bibkey: liu2026quasirhmixedsixth
authors: Baiying Liu
year: 2026
title: Actual mixed-sixth coefficient and conditional complete-bin interface
doi: null
url: https://arxiv.org/abs/2610.12234v2
claim: Repo-derived conditional paper applications pay the exact joint-Euler short-cofactor Type-I component and the modified large-prime collision term, identify the complete remaining coefficient, conditionally pay its original-row-conductor sector through Q^(1/2), above every M4 plain length, with one explicit common-cost budget, rule out a coefficient-norm power saving and two specified coefficient simplifications, and quantify the implication of an unproved neighborhood mixed-sixth moment for the original-W0 bare nonunit C93-C97 consumer. The original detector and annular A=A_R+E route are conditionally composed with the supplied fixed-local C118 parent to bound the final aggregate profile cost; the absolute-line R identity is not an arithmetic supplier. Full mixed moment, C68/J40, signed Robin and RH remain open; no source-proof, Lean or priority certification is supplied.
strata_touched: []
license: citation-only
triage: anchor

Actual mixed-sixth coefficient and conditional complete-bin interface

This repo-derived application companion uses the Liu source card and the reduced-frequency owner. The source metadata locates that research line; the deductions below are not statements attributed to Liu’s paper. They are conditional paper mathematics. Analytic source proofs and their transports remain named, unverified premises. The fixed-local envelope application below uses those printed laws; larger component-family availability remains a separate premise. There is no new analytic supplier, Lean certification or originality claim.

The finite-height application M41–M46 uses the fixed cutoff $A=Q^{3/20}$ and specifies its complete unpaid remainder. The small original-row-conductor application M47–M56 pays only its stated original-row sector when the common height and envelope costs fit the explicit gap; the complementary complete remainder stays open. The bounded-valuation cap application M57–M62 uses the physical-row count, full original inverse and fixed plain reflection to include $C_u\le Q^{1/2}$ under one stated common-cost condition, retaining the same coefficient and full complement.

The original-generator composition M63–M68 bounds the actual fixed-local coupled envelope through the original annular arithmetic inputs and their printed laws; it supplies no full mixed-moment saving.

Source correspondence and exact reuse

The primary source used here is exactly openai/math at adc7f1241b42e322a6451854ab7e4b4c146bf78a. Its lem:buffered-bins, lem:detector-dyads, prop:detector-witness, lem:inverse-amplification, zero-slot lem:plain, and lem:smooth-calculus supply the detector, marginal moments and smooth calculus used below. Its lem:fixed-gauss-phase, eq:signal, eq:sextic-sieve-pair-gauss, eq:sextic-sieve-pair-poisson and lem:ray-prime-normalizer supply the local identities and fixed-field prime counts. The main theorem and lem:logarithmic-control are retained only as premises of the existing C25–C31 and C77–C97 applications and the short-cofactor payment below, including both $L$ and $1/L$ control. This pin is not identified with the earlier fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb used in parts of the fixed-order owner.

The application consequences retained here are the quantitative collision and joint-Euler short-cofactor mixed-moment payments, complete modified-remainder equivalence, the exact four-prime coefficient test, two restricted diagnostic failures, and the quantified neighborhood-moment interface to that existing complete consumer. Old degree-two applications and the generic quadratic/cubic obstruction are not counted as additional results. The existing actual sextic experiment is already linked by the Liu source card; none of the implications here requires new experimental code or data.

The complete coefficient and remaining target

Work over $F=\mathbb{Q}(\sqrt{-3})$ with the original primary convention, fixed arithmetic exclusions $S$, and $q_h=Nh$. Ideals are counted once; physical element rows retain all original unit and $S$ labels. The rows are the complete retained nonunit sixth-power-free physical rows $q_u\asymp Q$, including primitive inducing orders properly dividing six. The source’s finite exceptional-row treatment remains separate. $Q$ is physical norm, never the norm of a primitive conductor.

Fix an original presentation ${\psi }_u(h)=\nu (h){\chi }_h(u{)}^{\pm 1}$, with conjugation as the meaning of the negative orientation even at zero. Use the same presentation and selected norm height in all three slots. For original smooth annular profiles, put

$$\begin{array}{rl}D& =Q^r,N=Q^m,L=D{N}^2=Q^x,x=r+2m,\\ M_u& =D^{-1/2}\sum _d\mu (d)V_M(q_d/D){\psi }_u(d),\\ S_u& =N^{-1/2}\sum _n{V}_S(q_n/N){\psi }_u(n),\\ c(h)& =\sum _{d{n}_1{n}_2=h}\mu (d)V_M(q_d/D)V_S(q_{n_1}/N)V_S(q_{n_2}/N),\\ A_u& =M_u{S}_u^2=L^{-1/2}\sum _{h}c(h){\psi }_u(h).\end{array}\text{\hspace{1em}(M1)}$$

The last equality is exact zero-extended multiplicativity. There is no coprimality condition between the slots. All original row-collision zeros, small-prime overlaps, nonsquarefree columns and sixth-power tails remain. Annular weights prevent using an unrestricted $\mu \ast 1\ast 1$ identity to replace $c$. Fixed finite multiplicative column corrections may be absorbed as in the owner; they do not change the three-slot convolution.

Write

$$r_{\ast }=\frac{\sqrt{6}}{3},m_{\ast }=\frac{\sqrt{6}}{6},x_{\ast }=\frac{2\sqrt{6}}{3},T_{\ast }=\frac{5\sqrt{6}-4}{6}.\text{\hspace{1em}(M2)}$$

The missing estimate is a strict saving below $Q^{T_{\ast }}$ for ${\sum }_u|A_u{|}^2$, on an open neighborhood of $(r_{\ast },m_{\ast })$, with the original finite labels, smooth parameters and polynomial height control. Every upper estimate below carries one original finite seminorm envelope ${\mathcal{P}}_J^2$ and one polynomial height factor $(1+H{)}^C$. The constants and finite orders are uniform on the stated compact length ranges. They are fixed before the small source height exponent is chosen; they are not separate reserves per slot, column decomposition or label.

