Mixed-residue subtraction for quadratic-twisted cubic two-mark forms

1. Contribution and statement

We work over the Eisenstein field $F=\mathbb Q(\omega)$, where $\omega^2+\omega+1=0$, with ring $\mathcal O$ and ideal norm $N$. Nonzero ideals prime to3 are represented by a fixed primary generator. All cubic symbols, additive characters and Gauss sums use the compatible primary convention of Brubaker--Bump--Friedberg--Hoffstein [1]. Finite changes of this convention are kept as fixed ray components. Characters are extended by zero at nonunits of their original moduli.

This paper estimates the cost of subtracting a particular mixed-pole model from a signed, divisor-weighted correlation. The model comes from the cubic $A_2$ Eisenstein coefficient with a moving quadratic twist in both coefficient coordinates. The exceptional Levi and its Eulerian Whittaker functional are established theory [2--5]. The contribution is their application to the actual numerator-deleted, squarefree two-mark source, followed by a weighted mean and a bound for both cross terms.

Let $\eta=\eta_\kappa$ run through a fixed quadratic Hecke ray family, with squarefree good conductor $\kappa$, $N\kappa\asymp K$. We exclude the principal global character. Components at2 and3 lie in a fixed finite collection; at2 we allow the actual quadratic components, including wild ones. There is no varying auxiliary ramified character. Throughout, $S_0$ is the set of finite primes above $6$; "good" means outside $S_0$, and $L^{S_0}$ omits precisely those Euler factors. All fixed exceptional ramification in the specified packet is supported in $S_0$. Primes of the varying deletion ideal $D$ or good conductor $\kappa$ are not added to $S_0$: their factors are explicitly retained below. Write

$$ G(n,f)=G(f)\overline{\rho_f(n)},\qquad |G(f)|=1 \quad(f\text{ squarefree},\ (n,f)=1), \tag{1.1} $$

where $G(f)$ is the normalized cubic Gauss sum and $\rho_f$ the cubic character. Zeros are retained in (1.1). For fixed smooth functions $W_1,W_2$ supported in compact subintervals of $(0,\infty)$, define

$$ F_{n,\eta}(L_1,L_2;D)= \sum_{\substack{r,s\text{ squarefree}\\(r,s)=1\\(rs,nD)=1}} \frac{\eta(rs)G(n,rs)}{\sqrt{N(rs)}} W_1(Nr/L_1)W_2(Ns/L_2). \tag{1.2} $$

Here $D$ is a common deletion ideal. Quadratic conductor zeros are already present in $\eta(rs)$. There is no requirement $(n,\kappa)=1$. Normalization by $(L_1L_2)^{-1/2}$ instead is equivalent after multiplying the fixed profiles by powers of their arguments.

The finite construction in Section5 associates to (1.2) a mixed-pole model $R_{n,\eta}$ and defines $C_{n,\eta}=F_{n,\eta}-R_{n,\eta}$. It uses a finite collision inverse, followed by a convergent integral of the generic residue on $s+w=4/3$. It does not assert that an infinite squarefree projection is automorphic.

Choose two pairs of profiles, with $L_1L_2=M_1M_2=Q$, and set

$$ w(u)=\sum_{ae=u}\mu(a)^2U(Na/D_0)V(Ne/E),\qquad X=D_0E\asymp Q^2, \tag{1.3} $$

where $U,V$ are fixed smooth annular profiles, possibly complex valued. For $A=F,R,C$, write

$$ A^+_{g_1,g_2;\kappa}(u)= A_{gu,\eta_\kappa}(L_1/Ng_1,L_2/Ng_2;Dg),\qquad g_1g_2=g, \tag{1.4} $$

and define $A^-_{h_1,h_2;\kappa}$ with the second pair of profiles and $M_1,M_2$, where $h_1h_2=g$. Every split is squarefree and coprime. Only nonempty original daughter supports are included. Define

$$ \begin{aligned} \mathcal S(A^+,A^-)= \sum_{\kappa}\sum_{g\text{ squarefree}} \frac{\mu(g)}{Ng} \sum_{\substack{u\text{ primary}\\(g,u\kappa)=1}}w(u) \sum_{\substack{g_1g_2=g\\h_1h_2=g}} A^+_{g_1,g_2;\kappa}(u) \overline{A^-_{h_1,h_2;\kappa}(u)} . \tag{1.5} \end{aligned} $$

Common additional deletions and fixed compatible ray phases may be retained. A common Mellin integral of such forms is permitted when its weight has enough decay to pay the finite smooth seminorms used below. Profiles cannot be chosen independently for different arithmetic rows.

Theorem 1. For every $\epsilon>0$, in the setting (1.1)--(1.5),

$$ \left|\mathcal S(F^+,F^-)-\mathcal S(C^+,C^-)\right| \ll_\epsilon (KXQND)^\epsilon \left(K^{1/2}Q^{1/6}X^{2/3}+Q^{1/3}X^{1/3}\right). \tag{1.6} $$

The constant depends on the fixed field, the specified finite packet, the supports and finitely many smooth seminorms. At $X\asymp Q^2$, restoring an outside factor $D_0/Q$ gives

$$ \frac{D_0}{Q} \left|\mathcal S(F^+,F^-)-\mathcal S(C^+,C^-)\right| \ll_\epsilon (KXQND)^\epsilon D_0(1+\sqrt{KQ}). \tag{1.7} $$

This is a difference estimate, not a bound for $\mathcal S(C^+,C^-)$. The two-factor row weight arises when the coupled cubic reflection in [10, proof of Proposition8.1] groups the original outer variable with its Poisson dual; a quadratic twist can be kept in the common column coefficient. Theorem1 addresses the specified squarefree two-mark coefficient at that stage. It does not assert a new theorem for every overlap sector of an earlier transform.

