bibkey: liu2026quasirhreduced
authors: Baiying Liu
year: 2026
title: Reduced-frequency applications to the actual C68 columns
doi: null
url: https://arxiv.org/abs/2610.12234v2
claim: Repo-derived source-conditional applications to actual bare-J4 C68 columns give the original-W0 endpoint for all nonunit reduced frequencies and strict bounds on physical reduced frequencies with a small valuation-one good-prime product, retaining all sixth-power repetitions and one original coupled envelope. The powerful sector retains its U^(27/13+epsilon) raw bound; the parameterized extension has raw exponent E((1+eta)/2) for each fixed 0<=eta<(13-5sqrt6)/3. The sharper dominant-annulus dependence and the small sector up to U^(1/5) for both original and exact redesigned cutoffs remain valid. An explicit mixed planar/log Fourier producer supplies the common envelope for fixed original local profiles; any preceding signed component/height aggregate still requires its own bound. Full C68, other branches, J40, signed Robin restoration and RH remain unresolved.
strata_touched: []
license: citation-only
triage: anchor

Reduced-frequency applications to the actual C68 columns

This repo-derived companion uses the notation and source conventions of the Liu source card, the complete-dual application, and the joint mixed-Mobius and residue companion. C1–C50 remain in the source card, C51–C76 in the complete-dual owner, and R1–R46 in the residue companion. C77–C119 belong only to this note. The deductions are conditional paper mathematics, without a source-proof audit or Lean certification. The actual mixed-sixth companion gives the complete-coefficient reductions and the quantified conditional neighborhood-moment interface to C93–C97.

A small reduced-frequency sector of the actual C68 columns

This is a conditional application of existing suppliers to the bare (J4)/(C57) columns in (C68). Retain the corefree/common-$e=1$ branch, $C=1$, source coprime-projector $\mathfrak{b}=1$, $q=L=U$ and $D_{\mathrm{src}}=1$. Thus $g=f_i=1$, $N_i=X=U$ and $K_{\mathrm{row}}\asymp RU$. Fix precisely R43’s genuine ideal-ray quotient $H$, its original total class, ${\zeta }_i=1$, and no additional angular multiplier. The original $\xi$, signs, squarefree/collision masks, profiles and norm heights remain. This does not allow the arbitrary coefficient vectors admitted by a generic large sieve.

The analytic premises are the pinned primary manuscript, thm:main and lem:logarithmic-control, in their full uniform finite-order form. In particular, for every fixed $\delta ,\eta >0$, primitive nonprincipal $\rho$ of conductor norm $Q$ satisfies

$$|L(s,\rho )|+|L(s,\rho {)}^{-1}|{\ll }_{\delta ,\eta }\{2Q(3+|\Im s|{)}^2{\}}^{\eta },\Re s\ge 7/8+\delta .\text{\hspace{1em}(C77)}$$

Both directions of this bound, including its conductor uniformity, are assumed; fixed-conductor polynomial bounds alone are not substituted. Principal functions have the source’s different regularization and pole convention. They will not occur in the two slots below. These are source premises, not an audit of the primary’s proofs.

The actual coefficients and their primitive characters. Treat the original $W_0$ and the exact redesigned $W_U$ separately. Write $b^{(0)}=b_A$ and $b^{(U)}=b_U$, respectively, on the original inverse annulus, with $A=U^{1/12}$. Their identities are C33–C38 and R34–R37:

$$b^{(\circ )}(d)=\mu (d)+K_{A;\circ }(d),K_{A;\circ }(d)=\sum _{e\mid d}\mu (e)2^{\omega (e)}{W}_{\circ }(q_e/A),\circ \in \{0,U\}.\text{\hspace{1em}(C78)}$$

For $W_U$ the law, clamp and coefficient are fixed at this same $U$; no law is reselected at a cofactor scale, and no comparison of the two signed realizations is asserted. R35/R41’s fixed real and logarithmic moments pay the dilation integrals. There is no maximum-support power of the clamp length.

Every nonzero element frequency has the unique factorization

$$k=ϵe_S(\mathbf{j})r{a}^6,ϵ\in {\mathcal{O}}^{\times },0\le j_p\le 5(p\in S),r\text{ good sixth-power-free},a\text{ an integral ideal}.\text{\hspace{1em}(C79)}$$

Use the same fixed generators as C58. The ideal $a$ includes all its $S$ factors and may overlap $r$; there is no $(a,r)=1$ restriction. The sector here is $1<q_r\le U^t$, with fixed $t>0$. Sixth powers give exactly ${\chi }_n(a^6)=1_{(n,a)=1}$, including collisions; their conductor triviality does not remove this mask. Units and $S$ labels remain separate from the ideal-ray quotient.

Let $B_{i,\circ }(k;{\Pi }_i)$ be the literal unnormalized inner column in (C68) on this frozen branch, with the actual compressed coefficient from (C57) and its original external profile. Opening that coefficient expresses it, for each original component, as

$$\sum _{\begin{array}{c}{d}_1{d}_2\mathrm{sf}\\ (d_1{d}_2,S{\Pi }_{i}ra)=1\end{array}}{b}^{(\circ )}(d_1)b^{(\circ )}(d_2){\vartheta }_{i,r,ϵ,\mathbf{j}}(d_1{d}_2)1_{[d_1{d}_2]={\sigma }_i}{F}_i(q_{d_1}/D,q_{d_2}/D).\text{\hspace{1em}(C80)}$$

Here $F_i$ is the given annular two-factor profile, including the original external product profile and separated norm heights. $\vartheta$ is the original orientation of ${\chi }_{\bullet }(r)$ times its fixed finite-ray twists, including $\xi$, the C56 change-of-form factor and the unit/$S$ label. This is the raw-column orientation of (C68); C56 already accounts for the original Gauss multiplier. No additional inverse Gauss factor is inserted. ${\Pi }_i$ denotes any of the original polynomial-size deletion masks. The argument is uniform in these masks, even when they depend on frozen row data; it does not choose them anew or differentiate their coefficients.

Expand only the actual total-class selector: $\mathbf1_{[n]=\sigma_i}=h^{-1}\sum_{\chi\in\widehat H} \overline{\chi(\sigma_i)}\chi(n)$. Its absolute coefficient mass is $1$. Each resulting presentation $\rho =\vartheta \chi$ has a primitive inducer ${\rho }^{\ast }$ with $Q_{\rho }{\ll }_{H,S}{q}_r$. Indeed, at any good prime with $v_p(r)=j\in \{1,\dots ,5\}$ the local sextic character has order $6/\gcd (6,j)>1$; every fixed ray twist is unramified there. Since $r\ne 1$, none of these inducers is principal. This uses the primary’s physical-row ramification and defining-modulus convention, not equality of conductor norms or a new lower bound on $Q_{\rho }$. It applies equally to the conjugate orientation. The complete original defining-modulus zeros, including redundant deletions, remain attached.

Two slots of the literal joint product. Reuse R3–R8 and R44 with two slots. For each of the four actual $\mu /K$ choices, let $M\subseteq \{1,2\}$, $T=\{1,2\}\setminus M$ and $P=\mathrm{rad}({\Pi }_{i}ra){|}_{p\notin S}$. Put $x_j={\rho }^{\ast }(p)q_p^{-s_j}$ and $y_j=q_p^{-z_j}$. The joint series is

$$\begin{array}{rl}{F}_{M,P,\rho }(\mathbf{s},\mathbf{z})& =\prod _{p\nmid SP}\left(1-\sum _{j\in M}{x}_j+\sum _{j\in T}{x}_j(1-2y_j)\right)\\ & =H_{M,P,\rho }(\mathbf{s},\mathbf{z})\prod _{j\in M}L(s_j,{\rho }^{\ast }{)}^{-1}\prod _{j\in T}L(s_j,{\rho }^{\ast })L(s_j+z_j,{\rho }^{\ast }{)}^{-2}.\end{array}\text{\hspace{1em}(C81)}$$

At an unexcluded prime, $H_p$ is the parenthesis times $\prod_{j\in M}(1-x_j)^{-1} \prod_{j\in T}(1-x_j)(1-x_jy_j)^{-2}$. At $p\mid SP$ replace only the parenthesis by $1$, retaining primitive local values, including zeros at ramified primes. Thus the full squarefree product and every deletion are encoded once. No local numerator is inverted. R4–R5 give normal convergence after cancellation of all linear terms whenever $\Re s_j>1/2$ and $\Re (s_j+z_j)>1/2$, and $H{\ll }_{\eta }{q}_P^{\eta }$ on closed substrips, uniformly in the character phases and heights.

