Smoothed Biorthogonal Tail Extensions and Multi-Zero Step-Space Superposition in the Shifted Nyman-Beurling Criterion: Two-Sided Divisor-Sum Operator Bounds, Cutoff Jump Elimination, and Interval-Certified Collective Zero Exclusion

Abstract

In Beurling's shifted Nyman criterion on $L^2((0, 1], d\mu_\sigma)$ ($d\mu_\sigma(x) = 2\sigma x^{2\sigma-1}\,dx$, $\sigma \in [1/2, 1)$), $\zeta(s)$ is zero-free in $\operatorname{Re}(s) > \sigma$ if and only if $E = \mathbf{1}_{(0,1]}$ lies in the closed span of the pole-canceled dilates $\tilde{\rho}_{1/k}(x) = -\{\lfloor 1/x \rfloor / k\}$ ($k \ge 2$) on $I_m = (1/(m+1), 1/m]$. Building on the Vasyunin-Balazard backward-difference basis $\{\mathcal{C}_n\}_{n\ge 1}$ and the step-space annihilator duality of riemann_hypothesis_9, we prove four results. First, for the head-correction Gram operator $H_N^{(\sigma)}$ of $\mathcal{H}_N = \sum_{k=2}^N \gamma_k \mathcal{C}_k$ relative to $N\operatorname{diag}(k^{2\sigma})$, localized alternating vectors prove $\liminf_{N\to\infty}\lambda_{\max}^{(N)}(\sigma) \ge 2/\sigma$, while the exact finite-$N$ Cauchy-Schwarz column weight $\widetilde{\Psi}_{\sigma, N}(k) = \sum_{j\mid k}\mu(j)^2 j^{-2\sigma}\vartheta_{k/j}^{(\sigma)} Q(\lfloor Nj/k\rfloor)/(Nj/k)$ yields $\lambda_{\max}^{(N)}(\sigma) \le (2/\sigma)\max_{k\le N}\widetilde{\Psi}_{\sigma, N}(k) \le \frac{2}{1-2^{-2\sigma}}\prod_{i=1}^{r(N)}(1+p_i^{-2\sigma}) < \frac{2}{1-2^{-2\sigma}}\zeta(2\sigma)/\zeta(4\sigma)$, remaining finite for each $N$ at $\sigma = 1/2$. Second, we show that sharp truncation at $k=N$ creates an $m=N$ jump of norm squared $|\gamma_N|^2/w_N^{(\sigma)} \asymp N^{2\sigma+1}|\gamma_N|^2$, and eliminate it via smoothed tail-extended corrections $\mathcal{H}_{N_\delta}^{\mathrm{smooth}} = \sum_{n=2}^{N_\delta} \gamma_n^{(M; W_{r+1,\theta})}(\rho_0) W_{2, 1/(1+\delta)}(n/(N_\delta+1))\mathcal{C}_n$ ($N_\delta = \lfloor(1+\delta)N\rfloor$), proving $\|\mathcal{H}_{N_\delta}^{\mathrm{smooth}}\|_{S_\sigma^*}^2 \ll_\sigma \zeta(3-2\sigma)N^{2r+2}/(M+1)^{2\beta_0+2r}$ (saving $N^{2\sigma}$) and $|\eta_1^{\mathrm{smooth}}| \ll N^{r+2}/(M+1)^{\beta_0+r+1}$, which reduces the unconditional support exponent to $a(\sigma, \beta_0, r) = \max\!\big(\frac{r+1}{\sigma+r},\, \frac{r+2}{\beta_0+r+1}\big) = 1 + \frac{1}{2r+3}$ at $\beta_0 = 1/2$ (halving the exponent excess over riemann_hypothesis_9). Third, we prove asymptotic factorization of $2J$-dimensional smoothed multi-zero annihilator superpositions via the continuum Mellin-Sobolev kernel $\mathcal{K}_W^{(\sigma)}(\rho_j, \rho_\ell)$, certifying $3.55\times$ gains at $N=40$ in 60-digit mpmath.iv arithmetic. Fourth, extending Burnol's Blaschke formula and riemann_hypothesis_1, we certify collective multi-pair step-space exclusions at $N=40, 160$ in 60-digit interval arithmetic.

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1. Introduction, Single Consistent Backward-Difference Convention, and Structural Synthesis

1.1 The Shifted Nyman-Beurling Criterion and Step-Space Annihilator Duality

Let $\sigma \in [1/2, 1)$, and let $\mathcal{H}_\sigma := L^2((0, 1], d\mu_\sigma)$ denote the probability-normalized Hilbert space with measure $d\mu_\sigma(x) := 2\sigma x^{2\sigma-1}\,dx$, inner product $\langle f, g \rangle_\sigma := 2\sigma \int_0^1 f(x)\overline{g(x)}\,x^{2\sigma-1}\,dx$ (linear in the first argument, conjugate-linear in the second), and norm $\|f\|_\sigma = \langle f, f \rangle_\sigma^{1/2}$, so that the constant indicator $E(x) := \mathbf{1}_{(0, 1]}(x)$ has $\|E\|_\sigma^2 = 1$ (in the unnormalized convention $\|f\|_{\mathrm{unnorm}, \sigma}^2 = \int_0^1 |f(x)|^2 x^{2\sigma-1}\,dx = (2\sigma)^{-1}\|f\|_\sigma^2$). Following Nyman [1], Beurling [2], and Báez-Duarte [3], define the pole-canceled fractional-part dilates $$\tilde{\rho}_{1/k}(x) := \frac{1}{k}\left\{\frac{1}{x}\right\} - \left\{\frac{1}{kx}\right\} = -\left\{\frac{\lfloor 1/x \rfloor}{k}\right\} \qquad (k \in \mathbb{N}_{\ge 2},\; x \in (0, 1]). \tag{1}$$ For $\operatorname{Re}(s) > 0$, the Mellin transform of $\tilde{\rho}_{1/k}$ is $\int_0^1 \tilde{\rho}_{1/k}(x)x^{s-1}\,dx = -\frac{\zeta(s)}{s}\big(k^{-1} - k^{-s}\big)$, in which the simple pole of $\zeta(s)$ at $s = 1$ cancels identically for every $k \ge 2$. Both $E$ and $\tilde{\rho}_{1/k}$ ($k \ge 2$) are constant on each harmonic step interval $I_m := (1/(m+1), 1/m]$ ($m \ge 1$) with interval measures $$w_m^{(\sigma)} := \mu_\sigma(I_m) = 2\sigma \int_{1/(m+1)}^{1/m} x^{2\sigma-1}\,dx = m^{-2\sigma} - (m+1)^{-2\sigma} = \frac{2\sigma}{m^{2\sigma+1}}\left(1 + O\left(\frac{1}{m}\right)\right). \tag{2}$$ Consequently, $\operatorname{span}\{\tilde{\rho}_{1/k} : k \ge 2\}$ lies inside the harmonic step subspace $S_\sigma := \overline{\operatorname{span}}\{\mathbf{1}_{I_m} : m \ge 1\} \subsetneq \mathcal{H}_\sigma$. Writing $f_m := f|_{I_m}$ for the constant value of $f \in S_\sigma$ on $I_m$ (so $E_m = 1$ and $(\tilde{\rho}_{1/k})_m = -\{m/k\}$ for all $m \ge 1$), the norm on $S_\sigma \simeq \ell^2(\mathbb{N}, w^{(\sigma)})$ and the Riesz dual norm on $S_\sigma^*$ are given by $$\|f\|_\sigma^2 = \sum_{m=1}^\infty |f_m|^2 w_m^{(\sigma)}, \qquad \|\mathcal{L}\|_{S_\sigma^*}^2 = \sum_{m=1}^\infty \frac{|c_m(\mathcal{L})|^2}{w_m^{(\sigma)}}, \qquad \mathcal{L}(f) = \sum_{m=1}^\infty c_m(\mathcal{L})\,f_m, \tag{3}$$ and the dual inner product is $\langle \mathcal{L}_1, \mathcal{L}_2 \rangle_{S_\sigma^*} := \sum_{m=1}^\infty \frac{c_m(\mathcal{L}_1)\overline{c_m(\mathcal{L}_2)}}{w_m^{(\sigma)}}$.

At truncation dimension $N \ge 2$, the shifted Nyman-Beurling-Báez-Duarte squared approximation distance in $\mathcal{H}_\sigma$ is $$d_N(\sigma)^2 := \inf_{b_2, \dots, b_N \in \mathbb{C}} \left\| E - \sum_{k=2}^N b_k \tilde{\rho}_{1/k} \right\|_\sigma^2 = 1 - b_{\mathrm{BD}}(\sigma)^T A_N(\sigma)^{-1} b_{\mathrm{BD}}(\sigma), \tag{4}$$ where $(A_N(\sigma))_{j,k} = \langle \tilde{\rho}_{1/j}, \tilde{\rho}_{1/k} \rangle_\sigma$ and $(b_{\mathrm{BD}}(\sigma))_k = \langle E, \tilde{\rho}_{1/k} \rangle_\sigma$ ($2 \le j, k \le N$). By the Hahn-Banach theorem in $S_\sigma$, $d_N(\sigma)^2$ satisfies the exact step-space annihilator duality identity (established in riemann_hypothesis_7 [15] and riemann_hypothesis_9 [17]): $$d_N(\sigma)^2 = \sup\left\{ \frac{|\mathcal{L}(E)|^2}{\|\mathcal{L}\|_{S_\sigma^*}^2} : \mathcal{L} \in \operatorname{Ann}_{S_\sigma^*}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}) \setminus \{0\} \right\}, \tag{5}$$ where $\operatorname{Ann}_{S_\sigma^*}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}) := \{\mathcal{L} \in S_\sigma^* : \mathcal{L}(\tilde{\rho}_{1/k}) = 0 \text{ for all } 2 \le k \le N\}$.

Following Vasyunin [6], Balazard and de Roton [7], and Balazard [18] (see also Bettin, Conrey, and Farmer [8], Ehm [19], Alouges, Darses, and Hillion [20], and our earlier papers [15, 17]), we use throughout this paper the backward-difference convention $\mathcal{D}_m(f) := f_m - f_{m-1}$ for $m \ge 1$, with the boundary convention $f_0 := 0$. Since $(\tilde{\rho}_{1/k})_m = -\{m/k\} = \lfloor m/k \rfloor - m/k$ and $(\tilde{\rho}_{1/k})_0 = 0$, we have $\lfloor m/k \rfloor - \lfloor (m-1)/k \rfloor = \mathbf{1}_{k \mid m}$, so $$\mathcal{D}_m(\tilde{\rho}_{1/k}) = \mathbf{1}_{k \mid m} - \frac{1}{k} \quad (m \ge 1,\; k \ge 2), \qquad \mathcal{D}_m(E) = \delta_{m,1} \quad (m \ge 1). \tag{6}$$ The Vasyunin-Balazard functionals $\mathcal{C}_n \in S_\sigma^*$ ($n \ge 1$) are defined by Möbius inversion of $\{\mathcal{D}_d\}_{d \mid n}$: $$\mathcal{C}_n(f) := \sum_{d \mid n} \mu\!\left(\frac{n}{d}\right) \mathcal{D}_d(f) = \sum_{d \mid n} \mu\!\left(\frac{n}{d}\right) (f_d - f_{d-1}) = \sum_{m=1}^n c_{n,m}\,f_m, \qquad c_{n,m} := \mu\!\left(\frac{n}{m}\right)\mathbf{1}_{m \mid n} - \mu\!\left(\frac{n}{m+1}\right)\mathbf{1}_{(m+1) \mid n}. \tag{7}$$ Applying Möbius inversion $\sum_{d \mid n} \mu(n/d)\mathbf{1}_{k \mid d} = \delta_{n,k}$ ($n, k \ge 2$) and $\sum_{d \mid n} \mu(d) = \delta_{n,1}$ to (6) yields the exact algebraic biorthogonality and target evaluation identities ([15, Theorem 2], [17, Section 2.1]): $$\mathcal{C}_1(\tilde{\rho}_{1/k}) = -\frac{1}{k} \quad (k \ge 2), \qquad \mathcal{C}_n(\tilde{\rho}_{1/k}) = \delta_{n,k} \quad (n, k \ge 2), \qquad \mathcal{C}_n(E) = \mu(n) \quad (n \ge 1). \tag{8}$$ More generally, any finitely supported step functional $\mathcal{L}(f) = \sum_{d=1}^M \beta_d \mathcal{D}_d(f) = \sum_{m=1}^M (\beta_m - \beta_{m+1}) f_m$ ($\beta_{M+1} := 0$) has step coefficients $c_m(\mathcal{L}) = \beta_m - \beta_{m+1}$, target evaluation $\mathcal{L}(E) = \sum_{d=1}^M \beta_d \delta_{d,1} = \beta_1$, Vasyunin-Balazard coordinates $\alpha_n := \sum_{j=1}^{\lfloor M/n \rfloor} \beta_{nj}$ ($\mathcal{L} = \sum_{n=1}^M \alpha_n \mathcal{C}_n$, $\beta_d = \sum_{j=1}^{\lfloor M/d \rfloor} \alpha_{dj}\mu(j)$), and dilate evaluations $$\gamma_k := \mathcal{L}(\tilde{\rho}_{1/k}) = \alpha_k - \frac{1}{k}\alpha_1 \qquad (k \ge 2). \tag{9}$$

1.2 Relation to Prior Literature and Summary of Main Results

In riemann_hypothesis_9 [17], we proved the power-saving asymptotic norm law $\|\mathcal{F}_N\|_{S_\sigma^*}^2 = \frac{\zeta(2\sigma+1)}{2\sigma\,\zeta(4\sigma+2)}N^{2\sigma-1} + O_\sigma(N^{\sigma-1/2})$ for the harmonic head functional $\mathcal{F}_N := \sum_{k=1}^N k^{-1}\mathcal{C}_k$ (whose step-level form $\sum_{k=1}^N f_k/k$ at $\sigma = 1/2$ was studied by Balazard [18], Propositions 5-6), introduced the flat-top Hermite spline zero annihilators $\widetilde{\mathcal{L}}_{\rho_0, N}^{(M; W_{r,\theta})}$ to constructively realize the existential rate exponent $d_N(\sigma)^2 = N^{-2(\sigma-\Theta)^+ + o(1)}$ of riemann_hypothesis_6 [14] (v3), and introduced the hybrid step-head + continuous-tail kernels $k_{\rho, M}^{(\sigma)}$ extending the step-space codimension framework of riemann_hypothesis_1 [12] (v2) and Burnol's continuous Blaschke projection formula [21, Lemma 2.4].

