Smoothed and Variational Harmonic Head Functionals in the Shifted Nyman-Beurling Criterion: Universal Tail-Extension Lower Bounds, Quotient-Rule Möbius Cancellation, Full $N^1$ Power-Saving Norm Asymptotics, and Multi-Scale Zero-Independent Annihilator Superposition

Abstract

In Beurling's shifted Nyman criterion on $\mathcal{H}_\sigma = L^2((0, 1], 2\sigma x^{2\sigma-1}\,dx)$ ($\sigma \in [1/2, 1)$), the Vasyunin-Balazard backward-difference basis $\{\mathcal{C}_n\}_{n \ge 1}$ yields a unique $N$-step annihilator $\mathcal{F}_N = \sum_{n=1}^N n^{-1}\mathcal{C}_n \in \operatorname{Ann}_{S_\sigma^*}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$, whose dual norm squared satisfies $\|\mathcal{F}_N\|_{S_{1/2}^*}^2 \ge 1 + 1/N$ at $\sigma = 1/2$ and grows as $\|\mathcal{F}_N\|_{S_\sigma^*}^2 \sim \frac{\zeta(2\sigma+1)}{2\sigma\,\zeta(4\sigma+2)}N^{2\sigma-1}$ for $\sigma > 1/2$ (riemann_hypothesis_9), losing a full factor of $N^1$ in Hahn-Banach lower bounds. We resolve this $N^1$ norm loss within the arithmetic harmonic head basis by extending $\mathcal{F}_N$ across $N < n \le N_\delta := \lfloor(1+\delta)N\rfloor$. First, for $\delta \in (0, 1)$ ($N + 1 \le N_\delta < 2N$), we prove that every tail extension $\mathcal{L} = \mathcal{F}_N + \sum_{n=N+1}^{N_\delta} \alpha_n \mathcal{C}_n$ has fixed boundary coordinates $\beta_N(\mathcal{L}) = 1/N$ and $\beta_{N_\delta+1}(\mathcal{L}) = 0$, enforcing the universal tail-segment Cauchy-Schwarz bound $\sum_{m=N}^{N_\delta}|c_m(\mathcal{L})|^2/w_m^{(\sigma)} \ge \frac{N^{-2}}{N^{-2\sigma} - (N_\delta+1)^{-2\sigma}} \sim \frac{N^{-2(1-\sigma)}}{1 - (1+\delta)^{-2\sigma}}$ (attained on $N \le m \le N_\delta$ uniquely by $\alpha_n^{\mathrm{var}}$), while at the coupled endpoint $\delta = 1$ ($N_\delta = 2N$), $\beta_N(\mathcal{L}) + \beta_{2N}(\mathcal{L}) = 1/N$ enforces the exact bound $\frac{N^{-2}}{N^{-2\sigma} - (2N)^{-2\sigma} + 4w_{2N}^{(\sigma)}}$ ($5/9$ at $N = 2, \sigma = 1/2$). Second, for the smoothed harmonic head functional $\mathcal{F}_{N, \delta}^{\mathrm{smooth}} = \sum_{n=1}^{N_\delta} n^{-1}\Omega_\delta\big(\frac{n}{N_\delta+1}\big)\mathcal{C}_n$ with fixed knot $\theta = \frac{1}{1+\delta}$ and $\Omega_\delta \in C^{2,1}([0, \infty))$, we prove the exact two-point Möbius identity $\eta_m^{\mathcal{F}} - \eta_{m+1}^{\mathcal{F}} = \frac{1}{m(m+1)}M_{V_{m,\delta}}\big(\frac{N_\delta+1}{m}\big)$, the Fubini-Abel bound $|\eta_m^{\mathcal{F}} - \eta_{m+1}^{\mathcal{F}}| \le C_\delta/[m(m+1)]$ with $C_\delta := \int_\theta^1 t|\Omega_\delta''(t)|\,dt$, the Lipschitz second-derivative bound $\|V'_{m,\delta} - \widetilde{\Omega}'_\delta\|_{L^\infty} \le m^{-1}\big(\frac{1+1/m}{2}\|\Omega_\delta'''\|_{L^\infty} + \|\Omega_\delta''\|_{L^\infty}\big)$, and via Abel summation and the Prime Number Theorem that $\lim_{N\to\infty}\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\|_{S_\sigma^*}^2 = 0$ unconditionally for all $\sigma \in [1/2, 1)$ (while noting that $\mathcal{F}_{N,\delta}^{\mathrm{smooth}}(E) \to 0$ as well, so strong vanishing of the unnormalized dual norm does not imply $d_N(1/2) \to 0$). Third, for $\sigma \in (\Theta, 1)$ (where $\Theta := \sup\{\operatorname{Re}(\rho) : \zeta(\rho) = 0\}$), we prove the exact $N^1$-saving asymptotic law $\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\|_{S_\sigma^*}^2 = \mathcal{C}_{\sigma, \delta}N^{-2(1-\sigma)} + O_{\sigma,\delta}(N^{-2(1-\sigma)-(\sigma-\Theta)/2})$, establish for every generic real scale $\delta \in (0, 1] \setminus \mathcal{E}_r$ the infinitely-often ($\limsup_{N\to\infty}$) rate bound $d_N(\sigma)^2 = \Omega(N^{-2(\sigma-\Theta)-\varepsilon})$ at linear support $N_\delta$, and certify at the explicit rational scale $\delta = 1/4 \in \mathbb{Q}$ ($\theta = 4/5$) via Rouché-Euler-Maclaurin zero bracketing and 60-digit interval arithmetic that $|\widehat{W}_{3, 4/5}(\rho_1 - 1)| \in [0.062251, 0.062252] > 0$, yielding $d_N(\sigma)^2 = \Omega(N^{-2(\sigma-1/2)-\varepsilon})$. Fourth, using the fixed knots $\theta_j = \frac{1}{1+\delta_j}$ consistently, we certify in exact rational + 60-digit mpmath.iv arithmetic that nested 8-scale ($M = 2N$) and 16-scale ($M = 4N$) zero-independent superpositions (constructed without specifying zeta zeros) reach $23.7\%$–$53.8\%$ ($K = 8$) and $56.2\%$–$63.8\%$ ($K = 16$) of the numerical reference distance $d_N(0.75)^2$ at $N \in \{40, 80, 160\}$.

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1. Introduction, Inherited Background, and Summary of New Contributions

1.1 The Shifted Nyman-Beurling Criterion and Inherited Biorthogonal Framework

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 $\|f\|_{\mathrm{unnorm}, \sigma}^2 = \int_0^1 |f(x)|^2 x^{2\sigma-1}\,dx = (2\sigma)^{-1}\|f\|_\sigma^2$; note that at $\sigma = 1/2$, $2\sigma = 1$ so the two conventions coincide identically). Following Nyman [1], Beurling [2], and Báez-Duarte [3], define the pole-canceled fractional-part dilates $$\tilde{\rho}_{1/k}(x) := \frac{1}{k}\left\{\frac{1}{x}\right\} - \left\{\frac{1}{kx}\right\} = -\left\{\frac{\lfloor 1/x \rfloor}{k}\right\} \qquad (k \in \mathbb{N}_{\ge 2},\; x \in (0, 1]). \tag{1}$$ 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 $$w_m^{(\sigma)} := \mu_\sigma(I_m) = m^{-2\sigma} - (m+1)^{-2\sigma} = \frac{2\sigma}{m^{2\sigma+1}}\left(1 + O\left(\frac{1}{m}\right)\right), \tag{2}$$ so $\operatorname{span}\{\tilde{\rho}_{1/k} : k \ge 2\}$ 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 $I_m$ (with boundary convention $f_0 := 0$, so $E_m = 1$ and $(\tilde{\rho}_{1/k})_m = \lfloor m/k \rfloor - m/k$ 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 $$\|f\|_\sigma^2 = \sum_{m=1}^\infty |f_m|^2 w_m^{(\sigma)}, \qquad \|\mathcal{L}\|_{S_\sigma^*}^2 = \sum_{m=1}^\infty \frac{|c_m(\mathcal{L})|^2}{w_m^{(\sigma)}}, \qquad \mathcal{L}(f) = \sum_{m=1}^\infty c_m(\mathcal{L})\,f_m. \tag{3}$$ At truncation dimension $N \ge 2$, the shifted Nyman-Beurling-Báez-Duarte squared distance $d_N(\sigma)^2 := \inf_{b_2, \dots, b_N \in \mathbb{C}} \|E - \sum_{k=2}^N b_k \tilde{\rho}_{1/k}\|_\sigma^2$ satisfies the Hahn-Banach step-space annihilator duality identity (riemann_hypothesis_7 [15], riemann_hypothesis_9 [17]): $$d_N(\sigma)^2 = \sup\left\{ \frac{|\mathcal{L}(E)|^2}{\|\mathcal{L}\|_{S_\sigma^*}^2} : \mathcal{L} \in \operatorname{Ann}_{S_\sigma^*}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}) \setminus \{0\} \right\}, \tag{4}$$ 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 [8, 15, 17, 19, 20, 23]), we use the backward-difference functionals $\mathcal{D}_m(f) := f_m - f_{m-1}$ ($m \ge 1$, $f_0 := 0$), for which $\mathcal{D}_m(\tilde{\rho}_{1/k}) = \mathbf{1}_{k \mid m} - 1/k$ ($k \ge 2$) and $\mathcal{D}_m(E) = \delta_{m,1}$. The Vasyunin-Balazard functionals $\mathcal{C}_n(f) := \sum_{d \mid n} \mu(n/d)\mathcal{D}_d(f)$ ($n \ge 1$) satisfy the exact algebraic biorthogonality identities ([15, Theorem 2], [17, Section 2.1]): $$\mathcal{C}_1(\tilde{\rho}_{1/k}) = -\frac{1}{k} \quad (k \ge 2), \qquad \mathcal{C}_n(\tilde{\rho}_{1/k}) = \delta_{n,k} \quad (n, k \ge 2), \qquad \mathcal{C}_n(E) = \mu(n) \quad (n \ge 1). \tag{5}$$ Any finitely supported step functional $\mathcal{L}(f) = \sum_{d=1}^M \beta_d \mathcal{D}_d(f) = \sum_{m=1}^M (\beta_m - \beta_{m+1}) f_m$ ($\beta_{M+1} := 0$) supported on the first $M$ harmonic step intervals $I_1, \dots, I_M$ has step coefficients $c_m(\mathcal{L}) = \beta_m - \beta_{m+1}$, target evaluation $\mathcal{L}(E) = \beta_1$, 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 $$\mathcal{L}(\tilde{\rho}_{1/k}) = \alpha_k - \frac{1}{k}\alpha_1 \qquad (k \ge 2). \tag{6}$$ As established in riemann_hypothesis_9 [17, Section 2.1], for any $M \ge N$, the finite-support annihilator subspace $$\operatorname{Ann}_{S_\sigma^*}^{(M)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}) := \left\{\mathcal{L} \in \operatorname{Ann}_{S_\sigma^*}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}) : c_m(\mathcal{L}) = 0 \text{ for all } m > M\right\} = \operatorname{span}\!\big\{\mathcal{F}_N,\; \mathcal{C}_{N+1},\; \dots,\; \mathcal{C}_M\big\} \tag{7}$$ has exact algebraic dimension $M - N + 1$, where $$\mathcal{F}_N := \sum_{n=1}^N \frac{1}{n}\,\mathcal{C}_n \in \operatorname{Ann}_{S_\sigma^*}^{(N)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}), \qquad \mathcal{F}_N(E) = m_N := \sum_{n=1}^N \frac{\mu(n)}{n}, \tag{8}$$ is the unsmoothed harmonic head functional (the unique annihilator up to scaling supported on $I_1, \dots, I_N$).