Start the two $s$ lines at $2$ and all present $z$ lines at a fixed small $\delta >0$. Move both $s$ lines to $7/8+\delta$, leaving those $z$ lines at $\delta$. All numerator $L$ functions are nonprincipal and entire, so there are no $s=1$ residues; no cutoff pole is crossed. All reciprocal arguments remain in C77’s strip, and the whole joint correction stays in R5’s domain. C77 bounds the numerator as well as the reciprocal factors with arbitrarily small conductor/height powers. The scale cost, before allocating losses, is $D^{7/4+2\delta }{A}^{|T|\delta }$, at most $U^{7/8+7\delta /6}$. This avoids paying the polynomial $Q^{3/5}$ numerator bound below the uniform right strip.

Use the same finite Mellin/seminorm argument as R6–R8, with at most four variables. For $1\le R\le U$, $q_k\ll RU$, $q_r\le U^t$ and $q_{{\Pi }_i}\le U^{C_{\Pi }}$, both $Q_{\rho }$ and $q_P$ are polynomial in $U$ (in particular $q_a\ll U^{1/3}$). Choose the internal $\delta ,\eta$ small enough for this fixed polynomial range and the final $\epsilon$. For $W_U$, Mellin inversion first uses $W_0(t/x)$ under the actual fixed-$U$ product law. Each present $K$ slot costs $x^{\delta }$, and R35’s fixed real/log moments pay the same integrations and horizontal joins. Consequently both actual coefficient realizations satisfy

$$|B_{i,\circ }(k;{\Pi }_i)|{\ll }_{H,S,t,C_{\Pi },\epsilon }{\mathcal{H}}_{i,J}{U}^{7/8+\epsilon },1\le R\le U,1<q_r\le U^t.\text{\hspace{1em}(C82)}$$

Here ${\mathcal{H}}_{i,J}$ is its original side seminorm, with R41’s single moment enlargement for $W_U$. The finite Fourier sum is paid by Minkowski with its actual coefficient mass. Choose one finite $J$ for all four choices, masks, units/$S$ labels and shells. Original norm-height costs remain polynomial and are integrated against the original sufficiently rapid fixed log-Fourier decay. For the actual coupled component/height measure $dc$, require $\int|dc|,\mathcal H_{1,J}\mathcal H_{2,J} \le\mathcal N_{{\rm base},J}$,fixedorpolylogarithmicin$U$, with the original Fourier shell seminorms included. This is the common base-envelope contract of R43–R46. An envelope supplied only for its $\eta =h^{-2}$ component costs the fixed factor $h^2$ to recover this base envelope. No allowance restarts per $r,a$, mask or component; no derivative acts on $b^{(\circ )}$. For the fixed original-$W_0$ local inputs specified in C114–C119 below, that construction supplies this envelope; it does not supply an unspecified preceding signed aggregate.

Counting every repetition and restoring the original scale. Ordinary ideal counting, with all six units and the original finite $S$ labels, gives for the ball $0<q_k\ll RU$

$$\begin{array}{rl}\#\{k:1<q_r\le U^t\}& \ll 6\sum _{\mathbf{j}}{q}_{e_S(\mathbf{j})}^{-1/6}(RU{)}^{1/6}\sum _{\begin{array}{c}r\text{ as in C79}\\ 1<q_r\le U^t\end{array}}{q}_r^{-1/6}\\ & {\ll }_{K,S}{R}^{1/6}{U}^{1/6+5t/6}.\end{array}\text{\hspace{1em}(C83)}$$

This counts each full frequency once, including every admissible sixth-power repetition, overlaps with $r$, and the $S$ factors of $a$. It does not count each primitive inducer once and discard its repetitions. Squaring C82 with an internally smaller loss therefore yields the following restriction of the literal left side of C68:

$$\begin{array}{rl}{M}_{i,\circ }^{\le t}(R)& :=\sum _{\begin{array}{c}0<q_k\ll RU\\ k\text{ as in C79},1<q_r\le U^t\end{array}}|B_{i,\circ }(k;{\Pi }_i){|}^2\\ & \ll {\mathcal{H}}_{i,J}^2{R}^{1/6}{U}^{23/12+5t/6+\epsilon }\le {\mathcal{H}}_{i,J}^{2}R{U}^{2+{\lambda }_t+\epsilon },{\lambda }_t=-1/12+5t/6.\end{array}\text{\hspace{1em}(C84)}$$

Every character in this sum satisfies C68’s ${\psi }_k^{\ast }\ne 1$ test. This is a sector estimate, not $W({\lambda }_t)$ for its full row. At $t=1/5$ it has exponent $25/12$ and restricted $\lambda =1/12$.

Apply Cauchy to the two original columns on this same frequency set, then integrate their costs against the same $dc$. There are only the original fixed labels in this $C=\mathfrak{b}=g=1$ branch, and C68’s source relation bounds its frozen scalar. The source multiplier is exactly $q/L^3=U^{-2}$. Restoring the original unnormalized Gauss pairing contributes exactly $U^2$ as in C57–C59; these cancel once. Thus its shell/ball contribution costs ${\mathcal{N}}_{\mathrm{base},J}{R}^{1/6}{U}^{23/12+5t/6+\epsilon }$, before the original $D^{-2}$ restoration. No second $U^2$ is present. C56 fixes the self-dual planar measure; ideals are counted once, with no extra area factor or division by six.

The complete original Fourier-weighted shell sum is also paid. On a shell $q_k/U\asymp R\ge 1$, the same separated profile envelope has arbitrary fixed decay $R^{-B}$, including all needed derivatives. For $R\le U$, sum C84 against this decay. For $R>U$ use the original divisor bound $|b^{(\circ )}(d)|\ll {\tau }_3(d)$ instead of a nonuniform puncture estimate: an unnormalized two-factor column has absolute size $O(U^{1+\epsilon })$. C83 and the same decay bound its full energy tail by

$${\mathcal{N}}_{\mathrm{base},J}{U}^{13/6+5t/6+\epsilon }\sum _{R>U\mathrm{dyadic}}{R}^{1/6-B}\ll {\mathcal{N}}_{\mathrm{base},J}{U}^{7/3+5t/6-B+\epsilon }.\text{\hspace{1em}(C85)}$$

A single fixed $B\ge 4$ makes this negligible in the stated range; the ball $q_k/U\le 1$ uses $R=1$. Choose the derivative order once to pay $B$, the Mellin integrals and all original height costs. There is no unweighted infinite mean-square claim and no discarded large-$a$ mask.

Writing ${\mathcal{E}}_{\circ }^{\le t}$ for precisely this local signed Poisson contribution with its original Fourier profiles, the result is

$$\begin{array}{rl}|{\mathcal{E}}_{\circ }^{\le t}|& \ll {\mathcal{N}}_{\mathrm{base},J}{U}^{23/12+5t/6+\epsilon },\\ 0<t& <\frac{51-20\sqrt{6}}{10},\\ (1+B_{\ast })-\left(\frac{23}{12}+\frac{5t}{6}\right)& =\frac{51-20\sqrt{6}-10t}{12}>0,\\ t=1/5:|{\mathcal{E}}_{\circ }^{\le 1/5}|& \ll {\mathcal{N}}_{\mathrm{base},J}{U}^{25/12+\epsilon },\\ D^{-2}|{\mathcal{E}}_{\circ }^{\le 1/5}|& \ll {\mathcal{N}}_{\mathrm{base},J}{U}^{13/12+\epsilon },(1+B_{\ast })-25/12=\frac{49-20\sqrt{6}}{12}>0.\end{array}\text{\hspace{1em}(C86)}$$

For the last strict inequality, $49^2-6\cdot 20^2=1$. Take the final loss and absorption of the stipulated polylogarithms smaller than this reserve. The cutoff $q_r\le U^{1/5}$ is unbounded as $U$ grows; it is not a finite-conductor application. The conclusion holds separately for $b_A$ with $W_0$ and for the exact $b_U$ with $W_U$.