A paid large-prime collision term

Let $P_0=Q^{1/3}$ and define the internal coefficient split

$$c_{\mathrm{hi}}(h)=c(h)1_{\exists p\notin S:q_p\ge P_0,p^2\mid h},c_0=c-c_{\mathrm{hi}},A_{b,u}=L^{-1/2}\sum _h{c}_b(h){\psi }_u(h),b\in \{0,\mathrm{hi}\}.\text{\hspace{1em}(M3)}$$

Conditional on de Faveri’s theorem and the owned transport, throughout

$$|r-r_{\ast }|<10^{-3},|m-m_{\ast }|<10^{-3},\text{\hspace{1em}(M4)}$$

one has, for every prescribed $\epsilon >0$,

$$\begin{array}{rl}\sum _u|A_{\mathrm{hi},u}{|}^2& {\ll }_{\epsilon }{Q}^{-1/3+\epsilon }{F}_6(Q,L){\mathcal{P}}_J^2(1+H{)}^C\\ & {\ll }_{\epsilon }{Q}^{T_{\ast }-1/100+\epsilon }{\mathcal{P}}_J^2(1+H{)}^C,\\ F_6(Q,L)& =Q+L+Q^{5/6}{L}^{1/3}+Q^{1/3}{L}^{5/6}.\end{array}\text{\hspace{1em}(M5)}$$

Proof. The actual coefficient is supported on $q_h\asymp L$ and is bounded by a fixed profile constant times $\tau_3(h)\ll_\varepsilon Q^\varepsilon$.Write$h=ab^6$uniquelywith$a$ sixth-power-free, without imposing $(a,b)=1$. On M4 one has $1<x<2$. For sufficiently large $Q$, a prime of norm at least $P_0$ cannot divide $b$, since then $q_h\ge P_0^6=Q^2>CL$. Thus the collision survives in $a$. For $B=CL/q_b^6$, elementary ideal counting gives

$$\#\{a:q_a\le B,\exists p,q_p\ge P_0,p^2\mid a\}\ll B\sum _{q_p\ge P_0}{q}_p^{-2}\ll B/P_0.\text{\hspace{1em}(M6)}$$

Only primes with $q_p^2\le B$ enter the count. The bound uses ordinary ideal counting and its convergent tail; no new rough-prime sieve is invoked. Consequently, for each fixed $b$,

$$\sum _a|L^{-1/2}{c}_{\mathrm{hi}}(a{b}^6){|}^2{\ll }_{\epsilon }{Q}^{\epsilon }{P}_0^{-1}{q}_b^{-6}{\mathcal{P}}_J^2.$$

The exact character at this column is ${\chi }_a(u{)}^{\pm 1}{1}_{(b,u)=1}$, with its remaining original zeros and fixed twist $\nu (b^6)$. Apply the owned finite-family transport and Theorem 1.1 at row norm $Q$ and column norm at most $CL/q_b^6$. Enlarge $(b,u)=1$ only within this fixed-$b$ positive squared norm. The four sieve terms are monotone in column length, hence bounded by a fixed multiple of $F_6(Q,L)$. Minkowski sums the row norms with weight $q_b^{-3}$; ${\sum }_b{q}_b^{-3}<\infty$. This proves the first bound.

For $1<x<2$ the largest exponent in $F_6(Q,Q^x)$ is $J_6(x)=1/3+5x/6$, so the collision exponent is $5x/6$. At $x_{\ast }$ its margin below $T_{\ast }$ is

$$T_{\ast }-5x_{\ast }/6=\frac{5\sqrt{6}-12}{18}.$$

On M4, $x<x_{\ast }+3/1000$ and

$$\frac{5\sqrt{6}-12}{18}-\frac{1}{400}-\frac{1}{100}>0.$$

This proves M5, including fixed support constants and sufficiently large $Q$; bounded remaining $Q$ is absorbed in the constant. The indicator in M3 commutes with all original profile derivatives. Apply the owned finite parameter Sobolev argument after the same estimates for those derivatives. It preserves the original envelope and one polynomial height cost.

The split includes collisions between every pair of slots and triple collisions. On M4 each plain slot contains at most one copy of a prime $q_p\ge P_0$, since $2/3>m$; the Möbius slot is squarefree. There is no roughness condition on the other prime factors. In particular, $c_0$ retains every squarefree column and every column whose squared prime divisors are smaller than $P_0$, with all small-prime gcds and sixth-power tails. It is not a total-squarefree projection or a replacement row family.

Exact remainder equivalence and interference

On the identical complete physical row space, $A=A_0+A_{\mathrm{hi}}$. Both triangle inequalities give

$$\Vert A{\Vert }_2\le \Vert A_0{\Vert }_2+\Vert A_{\mathrm{hi}}{\Vert }_2,\Vert A_0{\Vert }_2\le \Vert A{\Vert }_2+\Vert A_{\mathrm{hi}}{\Vert }_2.\text{\hspace{1em}(M7)}$$

Under M5, existence of a strict exponent saving below $T_{\ast }$ for $A$ is equivalent to existence of one for $A_0$, after decreasing the saving below $1/100$. This statement includes the original uniformity and costs; it is not a proof of either saving. In particular,

$$\Vert A{\Vert }_2^2=\Vert A_0{\Vert }_2^2+\Vert A_{\mathrm{hi}}{\Vert }_2^2+2\Re ⟨A_0,A_{\mathrm{hi}}⟩.$$

If $\Vert A_0{\Vert }_2^2\ll Q^{T_{\ast }-\eta +\epsilon }$ with the common costs, Cauchy bounds the cross term by those costs times $Q^{T_{\ast }-(\eta +1/100)/2+\epsilon }$. Without that remainder estimate this interference has not independently been paid below $T_{\ast }$.

The fixed-order owner’s lower-order-block proof uses only divisor bounds, the owned family/profile transport and row/column scales at least one. It therefore applies to $c_0$ at column scale $L$, including its original finite corrections ${\omega }_j(h)$. In its notation each of $H_{2,j}/L,H_{3,j}/L,H_{6,j}/L$ is $\ll Q^{1+\epsilon }$ with the same costs. The exact identity is still $L{\sum }_u|A_{0,u}{|}^2=H_{2,j}+H_{3,j}-H_{6,j}+E_{6,j}$ on each label, where the $H$ on the right are unnormalized. Thus the unpaid part can also be expressed as the complete Hermitian actual-order-six contraction $E_{6,j}/L$. Its original ratio-conductor zeros remain. The individual signed lower-order contractions need not be positive, and none of the owner’s special degree-two pure-power consumers is asserted for this triple coefficient.

A lower bound for the complete remainder coefficient

For every fixed nonzero complex smooth annular $V_M,V_S$, on any compact subset of $0<m<r<3m$ containing M4, for sufficiently large $Q$,

$$\sum _h|c_0(h){|}^2{\gg }_{V_M,V_S}\frac{L}{(\log Q{)}^4}.\text{\hspace{1em}(M8)}$$

The constants may also depend on the compact set and fixed arithmetic data. This is an assertion about the actual coefficient, not the row moment or a sixth-power-compressed coefficient with cancellations between $b$ values.

Proof. Set $P_1=Q^{(r+m)/2}$ and $P_2=Q^{(r-m)/2}$. The strict wedge gives $P_2\ll N\ll P_1$, with fixed power gaps on the compact set. Choose four distinct good primes with norms $P_1,P_2,N,N$ up to fixed relative factors. Choose their small normalized intervals so that $q_{p_1}{q}_{p_2}/D$ lies in an interval on which $|V_M|$ is bounded below, and both $q_{p_3}/N,q_{p_4}/N$ lie where $|V_S|$ is bounded below. Such intervals exist even for complex profiles. The last two can be chosen disjoint inside a nonzero interval.