In the present paper, we resolve three structural questions raised after riemann_hypothesis_9 [17]:

1. Two-Sided Divisor-Sum Operator Norm Bounds (Section 2, Theorem 1): In Lemma 1 of riemann_hypothesis_9 [17], we bounded the dual norm squared of any head correction $\mathcal{H}_N = \sum_{k=2}^N \gamma_k \mathcal{C}_k$ by $\frac{3^{2\sigma+1}}{\sigma}\zeta(2\sigma) N \sum_{k=2}^N k^{2\sigma}|\gamma_k|^2$, which diverges as $(2\sigma - 1)^{-1}$ as $\sigma \downarrow 1/2$. Let $H_N^{(\sigma)}$ be the $(N-1) \times (N-1)$ Gram matrix of $\{\mathcal{C}_2, \dots, \mathcal{C}_N\}$ in $S_\sigma^*$, and let $\lambda_{\max}^{(N)}(\sigma) := \sup_{\gamma \ne 0} \frac{\|\sum_{k=2}^N \gamma_k \mathcal{C}_k\|_{S_\sigma^*}^2}{N \sum_{k=2}^N k^{2\sigma}|\gamma_k|^2}$.

   • Asymptotic Lower Bound: On $m \in (N/2, N]$, $\lfloor N/m \rfloor = 1$, so $\eta_m = \sum_{j=1}^{\lfloor N/m \rfloor} \gamma_{mj}\mu(j) = \gamma_m \mu(1) = \gamma_m$ consists exclusively of the $j = 1$ divisor term (showing in particular that the asymptotic density $6/\pi^2$ of squarefree integers cannot multiply the $j = 1$ term). Testing localized alternating vectors $\gamma_k = (-1)^k k^{-\sigma}\mathbf{1}_{(1-\delta)N < k \le N}$ proves $\liminf_{N\to\infty}\lambda_{\max}^{(N)}(\sigma) \ge \frac{2}{\sigma}$ for all $\sigma \in [1/2, 1)$.
   • Exact Finite-$N$ Cauchy-Schwarz Column Weight and Primorial/Euler Upper Bounds: Retaining the exact squarefree counting function $Q(L) := \sum_{\ell=1}^L \mu(\ell)^2$ ($L = \lfloor N/m \rfloor = \lfloor Nj/k \rfloor$) and the exact step-weight ratio $\vartheta_m^{(\sigma)} := \frac{\sigma}{2m^{2\sigma+1}}\big(\frac{2}{w_m^{(\sigma)}} + \frac{2\mathbf{1}_{m\ge 2}}{w_{m-1}^{(\sigma)}}\big)$ (which equals $1$ identically for all $m \ge 1$ at $\sigma = 1/2$ and satisfies $1 < \vartheta_m^{(\sigma)} \le \vartheta_1^{(\sigma)} = \frac{\sigma}{1 - 2^{-2\sigma}}$ for $\sigma \in (1/2, 1)$) proves the exact finite-$N$ hierarchy $$\lambda_{\max}^{(N)}(\sigma) \le C_{\mathrm{col}}^{(N)}(\sigma) := \frac{2}{\sigma}\max_{2 \le k \le N}\widetilde{\Psi}_{\sigma, N}(k) \le C_{\mathrm{prim}}^{(N)}(\sigma) := \frac{2}{1 - 2^{-2\sigma}}\prod_{i=1}^{r(N)}\left(1 + p_i^{-2\sigma}\right) < C_{\mathrm{Euler}}(\sigma) := \frac{2}{1 - 2^{-2\sigma}}\frac{\zeta(2\sigma)}{\zeta(4\sigma)}, \tag{10}$$ where $\widetilde{\Psi}_{\sigma, N}(k) := \sum_{j \mid k}\mu(j)^2 j^{-2\sigma}\vartheta_{k/j}^{(\sigma)}\frac{Q(\lfloor Nj/k\rfloor)}{Nj/k}$ and $P_{r(N)} = p_1\cdots p_{r(N)} \le N < P_{r(N)+1}$. Both $C_{\mathrm{col}}^{(N)}(\sigma)$ and $C_{\mathrm{prim}}^{(N)}(\sigma)$ are finite for each fixed $N$ right down to $\sigma = 1/2$ (while growing slowly with $N$ as $\prod_{i=1}^{r(N)}(1+p_i^{-1}) \ll \log\log N$ at $\sigma = 1/2$, and remaining uniformly bounded in $N$ by $C_{\mathrm{Euler}}(\sigma)$ for every $\sigma > 1/2$).

2. Sharp-Cutoff Boundary Obstruction vs. Smoothed Tail-Extended Biorthogonal Corrections (Section 3, Theorem 2):

   • The $m = N$ Cutoff Jump Obstruction: When the head correction $\mathcal{H}_N = \sum_{k=2}^N \gamma_k \mathcal{C}_k$ is truncated sharply at $k = N$, the single interval $m = N$ has $\eta_N = \gamma_N$ and $\eta_{N+1} = 0$, contributing $\frac{|\gamma_N|^2}{w_N^{(\sigma)}} \ge \frac{N^{2\sigma+1}}{2\sigma}|\gamma_N|^2$, which grows as $N^1$ relative to $\|\mathcal{L}_{\rho_0}^{(M; W)}\|_{S_\sigma^*}^2$ at fixed linear support ratio $M = \lfloor \Lambda N \rfloor$.
   • Smoothed Tail Extension, Uniform $X_n$-Derivative Proof, and Combined Support Scaling: Because $\mathcal{C}_n(\tilde{\rho}_{1/k}) = \delta_{n,k} = 0$ for all $n > N \ge k \ge 2$, extending the correction smoothly to $N_\delta := \lfloor(1+\delta)N\rfloor$ ($\delta \in (0, 1]$) via $$\mathcal{H}_{N_\delta}^{\mathrm{smooth}} := \sum_{n=2}^{N_\delta} \tilde{\gamma}_n\,\mathcal{C}_n, \qquad \tilde{\gamma}_n := \gamma_n^{(M; W_{r+1,\theta})}(\rho_0)\,\Omega_\delta\!\left(\frac{n}{N_\delta + 1}\right), \qquad \Omega_\delta := W_{2, \frac{1}{1+\delta}}, \tag{11}$$ preserves exact annihilation of $\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}$ while eliminating all cutoff jumps ($\tilde{\gamma}_{(m+1)j} = 0$ for $j > \lfloor N_\delta/(m+1)\rfloor$). Using the exact Euler-Maclaurin subtraction identity in $X_n := \frac{M+1}{n}$ from riemann_hypothesis_9 [17]—where subtracting $t^{-\rho_0}$ on $[1, \infty)$ from $g_X(t) = t^{-\rho_0}W_{r+1,\theta}(t/X)$ leaves $h_X(t) := t^{-\rho_0}(W_{r+1,\theta}(t/X) - 1) \in C^r([1, \infty))$ vanishing identically on $[1, \theta X]$ with no dependence on $\lfloor \theta X \rfloor$—and integrating by parts once on $y = t/X \in [\theta, 1]$, we prove uniformly for all $2 \le n \le \frac{\theta}{2}(M+1)$ that $|\gamma_n| \ll \frac{n^{r+1}}{(M+1)^{\beta_0+r+1}}$ and $|\gamma_n - \gamma_{n+1}| \ll \frac{n^{r-1}}{(M+1)^{\beta_0+r}}$. This yields $|\eta_1^{\mathrm{smooth}}| \le \sum_{n=2}^{N_\delta}|\tilde{\gamma}_n| \ll \frac{N^{r+2}}{(M+1)^{\beta_0+r+1}}$, $|\eta_m^{\mathrm{smooth}} - \eta_{m+1}^{\mathrm{smooth}}| \ll \frac{N_\delta^{r+1}}{m^2(M+1)^{\beta_0+r}}$, and the $N^{2\sigma}$-saving norm bound $$\left\|\mathcal{H}_{N_\delta}^{\mathrm{smooth}}\right\|_{S_\sigma^*}^2 \ll_{\sigma,\rho_0,r,\theta,\delta} \frac{N^{2r+2}}{(M+1)^{2\beta_0+2r}}\sum_{m=1}^{N_\delta} m^{2\sigma-3} \le \zeta(3 - 2\sigma)\,\frac{N^{2r+2}}{(M+1)^{2\beta_0+2r}}. \tag{12}$$ Combining the relative norm condition $\|\mathcal{H}_{N_\delta}^{\mathrm{smooth}}\|_{S_\sigma^*} / \|\mathcal{L}_{\rho_0}^{(M; W_{r+1,\theta})}\|_{S_\sigma^*} \ll \frac{N^{r+1}}{(M+1)^{\sigma+r}}$ and the absolute target-perturbation condition $|\eta_1^{\mathrm{smooth}}| \ll \frac{N^{r+2}}{(M+1)^{\beta_0+r+1}}$ yields the unconditional constructive support exponent $a(\sigma, \beta_0, r) := \max\!\big(\frac{r+1}{\sigma+r},\, \frac{r+2}{\beta_0+r+1}\big)$ (which equals $\frac{r+2}{r+3/2} = 1 + \frac{1}{2r+3}$ at $\beta_0 = 1/2$, halving the exponent excess $1 + \frac{1}{r+1/2}$ of riemann_hypothesis_9 [17], e.g., $8/7 \approx 1.1429$ vs. $7/5 = 1.40$ at $r = 2$).

3. Multi-Zero Step-Space Annihilator Superposition (Section 4, Theorem 3): For $d = 2J$ distinct zeros $\rho_1, \dots, \rho_d$ in $\operatorname{Re}(s) \in [1/2, \sigma)$, we prove that the projected Gram matrix $\widetilde{G}_{\mu,\nu}^{(N, M)} = \langle \widetilde{\mathcal{L}}_{\mu, N}^{(M)}, \widetilde{\mathcal{L}}_{\nu, N}^{(M)} \rangle_{S_\sigma^*}$ satisfies $\widetilde{G}_{\mu,\nu}^{(N, M)} = 2\sigma (M+1)^{2\sigma-\rho_\mu-\overline{\rho_\nu}}\mathcal{K}_W^{(\sigma)}(\rho_\mu, \rho_\nu) + O\big(1 + \frac{N^{r+1}}{(M+1)^{\beta_\mu+\beta_\nu-\sigma+r}}\big)$, where $\mathcal{K}_W^{(\sigma)}(\rho, \rho')$ is the continuum Mellin-Sobolev kernel, and we certify $3.55\times$ multi-zero gains at $N = 40, M = 320$ ($N_\delta = 45, W_3$) in 60-digit mpmath.iv arithmetic.

4. Multi-Pair Hybrid Step-Space Reproducing Kernels (Section 5, Theorem 4): Specializing Burnol's continuous Blaschke distance formula [21, Lemma 2.4] to $J$ conjugate zero pairs $Z_J = \{\rho_1, \bar{\rho}_1, \dots, \rho_J, \bar{\rho}_J\}$ at $M = 0$ and extending the finite-set step-space Gram-inverse bounds of riemann_hypothesis_1 [12] (Theorem C and Corollary D) and riemann_hypothesis_9 [17] (Theorem 4) to $2J \times 2J$ hybrid kernels $K_{Z_J, M}^{(\sigma)}$, we prove strict Loewner monotonicity $K_{Z_J, M+1}^{(\sigma)} \prec K_{Z_J, M}^{(\sigma)} \prec K_{Z_J, 0}^{(\sigma)}$ and certify in 60-digit mpmath.iv arithmetic ($M = 500$) that collective $J \in \{2, 3\}$ step-space bounds exclude test configurations at $N = 40$ where $J = 1$ and continuous $H^2$ Blaschke bounds fail.

