlet factors are holomorphic in \(0<\Re s<1\). A zero of \(L(s,\chi)\) there would therefore give a zero of the Hecke factor, with no cancellation by a pole, and the assumed Hecke half-plane excludes it when \(\Re s>\sigma_0\). Absolute Euler convergence handles \(\Re s>1\). On \(s=1+it\) with \(t\ne0\), neither Dirichlet factor has a pole, so the same product argument applies.

At \(s=1\), a pole-zero cancellation could occur only if one primitive inducing character among \(\chi\) and \(\chi\chi_{-3}\) were principal. The other would then be \(\chi_{-3}\). Its Dirichlet series converges at one by bounded partial sums of the nonprincipal periodic character, and grouping consecutive terms gives \[\begin{split} L(1,\chi_{-3}) &=\sum_{n\ge0}\left(\frac1{3n+1}-\frac1{3n+2}\right)\\ &=\int_0^1\frac{1-x}{1-x^3}\,dx =\int_0^1\frac{dx}{1+x+x^2} =\frac{\pi}{3\sqrt3}>0. \end{split}\] The first integral follows by monotone convergence of the nonnegative paired integrands. Thus there is no zero in this last case either; a principal pole is allowed. This proves the strict half-plane assertion with no claim on its boundary. ◻

Proof of Theorem 4. Suppose that \(\beta_*>11/12\). Lemma 26 and Equation (110) give the required holomorphic, nonzero \(H_\eta\) on \(\Re s>11/12\) for every primitive target. The low and high bounds established in Proposition 39 and its proof verify Proposition 3 with \[\sigma_0=\frac{11}{12},\qquad C=C_{\mathrm I},\qquad \omega=\frac{\Delta_1}{2},\qquad \sigma=\frac1{4800}.\] These two positive losses are independent of the target, while the allowed constants, excluded sets, and thresholds may depend on it. The continuation criterion contradicts the supposition. Therefore \(\beta_*\le11/12\). By its definition and absolute Euler convergence in \(\Re s>1\), every primitive finite-order Hecke \(L\)-function over \(F\) is zero-free in the strict half-plane \(\Re s>11/12\), with the principal pole allowed.

Proposition 40, applied at the same boundary, extends this assertion to all finite-order Hecke characters and all Dirichlet \(L\)-functions, including \(\zeta(s)\). It preserves the strict half-plane and the allowed principal pole. This proves the theorem. ◻

Proof. For the fixed abelian extension of \(F\) corresponding to \(T\), the prime ideal theorem in a fixed Frobenius class gives \[\pi_{1_T}(x):=\#\{p\in1_T:q_p\le x\} \sim\frac{\operatorname{Li}(x)}{|T|}.\] This is the fixed-extension consequence of Chebotarev in [27]; the extension and its conductor are fixed as \(x\to\infty\). If \(\operatorname{supp}W\subset[a,b]\subset(0,\infty)\), the error \(o(x/\log x)\) is uniform for \(aP\le x\le bP\) as \(P\to\infty\). Stieltjes integration by parts therefore changes the left side of Equation (143) by \(o(P^{1/6}/\log P)\) when \(d\pi_{1_T}(x)\) is replaced by \(dx/(|T|\log x)\). The resulting integral is \[\frac{P^{1/6}}{|T|\log P} \int_a^b W(y)y^{-5/6}\frac{\log P}{\log(Py)}\,dy.\] The last ratio tends uniformly to one. This proves the formula and its positivity. A fixed finite set is eventually outside the annular window. No error exponent uniform in the ray conductor is used. ◻

Proof. Write a residue modulo \(p^a\) as \(x_0+py\), with \(x_0\bmod p\) and \(y\bmod p^{a-1}\). The sum over \(y\) is zero unless \(p^{a-1}\mid k\), and equals \(P^{a-1}\) otherwise. In the latter case write \(k=p^{a-1}k_0\). The remaining normalized sum is \[P^{a/2-1}\sum_{x_0\bmod p}\chi_p(x_0)^a e(k_0x_0/p).\] If \(6\nmid a\), its inner sum vanishes when \(p\mid k_0\) and otherwise has absolute value \(\sqrt P\). If \(6\mid a\), the inner sum is the sum of the additive character over the units, namely \(P1_{p\mid k_0}-1\). These are exactly the two cases in the statement. ◻

Proof. Expanding both Gauss sums in the displayed Fourier transform leaves \[\frac1{q_uq_v}\sum_{x\bmod u,y\bmod v}\chi_u(x)\overline{\chi_v(y)} \sum_{h\bmod uv}e\bigl(h(vx-uy-j)/(uv)\bigr).\] Additive orthogonality makes the inner sum \(q_uq_v\) when the congruence holds and zero otherwise. If \(j=0\) and a summand is nonzero, \(x\) and \(y\) are units modulo \(u\) and \(v\). Reducing the congruence modulo \(u\) gives \(u\mid v\), and reducing modulo \(v\) gives \(v\mid u\). Since the generators are primary, \(u=v\). The congruence then says \(x=y\bmod u\) and gives \(\varphi(u)\). This argument uses the zero masks also when a local character power is principal.

The divisibility by \(C\) is immediate. After division by \(C\), the congruence for \(j=Ck\) is \[n_2x-n_1y\equiv k\pmod{Cn_1n_2}.\] Reduction modulo \(n_1\) and \(n_2\) gives, with zero values retained, \[\chi_{n_1}(x)=\chi_{n_1}(k)\overline{\chi_{n_1}(n_2)},\qquad \overline{\chi_{n_2}(y)} =\overline{\chi_{n_2}(-k)}\chi_{n_2}(n_1).\] The product of the two unit factors is \(\mathcal R(n_1,n_2)\). It remains to identify the multiplicity of lifts of the congruence modulo \(C\). This can be checked at each prime. If \(c=v_p(C)\) and neither residual modulus contains \(p\), there is no additional lift. If, say, \(d=v_p(n_1)>0\) and \(p\nmid n_2\), then \(y\bmod p^c\) is free and the congruence uniquely determines \(x\bmod p^{c+d}\) from it. Reduction modulo \(p^c\) is exactly the common congruence in Equation (147). The case \(p\mid n_2\) is symmetric, and this argument includes \(c=0\). The Chinese remainder theorem thus gives a bijection with the common solutions used in \(L_C\), even when \(C\) meets one residual modulus.

For the local calculation put \(k_p=\mathcal O/(p)\) and \(A=\chi_p^c\), a character of \(k_p^\times\) extended by zero. Since at least one of \(n_1,n_2\) is a unit at \(p\), each solution modulo \(p\) has \(P^{c-1}\) lifts modulo \(p^c\). If both are units and \(p\mid k\), the unit variables are proportional and the field sum is \(A(n_1/n_2)(P-1)\). If \(p\nmid n_1n_2k\), put \(y=(k/n_1)t\) and \(x=(k/n_2)(1+t)\). The field sum becomes \[A(n_1/n_2)\sum_{t\ne0,-1}A((1+t)/t) =A(n_1/n_2)\sum_{z\in k_p^\times\setminus\{1\}}A(z).\] It is \(-A(n_1/n_2)\) for nonprincipal \(A\) and \((P-2)A(n_1/n_2)\) for principal \(A\). If \(p\mid n_1\) and \(p\nmid n_2\), the field equation forces \(x=k/n_2\). For \(p\mid k\) its character is zero. For \(p\nmid k\) the remaining sum over \(y\) is zero unless \(A\) is principal, in which case it is \(P-1\). The other case is symmetric. This proves all local formulas on the genuine residual locus. Extending them by the stipulated zero when both residuals meet \(p\) gives functions of the residual columns modulo \(p\), each of absolute value at most \(P^c\). Their product proves the final assertions for the artificial extension as well; no congruence-sum identity is asserted at a newly added pair. ◻