For $h=p_1{p}_2{p}_3{p}_4$, the only divisors in the plain annulus are $p_3,p_4$: the unit and $p_2$ are too small, $p_1$ is too large, and every divisor with at least two primes is too large. Hence the complete original coefficient has exactly two summands, both with $d=p_1{p}_2$ and $\mu (d)=1$:

$$c_0(p_1{p}_2{p}_3{p}_4)=2V_M(q_{p_1}{q}_{p_2}/D)V_S(q_{p_3}/N)V_S(q_{p_4}/N).\text{\hspace{1em}(M9)}$$

There are no other divisor terms available to cancel this value. The fixed-field prime-ideal theorem, in the form of the pinned primary’s lem:ray-prime-normalizer, gives respectively $\gg P_1/\log Q,P_2/\log Q,N/\log Q,N/\log Q$ primes in these fixed relative intervals. Unique ideal factorization gives bounded multiplicity for $h$. There are therefore $\gg L/(\log Q{)}^4$ such squarefree $h$, proving M8. If fixed ray classes are needed, choose compatible fixed classes and use the corresponding fixed extension; no growing-ray prime theorem is invoked. The lower constant is for fixed profiles, not arbitrary profiles varying with $Q$ or height.

Multiplication by an allowed good-ideal ray character has modulus one; negative orientation has the same norm. In particular, ${\sum }_h|c_0(h){|}^2\ll L{Q}^{-\delta }$ is false for every fixed $\delta >0$. At $x_{\ast }$ the generic sieve exponent is

$$J_6(x_{\ast })=\frac{1}{3}+\frac{5\sqrt{6}}{9},J_6(x_{\ast })-T_{\ast }=1-\frac{5\sqrt{6}}{18}>0.\text{\hspace{1em}(M10)}$$

One cannot pay that gap by a power saving in the complete coefficient norm. M8 does not prove sharpness of the generic sieve on $c_0$, give a row-moment lower bound, exclude inter-column cancellation, or refute the mixed-moment target. The four-prime packet diagnoses the full coefficient; it is not substituted for the complete physical rows or polynomial.

The actual quadratic residual after pair Poisson

Reuse the generic Poisson construction in the fixed-order owner. First majorize the complete positive row norm by its fixed smooth nonnegative annular majorant, and only then expand it. For an actual ratio pair let $\xi$ be its primitive inducing character with full conductor $f$, and let $R$ contain every original zero-support prime absent from $f$. The ratio is $\xi (v)1_{(v,R)=1}$. The owner’s finite sixth-power-free expansion in $\ell$, divisor sum over $r\mid R$, primitive ramified factors and entire nonzero dual sum remain unchanged. There is no comparison of signed contractions by enlarging rows.

For a diagnostic pair take squarefree coprime $h,k$ in the same original primary ray class. The existing formulas give $f=hk$ and root number $\eta {\gamma }_1(h){\gamma }_{-1}(k)$, where ${\gamma }_{-1}(k)={\chi }_k(-1)\overline{{\gamma }_1(k)}$ and $\eta$ is the retained fixed-class phase. For positive orientation the actual paired coefficient, apart from the unchanged transform and $\ell ,r$ factors, is $\eta {\chi }_k(-1)B_a(h)\overline{B_a(k)}$, with

$$B_a(h)={\omega }_j(h)c_0(h){\gamma }_1(h)\overline{{\chi }_h(a)}.\text{\hspace{1em}(M11)}$$

It vanishes when $(a,h)\ne 1$. On M9, writing $V=V_M(q_{p_1}{q}_{p_2}/D)V_S(q_{p_3}/N)V_S(q_{p_4}/N)$,

$$\begin{array}{rl}{B}_a(h)& =2V{\omega }_j(h){\gamma }_1(h)\prod _{i=1}^4\overline{{\chi }_{p_i}(a)},\\ {\gamma }_1(h)& =\prod _i{\gamma }_1(p_i)\prod _{i<j}{\chi }_{p_i}(p_j){\chi }_{p_j}(p_i).\end{array}\text{\hspace{1em}(M12)}$$

Both allowed plain-slot assignments give the same value, including every Gauss and reciprocity phase. Their quadratic residual does not cancel. No scalar ${\gamma }_1(h)$ formula is being asserted for nonsquarefree $h$. For the $\ell =1$ term and a fixed $r\mid \mathrm{rad}S$, the natural dual scale is $q_a\asymp L^2/Q$; at M2 its $Q$ exponent is $2x_{\ast }-1$. This observation discards none of the other $\ell ,r,a$ terms.

Here is an exact finite coefficient obstruction. Fix such an $h$ and a disjoint same-class squarefree $k$ with nonzero original coefficient. Let $G$ be the full unit group modulo $hk$ and the fixed bad modulus. By CRT there is $z\in G$ with ${\chi }_{p_1}(z)=-1$, and $z=1$ at every other good prime in $hk$ and at the bad modulus. A good sextic local character is surjective. Multiplication $a\mapsto az$ preserves all local cubic data ${\chi }_p(a{)}^2$ and fixed bad-ray labels, but

$$F(a):=B_a(h)\overline{B_a(k)},F(az)=-F(a).\text{\hspace{1em}(M13)}$$

Thus the orthogonal projection of $F$ onto cubic-data-only functions on the uniform finite group $G$ is zero. Indeed, every such function $g$ is invariant under $z$, so averaging and translating by $z$ gives $⟨F,g⟩=-⟨F,g⟩=0$. In particular,

$${\mathbb{E}}_{a\in G}|F(a)-g(a){|}^2=|{\omega }_j(h)c_0(h){\omega }_j(k)c_0(k){|}^2+{\mathbb{E}}_{a\in G}|g(a){|}^2.\text{\hspace{1em}(M14)}$$

This rejects an exact cubic-only coefficient replacement and a small mean-square coefficient error on that complete finite group. It gives no lower bound on the actual weighted dual annulus or physical-row moment. Multiplication by $z$ need not preserve that annulus.

The general moving quadratic/cubic coupling is already in the owner. In its complete unpaid order-six block, the good conductor of ${\xi }^3$ is the concrete parity difference

$$D_2(h,k)=\prod _{v_p(h)-v_p(k)\mathrm{odd}}p=\frac{r_2(h)r_2(k)}{\gcd (r_2(h),r_2(k){)}^2}>1.$$

Small-prime gcds and sixth-power tails do not remove this component. On the disjoint squarefree test its conductor is $hk$. This identifies the part that must be estimated jointly, rather than supplying its bound.