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2. Two-Sided Divisor-Sum Operator Norm Theory for the Biorthogonal Head Basis

For any coefficient vector $\gamma = (\gamma_2, \dots, \gamma_N)^T \in \mathbb{C}^{N-1}$ (setting $\gamma_1 := 0$), consider the biorthogonal head combination $\mathcal{H}_N := \sum_{k=2}^N \gamma_k \mathcal{C}_k \in S_\sigma^*$. By (7), $\mathcal{H}_N(f) = \sum_{d=1}^N \eta_d \mathcal{D}_d(f) = \sum_{m=1}^N (\eta_m - \eta_{m+1}) f_m$, where $$\eta_m := \sum_{\substack{j=1 \\ mj \ge 2}}^{\lfloor N/m \rfloor} \gamma_{mj}\,\mu(j) \quad (1 \le m \le N), \qquad \eta_{N+1} := 0. \tag{13}$$ Thus the $(N-1) \times (N-1)$ Gram matrix $H_N^{(\sigma)}$ of $\{\mathcal{C}_2, \dots, \mathcal{C}_N\}$ in $S_\sigma^*$ satisfies $$\left\|\sum_{k=2}^N \gamma_k \mathcal{C}_k\right\|_{S_\sigma^*}^2 = \gamma^* H_N^{(\sigma)} \gamma = \sum_{m=1}^N \frac{|\eta_m - \eta_{m+1}|^2}{w_m^{(\sigma)}}. \tag{14}$$ Let $D_N^{(\sigma)} := \operatorname{diag}(2^{2\sigma}, 3^{2\sigma}, \dots, N^{2\sigma})$ and define the generalized maximum eigenvalue $$\lambda_{\max}^{(N)}(\sigma) := \lambda_{\max}\!\left(H_N^{(\sigma)}, N D_N^{(\sigma)}\right) = \sup_{\gamma \in \mathbb{C}^{N-1} \setminus \{0\}} \frac{\left\|\sum_{k=2}^N \gamma_k \mathcal{C}_k\right\|_{S_\sigma^*}^2}{N \sum_{k=2}^N k^{2\sigma}|\gamma_k|^2}. \tag{15}$$

Theorem 1 (Two-Sided Divisor-Sum Operator Norm Bounds and Critical-Line Primorial Enclosures). Let $\sigma \in [1/2, 1)$ and $N \ge 2$.

1. (Exact Asymptotic Lower Bound from the $j = 1$ Regime): For every $\sigma \in [1/2, 1)$, $$\liminf_{N \to \infty} \lambda_{\max}^{(N)}(\sigma) \ge \frac{2}{\sigma}. \tag{16}$$
2. (Exact Finite-$N$ Cauchy-Schwarz Column Weight): For each $m \ge 1$, define $$\widetilde{W}_m^{(\sigma)} := \frac{2}{w_m^{(\sigma)}} + \frac{2\,\mathbf{1}_{m \ge 2}}{w_{m-1}^{(\sigma)}}, \qquad \vartheta_m^{(\sigma)} := \frac{\sigma\,\widetilde{W}_m^{(\sigma)}}{2m^{2\sigma+1}} = 1 + \frac{4\sigma^2 - 1}{12 m^2} + O(m^{-4}). \tag{17}$$ Then $\vartheta_m^{(1/2)} = 1$ identically for all $m \ge 1$, and for every $\sigma \in (1/2, 1)$, $m \mapsto \vartheta_m^{(\sigma)}$ is strictly decreasing on $m \ge 1$ with $1 < \vartheta_m^{(\sigma)} \le \vartheta_1^{(\sigma)} = \frac{\sigma}{1 - 2^{-2\sigma}}$. Let $Q(L) := \sum_{\ell=1}^L \mu(\ell)^2$ denote the number of squarefree integers in $\{1, \dots, L\}$, and for each $2 \le k \le N$ define the exact finite-$N$ column weight (and its principal part $\Psi_{\sigma, N}(k)$) $$\widetilde{\Psi}_{\sigma, N}(k) := \sum_{j \mid k} \mu(j)^2 j^{-2\sigma}\,\vartheta_{k/j}^{(\sigma)}\,\frac{Q(\lfloor Nj/k \rfloor)}{Nj/k}, \qquad \Psi_{\sigma, N}(k) := \sum_{j \mid k} \mu(j)^2 j^{-2\sigma}\,\frac{Q(\lfloor Nj/k \rfloor)}{Nj/k}. \tag{18}$$ Then for every vector $\gamma = (\gamma_2, \dots, \gamma_N)^T \in \mathbb{C}^{N-1}$, $$\left\|\sum_{k=2}^N \gamma_k \mathcal{C}_k\right\|_{S_\sigma^*}^2 \le \frac{2}{\sigma} N \sum_{k=2}^N \widetilde{\Psi}_{\sigma, N}(k)\,k^{2\sigma}|\gamma_k|^2 \le C_{\mathrm{col}}^{(N)}(\sigma)\,N \sum_{k=2}^N k^{2\sigma}|\gamma_k|^2, \qquad C_{\mathrm{col}}^{(N)}(\sigma) := \frac{2}{\sigma}\max_{2 \le k \le N}\widetilde{\Psi}_{\sigma, N}(k). \tag{19}$$
3. (Finite-$N$ Primorial Bound and Uniform Squarefree Euler Bound): Let $p_1 < p_2 < \cdots$ denote the primes, and let $r(N)$ be the unique integer such that the primorial $P_{r(N)} := \prod_{i=1}^{r(N)} p_i \le N < P_{r(N)+1}$. Then for every $2 \le k \le N$, $$\Psi_{\sigma, N}(k) \le \widetilde{\Psi}_{\sigma, N}(k) \le \vartheta_1^{(\sigma)} \Psi_{\sigma, N}(k) \le \vartheta_1^{(\sigma)} \prod_{p \mid k}\left(1 + p^{-2\sigma}\right) \le \vartheta_1^{(\sigma)} \prod_{i=1}^{r(N)}\left(1 + p_i^{-2\sigma}\right), \tag{20}$$ and consequently, for all $\sigma \in [1/2, 1)$ (and for $\sigma > 1/2$ in the $N$-uniform Euler bound), $$\lambda_{\max}^{(N)}(\sigma) \le C_{\mathrm{col}}^{(N)}(\sigma) \le C_{\mathrm{prim}}^{(N)}(\sigma) := \frac{2}{1 - 2^{-2\sigma}}\prod_{i=1}^{r(N)}\left(1 + p_i^{-2\sigma}\right) < C_{\mathrm{Euler}}(\sigma) := \frac{2}{1 - 2^{-2\sigma}}\frac{\zeta(2\sigma)}{\zeta(4\sigma)} < C_{\mathrm{unw}}(\sigma) := \frac{2}{1 - 2^{-2\sigma}}\zeta(2\sigma). \tag{21}$$

Proof. Part 1 (Asymptotic Lower Bound $2/\sigma$): Let $\delta \in (0, 1/2)$ and $N > 2/(1-\delta)$, and test the localized alternating vector $\gamma_k := (-1)^k k^{-\sigma}\,\mathbf{1}_{(1-\delta)N < k \le N}$ ($2 \le k \le N$). For every step index $m \in ((1-\delta)N, N]$, we have $m > N/2$, so $\lfloor N/m \rfloor = 1$ in (13) and all $j \ge 2$ divisor terms vanish identically, leaving $\eta_m = \gamma_m \mu(1) = (-1)^m m^{-\sigma}$. For $(1-\delta)N < m \le N - 1$, $\gamma_m$ and $\gamma_{m+1}$ have opposite signs, so $|\eta_m - \eta_{m+1}|^2 = (m^{-\sigma} + (m+1)^{-\sigma})^2 = 4m^{-2\sigma}(1 + O(m^{-1}))$. Since $(w_m^{(\sigma)})^{-1} = \frac{m^{2\sigma+1}}{2\sigma}(1 + O(m^{-1}))$, summing over $(1-\delta)N < m \le N - 1$ gives $$\left\|\sum_{k=2}^N \gamma_k \mathcal{C}_k\right\|_{S_\sigma^*}^2 \ge \sum_{(1-\delta)N < m \le N-1} \frac{|\eta_m - \eta_{m+1}|^2}{w_m^{(\sigma)}} = \frac{2}{\sigma}\frac{1 - (1-\delta)^2}{2}N^2 + O_\sigma(N). \tag{22}$$ Dividing by $N \sum_{k=2}^N k^{2\sigma}|\gamma_k|^2 = \delta N^2 + O(N)$ and letting $N \to \infty$ then $\delta \downarrow 0$ yields (16).

Part 2 (Exact Finite-$N$ Cauchy-Schwarz Column Weight): Using $|\eta_m - \eta_{m+1}|^2 \le 2|\eta_m|^2 + 2|\eta_{m+1}|^2$ in (14) and shifting $m + 1 \mapsto m$ in the second sum (with $\eta_{N+1} = 0$) gives the exact inequality $$\sum_{m=1}^N \frac{|\eta_m - \eta_{m+1}|^2}{w_m^{(\sigma)}} \le \sum_{m=1}^N \widetilde{W}_m^{(\sigma)} |\eta_m|^2 = \frac{2}{\sigma}\sum_{m=1}^N m^{2\sigma+1}\vartheta_m^{(\sigma)} |\eta_m|^2. \tag{23}$$ At $\sigma = 1/2$, $w_m^{(1/2)} = \frac{1}{m(m+1)}$, so $\widetilde{W}_1^{(1/2)} = 2(2) = 4 = 4 \cdot 1^2$ and for $m \ge 2$, $\widetilde{W}_m^{(1/2)} = 2m(m+1) + 2(m-1)m = 4m^2$, whence $\vartheta_m^{(1/2)} = \frac{(1/2)(4m^2)}{2m^2} = 1$ identically for all $m \ge 1$. For $\sigma \in (1/2, 1)$, let $\phi(t) := \frac{1 - (1+t)^{-2\sigma}}{2\sigma} = \int_0^t (1+u)^{-2\sigma-1}\,du$. Since $(1+u)^{-2\sigma-1}$ is strictly convex on $u \in (-1, \infty)$, Jensen's inequality gives $\phi(t) > t(1+t/2)^{-2\sigma-1}$, and for $m \ge 2$ we have $\vartheta_m^{(\sigma)} = \frac{1}{2}\big(\frac{1}{\phi(1/m)} + \frac{1}{-\phi(-1/m)}\big)$. Because $t \mapsto \frac{1}{2}\big(\frac{1}{\phi(t)} - \frac{1}{\phi(-t)}\big) = 1 + \frac{4\sigma^2-1}{12}t^2 + \cdots$ has strictly positive even Taylor coefficients on $|t| < 1$, $\vartheta_m^{(\sigma)}$ is strictly decreasing for $m \ge 2$ with $\vartheta_2^{(\sigma)} < \vartheta_1^{(\sigma)} = \frac{\sigma}{1 - 2^{-2\sigma}}$.

Next, for each $1 \le m \le N$, since $\mu(j) \in \{-1, 0, 1\}$, we have $|\mu(j)| = \mu(j)^2$. Applying Cauchy-Schwarz over the squarefree integers $1 \le j \le L_m := \lfloor N/m \rfloor$ gives $$|\eta_m|^2 = \left|\sum_{\substack{j=1 \\ mj \ge 2}}^{L_m} \mu(j)\,\gamma_{mj}\right|^2 \le \left(\sum_{j=1}^{L_m} \mu(j)^2\right)\left(\sum_{\substack{j=1 \\ mj \ge 2}}^{L_m} \mu(j)^2 |\gamma_{mj}|^2\right) = Q\!\left(\left\lfloor \frac{N}{m}\right\rfloor\right)\sum_{\substack{j=1 \\ mj \ge 2}}^{\lfloor N/m \rfloor} \mu(j)^2 |\gamma_{mj}|^2. \tag{24}$$ Substituting (24) into (23) and reindexing by $k = mj$ ($2 \le k \le N$, $m = k/j$, $\lfloor N/m \rfloor = \lfloor Nj/k \rfloor$), the coefficient of $|\gamma_k|^2$ is $$\frac{2}{\sigma}\sum_{j \mid k} \mu(j)^2 \left(\frac{k}{j}\right)^{2\sigma+1} \vartheta_{k/j}^{(\sigma)}\,Q\!\left(\left\lfloor \frac{Nj}{k}\right\rfloor\right) = \frac{2}{\sigma} N k^{2\sigma} \sum_{j \mid k} \mu(j)^2 j^{-2\sigma}\,\vartheta_{k/j}^{(\sigma)}\,\frac{Q(\lfloor Nj/k \rfloor)}{Nj/k} = \frac{2}{\sigma} N k^{2\sigma}\widetilde{\Psi}_{\sigma, N}(k), \tag{25}$$ establishing (18)-(19). (For example, at $N = k = 2$, $\sigma = 1/2$, $\gamma_2 = 1$, $\mathcal{C}_2(f) = -f_1 + 2f_2 - f_3$ has $\eta_1 = -1, \eta_2 = 1, \eta_3 = 0$ and $\|\mathcal{C}_2\|_{S_{1/2}^*}^2 = \frac{(-2)^2}{1/2} + \frac{1^2}{1/6} = 8 + 6 = 14$, while the right-hand side of (19) is $4 \cdot 2 \cdot \widetilde{\Psi}_{1/2, 2}(2) \cdot 2 = 16 \cdot \frac{3}{2} = 24 > 14$.)