Scope beyond the small sector. C84 alone supplies no full C68 $W(1/12)$. For original $W_0$, C93–C97 below give a bare endpoint on every nonunit reduced-frequency annulus, including $q_r>U^{1/5}$; the exact redesigned $W_U$ application here remains restricted to this small sector. The $r=1$ frequencies are C58’s exceptional sector: R46 already pays their specified redesigned bare-J4 realization, whereas the original-$W_0$ principal scalar retains its stated obstruction. This application neither duplicates that exceptional bound nor supplies a signed cutoff-change estimate. The $k=0$ term, other source $\mathfrak{b}/C$, common-$e$, cores/gcds, bad duals, generalized additional finite/angular factors and their full measure, physical-row restoration, and same-common-beta signed detector/$\rho$ recombination remain unpaid in their original places. A selected Poisson contribution has no positivity or monotonicity under physical-row selection. Full C68, J40, signed Robin restoration and RH remain open. This is a repo-derived paper application of C56–C68, C33–C38, R3–R9/R34–R46 and the explicit source premises; no new generic theorem, Lean certification, source-proof audit or literature-priority claim is made.

A source-conditional dominant endpoint for original W0

Retain exactly C80 with the original $W_0$: write $B_i=B_{i,0}$ and $b_A=\mu +K_{A;0}$, with $D=U^{1/2}$ and $A=U^{1/12}$. Keep the bare corefree/common-$e=1$, $C=\mathfrak{b}=D_{\mathrm{src}}=1$, $q=L=U$ branch, R43’s original genuine fixed ideal-ray quotient and total-class selector, ${\zeta }_i=1$, and no additional angular multiplier. Fix $0<c<C_1$. In C79 retain every frequency with $cU\le q_r\le C_{1}U$, including $a=1$, all six units, the original $S$ labels, every sixth-power repetition $a$, its $S$ factors and its overlaps with $r$. All actual $\mu /K$ terms, signs, twists, squarefree/collision zeros and original deletion masks remain.

For sufficiently large $U$, conditional on the local source premises specified below, for every final $\epsilon >0$ one finite original side-seminorm order $J$ gives

$$\begin{array}{rl}{M}_i^{\mathrm{dom}}(R)& :=\sum _{\begin{array}{c}0<q_k\ll RU\\ k\text{ as in C79},cU\le q_r\le C_{1}U\end{array}}|B_i(k;{\Pi }_i){|}^2\\ & {\ll }_{c,C_1,H,S,C_{\Pi },\epsilon }{\mathcal{H}}_{i,J}(R{)}^2{R}^{1/6}{U}^{\kappa +\epsilon },1\le R\le U,\\ \kappa & =\frac{37-10\sqrt{6}}{6}=1+B_{\ast }.\end{array}\text{\hspace{1em}(C87)}$$

Here ${\mathcal{H}}_{i,J}(R)$ is C82’s seminorm of the original side profile on that Fourier shell (the ball uses $R=1$). Its required weighted shell seminorm and the original component/height measure are part of the one base-envelope premise below. This is an application to the prescribed coefficient, rather than the prime coefficient of Xu–Sang v2, Theorem 1.2, which already has the same numerical exponent. C93–C97 below cover the other original-W0 nonunit annuli at the bare endpoint; C87 retains the sharper $R^{1/6}$ dependence on this dominant annulus. Neither application supplies strict/full C68.

Source correspondence and common bins. Reuse C80–C81 and R4–R5/R44 for the literal two-slot series, including the complete defining-modulus zeros. The additional local input is the pinned primary’s lem:logarithmic-control, lines 1531–1551. For $\alpha \in [1/2,1]$ and small $e>0$, absence of zeros in the open disk centered at $2+it$ of radius $2-\alpha -2e$ supplies $|L|+|1/L|\ll \{2Q_{\rho }(3+|t|{)}^2{\}}^{\eta }$ on the closed concentric disk of radius $2-\alpha -6e$, for every prescribed $\eta >0$. This explicitly supplies each numerator and reciprocal below $7/8$ on its local disk. Neither $L^{\prime }/L$ control nor C77’s global right-strip bound substitutes for it. This lemma and thm:main remain unaudited source premises; no validity of their major proofs is certified.

Use Xu–Sang v2, §6.1, (6.1), Lemma 6.2 and Proposition 6.3 on the original good sixth-power-free physical rows $r$, including inducing orders properly dividing six. Cover $[cU,C_{1}U]$ by finitely many source dyads $Q\asymp U$; $Q$ in that source is this row scale, not the primitive conductor $Q_{\rho }$. Put all the original fixed ray, unit/$S$, $\xi$ and C56 orientation twists of both sides and their conjugates into the single fixed finite group $\Theta$, supported on $S$, before assigning bins. Then every C80 presentation is in the source family ${\mathcal{X}}_r=\{\nu (n){\chi }_n(r{)}^{\pm 1}:\nu \in \Theta \}$. The original total-class selector has absolute Fourier mass $1$, as in C80. C80’s good-prime ramification proves every primitive inducer nonprincipal and $Q_{\rho }{\ll }_{H,S}{q}_r\ll U$; no conductor lower bound or principal regularizer is needed.

On each source dyad choose $H_1=Q^{\tau }\asymp U^{\tau }$ in the order required by Lemma 6.2: fix the final losses, $e$, and finite internal moment/parameter-Sobolev orders, then a sufficiently small positive $\tau \le 1/100$, and only then the external Mellin-tail orders. There are $O_e(1)$ common bins $(\ell ,\alpha )$ with $\alpha \le M_{\ell }(r)<\alpha +e$ and $M_{\ell +1}(r)<\alpha +2e$. The count exponent is exactly $\mathcal{R}(\alpha )$ of §459, (459.7); the floor ${\alpha }_0=51/100$ has $O(U)$ rows. The primary’s lem:buffered-bins, lines 4281–4353 places all needed zero-free disks inside the common height rectangle $|t|\le (3\ell +2)H_1$, giving both local $L$ directions for $\Re s\ge \alpha +6e$. The extra ${\Pi }_i$ and $a$ deletions enter C81’s correction, not $\Theta$ or the zeros defining these bins. In particular the count is never applied to a newly row-masked witness polynomial or to the actual bilinear coefficient as a witness.

The finite joint contour of the actual coefficient. For each $M\subseteq \{1,2\}$ and $T=\{1,2\}\setminus M$, the exact C81 inversion, with $P=\mathrm{rad}({\Pi }_{i}ra){|}_{p\notin S}$, has integrand

$${\tilde{F}}_i(s_1,s_2)D^{s_1+s_2}\prod _{j\in T}[{\hat{W}}_0(z_j)A^{z_j}]H_{M,P,\rho }(\mathbf{s},\mathbf{z})\prod _{j\in M}L(s_j,{\rho }^{\ast }{)}^{-1}\prod _{j\in T}L(s_j,{\rho }^{\ast })L(s_j+z_j,{\rho }^{\ast }{)}^{-2},\frac{d\mathbf{s}d\mathbf{z}}{(2\pi i{)}^{2+|T|}}.\text{\hspace{1em}(C88)}$$

Thus the four choices have respectively two reciprocal factors, one reciprocal and one numerator with its squared reciprocal, or two numerators with their two squared reciprocals. At most six $L$ or reciprocal factors occur, counted with multiplicity. This is C81’s exclusive-prime identity, with $\mu$ in $M$ and $K_{A;0}$ in $T$, not independently coprime slot sums. In particular neither a sign nor a cross-slot collision zero has been replaced.

Start on $\Re s_1=\Re s_2=2$ and $\Re z_j=e>0$. Before moving any line truncate each imaginary coordinate at $T_0=H_1/16$, common to the four choices, both original sides and all repetitions on this source dyad. Move the two finite $s$ segments successively to $\alpha +8e$, keeping every $z$ line at $e$. Throughout the rectangles, $|\Im s_j|\le T_0$ and $|\Im (s_j+z_j)|\le 2T_0$. Every numerator argument has real part at least $\alpha +8e$, and every $K$ reciprocal argument at least $\alpha +9e$. These arguments and all horizontal faces lie in the same buffer. Centering the local disk at each argument’s height places its part with real part at most $2$ in the controlled smaller disk; the part farther right uses absolute Euler convergence. Each factor in C88 is therefore $U^{\eta }$ for arbitrarily small internal loss. The numerator is entire because its inducer is nonprincipal. No $s=1$ pole, reciprocal zero, or $z=0$ cutoff pole is crossed. R5 applies throughout since both $\Re s_j$ and $\Re (s_j+z_j)$ exceed $1/2$; no local numerator of $H$ is inverted.