Effective numerator in the direct plain-slot action

Fix squarefree pairwise coprime $d,n_1,n_2$ on the actual coefficient support and a dual $a$ coprime to their product. After the owned Gauss CRT and fixed ray decomposition, the $n_1$ dependence is exactly

$${\gamma }_1(n_1)\overline{{\chi }_{n_1}(a)}{\chi }_{n_1}(d{n}_2{)}^2={\gamma }_1(n_1)\overline{{\chi }_{n_1}(A_{\mathrm{eff}})},A_{\mathrm{eff}}=a(d{n}_2{)}^4.\text{\hspace{1em}(M15)}$$

The equality is $-4\equiv 2(\mathrm{mod}6)$. Fixed finite-class factors are retained before summation. If the primary/E-primary transport contributes a fixed bad-prime numerator $b_j$, the literal supplier numerator is $b_j{A}_{\mathrm{eff}}$, not $A_{\mathrm{eff}}$ with that factor discarded. This bounded factor does not change any exponent below.

Gao–Zhao v2 Lemma 4.1, followed by the existing finite-class transport, puncture inclusion-exclusion and annular partial summation, supplies

$${\ll }_{\epsilon }{\mathcal{P}}_J\left(q_{A_{\mathrm{eff}}}^{1/6+\epsilon }{N}^{2/3+\epsilon }+q_{A_{\mathrm{eff}}}^{1/14+\epsilon }{N}^{6/7+\epsilon }\right),\text{\hspace{1em}(M16)}$$

with the owned divisor/subpower puncture costs. This is a sum over the complete squarefree plain $n_1$ annulus for fixed outer data, not a theorem for a prime-restricted $n_1$ subset, arbitrary modulus-dependent weights or the Möbius-weighted inverse slot. Both displayed terms require $q_{A_{\mathrm{eff}}}<N^{2-\delta }$ for a fixed power saving over $N$.

On M9 take $d=p_1{p}_2,n_2=p_4$. The three cofactor primes occur to exponent four in $A_{\mathrm{eff}}$, so even reducing sixth powers of $a$ does not remove them when $(a,h)=1$. At the critical lengths, even bounded $q_a$ gives

$$q_{A_{\mathrm{eff}}}\asymp (DN{)}^4=Q^{2\sqrt{6}},N^2=Q^{\sqrt{6}/3}.$$

Writing $q_a=Q^y$, the two exponents supplied for that inner sum are

$$\frac{2x_{\ast }}{3}+\frac{y}{6},\frac{2r_{\ast }+8m_{\ast }}{7}+\frac{y}{14}.\text{\hspace{1em}(M17)}$$

At $y=0$ both exceed $m_{\ast }$: their ratios to $m_{\ast }$ are respectively $8/3$ and $12/7$. At the natural $y=2x_{\ast }-1$ they are larger still. The strict failure persists on an open critical neighborhood. Thus this direct invocation provides no saving even before paying outer variables, height costs or the remaining Poisson terms. It is a failure of the supplied bound for this action, not a lower bound or a counterexample to the complete moment. Writing the fourth power as an inverse square modulo each varying $n_1$ would change a fixed integral numerator theorem into a denominator-uniform one; no such theorem has been supplied.

The inverse slot has the legitimate owned cubic conversion, but retains a quadratic factor. With $s=n_1{n}_2$, ${\kappa }_d={\chi }_d^2$, ${\lambda }_d={\chi }_d^3$ and the pinned primary’s phase functions $\alpha (d),G(d),\mathcal{R}(d,s)$, on $(a,ds)=1$ its M11 value at M9 is

$$2V{\omega }_j(h)\mathcal{R}(d,s)\alpha (d)G(d){\gamma }_1(s)\overline{{\chi }_s(a)}{\lambda }_d(a)\overline{{\tilde{g}}_3(-3as,d)}.\text{\hspace{1em}(M18)}$$

This uses the existing identities $\mu (d){\gamma }_1(d)=\alpha (d)G(d)\overline{{\gamma }_2(d)}$ and the exact additive-character transport $-3as$; here ${\tilde{g}}_3$ has DDDS’s own additive convention. This transport is reused as documented in the owner with its v2 citation; it is not a nonsquarefree-column identity. The separate comparison with DDDS v3 §6 below does not identify the two revisions. That section estimates its particular cubic Gauss series, with $1<\Re s\le 5/4$ and mean square $q_h^{1/2}{M}^{1+\epsilon }(1+|\Im s|{)}^2$. It does not remove ${\lambda }_d(a)$ or its moving quadratic conductor. The fixed-order owner’s B2 growing-ray bound retains an explicit extra conductor cost; that cost cannot be replaced by a constant as $a$ varies. These are supplier boundaries, not additional analytic estimates.

Conditional neighborhood moment and every detector witness

Assume that there exist fixed $\eta ,\rho >0$ such that, uniformly for

$$|r-r_{\ast }|<\rho ,|m-m_{\ast }|<\rho ,$$

every original fixed label and orientation and the entire original annular profile/height class satisfy

$$\sum _{u\mathrm{complete}\mathrm{physical}\mathrm{rows}}|M_u(Q^r){|}^2|S_u(Q^m){|}^4{\ll }_{\epsilon }{Q}^{T_{\ast }-\eta +\epsilon }{\mathcal{P}}_J^2(1+H{)}^C.\text{\hspace{1em}(H)}$$

The hypothesis includes quantitative uniformity for the finitely many parameter derivatives required by the source smooth calculus. $J,C$ are fixed independently of $Q$, row, label and length pair in this neighborhood. H is unproved. A pointwise bound for one fixed profile, one pair of lengths, a primitive-conductor family or a sparse physical sector does not satisfy H.

If instead a bound of this form is assumed only for the complete $A_0$ from M3, with parameters ${\eta }_0,{\rho }_0>0$, first set

$$\rho =\min \{{\rho }_0,1/2000\},\eta =\min \{{\eta }_0,1/200\}.\text{\hspace{1em}(H0)}$$

This is a smaller open intersection with M4’s paid collision neighborhood, not an extension of M5 to ${\rho }_0$. M5 and M7 then give H on this smaller neighborhood with the displayed $\eta$, retaining the cross term and common costs. All consumer choices below must use these reduced parameters. A remainder assertion outside that intersection alone has no such implication.