Part 3 (Primorial and Euler Bounds): Since $1 \le \vartheta_{k/j}^{(\sigma)} \le \vartheta_1^{(\sigma)} = \frac{\sigma}{1 - 2^{-2\sigma}}$ and $Q(L) \le L$ for all $L \ge 1$, we have $\frac{Q(\lfloor Nj/k \rfloor)}{Nj/k} \le \frac{Q(\lfloor Nj/k \rfloor)}{\lfloor Nj/k \rfloor} \le 1$, so $\widetilde{\Psi}_{\sigma, N}(k) \le \vartheta_1^{(\sigma)} \sum_{j \mid k}\mu(j)^2 j^{-2\sigma} = \vartheta_1^{(\sigma)} \prod_{p \mid k}(1 + p^{-2\sigma})$. Since the number of distinct prime factors $\omega(k)$ of any $k \le N$ satisfies $p_1\cdots p_{\omega(k)} \le k \le N$, we have $\omega(k) \le r(N)$, and replacing the prime factors of $k$ by the smallest $\omega(k)$ primes gives $\prod_{p \mid k}(1 + p^{-2\sigma}) \le \prod_{i=1}^{r(N)}(1 + p_i^{-2\sigma}) < \prod_p (1 + p^{-2\sigma}) = \frac{\zeta(2\sigma)}{\zeta(4\sigma)}$, proving (20)-(21). $\square$

2.1 Interval-Certified Rayleigh Lower Bounds and Column/Primorial Upper Bounds

Table 1 reports 60-digit mpmath.iv interval-certified Rayleigh-quotient lower bounds on $\lambda_{\max}^{(N)}(\sigma)$ for $N \in \{100, 400, 1600\}$ (obtained via explicit integer-rounded Rayleigh-quotient witnesses $\tilde{\gamma} \in 10^{-8}\mathbb{Z}^{N-1}$, extending the $\sigma \in \{0.60, 0.75, 0.90\}$ eigenvalue calculations of span_of_dilates on the r/riemann_hypothesis forum down to $\sigma = 0.55$ and $\sigma = 0.50$) together with the 60-digit mpmath.iv interval-certified upper bounds $C_{\mathrm{col}}^{(1600)}(\sigma)$, $C_{\mathrm{prim}}^{(1600)}(\sigma)$ ($r(1600) = 4$), $C_{\mathrm{Euler}}(\sigma)$, and $C_{\mathrm{unw}}(\sigma)$. All lower bounds are rounded strictly downward (ROUND_FLOOR) and all upper bounds are rounded strictly upward (ROUND_CEILING).

| $\sigma$ | Asymptotic Lower $2/\sigma \ge$ | Certified Rayleigh Lower $\lambda_{\max}^{(100)}(\sigma) \ge$ | Certified Rayleigh Lower $\lambda_{\max}^{(400)}(\sigma) \ge$ | Certified Rayleigh Lower $\lambda_{\max}^{(1600)}(\sigma) \ge$ | Exact Column Upper $C_{\mathrm{col}}^{(1600)}(\sigma) \le$ ($\arg\max k$) | Primorial Upper $C_{\mathrm{prim}}^{(1600)}(\sigma) \le$ ($r=4$) | Squarefree Euler $C_{\mathrm{Euler}}(\sigma) \le$ | Unweighted $C_{\mathrm{unw}}(\sigma) \le$ | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $0.50$ | $4.000000$ | $4.269010$ | $4.450491$ | $4.527602$ | $9.342167$ ($k=1470$) | $10.971429$ ($= 384/35$) | $\infty$ | $\infty$ | | $0.55$ | $3.636363$ | $3.803289$ | $3.948459$ | $4.019124$ | $7.849941$ ($k=1560$) | $9.338114$ | $26.621508$ | $39.680509$ | | $0.60$ | $3.333333$ | $3.429664$ | $3.556685$ | $3.616714$ | $6.706303$ ($k=1560$) | $8.091347$ | $14.315223$ | $19.802861$ | | $0.75$ | $2.666666$ | $2.655197$ | $2.747755$ | $2.790196$ | $4.546628$ ($k=1590$) | $5.733976$ | $6.723694$ | $8.082262$ | | $0.90$ | $2.222222$ | $2.171883$ | $2.245616$ | $2.278942$ | $3.360442$ ($k=1590$) | $4.468914$ | $4.732161$ | $5.281040$ |

Table 1: Two-sided divisor-sum operator norm bounds across $\sigma \in \{0.50, 0.55, 0.60, 0.75, 0.90\}$ (results/multizero_smoothed_step_duality.json). All lower bounds ($2/\sigma$ and the certified Rayleigh quotients for $\lambda_{\max}^{(N)}(\sigma)$ at $N \in \{100, 400, 1600\}$) are rounded strictly downward (ROUND_FLOOR), and all upper bounds ($C_{\mathrm{col}}^{(1600)}(\sigma)$, $C_{\mathrm{prim}}^{(1600)}(\sigma)$, $C_{\mathrm{Euler}}(\sigma)$, $C_{\mathrm{unw}}(\sigma)$) are rounded strictly upward (ROUND_CEILING) from 60-digit mpmath.iv interval enclosures.

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3. Sharp-Cutoff Boundary Obstruction vs. Smoothed Tail-Extended Biorthogonal Corrections

We now prove that extending the biorthogonal head correction smoothly across $N < n \le N_\delta := \lfloor(1+\delta)N\rfloor$ eliminates the $m = N$ cutoff jump and saves a full factor of $N^{2\sigma}$ in the dual norm squared.

Recall from riemann_hypothesis_9 [17] (Theorem 2(2)-(3)) that for every integer $q \ge 2$ and knot $\theta \in (0, 1)$, the flat-top Hermite spline $W_{q,\theta} \in C^{q-1}([0, \infty))$ is defined by $W_{q,\theta}(u) = 1$ on $[0, \theta]$, $W_{q,\theta}(u) = (1-\tau)^q \sum_{j=0}^{q-1}\binom{q-1+j}{j}\tau^j$ ($\tau = \frac{u-\theta}{1-\theta}$) on $(\theta, 1)$, and $W_{q,\theta}(u) = 0$ on $[1, \infty)$. For any nontrivial zero $\rho_0 = \beta_0 + i\gamma_0$ of $\zeta(s)$ with $\beta_0 \in [1/2, \sigma)$, the unprojected spline functional $\mathcal{L}_{\rho_0}^{(M; W_{q,\theta})}(f) := \sum_{d=1}^M \beta_d^{(M; W_{q,\theta})}\mathcal{D}_d(f)$ with $\beta_d^{(M; W_{q,\theta})} = \frac{2\sigma}{\rho_0}d^{-\rho_0}W_{q,\theta}\big(\frac{d}{M+1}\big)$ has coordinates $\alpha_n^{(M; W_{q,\theta})}(\rho_0) = \frac{2\sigma}{\rho_0}n^{-\rho_0}\sum_{\ell=1}^{\lfloor X_n \rfloor} \ell^{-\rho_0}W_{q,\theta}(\ell/X_n)$, where $X_n := \frac{M+1}{n}$.

Theorem 2 (Sharp-Cutoff Boundary Obstruction at $m = N$ vs. $N^{2\sigma}$-Saving Smoothed Tail Extensions). Let $\sigma \in (1/2, 1)$ and let $\rho_0 = \beta_0 + i\gamma_0$ be a nontrivial zero of $\zeta(s)$ with $\beta_0 \in [1/2, \sigma)$.

1. (Linear Growth Obstruction of the Sharp Cutoff at $m = N$): For the sharp-cutoff head correction $\mathcal{H}_N := \sum_{k=2}^N \gamma_k^{(M; W_{q,\theta})}(\rho_0)\,\mathcal{C}_k$, the single step interval $m = N$ has $\eta_N = \gamma_N^{(M; W_{q,\theta})}(\rho_0)$ and $\eta_{N+1} = 0$, contributing $$\left\|\mathcal{H}_N\right\|_{S_\sigma^*}^2 \ge \frac{|\eta_N - \eta_{N+1}|^2}{w_N^{(\sigma)}} = \frac{\big|\gamma_N^{(M; W_{q,\theta})}(\rho_0)\big|^2}{w_N^{(\sigma)}} \ge \frac{N^{2\sigma+1}}{2\sigma}\big|\gamma_N^{(M; W_{q,\theta})}(\rho_0)\big|^2. \tag{26}$$ At fixed linear support scaling $M = \lfloor \Lambda N \rfloor$ ($\Lambda \ge 2/\theta$), $|\gamma_N^{(M; W_{q,\theta})}(\rho_0)| \asymp_{\rho_0,q,\theta,\Lambda} N^{-\beta_0}$ for generic $\Lambda$, so $\frac{|\gamma_N|^2}{w_N^{(\sigma)}} \asymp N^{2(\sigma-\beta_0)+1} \asymp N^1 \|\mathcal{L}_{\rho_0}^{(M; W_{q,\theta})}\|_{S_\sigma^*}^2$.
2. (Smoothed Tail-Extended Biorthogonal Corrections, Target Perturbation, and Dual Norm Bound): Fix $r \ge 2$, $\theta \in (0, 1)$, and $\delta \in (0, 1]$, set $N_\delta := \lfloor(1+\delta)N\rfloor$, and define the smoothed tail-extended correction using the $C^r([0, \infty))$ spline $W_{r+1,\theta}$: $$\mathcal{H}_{N_\delta}^{\mathrm{smooth}} := \sum_{n=2}^{N_\delta} \tilde{\gamma}_n\,\mathcal{C}_n, \qquad \tilde{\gamma}_n := \gamma_n^{(M; W_{r+1,\theta})}(\rho_0)\,\Omega_\delta\!\left(\frac{n}{N_\delta + 1}\right), \qquad \Omega_\delta := W_{2, \frac{1}{1+\delta}}. \tag{27}$$ Then $\widetilde{\mathcal{L}}_{\rho_0, N}^{(M; W_{r+1,\theta}, \delta)} := \mathcal{L}_{\rho_0}^{(M; W_{r+1,\theta})} - \mathcal{H}_{N_\delta}^{\mathrm{smooth}}$ belongs to $\operatorname{Ann}_{S_\sigma^*}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$ and satisfies $\widetilde{\mathcal{L}}_{\rho_0, N}^{(M; W_{r+1,\theta}, \delta)}(E) = \beta_1^{(M; W_{r+1,\theta})} - \eta_1^{\mathrm{smooth}}$. Moreover, for all $M + 1 \ge \frac{2}{\theta}N_\delta$, the residuals $\gamma_n := \gamma_n^{(M; W_{r+1,\theta})}(\rho_0)$ and their discrete differences satisfy uniformly for all $2 \le n \le N_\delta$: $$\big|\gamma_n\big| \ll_{\sigma,\rho_0,r,\theta} \frac{n^{r+1}}{(M+1)^{\beta_0 + r + 1}}, \qquad \big|\gamma_n - \gamma_{n+1}\big| \ll_{\sigma,\rho_0,r,\theta} \frac{n^{r-1}}{(M+1)^{\beta_0 + r}}, \tag{28}$$ so the target perturbation $\eta_1^{\mathrm{smooth}} = \sum_{n=2}^{N_\delta}\mu(n)\tilde{\gamma}_n$ and the dual norm squared of $\mathcal{H}_{N_\delta}^{\mathrm{smooth}}$ satisfy $$\big|\eta_1^{\mathrm{smooth}}\big| \ll_{\sigma,\rho_0,r,\theta,\delta} \frac{N^{r+2}}{(M+1)^{\beta_0 + r + 1}}, \qquad \left\|\mathcal{H}_{N_\delta}^{\mathrm{smooth}}\right\|_{S_\sigma^*}^2 \ll_{\sigma,\rho_0,r,\theta,\delta} \frac{N^{2r+2}}{(M+1)^{2\beta_0 + 2r}}\sum_{m=1}^{N_\delta} m^{2\sigma-3} \le \zeta(3 - 2\sigma)\,\frac{N^{2r+2}}{(M+1)^{2\beta_0 + 2r}}. \tag{29}$$
3. (Unconditional Constructive Support Scaling Exponent): Fix $\theta \in [1/2, 1)$ such that $I_{W_{r+1,\theta}}(\rho_0) \ne 0$, and define the combined support exponent $$a(\sigma, \beta_0, r) := \max\!\left(\frac{r+1}{\sigma + r},\; \frac{r+2}{\beta_0 + r + 1}\right) \qquad \left(= \frac{r+2}{r + 3/2} = 1 + \frac{1}{2r + 3} \text{ when } \beta_0 = \frac{1}{2}\right). \tag{30}$$ Under the support scaling $M = \big\lfloor \Lambda N^{a(\sigma, \beta_0, r)} \big\rfloor$, for $\Lambda = \Lambda(\sigma, \rho_0, r, \theta, \delta) > 0$ sufficiently large, both $\|\mathcal{H}_{N_\delta}^{\mathrm{smooth}}\|_{S_\sigma^*} \le \frac{1}{2}\|\mathcal{L}_{\rho_0}^{(M; W_{r+1,\theta})}\|_{S_\sigma^*}$ and $|\eta_1^{\mathrm{smooth}}| \le \frac{\sigma}{2|\rho_0|} = \frac{1}{4}|\beta_1^{(M; W_{r+1,\theta})}|$ hold simultaneously for all $N \ge 2$, yielding the unconditional constructive power-law lower bound $$d_N(\sigma)^2 \ge \frac{\big|\widetilde{\mathcal{L}}_{\rho_0, N}^{(M; W_{r+1,\theta}, \delta)}(E)\big|^2}{\big\|\widetilde{\mathcal{L}}_{\rho_0, N}^{(M; W_{r+1,\theta}, \delta)}\big\|_{S_\sigma^*}^2} \gg_{\sigma,\rho_0,r,\theta,\delta} M^{-2(\sigma - \beta_0)} \asymp N^{-2(\sigma - \beta_0)\,a(\sigma, \beta_0, r)}. \tag{31}$$

Proof. Part 1: Immediate from (13)-(14) since $\eta_N = \gamma_N \mu(1) = \gamma_N$, $\eta_{N+1} = 0$, and $w_N^{(\sigma)} \le 2\sigma N^{-2\sigma-1}$.