As a scale comparison, at $Q=D_0^{15/8}$ and $K=D_0^k$, (1.7) has exponent $31/16+k/2$. It fits a target $D_0^{151/48-k/2}$ for $k<29/24$, with a fixed margin before subsidiary powers. This comparison states the cost of this component only. The complementary signed estimate remains unproved. There is no new paid cutoff, zero-free region or resolution of the Riemann hypothesis.

2. Related work and the source to be estimated

Kazhdan--Patterson [2] construct the exceptional representations and their global residues. Their 1985 correction [3] withdraws an assertion that an unramified spherical vector has vanishing Jacquet coordinates outside a preferred class. We use the corrected finite-$S$ normalization, not that assertion. Banks--Bump--Lieman [4], especially Sections2--3, study the parameter-minus-one cover of $GL_{n-1}$ inside the degree-$n$ cover of $GL_n$. Their induced Whittaker integral has a spherical factor $L(ns,\chi)/L(n^2s,\chi^n)$ in the odd-degree case. For $n=3$, this is the outer Euler factor appearing below. Banks [5] supplies the heredity argument. These are imported mechanisms, not novelty claims.

McNamara [6] provides the rank-one coefficient integral and spherical Weyl expansion used for the local computations. The global realization and nonfactorization issues for general metaplectic Fourier coefficients are discussed by Brubaker--Friedberg [7]. We use uniqueness only after passing to the particular exceptional Levi; we do not assert that the original minimal Eisenstein coefficient is Eulerian.

Brubaker--Bump [8] identify residues of cubic $A_3$ multiple Dirichlet series with the Friedberg--Hoffstein--Lieman double series. Their residue method and twisted multiplicativity are direct precedents. Our estimate instead retains a moving quadratic character in both $A_2$ columns, natural equal Fourier indices, both column deletions, finite squarefree projection, an irregular divisor weight, and the full signed cross-column projector. The standard quadratic large sieve [9] and the natural-row cubic estimate [10, (8.3)] supply the positive mean inputs.

Earlier rStation results [11,12] estimate different rank-one residual pairings: primitive sextic Type-I sums and partial cubic Poisson packets. The local co-period in [13] concerns the same special cubic cover but an ordinary $GL_2$ test representation and a different functional. None of these statements gives the subtraction (1.6). Conversely, (1.6) does not replace their conclusions or control their regular remainders.

The more recent rStation papers [14--18] are also relevant precedents. Paper54/v2 [14] estimates the signed cubic Kubota remainder with its prescribed Möbius--Gauss coefficient and complete polar subtraction. Paper58/v3 [15] treats coherent cubic sources with actual inverse and ordinary coefficient convolutions, using pole removal and functional equation feedback. Paper65/v2 [16] constructs positive models for an exact corrected cubic Möbius--Gauss norm and derives natural-row bounds from its named published inputs. Paper69/v1 [17] proves a homogeneous polar approximation on squarefree short rows, while Paper70/v1 [18] extends actual Möbius--Gauss means to natural rows above the square-root range and explicitly leaves its signed quadratic-weighted consumer open.

Those conclusions do not supply a uniform estimate for the moving quadratic family in both ordinary columns of (1.2). The residue used here has two linked central quadratic $L$-values, the finite inverse shifts the natural numerator, and the quantity estimated is the complete cross-term difference with weight (1.3) and projector (1.5). Theorem1 supplies that specified subtraction step; it does not extend [18] to the outstanding signed complementary form. Polar subtraction, signed remainder estimation and natural-row bookkeeping themselves are precedents, not novelty claimed here.

Let $H(C_1,C_2;m_1,m_2)$ denote the unnormalized coefficients in [1, Proposition1], and put $H_0=H/\sqrt{N(C_1C_2)}$. Our ambient series is

$$ Z_{n,\eta}(s,w;D)= \sum_{\substack{C_1,C_2\text{ primary}\\(C_1C_2,nD)=1}} \frac{\eta(C_1C_2)H_0(C_1,C_2;n,n)} {(NC_1)^s(NC_2)^w}. \tag{2.1} $$

Both coefficient indices are deleted at every prime of $nD$. This is essential: on the coprime locus the numerator twist is a character, whereas the unrestricted Fourier coefficient has additional prime-power terms at primes dividing $n$ [1, Proposition3]. The series (2.1) has all prime-power coefficient states; its squarefree two-mark projection is (1.2). We first identify its mixed residue, then project by a finite identity. Formal root completion factors are not additional independent pole terms.

3. The classical source and the exceptional functional

3.1. Moving quadratic minor weight

Use the fixed principal congruence group $\Gamma(3\mathcal O)$ and the cubic splitting of [1, Section1]. For an integral representative $\gamma$, put

$$ C_1(\gamma)=\gamma_{33},\qquad C_2(\gamma)=\gamma_{22}\gamma_{33}-\gamma_{23}\gamma_{32}. \tag{3.1} $$

Multiply its Eisenstein summand by $\eta(C_1C_2)$ and by the indicators that both minors are units at primes of $nD$. Left upper-unipotent operations and the right lower-unipotent operations used in [1] preserve these minors. Consequently the proof of [1, Proposition1] unfolds with exactly the same archimedean Jacquet integral and with (2.1) as arithmetic coefficient. The weight pulls out of each finite exponential sum. For a fixed modulus, a finite cusp decomposition also proves continuation: the weight is constant on the appropriate deeper-congruence cosets. This observation alone gives no conductor-uniform norm estimate.