The original separated norm twists are kept in $F_i$ before this inversion. Their actual polynomial derivative costs belong to ${\mathcal{H}}_{i,J}(R)$, as in C82; this introduces neither a new character nor an unbudgeted infinity type. For $q_k\ll RU$ on the current annulus, $q_a\ll R^{1/6}$. With the original $q_{{\Pi }_i}\le U^{C_{\Pi }}$ and $1\le R\le U$, even if $a$ overlaps $r$ or contains $S$ factors, $q_P\ll U^{C_{\Pi }+1}{R}^{1/6}\ll U^{C_{\Pi }+7/6}$. R4–R5 and the primary’s lem:deleted-euler-factors, lines 1602–1620 absorb $q_P^{{\eta }_P}$ by an internally small ${\eta }_P$, uniformly in every original frozen polynomial-size mask. On the moved lines, $D^{2\alpha +16e}{A}^{|T|e}\le U^{\alpha +49e/6}$. Allocating $e$, the at most six $L$ losses and the deletion loss inside the final $\epsilon$ gives, on the same bin,

$$|B_i(k;{\Pi }_i)|\ll {\mathcal{H}}_{i,J}(R)U^{\alpha +\epsilon },cU\le q_r\le C_{1}U,q_k\ll RU,1\le R\le U.\text{\hspace{1em}(C89)}$$

Summing the four actual terms and the original selector uses only fixed constants and its actual mass $1$.

For completeness the joins use the primary’s lem:smooth-calculus, lines 1123–1177, not an infinite local-contour assertion. On the original absolute lines the entire joint series is absolutely bounded, uniformly in its masks. The joint annular ${\tilde{F}}_i$ and each ${\hat{W}}_0(e+it)$ have arbitrarily rapid Mellin decay. Hence the omitted tails are $O({\mathcal{H}}_{i,J}(R)U^{C_0}{T}_0^{-N})$, with a fixed $C_0$ independent of $N$. On an $s_j$ horizontal face, $|\Im s_j|=T_0$; pointwise logarithmic integration by parts in that coordinate of $F_i$ gives the same bound after integrating the other coordinates. All $L$ arguments on that face stay inside the buffer and R5 still bounds $H$. Intersections of faces have the same estimates. The $z$ tails remain on absolute lines and no $z$ contour is shifted. After $\tau$ is fixed choose one finite external order with $N\tau >C_0+10$, enlarging $J$ once to make every tail and join $O({\mathcal{H}}_{i,J}(R)U^{-10})$. This does not change the previously fixed internal height order. The moved vertical segments are never extended beyond the buffer, and no derivative acts on $b_A$.

Bin energy and all original repetitions. For each fixed original unit/$S$ label and each $a$, apply Proposition 6.3 to the same physical $r$ family. The restriction $q_k\ll RU$ can only decrease this positive bin energy. Uniformity of C89 in its original masks therefore gives

$$\sum _{\begin{array}{c}r\in {\mathcal{B}}_{\ell ,\alpha }\\ cU\le q_r\le C_{1}U,q_k\ll RU\end{array}}|B_i(k;{\Pi }_i){|}^2\ll {\mathcal{H}}_{i,J}(R{)}^2{U}^{\mathcal{R}(\alpha )+2\alpha +\epsilon }.\text{\hspace{1em}(C90)}$$

Reuse the evaluated maximum (459.12), with $\delta =2\alpha -1$: $\max (\mathcal{R}(\alpha )+2\alpha )=1+B_{\ast }=\kappa$, at ${\alpha }_{\ast }=(21-5\sqrt{6})/12$. The second branch gives $2$ and the floor gives $1+2{\alpha }_0=101/50<\kappa$. This is the existing count and maximization, also used for the different prime coefficient in Xu–Sang v2, Theorem 1.2; neither is a new supplier here. There are only $O_e(1)$ bins and finitely many source dyads. Ordinary ideal counting gives $O(R^{1/6})$ choices of $a$ for this annulus, rather than $O(U^{1/6})$ at $R=1$. Summing these choices with all six units and the finite $S$ labels proves C87. No $(a,r)=1$ restriction, squarefree restriction on $a$, or omission of its $S$ factors is made.

For $R>U$ use the original absolute divisor bound $|B_i|\ll {\mathcal{H}}_{i,J}(R)U^{1+\epsilon }$, rather than extrapolating the polynomial-puncture range. The number of frequencies on this same dominant $r$ annulus is $O(U{R}^{1/6})$. The original pair-profile shell decay $R^{-B}$, with one fixed $B\ge 4$, consequently pays its entire large-$a$ tail:

$${\mathcal{N}}_{\mathrm{base},J}{U}^{3+\epsilon }\sum _{R>U\mathrm{dyadic}}{R}^{1/6-B}\ll {\mathcal{N}}_{\mathrm{base},J}{U}^{19/6-B+\epsilon }.\text{\hspace{1em}(C91)}$$

For $1\le R\le U$, sum C87 with the same decay; the ball uses $R=1$. Thus every repetition of this $r$ annulus is restored in the original Fourier-weighted sum, without an unweighted infinite-energy claim.

One original measure and one normalization. Choose $J$ once for all four terms, bins, masks, classes, units/$S$ labels, repetitions, components and shells. Use exactly C82 and R43–R46’s original coupled component/height measure $dc$. Including the single chosen weighted shell seminorm in that same envelope means $\int|dc|,\mathcal H_{1,J}(R)\mathcal H_{2,J}(R) \ll\mathcal N_{{\rm base},J}(1+R)^{-B}$, with ${\mathcal{N}}_{\mathrm{base},J}$ fixed or polylogarithmic in $U$. All original polynomial norm-height derivative costs are included there. If only the $\eta =h^{-2}$ component envelope is supplied, recovering the base envelope costs the same fixed $h^2$ as R43–R46. No allowance restarts per mask, $r$, $a$, bin, class, component, cofactor or height. This is a retained applicability premise, not certification of an unspecified larger growing-height allowance or of the full J40 component table.

Apply Cauchy to the two original columns on the same $k$ set, then integrate against that same $dc$. In the bare branch, $N_i=U$, $K_{\mathrm{row}}\asymp RU$ and $D_{\mathrm{src}}=1$; C87 has the restricted C68 scaling $\lambda =\kappa -2=B_{\ast }-1$, using $R^{1/6}\le R$. Restore C68’s $q/L^3=U^{-2}$ and C57–C59’s raw Gauss-pairing factor $U^2$ exactly once; they cancel. C91 and the small shells then give for precisely this signed local contribution ${\mathcal{E}}_0^{\mathrm{dom}}$

$$|{\mathcal{E}}_0^{\mathrm{dom}}|\ll {\mathcal{N}}_{\mathrm{base},J}{U}^{\kappa +\epsilon },D^{-2}|{\mathcal{E}}_0^{\mathrm{dom}}|\ll {\mathcal{N}}_{\mathrm{base},J}{U}^{B_{\ast }+\epsilon }.\text{\hspace{1em}(C92)}$$

The last factor is the original $D^{-1}$ on each side, $D^{-2}=U^{-1}$. C56 fixes the self-dual planar measure; there is no extra area, inverse Gauss, division by six or second $U^2$.

Scope and remaining bound. The added deduction is this finite buffered-contour application to literal original-W0 C80, with its same-annulus repetitions and common costs. It is not a source theorem for these coefficients or a new general $L$-function estimate. Under its stated premises it improves the generic raw $U^{13/6+\epsilon }$ certificate to $U^{\kappa +\epsilon }$ on this annulus. Accordingly, with the same fixed/polylog envelope an actual-coefficient lower bound $U^{13/6-o(1)}$ here would contradict C87, since $13/6-\kappa =(5\sqrt{6}-12)/3>0$. Generic metaplectic operator sharpness by itself does not identify these prescribed columns with an operator witness. No realizable coherent lower-bound example is asserted; coherent raw energy of size $U^{\kappa -o(1)}$ remains compatible with C87.