Proof. At primes of \(a\) and \(b\), solving the congruence in Equation (146) gives respectively \(\chi_a(j)\overline{\chi_a(Eb)}\) and \(\overline{\chi_b(-j)}\chi_b(Da)\). These statements remain valid when a character of \(j\) is zero; only the displayed unit factors are inverted. At primes of \(DE\) change variables \(x'=bx\bmod D\), \(y'=ay\bmod E\). The congruence becomes \(Ex'-Dy'\equiv j\pmod{DE}\) and its character factor changes by \(\overline{\chi_D(b)}\chi_E(a)\). The remaining unit factor is \[\frac{\chi_E(a)}{\chi_a(E)} \frac{\chi_b(D)}{\chi_D(b)} \frac{\chi_b(a)}{\chi_a(b)} =\mathcal R(a,E)\overline{\mathcal R(b,D)}\mathcal R(a,b).\] All denominators here are symbols of units by the hypotheses. The Chinese remainder theorem and the definition of \(F(D,E;j)\) complete the proof. ◻

Proof. For every good prime and every \(A\in\mathcal O\), the zero convention gives the pointwise identity \(1_{q\mid A}=1-\chi_q(A)^0\). Expand its product over \(\mathcal Q\), and apply Proposition 15 to each of the resulting finitely many completed sums. For a fixed \(h_0\) and active radical, the Proposition uses the same \(c_F,c,d,\vartheta,\kappa_F\), other local phases, and kernel whether an absent prime is omitted from the local set or included with exponent zero in its inactive branch. Thus the two inactive contributions at \(q\) combine with coefficient \(1-(1-q_q^{-1})=q_q^{-1}\). The active contribution comes only from the subtracted zero-mask transform and has local factor \(q_q^{-1/2}\chi_q(\lambda^4\mu)^{-2}\) and scalar \(-\omega_{q,0}=\tau_{q,2}^{+}\chi_q(\epsilon_q)^{-2}\). Collecting these choices at all marked primes proves Equations (152) and (153). The inactive prime is absent from \(r\), hence from that branch’s conductor and kernel scale. This is finite inclusion-exclusion between compatible branches, not a new theta identity.

If a marked prime is a base local prime, the mark forces it to divide \(nb^3\), while the zero-extended base factor at that prime then vanishes, including for exponent zero. This proves the last assertion without any division of a zero symbol. It also covers the case where the primal squarefree and cube ideals share the marked prime. ◻

Proof. Write \(\mathcal H\) for the left side of Equation (156). First remove the complete moving row-mark mask: \[1_{(P,k)=1}=\sum_{D\mid(P,k)}\mu(D).\] For each row the number of such divisors is at most \(d_{\mathcal O}(k)\). Rowwise divisor Cauchy, followed by the positive sum over rows, therefore costs a permitted factor \((KNBL)^\epsilon\). In a fixed \(D\)-summand write \(P=DP'\) and \(k=Dk'\). Squarefreeness gives the fixed restrictions \((D,P')=(D,k')=1\). Define \[\begin{split} a_D(P')&=a(DP')1_{(D,P')=1},\\ \beta_D(n,b)&=\beta(n,b)\chi_D(nb)^3\chi_D(n)^{-2}1_{(D,b)=1}. \end{split}\] All these factors retain their zeros and \(|\beta_D|\le1\). Factoring \(q_P^{-1/2}=q_D^{-1/2}q_{P'}^{-1/2}\) gives \[ \begin{split} \mathcal H\ll (KNBL)^\epsilon\sum_D\frac1{q_D} \sum_{k'}\left| \sum_{n,b}\beta_D(n,b)\chi_{k'}(nb)^3 \sum_{P'}\frac{a_D(P')}{\sqrt{q_{P'}}} \chi_{P'}(n)^{-2}1_{(P',b)=1} \right|^2. \end{split} \tag{157}\] Here \(q_{k'}\ll K/q_D\), \(q_{P'}\asymp L/q_D\), and all inherited fixed restrictions are retained. The outer \(D\)’s are the squarefree good divisors allowed by the split, so \(q_D\ll\min(K,L)\). In particular, no \((P',k')\) mask is reinstated inside an individual \(D\)-summand. Its cancellation belongs to the complete Möbius sum already bounded by rowwise Cauchy.

For fixed \(D\), the product of \(\beta_D(n,b)\) and the inner \(P'\)-sum is an arbitrary complex coefficient independent of \(k'\). Apply Equation (49) with row length \(K/q_D\). Use its notation \[b=gt^2,\qquad c=(n,g),\qquad n=cm,\qquad g=ch, \qquad q_c\asymp C,\quad q_g\asymp G,\quad q_t\asymp T,\] so \(B\asymp GT^2\), and write \(c=rc'\) for its complete \(c\)-mask divisor. The ideals \(c,m,h\) are pairwise coprime and squarefree; no coprimality between \(g\) and \(t\) is imposed. The cited reduction already includes the natural \(k'\)-\(t\) restriction and the complete Möbius expansion of the \(k'\)-\(c\) mask. It does not reinstate a mask between the remaining row and \(c'\).

For one \(C,G,T\) block its positive output, inserted in Equation (157), is at most \[ \begin{split} (KNBL)^\epsilon\sum_D\frac1{q_D}T\sum_{q_t\asymp T} \sum_{\substack{r\ {\rm sf},\ (r,S)=1\\q_r\ll\min(K/q_D,C)}} &\left\{\frac K{q_Dq_r}+\frac{NG}{C^2}\right\}\frac C{q_r} \sum_{c',m,h}|\beta_D(rc'm,rc'ht^2)|^2\\ &\quad\cdot\left| \sum_{P'}\frac{a_{D,r}(P')}{\sqrt{q_{P'}}} \chi_{P'}(c'm)^{-2}1_{(P',ht)=1} \right|^2, \end{split} \tag{158}\] where the positive \(c',m,h\)-sum keeps the original support and \[a_{D,r}(P')=a_D(P')\chi_{P'}(r)^{-2}.\] The factorization used here is, including all zeros, \[\chi_{P'}(rc'm)^{-2}1_{(P',rc'ht^2)=1} =\chi_{P'}(r)^{-2}\chi_{P'}(c'm)^{-2}1_{(P',ht)=1}.\] The character already supplies the missing zeros at \(r,c'\), and \(m\). Thus the \(r\)-factor is tuple-only, including its zero when \((P',r)\ne1\). The fixed \(D\)-phases remain inside \(\beta_D\) until this positive bound. The quadratic reduction has already separated the finitely many squarefree \(S\)-parts of \(m,h\) before sieving their good product; \(m,h\) in Equation (158) are the original ideals.

We may now discard \(|\beta_D|^2\le1\). For fixed \(h,t,D,r\), the \(P'\)-coefficients are independent of \(j=c'm\), and ideal counting gives \[\sum_{P'}\frac{|a_{D,r}(P')|^2\,1_{(P',ht)=1}}{q_{P'}} \ll_\epsilon (KNBL)^\epsilon.\] The product \(j=c'm=n/r\) is squarefree primary and has norm \(O(N/q_r)\). It may contain a permitted prime of \(S\) other than \(\lambda\), since only the quadratic column was stripped of its fixed \(S\)-part. Grouping \(c',m\) by \(j\) costs at most a divisor factor. Enlarge only this positive squarefree \(j\)-range. Cubic reciprocity, conjugation, and Equation (155), followed by the \(O(G/C)\) choices of \(h\), give, on \(q_D\asymp D_0,\ q_r\asymp r_0\), \[ (KNBL)^\epsilon\frac GC \left\{\frac N{r_0}+\frac L{D_0} +\left(\frac{NL}{r_0D_0}\right)^{2/3}\right\}. \tag{159}\] The fixed mask at \(ht\) is a coefficient restriction in this cubic sieve, not a row-dependent mask.