Use the existing common buffered bins from C93–C94, with the same fixed $\Theta$ for both actual C80 sides, all labels, orientations and conjugates. Fix $\alpha >{\alpha }_0=51/100$ and put

$$\begin{array}{rlrlrl}v& =2\alpha -1,& c_{\alpha }& =5/6-v,& t_0(\alpha )& =\frac{5}{5-2v},\\ r_0& =\frac{2t_0}{3},& m_0& =\frac{t_0}{3},& d(\alpha )& =\frac{31-32\alpha }{21-12\alpha }=\frac{15-16v}{3(5-2v)}\\ & & & =1-v{r}_0=\frac{1}{6}+c_{\alpha }{t}_0.& & \end{array}\text{\hspace{1em}(M19)}$$

Let

$$\begin{array}{rlrlrl}{\alpha }_{\ast }& =\frac{21-5\sqrt{6}}{12},& d_{\ast }& =\frac{16-5\sqrt{6}}{6},& t_0({\alpha }_{\ast })& =\frac{\sqrt{6}}{2},\\ \kappa & =d_{\ast }+2{\alpha }_{\ast }=\frac{37-10\sqrt{6}}{6},& B_{\ast }& =\kappa -1,& T_{\ast }& =2-d_{\ast }.\end{array}$$

Fix the explicit positive parameters

$$h=\min \{1/100,\rho /100,\eta /100\},\delta =h/16,t(\alpha )=t_0(\alpha )-h,|\alpha -{\alpha }_{\ast }|\le h.\text{\hspace{1em}(M20)}$$

On this interval $2/5\le v\le 1/2$, $1/3\le c_{\alpha }\le 13/30$, $25/21\le t_0\le 5/4$ and $7/12\le d(\alpha )\le 43/63$. In particular $1<t(\alpha )<3/2$ is a permitted detector setting, and $d(\alpha )-\delta >1/6$. Differentiating M19 on these compact ranges gives

$$|d(\alpha )-d_{\ast }|\le \frac{25}{12}h,|r_0-r_{\ast }|\le \frac{5}{6}h,|m_0-m_{\ast }|\le \frac{5}{12}h.\text{\hspace{1em}(M21)}$$

We show that H conditionally bounds every such complete bin by $Q^{d(\alpha )-\delta }$ after fixing sufficiently small internal losses. Fix an original label and a dyadic witness pair. The source supplies the same row, presentation, zero, height and simultaneous individual saturation for $M,S$. With aggregate geometric and counting errors allocated at most $\lambda \le h/4096$, Markov and the old moments give count exponents

$$\begin{array}{rlrl}A(r)& =E(r)-vr+\lambda ,& E(r)& =\max \{1,(1+5r)/6\},\\ B(m)& =1-2vm+\lambda ,& C(r,m)& =T_{\ast }-\eta -v(r+2m)+\lambda ,\end{array}\text{\hspace{1em}(M22)}$$

where $C$ is available only inside H’s neighborhood. Its denominator is justified by the simultaneous lower bound $|M{|}^2|S{|}^4\gg Q^{v(r+2m)-3ϵ}$; witnesses from different rows or different presentations have not been multiplied. The original length conditions imply

$$r\le t_0-h+\lambda ,r+m\ge t_0-h-\lambda .\text{\hspace{1em}(M23)}$$

The source also retains $m\le 1/2+O(ϵ)$ and the lower $r$ boundary; no equality $r+m=t$ or relation $m=r/2$ is imposed.

Long witnesses, including $r=1$. For $r\ge 1$, $E(r)-vr=1/6+c_{\alpha }r$. By M19 and M23,

$$A(r)\le d(\alpha )-c_{\alpha }h+(1+c_{\alpha })\lambda \le d(\alpha )-h/4.$$

The finite union loss of at most another $\lambda$ leaves more than $\delta$ saving. There is no condition on $m$ in this branch.

Short witnesses paid by one marginal. If $r<1$ and either

$$1-vr\le d(\alpha )-2\delta or1-2vm\le d(\alpha )-2\delta ,\text{\hspace{1em}(M24)}$$

the appropriate old moment pays the witness. Its two $\lambda$ losses leave at least $\delta$ saving. Equality in M24 is included.

Localization of every remaining short witness. Failure of both inequalities gives the separate upper bounds

$$r<r_0+2\delta /v,m<m_0+\delta /v.$$

Combine each with M23 and the other upper bound to get

$$r>r_0-h-\delta /v-\lambda ,m>m_0-h-2\delta /v-\lambda .\text{\hspace{1em}(M25)}$$

Using $v\ge 2/5$, $\delta =h/16$ and M21 proves $|r-r_{\ast }|<2h$ and $|m-m_{\ast }|<2h$. Since $2h\le \rho /50<\rho$, every remaining pair lies strictly within H. This step uses both marginal failures separately, on the full allowed two-dimensional length region.

Mixed payment of the remaining witnesses. The identity $r+2m=2(r+m)-r$ and the preceding upper bound for $r$ imply

$$r+2m\ge \frac{4t_0}{3}-2h-2\delta /v-2\lambda .$$

Since $v(4t_0/3)=2(1-d(\alpha ))$, M22 yields

$$C(r,m)\le 2d(\alpha )-d_{\ast }-\eta +2vh+2\delta +(1+2v)\lambda .$$

Including the finite union, this is at most $d(\alpha )-\delta$ if

$$\eta >d(\alpha )-d_{\ast }+2vh+3\delta +(2+2v)\lambda .$$

The right side is bounded above by

$$\left(\frac{25}{12}+1+\frac{3}{16}+\frac{3}{4096}\right)h<4h\le \eta /25<\eta .$$

This proves the asserted complete-bin saving with unused reserve.

Boundaries and actual witness selection. The long case contains $r=1$, where $E$ is continuous. At $m\ge 1/2-\lambda$ the plain moment gives at most $1-v+2v\lambda$ before its counting error, while

$$d(\alpha )-(1-v)=v(1-r_0)\ge 1/15.$$

Thus the $m=1/2$ boundary and its original small error strip are paid; the zero-slot plain fourth moment has no bounded-nonnegative-length restriction. With $m=0$, a short $r<1$ cannot satisfy M23, since $t_0-h-\lambda >1$. The lower boundary $r=t-1/2-O(ϵ)$, the upper boundary $r=t+O(1/\log Q)$ and every intermediate pair are covered by the same cases.

Partition each bin by its original finite label and orientation and its $O((\log Q{)}^2)$ dyadic length pairs. Rowwise profile and height choices are handled before summing rows: apply the source parameter Sobolev inequality to $F_u(\theta )=M_u(\theta )S_u(\theta {)}^2$. Every derivative is a finite sum of $M_{u,j}{S}_{u,k}{S}_{u,l}$, and

$$|M_{u,j}{S}_{u,k}{S}_{u,l}{|}^2\le \frac{1}{2}|M_{u,j}{|}^2(|S_{u,k}{|}^4+|S_{u,l}{|}^4).$$

H for the original derivative-stable annular class controls these terms with the same $Q$ exponent. Derivatives change profiles, not arithmetic coefficients. The inverse cutoff parameter lies in a fixed compact interval, the real exponent lies in a compact interval, and the original height parameters range over intervals of size $O_e(H_1)$. The finite Sobolev covers cost one fixed polynomial in $1+H_1$. Analogous owned arguments apply to the marginal moments. Pointwise profile estimates without these quantitative derivative costs would leave this step open. No arbitrary row-dependent arithmetic weights or uncontrolled nets occur.