Part 2: For every $2 \le k \le N$, $\frac{k}{N_\delta + 1} \le \frac{N}{(1+\delta)N} = \frac{1}{1+\delta}$, where $\Omega_\delta = W_{2, \frac{1}{1+\delta}} \equiv 1$, so $\tilde{\gamma}_k = \gamma_k$. By (8)-(9), for every $2 \le k \le N$, $$\widetilde{\mathcal{L}}_{\rho_0, N}^{(M; W_{r+1,\theta}, \delta)}(\tilde{\rho}_{1/k}) = \mathcal{L}_{\rho_0}^{(M; W_{r+1,\theta})}(\tilde{\rho}_{1/k}) - \sum_{n=2}^{N_\delta} \tilde{\gamma}_n \delta_{n,k} = \gamma_k - \tilde{\gamma}_k = 0, \tag{32}$$ and $\widetilde{\mathcal{L}}_{\rho_0, N}^{(M; W_{r+1,\theta}, \delta)}(E) = \beta_1^{(M; W_{r+1,\theta})} - \sum_{n=2}^{N_\delta} \tilde{\gamma}_n \mu(n) = \beta_1^{(M; W_{r+1,\theta})} - \eta_1^{\mathrm{smooth}}$.

To prove the uniform bounds (28) without any endpoint-jump artifact at $\lfloor \theta X \rfloor$, we use the exact Euler-Maclaurin subtraction method of riemann_hypothesis_9 [17] (equation (249)) at order $r + 1$. For any real $X \ge 2/\theta > 2$, consider the two functions $g_X(t) := t^{-\rho_0}W_{r+1,\theta}(t/X)$ and $g_\infty(t) := t^{-\rho_0}$ on $[1, \infty)$. Their difference $$h_X(t) := g_X(t) - g_\infty(t) = t^{-\rho_0}\left(W_{r+1,\theta}\!\left(\frac{t}{X}\right) - 1\right) \tag{33}$$ belongs to $C^r([1, \infty))$ (since $t^{-\rho_0} \in C^\infty([1, \infty))$ and $W_{r+1,\theta} \in C^r([0, \infty))$) and vanishes identically on $[1, \theta X]$ because $W_{r+1,\theta}(u) = 1$ on $[0, \theta]$ and $\theta X \ge 2 > 1$. In particular, $h_X^{(m)}(1) = 0$ and $h_X^{(m)}(\theta X) = 0$ for all $0 \le m \le r$, and for $t \ge X$, $h_X(t) = -t^{-\rho_0}$. Applying Euler-Maclaurin summation of order $r + 1$ on $[1, \infty)$ separately to $g_X(t)$ and $g_\infty(t)$ and subtracting (using $\sum_{\ell=1}^{\lfloor X \rfloor} g_X(\ell) = \sum_{\ell=1}^\infty g_X(\ell)$ and the analytic continuation identity for $\zeta(\rho_0) = 0$), all lower-endpoint boundary terms at $t = 1$ and the remainder integral on $[1, \theta X]$ cancel identically, leaving the exact identity for all real $X \ge 2/\theta$: $$\sum_{\ell=1}^{\lfloor X \rfloor} \ell^{-\rho_0}W_{r+1,\theta}\!\left(\frac{\ell}{X}\right) = I_{W_{r+1,\theta}}(\rho_0)\,X^{1-\rho_0} + \mathcal{R}_{r+1}(X), \qquad \mathcal{R}_{r+1}(X) := \frac{(-1)^{r+2}}{(r+1)!}\int_{\theta X}^\infty \widetilde{B}_{r+1}(t)\,h_X^{(r+1)}(t)\,dt. \tag{34}$$ Notice that $\mathcal{R}_{r+1}(X)$ is a smooth integral over $t \in [\theta X, \infty)$ with no dependence on the integer part $\lfloor \theta X \rfloor$! Changing variables to $y = t/X \in [\theta, \infty)$ and setting $H_0(y) := y^{-\rho_0}(W_{r+1,\theta}(y) - 1)$, we have $h_X^{(r+1)}(X y) = X^{-\rho_0-r-1} H_0^{(r+1)}(y)$, where $H_0^{(r+1)}(y)$ is smooth on $[\theta, 1]$ and equals $-(\rho_0)_{r+1}^{\downarrow} y^{-\rho_0-r-1}$ on $(1, \infty)$ (with $(s)_m^{\downarrow} := s(s+1)\cdots(s+m-1)$). Thus the rescaled remainder $\Phi_{r+1}(X) := X^{\rho_0}\mathcal{R}_{r+1}(X)$ equals $$\Phi_{r+1}(X) = \frac{(-1)^{r+2}}{(r+1)!}\,X^{-r}\left[\int_\theta^1 \widetilde{B}_{r+1}(Xy)\,H_0^{(r+1)}(y)\,dy - (\rho_0)_{r+1}^{\downarrow}\int_1^\infty \widetilde{B}_{r+1}(Xy)\,y^{-\rho_0-r-1}\,dy\right]. \tag{35}$$ Since $r + 1 \ge 3$, $\widetilde{B}_{r+1}'(t) = (r+1)\widetilde{B}_r(t)$ and $\widetilde{B}_{r+2}'(t) = (r+2)\widetilde{B}_{r+1}(t)$ are continuous and bounded on $\mathbb{R}$. Writing $\widetilde{B}_{r+1}(Xy) = \frac{1}{(r+2)X}\frac{d}{dy}\widetilde{B}_{r+2}(Xy)$ and integrating by parts once on $[\theta, 1]$ and on $[1, \infty)$ gains a factor of $X^{-1}$, proving $$\big|\Phi_{r+1}(X)\big| \ll_{\rho_0,r,\theta} X^{-r-1} \qquad (\forall X \ge 2/\theta). \tag{36}$$ Moreover, since $\widetilde{B}_{r+1}$ is $C^1$ with derivative $(r+1)\widetilde{B}_r$, we may differentiate (35) under the integral sign with respect to $X \ge 2/\theta$: $$\Phi_{r+1}'(X) = -\frac{r}{X}\Phi_{r+1}(X) + \frac{(-1)^{r+2}}{r!}\,X^{-r}\left[\int_\theta^1 \widetilde{B}_r(Xy)\,y H_0^{(r+1)}(y)\,dy - (\rho_0)_{r+1}^{\downarrow}\int_1^\infty \widetilde{B}_r(Xy)\,y^{-\rho_0-r}\,dy\right]. \tag{37}$$ Writing $\widetilde{B}_r(Xy) = \frac{1}{(r+1)X}\frac{d}{dy}\widetilde{B}_{r+1}(Xy)$ and integrating by parts once on $[\theta, 1]$ and on $[1, \infty)$ (since $\int_1^\infty y^{-\beta_0-r-1}\,dy < \infty$ for $r \ge 2$) again gains a factor of $X^{-1}$, proving $$\big|\Phi_{r+1}'(X)\big| \ll_{\rho_0,r,\theta} X^{-r-1} \qquad (\forall X \ge 2/\theta). \tag{38}$$ Now substitute $X_n = \frac{M+1}{n}$ into $\alpha_n = \frac{2\sigma}{\rho_0}n^{-\rho_0}\big[I_{W_{r+1,\theta}}(\rho_0)X_n^{1-\rho_0} + \mathcal{R}_{r+1}(X_n)\big]$: in $\gamma_n = \alpha_n - n^{-1}\alpha_1$, the main term $\frac{2\sigma}{\rho_0}I_{W_{r+1,\theta}}(\rho_0)\frac{(M+1)^{1-\rho_0}}{n}$ cancels identically, leaving the exact formula $$\gamma_n = \frac{2\sigma}{\rho_0}(M+1)^{-\rho_0}\left(\Phi_{r+1}(X_n) - \frac{\Phi_{r+1}(M+1)}{n}\right) \qquad (2 \le n \le N_\delta). \tag{39}$$ By (36), $|\gamma_n| \ll (M+1)^{-\beta_0}\big(X_n^{-r-1} + n^{-1}(M+1)^{-r-1}\big) \ll \frac{n^{r+1}}{(M+1)^{\beta_0+r+1}}$. And for $2 \le n < N_\delta$, by the mean value theorem on $[X_{n+1}, X_n]$ where $|X_n - X_{n+1}| = \frac{M+1}{n(n+1)} \le \frac{X_n}{n}$, (38) gives $$\big|\Phi_{r+1}(X_n) - \Phi_{r+1}(X_{n+1})\big| \le \sup_{\xi \in [X_{n+1}, X_n]} |\Phi_{r+1}'(\xi)|\,\frac{X_n}{n} \ll_{\rho_0,r,\theta} X_n^{-r-1}\,\frac{X_n}{n} = \frac{n^{r-1}}{(M+1)^r}. \tag{40}$$ Multiplying by $(M+1)^{-\beta_0}$ in (39) yields $|\gamma_n - \gamma_{n+1}| \ll_{\sigma,\rho_0,r,\theta} \frac{n^{r-1}}{(M+1)^{\beta_0+r}}$, completing the proof of (28). Summing $|\tilde{\gamma}_n| \le |\gamma_n| \ll \frac{n^{r+1}}{(M+1)^{\beta_0+r+1}}$ over $2 \le n \le N_\delta$ yields $|\eta_1^{\mathrm{smooth}}| \le \sum_{n=2}^{N_\delta}|\tilde{\gamma}_n| \ll_\delta \frac{N^{r+2}}{(M+1)^{\beta_0+r+1}}$.

Finally, multiplying $\gamma_n$ by $\Omega_\delta\big(\frac{n}{N_\delta+1}\big)$, the discrete product rule gives $|\tilde{\gamma}_n - \tilde{\gamma}_{n+1}| \le |\gamma_n - \gamma_{n+1}| + |\gamma_{n+1}|\frac{\|\Omega_\delta'\|_\infty}{N_\delta+1} \ll_\delta \frac{n^{r-1}}{(M+1)^{\beta_0+r}}$ for all $2 \le n \le N_\delta$ (with $\tilde{\gamma}_{N_\delta+1} = 0$ since $\Omega_\delta(1) = \Omega_\delta'(1) = 0$ and $\frac{n^{r+1}}{(N_\delta+1)(M+1)^{\beta_0+r+1}} \le \frac{n^{r-1}}{(M+1)^{\beta_0+r}}$). Because $\tilde{\gamma}_n = 0$ for all $n \ge N_\delta + 1$, for every $1 \le m \le N_\delta$ we have $\eta_m^{\mathrm{smooth}} - \eta_{m+1}^{\mathrm{smooth}} = \sum_{j=1}^{\lfloor N_\delta/m \rfloor} \mu(j)(\tilde{\gamma}_{mj} - \tilde{\gamma}_{(m+1)j})$ with no cutoff jump at $j > \lfloor N_\delta/(m+1)\rfloor$. Since $|\tilde{\gamma}_{mj} - \tilde{\gamma}_{(m+1)j}| \le \sum_{\ell=0}^{j-1}|\tilde{\gamma}_{mj+\ell} - \tilde{\gamma}_{mj+\ell+1}| \ll \frac{(mj)^{r-1}j}{(M+1)^{\beta_0+r}}$, summing over $1 \le j \le \lfloor N_\delta/m \rfloor$ gives $$\big|\eta_m^{\mathrm{smooth}} - \eta_{m+1}^{\mathrm{smooth}}\big| \ll_{\sigma,\rho_0,r,\theta,\delta} \sum_{j=1}^{\lfloor N_\delta/m \rfloor} \frac{m^{r-1} j^r}{(M+1)^{\beta_0+r}} \ll \frac{N_\delta^{r+1}}{m^2(M+1)^{\beta_0+r}} \qquad (1 \le m \le N_\delta). \tag{41}$$ Squaring (41) and summing against $(w_m^{(\sigma)})^{-1} \ll_\sigma m^{2\sigma+1}$ gives (29).

Part 3: Since $\|\mathcal{L}_{\rho_0}^{(M; W_{r+1,\theta})}\|_{S_\sigma^*} \asymp (M+1)^{\sigma-\beta_0}$ and $|\beta_1^{(M; W_{r+1,\theta})}| = \frac{2\sigma}{|\rho_0|}$, the two bounds in (29) yield $$\frac{\big\|\mathcal{H}_{N_\delta}^{\mathrm{smooth}}\big\|_{S_\sigma^*}}{\big\|\mathcal{L}_{\rho_0}^{(M; W_{r+1,\theta})}\big\|_{S_\sigma^*}} \ll_{\sigma,\rho_0,r,\theta,\delta} \frac{N^{r+1}}{(M+1)^{\sigma + r}}, \qquad \big|\eta_1^{\mathrm{smooth}}\big| \ll_{\sigma,\rho_0,r,\theta,\delta} \frac{N^{r+2}}{(M+1)^{\beta_0 + r + 1}}. \tag{42a}$$ Under $M = \big\lfloor \Lambda N^{a(\sigma, \beta_0, r)} \big\rfloor$ with $a(\sigma, \beta_0, r) = \max\!\big(\frac{r+1}{\sigma+r},\, \frac{r+2}{\beta_0+r+1}\big)$, we have $(M+1)^{\sigma+r} \ge \Lambda^{\sigma+r} N^{r+1}$ and $(M+1)^{\beta_0+r+1} \ge \Lambda^{\beta_0+r+1} N^{r+2}$. Thus $\frac{\|\mathcal{H}_{N_\delta}^{\mathrm{smooth}}\|_{S_\sigma^*}}{\|\mathcal{L}_{\rho_0}^{(M; W_{r+1,\theta})}\|_{S_\sigma^*}} \ll \Lambda^{-(\sigma+r)}$ and $|\eta_1^{\mathrm{smooth}}| \ll \Lambda^{-(\beta_0+r+1)}$ uniformly for all $N \ge 2$. Choosing $\Lambda$ sufficiently large guarantees $\|\mathcal{H}_{N_\delta}^{\mathrm{smooth}}\|_{S_\sigma^*} \le \frac{1}{2}\|\mathcal{L}_{\rho_0}^{(M; W_{r+1,\theta})}\|_{S_\sigma^*}$ and $|\eta_1^{\mathrm{smooth}}| \le \frac{\sigma}{2|\rho_0|}$, so $|\widetilde{\mathcal{L}}_{\rho_0, N}^{(M; W_{r+1,\theta}, \delta)}(E)| \ge |\beta_1| - |\eta_1^{\mathrm{smooth}}| \ge \frac{3\sigma}{2|\rho_0|} > \frac{\sigma}{|\rho_0|}$, yielding (31). $\square$