1.2 The $N^1$ Sharp-Cutoff Bottleneck of $\mathcal{F}_N$ and Summary of New Contributions

At $\sigma = 1/2$ (where $w_m^{(1/2)} = \frac{1}{m(m+1)}$), Balazard [18, Propositions 5-6] proved that the Riesz dual partial sums associated with $\mathcal{F}_N$ have uniformly bounded $L^2((0, 1), dx)$ norm and converge weakly to $0$ as $N \to \infty$. However, $\mathcal{F}_N$ never converges to $0$ in the strong dual norm $\|\cdot\|_{S_{1/2}^*}$: because $\beta_N(\mathcal{F}_N) = 1/N$ and $\beta_{N+1}(\mathcal{F}_N) = 0$, the single boundary jump $c_N(\mathcal{F}_N) = 1/N$ at $m = N$ already forces ([18, Proposition 5]) $$\|\mathcal{F}_N\|_{S_{1/2}^*}^2 \ge \frac{|c_N(\mathcal{F}_N)|^2}{w_N^{(1/2)}} = \frac{N^{-2}}{1/[N(N+1)]} = 1 + \frac{1}{N} > 1 \qquad (\forall N \ge 1), \tag{9}$$ so $\liminf_{N\to\infty} \|\mathcal{F}_N\|_{S_{1/2}^*}^2 \ge 1 > 0$. Moreover, for $\sigma \in (1/2, 1)$, in Theorem 1 of riemann_hypothesis_9 [17] we proved the unconditional strong-norm asymptotic $$\|\mathcal{F}_N\|_{S_\sigma^*}^2 = \frac{\zeta(2\sigma+1)}{2\sigma\,\zeta(4\sigma+2)}\,N^{2\sigma-1} + O_\sigma(N^{\sigma-1/2}) \qquad (\sigma \in (1/2, 1)). \tag{10}$$ Because $\|\mathcal{F}_N\|_{S_\sigma^*}^2 \asymp N^{2\sigma-1}$ grows with $N$ for $\sigma > 1/2$, while $\limsup_{N\to\infty} \frac{\log(|m_N|^2)}{\log N} = -2(1-\Theta)$ (where $\Theta := \sup\{\operatorname{Re}(\rho) : \zeta(\rho) = 0\} \in [1/2, 1]$), the Hahn-Banach Rayleigh quotient of $\mathcal{F}_N$ satisfies $$\limsup_{N \to \infty} \frac{\log\big(|\mathcal{F}_N(E)|^2 / \|\mathcal{F}_N\|_{S_\sigma^*}^2\big)}{\log N} = \limsup_{N \to \infty} \frac{\log\big(|m_N|^2 / \|\mathcal{F}_N\|_{S_\sigma^*}^2\big)}{\log N} = -2(\sigma - \Theta) - 1, \tag{11}$$ which loses a full factor of $N^1$ in its limsup rate compared to the optimal Nyman-Beurling rate exponent $d_N(\sigma)^2 = N^{-2(\sigma-\Theta)^+ + o(1)}$ established existentially by bramble_trace in riemann_hypothesis_6 [14] (v3). In our prior work riemann_hypothesis_9 [17] and riemann_hypothesis_11 [23], recovering the rate exponent $N^{-2(\sigma-\Theta)}$ constructively required introducing zero-dependent complex spline functionals $\widetilde{\mathcal{L}}_{\rho_0, N}^{(M; W)}$ supported up to a superlinear cutoff $M = \lfloor \Lambda N^{a(\sigma, \beta_0, r)}\rfloor \gg N^a$ with $a(\sigma, \beta_0, r) = \max\!\big(\frac{r+1}{\sigma+r},\, \frac{r+2}{\beta_0+r+1}\big) > 1$.

In the present paper, building on the inherited annihilator parameterization (7) from [17] and the flat-top Hermite spline smoothing and Gram-inverse superposition framework from [23], we show that the entire $N^{2\sigma-1}$ growth of $\|\mathcal{F}_N\|_{S_\sigma^*}^2$ in (10)—and the $1 + 1/N$ boundary obstruction (9) at $\sigma = 1/2$—is an artifact of the sharp truncation of $\mathcal{F}_N$ at $n = N$. By applying variational optimization and flat-top smoothing directly to the arithmetic harmonic head $\mathcal{F}_N$ across a strictly linear tail $N < n \le N_\delta := \lfloor(1+\delta)N\rfloor$, we establish four new results:

1. Universal Tail-Segment Cauchy-Schwarz Lower Bounds and Exact Variational Extremizers for Interior $\delta \in (0, 1)$ and the Coupled Endpoint $\delta = 1$ (Section 2, Theorem 1):

   • For any interior $\delta \in (0, 1)$ with $N + 1 \le N_\delta < 2N$ and every tail-extended annihilator $\mathcal{L} = \mathcal{F}_N + \sum_{n=N+1}^{N_\delta} \alpha_n \mathcal{C}_n \in \operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$, the strict inequality $\lfloor N_\delta / 2 \rfloor \le N - 1 < N$ forces $\beta_N(\mathcal{L}) = 1/N$ while $\beta_{N_\delta+1}(\mathcal{L}) = 0$. Cauchy-Schwarz over the tail segment $N \le m \le N_\delta$ yields the universal lower bound $$\mathcal{E}_{\mathrm{tail}}(\mathcal{L}) := \sum_{m=N}^{N_\delta} \frac{|\beta_m(\mathcal{L}) - \beta_{m+1}(\mathcal{L})|^2}{w_m^{(\sigma)}} \ge \frac{N^{-2}}{N^{-2\sigma} - (N_\delta + 1)^{-2\sigma}} = \frac{N^{-2(1-\sigma)}}{1 - (1+\delta)^{-2\sigma}}\big(1 + O_\sigma(N^{-1})\big), \tag{12}$$ with exact algebraic equality in $\mathcal{E}_{\mathrm{tail}}(\mathcal{L})$ (and in the two-segment energy $\mathcal{E}_{\mathrm{two\text{-}seg}}(\mathcal{L})$ on $\lfloor N_\delta/2\rfloor + 1 \le m \le N_\delta$) attained uniquely by the variational harmonic head functional $\mathcal{F}_{N, \delta}^{\mathrm{var}}$ with $\alpha_n^{\mathrm{var}} := \frac{1}{N}\frac{n^{-2\sigma} - (N_\delta+1)^{-2\sigma}}{N^{-2\sigma} - (N_\delta+1)^{-2\sigma}}$ ($N < n \le N_\delta$), while the full norm $\|\mathcal{L}\|_{S_\sigma^*}^2 = \sum_{m=1}^{\lfloor N_\delta/2\rfloor} \frac{|c_m(\mathcal{L})|^2}{w_m^{(\sigma)}} + \mathcal{E}_{\mathrm{two\text{-}seg}}(\mathcal{L})$ is strictly larger by the positive prefix sum over $1 \le m \le \lfloor N_\delta/2\rfloor$.
   • At the coupled endpoint $\delta = 1$ ($N_\delta = 2N$), Möbius inversion at $m = N$ gives $\beta_N(\mathcal{L}) = \frac{1}{N} - \alpha_{2N}$ and $\beta_{2N}(\mathcal{L}) = \alpha_{2N}$, so $\beta_N(\mathcal{L}) + \beta_{2N}(\mathcal{L}) = \sum_{m=N}^{2N-1} c_m(\mathcal{L}) + 2\,c_{2N}(\mathcal{L}) = \frac{1}{N}$, yielding the exact universal tail-segment bound $$\mathcal{E}_{\mathrm{tail}}(\mathcal{L}) = \sum_{m=N}^{2N} \frac{|\beta_m(\mathcal{L}) - \beta_{m+1}(\mathcal{L})|^2}{w_m^{(\sigma)}} \ge \frac{N^{-2}}{N^{-2\sigma} - (2N)^{-2\sigma} + 4\,w_{2N}^{(\sigma)}}, \tag{12a}$$ attained uniquely on $N \le m \le 2N$ by $\alpha_n^{\mathrm{var},(1)}$ in (23b) (giving $\frac{5}{9} < \frac{14}{25} < \frac{29}{50}$ at $N = 2, \sigma = 1/2$).