The original common $\Theta$ and buffered bins are fixed once, before choosing $t=t(\alpha )$. The original buffered maximum is independent of $t$; changing that permitted detector setting in finitely many bins only changes its cutoff/smooth parameters. Extra ${\Pi }_i,a$ deletions remain in C81–C94, and do not redefine $\Theta$ or the detector zeros.

Quantified complete C93–C97 consumer

Continue to assume H, the old detector and moments, C77–C97’s source premises and the actual original coupled-envelope premise. Retain precisely C87’s bare branch: original $W_0$, corefree/common-$e=1$, $C=1$, source projector $\mathfrak{b}=1$, $q=L=U$, $D_{\mathrm{src}}=1$, $g=f_i=1$, $N_i=X=U$, ${\zeta }_i=1$ and no additional angular multiplier, with R43’s genuine fixed ideal-ray quotient and original total class. These capital $D=U^{1/2},A=U^{1/12}$ consumer scales are distinct from the detector lengths in M1. Use the literal $B_i=B_{i,0}$ of C80–C81, $b_A=\mu +K_{A;0}$, and all four $\mu /K$ choices on both sides. Keep the original $\xi$, signs, phases, product squarefreeness, class-selector mass, defining-modulus and collision zeros, and every original ${\Pi }_i$ mask.

Define the positive reserves

$$\begin{array}{rl}G=\min \{& \delta ,(8/5)h^2,\kappa -25/12,\kappa -101/50,\kappa -57/28,\\ & \kappa -2,\kappa -23/12,\kappa \},\Gamma =G/4.\end{array}\text{\hspace{1em}(M26)}$$

Every entry is strictly positive; in particular $\kappa -25/12=(49-20\sqrt{6})/12>0$. After the finite loss choices below, for every final $\epsilon >0$ one finite original order $J$ satisfies

$$M_i^{>1}(R)\ll {\mathcal{H}}_{i,J}(R{)}^{2}R{U}^{\kappa -\Gamma +\epsilon },1\le R\le U.\text{\hspace{1em}(M27)}$$

This is a conditional consequence for the complete specified nonunit branch, not the missing mixed estimate itself.

Counts and every repeated frequency. Set ${\tilde{d}}_{\alpha }=d(\alpha )-\delta$ for $|\alpha -{\alpha }_{\ast }|\le h$, retain the old $\mathcal{R}(\alpha )$ elsewhere, and retain ${\tilde{d}}_{{\alpha }_0}=1$ at the floor. The old high branch is still $\mathcal{R}(\alpha )=2(1-\alpha )$ for $\alpha \ge 11/12$. For the lower density branch the exact complement identity is

$$\kappa -[d(\alpha )+2\alpha ]=\frac{8(\alpha -{\alpha }_{\ast }{)}^2}{7-4\alpha },{\alpha }_0<\alpha \le 11/12.\text{\hspace{1em}(M28)}$$

This follows by substitution into M19; the numerator is the square of $\alpha -{\alpha }_{\ast }$ with the displayed coefficient. Since $7-4\alpha <5$, outside the improved interval it gives a reserve at least $(8/5)h^2$. Inside, M19 and M28 give ${\tilde{d}}_{\alpha }+2\alpha \le \kappa -\delta$. No assumption that all rows have critical $\alpha$ is made.

Use C95 unchanged except for this count exponent. It counts every

$$k=ϵ\mathrm{gen}(e_{S}r{a}^6)$$

from C79, with all six units, original finite $S$ labels, every integral ideal $a$, its $S$ factors and every overlap with $r$. There is no restriction $a=1$, $(a,r)=1$ or squarefreeness of $a$. The physical reduced part $r$ is good sixth-power-free; lower inducing orders remain. For $U^{1/5}\ll Q\ll RU\ll U^2$, the source row norm is $Q$ and the consumer columns stay at $U^{1/2}$, not $Q^{1/2}$. The resulting C95 bound is

$${\mathcal{H}}_{i,J}(R{)}^2(RU{)}^{1/6}{Q}^{{\tilde{d}}_{\alpha }-1/6}{U}^{2\min (\alpha ,7/8)+\text{small loss}}.\text{\hspace{1em}(M29)}$$

For ${\tilde{d}}_{\alpha }\ge 1/6$, use $Q\ll RU$ and ${\tilde{d}}_{\alpha }\le 1$ to replace this by $\mathcal H_{i,J}(R)^2R^{\widetilde d_\alpha} U^{\widetilde d_\alpha+2\min(\alpha,7/8)+\text{small loss}}$, with $R^{{\tilde{d}}_{\alpha }}\le R$. The remaining branches are all strictly below $\kappa$ before losses:

• The floor has exponent $1+2{\alpha }_0=101/50$.
• The cap starts at $7/8$, outside M20. There $d(\alpha )+7/4$ is maximized at $7/8$ and equals $2/7+7/4=57/28$.
• The other density branch has $\mathcal{R}(\alpha )+2\alpha =2$.
• If ${\tilde{d}}_{\alpha }<1/6$, retain C95’s negative-$Q$ argument, giving at most ${\mathcal{H}}_{i,J}(R{)}^2{R}^{1/6}{U}^{23/12+\text{small loss}}$; substituting an upper endpoint for a negative power would be invalid.
• For $1<q_r\le U^{1/5}$, retain C84’s ${\mathcal{H}}_{i,J}(R{)}^2{R}^{1/6}{U}^{25/12+\text{small loss}}$.

M28 and these reserves are precisely the entries of M26. Thus summing all original bins and physical dyads gives M27 once aggregate losses are at most $G/4$; the chosen $\Gamma =G/4$ leaves further unused reserve. The argument changes neither C94’s actual-coefficient contour nor any source density count outside the fixed critical interval.

One finite loss and height order. The following order of choices makes the count, suprema and complete consumer simultaneous. First fix H’s $\eta ,\rho$ (using H0 if applicable), then M20 and M26. Given a desired final exponent allowance $\xi >0$, choose

$$0<\lambda \le \min \{h/4096,G/10000,\xi /10000\}.$$

Let $C_{\mathrm{err}}\ge 1$ dominate the finitely many absolute detector error constants and loss coefficients on the compact ranges. Choose the preliminary detector/moment powers at most $\lambda /(1000C_{\mathrm{err}})$, reducing further for their fixed finite sums, so each aggregate in M22–M23 is at most $\lambda$. Choose one bin mesh

$$0<e<\min \{10^{-3},h/100,\lambda /(100000C_{\mathrm{err}}),e_0/2\},I=\lceil 1/e\rceil +2,$$

where $e_0$ is the source detector’s admissibility threshold for those powers. Fix all internal old/mixed moment and parameter-Sobolev orders now. The finite label/dyadic union costs at most one further $Q^{\lambda }$ for sufficiently large $Q$.