Corollary 1 (Intermediate-Scale Decomposition and Linear-Support Bound on the Macroscopic Step Range). In the setting of Theorem 2(2), for any intermediate scale $K_N := \lfloor \varepsilon_0 N^\kappa \rfloor$ with $\kappa \in (0, 1]$ and $\varepsilon_0 \in (0, 1]$, summing (41) over $K_N < m \le N_\delta$ gives unconditionally $$\sum_{m=K_N+1}^{N_\delta} \frac{\big|\eta_m^{\mathrm{smooth}} - \eta_{m+1}^{\mathrm{smooth}}\big|^2}{w_m^{(\sigma)}} \ll_{\sigma,\rho_0,r,\theta,\delta,\varepsilon_0} \frac{N^{2r+2} K_N^{2\sigma-2}}{(M+1)^{2\beta_0+2r}} \asymp \frac{N^{2r + 2 - 2\kappa(1-\sigma)}}{(M+1)^{2\beta_0+2r}}. \tag{42b}$$ In particular, at $\kappa = 1$ ($K_N = \lfloor \varepsilon_0 N \rfloor$, e.g., $m > N/2$), the macroscopic tail satisfies $\sum_{m > \varepsilon_0 N}^{N_\delta} \frac{|\eta_m^{\mathrm{smooth}} - \eta_{m+1}^{\mathrm{smooth}}|^2}{w_m^{(\sigma)}} \ll \frac{N^{2\sigma+2r}}{(M+1)^{2\beta_0+2r}}$, which at strictly linear support scaling $M = \lfloor \Lambda N \rfloor$ is already $\ll \Lambda^{-(2\beta_0+2r)} N^{2(\sigma-\beta_0)} \asymp \Lambda^{-(2\sigma+2r)}\|\mathcal{L}_{\rho_0}^{(M; W_{r+1,\theta})}\|_{S_\sigma^*}^2$ unconditionally.

3.1 Numerical Verification of Cutoff Jump Elimination and Support Scaling

Table 2 compares the sharp-cutoff head correction $\mathcal{H}_N$ against the smoothed tail-extended corrections $\mathcal{H}_{\lfloor 9N/8\rfloor}^{\mathrm{smooth}}$ ($\delta = 1/8$) and $\mathcal{H}_{2N}^{\mathrm{smooth}}$ ($\delta = 1$) using $W_3 = W_{3, 1/2}$ ($r = 2$) at fixed linear ratio $M = 8N$ across $N \in \{40, 80, 160, 320, 640, 1280, 2560\}$ at $\sigma = 0.75$ ($\rho_1 = 1/2 + 14.134725i$), and verifies the Theorem 2(3) combined scaling $M = \lfloor 4 N^{a(0.75, 1/2, 2)}\rfloor = \lfloor 4 N^{8/7}\rfloor$ ($r = 2$, $a = \max(12/11, 8/7) = 8/7 \approx 1.1429$).

| $N$ | $M = 8N$ | Sharp Single $m=N$ Jump $\frac{|\gamma_N|^2}{w_N^{(\sigma)}\|\mathcal{L}\|^2}$ | Sharp Total Ratio $\frac{\|\mathcal{H}_N\|^2}{\|\mathcal{L}\|^2}$ ($\propto N^1$) | Smoothed ($\delta = 1/8$) Ratio $\frac{\|\mathcal{H}_{\lfloor 9N/8\rfloor}^{\mathrm{smooth}}\|^2}{\|\mathcal{L}\|^2}$ | Smoothed ($\delta = 1$) Ratio $\frac{\|\mathcal{H}_{2N}^{\mathrm{smooth}}\|^2}{\|\mathcal{L}\|^2}$ | Smoothed ($\delta = 1/8$) Target Ratio $\frac{|\eta_1^{\mathrm{smooth}}|}{|\beta_1|}$ | Theorem 2(3) $M = \lfloor 4N^{8/7}\rfloor$ Ratio $\frac{\|\mathcal{H}^{\mathrm{smooth}}\|^2}{\|\mathcal{L}\|^2}$ | Theorem 2(3) Scaled Bound $d_N(0.75)^2 N^{4/7}$ | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $40$ | $320$ | $1.36 \times 10^{-6}$ | $4.06 \times 10^{-6}$ | $8.01 \times 10^{-6}$ | $4.18 \times 10^{-4}$ | $2.02 \times 10^{-3}$ | $3.76 \times 10^{-5}$ ($M=271$) | $0.002722$ | | $80$ | $640$ | $2.60 \times 10^{-6}$ | $5.90 \times 10^{-6}$ | $8.05 \times 10^{-6}$ | $4.16 \times 10^{-4}$ | $1.09 \times 10^{-3}$ | $1.75 \times 10^{-5}$ ($M=598$) | $0.002545$ | | $160$ | $1280$ | $5.03 \times 10^{-6}$ | $9.30 \times 10^{-6}$ | $8.21 \times 10^{-6}$ | $4.25 \times 10^{-4}$ | $1.16 \times 10^{-3}$ | $5.03 \times 10^{-6}$ ($M=1321$) | $0.002433$ | | $320$ | $2560$ | $9.82 \times 10^{-6}$ | $1.57 \times 10^{-5}$ | $8.39 \times 10^{-6}$ | $4.28 \times 10^{-4}$ | $1.56 \times 10^{-3}$ | $1.41 \times 10^{-6}$ ($M=2918$) | $0.002366$ | | $640$ | $5120$ | $1.93 \times 10^{-5}$ | $2.82 \times 10^{-5}$ | $8.49 \times 10^{-6}$ | $4.32 \times 10^{-4}$ | $2.32 \times 10^{-3}$ | $7.19 \times 10^{-7}$ ($M=6443$) | $0.002322$ | | $1280$ | $10240$ | $3.81 \times 10^{-5}$ | $5.25 \times 10^{-5}$ | $8.57 \times 10^{-6}$ | $4.36 \times 10^{-4}$ | $8.49 \times 10^{-4}$ | — | — | | $2560$ | $20480$ | $7.56 \times 10^{-5}$ | $1.01 \times 10^{-4}$ | $8.69 \times 10^{-6}$ | $4.39 \times 10^{-4}$ | $2.61 \times 10^{-3}$ | — | — |

Table 2: Comparison of the sharp-cutoff head correction $\mathcal{H}_N$ vs. the smoothed tail-extended corrections $\mathcal{H}_{\lfloor 9N/8\rfloor}^{\mathrm{smooth}}$ ($\delta = 1/8$) and $\mathcal{H}_{2N}^{\mathrm{smooth}}$ ($\delta = 1$) using $W_3 = W_{3, 1/2}$ ($r = 2$) at $\sigma = 0.75$ (results/multizero_smoothed_step_duality.json). Under the Theorem 2(3) combined scaling $M = \lfloor 4N^{8/7}\rfloor$, both $\frac{\|\mathcal{H}^{\mathrm{smooth}}\|^2}{\|\mathcal{L}\|^2}$ (from $3.76 \times 10^{-5}$ to $7.19 \times 10^{-7}$) and $\frac{|\eta_1^{\mathrm{smooth}}|}{|\beta_1|}$ (from $3.01 \times 10^{-3}$ to $3.14 \times 10^{-4}$) decay rapidly as $N$ increases.

Figure 1: (a) Two-sided divisor-sum operator norm bounds across sigma in [0.50, 0.90] (Theorem 1). (b) Elimination of the m=N sharp-cutoff jump via smoothed tail-extended biorthogonal corrections H_{N_delta}^{smooth} for delta in {1/8, 1} across N in [40, 2560] (Theorem 2).

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4. Multi-Zero Step-Space Annihilator Superposition and the Continuum Mellin-Sobolev Kernel

We now analyze the superposition of $d = 2J$ smoothed zero-annihilators $\widetilde{\mathcal{L}}_{\mu, N}^{(M)} := \mathcal{L}_{\rho_\mu}^{(M; W)} - \mathcal{H}_{N_\delta}^{\mathrm{smooth}}(\rho_\mu) \in \operatorname{Ann}_{S_\sigma^*}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$ ($W = W_{r+1,\theta}$), establishing their asymptotic Gram factorization via the continuum Mellin-Sobolev kernel on $L^2((0, 1], u^{2\sigma-1}\,du)$: $$\mathcal{K}_W^{(\sigma)}(\rho, \rho') := \int_0^1 u^{2\sigma - \rho - \overline{\rho'} - 1}\left(W(u) - \frac{u}{\rho}W'(u)\right)\overline{\left(W(u) - \frac{u}{\rho'}W'(u)\right)}\,du. \tag{43}$$

Theorem 3 (Multi-Zero Step-Space Annihilator Superposition and Mellin-Sobolev Gram Factorization). Let $\sigma \in (1/2, 1)$, let $\rho_1, \dots, \rho_d$ be $d$ distinct nontrivial zeros of $\zeta(s)$ in $\operatorname{Re}(s) \in [1/2, \sigma)$, and let $W = W_{r+1,\theta}$ ($r \ge 2, \theta \in (0, 1)$). For each $1 \le \mu \le d$, let $\widetilde{\mathcal{L}}_{\mu, N}^{(M)} := \mathcal{L}_{\rho_\mu}^{(M; W)} - \mathcal{H}_{N_\delta}^{\mathrm{smooth}}(\rho_\mu) \in \operatorname{Ann}_{S_\sigma^*}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$.

1. (Continuum Mellin-Sobolev Gram Factorization and Explicit Projection Remainder): As $M \to \infty$, the $d \times d$ unprojected dual Gram matrix $G_{\mu,\nu}^{(M)} := \langle \mathcal{L}_{\rho_\mu}^{(M; W)}, \mathcal{L}_{\rho_\nu}^{(M; W)} \rangle_{S_\sigma^*}$ satisfies $$G_{\mu,\nu}^{(M)} = 2\sigma (M+1)^{2\sigma - \rho_\mu - \overline{\rho_\nu}}\,\mathcal{K}_W^{(\sigma)}(\rho_\mu, \rho_\nu) + O_{\sigma,\rho_\mu,\rho_\nu,W}(1). \tag{44}$$ Moreover, for all $M + 1 \ge \frac{2}{\theta}N_\delta$, Theorem 2(2) yields the explicit projection remainders $$\frac{\big|\widetilde{G}_{\mu,\nu}^{(N, M)} - G_{\mu,\nu}^{(M)}\big|}{(M+1)^{2\sigma - \beta_\mu - \beta_\nu}} \ll \frac{N^{r+1}}{(M+1)^{\sigma+r}}, \qquad b_\mu^{(N, M)} := \widetilde{\mathcal{L}}_{\mu, N}^{(M)}(E) = \frac{2\sigma}{\rho_\mu} + O\!\left(\frac{N^{r+2}}{(M+1)^{\beta_\mu+r+1}}\right), \tag{44a}$$ both of which are $O\big(\Lambda^{-(\sigma+r)} + \Lambda^{-(\beta_\mu+r+1)}\big)$ uniformly in $N \ge 2$ at $M = \lfloor \Lambda N^{a(\sigma, \beta_0, r)}\rfloor$, and vanish as $N^{-\varepsilon}$ as $N \to \infty$ for any $M \asymp N^a$ with $a > \max_\mu a(\sigma, \beta_\mu, r)$.
2. (Multi-Zero Superposition Lower Bound in $\operatorname{Ann}_{S_\sigma^*}$): For every coefficient vector $\lambda = (\lambda_1, \dots, \lambda_d)^T \in \mathbb{C}^d \setminus \{0\}$, the linear combination $\widetilde{\mathcal{L}}_{\lambda, N}^{(M)} := \sum_{\mu=1}^d \lambda_\mu \widetilde{\mathcal{L}}_{\mu, N}^{(M)}$ belongs to $\operatorname{Ann}_{S_\sigma^*}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$ and certifies $$d_N(\sigma)^2 \ge \frac{\big|\lambda^T b^{(N, M)}\big|^2}{\lambda^T \widetilde{G}^{(N, M)} \bar{\lambda}}, \qquad \sup_{\lambda \in \mathbb{C}^d \setminus \{0\}} \frac{\big|\lambda^T b^{(N, M)}\big|^2}{\lambda^T \widetilde{G}^{(N, M)} \bar{\lambda}} = \big(b^{(N, M)}\big)^* \big(\widetilde{G}^{(N, M)}\big)^{-1} b^{(N, M)}. \tag{45}$$ When all $\rho_\mu = \frac{1}{2} + i\gamma_\mu$ lie on the critical line, defining $D_M := \operatorname{diag}\big((M+1)^{-i\gamma_1}, \dots, (M+1)^{-i\gamma_d}\big)$ and $(b_0)_\mu := 2\sigma/\rho_\mu$, the projected multi-zero lower bound satisfies $$\big(b^{(N, M)}\big)^* \big(\widetilde{G}^{(N, M)}\big)^{-1} b^{(N, M)} = \frac{(M+1)^{-(2\sigma-1)}}{2\sigma}\left[(D_M^* b_0)^* \big(\mathcal{K}_W^{(\sigma)}\big)^{-1} (D_M^* b_0) + O\!\left(M^{-(2\sigma-1)} + \frac{N^{r+1}}{M^{\sigma+r}} + \frac{N^{r+2}}{M^{r+3/2}}\right)\right]. \tag{46}$$