The coefficient correspondence is checked by the four C81 identities for literal C80 and the displayed finite rectangles, with ${\Pi }_i,r,a$ left symbolic. It fails if an actual numerator or reciprocal leaves the common buffer, a primitive presentation escapes fixed $\Theta$, a horizontal face needs an unavailable original seminorm, or the actual $dc$ fails the stated common-envelope premise. No generic operator witness or norm-only lower bound repairs any of these gaps.

There is zero strict reserve: local C68 sufficiency asks for $\lambda <B_{\ast }-1$, whereas C87 gives equality. A coefficient-sensitive positive-power improvement at the critical ${\alpha }_{\ast }$ bins, or cancellation in the original common-beta signed aggregate, is still missing. The all-annulus original-W0 endpoint C93–C97 below closes the nonunit annular coverage gap at this same endpoint power. The original-W0 exceptional principal scalar, $k=0$, other cores/common-$e$ and common-column branches, other source $C/\mathfrak{b}$, generalized finite/angular factors and their actual measure, bad duals, physical-row restoration and the original signed common-beta Mobius detector/$\rho$ aggregation remain unresolved. R46’s redesigned exceptional payment is not transferred to W0. Full C68, J40, Robin and RH remain open. All conclusions here are source-conditional paper deductions, without source-proof audit, Lean certification or originality certification.

A source-conditional all-annulus endpoint for original W0

Retain the literal $B_i=B_{i,0}$ of C80 and exactly C87’s bare branch, original $W_0$, $b_A=\mu +K_{A;0}$, $D=U^{1/2}$ and $A=U^{1/12}$. All four $\mu /K$ terms, product squarefreeness, original signs and phases, class selectors and defining-modulus/collision zeros remain. For sufficiently large $U$, under C77–C92’s source premises and the same original coupled-envelope premise, every final $\epsilon >0$ admits one finite original seminorm order $J$ such that

$$\begin{array}{rl}{M}_i^{>1}(R)& :=\sum _{\begin{array}{c}0<q_k\ll RU\\ k\text{ as in C79},q_r>1\end{array}}|B_i(k;{\Pi }_i){|}^2\\ & {\ll }_{H,S,C_{\Pi },\epsilon }{\mathcal{H}}_{i,J}(R{)}^{2}R{U}^{\kappa +\epsilon },1\le R\le U,\kappa =1+B_{\ast }=\frac{37-10\sqrt{6}}{6}.\end{array}\text{\hspace{1em}(C93)}$$

There is no restriction $q_r\asymp U$, $a=1$, $(a,r)=1$, or squarefreeness of $a$. Each frequency includes its six-unit label, original $S$ label and complete ideal sixth power, with $S$ factors and overlaps retained. C87 remains the sharper $R^{1/6}$ estimate when $q_r\asymp U$. C93 is an application of the existing density count and maximization to these actual columns, not a new density theorem or a use of Xu–Sang’s prime-coefficient estimate.

Independent row and column scales. Set $\theta =1/5$. C84 with $t=\theta$ already pays $1<q_r\le U^{\theta }$, for original $W_0$, by ${\mathcal{H}}_{i,J}(R{)}^2{R}^{1/6}{U}^{25/12+\epsilon }$. This is within C93 since $25/12<\kappa$. For the remaining rows use disjoint physical dyads $q_r\asymp Q$, with $U^{\theta }\ll Q\ll RU\ll U^2$. The source row parameter is $Q$; the column scales stay $D=U^{1/2}$ and $A=U^{1/12}$. Neither is reset to a power of $Q$, nor is $Q$ identified with $Q_{\rho }$.

Use precisely C87–C90’s common fixed $\Theta$ and ${\mathcal{X}}_r=\{\nu (n){\chi }_n(r{)}^{\pm 1}:\nu \in \Theta \}$ for both original sides, all fixed ray/orientation twists and their conjugates. The rows are the original good sixth-power-free nonunits, including inducing orders properly dividing six. Good-prime ramification still makes every primitive inducer nonprincipal. Xu–Sang §6.1 (6.1) gives $Q_{\rho }{\ll }_{H,S}Q$ and the original row-deletion radical of norm $O_{H,S}(Q)$; its zero extension is retained. The extra ${\Pi }_i,a$ deletions enter C81’s $H$, not a growing $\Theta$, a new detector polynomial or a different family of zeros.

Apply the same Lemma 6.2 and Proposition 6.3 on each $Q$ dyad, with ${\alpha }_0=51/100$, fixed $0<e<10^{-3}$ sufficiently small and $I=\lceil 1/e\rceil +2$. The common bins have $1\le \ell <I$, $\alpha \in ({\alpha }_0+e{\mathbb{Z}}_{\ge 0})\cap [{\alpha }_0,1]$, $\alpha \le M_{\ell }(r)<\alpha +e$ and $M_{\ell +1}(r)<\alpha +2e$, where $M_j$ is the common primitive-zero maximum through height $3j{H}_1$, floored at ${\alpha }_0$. Use $H_1=Q^{\tau }$, $0<\tau \le 1/100$. Fix the final losses, small $e$ and finite internal moment/detector/parameter-Sobolev orders first; then choose $\tau$ for those height costs and detector hypotheses, and only then choose the external Mellin-tail orders. All these choices are uniform over $U^{\theta }\ll Q\ll U^2$. Floor bins have $O(Q)$ rows and above-floor bins have $O(Q^{\mathcal{R}(\alpha )+\eta })$ rows, with exactly (459.7)’s $\mathcal{R}$ and arbitrarily small prescribed $\eta >0$. This reuses the source’s permitted witness lengths, finite smooth parameter dependence and polynomial height costs; no moment estimate for the actual bilinear coefficient is substituted for that count.

Uniform finite contours when $Q\ne U$. Use C88’s exact joint integrand separately for all four actual choices. Truncate every imaginary coordinate at $T_0=Q^{\tau }/16$ before shifting either $s$ segment from $2$ to $\alpha +8e$, with each $z$ line fixed at $e$. The numerator and reciprocal arguments, including every $s_j+z_j$, stay in C89’s common buffered disks: their heights are at most $2T_0$ and their real parts are at least $\alpha +8e$ or $\alpha +9e$. The local logarithmic-control premise supplies both $L$ directions; C77 alone is not used below its right strip. Nonprincipality and positive $z$ lines exclude principal or cutoff residues, and R5 controls the joint correction throughout. All horizontal faces and intersections use that same finite buffer.

On these dyads $q_a\ll (RU/Q{)}^{1/6}$, and the actual puncture satisfies

$$q_P\ll U^{C_{\Pi }}Q(RU/Q{)}^{1/6}\ll U^{C_{\Pi }+2}.$$

Thus deletion costs and the at most six numerator/reciprocal losses fit inside a prescribed $U^{\epsilon }$, uniformly in $Q,r,a$ and the original masks. C89’s scale cost is still $D^{2\alpha +16e}{A}^{|T|e}\le U^{\alpha +49e/6}$. The original absolute-line tails and all horizontal joins have C89’s bound $O({\mathcal{H}}_{i,J}(R)U^{C_0}{T}_0^{-N})$, with one fixed $C_0$ independent of the external order $N$. Since $Q\gg U^{1/5}$, choose one finite $N$ with $N\tau /5>C_0+10$ and enlarge $J$ once. Every tail and join is then $O({\mathcal{H}}_{i,J}(R)U^{-10})$, uniformly even at the smallest retained $Q$. This does not increase the already fixed internal source height order. No moved vertical segment is extended to infinity, and no derivative acts on $b_A$.