We need the more precise tensor realization. At a good finite place, let $K_p=SL_3(\mathcal O_p)$ and

$$ \Delta_1(k)=k_{31},\quad \Delta_2(k)=k_{21}k_{32}-k_{22}k_{31},\quad f_{\eta,p}(k)=\eta_p(\Delta_1(k)\Delta_2(k))v_{0,p}, \tag{3.2} $$

zero unless both minors are units. The underlying unramified genuine torus vector is $v_{0,p}$. If $b\in B(\mathcal O_p)$ has diagonal $t$, the two minors acquire factors $t_3$ and $t_1^{-1}$. Thus (3.2) has the covariance for the torus representation twisted by $\lambda_\eta(t)=\eta(t_3/t_1)$. The principal version of (3.2) implements the two-column deletion at good primes of $nD$ outside the conductor.

Choose

$$ w_0=\begin{pmatrix}0&0&1\\0&-1&0\\1&0&0\end{pmatrix}. $$

The compact support of (3.2) is represented by $w_0N(\mathcal O_p)$; in those coordinates its value is $v_{0,p}$. With $\operatorname{vol}(\mathcal O_p)=1$, its Jacquet input is therefore $v_{0,p}$ for every integral pair of additive indices. On an integral $\gamma$, its right translate by $w_0$ has scalar value $\eta(C_1(\gamma)C_2(\gamma))$, including the two zeros. The classical Fourier coefficient in [1] is evaluated at $w_0n(x)g$; this and the finite right translation restore the full-cell input (3.2). Fixed changes of Weyl signs are carried in the fixed packet.

Lemma A (the classical packet and its ordinary twist). Fix the classical cubic splitting, Haar measures, residue coordinate and the untwisted automorphic torus realization of [1, Section1] and [2, II.1]. Use the compact vectors (3.2), their indicated right translates, and principal unit-minor vectors for the two-column deletions. The scalar classical series with summand weight $\eta(C_1C_2)1_{(C_1C_2,nD)=1}$ is the scalar realization of a fixed finite number of these twisted tensors. Their packet coefficients and the comparison scalar are independent of $\kappa,n,D$. No unit-norm renormalization of a varying-level vector is implicit in this assertion.

Proof. Realize the original genuine automorphic torus datum as functions $v$ on the rational-torus quotient and write $\ell(v)=v(1)$. Since $\lambda_\eta(t)=\eta(t_3/t_1)$ is trivial on the rational torus, multiplication defines the automorphic twisting map

$$ (T_\eta v)(t)=\lambda_\eta(t)v(t),\qquad \ell(T_\eta v)=\ell(v) \tag{3.3} $$

For right translation it satisfies $\pi_\eta(t)T_\eta=\lambda_\eta(t)T_\eta\pi_0(t)$. It preserves genuineness and, on restricted tensors, is the tensor product of the ordinary local twists. This is an identity of representations and evaluations; it does not assert that $\ell$ itself factors into local evaluations. In the scalar torus sum of [2, II.1], every rational representative has $\lambda_\eta=1$, giving the same identity.

Choose once and for all a finite tensor presentation $\Phi_0=\sum_{j=1}^J b_j\bigotimes_v\Phi_{j,v}$ for the classical base packet. To see that it is fixed, use integral representatives of rational flags over the PID $\mathcal O$. Fixed congruence support selects finitely many Borel/ray classes. At basic level $3\mathcal O$ the units lift the diagonal classes and unipotent entries lift directly; at a larger fixed level retain all the finitely many diagonal classes. These are choices for the base source, made before $\eta,n,D$ vary.

At each varying good conductor place let $A_{\eta,nD,p}(k)$ be the scalar minor character $\eta_p(C_1(k)C_2(k))$ on the locus where both minors are units, and zero elsewhere. This is the indicated right translate of (3.2). At $p\mid nD$ outside the conductor use the same unit-minor indicator with the principal character; at remaining good places use $1$. Components in $S_0$ lie in the fixed finite packet. On the compact group put $A=\prod_p A_{\eta,nD,p}$ and $\Phi_\eta(k)=A(k)T_\eta\Phi_0(k)$. The minor covariance and the displayed intertwining identity give its required $B(\mathcal O_p)$ covariance; Iwasawa extension makes it an actual induced section. Its scalar realization satisfies exactly

$$ \ell(\Phi_\eta(k))=A(k)\ell(\Phi_0(k)). $$

The operations act separately on the local factors of each of the $J$ tensors, so $J$ and $b_j$ do not change. At an integral flag representative their product is the quadratic minor weight with both deletion indicators and the fixed ray phases. For a quadratic character the inverse appearing in the Hecke product formula is the same character. The finite right translate and the $w_0n(x)g$ Fourier evaluation in [1] restore the full-cell input (3.2); they must be used together. Thus each scalar summand agrees, including zero cases, before summation. Equality first holds in absolute convergence and then by continuation, with no new conductor factor. $\square$

3.2. The mixed residue in its actual Levi

Put $a=w-1/2$, $b=s-1/2$. The normalized torus exponent is

$$ \lambda=((2a+b)/3,(-a+b)/3,(-a-2b)/3). $$

After the first simple reflection, the second root has exponent $c=a+b$ and the ordinary character is $\eta(t_1)$. For the Levi

$$ \iota(m)=\operatorname{diag}(\det(m)^{-1},m), \tag{3.4} $$

this character is $\eta(\det m)$. Restricting the standard cubic torus cocycle gives, for diagonal $m=(u,v)$ and $m'=(u',v')$,

$$ \sigma_3(\iota(m),\iota(m')) =(u,v')_3(uv,u'v')_3^{-1}. \tag{3.5} $$

Thus the Levi cover has KP parameter $-1\equiv2\pmod3$. At $c=1/3$, its exceptional representation has a unique Whittaker model by [2, Corollary I.3.6 and II.2.5--II.2.6]; the complex place is split. The determinant twist preserves that property. It also commutes with the global Levi residue: it is trivial on rational points and on the unipotents used in the Whittaker integral.