Proof. Part 1: For each $1 \le m \le M$, set $u_m := \frac{m}{M+1}$. Since $W(u) = 1$ on $[0, \theta]$, for $1 \le m < \theta(M+1)$ we have $c_m(\mathcal{L}_{\rho_\mu}^{(M; W)}) = \frac{2\sigma}{\rho_\mu}(m^{-\rho_\mu} - (m+1)^{-\rho_\mu}) = 2\sigma m^{-\rho_\mu-1} + O(m^{-\beta_\mu-2})$, while on $\theta(M+1) \le m \le M$, the mean value theorem gives $c_m(\mathcal{L}_{\rho_\mu}^{(M; W)}) = 2\sigma m^{-\rho_\mu-1}\big(W(u_m) - \frac{u_m}{\rho_\mu}W'(u_m)\big) + O(M^{-\beta_\mu-2})$. Multiplying $c_m(\mathcal{L}_{\rho_\mu}^{(M; W)})\overline{c_m(\mathcal{L}_{\rho_\nu}^{(M; W)})}$ by $(w_m^{(\sigma)})^{-1} = \frac{m^{2\sigma+1}}{2\sigma}(1 + O(m^{-1}))$, the accumulated error over $1 \le m \le M$ is $\sum_{m=1}^M O(m^{2\sigma-\beta_\mu-\beta_\nu-2}) = O(1)$ (since $2\sigma - \beta_\mu - \beta_\nu < 1$, so $2\sigma - \beta_\mu - \beta_\nu - 2 < -1$), while by Euler-Maclaurin summation on $1 \le m \le \theta(M+1)$ and Riemann-sum comparison on $\theta(M+1) < m \le M$, the difference between $\sum_{m=1}^M m^{2\sigma-\rho_\mu-\overline{\rho_\nu}-1}\psi_\mu(u_m)\overline{\psi_\nu(u_m)}$ ($\psi_\mu(u) := W(u) - \frac{u}{\rho_\mu}W'(u)$) and $(M+1)^{2\sigma-\rho_\mu-\overline{\rho_\nu}}\int_0^1 u^{2\sigma-\rho_\mu-\overline{\rho_\nu}-1}\psi_\mu(u)\overline{\psi_\nu(u)}\,du$ is the lower-endpoint constant $\zeta(1 - 2\sigma + \rho_\mu + \overline{\rho_\nu}) + O\big((M+1)^{2\sigma-\beta_\mu-\beta_\nu-1}\big) = O(1)$, proving (44). By (42a) and Cauchy-Schwarz, $|\widetilde{G}_{\mu,\nu}^{(N, M)} - G_{\mu,\nu}^{(M)}| \ll (M+1)^{2\sigma-\beta_\mu-\beta_\nu}\frac{N^{r+1}}{(M+1)^{\sigma+r}}$ and $|b_\mu^{(N, M)} - \frac{2\sigma}{\rho_\mu}| = |\eta_1^{\mathrm{smooth}}(\rho_\mu)| \ll \frac{N^{r+2}}{(M+1)^{\beta_\mu+r+1}}$, proving (44a).

Part 2: Since our inner product $\langle \cdot, \cdot \rangle_{S_\sigma^*}$ is linear in the first argument and conjugate-linear in the second, for any $\lambda = (\lambda_1, \dots, \lambda_d)^T \in \mathbb{C}^d$, $$\left\|\sum_{\mu=1}^d \lambda_\mu \widetilde{\mathcal{L}}_{\mu, N}^{(M)}\right\|_{S_\sigma^*}^2 = \sum_{\mu,\nu=1}^d \lambda_\mu \overline{\lambda_\nu}\,\big\langle \widetilde{\mathcal{L}}_{\mu, N}^{(M)}, \widetilde{\mathcal{L}}_{\nu, N}^{(M)} \big\rangle_{S_\sigma^*} = \sum_{\mu,\nu=1}^d \lambda_\mu \widetilde{G}_{\mu,\nu}^{(N, M)} \overline{\lambda_\nu} = \lambda^T \widetilde{G}^{(N, M)} \bar{\lambda}. \tag{47}$$ Since $\widetilde{\mathcal{L}}_{\lambda, N}^{(M)}(E) = \lambda^T b^{(N, M)}$, Hahn-Banach duality (5) yields the first inequality in (45). Setting $w := \bar{\lambda}$, we have $\lambda^T \widetilde{G}^{(N, M)} \bar{\lambda} = w^T \widetilde{G}^{(N, M)} \bar{w} = w^* \big(\widetilde{G}^{(N, M)}\big)^T w$ and $|\lambda^T b^{(N, M)}|^2 = |w^* b^{(N, M)}|^2 = |\bar{w}^T b^{(N, M)}|^2$; Equivalently, writing $u := \bar{\lambda}$ so that $\lambda^T b^{(N, M)} = u^* b^{(N, M)}$, or writing $v := \lambda$ so that $\lambda^T \widetilde{G}^{(N, M)} \bar{\lambda} = (\bar{\lambda})^* \widetilde{G}^{(N, M)} \bar{\lambda} = u^* \widetilde{G}^{(N, M)} u$ (because $\sum_{\mu,\nu} \lambda_\mu \widetilde{G}_{\mu,\nu} \overline{\lambda_\nu} = \sum_{\mu,\nu} \overline{u_\mu} \widetilde{G}_{\mu,\nu} u_\nu = u^* \widetilde{G}^{(N, M)} u$), Cauchy-Schwarz for the positive-definite Hermitian matrix $\widetilde{G}^{(N, M)}$ gives directly $$\sup_{\lambda \in \mathbb{C}^d \setminus \{0\}} \frac{\big|\lambda^T b^{(N, M)}\big|^2}{\lambda^T \widetilde{G}^{(N, M)} \bar{\lambda}} = \sup_{u \in \mathbb{C}^d \setminus \{0\}} \frac{\big|u^* b^{(N, M)}\big|^2}{u^* \widetilde{G}^{(N, M)} u} = \big(b^{(N, M)}\big)^* \big(\widetilde{G}^{(N, M)}\big)^{-1} b^{(N, M)}, \tag{47a}$$ attained at $u_{\mathrm{opt}} = \bar{\lambda}_{\mathrm{opt}} = \big(\widetilde{G}^{(N, M)}\big)^{-1} b^{(N, M)}$, i.e., $\lambda_{\mathrm{opt}} = \Big(\big(\widetilde{G}^{(N, M)}\big)^T\Big)^{-1}\overline{b^{(N, M)}}$. Finally, when $\rho_\mu = \frac{1}{2} + i\gamma_\mu$, (44) reads $G^{(M)} = 2\sigma (M+1)^{2\sigma-1} D_M \mathcal{K}_W^{(\sigma)} D_M^* + O(1)$, whose inverse is $\frac{(M+1)^{-(2\sigma-1)}}{2\sigma} D_M \big(\mathcal{K}_W^{(\sigma)}\big)^{-1} D_M^* \big(1 + O(M^{-(2\sigma-1)})\big)$, so combining with (44a) yields (46). $\square$

4.1 Multi-Zero Superposition Progression and 60-Digit Interval Certification

Equation (46) shows that as $M$ varies, the phase factors $(M+1)^{i\gamma_\mu}$ in $D_M^* b_0$ modulate the off-diagonal terms of $\big(\mathcal{K}_W^{(\sigma)}\big)^{-1}$ by $\cos\big((\gamma_j \mp \gamma_\ell)\log(M+1) + \phi_{j,\ell}\big)$, providing a natural heuristic model for why the empirical optimal distance $F_{\mathrm{unnorm}, \mathrm{opt}}(N) = d_{\mathrm{unnorm}, N}(0.75)^2 N^{0.5}$ (evaluated in double precision on a 16-point dense grid $N \in [160, 320]$ in results/multizero_smoothed_step_duality.json) fluctuates between $0.012582$ ($N = 193$) and $0.010425$ ($N = 233$).

Table 3 reports the progression of the smoothed multi-zero lower bound $F_{\mathrm{unnorm}, \mathrm{ann}}^{(2J)}(N)$ using $W_3 = W_{3, 1/2}$ ($r = 2$) and the smoothed tail-extended correction $\mathcal{H}_{\lfloor 9N/8\rfloor}^{\mathrm{smooth}}$ ($\delta = 1/8$) across $J \in \{1, 2, 3, 5, 10, 15\}$ critical-line zero pairs ($d = 2J \in \{2, 4, 6, 10, 20, 30\}$) for $N \in \{40, 80, 160, 320, 640, 1280, 2560\}$ at $\sigma = 0.75$, together with our 60-digit mpmath.iv interval certificates at $N = 40, M = 320$ ($N_\delta = 45, W_3$) for $J \in \{1, 2, 3\}$.

| $N$ ($M=8N$) | $1\text{D}$ ($\rho_1$ only) | $J=1$ Pair ($2\text{D}$) | $J=2$ Pairs ($4\text{D}$) | $J=3$ Pairs ($6\text{D}$) | $J=5$ Pairs ($10\text{D}$) | $J=10$ Pairs ($20\text{D}$) | $J=15$ Pairs ($30\text{D}$) | Empirical $F_{\mathrm{unnorm}, \mathrm{opt}}(N)$ | Ratio $30\text{D} / \mathrm{opt}$ | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $40$ | $0.001261$ | $0.00248815^*$ | $0.00372948^*$ | $0.00447187^*$ | $0.005386$ | $0.006437$ | $0.007238$ | $0.014930$ | $48.5\%$ | | $80$ | $0.001192$ | $0.002364$ | $0.003654$ | $0.004680$ | $0.005690$ | $0.007037$ | $0.007632$ | $0.013250$ | $57.6\%$ | | $160$ | $0.001147$ | $0.002281$ | $0.003286$ | $0.003918$ | $0.005367$ | $0.006783$ | $0.007232$ | $0.012333$ | $58.6\%$ | | $320$ | $0.001119$ | $0.002230$ | $0.003076$ | $0.004037$ | $0.004950$ | $0.005851$ | $0.006653$ | $0.010927$ | $60.9\%$ | | $640$ | $0.001100$ | $0.002193$ | $0.003231$ | $0.003713$ | $0.004380$ | $0.005341$ | $0.005846$ | — | — | | $1280$ | $0.001087$ | $0.002169$ | $0.003325$ | $0.004209$ | $0.005190$ | $0.006629$ | $0.007006$ | — | — | | $2560$ | $0.001080$ | $0.002156$ | $0.003153$ | $0.003804$ | $0.004553$ | $0.005439$ | $0.006227$ | — | — |

Table 3: Smoothed multi-zero step-space annihilator superposition $F_{\mathrm{unnorm}, \mathrm{ann}}^{(2J)}(N) = d_{\mathrm{unnorm}, N}(0.75)^2 N^{0.5}$ using $W_3 = W_{3, 1/2}$ ($r = 2$) and $\mathcal{H}_{\lfloor 9N/8\rfloor}^{\mathrm{smooth}}$ ($\delta = 1/8$) across $J \in \{1, 2, 3, 5, 10, 15\}$ critical-line zero pairs at $\sigma = 0.75$ (results/multizero_smoothed_step_duality.json). Entries marked with $*$ at $N = 40, M = 320$ ($N_{\delta} = 45, W_3$) are certified lower bounds (rounded strictly downward) in 60-digit mpmath.iv interval arithmetic using explicit rational witness vectors $\tilde{\lambda} \in 10^{-8}\mathbb{Z}[i]^{2J}$:

• For $J = 1$ ($2\text{D}$): $\tilde{\lambda} = 10^{-8}(12855 + 278297i,\, 12855 - 278297i)^T$ certifies $d_{40}(0.75)^2 \ge 0.0005901171$ and $F_{\mathrm{unnorm}, \mathrm{ann}}^{(2)}(40) \ge 0.00248815$;
• For $J = 2$ ($4\text{D}$): $\tilde{\lambda} = 10^{-8}(23327 + 281057i,\, 23327 - 281057i,\, -20996 + 201052i,\, -20996 - 201052i)^T$ certifies $d_{40}(0.75)^2 \ge 0.0008845248$ and $F_{\mathrm{unnorm}, \mathrm{ann}}^{(4)}(40) \ge 0.00372948$;
• For $J = 3$ ($6\text{D}$): $\tilde{\lambda} = 10^{-8}(19378 + 278587i,\, 19378 - 278587i,\, -41535 + 201000i,\, -41535 - 201000i,\, 43414 + 155349i,\, 43414 - 155349i)^T$ certifies $d_{40}(0.75)^2 \ge 0.0010605991$ and $F_{\mathrm{unnorm}, \mathrm{ann}}^{(6)}(40) \ge 0.00447187$ ($3.55\times$ the $1\text{D}$ bound). (Under the Theorem 2(3) combined scaling $M = \lfloor 4N^{8/7}\rfloor$, superposing $J = 15$ pairs yields $d_{\mathrm{unnorm}, N}(0.75)^2 N^{4/7} \ge 0.010297, 0.010623, 0.010275, 0.009589, 0.008532$ across $N \in \{40, 80, 160, 320, 640\}$.)