2. Exact Two-Point Quotient-Rule Möbius Identity, Fubini-Abel Step Bound, $C^{2,1}$ Derivative Bound, and Unconditional Strong-Norm Vanishing on $[1/2, 1)$ (Section 3, Theorem 2): For any $\delta > 0$ and any flat-top taper $\Omega_\delta \in C^{2,1}([0, \infty))$ (twice continuously differentiable with Lipschitz second derivative, i.e., $\Omega_\delta''' \in L^\infty([0, \infty))$, such as the Hermite spline $\Omega_\delta = W_{r+1, \theta}$ for any $r \ge 2$) with fixed knot $\theta := \frac{1}{1+\delta}$ satisfying $\Omega_\delta(u) = 1$ on $[0, \theta]$ and $\Omega_\delta(u) = 0$ on $[1, \infty)$, define the smoothed harmonic head functional $$\mathcal{F}_{N, \delta}^{\mathrm{smooth}} := \sum_{n=1}^{N_\delta} \frac{1}{n}\,\Omega_\delta\!\left(\frac{n}{N_\delta + 1}\right)\,\mathcal{C}_n \in \operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}). \tag{13}$$ Defining the two-point profile $V_{m,\delta}(u) := (m+1)\Omega_\delta(u) - m\Omega_\delta\big(\frac{m+1}{m}u\big)$ and the quotient-rule derivative profile $\widetilde{\Omega}_\delta(u) := -u^2\frac{d}{du}\big(\frac{\Omega_\delta(u)}{u}\big) = \Omega_\delta(u) - u\,\Omega_\delta'(u)$, we prove the exact algebraic identity $\eta_m^{\mathcal{F}} - \eta_{m+1}^{\mathcal{F}} = \frac{1}{m(m+1)}M_{V_{m,\delta}}(X_m)$ ($X_m := \frac{N_\delta+1}{m}$, $M_V(X) := \sum_{j \le X}\frac{\mu(j)}{j}V(j/X)$), the exact derivative identity $V'_{m,\delta}(u) - \widetilde{\Omega}'_\delta(u) = (m+1)\big[\Omega_\delta'(u) - \Omega_\delta'\big(u + \frac{u}{m}\big)\big] + u\,\Omega_\delta''(u)$, and the uniform Lipschitz-$C^{2,1}$ Abel-summation bound $$\big|M_{V_{m,\delta}}(X_m) - M_{\widetilde{\Omega}_\delta}(X_m)\big| \le \frac{1 - \theta/2}{m}\left(\frac{1 + 1/m}{2}\big\|\Omega_\delta'''\big\|_{L^\infty([\theta, 1])} + \big\|\Omega_\delta''\big\|_{L^\infty([\theta, 1])}\right)\sup_{y \ge \frac{\theta}{2}X_m} |m_y| \qquad (1 \le m \le N_\delta). \tag{14}$$ Consequently, by Fubini-Tonelli on $V'_{m,\delta}(u) = -(m+1)\int_u^{(1+1/m)u} \Omega_\delta''(t)\,dt$, the step coefficients satisfy the unconditional bound $|\eta_m^{\mathcal{F}} - \eta_{m+1}^{\mathcal{F}}| \le \frac{C_\delta}{m(m+1)} < \frac{C_\delta}{m^2}$ with $C_\delta := \int_\theta^1 t|\Omega_\delta''(t)|\,dt = \|\widetilde{\Omega}'_\delta\|_{L^1([\theta, 1])}$ ($= \frac{135}{8} = 16.875$ for $\Omega_{1/4} = W_{3, 4/5}$), and the Prime Number Theorem implies unconditionally for every $\sigma \in [1/2, 1)$ (including strong dual-norm convergence to $0$ on the critical line $\sigma = 1/2$, in contrast to (9)): $$\lim_{N \to \infty} \big\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2 = 0, \qquad \big\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2 \ll_{\sigma, \delta} \exp\!\left(-c_1\sqrt{\log N}\right). \tag{15}$$ (Scope note: because the unnormalized target evaluation $\mathcal{F}_{N,\delta}^{\mathrm{smooth}}(E) = M_{\Omega_\delta}(N_\delta+1)$ also tends to $0$ as $N \to \infty$ by the Prime Number Theorem, strong vanishing (15) of the unnormalized dual norm does not imply $d_N(1/2) \to 0$.)

3. Full $N^1$ Power-Saving Asymptotic Norm Law and Generic Real / Certified Rational Infinitely-Often Rate Bounds (Section 4, Theorem 3): For every $\sigma \in (\Theta, 1)$ and $\delta \in (0, 1]$, using the multiplicative bound (14) and an explicit piecewise-$C^1$ jump-plus-derivative Riemann-sum comparison on $\big|M_{\widetilde{\Omega}_\delta}(X)\big|^2$ to control all small-$m$ and discretization error terms at scale $O\big(N^{-2(1-\sigma)-(\sigma-\Theta)/2}\big) = o\big(N^{-2(1-\sigma)}\big)$, we prove the exact $N^1$-saving asymptotic law $$\big\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2 = \mathcal{C}_{\sigma, \delta}\,N^{-2(1-\sigma)} + O_{\sigma,\delta}\!\left(N^{-2(1-\sigma)-(\sigma-\Theta)/2}\right), \qquad \mathcal{C}_{\sigma, \delta} := \frac{(1+\delta)^{-2(1-\sigma)}}{2\sigma}\int_1^\infty X^{1-2\sigma}\big|M_{\widetilde{\Omega}_\delta}(X)\big|^2\,dX. \tag{16}$$ Furthermore, for $\Omega_\delta = W_{r+1, \frac{1}{1+\delta}}$ ($r \ge 2$), we prove the exact Gauss hypergeometric Mellin identity $\widehat{\Omega}_\delta(s) = \frac{1}{s}\,{}_2F_1\big(-s, r+1; 2r+2; \frac{\delta}{1+\delta}\big)$ ($\operatorname{Re}(s) > -1$) and establish two complementary infinitely-often ($\limsup_{N\to\infty}$) rate bounds at strictly linear support $N_\delta = \lfloor(1+\delta)N\rfloor$:

   • (Generic Real Scale): For every fixed real $\delta \in (0, 1] \setminus \mathcal{E}_r$ outside a countable exceptional set $\mathcal{E}_r$, the zero-independent real functional $\mathcal{F}_{N, \delta}^{\mathrm{smooth}}$ satisfies $\limsup_{N\to\infty} N^{2(\sigma-\Theta)+\varepsilon} d_N(\sigma)^2 = \infty$ ($\forall \varepsilon > 0$);
   • (Explicit Rational Scale $\delta = 1/4$ at the First Critical-Line Zero $\rho_1$): At the explicit rational scale $\delta = 1/4 \in \mathbb{Q}$ ($\theta = 4/5, r = 2$, where $\mathcal{F}_{N, 1/4}^{\mathrm{smooth}}$ has purely rational coefficients $\alpha_n \in \mathbb{Q}$ and $-W'_{3, 4/5}(u) = 150(5u-4)^2(5-5u)^2 = 93750u^4 - 337500u^3 + 453750u^2 - 270000u + 60000$ on $[4/5, 1]$), 60-digit mpmath.iv Euler-Maclaurin and Rouché certification of $\rho_1 = \frac{1}{2} + i\gamma_1 \in D(s_0, r_0)$ ($s_0 = \frac{1}{2} + i\frac{14134725141735}{10^{12}}, r_0 = 10^{-11}$) propagated through the exact 5-term polynomial formula for $\widehat{W}_{3, 4/5}(s)$ certifies $\big|\widehat{W}_{3, 4/5}(\rho_1 - 1)\big| \in [0.06225173, 0.06225178] \subset [0.062251, 0.062252] > 0$, proving unconditionally that $\limsup_{N\to\infty} N^{1+\varepsilon}\big|M_{W_{3, 4/5}}(\lfloor 5N/4\rfloor+1)\big|^2 = \infty$ and hence $d_N(\sigma)^2 = \Omega\big(N^{-2(\sigma - 1/2) - \varepsilon}\big)$ for all $\sigma \in (\Theta, 1)$ (attaining $N^{-2(\sigma-\Theta)-\varepsilon}$ whenever $\Theta = 1/2$).

4. Consistent Fixed-Knot Multi-Scale Zero-Independent Superposition Certificates (Section 5, Theorem 4): Using the fixed knots $\theta_j = \frac{1}{1+\delta_j}$ consistently across all scales, we certify in exact rational arithmetic (fractions.Fraction) + 60-digit mpmath.iv interval arithmetic that nested $K = 8$ ($M = 2N$) and $K = 16$ ($M = 4N$) zero-independent rational superpositions (constructed without specifying zeta zeros) of $\big\{\mathcal{F}_{N, \delta_j}^{\mathrm{smooth}}\big\}$ satisfy exact rational annihilation $\mathcal{L}_{\tilde{\lambda}}(\tilde{\rho}_{1/k}) \equiv 0 \in \mathbb{Q}$ ($2 \le k \le N$) and reach $23.7\%$–$53.8\%$ ($K = 8$) and $56.2\%$–$63.8\%$ ($K = 16$) of the double-precision reference distance $d_N(0.75)^2$ at $N \in \{40, 80, 160\}$.