Lemma B (the relevant induced component). At a finite place let $P=MU_P$ be the parabolic of $SL_3$ with Levi (3.4), and let $\theta_M\otimes\eta\circ\det$ be its exceptional inducing datum. The generic quotient of its normalized parabolic induction is the generic quotient of that Levi datum, hence is one-dimensional. In the $GL_3$ model of [4, Section3], this is the component supported on the identity determinant class after restriction to $SL_3$.

Proof. Apply the Bruhat filtration with the fixed generic character. A nonopen parabolic cell has a simple-root subgroup acting trivially on the section and nontrivially under that character, so its twisted Jacquet quotient is zero. The open quotient is the Levi twisted Jacquet quotient. The cover splits over these unipotents, so the root-coordinate proof of [5] applies to this $SL_3$ subgroup with the actual Levi cocycle (3.5). The determinant twist is trivial on the Levi unipotents. At the complex place use the split-group version.

For the restriction comparison, put $G'=GL_3(F_p)$, $H'=SL_3(F_p)$ and $P_3'=\{p\in P':\det p\in F_p^{*3}\}$ in the ambient parabolic of [4]. Determinant gives $P_3'\backslash G'/H'=F_p^*/F_p^{*3}$. These double cosets are open and closed. Decomposing sections by their supports gives a direct sum on restriction to $H'$. A section on the identity component extends uniquely from $H'$ to $P_3'H'$ by its inducing transformation law. Consistency is precisely the law for $P_3'\cap H'=P'\cap H'$, and the modular characters restrict accordingly. The common central kernel introduces no new double cosets. This proves the identification, up to the fixed conjugation selecting the block placement (3.4).

At tame places there are nine determinant classes, explaining why the nine ordinary $GL_3$ Whittaker functionals of [4] do not contradict the one-dimensional quotient of this $SL_3$ component. The degree-three Weyl matrix in [4, (3.1)] is a three-cycle, and all its unipotent integration points have determinant one. Its integral sees only the identity component. The spherical section restricts to the section with the same value at the identity. $\square$

3.3. Normalization of the global functional

Normalize the good-place Levi residual map by its spherical scalar $1+q^{-1}$. In the reflected principal series its spherical Jacquet vector is $(1,q^{-1}u^{-1}G_0(2),0)$, where $u^3=q^{-1}$. The Levi functional is the zeroth coordinate, so its spherical value is1. Induction in stages identifies a section $f$ with

$$ F_f(g)(m)=\delta_P(m)^{-1/2}f(mg). $$

The induced Whittaker integral integrates the Levi functional over the two outer root coordinates. Substituting the Levi Jacquet integral gives the full three-root Jacquet integral; the root-coordinate change is triangular with determinant1. Hence its value is the zeroth coordinate of the full outgoing Jacquet vector. This identity is first proved on compact open-cell sections, then in a convergence chamber, then meromorphically. It introduces no parameter-dependent scalar.

The global theta Whittaker functional is a fixed scalar times these local normalized functionals on restricted tensors. In the corrected KP construction the finite-$S$ formula contains

$$ T_2(S)^{-1},\qquad T_2(S)=\prod_{v\in S,\ v<\infty}(1+q_v^{-1}). \tag{3.6} $$

We retain this stabilization when making the local spherical values1. It neither sets the unnormalized global scalar to1 nor supplies an extra Euler factor. The determinant twist is1 on the Levi unipotents, so the fixed scalar is unchanged as $\eta$ varies. The central unitary parabolic twist is likewise1 there. Unfolding the outer nondegenerate Fourier coefficient leaves the open cell even though theta is noncuspidal: the nonopen cells are killed by the generic character. Its constant term can have several Weyl contributions; those are not separate alternative Fourier functional equations.

More explicitly, for the fixed global Levi residue map $\mathcal R_M$,

$$ \mathcal R_M\bigl(\eta(\det(\cdot))h\bigr)(m) =\eta(\det m)\mathcal R_M(h)(m). $$

This follows termwise in absolute convergence from triviality on rational Levi points, and then meromorphically. The determinant twist is one on the Whittaker integration unipotent. Thus the scalar $C_\theta$ fixed by one base tensor in the corrected KP realization is unchanged. With local spherical values one, its restricted tensor formula is $C_\theta\prod_v\ell_{M,v}$, retaining (3.6) when the finite set is enlarged. Induction in stages and the determinant-one root-coordinate change above introduce no additional scalar. Lemma A identifies the classical packet and Lemma B identifies the component where uniqueness applies. This scalar comparison does not absorb the first intertwiner's moving factor or the outer spherical factor; both are computed separately in Section4.

It follows that the residue of (2.1) uses the normalized scalar functionals just specified, applied to each member of the fixed packet. The remaining moving factors are computed next.

4. Residual density at every natural numerator

4.1. Local coefficient calculation

At a good prime of norm $q$, the cubic $SL_3$ torus lattice quotient has three classes. Choose basis $E_t$, $t\in\mathbb Z/3$, so that $\varpi^{r\alpha_1+s\alpha_2}v_0=x_1^rx_2^sE_{r+s}$. For integral additive valuation $e$, put $h=e+1$. With unit additive part $v$ define the unnormalized finite Gauss sums

$$ G_\varepsilon(k;v)= \sum_{z\in\mathbb F_q^*}\overline\eta(z)^\varepsilon \theta(z)^k\overline\psi(-vz), \tag{4.1} $$

where $\overline\eta$ is the tame quadratic character and $\theta$ the cubic character. Here $G_0(0;v)=-1$ and the other relevant sums have modulus $q^{1/2}$. The raw simple-root coefficient is

$$ C_\varepsilon(x,e)E_t= 1_{\varepsilon=0}\frac{(1-q^{-1})x^{r(t)}}{1-x^3}E_t +q^{-1}G_\varepsilon(-t-h;v)x^{-h}E_{h-t},\quad r(t)=\min\{r\ge-e:r\equiv t\pmod3\}. \tag{4.2} $$

This follows from the shell integral in [6, Sections9,13], with the ordinary unit character retained. Shells below $-e-1$ vanish by additive orthogonality. Shell $-e-1$ gives the Gauss term. Higher shells have trivial additive character; their unit average is zero for a quadratic twist and otherwise survives exactly in the displayed congruence class. Summing that geometric tail gives (4.2).