Also retain C81–C82’s global right-strip contour in this fixed polynomial range $q_r\ll U^2$, $q_a\ll U^{1/3}$. C77 supplies its uniform $U^{7/8+\epsilon }$ bound, with the same masks and one common enlarged $J$. Taking the better of these two bounds gives

$$|B_i(k;{\Pi }_i)|\ll {\mathcal{H}}_{i,J}(R)U^{\min \{\alpha ,7/8\}+\epsilon },r\in {\mathcal{B}}_{\ell ,\alpha }(Q),q_k\ll RU.\text{\hspace{1em}(C94)}$$

Density with every repetition. Put $d_{{\alpha }_0}=1$ for floor bins and $d_{\alpha }=\mathcal{R}(\alpha )$ otherwise. For each fixed unit/$S$ label and ideal $a$, the source count applies to the same physical $r$ family; imposing the original frequency and collision restrictions only decreases its positive energy. Sum over every $a$ with $q_a\ll (RU/Q{)}^{1/6}$ by ordinary ideal counting. C94 and the existing bin count therefore give

$$\sum _{\begin{array}{c}0<q_k\ll RU,k\text{ as in C79}\\ r\in {\mathcal{B}}_{\ell ,\alpha }(Q)\end{array}}|B_i(k;{\Pi }_i){|}^2\ll {\mathcal{H}}_{i,J}(R{)}^2(RU{)}^{1/6}{Q}^{d_{\alpha }-1/6}{U}^{2\min \{\alpha ,7/8\}+\epsilon }.\text{\hspace{1em}(C95)}$$

The six units and finite $S$ labels cost only their original fixed sum, including $q_{e_S}^{-1/6}$. No primitive-inducer multiplicity is discarded. The bound is uniform in the $a$-dependent deletion mask, while the bins themselves are independent of that mask.

For $d_{\alpha }\ge 1/6$, use $Q\ll RU$ and $d_{\alpha }\le 1$ to bound C95 by $\mathcal H_{i,J}(R)^2R^{d_\alpha} U^{d_\alpha+2\min{\alpha,7/8}+\varepsilon}$. Reuse C90 and (459.12): above the floor, $\mathcal{R}(\alpha )+2\alpha \le \kappa$, with equality at ${\alpha }_{\ast }=(21-5\sqrt{6})/12<7/8$; the floor gives $1+2{\alpha }_0=101/50<\kappa$. Hence this part is at most ${\mathcal{H}}_{i,J}(R{)}^{2}R{U}^{\kappa +\epsilon }$. For $d_{\alpha }<1/6$, the negative $Q$ exponent in C95 and C77’s cap give instead $\mathcal H_{i,J}(R)^2R^{1/6}U^{1/6+7/4+\varepsilon} =\mathcal H_{i,J}(R)^2R^{1/6}U^{23/12+\varepsilon}$. In particular the high branch $\alpha >11/12$ does not create an unpaid repetition cost. Sum the $O_e(1)$ bins and $O(\log U)$ physical dyads, absorbing their losses internally. Together with the small-$Q$ C84 estimate this proves C93 without redoing the density or its optimization.

Original weighted tail and coupled restoration. For $R>U$, apply no bin estimate or polynomial-puncture extrapolation. The original divisor bound gives $|B_i|\ll {\mathcal{H}}_{i,J}(R)U^{1+\epsilon }$, uniformly in every deletion. There are $O(RU)$ full element frequencies in the ball, so all nonunit reduced parts and all their repetitions have energy $O({\mathcal{H}}_{i,J}(R{)}^{2}R{U}^{3+\epsilon })$. Use exactly C91–C92’s original coupled measure $dc$ and shell seminorms, with one fixed $B\ge 4$ and one finite $J$ large enough for all the preceding bounds:

$$\begin{array}{rl}\int |dc|{\mathcal{H}}_{1,J}(R){\mathcal{H}}_{2,J}(R)& \ll {\mathcal{N}}_{\mathrm{base},J}(1+R{)}^{-B},\\ {\mathcal{N}}_{\mathrm{base},J}{U}^{3+\epsilon }\sum _{R>U\mathrm{dyadic}}{R}^{1-B}& \ll {\mathcal{N}}_{\mathrm{base},J}{U}^{4-B+\epsilon }.\end{array}\text{\hspace{1em}(C96)}$$

The same fixed or polylogarithmic ${\mathcal{N}}_{\mathrm{base},J}$ pays all original components and norm heights, masks, classes, units/$S$ labels, bins, dyads and repetitions. An eta-only envelope still costs R43–R46’s fixed $h^2$ to recover this base envelope. No allowance is restarted at $Q$, $r$, $a$, or a component. Its actual finite-order availability remains an applicability premise.

For $1\le R\le U$, Cauchy uses the two actual columns on the same full nonunit $k$ set, followed by integration against this same $dc$. C93 costs $R{U}^{\kappa +\epsilon }$ before the shell decay; the ball uses $R=1$. Sum with C96’s decay and include its $R>U$ tail. In this bare branch $N_i=U$, $K_{\mathrm{row}}\asymp RU$ and $D_{\mathrm{src}}=1$, so C93 has C68’s endpoint scaling $\lambda =\kappa -2=B_{\ast }-1$. Restore $q/L^3=U^{-2}$ and C57–C59’s raw Gauss-pairing $U^2$ exactly once; they cancel. For precisely the original signed local nonunit contribution ${\mathcal{E}}_0^{>1}$,

$$|{\mathcal{E}}_0^{>1}|\ll {\mathcal{N}}_{\mathrm{base},J}{U}^{\kappa +\epsilon },D^{-2}|{\mathcal{E}}_0^{>1}|\ll {\mathcal{N}}_{\mathrm{base},J}{U}^{B_{\ast }+\epsilon }.\text{\hspace{1em}(C97)}$$

Here $D^{-2}=U^{-1}$ restores the original $D^{-1}$ on each side. C56’s self-dual planar measure, the six units and ideals counted once introduce no extra area, inverse Gauss, unit division or second $U^2$. There is no unweighted infinite-energy assertion.

Exact scope. C93–C97 close the original-W0 bare nonunit annular coverage gap, with zero strict reserve: local sufficiency still requires $\lambda <B_{\ast }-1$, while this application gives equality. The critical ${\alpha }_{\ast }$ bin is below C77’s cap, so that cap supplies no strict saving there. A coefficient-sensitive improvement or cancellation in the same original signed aggregate remains needed. The exact redesigned $W_U$ remains covered here only by C77–C86’s small nonunit sector and R46’s specified exceptional realization; no all-annulus $W_U$ bound or signed cutoff-change estimate is added.

The original-W0 $r=1$ principal scalar, $k=0$, other cores/common-$e$ and common-column branches, other source $C/\mathfrak{b}$, generalized finite/angular factors and their actual measure, bad duals, full physical-row restoration and the original common-beta signed Mobius detector/$\rho$ aggregation remain unresolved. No positivity under physical-row selection is assumed. All deductions retain the unaudited source main, local logarithmic-control and density inputs, and the actual one-envelope condition. There is no source-proof, Lean or originality certification, unconditional or strict-C68 claim, or completion of full C68, J40, signed Robin restoration or RH.

A source-conditional strict sector for powerful physical frequencies

Keep the literal original-$W_0$ columns $B_i=B_{i,0}$ of C80, on C87’s bare corefree/common-$e=1$, $C=\mathfrak{b}=D_{\mathrm{src}}=1$, $q=L=U$ branch, with $D=U^{1/2}$ and $A=U^{1/12}$. Define $\mathcal{P}$ to be the nonunit good sixth-power-free reduced parts $r$ of C79 for which every nonzero good-prime valuation belongs to $\{2,3,4,5\}$. Thus $(r)$ is powerful, and $q_r>1$. This restricts the actual physical reduced frequency, with no restriction $q_r\asymp U$ and no identification with a primitive conductor. For each original component retain its original frequency-independent ${\Pi }_i$, with $q_{{\Pi }_i}\le U^{C_{\Pi }}$, fixed before the frequency sum. The additional radical $P=\mathrm{rad}({\Pi }_{i}ra){|}_{p\notin S}$ in C81 still varies with the actual $r,a$.

All four terms of $b_A(d_1)b_A(d_2)=(\mu+K_{A;0})(d_1) (\mu+K_{A;0})(d_2)$, product squarefreeness, signs, fixed twists, total-class selection and defining-modulus/collision zeros are unchanged. Retain every C79 presentation $k=ϵe_S(\mathbf{j})r{a}^6$ with $r\in \mathcal{P}$: all six units, all original $S$ labels, every ideal $a$, its $S$ factors and its overlaps with $r$. In particular ${\chi }_n(a^6)$ remains the collision mask $1_{(n,a)=1}$.