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2. Universal Tail-Segment Lower Bounds and Variational Harmonic Head Extremizers

Let $N \ge 2$ and $M > N$. By (5)-(7), any step functional $\mathcal{L} = \sum_{n=1}^M \alpha_n \mathcal{C}_n \in \operatorname{Ann}_{S_\sigma^*}^{(M)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$ normalized by $\alpha_1 = 1$ has $\alpha_k = 1/k$ for all $1 \le k \le N$, while $\alpha_{N+1}, \dots, \alpha_M \in \mathbb{C}$ are free parameters: $$\mathcal{L} = \mathcal{F}_N + \sum_{n=N+1}^M \alpha_n\,\mathcal{C}_n, \qquad \beta_m(\mathcal{L}) = \frac{m_{\lfloor N/m \rfloor}}{m} + \sum_{j = \lfloor N/m \rfloor + 1}^{\lfloor M/m \rfloor} \alpha_{mj}\,\mu(j) \quad (1 \le m \le M), \tag{17}$$ with $\beta_{M+1}(\mathcal{L}) = 0$, where $m_L := \sum_{j=1}^L \mu(j)/j$. For any $M = N_\delta := \lfloor(1+\delta)N\rfloor$, define the tail-segment energy and two-segment energy: $$\mathcal{E}_{\mathrm{tail}}(\mathcal{L}) := \sum_{m=N}^{N_\delta} \frac{|\beta_m(\mathcal{L}) - \beta_{m+1}(\mathcal{L})|^2}{w_m^{(\sigma)}}, \qquad \mathcal{E}_{\mathrm{two\text{-}seg}}(\mathcal{L}) := \sum_{m=\lfloor N_\delta/2\rfloor + 1}^{N_\delta} \frac{|\beta_m(\mathcal{L}) - \beta_{m+1}(\mathcal{L})|^2}{w_m^{(\sigma)}}, \tag{18}$$ so that the full dual norm squared splits into three nonnegative parts: $$\|\mathcal{L}\|_{S_\sigma^*}^2 = \sum_{m=1}^{\lfloor N_\delta/2\rfloor} \frac{|\beta_m(\mathcal{L}) - \beta_{m+1}(\mathcal{L})|^2}{w_m^{(\sigma)}} + \mathcal{E}_{\mathrm{two\text{-}seg}}(\mathcal{L}) \ge \mathcal{E}_{\mathrm{two\text{-}seg}}(\mathcal{L}) \ge \mathcal{E}_{\mathrm{tail}}(\mathcal{L}). \tag{19}$$

Theorem 1 (Universal Tail-Segment Cauchy-Schwarz Lower Bounds and Variational Extremizers for $\delta \in (0, 1)$ and $\delta = 1$). Let $\sigma \in [1/2, 1)$ and $N \ge 2$.

1. (Fixed Boundary Coordinates for Interior $\delta \in (0, 1)$): Let $\delta \in (0, 1)$ be such that $N + 1 \le N_\delta := \lfloor(1+\delta)N\rfloor < 2N$. Then $\lfloor N_\delta / 2 \rfloor \le N - 1 < N$, and for every $\mathcal{L} = \mathcal{F}_N + \sum_{n=N+1}^{N_\delta} \alpha_n \mathcal{C}_n \in \operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$, the step-difference coordinates $\beta_m(\mathcal{L})$ satisfy $$\beta_m(\mathcal{L}) = \frac{1}{m} \quad (\lfloor N_\delta / 2 \rfloor < m \le N), \qquad \beta_m(\mathcal{L}) = \alpha_m \quad (N < m \le N_\delta), \qquad \beta_{N_\delta + 1}(\mathcal{L}) = 0. \tag{20}$$ In particular, $\beta_N(\mathcal{L}) = 1/N$ is independent of $\alpha_{N+1}, \dots, \alpha_{N_\delta}$, and on $\lfloor N_\delta/2 \rfloor + 1 \le m \le N - 1$ the step coefficients are fixed to $c_m(\mathcal{L}) = \frac{1}{m(m+1)}$.
2. (Universal Cauchy-Schwarz Lower Bounds for $\delta \in (0, 1)$): For $N + 1 \le N_\delta < 2N$, every $\mathcal{L} = \mathcal{F}_N + \sum_{n=N+1}^{N_\delta} \alpha_n \mathcal{C}_n$ satisfies the universal tail-segment and two-segment lower bounds $$\mathcal{E}_{\mathrm{tail}}(\mathcal{L}) \ge \frac{N^{-2}}{N^{-2\sigma} - (N_\delta + 1)^{-2\sigma}} = \frac{N^{-2(1-\sigma)}}{1 - (1+\delta)^{-2\sigma}}\left(1 + O_\sigma\!\left(\frac{1}{N}\right)\right), \tag{21}$$ $$\|\mathcal{L}\|_{S_\sigma^*}^2 \ge \mathcal{E}_{\mathrm{two\text{-}seg}}(\mathcal{L}) \ge \sum_{m=\lfloor N_\delta/2\rfloor+1}^{N-1}\frac{1}{m^2(m+1)^2 w_m^{(\sigma)}} + \frac{N^{-2}}{N^{-2\sigma} - (N_\delta + 1)^{-2\sigma}}. \tag{21a}$$
3. (Exact Segment Equality for the Variational Harmonic Head Functional when $N + 1 \le N_\delta < 2N$): For $N + 1 \le N_\delta < 2N$, equality holds in the tail-segment bound (21) (and in the second inequality of (21a) for $\mathcal{E}_{\mathrm{two\text{-}seg}}(\mathcal{L})$) if and only if $\alpha_{N+1}, \dots, \alpha_{N_\delta}$ are given by the variational harmonic head extension $\mathcal{F}_{N, \delta}^{\mathrm{var}} := \mathcal{F}_N + \sum_{n=N+1}^{N_\delta} \alpha_n^{\mathrm{var}}\mathcal{C}_n$, where $$\alpha_n^{\mathrm{var}} := \frac{1}{N}\,\frac{n^{-2\sigma} - (N_\delta + 1)^{-2\sigma}}{N^{-2\sigma} - (N_\delta + 1)^{-2\sigma}} \qquad (N < n \le N_\delta), \tag{22}$$ for which $c_m\big(\mathcal{F}_{N, \delta}^{\mathrm{var}}\big) = \frac{w_m^{(\sigma)}}{N(N^{-2\sigma} - (N_\delta+1)^{-2\sigma})}$ for all $N \le m \le N_\delta$ and $\mathcal{E}_{\mathrm{tail}}\big(\mathcal{F}_{N, \delta}^{\mathrm{var}}\big) = \frac{N^{-2}}{N^{-2\sigma} - (N_\delta + 1)^{-2\sigma}}$ identically. (The full norm squared $\big\|\mathcal{F}_{N, \delta}^{\mathrm{var}}\big\|_{S_\sigma^*}^2$ exceeds $\mathcal{E}_{\mathrm{two\text{-}seg}}\big(\mathcal{F}_{N, \delta}^{\mathrm{var}}\big)$ by the strictly positive prefix sum $\sum_{m=1}^{\lfloor N_\delta/2\rfloor} |c_m|^2/w_m^{(\sigma)} > 0$.)
4. (Coupled Boundary Constraint and Exact Universal Lower Bound at the Endpoint $\delta = 1$, $N_\delta = 2N$): When $\delta = 1$ ($N_\delta = 2N$), every $\mathcal{L} = \mathcal{F}_N + \sum_{n=N+1}^{2N} \alpha_n \mathcal{C}_n \in \operatorname{Ann}_{S_\sigma^*}^{(2N)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$ satisfies $\beta_m(\mathcal{L}) = \alpha_m$ for $N < m \le 2N$, $\beta_{2N+1}(\mathcal{L}) = 0$, and at $m = N$ satisfies the coupled boundary relation $$\beta_N(\mathcal{L}) = \alpha_N \mu(1) + \alpha_{2N}\mu(2) = \frac{1}{N} - \alpha_{2N} = \frac{1}{N} - \beta_{2N}(\mathcal{L}), \qquad \sum_{m=N}^{2N-1} c_m(\mathcal{L}) + 2\,c_{2N}(\mathcal{L}) = \beta_N(\mathcal{L}) + \beta_{2N}(\mathcal{L}) = \frac{1}{N}. \tag{23}$$ Consequently, at $\delta = 1$ ($N_\delta = 2N$), the tail-segment energy obeys the exact universal Cauchy-Schwarz lower bound $$\mathcal{E}_{\mathrm{tail}}(\mathcal{L}) = \sum_{m=N}^{2N} \frac{|\beta_m(\mathcal{L}) - \beta_{m+1}(\mathcal{L})|^2}{w_m^{(\sigma)}} \ge \frac{N^{-2}}{\sum_{m=N}^{2N-1} w_m^{(\sigma)} + 4\,w_{2N}^{(\sigma)}} = \frac{N^{-2}}{N^{-2\sigma} - (2N)^{-2\sigma} + 4\,w_{2N}^{(\sigma)}}, \tag{23a}$$ with exact equality in (23a) attained uniquely by $$\alpha_n^{\mathrm{var}, (1)} := \frac{n^{-2\sigma} - (2N)^{-2\sigma} + 2\,w_{2N}^{(\sigma)}}{N\big(N^{-2\sigma} - (2N)^{-2\sigma} + 4\,w_{2N}^{(\sigma)}\big)} \quad (N < n \le 2N - 1), \qquad \alpha_{2N}^{\mathrm{var}, (1)} := \frac{2\,w_{2N}^{(\sigma)}}{N\big(N^{-2\sigma} - (2N)^{-2\sigma} + 4\,w_{2N}^{(\sigma)}\big)}. \tag{23b}$$

Proof. Part 1: Since $N + 1 \le N_\delta < 2N$, we have $\lfloor N_\delta / 2 \rfloor \le N - 1$. For any $\lfloor N_\delta / 2 \rfloor < m \le N$, we have $2m \ge 2(\lfloor N_\delta/2\rfloor + 1) > N_\delta \ge N$, so $\lfloor N/m \rfloor = \lfloor N_\delta / m \rfloor = 1$. Thus the second sum in (17) is empty and $\beta_m(\mathcal{L}) = m_1 / m = 1/m$ (including at $m = N$), so $c_m(\mathcal{L}) = \frac{1}{m} - \frac{1}{m+1} = \frac{1}{m(m+1)}$ for $\lfloor N_\delta/2 \rfloor + 1 \le m \le N - 1$. For $N < m \le N_\delta$, $\lfloor N/m \rfloor = 0$ and $\lfloor N_\delta / m \rfloor = 1$, so (17) reduces to the single $j = 1$ term $\alpha_m \mu(1) = \alpha_m$, establishing (20).