The reduction supplies \(T\sum_t\), with \(O(T)\) possible \(t\), and the factor \(C/q_r\). Together with the \(h\)-count, these contribute \[T^2(C/r_0)(G/C)\asymp B/r_0.\] There are \(O(r_0)\) possible \(r\) in its dyad, while the sum of \(q_D^{-1}\) over its dyad is \(O(1)\), up to a permitted small power. Both sieve factors increase when \(D_0,r_0\) are replaced by \(1\), and \(NG/C^2\ll NB\). Summing the logarithmically many dyads proves Equation (156). Subunit quotient ranges are empty, and bounded ranges are included by changing the fixed annular constants. This argument has introduced neither a \((g,t)=1\) condition nor a new pair-dependent mask. ◻

Proof. Put \(A=cn^3\) and \(\varrho_i=q_{p_i}/P_i\) for \(i\notin J\). Since \(q_A=q_cq_n^3\) even when \(c,n\) share primes, and \(q_{p_{J^c}}=Z^{\ell'}\prod_{i\notin J}\varrho_i\), Mellin inversion of the displayed integral defining \(B^J\) gives the physical smooth factor \[\begin{gathered} V\!\left(\frac{q_A}{Zq_{p_{J^c}}}\right) =W_{\mathrm G}\!\left(\frac{r}{\prod_{i\notin J}\varrho_i}\right),\\ r=\frac{q_A}{Z^{1+\ell'}}. \end{gathered}\] Here \(V=W_{\mathrm G}\) is the Gaussian of Lemma 10. Empty products are one.

Use the fixed translate partition \(\chi\) from that lemma on \(r\). On the annulus \(r=e^kx\), retain the single joint profile \[w_{k,J}(x,\boldsymbol\varrho) =\chi(\log x)W_{\mathrm G}\!\left( \frac{e^kx}{\prod_{i\notin J}\varrho_i}\right) \prod_{i\notin J}W_i(\varrho_i).\] The logarithms of \(x\) and of all the \(\varrho_i\) lie in fixed compact sets on this support. Thus the logarithm of the Gaussian argument is \(k+O_{\rm fixed}(1)\). The derivative estimate in Lemma 10, with this bounded shift and the fixed slot windows, gives for every fixed \(\kappa_{\mathrm G}>0\), \(A_{\rm cnt},B_{\rm cnt},C_{\rm ann}\ge0\), \(D_{\rm tail}>0\), and fixed seminorm order \(j\), \[\begin{gathered} \sum_{k\in\mathbb Z}e^{A_{\rm cnt}|k|}p_j(w_{k,J})<\infty, \\ Z^{B_{\rm cnt}} \sum_{|k|>\kappa_{\mathrm G}\log Z-C_{\rm ann}} e^{A_{\rm cnt}|k|}p_j(w_{k,J}) \ll Z^{-D_{\rm tail}}. \end{gathered}\] The constants may depend on the fixed slot system, but not on its moving prime labels. On an annulus \(\log r=k+O(1)\), an absolute bound for the row \(\ell^2\) norm is at most \(Z^{B_{\rm cnt}}e^{A_{\rm cnt}|k|}p_j(w_{k,J})\) for some fixed exponents, by the elementary row, slot, and ideal counts. Choose \(C_{\rm ann}\) larger than the fixed logarithmic radius of \(\chi\) and discard only whole annuli with \(|k|>\kappa_{\mathrm G}\log Z+C_{\rm ann}\). Summing their norm bounds and then squaring gives less than any prescribed power of \(Z^{-1}\). Every annulus meeting \(|\log_Z r|\le\kappa_{\mathrm G}\) is retained.

For each remaining annulus, keep fixed individual annular cutoffs equal to one on the support of \(w_{k,J}\) and apply Lemma 9 to this whole profile. Its logarithmic Fourier inversion uses one coefficient density common to every row and every surviving slot tuple. The normalized slot ratios remain variables of the joint profile until separation; they do not index separately chosen densities. For fixed Fourier variables, each resulting slot coefficient is a product of the original arithmetic coefficient \(\overline{\eta(p_i)}\) and an individual annular cutoff and norm power. It remains bounded and independent of the row and of every other slot. The completed factor is an annular test at length \[N_*=1+\ell'+\theta_N,\qquad \theta_N=\frac{k}{\log Z},\qquad |\theta_N|\le\kappa_{\mathrm G}+O_{\rm fixed}(1/\log Z).\] These retained lengths lie in a fixed bounded range. The weighted Fourier norms have a summable total by the displayed Gaussian estimate; the small annular weight stays in this homogeneous single-profile Fourier norm, not in an inhomogeneous tuple seminorm. Minkowski’s inequality passes the separated row norm through this common density. The existing \(q_c^{-1/2}q_n^{-1}\) normalization stays unchanged when the test scale is recentered, so no additional power of \(e^k\) is introduced. Choose \(\kappa_{\mathrm G}\) within the final \(\epsilon\) allowance. For each fixed \(\theta\), Lemma 16 supplies the row-sector reduction of \(\nu_\sigma\theta\chi_\bullet(m)\) with every original zero mask. It covers all nonzero element rows, including those meeting \(S\); no factor \(\xi(m)\) is used in this row norm. The row ball depends only on the fixed rescaled tuple \(p_J\), through \(Q=q_{b_*}X'Y'\), and is independent of the surviving marked slots and dual variables. The physical slot sets remain disjoint and their separated coefficients are product-form. Finite triangle over \(\theta\) and Lemma 47 therefore give the exponent in Equation (163) at this actual \(N_*\), taking the retention parameter \(\tau_{\rm ref}\) and the independently prescribed output and separation losses all at most a small \(\epsilon_0>0\). We use the exponent \(E_{\rm ref}\) after the fixed row parts are summed; their count is already included in \(E_{\rm ref}+\epsilon_0\), with principal exponent \(O/2\), and is not counted again.

Here \(f=1\) and the puncture is \(\rho=1\), so the only moving non-slot primes are those of the row. After the finite unit and reflection-sector splits in Lemma 47, freeze its powerful part of norm length \(O\) and its valuation-one part supported on \(S\), and split its residual squarefree good product into actual dyads of norm \(\asymp Z^H\). Let \(\epsilon_Z=O_{\rm fixed}(1/\log Z)\) be nonnegative and large enough to contain all fixed annular and fixed-conductor logarithmic offsets in this calculation. Then \[H\le M'-O+\epsilon_Z,\qquad \Delta_H=M'-O-H\ge-\epsilon_Z,\qquad O\ge-\epsilon_Z.\] The actual row and dual annular cutoffs are kept in one joint profile while its common Fourier density is separated. The small kernel amplitude is extracted from that homogeneous density before the row norm is squared, and only then may the positive sieve sum be enlarged, as in Equation (46). In particular the kernel saving in Equation (163) uses this \(H\); it is not reevaluated on any shorter rows added by the positive enlargement.