Conditional on C77–C96’s source main, local logarithmic-control and physical-row density premises, and on the same original finite coupled profile/height/shell envelope at the order used below, for every final $\epsilon >0$ one finite $J$ gives, for sufficiently large $U$,

$$\begin{array}{rl}{M}_i^{\mathcal{P}}(R)& :=\sum _{\begin{array}{c}0<q_k\ll RU,k\text{ as in C79}\\ r\in \mathcal{P}\end{array}}|B_i(k;{\Pi }_i){|}^2\\ & {\ll }_{H,S,C_{\Pi },\epsilon }{\mathcal{H}}_{i,J}(R{)}^2{R}^{1/2}{U}^{27/13+\epsilon },1\le R\le U.\end{array}\text{\hspace{1em}(C98)}$$

Here $R$ is the original Fourier ball/shell scale and $U$ is the column scale. The source’s physical-row dyad $Q\asymp q_r$ is a third parameter; neither $D$ nor $A$ is reset to a power of $Q$. The seminorm and measure are exactly those of C94–C96, with one common finite enlargement described below.

Intersect the existing physical-row counts. The fixed primary, powerful-ideal count, lines 5272–5282 already supplies $\#\{r\in \mathcal{P}:q_r\le Y\}{\ll }_{K,S}{Y}^{1/2}$ for $Y\ge 1$. Reuse that classical count; no new counting or multiplicative-sieve theorem is asserted. C94–C95 use the common physical-row bins from Xu–Sang v2, §6.1, Lemma 6.2 and Proposition 6.3 with the owned exponent (459.7). Set $d_{{\alpha }_0}=1$ at the floor ${\alpha }_0=51/100$, and above the floor set

$$d_{\alpha }=\mathcal{R}(\alpha )=\{\begin{array}{ll}(31-32\alpha )/(21-12\alpha ),& \alpha \le 11/12,\\ 2(1-\alpha ),& \alpha \ge 11/12.\end{array}$$

For every one of those same bins and every prescribed small $\eta >0$, the two upper counts give

$$\#({\mathcal{B}}_{\ell ,\alpha }(Q)\cap \mathcal{P})\ll Q^{m_{\alpha }+\eta },m_{\alpha }=\min \{d_{\alpha },1/2\},m_{{\alpha }_0}=1/2.\text{\hspace{1em}(C99)}$$

The restriction is imposed after forming the original bins. The fixed common $\Theta$ still contains both sides’ original ray, unit/$S$, $\xi$, orientation twists and conjugates; the presentations remain $\nu (n){\chi }_n(r{)}^{\pm 1}$. Good-prime ramification in C80 makes every primitive inducer nonprincipal even for valuations $2,3,4,5$, including inducing orders properly dividing six. The primitive conductor satisfies only the existing $Q_{\rho }{\ll }_{H,S}Q$. Neither $a$ nor ${\Pi }_i$ changes the family of zeros used to define a bin. In particular C99 does not assert a new density theorem for a masked witness or for the actual bilinear coefficient.

The small physical dyads. Set $\theta =1/5$ as in C93–C95. Partial summation of the supplied powerful count gives ${\sum }_{r\in \mathcal{P},q_r\le Y}{q}_r^{-1/6}\ll Y^{1/3}$. Counting all ideal sixth powers exactly as in C83, with the original finite sum of $6q_{e_S(\mathbf{j})}^{-1/6}$, consequently bounds the number of frequencies with $q_r\le U^{\theta }$ by $(RU{)}^{1/6}{U}^{\theta /3}$. C82, with the same original masks, therefore gives

$$\sum _{\begin{array}{c}0<q_k\ll RU,k\text{ as in C79}\\ r\in \mathcal{P},q_r\le U^{1/5}\end{array}}|B_i(k;{\Pi }_i){|}^2\ll {\mathcal{H}}_{i,J}(R{)}^2{R}^{1/6}{U}^{119/60+\epsilon },\frac{119}{60}<\frac{27}{13}.\text{\hspace{1em}(C100)}$$

This directly pays every small-$Q$ row at the stronger target power. C84’s $25/12$ bound for its larger row family remains valid; it is not the small-$Q$ input for C98.

Every remaining dyad, bin and repetition. On disjoint physical dyads $U^{1/5}\ll Q\ll RU\ll U^2$, use C88’s exact joint series and C94’s pointwise estimate. Its finite contours have $T_0=Q^{\tau }/16$, both $s$ lines moved to $\alpha +8e$, and all present $z$ lines kept at $e>0$. Every actual numerator and reciprocal, including $L(s_j+z_j,{\rho }^{\ast }{)}^{-2}$, remains in the same source buffer. C81’s joint correction retains the product squarefree condition and all deletions, with $q_P\ll U^{C_{\Pi }+2}$. The global right-strip contour supplies the simultaneous cap at $7/8$. Thus C94 applies without a change to either $\mu /K$ slot or to the original profiles.

Keep the same choice order: final losses, $e$ and internal source orders, then sufficiently small $\tau >0$, then a finite external Mellin-tail order with $N\tau /5>C_0+10$. C94’s tails and joins are uniformly $O({\mathcal{H}}_{i,J}(R)U^{-10})$ on these dyads; no moved segment is extended beyond its finite buffer and no derivative acts on $b_A$.

For each $r$ there are $O((RU/Q{)}^{1/6})$ ideal choices of $a$. C99 and C94, or equivalently C95 with its count intersected before summing, now give

$$\sum _{\begin{array}{c}0<q_k\ll RU,k\text{ as in C79}\\ r\in {\mathcal{B}}_{\ell ,\alpha }(Q)\cap \mathcal{P}\end{array}}|B_i(k;{\Pi }_i){|}^2\ll {\mathcal{H}}_{i,J}(R{)}^2(RU{)}^{1/6}{Q}^{m_{\alpha }-1/6}{U}^{2\min \{\alpha ,7/8\}+\epsilon }.\text{\hspace{1em}(C101)}$$

The positive count includes every admissible $a$, without $(a,r)=1$ or squarefreeness of $a$. Original collision restrictions can only reduce that count. Uniformity in the actual $a$-dependent radical pays its full energy; no primitive-inducer multiplicity or sixth-power repetition is discarded.

If $m_{\alpha }\ge 1/6$, use $Q\ll RU$ in C101 to obtain $\mathcal H_{i,J}(R)^2R^{m_\alpha} U^{m_\alpha+2\min{\alpha,7/8}+\varepsilon}$. Here $R^{m_{\alpha }}\le R^{1/2}$. If $m_{\alpha }<1/6$, use $Q\ge 1$ and the cap instead, giving ${\mathcal{H}}_{i,J}(R{)}^2{R}^{1/6}{U}^{23/12+\epsilon }$. This handles in particular every high-branch bin above $11/12$.

For the remaining bins, the continuous upper bound over the source range has the exact maximum

$$\underset{\begin{array}{c}\alpha \in [{\alpha }_0,1]\\ m_{\alpha }\ge 1/6\end{array}}{\max }(m_{\alpha }+2\min \{\alpha ,7/8\})=\frac{27}{13},{\alpha }_c=\frac{41}{52},\mathcal{R}({\alpha }_c)=\frac{1}{2}.\text{\hspace{1em}(C102)}$$

Indeed, up to ${\alpha }_c$ the intersected exponent is $1/2+2\alpha$, increasing to $27/13$; the floor contributes $1/2+2{\alpha }_0=38/25$. From ${\alpha }_c$ to $7/8$ the derivative of $\mathcal{R}(\alpha )+2\alpha$ is $2-300/(21-12\alpha {)}^2\le -19/75<0$. From $7/8$ to $11/12$, $\mathcal{R}(\alpha )+7/4$ decreases from $57/28<27/13$. At $11/12$ the count exponent is $1/6$, and above it the $23/12$ bound already applies. Thus neither the floor nor the second density branch is omitted. The $O_e(1)$ bins and $O(\log U)$ dyads, together with C100, prove C98 after allocating all internal losses inside the final $\epsilon$.