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5. Multi-Pair Hybrid Step-Space Reproducing Kernels and Interval-Certified Collective Zero Exclusions

We now establish the multi-pair hybrid step-space reproducing-kernel bounds for $J \ge 1$ hypothetical off-line zero pairs $Z_J := \{\rho_1, \bar{\rho}_1, \dots, \rho_J, \bar{\rho}_J\}$ ($\rho_j = \beta_j + i\gamma_j$ with $\beta_j > \sigma$ and $\gamma_j > 0$). In the continuous setting $M = 0$, Burnol [21, Lemma 2.4] established the classical Blaschke-product distance formula in $\mathcal{H}_\sigma$; in the step-space setting, riemann_hypothesis_1 [12, Theorem C and Corollary D] established finite-set step-space Gram-inverse bounds, and riemann_hypothesis_9 [17, Theorem 4] introduced the hybrid step-head + continuous-tail kernels $k_{s, M}^{(\sigma)} := P_{S_{\sigma, M}} k_s^{(\sigma)}$ on $S_{\sigma, M} := \operatorname{span}\{\mathbf{1}_{I_1}, \dots, \mathbf{1}_{I_M}\} \oplus L^2((0, 1/(M+1)], d\mu_\sigma)$ with inner products $$\big\langle k_{s_1, M}^{(\sigma)}, k_{s_2, M}^{(\sigma)} \big\rangle_\sigma = \sum_{m=1}^M \frac{c_m^{(\sigma)}(s_1)\,\overline{c_m^{(\sigma)}(s_2)}}{w_m^{(\sigma)}} + \frac{2\sigma}{\bar{s}_1 + s_2 - 2\sigma}\,(M+1)^{-(\bar{s}_1 + s_2 - 2\sigma)}, \qquad c_m^{(\sigma)}(s) := \frac{2\sigma}{\bar{s}}\big(m^{-\bar{s}} - (m+1)^{-\bar{s}}\big). \tag{48}$$

Theorem 4 (Multi-Pair Hybrid Step-Space Loewner Monotonicity and Interval-Certified Collective Zero Exclusions). Let $\sigma \in (1/2, 1)$ and let $Z_J = \{\rho_1, \bar{\rho}_1, \dots, \rho_J, \bar{\rho}_J\}$ be $J \ge 1$ distinct conjugate pairs of hypothetical zeros of $\zeta(s)$ in $\operatorname{Re}(s) > \sigma$ ($\rho_j = \beta_j + i\gamma_j$, $\beta_j > \sigma$, $\gamma_j > 0$). Enumerate $Z_J$ as $(s_1, s_2, \dots, s_{2J}) = (\rho_1, \bar{\rho}_1, \dots, \rho_J, \bar{\rho}_J)$, and for every $M \in \mathbb{N}_{\ge 0}$ define the $2J \times 2J$ Hermitian Gram matrix $K_{Z_J, M}^{(\sigma)} := \big(\langle k_{s_\mu, M}^{(\sigma)}, k_{s_\nu, M}^{(\sigma)} \rangle_\sigma\big)_{1 \le \mu, \nu \le 2J}$ and target vector $b_{Z_J} := \big(\frac{2\sigma}{s_1}, \dots, \frac{2\sigma}{s_{2J}}\big)^T$.

1. (Continuous $2J$-Point Cauchy-Blaschke Baseline at $M = 0$ [21, Lemma 2.4]): At $M = 0$, the $2J \times 2J$ Cauchy Gram projection equals the Blaschke product complement $$B_{2J\text{-pt}, 0}^{(H^2)}(\sigma; Z_J) := b_{Z_J}^* \Big(\big(K_{Z_J, 0}^{(\sigma)}\big)^T\Big)^{-1} b_{Z_J} = 1 - \prod_{j=1}^J \left(1 - \frac{4\sigma(\beta_j - \sigma)}{\beta_j^2 + \gamma_j^2}\right)^2. \tag{49}$$
2. (Strict Loewner Monotonicity in $M$ and Rayleigh-Quotient Certification): For all $M \ge 0$, $K_{Z_J, M+1}^{(\sigma)} \prec K_{Z_J, M}^{(\sigma)} \prec K_{Z_J, 0}^{(\sigma)}$ in the strict Loewner order, so the hybrid step-space projection norm squared $$B_{2J\text{-pt}, M}^{(S_\sigma)}(\sigma; Z_J) := b_{Z_J}^* \Big(\big(K_{Z_J, M}^{(\sigma)}\big)^T\Big)^{-1} b_{Z_J} = \sup_{\lambda \in \mathbb{C}^{2J} \setminus \{0\}} \frac{\big|\lambda^* b_{Z_J}\big|^2}{\lambda^* \big(K_{Z_J, M}^{(\sigma)}\big)^T \lambda} \tag{50}$$ is strictly increasing in both $M \ge 0$ and $J \ge 1$, and for every $N \ge 2$ and every test vector $\tilde{\lambda} \in \mathbb{C}^{2J} \setminus \{0\}$, $$d_N(\sigma)^2 \ge \operatorname{dist}_\sigma\big(E, \operatorname{span}\{\tilde{\rho}_{1/k} : k \ge 2\}\big)^2 \ge B_{2J\text{-pt}, M}^{(S_\sigma)}(\sigma; Z_J) \ge \frac{\big|\tilde{\lambda}^* b_{Z_J}\big|^2}{\tilde{\lambda}^* \big(K_{Z_J, M}^{(\sigma)}\big)^T \tilde{\lambda}}. \tag{51}$$

Proof. Part 1: By Burnol [21, Lemma 2.4], under the Mellin-Plancherel isomorphism from $\mathcal{H}_\sigma$ onto $H^2(\operatorname{Re}(w) > \sigma)$, the squared distance from the unit vector $E$ ($w_0 = 2\sigma$) to $\operatorname{span}\{k_{s_1}^{(\sigma)}, \dots, k_{s_{2J}}^{(\sigma)}\}$ is $\prod_{\mu=1}^{2J}\big|\frac{2\sigma - s_\mu}{2\sigma + \bar{s}_\mu - 2\sigma}\big|^2$. Grouping each conjugate pair $\{\rho_j, \bar{\rho}_j\}$ gives $\big|\frac{(2\sigma-\beta_j)^2+\gamma_j^2}{\beta_j^2+\gamma_j^2}\big|^2 = \big(1 - \frac{4\sigma(\beta_j-\sigma)}{\beta_j^2+\gamma_j^2}\big)^2$, yielding (49).

Part 2: On $I_{M+1} = (1/(M+2), 1/(M+1)]$, the functions $k_{s_1}^{(\sigma)}, \dots, k_{s_{2J}}^{(\sigma)}$ are linearly independent powers $x^{\bar{s}_\mu-2\sigma}$, so no non-zero combination $\sum_{\mu=1}^{2J}\bar{\lambda}_\mu k_{s_\mu}^{(\sigma)}$ is constant on $I_{M+1}$. Thus $\lambda^*(K_{Z_J, M}^{(\sigma)} - K_{Z_J, M+1}^{(\sigma)})\lambda > 0$ for all $\lambda \in \mathbb{C}^{2J}\setminus \{0\}$, proving $K_{Z_J, M+1}^{(\sigma)} \prec K_{Z_J, M}^{(\sigma)} \prec K_{Z_J, 0}^{(\sigma)}$ and (51). $\square$

5.1 60-Digit Complex Interval Certification of Collective Multi-Pair Zero Exclusions

Table 4 reports our 60-digit mpmath.iv complex interval certifications of Theorem 4 at $M = 500$ for $J \in \{1, 2, 3\}$ benchmark hypothetical conjugate zero pairs at ordinates $(\gamma_1, \gamma_2, \gamma_3) = (14.13472514, 21.02203964, 25.01085758)$ across $(\sigma, \beta) \in \{(0.55, 0.75), (0.55, 0.85), (0.65, 0.75), (0.65, 0.85), (0.75, 0.85)\}$, compared against the 60-digit certified upper bounds $d_{40}^{\mathrm{cert}}(\sigma)^2$ and $d_{160}^{\mathrm{cert}}(\sigma)^2$ from Theorem 3 of riemann_hypothesis_9 [17]. (We note explicitly that since $\gamma_1, \gamma_2, \gamma_3 \le 26$ lie well inside the height range $T \le 3 \cdot 10^{12}$ where the Riemann Hypothesis has already been verified unconditionally by Platt and Trudgian [22], these exclusions are not new unconditional zero-free regions for $\zeta(s)$, but rather rigorous quantitative benchmarks demonstrating that finite-$N$ Nyman-Beurling approximation certificates at $N = 40$ and $N = 160$ exclude collective off-line configurations that neither continuous $H^2$ Blaschke bounds nor single-pair step-space bounds can exclude.)

| $\sigma$ | $\beta$ | $J$ Pairs ($2J \times 2J$) | Exact $H^2$ Blaschke Bound $B_{2J\text{-pt}, 0}^{(H^2)}$ | Certified Step-Space Lower Bound $B_{2J\text{-pt}, 500}^{(S_\sigma)}$ | Step-Space Gain Ratio | Certified $d_{40}^{\mathrm{cert}}(\sigma)^2$ Upper Bound | Excluded at $N = 40$? ($S_\sigma$ / $H^2$) | Certified $d_{160}^{\mathrm{cert}}(\sigma)^2$ Upper Bound | Excluded at $N = 160$? ($S_\sigma$) | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $0.55$ | $0.75$ | $J = 1$ ($2\times 2$) | $0.00438743$ | $0.00787546$ | $1.7950\times$ | $0.00973298$ | No / No | $0.00680891$ | Yes (Cert.) | | $0.55$ | $0.75$ | $J = 2$ ($4\times 4$) | $0.00636647$ | $0.01131851$ | $1.7778\times$ | $0.00973298$ | Yes (Cert.) / No | $0.00680891$ | Yes (Cert.) | | $0.55$ | $0.75$ | $J = 3$ ($6\times 6$) | $0.00776254$ | $0.01412607$ | $1.8198\times$ | $0.00973298$ | Yes (Cert.) / No | $0.00680891$ | Yes (Cert.) | | $0.55$ | $0.85$ | $J = 1$ ($2\times 2$) | $0.00657228$ | $0.01538571$ | $2.3410\times$ | $0.00973298$ | Yes (Cert.) / No | $0.00680891$ | Yes (Cert.) | | $0.55$ | $0.85$ | $J = 2$ ($4\times 4$) | $0.00953252$ | $0.02198966$ | $2.3068\times$ | $0.00973298$ | Yes (Cert.) / No | $0.00680891$ | Yes (Cert.) | | $0.55$ | $0.85$ | $J = 3$ ($6\times 6$) | $0.01161906$ | $0.02788268$ | $2.3997\times$ | $0.00973298$ | Yes (Cert.) / Yes | $0.00680891$ | Yes (Cert.) | | $0.65$ | $0.75$ | $J = 1$ ($2\times 2$) | $0.00259374$ | $0.00350650$ | $1.3519\times$ | $0.00548802$ | No / No | $0.00326480$ | Yes (Cert.) | | $0.65$ | $0.75$ | $J = 2$ ($4\times 4$) | $0.00376551$ | $0.00506703$ | $1.3456\times$ | $0.00548802$ | No / No | $0.00326480$ | Yes (Cert.) | | $0.65$ | $0.75$ | $J = 3$ ($6\times 6$) | $0.00459275$ | $0.00624248$ | $1.3592\times$ | $0.00548802$ | Yes (Cert.) / No | $0.00326480$ | Yes (Cert.) | | $0.65$ | $0.85$ | $J = 1$ ($2\times 2$) | $0.00517997$ | $0.00929212$ | $1.7939\times$ | $0.00548802$ | Yes (Cert.) / No | $0.00326480$ | Yes (Cert.) | | $0.65$ | $0.85$ | $J = 2$ ($4\times 4$) | $0.00751592$ | $0.01335086$ | $1.7763\times$ | $0.00548802$ | Yes (Cert.) / Yes | $0.00326480$ | Yes (Cert.) | | $0.65$ | $0.85$ | $J = 3$ ($6\times 6$) | $0.00916339$ | $0.01665793$ | $1.8179\times$ | $0.00548802$ | Yes (Cert.) / Yes | $0.00326480$ | Yes (Cert.) | | $0.75$ | $0.85$ | $J = 1$ ($2\times 2$) | $0.00299009$ | $0.00404142$ | $1.3516\times$ | $0.00298277$ | Yes (Cert.) / Yes | $0.00147346$ | Yes (Cert.) | | $0.75$ | $0.85$ | $J = 2$ ($4\times 4$) | $0.00434105$ | $0.00583983$ | $1.3453\times$ | $0.00298277$ | Yes (Cert.) / Yes | $0.00147346$ | Yes (Cert.) | | $0.75$ | $0.85$ | $J = 3$ ($6\times 6$) | $0.00529473$ | $0.00719413$ | $1.3587\times$ | $0.00298277$ | Yes (Cert.) / Yes | $0.00147346$ | Yes (Cert.) |

Table 4: 60-digit mpmath.iv complex interval certification of the $2J \times 2J$ hybrid step-space lower bounds $B_{2J\text{-pt}, 500}^{(S_\sigma)}(\sigma; Z_J)$ (results/multizero_smoothed_step_duality.json, rounded strictly downward). Complete rational witness vectors $\tilde{\lambda} \in 10^{-8}\mathbb{Z}[i]^{2J}$ for all 15 rows are archived in results/multizero_smoothed_step_duality.json.

Figure 2: (a) Multi-zero step-space annihilator superposition across J in {1, 2, 3, 5, 10, 15} zero pairs and dense-grid oscillations of F_opt(N) on [160, 320] at sigma = 0.75 (Theorem 3). (b) Interval-certified collective multi-pair step-space zero exclusion at sigma = 0.55, beta = 0.75 across J in {1, 2, 3} (Theorem 4).

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6. Verification & Reproduction

All analytical identities, generalized eigenvalue computations, smoothed tail-extension progressions, multi-zero annihilator superpositions, and 60-digit mpmath.iv interval certificates reported in this paper are implemented and reproducible via the self-contained verification script src/verify_multizero_smoothed_step_duality.py, with full numerical outputs and rational witness vectors archived in results/multizero_smoothed_step_duality.json.

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