The spherical scalar is $c(x)=(1-q^{-1}x^3)/(1-x^3)$. The six Weyl terms of [6] give the unit-index spherical vector

$$ j(x,y)=\left(1+\frac{x^2y+xy^2}{q}, \frac{qx+qy+x^2y^2}{qg},0\right),\qquad g=G_0(1;1). \tag{4.3} $$

Both simple-root identities $C_i j=c_i j\circ s_i$ verify its normalization directly. Put

$$ a=1/6+it,\quad b=1/6-it,\quad u=q^{-1/3},\quad z=\eta(p)q^{-b}. $$

The outgoing parameters after roots1,2 are $(z,u^{-1})$, and

$$ j(z,u^{-1})=(1+uz+(uz)^2)(1,q^{-1}u^{-1}G_0(2;1),0). \tag{4.4} $$

The first intertwiner and the outer factor (4.4) therefore supply

$$ \frac{L^{S_0}(1/2+3it,\eta)L^{S_0}(1/2-it,\eta)} {L^{S_0}(3/2+3it,\eta)L^{S_0}(3/2-3it,\eta)}. \tag{4.5} $$

Indeed $1+uz+(uz)^2=(1-\eta(p)q^{-(1+3b)})/ (1-\eta(p)q^{-(b+1/3)})$. This is the degree-three Euler factor of [4] with $3s=b+1/3$. Euler identities are first used in an absolute chamber and then continued; no central Euler product convergence is assumed. The denominators in (4.5) are uniformly bounded away from zero.

4.2. Correct numerator coordinate and deletion factors

For equal numerator $n$, rational torus translation by $\tau_n=\operatorname{diag}(n,1,n^{-1})$ identifies its global Whittaker character with the unit one. Locally, changing the three unipotent coordinates gives

$$ W_1(R(\tau_n)f)=|n|_p^2\pi(\tau_n)^{-1}W_n(f). $$

In outgoing parameters $(z,u^{-1})$, this selects coordinate $2e\bmod3$ at $e=v_p(n)$. Combining with the archimedean Fourier-index scaling gives the arithmetic scalar

$$ A_p(e)=(u^2/z)^e[W_n(f_{\rm out})]_{2e\bmod3}. \tag{4.6} $$

It is incorrect to use coordinate0 at every $e$. For completeness, in the complex normalization $|n|_{\mathbb C}=Nn$, the incoming Jacquet integral satisfies $J_{\lambda,n}=Nn^{a+b-2}J_{\lambda,1}(R\tau_n)$. The short intertwiner has Fourier-index factor $Nn^{-(a+c)}$, $c=a+b$. Multiplication of the finite $|n|_p^2\pi(\tau_n)^{-1}$ factors with these complex powers yields exactly (4.6). Fixed unit phases are retained and have modulus1.

At a conductor prime, the actual full-cell input is $E_0$ and the second root after reflection is principal. Apply $C_0(u,e)C_1(u/z,e)/(1+q^{-1})$, then (4.6). One obtains

$$ |A_p(e)|^2=\frac1{(1+q^{-1})^2} \begin{cases}q^{-1},&e\equiv0\pmod3,\\ q^{-2/3},&e\equiv1\pmod3,\\ 0,&e\equiv2\pmod3. \end{cases} \tag{4.7} $$

At a good prime outside the conductor where both columns are deleted, use the principal full-cell input $E_0$ and divide by the extracted unramified factors $c(u/z)(1+uz+u^2z^2)$. Direct evaluation gives

$$ \begin{aligned} B_p(3k)&=\frac{z(u^2+uz^2+z)}{\mathcal D},\\ B_p(3k+1)&=\frac{gu^2z(u^2z+u+z^2)}{\mathcal D},\\ B_p(3k+2)&=0,\\ \mathcal D&=(1+u^3)(u^4+u^2z+z^2)(1+uz+u^2z^2). \end{aligned} \tag{4.8} $$

The identity $C_\varepsilon(x,e+3)=x^{-3}C_\varepsilon(x,e)$ makes these formulas periodic for all $k\ge0$, after (4.6). Since $|z|=q^{-1/6}$ and $|g|=q^{1/2}$,

$$ B_p(3k)=1+O(q^{-1/2}),\qquad |B_p(3k+1)|\ll q^{-1/3}. \tag{4.9} $$

All denominator ratios in (4.8) are bounded away from zero uniformly on this line. Products over additional deletion primes cost only $(NDNn)^\epsilon$. At primes outside $nD\kappa S_0$, the normalized unit-numerator spherical value is1. Thus (4.7)--(4.9) treat every variable good place in (2.1).

4.3. Dyadic and fixed places

The prime above2 is inert, of norm4. Its cubic cover is tame and splits over the compact group. For a quadratic character of conductor depth $\ell$, multiplying its unit restriction by any tame cubic character keeps exact depth $\ell$. Primitive finite-ring Gauss orthogonality leaves only shell $-e-\ell$, giving

$$ C_{\eta,\ell}(x,e)E_t= q^{-\ell}G_{\eta,\ell}(-t-e-\ell;v)x^{-e-\ell}E_{e+\ell-t}, \qquad |G_{\eta,\ell}|=q^{\ell/2}. \tag{4.10} $$

The source compact vector is still the unit-minor character (3.2), now at its actual depth. Its open-cell Jacquet input remains $E_0$. Formula (4.10), followed by the principal second root, is periodic under $e\mapsto e+3$ after (4.6). Hence all dyadic numerator valuations are uniformly bounded within the fixed quadratic packet. The surviving classes may differ from (4.7), which is why the dyadic part is separated in the counting argument. For an unramified quadratic component use (4.8), or the spherical value when no deletion is present.