In the notation of that energy estimate, every moving non-slot prime counted in \(A_0\) divides the powerful row part to exponent at least two. The comparison of its exact norm with the powerful dyad, and any fixed-conductor contribution, therefore gives \[2A_0\le O+\epsilon_Z,\qquad N_0\le A_0,\qquad z_a\le\ell'.\] The available dual length at the actual completed scale is \[T_d=2H+2A_0+2z_a-1-\ell'-\theta_N-N_0-3B_0.\] Since \(M'+\ell'-1=-3d\), direct subtraction gives \[T_d-(H-3d) =H-M'+2A_0+2(z_a-\ell')-N_0-3B_0-\theta_N \le 2\epsilon_Z+|\theta_N|.\] As in the proof of the reflected energy, dual ranges below a fixed negative length are negligible after a positive Mellin shift, and the bounded boundary range costs an arbitrarily small power. The source dual ideals \(n,b\) in these dyads retain the support of Equation (24): they may share primes and may contain the permitted primes of \(S\); no additional \(S\)-mask or coprimality condition between \(n\) and \(b\) is imposed. On each remaining dual dyad, put \(y=v+3\ell_b+e_\lambda\). Retention gives \(y\le T_d+\epsilon_0\), while the nonzero unit ranges give \(v,\ell_b,e_\lambda\ge-\epsilon_Z\) after increasing its fixed constant. It follows that \[\max(H,v+\ell_b)=H+O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z).\]

Write \(s_{\rm hyb}=\min\{v,z_a,(v+z_a)/3\}\) for the quantity called \(u\) in Equation (162). If \(s_{\rm hyb}=z_a\), then \(-S_0-B_0+z_a-s_{\rm hyb}\le0\), the kernel term is nonpositive, and \[-\ell_b-\frac{2e_\lambda}{3}\le\frac53\epsilon_Z.\] Here the standalone \(O\) is the powerful-row logarithmic length. It follows that \[\begin{aligned} E_{\rm ref}&\le O/2+H+O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z)\\ &=M'-O/2-\Delta_H+O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z). \end{aligned}\] Otherwise \(s_{\rm hyb}\ge v/2-O(\epsilon_Z)\), and the bounded unit ranges imply \[s_{\rm hyb}+\ell_b+\frac{2e_\lambda}{3} +\frac{(T_d-y)_+}{2} \ge \frac y4+\frac{(T_d-y)_+}{2}-O(\epsilon_Z) \ge \frac{T_d}{4}-O(\epsilon_Z).\] The last inequality holds separately for \(y\le T_d\) and \(y\ge T_d\). Here the squared kernel saving is obtained by first extracting \(\min\{1,Z^{(y-T_d)/4}\}\) from the common Fourier density and only then squaring the row norm. Relative to \(M'\), the saving before the remaining \(+z_a\) is exactly \[ \begin{split} O/2+\Delta_H+S_0+B_0+T_d/4 ={}&\frac{2M'+2z_a-1-\ell'+2\Delta_H-\theta_N}{4}\\ &+\frac{2A_0-N_0+B_0+4S_0}{4}. \end{split} \tag{170}\] The second numerator is nonnegative. Since \(z_a\le\ell'\), the energy in this branch is at most \[\begin{aligned} &M'+\frac{1+3\ell'-2M'-2\Delta_H+\theta_N}{4} +O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z)\\ &\qquad=M'+\frac{d-1/6-2\Delta_H+\theta_N}{4} +O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z). \end{aligned}\] After using \(O\ge-\epsilon_Z\) and absorbing \(\theta_N\), the two upper bounds are \[\begin{gathered} M'-\Delta_H+O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z),\\ M'+\frac{d-1/6-2\Delta_H}{4} +O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z). \end{gathered}\] They are nonincreasing in \(\Delta_H\). Since \(\Delta_H\ge-\epsilon_Z\), their formal \(\Delta_H=0\) values bound every actual dyad up to \(O(\epsilon_Z)\), whether or not an endpoint dyad occurs. Their maximum is therefore at most \(M'+((d-1/6)/4)_++O(\epsilon_0+\kappa_{\mathrm G}+\epsilon_Z)\). At \(d=1/6\) one has \(\ell'=z_a=0\) and the clipped unit range falls in the first branch; bounded negative unit dyads change only \(\epsilon_Z\). Choose \(\epsilon_0\) and \(\kappa_{\mathrm G}\) within the prescribed \(\epsilon\) allowance, and then increase the fixed-data lower threshold so that \(\epsilon_Z\) also lies within it. Summing the logarithmically many row and dual dyads proves the lemma. ◻

Proof. Majorize the row ball by a fixed nonnegative radial Schwartz function. Put \(r_s=q_s/Y'\) and, on the primary elements prime to \(S\), \(c_\sigma(s)=1_{s\in\sigma}\chi_s(b_*)/(\tau\xi(s))\), extended by zero elsewhere before the inverse character is evaluated. The exact annular rewriting of the polynomial is \[A_m=Y'^{-3/2}\sum_s c_\sigma(s)W_1(r_s)r_s^{-1+i\nu} g_{\chi_s}(s,-m).\] Thus the expanded square has the common normalization \(Y'^{-3}\) and the joint annular profile contains the factors \(W_1(r_{s_1})r_{s_1}^{-1+i\nu}W_1(r_{s_2})r_{s_2}^{-1-i\nu}\). Write \(s_1=Cn_1\), \(s_2=Cn_2\), with \((n_1,n_2)=1\). Poisson in the row variable has factor \(Q/(q_{s_1}q_{s_2})\), whereas the complete transform of the two unnormalized Gauss sums is \(q_{s_1}q_{s_2}F(s_1,s_2;j)\). It therefore gives the exact prefactor \(Q/Y'^3\) and the correlation \(F(Cn_1,Cn_2;Ck)\) of Equation (147). Here \(F\) again denotes the full correlation of Equation (146), not the local coefficient of Equation (64). The first Fourier kernel has argument comparable to \(q_Cq_k/P_a\).

On coprime residuals the factor \(L_C\) in that correlation has \(|L_C|\le q_C\). For the subsequent signed extension use its specified bounded extension, periodic modulo \(\operatorname{rad}C\) in both columns, including when \(C\) shares primes with a column. Its value is defined to be zero when a prime of \(C\) divides both columns; this is a definition of the extension, not a claim about the genuine correlation there. Before any factorwise estimate, insert the complete Möbius identity \[1_{(n_1,n_2)=1}=\sum_{d\mid n_1,\ d\mid n_2}\mu(d)\] and write \(n_i=dm_i\). No coprimality condition on \(m_1,m_2\) is then imposed. Nonzero terms have \((d,Ck)=1\). Throughout this step use the fixed ray extension of \(\mathcal R(n_1,n_2)\); a quotient of zero residue symbols would not be defined.

We specify a common fixed modulus for that coefficient. For \(k\ne0\) write \(k=\zeta k_Sk_{\mathrm{good}}\) using the unit convention and the fixed \(S\)-prime generators of Lemma 5, with \(k_S=\prod_{\mathfrak p\in S}\pi_{\mathfrak p}^{e_{\mathfrak p}}\) and \(k_{\mathrm{good}}=\prod_{p\notin S}p^{e_p}\). For a primary residual ideal \(m\) outside \(S\), reciprocity gives \[\begin{split} \chi_m(k)&=\kappa_{\zeta,\boldsymbol e}(m) \mathcal R(k_{\mathrm{good}},m) \prod_{p\mid k_{\mathrm{good}}}\chi_p(m)^{e_p},\\ \kappa_{\zeta,\boldsymbol e}(m) &=\chi_m(\zeta)\prod_{\mathfrak p\in S} \chi_m(\pi_{\mathfrak p})^{e_{\mathfrak p}\bmod6}. \end{split}\] Every displayed good local power retains its zero on a shared prime, including when \(6\mid e_p\). Lemma 5 places the finitely many \(\kappa_{\zeta,\boldsymbol e}\) in a fixed ray group supported on \(S\). Also \(\mathcal R(k_{\mathrm{good}},m)\) belongs to a fixed finite family in \(m\).