Part 2: By (20), $\beta_N(\mathcal{L}) = 1/N$ and $\beta_{N_\delta+1}(\mathcal{L}) = 0$, so $\sum_{m=N}^{N_\delta} (\beta_m(\mathcal{L}) - \beta_{m+1}(\mathcal{L})) = \beta_N(\mathcal{L}) - \beta_{N_\delta+1}(\mathcal{L}) = 1/N$. Applying Cauchy-Schwarz with weights $w_m^{(\sigma)}$ gives $$\frac{1}{N^2} = \left|\sum_{m=N}^{N_\delta} \frac{\beta_m(\mathcal{L}) - \beta_{m+1}(\mathcal{L})}{\sqrt{w_m^{(\sigma)}}}\,\sqrt{w_m^{(\sigma)}}\right|^2 \le \mathcal{E}_{\mathrm{tail}}(\mathcal{L})\left(\sum_{m=N}^{N_\delta} w_m^{(\sigma)}\right) = \mathcal{E}_{\mathrm{tail}}(\mathcal{L})\big(N^{-2\sigma} - (N_\delta + 1)^{-2\sigma}\big), \tag{24}$$ which yields (21) and, upon adding the fixed sum over $\lfloor N_\delta/2 \rfloor + 1 \le m \le N - 1$, yields (21a).

Part 3: Equality holds in (24) if and only if $c_m(\mathcal{L}) = \beta_m(\mathcal{L}) - \beta_{m+1}(\mathcal{L}) = \kappa\,w_m^{(\sigma)}$ for all $N \le m \le N_\delta$ for some constant $\kappa \in \mathbb{C}$. Summing from $m = n$ to $N_\delta$ (with $\beta_{N_\delta+1} = 0$) gives $\beta_n(\mathcal{L}) = \kappa\big(n^{-2\sigma} - (N_\delta + 1)^{-2\sigma}\big)$ for $N \le n \le N_\delta$, and $\beta_N(\mathcal{L}) = 1/N$ uniquely determines $\kappa = \frac{1}{N(N^{-2\sigma} - (N_\delta + 1)^{-2\sigma})}$, giving (22).

Part 4: When $N_\delta = 2N$, at $m = N$ we have $\lfloor 2N / N \rfloor = 2$, so (17) gives $\beta_N(\mathcal{L}) = \alpha_N\mu(1) + \alpha_{2N}\mu(2) = \frac{1}{N} - \alpha_{2N}$, while for $N < m \le 2N$, $\lfloor 2N / m \rfloor = 1$ gives $\beta_m(\mathcal{L}) = \alpha_m$ and $\beta_{2N+1}(\mathcal{L}) = 0$. Hence $c_{2N}(\mathcal{L}) = \beta_{2N}(\mathcal{L}) = \alpha_{2N}$ and $\sum_{m=N}^{2N-1} c_m(\mathcal{L}) + 2\,c_{2N}(\mathcal{L}) = (\beta_N(\mathcal{L}) - \beta_{2N}(\mathcal{L})) + 2\,\beta_{2N}(\mathcal{L}) = \frac{1}{N}$, proving (23). Applying Cauchy-Schwarz to $\frac{1}{N} = \sum_{m=N}^{2N-1} \frac{c_m}{\sqrt{w_m^{(\sigma)}}}\sqrt{w_m^{(\sigma)}} + \frac{c_{2N}}{\sqrt{w_{2N}^{(\sigma)}}}\big(2\sqrt{w_{2N}^{(\sigma)}}\big)$ yields $$\frac{1}{N^2} \le \left(\sum_{m=N}^{2N} \frac{|c_m(\mathcal{L})|^2}{w_m^{(\sigma)}}\right)\left(\sum_{m=N}^{2N-1} w_m^{(\sigma)} + 4\,w_{2N}^{(\sigma)}\right),$$ establishing (23a), with equality if and only if $c_m = \kappa_1 w_m^{(\sigma)}$ ($N \le m \le 2N-1$) and $c_{2N} = 2\kappa_1 w_{2N}^{(\sigma)}$ where $\kappa_1 = \big[N(N^{-2\sigma} - (2N)^{-2\sigma} + 4w_{2N}^{(\sigma)})\big]^{-1}$, which sums to (23b). $\square$

2.1 Exact Rational Verification at $N = 2, \delta = 1$ and 60-Digit Interval Certification at $\delta = 1/2$

At $N = 2, \delta = 1, \sigma = 1/2$ ($N_\delta = 4$, $w_m^{(1/2)} = \frac{1}{m(m+1)}$), Theorem 1(4) gives the exact universal tail-segment lower bound (verified in fractions.Fraction arithmetic in src/verify_smoothed_harmonic_head_duality.py): $$\mathcal{E}_{\mathrm{tail}}(\mathcal{L}) \ge \frac{2^{-2}}{w_2^{(1/2)} + w_3^{(1/2)} + 4\,w_4^{(1/2)}} = \frac{1/4}{1/6 + 1/12 + 4/20} = \frac{5}{9} = 0.5555\dots,$$ which is attained on $2 \le m \le 4$ uniquely by $\alpha^{\mathrm{var}, (1)} = \big(1, \frac{1}{2}, \frac{11}{54}, \frac{1}{9}\big)$ (with $\mathcal{E}_{\mathrm{tail}} = \frac{5}{9}$ and full norm squared $\|\mathcal{F}_{2, 1}^{\mathrm{var}, (1)}\|_{S_{1/2}^*}^2 = \frac{835}{1458} \approx 0.5727$). For comparison, the test functional $\mathcal{L} = \mathcal{C}_1 + \frac{1}{2}\mathcal{C}_2 + \frac{1}{5}\mathcal{C}_3 + \frac{1}{10}\mathcal{C}_4$ has $\beta_2(\mathcal{L}) = \frac{1}{2} - \frac{1}{10} = \frac{2}{5} \ne \frac{1}{2}$ (showing why (21) requires $N_\delta < 2N$), step coefficients $\big(-\frac{1}{10}, \frac{1}{5}, \frac{1}{10}, \frac{1}{10}\big)$, tail-segment energy $\mathcal{E}_{\mathrm{tail}}(\mathcal{L}) = \frac{14}{25} = 0.56 > \frac{5}{9}$, and full norm squared $\|\mathcal{L}\|_{S_{1/2}^*}^2 = \frac{29}{50} = 0.58 > \frac{5}{9}$.

Table 1 reports our 60-digit mpmath.iv interval certifications of Theorem 1(1)-(3) at $\delta = 1/2$ ($N_\delta = 3N/2 < 2N$) and fixed knot $\theta = \frac{1}{1+\delta} = \frac{2}{3}$ across $\sigma \in \{0.50, 0.60, 0.75, 0.90\}$ and $N \in \{40, 160, 640\}$. All lower bounds are rounded strictly downward (ROUND_FLOOR) and all upper bounds are rounded strictly upward (ROUND_CEILING) from 60-digit mpmath.iv enclosures.

| $\sigma$ | $N$ ($N_\delta = 1.5N$) | Unsmoothed $\|\mathcal{F}_N\|_{S_\sigma^*}^2$ (Num.) | Certified Tail Lower Bound $N^{2(1-\sigma)}\frac{N^{-2}}{N^{-2\sigma}-(N_\delta+1)^{-2\sigma}} \ge$ | Certified Tail Equality Width $\big|\mathcal{E}_{\mathrm{tail}}(\mathcal{F}_{N, 1/2}^{\mathrm{var}}) - \mathrm{LB}\big| \le$ | Certified Full Variational Norm $N^{2(1-\sigma)}\|\mathcal{F}_{N, 1/2}^{\mathrm{var}}\|_{S_\sigma^*}^2 \le$ | Certified Smoothed $W_{3, 2/3}$ Norm $N^{2(1-\sigma)}\|\mathcal{F}_{N, 1/2}^{\mathrm{smooth}}\|_{S_\sigma^*}^2 \le$ | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $0.50$ | $40$ ($60$) | $1.500964$ | $2.904761$ | $6.02 \times 10^{-60}$ | $3.738270$ | $6.076598$ | | $0.50$ | $160$ ($240$) | $1.527359$ | $2.975308$ | $6.21 \times 10^{-60}$ | $4.053387$ | $7.205401$ | | $0.50$ | $640$ ($960$) | $1.516696$ | $2.993769$ | $6.28 \times 10^{-60}$ | $4.278957$ | $8.190882$ | | $0.60$ | $40$ ($60$) | $2.456374$ | $2.516769$ | $4.81 \times 10^{-59}$ | $3.125796$ | $4.872807$ | | $0.60$ | $160$ ($240$) | $3.253139$ | $2.575188$ | $8.82 \times 10^{-59}$ | $3.319679$ | $5.432965$ | | $0.60$ | $640$ ($960$) | $4.269867$ | $2.590479$ | $1.41 \times 10^{-58}$ | $3.432326$ | $5.818727$ | | $0.75$ | $40$ ($60$) | $5.538634$ | $2.132200$ | $2.04 \times 10^{-59}$ | $2.542642$ | $3.797497$ | | $0.75$ | $160$ ($240$) | $10.974665$ | $2.178395$ | $3.94 \times 10^{-59}$ | $2.653606$ | $4.014943$ | | $0.75$ | $640$ ($960$) | $21.844473$ | $2.190492$ | $7.91 \times 10^{-59}$ | $2.702387$ | $4.120035$ | | $0.90$ | $40$ ($60$) | $13.274428$ | $1.879198$ | $2.12 \times 10^{-58}$ | $2.174667$ | $3.167473$ | | $0.90$ | $160$ ($240$) | $39.533666$ | $1.917154$ | $8.34 \times 10^{-58}$ | $2.248173$ | $3.258660$ | | $0.90$ | $640$ ($960$) | $119.193984$ | $1.927098$ | $3.30 \times 10^{-57}$ | $2.274325$ | $3.290060$ |

Table 1: 60-digit mpmath.iv interval certification of the universal tail-segment lower bound (21), exact variational tail-segment equality $\mathcal{E}_{\mathrm{tail}}(\mathcal{F}_{N, 1/2}^{\mathrm{var}})$, and rescaled full dual norms $N^{2(1-\sigma)}\|\mathcal{F}_{N, 1/2}^{\mathrm{var}}\|_{S_\sigma^*}^2$ and $N^{2(1-\sigma)}\|\mathcal{F}_{N, 1/2}^{\mathrm{smooth}}\|_{S_\sigma^*}^2$ with fixed knot $\theta = 2/3$ (results/smoothed_harmonic_head_duality.json).