Let $C_H\ge 1$ dominate the sum of the now fixed polynomial height orders and source detector height requirements. Only now choose

$$0<\tau \le \min \{1/100,\lambda /(10^6{C}_{\mathrm{err}}{C}_H)\},H_1=Q^{\tau }.$$

Fixed multiples $C_e{H}_1$ satisfy the prescribed source height bounds for sufficiently large $Q$. Retain the original cumulative frequency fractions within the single source allowance; none is restarted at a later shift. Take $Q$ sufficiently large for all $O(1/\log Q)$ errors to be below $\lambda$ and all fixed/polylogarithmic count factors to fit their allocations. Since $U^{1/5}\ll Q\ll U^2$, the choices work uniformly over every retained physical dyad; $Q^{\lambda }\le O(U^{2\lambda })$. Bounded remaining $U$ is absorbed in the fixed constant.

Keep C94’s finite buffered contours for both literal columns and all four $\mu /K$ terms. Its actual puncture range remains $q_P\ll U^{C_{\Pi }}Q(RU/Q{)}^{1/6}\ll U^{C_{\Pi }+2}$. Allocate its contour-width, at-most-six $L$-factor, polynomial-deletion, old-count, bin and dyadic losses in total at most $G/4$. This is possible by reducing each primitive power by the fixed sum of its coefficients, as above. All constants may depend on $\eta ,\rho$ and the fixed data; no claim uniform as $\eta ,\rho \to 0$ is made.

Only after $\tau$ is fixed choose external Mellin/Fourier tail orders. For C94 take an integer $N_{\mathrm{tail}}$ with $N_{\mathrm{tail}}\tau /5>C_0+10$, where $C_0$ is its absolute-line exponent independent of that external order, and meet the original detector’s tail requirements as well. This does not increase the internal height order already used to choose $\tau$. Enlarge the final original seminorm order $J$ once for all these finite requirements and shell decay $B\ge 4$. Use the single original coupled measure $dc$ and its one base envelope ${\mathcal{N}}_{\mathrm{base},J}$ at this $J$. Its fixed or polylogarithmic size and actual availability at that finite order are premises, not consequences of H. No envelope is multiplied or reset per row, mask, $a$, label, bin, dyad, shell or component. An eta-only envelope still needs the existing R43–R46 conversion to this base envelope.

Entire weighted large-$R$ tail and restoration. For $R>U$, retain C96: the original divisor bound and all $O(RU)$ full frequencies give energy ${\mathcal{H}}_{i,J}(R{)}^{2}R{U}^{3+\epsilon }$. No extrapolation of density counts or polynomial-puncture ranges is made. The same original envelope has

$$\int |dc|{\mathcal{H}}_{1,J}(R){\mathcal{H}}_{2,J}(R)\ll {\mathcal{N}}_{\mathrm{base},J}(1+R{)}^{-B}.$$

Consequently the dyadic $R>U$ tail is $\ll {\mathcal{N}}_{\mathrm{base},J}{U}^{4-B+\epsilon }$, which for $B\ge 4$ is below the claimed $U^{\kappa -\Gamma +\epsilon }$ scale. For $1\le R\le U$, apply Cauchy to M27 for the two actual columns on the same complete nonunit frequency set, and integrate once against this same $dc$. Equal improvements on both sides preserve $\Gamma$, not $\Gamma /2$. The ball uses $R=1$, and shell decay sums every remaining $R$.

Finally reuse C97’s normalization exactly once: $q/L^3=U^{-2}$ cancels the raw Gauss-pairing $U^2$, and $D^{-2}=U^{-1}$ restores the two original $D^{-1}$ factors. Thus for precisely the original signed local bare nonunit contribution,

$$\begin{array}{rl}|{\mathcal{E}}_0^{>1}|& \ll {\mathcal{N}}_{\mathrm{base},J}{U}^{\kappa -\Gamma +\epsilon },\\ D^{-2}|{\mathcal{E}}_0^{>1}|& \ll {\mathcal{N}}_{\mathrm{base},J}{U}^{B_{\ast }-\Gamma +\epsilon },\\ {\lambda }_{\mathrm{new}}& =\kappa -2-\Gamma =B_{\ast }-1-\Gamma <B_{\ast }-1.\end{array}\text{\hspace{1em}(M30)}$$

Taking $\epsilon <\Gamma$ displays a strict reserve under all named premises. There is no extra area, inverse Gauss, division by six or second $U^2$ factor and no unweighted infinite-energy assertion.

A paid short-cofactor Type-I application

Under the named source premises, reuse C25’s exact smooth split on M1’s rows and presentation. Fix $\mathrm{supp}{V}_M\subset [a_V,b_V]$ with $0<a_V<b_V$, and the same smooth $0\le W\le 1$, equal to one up to one and zero from two onward. Set $A=Q^{1/24}$ and $z=(b_{V}D{)}^{1/2}$. Eventually $2A<z<a_{V}D$, uniformly on M4. Writing the reused coefficient explicitly at these scales gives

$$\begin{array}{rl}{\lambda }_{\mathrm{II}}(d)& =-{\mu }^2(d)\sum _{\begin{array}{c}{r}_0{s}_{0}c=d\\ q_{r_0},q_{s_0}\le z\end{array}}\mu (r_0)\mu (s_0)(1-W(q_{r_0{s}_0}/A)),\\ M_{\mathrm{I}}(u)& =-D^{-1/2}\sum _{q\mathrm{sf}}\mu (q)\tau (q){\psi }_u(q)W(q_q/A)\sum _{\begin{array}{c}c\mathrm{sf}\\ (c,q)=1\end{array}}{\psi }_u(c)V_M(q_q{q}_c/D),\\ M_{\mathrm{II}}(u)& =D^{-1/2}\sum _d{\lambda }_{\mathrm{II}}(d){\psi }_u(d)V_M(q_d/D),M_u=M_{\mathrm{I}}(u)+M_{\mathrm{II}}(u).\end{array}\text{\hspace{1em}(M31)}$$

Here $\tau$ counts ideal divisors. This is C25 at different lengths. The squarefree $c$ condition and $(c,q)=1$ are internal to the Möbius slot; they impose no new condition between the three M1 slots.