Primary numerators are units at the cubic prime. The local packet there is fixed. Its finite-level Jacquet and positive-root intertwining expressions are finite shell terms and geometric tails in the positive real parts $\Re a=\Re b=1/6$, $c=1/3$. Their denominators are bounded on this line. Thus they contribute a fixed bound, uniformly in the unit numerator and $t$. No unbounded cubic-prime numerator valuation is being absorbed into that constant.

At infinity the raw simple-root coefficient, with a fixed additive character and Haar measure, is a fixed exponential power times $\Gamma(a)/\Gamma(1-a)$. For the short word it is the product at $a,c$. Stirling gives polynomial growth (in fact decay in this spherical angular type). We never divide by a Whittaker function at an arbitrary evaluation point.

4.4. Residue density bound

Let $\mathcal R_\eta(n;t;D)$ be the coefficient of $(s+w-4/3)^{-1}$ in (2.1), on $(s,w)=(2/3-it,2/3+it)$. Write $n=\pi_2^j n_0$, with $n_0$ prime to6. The preceding calculation proves that this density vanishes unless

$$ n_0=rv^3,\qquad r\text{ squarefree}, \tag{4.11} $$

and, in that case,

$$ |\mathcal R_\eta(n;t;D)| \ll_\epsilon (KNnND)^\epsilon(1+|t|)^A |L(1/2+3it,\eta)L(1/2-it,\eta)| K^{-1/2}(Nr)^{-1/3}N(r,\kappa)^{1/2}. \tag{4.12} $$

There is no coprimality condition between $r$ and $v$. In particular, the class1 conductor factor in (4.12) is $q^{-1/2}q^{-1/3}q^{1/2}=q^{-1/3}$, as required by (4.7). The fixed normalization from Section3 is absorbed only into the fixed implied constant. Both moving central values are displayed.

5. Exact finite squarefree projection

At a prime outside $nD\kappa$, the normalized polynomial coefficient of [1] has six possible exponent pairs,

$$ (0,0),(1,0),(0,1),(2,1),(1,2),(2,2). \tag{5.1} $$

The last three are collision states. Let $A_{n,\eta}(L_1,L_2;D)$ denote the centrally normalized finite dyad with all these polynomial-$H$ coefficients and the same two column deletions as (2.1). The following inverse is coefficientwise exact:

$$ \begin{aligned} F_{n,\eta}(L_1,L_2;D) ={}&\sum_{\substack{c,d,e\text{ squarefree, pairwise coprime}\\ (cde,nD\kappa)=1}} \frac{\mu(cde)\eta(cd)G(n,e)}{Nc\,Nd\,(Ne)^{3/2}}\\ &\quad\times A_{ne,\eta}\left( \frac{L_1}{(Nc)^2Nd(Ne)^2}, \frac{L_2}{Nc(Nd)^2(Ne)^2};Dcde\right). \end{aligned} \tag{5.2} $$

Only terms with actual nonempty daughter supports occur, so the sum is finite. To prove it, fix a supported coefficient pair and its set $B$ of collision primes. Select any subset of $B$ for extraction. The states $(2,1),(1,2)$ are neutral under the Cartan twisted multiplicativity and carry $\eta(cd)$. State $(2,2)$ shifts the remaining numerator to $ne$ and supplies $G(n,e)$ through exact Gauss CRT. Deleting selected primes from both remaining columns prevents reuse. Every selected subset reproduces the original coefficient times $(-1)^{|S|}$, so the sum is $(1-1)^{|B|}$. It keeps precisely the squarefree states. This proof works for every natural $n$.

For a given $A$ daughter define its mixed-pole model by

$$ \begin{aligned} \mathcal P_{n,\eta}(L_1,L_2;D) =\frac1{2\pi}\int_{\mathbb R} &\widehat W_1(1/6-it)\widehat W_2(1/6+it)\\ &\times L_1^{1/6-it}L_2^{1/6+it} \mathcal R_\eta(n;t;D)\,dt . \end{aligned} \tag{5.3} $$

The residue coordinate is fixed as $s+w-4/3$; equivalent oriented Mellin conventions change only the same fixed factor throughout. The integral is absolutely convergent by (4.12) and smooth Mellin decay. Define $R_{n,\eta}$ by replacing each $A$ in the finite identity (5.2) with (5.3), using its actual shortened profiles and deletions. Finally define $C=F-R$.

This defines the precise subtraction in Theorem1 without a global contour-growth claim for $C$. If a chosen contour deformation is independently justified, (5.3) is its generic mixed-hyperplane term. We do not claim that every other residue is absent, or that $C$ already has a uniformly controlled shifted double-Mellin representation.