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3. Exact Two-Point Quotient-Rule Möbius Identity and Unconditional Norm Vanishing on $[1/2, 1)$

We now analyze the smoothed harmonic head functional $\mathcal{F}_{N, \delta}^{\mathrm{smooth}}$ defined in (13) for any flat-top taper $\Omega_\delta \in C^{2,1}([0, \infty))$ (twice continuously differentiable with Lipschitz second derivative, i.e., $\Omega_\delta''' \in L^\infty([0, \infty))$) with fixed knot $\theta := \frac{1}{1+\delta}$ satisfying $\Omega_\delta(u) = 1$ on $[0, \theta]$ and $\Omega_\delta(u) = 0$ on $[1, \infty)$. Every flat-top Hermite spline $\Omega_\delta = W_{r+1, \theta}$ ($r \ge 2$) belongs to $C^r([0, \infty)) \cap C^{r, 1}([0, \infty)) \subset C^{2,1}([0, \infty))$ because it is a polynomial on $[\theta, 1]$ whose derivatives up to order $r$ vanish at $u = \theta$ and $u = 1$.

Theorem 2 (Exact Two-Point Möbius Identity, Lipschitz-$C^{2,1}$ Derivative Bound, and Unconditional Norm Vanishing on $[1/2, 1)$). Let $\sigma \in [1/2, 1)$ and $\delta > 0$, set $\theta := \frac{1}{1+\delta}$ and $N_\delta := \lfloor(1+\delta)N\rfloor$, and let $\Omega_\delta \in C^{2,1}([0, \infty))$ satisfy $\Omega_\delta(u) = 1$ for $0 \le u \le \theta$ and $\Omega_\delta(u) = 0$ for $u \ge 1$. For each integer $m \ge 1$, define the two-point profile and the quotient-rule derivative profile on $[0, \infty)$: $$V_{m,\delta}(u) := (m+1)\,\Omega_\delta(u) - m\,\Omega_\delta\!\left(\frac{m+1}{m}\,u\right), \qquad \widetilde{\Omega}_\delta(u) := -u^2\frac{d}{du}\!\left(\frac{\Omega_\delta(u)}{u}\right) = \Omega_\delta(u) - u\,\Omega_\delta'(u). \tag{25}$$

1. (Exact Annihilation, Exact Two-Point Identity, and Lipschitz-$C^{2,1}$ Abel Bound): For every $N \ge 2$, $\mathcal{F}_{N, \delta}^{\mathrm{smooth}} := \sum_{n=1}^{N_\delta} \frac{1}{n}\Omega_\delta\big(\frac{n}{N_\delta+1}\big)\mathcal{C}_n$ belongs to $\operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$ and has target evaluation $\mathcal{F}_{N, \delta}^{\mathrm{smooth}}(E) = M_{\Omega_\delta}(N_\delta + 1)$. With $X_m := \frac{N_\delta + 1}{m}$ and $M_V(X) := \sum_{j=1}^{\lfloor X \rfloor} \frac{\mu(j)}{j}V\big(\frac{j}{X}\big)$, its step coefficients $c_m\big(\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\big) = \eta_m^{\mathcal{F}} - \eta_{m+1}^{\mathcal{F}}$ satisfy the exact algebraic identity for all $1 \le m \le N_\delta$: $$\eta_m^{\mathcal{F}} - \eta_{m+1}^{\mathcal{F}} = \frac{1}{m(m+1)}\,M_{V_{m,\delta}}(X_m). \tag{26}$$ Moreover, the difference profile $W_{m,\delta}(u) := V_{m,\delta}(u) - \widetilde{\Omega}_\delta(u)$ has exact derivative $$W'_{m,\delta}(u) = (m+1)\left[\Omega_\delta'(u) - \Omega_\delta'\!\left(u + \frac{u}{m}\right)\right] + u\,\Omega_\delta''(u), \tag{26a}$$ which obeys the explicit pointwise bound $\big|W'_{m,\delta}(u)\big| \le \frac{1}{m}\left(\frac{1 + 1/m}{2}u^2\big\|\Omega_\delta'''\big\|_{L^\infty([\theta, 1])} + u\big\|\Omega_\delta''\big\|_{L^\infty([\theta, 1])}\right)$ for all $u \in [0, 1]$, yielding the uniform multiplicative Abel-summation bounds for all $1 \le m \le N_\delta$: $$\big|M_{V_{m,\delta}}(X_m)\big| \ll_\delta \sup_{y \ge \frac{\theta}{2}X_m} |m_y|, \qquad \big|M_{\widetilde{\Omega}_\delta}(X_m)\big| \ll_\delta \sup_{y \ge \frac{\theta}{2}X_m} |m_y|, \qquad \big|M_{V_{m,\delta}}(X_m) - M_{\widetilde{\Omega}_\delta}(X_m)\big| \ll_\delta \frac{1}{m}\sup_{y \ge \frac{\theta}{2}X_m} |m_y|. \tag{27}$$
2. (Unconditional Pointwise $O(m^{-2})$ Fubini-Abel Bound and Uniform Dual Norm Boundedness): For every $1 \le m \le N_\delta$ and every $N \ge 2$, the step coefficients satisfy the unconditional bound $$\big|\eta_m^{\mathcal{F}} - \eta_{m+1}^{\mathcal{F}}\big| = \frac{\big|M_{V_{m,\delta}}(X_m)\big|}{m(m+1)} \le \frac{C_\delta}{m(m+1)} < \frac{C_\delta}{m^2}, \qquad C_\delta := \int_\theta^1 t\,\big|\Omega_\delta''(t)\big|\,dt = \big\|\widetilde{\Omega}'_\delta\big\|_{L^1([\theta, 1])} \le (1-\theta)\big\|\Omega_\delta''\big\|_{L^\infty([\theta, 1])}, \tag{28}$$ and consequently $\big\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2 \le \frac{C_\delta^2}{1 - 2^{-2\sigma}}\,\zeta(3 - 2\sigma) < \infty$ uniformly for all $N \ge 2$ and each $\sigma \in [1/2, 1)$. (In particular, for $\Omega_\delta = W_{3, \theta}$, symmetry of $|W''_{3, \theta}|$ about $t = \frac{1+\theta}{2}$ gives $C_\delta = \frac{1+\theta}{2}\int_\theta^1 |W''_{3, \theta}(t)|\,dt = \frac{15(1+\theta)}{8(1-\theta)}$, so $C_{1/4} = \frac{135}{8} = 16.875$ at $\delta = 1/4, \theta = 4/5$.)
3. (Unconditional Strong Dual-Norm Vanishing on $[1/2, 1)$, Including $\sigma = 1/2$): Unconditionally for every $\sigma \in [1/2, 1)$ (and in particular in the strong dual norm at $\sigma = 1/2$, in contrast to (9)), $$\lim_{N \to \infty} \big\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2 = 0, \qquad \big\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2 \ll_{\sigma, \delta} \exp\!\left(-c_1\sqrt{\log N}\right) \quad (c_1 > 0). \tag{29}$$

Proof. Part 1: Since $N_\delta + 1 > (1+\delta)N$, for every $1 \le k \le N$ we have $\frac{k}{N_\delta + 1} < \frac{N}{(1+\delta)N} = \theta$, where $\Omega_\delta \equiv 1$. Hence $\alpha_k\big(\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\big) = 1/k$ for all $1 \le k \le N$, so (5)-(6) give $\mathcal{F}_{N, \delta}^{\mathrm{smooth}}(\tilde{\rho}_{1/k}) = 0$ for all $2 \le k \le N$ and $\mathcal{F}_{N, \delta}^{\mathrm{smooth}}(E) = M_{\Omega_\delta}(N_\delta + 1)$.