Choose \(\mathfrak l_{\rm fixed}\) once to contain the primary-class modulus and the \(S\) zero masks, the moduli of every fixed ray coefficient in \(A_m\), a period for \(\mathcal R\) in each variable, and the conductors of all the preceding \(\kappa_{\zeta,\boldsymbol e}\) characters. The extended coefficient is zero off the primary class or at a fixed nonunit before any inverse character is evaluated. This modulus is independent of \(C,d,k\), and may have higher powers at \(S\) than \(\operatorname{rad}(k)\). After reciprocity the joint arithmetic coefficient in \(m_1,m_2\) is periodic modulo \[\mathfrak r=\operatorname{lcm} (\mathfrak l_{\rm fixed},\operatorname{rad}C,\operatorname{rad}(k)), \qquad q_{\mathfrak r}\ll_{\rm fixed}q_Cq_k.\] If \(k\) has a valuation \(e_p\) not divisible by six at a prime outside \(C\) and \(\mathfrak l_{\rm fixed}\), call \(k\) nonexceptional. At that prime the first residual column has the nonprincipal zero-extended factor \(\chi_p(m_1)^{e_p}\) up to a unit scalar. The lift, the fixed phases, and the other prime factors are independent of \(m_1\) modulo \(p\). Its complete mean, and hence the complete joint mean by the Chinese remainder theorem, is zero. The substitution by \(d\) does not change this conclusion because \((d,k)=1\). Principal zero masks at valuations divisible by six remain present in the exceptional case.

The frequency \(k=0\) contributes only the diagonal: the correlation forces \(n_1=n_2=1\) and equals \(\varphi(C)\). There are \(O(Y')\) possible \(C\) of norm comparable to \(Y'\), each with \(\varphi(C)\le q_C\ll Y'\). Its contribution is therefore \(O(Q/Y')\).

For \(k\ne0\), split the Fourier argument into dyadic shells \(q_Cq_k/P_a\asymp R\), with \(R=1,2,4,\ldots\) and the first shell including all arguments at most two. Choose dyad centers \(C_0,D_0\in\{1,2,4,\ldots\}\) for \(q_C\asymp C_0\) and \(q_d\asymp D_0\), with the first dyads containing the bounded unit ranges. Use the nominal positive residual scale \[N=Y'/(C_0D_0),\qquad \mathcal Q=q_{\mathfrak r}\ll_{\rm fixed}RP_a.\] The actual residual norms lie in fixed multiples of \(N\); the nominal value \(N\) is allowed to be below one. Fix arbitrary \(A>2\) and \(B>2\). For each fixed \(C,d,k\), the one joint annular profile \(W_R\) for the two columns has the finitely many homogeneous single-profile seminorms used below bounded by \(O_{B,J_{\mathrm{Gr}}}((1+|\nu|)^{J_{\mathrm{Gr}}} R^{-B})\), with one fixed \(J_{\mathrm{Gr}}\) chosen after these derivative orders and \(A,B\). This follows by differentiating the first Fourier kernel, whose Schwartz decay absorbs every resulting power of \(R\); derivatives of the norm powers cost a fixed power of \(1+|\nu|\). This factor \(R^{-B}\) remains in the single-profile Fourier estimate and occurs once in the linear lattice sum below; no small inhomogeneous tuple seminorm is used.

Let \(a_{C,d,k}\) be this periodic arithmetic coefficient. For a nonexceptional frequency its normalized complete Fourier transform, with normalization \(\mathcal Q^{-2}\) on the two residue variables, has absolute value at most \(q_C\) and vanishes at \((0,0)\). Since every ideal of \(\mathcal O\) is principal, the period lattice has linear scale \(\mathcal Q^{1/2}\). Four-dimensional lattice Poisson therefore gives, for this \(A\), \[ \left|\sum_{\mathbf m\in\mathcal O^2} a_{C,d,k}(\mathbf m)W_R(\mathbf m/\sqrt N)\right| \ll (1+|\nu|)^{J_{\mathrm{Gr}}} R^{-B}q_C N^2 \min\{1,(\mathcal Q/N)^A\}. \tag{172}\] For \(N\ge\mathcal Q\), the zero frequency vanishes and the other dual vectors have linear scale \((N/\mathcal Q)^{1/2}\). Fourier decay of order \(2A>4\) bounds their sum by \((\mathcal Q/N)^A\). For \(N<\mathcal Q\), use the point count in the two annuli. A nonempty subunit annulus has \(N\) bounded below by a fixed positive constant, so it still contains \(O(N)\) lattice points; below that constant it is empty. This proves Equation (172) without any primitivity or tensor-product assumption on \(a_{C,d,k}\).

Put \(K_R=RP_a/C_0\). The fixed shell comparison constants give \(q_k\le C_{\rm sh}K_R\) for a fixed \(C_{\rm sh}>0\). Hence the nonzero frequency range is empty when \(K_R<C_{\rm sh}^{-1}\). On a nonempty range, elementary lattice counting gives \[\#\{k:0<q_k\le C_{\rm sh}K_R\} \ll_{\rm fixed}K_R+\sqrt{K_R}+1\ll_{\rm fixed}K_R.\] There are therefore \(O(C_0D_0K_R)\) choices of \(C,d,k\) on these dyads. Multiplying Equation (172) by this count and by the common prefactor gives at most \[(1+|\nu|)^{J_{\mathrm{Gr}}} R^{-B}\frac Q{Y'}\frac{RP_a}{C_0D_0} \min\left\{1,\left(\frac{RP_aC_0D_0}{Y'}\right)^A\right\}.\] For \(\Lambda=Y'/(RP_a)\) the required positive dyadic sum is \[\sum_{i,j\ge0}2^{-i-j}\min\{1,(2^{i+j}/\Lambda)^A\} =\sum_{n\ge0}(n+1)2^{-n}\min\{1,(2^n/\Lambda)^A\} \ll_A\frac{1+\log(2+\Lambda)}{\Lambda}.\] For \(\Lambda\ge1\), split the two geometric tails at \(2^n\asymp\Lambda\); for \(\Lambda<1\), the left side is bounded and the displayed right side is larger than a positive constant. Thus the nonexceptional frequencies contribute \[\ll (1+|\nu|)^{J_{\mathrm{Gr}}}\frac Q{Y'}\frac{P_a^2}{Y'}Z^\epsilon\] after the \(R\) sum, on choosing a fixed \(B>2\).

It remains to count the exceptional nonzero frequencies. They have \((k)=a_1^6e\), where \(e\) is sixth-power-free and supported on the primes of \(C\) and the fixed support. There are at most \(6^{\omega(C)+O_S(1)}\ll_\epsilon q_C^\epsilon\) possible \(e\). For each of them the shell bound \(q_k\le C_{\rm sh}K_R\) permits only \(q_{a_1}\le(C_{\rm sh}K_R/q_e)^{1/6}\). If this range is nonempty, its upper bound is at least one, and the ideal count gives \(O_{\rm fixed}(K_R^{1/6})\) choices; if \(q_e>C_{\rm sh}K_R\) it is empty. Unit factors cost only a fixed factor. Hence there are \(O_\epsilon(K_R^{1/6}q_C^\epsilon)\) such elements \(k\), including the bounded subunit range of \(K_R\).