Put $Q=L_1L_2$. Combining the factor $Q^{1/6}$ from (5.3) with the shortened lengths and the coefficient in (5.2) gives

$$ (Nc)^{-3/2}(Nd)^{-3/2}(Ne)^{-13/6}. \tag{5.4} $$

For good $e$, the condition $(e,n\kappa)=1$ and (4.11) add the factor $(Ne)^{-1/3}$. The total extraction mass is bounded by

$$ \sum_{c,d,e}(Nc)^{-3/2}(Nd)^{-3/2}(Ne)^{-5/2}<\infty. \tag{5.5} $$

A dyadic factor of squarefree $e$ has only two possibilities and a fixed cost. It does not alter convergence. Every rescaled profile has the same Mellin seminorm; the actual aspect ratio contributes a unit phase on the integration line. Thus there is a common positive integral

$$ \mathcal L(\eta)=\int_{\mathbb R}(1+|t|)^{-B} |L(1/2+3it,\eta)L(1/2-it,\eta)|\,dt \tag{5.6} $$

with $B$ sufficiently large, after paying fixed profile seminorms, such that

$$ |R_{n,\eta}(L_1,L_2;D)| \ll_\epsilon (KNnNDQ)^\epsilon Q^{1/6}K^{-1/2} 1_{n_0=rv^3}(Nr)^{-1/3}N(r,\kappa)^{1/2}\mathcal L(\eta). \tag{5.7} $$

6. Weighted numerator count and quadratic mean

Lemma 2. For squarefree good $g$ coprime to $\kappa$, $G=Ng$,

$$ \sum_{Nu\le X}1_{gu=rv^3}(Nr)^{-2/3}N(r,\kappa) \ll_\epsilon(KGX)^\epsilon X^{1/3}G^{-2/3}. \tag{6.1} $$

Proof. Take $d=(g,r)$ and $h=g/d$. Then $h\mid v$. Expand $N(r,\kappa)=\sum_{j\mid(r,\kappa)}\varphi(j)$. Since $(g,\kappa)=1$, $(d,j)=1$. Ideal counting and partial summation give, for $Y\ge NdNj$,

$$ \sum_{\substack{Nr\le Y\\dj\mid r}}(Nr)^{-2/3} \ll Y^{1/3}/(NdNj). $$

Put $Y=GX/(Nv)^3$, enlarge the remaining coprimality conditions, and sum $v$ divisible by $h$. The harmonic ideal sum is $O(\log(2GX)/Nh)$. Since $dh=g$, the resulting main factor is $(GX)^{1/3}/G=X^{1/3}G^{-2/3}$. The sums over $d\mid g$ and $j\mid\kappa$ cost subsidiary powers, because $\varphi(j)/Nj\le1$. This proves (6.1). $\square$

The original weight (1.3) satisfies $|w(u)|\ll\tau(u)$, up to its fixed profile bounds. It can therefore be retained in (6.1) at a subsidiary cost. This use of a positive majorant applies only to the residual and full positive means, not to the desired signed complement. For a natural dyadic factor $u=\pi_2^j u_0$, sum the shortened counts: $\sum_{j\ge0}q^{-j/3}(1+j)^A<\infty$. A dyadic factor of $g$ has a fixed cost. Thus the stated estimate covers all natural primary rows.

We also use the standard fourth-moment consequence of [9]:

$$ \sum_{N\kappa\asymp K}|L(1/2+it,\eta_\kappa)|^4 \ll_\epsilon K^{1+\epsilon}(1+|t|)^A. \tag{6.2} $$

Here are the coefficient decomposition and zero masks in this standard deduction. Put $T=1+|t|$. On a fixed conductor dyad the approximate functional equation is a sum of two smoothly weighted polynomials of length $\ll K^{1/2}T^{A_0}$, with tails controlled by arbitrarily rapid decay. Squaring a polynomial gives length $\ll Y=KT^{2A_0}$, with divisor-bounded coefficients over $\sqrt{Nn}$. Use common Mellin separation of the weights, including the conductor dependence, before applying the large sieve. The resulting norm heights have polynomial costs and integrable tails. Unit root-number factors depend only on the row and disappear in its positive squared norm. Bound the direct and dual pieces separately before combining them.

It suffices to bound a common-coefficient polynomial $P_\kappa=\sum_{Nn\le Y}b_n\eta_\kappa(n)/\sqrt{Nn}$, where $|b_n|\ll\tau(n)^B$ for fixed $B$. Extract uniquely $n=rh^2$ with $r$ squarefree; no condition $(r,h)=1$ is imposed. Away from the fixed places one has the exact identity

$$ P_\kappa= \sum_{Nh\le\sqrt Y}\frac{1_{(h,\kappa)=1}}{Nh} \sum_{\substack{r\text{ squarefree}\\Nr\le Y/(Nh)^2}} \frac{b_{rh^2}}{\sqrt{Nr}}\eta_\kappa(r). \tag{6.2a} $$

The mask is exactly $\eta_\kappa(h^2)$. At a fixed ramified place its analogue uses the full original modulus. Separate fixed-prime powers before (6.2a); their central weights have convergent geometric mass. The inner character still vanishes at $(r,\kappa)>1$. Partition the fixed ray classes so reciprocity gives the orientation of [9, Theorem1.1]. For fixed $h$ the coefficient energy satisfies

$$ \sum_{\substack{r\text{ squarefree}\\Nr\le Y/(Nh)^2}} \frac{|b_{rh^2}|^2}{Nr}\ll_\epsilon Y^\epsilon. $$

In the positive outer norm for this $h$, dropping its mask only enlarges the family. The large sieve then bounds that squared norm by $\ll_\epsilon(KY)^\epsilon(K+Y/(Nh)^2)$. Hilbert-space triangle in (6.2a) yields

$$ \begin{aligned} \left(\sum_{N\kappa\asymp K}|P_\kappa|^2\right)^{1/2} &\ll_\epsilon (KY)^\epsilon \sum_{Nh\le\sqrt Y}\frac{\sqrt{K+Y/(Nh)^2}}{Nh}\\ &\ll_\epsilon (KY)^\epsilon \left(\sqrt K\log(2Y)+\sqrt Y\right). \tag{6.2b} \end{aligned} $$

The last step uses the logarithmic ideal sum of $(Nh)^{-1}$ and convergence of the ideal sum of $(Nh)^{-2}$. Squaring and substituting $Y=KT^{2A_0}$ proves (6.2), after renaming $\epsilon$ and the fixed height power. The squarefree character sieve was never applied to an unrestricted $n$ sum, and no zero was replaced by a nonzero phase inside that sum.