For each $1 \le m \le N_\delta$, since $\Omega_\delta\big(\frac{(m+1)j}{N_\delta+1}\big) = 0$ whenever $j > \lfloor N_\delta / (m+1) \rfloor$, there is no endpoint cutoff jump between $m$ and $m + 1$: $$\eta_m^{\mathcal{F}} - \eta_{m+1}^{\mathcal{F}} = \sum_{j=1}^{\lfloor X_m \rfloor} \frac{\mu(j)}{j}\left[\frac{1}{m}\,\Omega_\delta\!\left(\frac{j}{X_m}\right) - \frac{1}{m+1}\,\Omega_\delta\!\left(\frac{m+1}{m}\frac{j}{X_m}\right)\right] = \frac{1}{m(m+1)}\sum_{j=1}^{\lfloor X_m \rfloor} \frac{\mu(j)}{j}\,V_{m,\delta}\!\left(\frac{j}{X_m}\right), \tag{30}$$ which is the exact identity (26). Next, observe that on $0 \le u \le \frac{m}{m+1}\theta$, both $u \le \theta$ and $\frac{m+1}{m}u \le \theta$, so $V_{m,\delta}(u) = (m+1) - m = 1 = \widetilde{\Omega}_\delta(u)$. Since $\frac{m}{m+1}\theta \ge \frac{\theta}{2}$ for all $m \ge 1$, the functions $V_{m,\delta}(u) - 1$, $\widetilde{\Omega}_\delta(u) - 1$, and $W_{m,\delta}(u) := V_{m,\delta}(u) - \widetilde{\Omega}_\delta(u) = m\big[\Omega_\delta(u) - \Omega_\delta(u + u/m)\big] + u\Omega_\delta'(u)$ all vanish identically on $[0, \theta/2]$ and at $u = 1$. Differentiating $V_{m,\delta}(u) = (m+1)\Omega_\delta(u) - m\Omega_\delta\big(\frac{m+1}{m}u\big)$ and $\widetilde{\Omega}_\delta(u) = \Omega_\delta(u) - u\Omega_\delta'(u)$ directly gives $$V'_{m,\delta}(u) = (m+1)\left[\Omega_\delta'(u) - \Omega_\delta'\!\left(u + \frac{u}{m}\right)\right], \qquad \widetilde{\Omega}'_\delta(u) = -u\,\Omega_\delta''(u), \tag{31}$$ whose difference is the exact identity (26a). Rewriting (26a) by adding and subtracting $(m+1)\frac{u}{m}\Omega_\delta''(u)$ gives $$W'_{m,\delta}(u) = -(m+1)\left[\Omega_\delta'\!\left(u + \frac{u}{m}\right) - \Omega_\delta'(u) - \frac{u}{m}\,\Omega_\delta''(u)\right] - \frac{u}{m}\,\Omega_\delta''(u) = -(m+1)\int_u^{u+u/m} \left(u + \frac{u}{m} - t\right)\Omega_\delta'''(t)\,dt - \frac{u}{m}\,\Omega_\delta''(u). \tag{32}$$ Since $\Omega_\delta \in C^{2,1}([0, \infty))$ has $\Omega_\delta'' = \Omega_\delta''' = 0$ a.e. outside $[\theta, 1]$ and $\int_u^{u+u/m} (u + u/m - t)\,dt = \frac{u^2}{2m^2}$, (32) yields for all $u \in [\theta/2, 1]$: $$\big|W'_{m,\delta}(u)\big| \le \frac{m+1}{2m^2}u^2\big\|\Omega_\delta'''\big\|_{L^\infty([\theta, 1])} + \frac{u}{m}\big\|\Omega_\delta''\big\|_{L^\infty([\theta, 1])}, \qquad \big\|W'_{m,\delta}\big\|_{L^\infty([\theta/2, 1])} \le \frac{1}{m}\left(\frac{1 + 1/m}{2}\big\|\Omega_\delta'''\big\|_{L^\infty([\theta, 1])} + \big\|\Omega_\delta''\big\|_{L^\infty([\theta, 1])}\right), \tag{32a}$$ while (31) similarly gives $\|V'_{m,\delta}\|_{L^\infty([\theta/2, 1])} \le \frac{m+1}{m}\|\Omega_\delta''\|_{L^\infty([\theta, 1])} \le 2\|\Omega_\delta''\|_{L^\infty([\theta, 1])}$ and $\|\widetilde{\Omega}'_\delta\|_{L^\infty([\theta/2, 1])} \le \|\Omega_\delta''\|_{L^\infty([\theta, 1])}$. For any piecewise $C^1$ function $V$ on $[0, 1]$ with $V(u) = V(\theta/2)$ on $[0, \theta/2]$ and $V(1) = 0$, summation by parts (Abel summation) gives $M_V(X) = -\int_{\theta/2}^1 m_{X u}\,V'(u)\,du$. Applying this identity to $V_{m,\delta}$, $\widetilde{\Omega}_\delta$, and $W_{m,\delta}$ and using (32a) establishes all three inequalities in (27).

Part 2: Since $m_y = 0$ for $0 < y < 1$ and the classical Möbius-Landau inequality gives $|m_y| = \big|\sum_{1 \le j \le y} \mu(j)/j\big| \le 1$ for all $y \ge 1$, the exact identity (26) and Abel summation $M_{V_{m,\delta}}(X_m) = -\int_{\frac{m}{m+1}\theta}^1 m_{X_m u}\,V'_{m,\delta}(u)\,du$ imply $$m(m+1)\big|\eta_m^{\mathcal{F}} - \eta_{m+1}^{\mathcal{F}}\big| = \big|M_{V_{m,\delta}}(X_m)\big| \le \int_{\frac{m}{m+1}\theta}^1 \big|V'_{m,\delta}(u)\big|\,du.$$ Writing $V'_{m,\delta}(u) = (m+1)\big[\Omega_\delta'(u) - \Omega_\delta'(u + u/m)\big] = -(m+1)\int_u^{(1+1/m)u} \Omega_\delta''(t)\,dt$ and noting that $\Omega_\delta''(t)$ is supported on $[\theta, 1]$, Fubini-Tonelli (since $u \le t \le (1+1/m)u \iff \frac{m}{m+1}t \le u \le t$) gives, for every $1 \le m \le N_\delta$: $$\int_{\frac{m}{m+1}\theta}^1 \big|V'_{m,\delta}(u)\big|\,du \le (m+1)\int_\theta^1 \big|\Omega_\delta''(t)\big|\left(t - \frac{m}{m+1}t\right)dt = \int_\theta^1 t\,\big|\Omega_\delta''(t)\big|\,dt = \big\|\widetilde{\Omega}'_\delta\big\|_{L^1([\theta, 1])} = C_\delta,$$ proving (28) (and at $N = 40, \delta = 1/4, N_\delta = 50, m = 45, \Omega = W_{3, 4/5}$, exact rational evaluation gives $45^2|c_{45}| = \frac{561034375}{65313422} \approx 8.589879 < 45\cdot 46|c_{45}| = \frac{112206875}{12778713} \approx 8.780765 \le C_{1/4} = \frac{135}{8} = 16.875$). Summing $|\eta_m^{\mathcal{F}} - \eta_{m+1}^{\mathcal{F}}|^2 / w_m^{(\sigma)}$ with $(w_m^{(\sigma)})^{-1} \le \frac{m^{2\sigma+1}}{1 - 2^{-2\sigma}}$ gives $\frac{C_\delta^2}{1 - 2^{-2\sigma}}\zeta(3 - 2\sigma)$.

Part 3: By the Prime Number Theorem with the de la Vallée Poussin remainder, $m_y \ll \exp(-c_0\sqrt{\log(y+2)})$ ($c_0 > 0$). Substituting this into (26) and the first inequality of (27) gives $|\eta_m^{\mathcal{F}} - \eta_{m+1}^{\mathcal{F}}| \ll_\delta m^{-2}\exp(-c_0\sqrt{\log(N/m + 2)})$ for all $1 \le m \le N_\delta$. Splitting $\sum_{m=1}^{N_\delta} \frac{|\eta_m^{\mathcal{F}} - \eta_{m+1}^{\mathcal{F}}|^2}{w_m^{(\sigma)}}$ at $m = \lfloor \sqrt{N} \rfloor$, the prefix $1 \le m \le \lfloor \sqrt{N} \rfloor$ is $O_{\sigma,\delta}(\exp(-c_0\sqrt{2\log N}))$ while the tail $m > \sqrt{N}$ is $O_{\sigma,\delta}\big(\sum_{m > \sqrt{N}} m^{2\sigma-3}\big) = O_{\sigma,\delta}(N^{-(1-\sigma)})$, proving (29). $\square$

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4. Full $N^1$ Power-Saving Asymptotic Norm Law and Infinitely-Often Linear-Support Rate Bounds

We now use the exact two-point identity (26), the Lipschitz-$C^{2,1}$ Abel error bound (27), and an explicit jump-plus-derivative Riemann-sum analysis to prove the $N^{-2(1-\sigma)}$ power-saving asymptotic law for $\sigma \in (\Theta, 1)$, together with the Gauss hypergeometric Mellin formula and both generic-real and certified-rational infinitely-often ($\limsup_{N\to\infty}$) rate bounds at strictly linear support $N_\delta = \lfloor(1+\delta)N\rfloor$.

Theorem 3 (Full $N^1$ Power-Saving Asymptotic Law and Generic Real / Certified Rational Infinitely-Often Rate Bounds). Let $\sigma \in (\Theta, 1)$ (where $\Theta := \sup\{\operatorname{Re}(\rho) : \zeta(\rho) = 0\} \in [1/2, 1)$) and $\delta \in (0, 1]$, and let $\Omega_\delta = W_{r+1, \theta} \in C^r([0, \infty)) \cap C^{r,1}([0, \infty))$ ($r \ge 2$) with fixed knot $\theta := \frac{1}{1+\delta}$.