Use \(K_R=RP_a/C_0\), \(|L_C|\le q_C\), and the trivial two-column point count. Before multiplication by \(Q/Y'^3\), the contribution on these dyads is \[\ll (1+|\nu|)^{J_{\mathrm{Gr}}} R^{-B}Z^\epsilon Y'^2(RP_a)^{1/6}C_0^{-1/6}D_0^{-1}.\] The sums over \(C_0,D_0\) converge after reducing the arbitrary small power, and the \(R\) sum converges for the same fixed \(B>2\). This gives the term \((Q/Y')P_a^{1/6}Z^\epsilon\). Adding the diagonal and nonexceptional terms proves the proposition. ◻

Proof. For a fixed rescaled subset and tuple, put \(Q=q_{b_*}X'Y'\). Its separation is Equation (63) with \(A_{m,\sigma,v}(Y')\) and \(B^J_{m,\sigma}(Z)\). Put \(r_J=q_{p_J}/Z^d\); the fixed annular supports give \(0<c_J\le r_J\le C_J<\infty\) uniformly in the tuple. Exactly, \[\begin{gathered} X'=Z^{l_x-d}/r_J,\qquad Y'=Z^{l_y-d}/r_J,\\ Q=q_{b_*}Z^{M'}/r_J^2,\qquad P_a=Y'^2/Q=q_{b_*}^{-1}Z^{1/8}. \end{gathered}\] In particular \(P_a\ge1\) for \(Z\ge q_{b_*}^8\). Enlarge the allowed fixed-data lower threshold so that \(Q,Y'\ge1\) uniformly over the tuple ratios as well; this is possible because \(M'\ge1/2\) and \(l_y-d\ge5/16\). The remaining length inequalities are \[l_x-d\ge3/16,\qquad l_y-d\ge5/16, \qquad l_y-d-11b/6\ge1/12.\] The last one gives \[\frac{Y'}{P_a^{11/6}} =q_{b_*}^{11/6}r_J^{-1}Z^{l_y-d-11b/6} \gg_{\rm fixed}Z^{1/12},\] and hence \(P_a^2/Y'\ll P_a^{1/6}\). Proposition 49 therefore gives \[ \sum_{q_m\ll Q}|A_{m,\sigma,v}(Y')|^2 \ll (1+|v|)^{J_{\mathrm{Gr}}}(Q/Y')P_a^{1/6}Z^\epsilon. \tag{174}\] Cauchy–Schwarz in the row sum, Lemma 48, and the integrability of \((1+|v|)^{J_{\mathrm{Gr}}/2}|\widehat W_0(iv)|\) now bound this tuple’s unscaled separation by \(Z^\epsilon\) times \[\begin{aligned} Q^{-1/2}\bigl[(Q/Y')P_a^{1/6}\bigr]^{1/2} \bigl[Z^{M'}\bigr]^{1/2} &=\bigl[Z^{M'}/Y'\bigr]^{1/2}P_a^{1/12}\\ &=r_J(X')^{1/2}P_a^{1/12}. \end{aligned}\] Since \(r_J\) is bounded, this is \(O((X')^{1/2}P_a^{1/12}Z^\epsilon)\).

For \(d=\sum_{i\in J}\ell_i\) there are at most \(Z^{d+\epsilon}\) rescaled tuples, the coefficient is \(O(Z^{-3d/2})\), and \((X')^{1/2}\asymp X^{1/2}Z^{-d/2}\). The total additional exponent is consequently \[ f(d)=d-3d/2-d/2=-d\le0. \tag{175}\] Choose the small powers in the component estimates so that their sum lies within the prescribed \(\epsilon\). Summing the finitely many subsets proves Equation (173). For the already defined \(C=C_{\mathrm{II}}\) of Equation (138), the normalization used for the principal signal satisfies \[ \begin{gathered} C_{\mathrm{II}}(s)=l_x/2+s-1+h/6=s-11/16,\\ C_{\mathrm{II}}(7/8)=3/16=l_x/2+b/12. \end{gathered} \tag{176}\] Thus, under the contradiction assumed in Part II, this low bound is smaller than \(Z^{C_{\mathrm{II}}(\beta_*)}\) by the exact power \(\Delta=\beta_*-7/8\), apart from the arbitrarily prescribed \(\epsilon\). ◻

Proof. In this region \(\vartheta=(-w_r)_+\le6e\). The estimates in the proof of Lemma 26 sharpen to \(H_p-1=O(Q^{-1-10e})\) for \(p\nmid u\) and \(H_p-1=O(Q^{-10e})\) for \(p\mid u\). Increase the fixed \(P_0\) so that \(|H_p-1|<1/2\) for all these primes. The holomorphic factor \(G_p\) in Equation (178), divided by \(H_p\), now defines \(\mathcal B_p\) throughout this region. It is \(H_p\), not the possibly singular raw factor \(P_p\), that is bounded away from zero when \(w_r<0\).

Consequently the tuple sum in Equation (179) factors as \(\mathcal H_{\eta,u}\prod_i\mathcal B_i\) on this dynamic region. Substituting \(\mathcal B_i=\mathcal Q_i+\mathcal D_i\) proves Equation (186). This factorization is not used to continue the correction outside the present region.

For \(p\nmid u\), Equation (184) and the estimate for \(\mathcal E_p\) give \[|H_p(\mathcal B_p+\overline{\chi_p(u)})| \ll Q^{-x_r+2\vartheta}+Q^{-6z_r+2\vartheta} +Q^{4-5x_r-6z_r+2\vartheta} +Q^{1-w_r-6z_r+\vartheta} \ll Q^{-51/100}.\] The four exponents are bounded respectively by \(-a-4e\), \(-51/25+12e\), \(49/25-5a-68e\), and \(a-51/25+12e\). Division by the bounded \(H_p\) preserves this estimate. The number of primes in a slot is \(O(P_i)\) by the ideal count; hence its total contribution from \(p\nmid u\), including \(Q^{z-1}\), is \(O(P_i^{z_r-51/100+\epsilon})\).

For \(p\mid u\) there are only divisor-many possible labels. A monomial in \(\mathcal E_p\) is multiplied by \(Q^{z-1+x}\) in \(Q^{z-1}\mathcal B_p\), so after separating \(Q^z\) its exponent is \(x_r-1\) plus the exponent of that monomial. For the non-tail boundary terms of the six-valuation table, these exponents at \(e=0\) are \[\begin{array}{c|ccccc} j&1&2&3&4&5\\\hline &a-2&1/2-2a&-a,\ 1-3a&1/2-2a&2-5a. \end{array}\] At positive \(e\) their changes are respectively \(+6e,-32e,(-26e,-48e),-42e,-80e\). They are all strictly below \(-1/2\) in the stated range. The common \(R\) term is smaller still: its exponent after separating \(Q^z\) is at most \(49/25-1-5a-80e<-1/2\). Additional valuation pairs and \(m\) values have ratios \(|R|\le Q^{-11/10}\) and \(|V|=Q^{-51/25}\), so their geometric sums preserve these bounds. The strict second-family term with an additional \(V\) is also smaller than \(Q^{-1/2}\) after this normalization.

The sole remaining term is the strict \((e_0,l,k,m)=(1,0,1,0)\) term, which occurs for \(j\ge2\) and is \(-\eta(p)(Q-1)Q^{-x-w}\). Its exponent after separating \(Q^z\) is \(-w_r\). The explicit rescaling term \(-Q^{-w}\) has exponent after separating \(Q^z\) \(-1-w_r<-1/2\), so it needs no further estimate.