One original envelope and the complete weighted tail. Choose one finite $J$ for all four terms, bins, dyads, masks, repetitions, units/$S$ labels, components, norm heights and shells. Require exactly C96’s original coupled measure and shell contract $\int|dc|,\mathcal H_{1,J}(R)\mathcal H_{2,J}(R) \ll\mathcal N_{{\rm base},J}(1+R)^{-B}$, with one fixed $B\ge 4$ and fixed or polylogarithmic ${\mathcal{N}}_{\mathrm{base},J}$. Its availability at this finite order is a premise. All polynomial norm-height derivative costs belong to it. An eta-only envelope costs the same fixed $h^2$ to recover the base envelope. No allowance restarts at a dyad, bin, mask, $r$, $a$ or component.

For $R>U$, restrict C96’s positive energy tail to this same powerful sector. The original absolute divisor bound and $O(RU)$ full-frequency count give the unchanged sufficient bound $\mathcal N_{{\rm base},J}U^{3+\varepsilon} \sum_{R>U\ {\rm dyadic}}R^{1-B} \ll\mathcal N_{{\rm base},J}U^{4-B+\varepsilon}$. This includes all large-$a$ masks and repetitions, without applying a bin or polynomial-puncture estimate beyond $R=U$.

For $1\le R\le U$ apply Cauchy on the same powerful $k$ set to the two actual columns, then integrate against that same $dc$. Sum C98 against the stipulated decay; the ball uses $R=1$. Restore the original $q/L^3=U^{-2}$ and the raw Gauss-pairing $U^2$ exactly once, as in C97. For precisely this original signed local sector contribution ${\mathcal{E}}_0^{\mathcal{P}}$ this yields

$$|{\mathcal{E}}_0^{\mathcal{P}}|\ll {\mathcal{N}}_{\mathrm{base},J}{U}^{27/13+\epsilon },D^{-2}|{\mathcal{E}}_0^{\mathcal{P}}|\ll {\mathcal{N}}_{\mathrm{base},J}{U}^{14/13+\epsilon }.\text{\hspace{1em}(C103)}$$

Here $D^{-2}=U^{-1}$ restores the original factor on each side. C56’s self-dual measure introduces no extra area, inverse Gauss, unit division or second $U^2$. Since $R^{1/2}\le R$, C98 has restricted C68 scaling

$${\lambda }_{\mathcal{P}}=\frac{1}{13}<B_{\ast }-1,\kappa -\frac{27}{13}=\frac{319-130\sqrt{6}}{78}>0,319^2-6\cdot 130^2=361.\text{\hspace{1em}(C104)}$$

Take the final loss and absorption of the stipulated polylogarithms smaller than this reserve. The unrestricted nonunit endpoint C93–C97, the sharper dominant-annulus $R^{1/6}$ dependence of C87, and C77–C86 for both original and exact redesigned cutoffs retain their stated strengths and scopes.

Exact scope and remaining critical family. C98–C104 are a repo-derived strict application to C79’s powerful physical reduced frequencies in the actual original-$W_0$ bare columns. Their suppliers are the cited classical count, existing density and C94’s actual-coefficient contour; none is asserted as a new count, large sieve, conductor-compression or multiplicity theorem. C9–C10 in the source card concern column squareparts, and PR1–PR10 in the reflection companion concern reflection choices; those distinct applications are not being counted as this physical-row result.

Within the bare nonunit complement of $\mathcal{P}$, at least one good prime necessarily has $v_p(r)=1$. This is a necessary condition for a remaining critical row in that complement, not a sufficient condition to be critical. No strict estimate for all such rows follows here. The original-$W_0$ $r=1$ principal scalar, $k=0$, other cores/common-$e$ and common-column branches, other source $C/\mathfrak{b}$, generalized angular/finite families and their actual measure, bad duals, full physical-row restoration and the original common-beta signed Mobius detector/$\rho$ aggregation remain unresolved. No positivity or monotonicity under physical-row selection, signed cutoff-change estimate, or all-annulus redesigned-$W_U$ extension is asserted. The unaudited source main/logarithmic-control/density inputs and the actual one-envelope premise remain conditions throughout. There is no source-proof audit, Lean or originality certification, or completion of full C68, J40, signed Robin restoration or RH.

A source-conditional strict sector with a small valuation-one product

Retain C98’s literal original-$W_0$ C80 columns and bare branch: $C=\mathfrak{b}=D_{\mathrm{src}}=1$, corefree/common-$e=1$, $q=L=U$, $D=U^{1/2}$ and $A=U^{1/12}$. For a nonunit good sixth-power-free physical reduced part $r$ of C79, define the ideals

$${\mathfrak{s}}_1(r)=\prod _{\begin{array}{c}p\notin S\\ v_p(r)=1\end{array}}p,\mathfrak{t}(r)=\prod _{\begin{array}{c}p\notin S\\ 2\le v_p(r)\le 5\end{array}}{p}^{v_p(r)},(r)=\mathfrak{t}(r){\mathfrak{s}}_1(r).\text{\hspace{1em}(C105)}$$

Thus ${\mathfrak{s}}_1$ is exactly the product of valuation-one good primes, not the radical of $r$; it is squarefree and coprime to the powerful ideal $\mathfrak{t}$. Fix, independently of $U$,

$$0\le \eta <{\eta }_{\star }:=\frac{13-5\sqrt{6}}{3},\theta =\frac{1+\eta }{2},{\mathcal{P}}_{\eta }=\{r\text{ as in C79}:q_r>1,N({\mathfrak{s}}_1(r))\le q_r^{\eta }\}.$$

This selects $r$, not the full frequency $k=ϵe_S(\mathbf{j})r{a}^6$. If $a$ overlaps a prime of ${\mathfrak{s}}_1(r)$, that prime has valuation $1+6v_p(a)$ in $k$, so the valuation-one product of the full $k$ need not equal ${\mathfrak{s}}_1(r)$. All ideals $a$, including overlaps and $S$ factors, remain in the frequency sum, together with every original unit and $S$ label.

Each component’s original ${\Pi }_i$ is frequency-independent, fixed before that sum, and satisfies $q_{{\Pi }_i}\le U^{C_{\Pi }}$. Retain literally $b_A=\mu +K_{A;0}$, all four $\mu /K$ terms in C80, the product-squarefree condition, original signs and phases, total-class selector, fixed twists and every defining-modulus/collision zero. In particular the original ${\chi }_n(a^6)=1_{(n,a)=1}$ stays in C81’s joint correction with $P=\mathrm{rad}({\Pi }_{i}ra){|}_{p\notin S}$. Neither the coefficient law nor the original column scales are reset at a physical dyad $Q\asymp q_r$.

Source correspondence and the intersected count. The pinned primary, physical ideal split and stratum count, lines 5238–5294 already contains this split and its $Q{V}^{-1/2}$ upper count. In that source’s notation, take $u=r$, its supported factor $\mathfrak{s}=1$ because $r$ is good, its powerful factor $\mathfrak{w}=\mathfrak{t}(r)$ and its residual ideal $\mathfrak{k}={\mathfrak{s}}_1(r)$; the source scale $U$ there is the present physical scale $Q$, not the column scale $U$ here. Its residual $k$ is also distinct from C79’s full frequency $k$. Use the original fixed generators/unit convention, leaving C79’s external six-unit and finite $S$ labels in their original sums.

For $q_r\asymp Q$ and $N(\mathfrak{t}(r))\asymp V$, the actual residual annulus is $N({\mathfrak{s}}_1(r))\asymp Q/V$. The sector condition forces $V\gg Q^{1-\eta }$. Sum the source’s existing $Q{V}^{-1/2}$ bound over these dyadic $V\ll Q$, including bounded unit ranges. This geometric sum, with all physical restrictions retained or enlarged only for a positive upper count, gives

$$\#\{r\in {\mathcal{P}}_{\eta }:q_r\asymp Q\}{\ll }_{K,S,\eta }{Q}^{\theta },\#({\mathcal{B}}_{\ell ,\alpha }(Q)\cap {\mathcal{P}}_{\eta })\ll Q^{m_{\alpha }+\delta },m_{\alpha }=\min \{d_{\alpha },\theta \}.\text{\hspace{1em}(C106)}$$