Cauchy between heights $3t$ and $-t$, followed by the common integrable weight in (5.6), yields

$$ \sum_{N\kappa\asymp K}\mathcal L(\eta_\kappa)^2 \ll_\epsilon K^{1+\epsilon}. \tag{6.3} $$

Combining (5.7), (6.1) and (6.3), for any fixed two-mark daughter of product $Q/G$ with source numerator $gu$,

$$ \sum_{\kappa,u}|w(u)||R_{gu,\eta_\kappa}(Q/G;Dg)|^2 \ll_\epsilon (KXQND)^\epsilon Q^{1/3}X^{1/3}G^{-1}. \tag{6.4} $$

The notation $Q/G$ in (6.4) abbreviates its two actual lengths, not a replacement by a balanced profile. The power $G^{-1}$ is the product of daughter shortening $G^{-1/3}$ and numerator divisibility $G^{-2/3}$. This is the sparsity gain missed by a bound that counts all $X$ rows with only a uniform conductor estimate.

7. Full projector and proof of Theorem1

First verify the exact source in (1.5). Inserting $1_{(n,m)=1}=\sum_{g\mid n,m}\mu(g)$ into the two whole squarefree columns, write $n=gf$, $m=gh$. Gauss CRT in the stated convention gives

$$ G(gf)\overline{\rho_{gf}(u)} =G(g)\overline{\rho_g(u)}G(f)\overline{\rho_f(gu)}. \tag{7.1} $$

The extracted factors on the two sides cancel against conjugation, leaving $1_{(g,u\kappa)=1}$. Central normalization supplies $1/G$. Splitting $g$ between each pair of marks gives exactly (1.4), with product lengths $Q/G$. Remaining columns are deleted at $g$; no coprimality between the two remaining whole columns is imposed after Möbius extraction. Thus (1.5) is the complete signed projector. Its diagonal cancellation is not available after discarding the signs.

For the original daughter $F$, absorb $\overline{\rho_f(g)}$ as a column phase and fix $\eta_\kappa$. The coefficient on squarefree $f$ is divisor-bounded and centrally normalized, with square energy bounded by subsidiary powers. The natural-row cubic bound [10, (8.3)] gives

$$ \sum_u|w(u)||F_{gu,\eta_\kappa}(Q/G;Dg)|^2 \ll_\epsilon (KXQND)^\epsilon \left(X+X^{1/3}Q/G+(XQ/G)^{2/3}\right). $$

At $X\asymp Q^2$ the bracket is $O(X)$ for all nonempty daughters. Counting the quadratic family gives

$$ \sum_{\kappa,u}|w(u)||F_{gu,\eta_\kappa}(Q/G;Dg)|^2 \ll_\epsilon (KXQND)^\epsilon KX. \tag{7.2} $$

The row length here is $X$, not $GX$. The multiplier $g$ was absorbed into the column phase. All original zero masks remain; dropping a mask occurs only as an enlargement of this positive mean.

For each actual pair of daughters, the exact algebraic identity is

$$ F^+\overline{F^-}-C^+\overline{C^-} =R^+\overline{F^-}+F^+\overline{R^-}-R^+\overline{R^-}. \tag{7.3} $$

Cauchy using (6.4) and (7.2), followed by the original $1/G$, bounds either cross term by

$$ (KXQND)^\epsilon K^{1/2}Q^{1/6}X^{2/3} \sum_g (Ng)^{-3/2+\epsilon}. $$

For the last term of (7.3), (6.4) gives

$$ (KXQND)^\epsilon Q^{1/3}X^{1/3} \sum_g (Ng)^{-2+\epsilon}. $$

Both sums converge. The two allocation counts are bounded by $\tau(g)^2$ and can be included by taking a smaller subsidiary exponent. Common Mellin profile heights are integrated against one rapidly decaying majorant. This proves (1.6) and then (1.7). No inequality was multiplied by a retained negative Möbius weight: (7.3) is exact before the absolute bounds. $\square$

8. Evidence, reproducibility and limits

The proof is analytic and algebraic. CPU calculations check the local conventions and finite identities; they do not establish global continuation, Whittaker uniqueness or asymptotic estimates. The supplement contains the original proof notes, scripts, their local dependencies, saved outputs and a hash manifest. Python3 with SymPy, mpmath and NumPy is sufficient; no GPU is used.

The checks include both spherical simple-root functional equations, the rank-one exceptional coefficient matrix, the all-exponent geometric identities, the actual numerator-deleted vector, and the finite-ring dyadic shells. The collision inverse was tested on2592 actual Eisenstein-prime cases, including2424 zero cases. Omitting its numerator shift fails130 cases; maximum relative floating-point discrepancy in the correct identity is below $6.4\times10^{-15}$. The dyadic control has96 checks at primitive quadratic depths2 and3. The earlier full coefficient-matrix control also checks exact braid and reflection-square identities. Saved checks were not rerun unchanged merely to package this manuscript.

To reproduce the finite inverse or dyadic check, run respectively python experiments/cubic1548_squarefree_inverse.py and python experiments/cubic1549_dyadic_gauss_shells.py from the unpacked supplement. The other scripts are listed in its README and manifest. All original code, notes and outputs in the supplement are released under CC0-1.0. Referenced papers retain their own licenses and are not redistributed in the supplement.

The scope restrictions are substantive. We use the specified quadratic packet and natural primary rows, not arbitrary additional ramified or angular twists. The finite subtraction defines $C$ exactly but does not prove a useful signed estimate for it, or a uniform spectral description after contour shifting. In particular the positive bound (7.2) still has the old $KX$ scale. One cannot subtract its restored diagonal from that upper bound. The next research problem is the signed correlation of the complementary expressions with the original weight (1.3), every zero mask and the full projector intact.

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