The joint squarefree-Hecke quotient retains the Möbius coupling. Take $\beta =7/8$ throughout this application. For each retained M1 row, the joint series on $\Re s,\Re t>1$ has C25–C26’s local factor:

$$\begin{array}{rl}{\mathcal{F}}_u(s,t)& :=\sum _{\begin{array}{c}q,c\mathrm{sf}\\ (q,c)=1\end{array}}\frac{\mu (q)\tau (q){\psi }_u(qc)}{q_q^s{q}_c^t}=\prod _p(1-2x_p+y_p)\\ & =\frac{L(t,{\psi }_u)}{L(s,{\psi }_u{)}^{2}L(2t,{\psi }_u^2)}{\tilde{\mathcal{H}}}_u(s,t),\\ {\tilde{\mathcal{H}}}_u(s,t)& =\prod _p{\tilde{\mathcal{H}}}_{u,p}(s,t),\\ {\tilde{\mathcal{H}}}_{u,p}(s,t)& =\frac{1-2x_p+y_p}{(1-x_p{)}^2(1+y_p)}=1-\frac{x_p^2}{(1-x_p{)}^2}+\frac{2x_p{y}_p}{(1-x_p{)}^2(1+y_p)},\\ x_p& ={\psi }_u(p)q_p^{-s},y_p={\psi }_u(p)q_p^{-t}.\end{array}\text{\hspace{1em}(M32)}$$

At each prime this is C26’s correction divided by $1-y_p^2$; the squarefree denominator $L(2t,{\psi }_u^2{)}^{-1}$ is exposed exactly. There are no pure $y_p$ monomials in the correction. The geometric denominators give a row-independent coefficient absolute-mass majorant on every compact subdomain of

$$\Re s>1/2,\Re t>0,\Re (s+t)>1,$$

since, writing $\sigma =\Re s,\theta =\Re t$,

$$|{\tilde{\mathcal{H}}}_{u,p}(s,t)-1|{\ll }_{\sigma ,\theta }{q}_p^{-2\sigma }+q_p^{-(\sigma +\theta )}.$$

The same bound holds for the coefficient absolute mass of the local deviation after geometric expansion. The Euler product is holomorphic in this tube and bounded uniformly in the row and imaginary parts when the real parts range over a compact subset of the displayed region. Its reciprocal is never used, so its zeros require no exclusion. At an original zero prime ${\psi }_u(p)=0$, every local factor is one, also for ${\psi }_u^2$; squaring does not restore a value on a nonunit. A norm twist of height $t_u$ squares to a twist of height $2t_u$.

The pinned primary’s detector conductor bounds give primitive conductor ${\ll }_{\mathcal{A}}Q$ and deletion radical ${\ll }_{\mathcal{A}}Q$ for the original presentation. The primitive conductor of its squared finite-order character divides that of the character before squaring; the original deletion primes remain. Thus the same global Hecke strip and lem:logarithmic-control apply to the numerator and the reciprocal denominator, with their actual heights and the full finite-order Hecke premise.

There is no principal numerator among retained rows: a good prime of $u$ with valuation $1,\dots ,5$ gives local order $6/\gcd (6,v_p(u))>1$, which a fixed $S$-supported twist cannot cancel. Rows supported only on $S$ form the original bounded exceptional set; M1 already leaves their treatment separate. They are not newly deleted here. All nonprincipal lower inducing orders remain; a principal square, including on quadratic rows, is allowed.

For small fixed $v>0$, put $b_{\beta }=\beta +v<1$. On the right line $b_{\beta }$, logarithmic control gives a subpower upper bound for the primitive nonprincipal numerator. Its functional equation gives conductor power $b_{\beta }-1/2$ on the reflected line $1-b_{\beta }$. Strip interpolation, using lem:hecke-strip-growth for the growth condition, therefore yields, after restoring the original deleted factors,

$$|L(\sigma +iy,{\psi }_u)|{\ll }_{v,\epsilon }{Q}^{(b_{\beta }-\sigma )/2+\epsilon }(1+|y|+H{)}^{C_0},1-b_{\beta }\le \sigma \le b_{\beta }.\text{\hspace{1em}(M33)}$$

The conductor bound does not replace the physical row scale. Use the full imprimitive quotients throughout. By lem:deleted-euler-factors, all original deletion factors and their reciprocals cost $Q^{\epsilon }$ on the fixed positive real lines used here, uniformly in height.

Write $\hat{W}(w)={\int }_0^{\infty }W(x)x^{w-1}dx$ for $\Re w>0$. It is holomorphic and rapidly decreasing vertically on each fixed positive line: integration by parts puts its derivative on the compact transition interval $[1,2]$. The annular Mellin transform ${\hat{V}}_M(t)$ is entire and rapidly decreasing. Exact double Mellin inversion of M31 on absolute lines gives

$$M_{\mathrm{I}}(u)=-\frac{D^{-1/2}}{(2\pi i{)}^2}{\int }_{(1)}{\int }_{(2)}\hat{W}(w){\hat{V}}_M(t)A^w{D}^t{\mathcal{F}}_u(w+t,t)dtdw.\text{\hspace{1em}(M34)}$$

Here $s=w+t$, so both variables retain their original joint weight. First move $\Re t$ from $2$ to $\beta /2+v$ with $\Re w=1$. Then move $\Re w$ from $1$ to $\beta /2$ with $\Re t=\beta /2+v$. Both moves stay in

$$\Re s>\beta ,\Re t>\beta /2,\Re (s+t)>1,\Re (s-t)>0.$$

In the first move $\Re (s+t)\ge 1+\beta +2v$; in the second it is at least $3\beta /2+2v>1$. Thus M32’s correction stays in its absolute-mass domain. The existing global strip makes both $L(s,{\psi }_u{)}^{-2}$ and $L(2t,{\psi }_u^2{)}^{-1}$ holomorphic and subpower throughout. A principal squared character’s pole at $2t=1$ is a zero of its reciprocal; the norm-twisted pole has the same property. The retained numerator is nonprincipal and entire. The pole of $\hat{W}$ at zero is not crossed. Rapid Mellin decay and the existing global polynomial growth pay the finite joins and passage to infinite contours, including the coupled heights $\Im (w+t),2\Im t$ and the original norm twists.

On the final plane $\Re s=\beta +v$ and $\Re t=\beta /2+v$. The latter lies between M33’s two lines, and its numerator conductor exponent is exactly $((\beta +v)-(\beta /2+v))/2=\beta /4$. Absorb $D^v$ into the prescribed $Q^{\epsilon }$ on the compact length ranges by choosing $v$ sufficiently small. The joint integral therefore gives

$$\begin{array}{rl}|M_{\mathrm{I}}(u)|& {\ll }_{\epsilon }{Q}^{\beta /4+\epsilon }{D}^{(\beta -1)/2}{A}^{\beta /2}{p}_{M,J}(1+H{)}^{C_1},\\ \sum _u|M_{\mathrm{I}}(u){|}^2|S_u{|}^4& {\ll }_{\epsilon }{Q}^{1+\beta /2+\epsilon }{D}^{\beta -1}{A}^{\beta }{\mathcal{P}}_J^2(1+H{)}^C.\end{array}\text{\hspace{1em}(M35)}$$