Let \(\mathfrak f_u\) be the conductor of \(\psi_u^*\). Write \(u=\epsilon_u\prod_{p\mid u}p^{j_p}\), with \(\epsilon_u\) a unit. For primary \(n\) coprime to \(uS\), reciprocity gives \[\chi_n(u)=\epsilon_u^{(q_n-1)/6} \prod_{p\mid u}\mathcal R(p,n)^{j_p}\chi_p(n)^{j_p}.\] The unit exponent is read modulo six. The prime formula extends to composite \(n\): when \(A\equiv B\equiv1\pmod6\), \[\frac{AB-1}{6}\equiv\frac{A-1}{6}+\frac{B-1}{6}\pmod6.\] Thus the unit factor depends only on \(q_n\) modulo \(36\), and the \(\mathcal R\) product depends only on \(n\) modulo \(4\). Together with the multiplicativity of \(\mathcal R\) and the local symbols, the displayed norm congruence makes the right side multiplicative on integral ideals prime to \(6u\); extend it to their fractional ideal group. A generator congruent to one modulo \((36)\operatorname{rad}(u)\) is already primary and makes every displayed factor one. This character therefore has a ray presentation with that fixed \(2,3\) modulus times \(\operatorname{rad}(u)\). It agrees with the Kummer character \(\chi_\bullet(u)\) of Lemma 5 on ideals avoiding \(S\). These characters induce the same primitive character: in the principal ideal ring \(\mathcal O\), the Chinese remainder theorem supplies a representative avoiding the additional finite set \(S\) in every ray class of a common modulus. At a good prime \(p\mid u\), hold the fixed class and all other good residues at one and vary the residue modulo \(p\) by the Chinese remainder theorem. The remaining local character \(\chi_p^{j_p}\) has exact order \(6/\gcd(6,j_p)>1\). Its conductor exponent at \(p\) is therefore exactly one: the displayed ray modulus has only the first power of \(p\), and the character cannot descend to a modulus omitting \(p\). This is a local assertion; the ideal character may still need the fixed normalization modulus at \(2,3\).

It follows that, for any set \(J_0\) of distinct selected ramified labels, \[ q_{\mathfrak f_u}\ll_S q_{\operatorname{rad}(u)} \le q_u\prod_{p\in J_0}q_p^{-(j_p-1)}. \tag{187}\] The functional equation for the primitive numerator [11], the assumed reflected bound, and Stirling give a cost \(q_{\mathfrak f_u}^{A_*}U^\epsilon(3+T_1)^C\), where \[A_*=1/2-w_r=a-1/2+6e>0.\] At a selected \(p\mid u\) the primitive character already has value zero, so restoring the original local factor there multiplies by exactly one. The remaining deleted Euler factors have radical \(O_S(U)\) and cost at most \(U^{6e+\epsilon}\), because \(w_r\ge-6e\).

Expand a product of error slots into their coprime-prime terms, their already bounded ramified terms, and their strict ramified terms. For each fixed tuple in this triangle expansion, let \(J_0\) be precisely its strict ramified labels. They are distinct because the slot supports are disjoint. Apply the single inequality Equation (187) to this entire \(J_0\) before summing the labels. The numerator together with these local factors, including \(q_p^{z-1}\) at each selected prime, is then bounded by \[U^{A_*+6e+\epsilon}(3+T_1)^C \prod_{p\in J_0}q_p^{z_r-w_r-(j_p-1)A_*} \le U^{a-1/2+12e+\epsilon}(3+T_1)^C \prod_{p\in J_0}q_p^{z_r-1/2},\] because \(j_p\ge2\) and \(-w_r-A_*=-1/2\). Thus different selected labels use different factors of the conductor deficit, and the bounds hold simultaneously. Summing the divisor-many ramified labels costs \(U^\epsilon\), while the coprime labels have the stronger exponent \(z_r-51/100\). This proves Equation (185).

This argument uses triangle only on error labels. It never changes the physical row or the coefficients and masks in any main slot. The tuple-independent factor \(\mathcal H_{\eta,u}\) separately costs \(U^\epsilon\) by Lemma 26. Relative to the central scale \(P_i^{z_r-1/2}\) of a main slot, an error slot thus has no positive amplitude exponent in the later row count. ◻

Proof. Both sets of lines lie in the first Euler region with \(\epsilon_0=3/8\). Its primewise defect estimates give \(H_p=O(1)\), while Equation (181) bounds the positive product of all unselected factors by \(O_\epsilon(U^\epsilon)\).

Consider first \(x_r=\beta_*+e\), \(w_r=1/2\), and \(z_r=17/50\). For a selected \(p\nmid u\), Equation (184) gives \[|G_p+\overline{\chi_p(u)}H_p| \ll Q^{-x_r}+Q^{-6z_r} +Q^{4-5x_r-6z_r}+Q^{1-w_r-6z_r}.\] The four exponents are at most \(-7/8-e,-51/25,-483/200-5e,-77/50\), respectively. Consequently \(G_p=O(1)\) off \(u\), without division by \(H_p\).

For a selected \(p\mid u\), one has \(W=D=0\) and \(\mathcal E_p=P_p^*\). Equation (178) becomes \[G_p=\overline{\eta(p)}Q^x(1-V)\mathcal E_p-Q^{-w}H_p.\] After multiplication by \(Q^{x_r}\), the strict term of the local table has exponent at most \(1-w_r=1/2\); for \(j=1\) it also contains the factor \(V\). The exponents of the boundary terms \(J_j\), in order \(j=1,\ldots,5\), are \[-\frac12,\quad \frac32-2x_r,\quad \left(\frac32-2x_r,\,2-3x_r\right),\quad 2-3x_r,\quad 3-5x_r.\] They are bounded above by \(-1/2,-1/4-2e,(-1/4-2e,-5/8-3e),-5/8-3e,-11/8-5e\), respectively. The common \(R\) term has exponent \(4-5x_r-6z_r\le-483/200-5e\). All further terms in the geometric families decrease these powers, since \(|R|\le Q^{-329/100-6e}\) and \(|V|=Q^{-51/25}\). Finally \(Q^{-w}H_p=O(Q^{-1/2})\). Hence \(G_p\ll Q^{1/2}\) on \(u\), also at a zero of \(H_p\).

The labels dividing \(u\) are divisor-many, while a slot contains \(O(P_i)\) ideals. Its positive sum is therefore \[ \sum_{p\in\mathcal P_i(Z)} |W_i(q_p/P_i)|q_p^{z_r-1}|G_p| \ll P_i^{z_r}+U^\epsilon P_i^{z_r-1/2} \ll U^\epsilon P_i^{z_r}. \tag{192}\] Using the positive majorant for the unselected product and distributing the requested \(\epsilon\) among the fixed number of slots proves Equation (191) on the first lines.

Now fix \(x_r=w_r=2\) and \(z_r=z_\infty>2\). For a selected prime off \(u\), the four exponents in Equation (184) are \(-2,-6z_\infty,-6-6z_\infty,-1-6z_\infty\), so \(G_p=O_{z_\infty}(1)\). On \(u\), the strict term after multiplication by \(Q^{x_r}\) has exponent at most \(1-w_r=-1\). The boundary exponents are \(-2,-5/2,(-4,-4),-11/2,-7\), the common \(R\) exponent is \(-6-6z_\infty\), and \(|R|=Q^{-8-6z_\infty}\), \(|V|=Q^{-6z_\infty}\). The rescaling term is \(O(Q^{-2})\). Thus \(G_p=O_{z_\infty}(1)\) on \(u\) as well. A positive slot sum is consequently \(O_{z_\infty}(P_i^{z_\infty})\). Together with Equation (181), this proves Equation (191) on the second lines. ◻

Proof. At a terminal vertex the assertion is its stated bound. Suppose a child has already been bounded by \(C Z^{E'}p_{J'}(\boldsymbol W_e)^{B'} \langle A_e\boldsymbol t+B_e\boldsymbol\xi\rangle^{H'}\). Since the linear maps are bounded, \[\langle A_e\boldsymbol t+B_e\boldsymbol\xi\rangle^{H'} \ll_e\langle\boldsymbol t\rangle^{H'} \langle\boldsymbol\xi\rangle^{H'}.\] The child-profile bound adds the power \(B'c'_e(J')\) to each of these two height exponents and replaces its profile factor by \(p_{m_e(J')}(\boldsymbol w)^{B'b'_e(J')}\). Use the coefficient-measure bound with \(H_e=H'+B'c'_e(J')\). The contribution of this edge is at most \[C_e Z^{a_e+E'} p_{\max\{m_e(J'),\ell_e(H_e)\}}(\boldsymbol w)^{B'b'_e(J')+b_e(H_e)} \langle\boldsymbol t\rangle^{H_e+c_e(H_e)}.\] Every displayed index is finite. Take the maximum of these indices and of the terminal indices over the finitely many outgoing edges, increasing the profile exponent when necessary because \(p_j(\boldsymbol w)\ge1\). This proves the parent bound with exponent equal to the maximum path exponent through that vertex. Reverse induction on the acyclic graph completes the proof. None of these choices mentions an external cutoff or a late derivative order.

For the product form, insert the already proved bound of each child and raise it to \(\theta_{e,r}\). The total power of \(\langle\boldsymbol\xi\rangle\) that the coefficient measure must integrate is \[H_e=\sum_{r=1}^{r_e}\theta_{e,r} \bigl(H_r+B_r c'_{e,r}(J_r)\bigr).\] The powers of the parent profile and of \(\langle\boldsymbol t\rangle\) are the corresponding finite sums, followed by the single coefficient-measure bound at \(H_e\). The powers of \(Z\) add as in the displayed recurrence for \(E_v\). Taking maxima over the finitely many terms proves this extension by the same reverse induction. ◻

Proof. Set \(\overline{\mathcal N}_v=Z^{-\Phi_v(\boldsymbol\ell)}\mathcal N_v\). Substitution in an edge and Equation (196) bound its norm power by \(Z^{\delta_e}\) times the product of the normalized children. Insert their inductive bounds. As in the product proof of Lemma 55, the required coefficient moment is \[H_e=\sum_r\theta_{e,r} \bigl(H_r+B_r c'_{e,r}(J_r)\bigr),\] which is finite and independent of \(\gamma\). Equation (197) at this order contributes \(Z^{\delta_e^{\mathrm{mass}}}\) and only finite profile and height orders. Backward induction over the finite edge types therefore gives exactly the stated normalized recurrence. A label averaged inside a child remains inside that child throughout this argument; it is not a second integration variable of \(\nu_e\). ◻

Proof. Expand the mask as \(\sum_{d\mid\operatorname{rad}R,\ d\mid k}\mu(d)\) and write \(k=dk'\). The scale becomes \(K/q_d\) and the character contributes \(\psi(d)\). Poisson summation in residue classes modulo \(m\) has prefactor \(K/(q_dq_m)\). Its finite transform is \[\sum_{x\bmod m}\psi(x)e(hx/m) =\sqrt{q_m}\gamma(\psi;m)\bar\psi(h).\] For unit \(h\) this follows by changing variables. For nonunit \(h\) the transform is zero by primitivity; for \(m=1\) the stated principal convention applies. This proves Equation (200). For a nonprincipal primitive character, finite Parseval gives \(\sum_h|\sum_x\psi(x)e(hx/m)|^2=q_m\sum_x|\psi(x)|^2\). Writing \(\phi(m)=|(\mathcal O/m)^\times|\), there are \(\phi(m)\) unit \(h\)’s, all with magnitude \(\sqrt{q_m}|\gamma(\psi;m)|\), and \(\sum_x|\psi(x)|^2=\phi(m)\). Thus \(|\gamma(\psi;m)|=1\).

A fixed ray restriction can first be expanded into finitely many characters. Each is then replaced by its primitive inducing character, with the removed local zero extensions kept in \(R\); the same formula applies term by term. For the principal modulus the absolute sum of the nonzero frequencies is \(O_\Phi(d_{\mathcal O}(\operatorname{rad}R))\). Indeed, for all \(A>0\), \[A\sum_{h\ne0}|\mathcal F\Phi(Aq_h)|\ll_\Phi1.\] Lattice counting proves this when \(A\le1\), and Schwartz decay proves it when \(A\ge1\). Sum this bound with \(A=K/q_d\) over \(d\). At every use below the relevant nonempty row scale has \(K\ge1\), so this is also \(O(KZ^\epsilon)\). When principal nonzero frequencies are added or removed while separating a nonprincipal sum, they will carry the same outer row mask as that sum; their absolute contribution is no larger than this full principal bound. ◻

Proof of Lemma 54. The order of the analytic choices is organized by Corollary 56 in Section 6.3. Its indexed conclusion will be used on each positive Cauchy-side sum separately; the proof below verifies the required common coefficient measures, fixed norm losses, and finite depth.

At each node, every joint \((f,k)\) Hilbert space uses the current label measure \(w(f)\), and that factor is retained exactly once in every parent \(f\)-sum until Equation (224) removes the old label.

At each node keep the parameters \(N,V,M\) fixed, and write \[F=N+V,\qquad Q=\log_Zq_{\mathfrak r_\rho}.\] Here \(F\) is the real length sum \(N+V\), so \(F=F_0\) at the starting node. The two hypotheses at a node with margin \(c_{\rm node}\) are exactly \[ F-M-Q-z_0\ge c_{\rm node},\qquad 4F-3M-6z_0\ge c_{\rm node}. \tag{201}\] In particular \(M+z_0+c_{\rm node}\le F\). The number \(Q\) is the actual logarithmic norm of the already fixed puncture radical, not a dyadic center. The parameter \(M\) is the fixed logarithmic row length; the first Poisson test has scale \(Z^M\).

Let \(M_{\max}\) bound the starting row lengths. Choose \(0<d\le c_*/200\) and put \[D=\left\lceil\frac{M_{\max}+2}{d}\right\rceil+1.\] We use positive parameters \(\eta,\tau,\pi,\tau_{\rm ref},\pi_{\rm ref}\), chosen in the quantified order at the end of the proof. Here \(\eta\) is a localization tolerance bounding only the logarithms of fixed annular ratios, \(\tau\) is the common tolerance in the two Poisson tail comparisons, and \(\pi\) bounds the aggregate freely chosen local small-power losses. At depth \(h\) set \[c_h=c_*-7h\eta,\qquad c_{\rm node}=c_h,\] and impose the row cap \(M\le M_{\max}-hd\). We prove the estimate by backwards induction on \(h\le D\). Every retained row parameter below is nonnegative. We shall show that a nonterminal passage decreases it by at least \(d\), loses at most \(7\eta\) in either required margin, and increases the energy exponent by at most \(40\eta+\tau+\pi\). No invariant will be transferred from the fixed \(F=N+V\) to an actual column norm.