1. (Exact $N^{-2(1-\sigma)}$ Power-Saving Asymptotic Law and Closed-Form Lower Bound): As $N \to \infty$, $\big\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2$ satisfies the asymptotic law $$\big\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2 = \mathcal{C}_{\sigma, \delta}\,N^{-2(1-\sigma)} + O_{\sigma,\delta}\!\left(N^{-2(1-\sigma) - (\sigma - \Theta)/2}\right), \qquad \mathcal{C}_{\sigma, \delta} := \frac{(1+\delta)^{-2(1-\sigma)}}{2\sigma}\int_1^\infty X^{1-2\sigma}\big|M_{\widetilde{\Omega}_\delta}(X)\big|^2\,dX, \tag{33}$$ where the asymptotic constant $\mathcal{C}_{\sigma, \delta} < \infty$ satisfies the strict closed-form polynomial lower bound $$\mathcal{C}_{\sigma, \delta} \ge \mathcal{C}_{\sigma, \delta}^{(1)} := \frac{(1+\delta)^{-2(1-\sigma)}}{2\sigma}\int_\theta^1 u^{2\sigma-3}\big|\widetilde{\Omega}_\delta(u)\big|^2\,du > \frac{1}{1 - (1+\delta)^{-2\sigma}}. \tag{34}$$ In particular, the ratio of the smoothed to the unsmoothed harmonic head norm squared decays as $\big\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2 / \|\mathcal{F}_N\|_{S_\sigma^*}^2 \sim \frac{2\sigma\,\zeta(4\sigma+2)}{\zeta(2\sigma+1)}\mathcal{C}_{\sigma, \delta}\,N^{-1}$.
2. (Gauss Hypergeometric Mellin Formula and Infinitely-Often Linear-Support Rate Realization for Generic Real Scales and the Certified Rational Scale $\delta = 1/4$): For $\operatorname{Re}(s) > -1$, the analytic continuation of the Mellin transform $\widehat{\Omega}_\delta(s) := -\frac{1}{s}\int_\theta^1 \Omega_\delta'(u)\,u^s\,du$ is given in closed form by Gauss's hypergeometric function: $$\widehat{\Omega}_\delta(s) = \frac{1}{s}\,{}_2F_1\!\left(-s,\; r+1;\; 2r+2;\; \frac{\delta}{1+\delta}\right) \qquad (\operatorname{Re}(s) > -1). \tag{35}$$
   • (a) Generic Real Scale: Define the countable exceptional set $\mathcal{E}_r := \big\{\delta \in (0, 1] : {}_2F_1(1-\rho, r+1; 2r+2; \frac{\delta}{1+\delta}) = 0 \text{ for some nontrivial zero } \rho \text{ of } \zeta(s)\big\}$. Then for every fixed real $\delta \in (0, 1] \setminus \mathcal{E}_r$ and every $\varepsilon > 0$, $\mathcal{F}_{N, \delta}^{\mathrm{smooth}} \in \operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$ satisfies the infinitely-often ($\limsup_{N\to\infty}$) rate bound $$\limsup_{N \to \infty} N^{2(\sigma - \Theta) + \varepsilon}\,d_N(\sigma)^2 \ge \limsup_{N \to \infty} N^{2(\sigma - \Theta) + \varepsilon}\,\frac{\big|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}(E)\big|^2}{\big\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2} = \infty \qquad \Big(\text{i.e., } d_N(\sigma)^2 = \Omega\!\left(N^{-2(\sigma - \Theta) - \varepsilon}\right)\Big). \tag{36}$$
   • (b) Certified Rational Scale $\delta = 1/4 \in \mathbb{Q}$ ($\theta = 4/5$, $r = 2$): For the explicit rational polynomial taper $\Omega_{1/4}(u) = W_{3, 4/5}(u) \in \mathbb{Q}[u]$ on $[4/5, 1]$, where $-W'_{3, 4/5}(u) = 150(5u - 4)^2(5 - 5u)^2 = 93750u^4 - 337500u^3 + 453750u^2 - 270000u + 60000$ ($\int_{4/5}^1 (-W'_{3, 4/5}(u))\,du = 1$), the Mellin transform $\widehat{W}_{3, 4/5}(s) = \frac{1}{s}\big(60000\frac{1-(4/5)^{s+1}}{s+1} - 270000\frac{1-(4/5)^{s+2}}{s+2} + 453750\frac{1-(4/5)^{s+3}}{s+3} - 337500\frac{1-(4/5)^{s+4}}{s+4} + 93750\frac{1-(4/5)^{s+5}}{s+5}\big)$ evaluated on the Rouché-Euler-Maclaurin certified enclosure $\rho_1 \in \frac{1}{2} + i[\gamma_0 - r_0, \gamma_0 + r_0]$ ($\gamma_0 := \frac{14134725141735}{10^{12}}, r_0 := 10^{-11}$) of the first critical-line zero of $\zeta(s)$ satisfies the 60-digit interval-certified non-vanishing bound $$\big|\widehat{W}_{3, 4/5}(\rho_1 - 1)\big| \in [0.06225173,\; 0.06225178] \subset [0.062251,\; 0.062252] > 0. \tag{36a}$$ Consequently, the purely rational functional $\mathcal{F}_{N, 1/4}^{\mathrm{smooth}}$ ($\alpha_n \in \mathbb{Q}$) unconditionally satisfies $\limsup_{N\to\infty} N^{1+\varepsilon}\big|M_{W_{3, 4/5}}(\lfloor 5N/4\rfloor+1)\big|^2 = \infty$ for every $\varepsilon > 0$, and for every $\sigma \in (\Theta, 1)$ certifies the unconditional critical-line rate bound $$\limsup_{N \to \infty} N^{2(\sigma - 1/2) + \varepsilon}\,d_N(\sigma)^2 \ge \limsup_{N \to \infty} N^{2(\sigma - 1/2) + \varepsilon}\,\frac{\big|\mathcal{F}_{N, 1/4}^{\mathrm{smooth}}(E)\big|^2}{\big\|\mathcal{F}_{N, 1/4}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2} = \infty \qquad \Big(\text{matching } \Omega\!\left(N^{-2(\sigma - \Theta) - \varepsilon}\right) \text{ whenever } \Theta = \frac{1}{2}\Big). \tag{36b}$$

Proof. Part 1: Since $\sigma > \Theta$, set $\varepsilon_0 := \frac{\sigma - \Theta}{4} > 0$. By Littlewood's equivalence between the zero-free half-plane $\operatorname{Re}(s) > \Theta$ and Möbius partial sums, $|m_y| \ll_{\varepsilon_0} y^{-(1-\Theta)+\varepsilon_0}$ for all $y \ge 1$. Substituting this bound into the multiplicative Lipschitz-$C^{2,1}$ Abel inequalities (27) gives, for all $1 \le m \le N_\delta$ (with $X_m = \frac{N_\delta + 1}{m}$): $$\big|M_{V_{m,\delta}}(X_m)\big| \ll_{\delta, \varepsilon_0} X_m^{-(1-\Theta)+\varepsilon_0}, \qquad \big|M_{\widetilde{\Omega}_\delta}(X_m)\big| \ll_{\delta, \varepsilon_0} X_m^{-(1-\Theta)+\varepsilon_0}, \qquad \big|M_{V_{m,\delta}}(X_m) - M_{\widetilde{\Omega}_\delta}(X_m)\big| \ll_{\delta, \varepsilon_0} \frac{1}{m}\,X_m^{-(1-\Theta)+\varepsilon_0}. \tag{37}$$ Combining the exact identity (26) with $\big[m^2(m+1)^2 w_m^{(\sigma)}\big]^{-1} = \frac{m^{2\sigma-3}}{2\sigma}\big(1 + O(m^{-1})\big)$ and writing $\big|M_{V_{m,\delta}}(X_m)\big|^2 - \big|M_{\widetilde{\Omega}_\delta}(X_m)\big|^2 = \big(M_{V_{m,\delta}}(X_m) - M_{\widetilde{\Omega}_\delta}(X_m)\big)\big(M_{V_{m,\delta}}(X_m) + M_{\widetilde{\Omega}_\delta}(X_m)\big)$ yields, via (37), for all $1 \le m \le N_\delta$: $$\frac{\big|\eta_m^{\mathcal{F}} - \eta_{m+1}^{\mathcal{F}}\big|^2}{w_m^{(\sigma)}} = \frac{m^{2\sigma-3}}{2\sigma}\big|M_{\widetilde{\Omega}_\delta}(X_m)\big|^2 + O_{\sigma,\delta}\!\left(m^{2\sigma-4}\,X_m^{-2(1-\Theta)+2\varepsilon_0}\right). \tag{38}$$ Summing the error term in (38) over all $1 \le m \le N_\delta$ gives $$\sum_{m=1}^{N_\delta} m^{2\sigma-4}\,X_m^{-2(1-\Theta)+2\varepsilon_0} = (N_\delta + 1)^{-2(1-\Theta)+2\varepsilon_0}\sum_{m=1}^{N_\delta} m^{2(\sigma - \Theta) - 2 - 2\varepsilon_0}. \tag{39}$$ Since $\sigma < 1$ and $\Theta \ge 1/2$, we have $\sigma - \Theta < 1/2$, so the exponent of $m$ in (39) satisfies $2(\sigma - \Theta) - 2 - 2\varepsilon_0 = \frac{3}{2}(\sigma - \Theta) - 2 < -\frac{5}{4} < -1$. Thus the series $\sum_{m=1}^\infty m^{\frac{3}{2}(\sigma-\Theta)-2} < \infty$ converges absolutely, and the total error sum (39) is $O_{\sigma,\delta}\big(N^{-2(1-\Theta)+2\varepsilon_0}\big) = O_{\sigma,\delta}\big(N^{-2(1-\sigma) - \frac{3}{2}(\sigma-\Theta)}\big)$.

Next, to compare the discrete sum $\sum_{m=1}^{N_\delta} \frac{m^{2\sigma-3}}{2\sigma}\big|M_{\widetilde{\Omega}_\delta}(X_m)\big|^2$ with the integral in (33), we verify the regularity of $M_{\widetilde{\Omega}_\delta}(X)$ explicitly. Since $\Omega_\delta = W_{r+1, \theta} \in C^r([0, \infty))$ with $r \ge 2$, $\widetilde{\Omega}_\delta(u) = \Omega_\delta(u) - u\Omega_\delta'(u) \in C^{r-1}([0, 1]) \subset C^1([0, 1])$. Moreover, at $u = 1$, because $\Omega_\delta(1) = \Omega_\delta'(1) = \Omega_\delta''(1) = 0$ ($r \ge 2$), we have both $\widetilde{\Omega}_\delta(1) = 0$ and $\widetilde{\Omega}'_\delta(1) = -\Omega_\delta''(1) = 0$. Consequently, the extension of $\widetilde{\Omega}_\delta$ by $0$ to $(0, \infty)$ is globally $C^1$ on $(0, \infty)$ with $\widetilde{\Omega}'_\delta(u) = -u\Omega_\delta''(u)$ supported on $[\theta, 1]$ and piecewise $C^1$ (since $\Omega_\delta \in C^{2,1}$). Therefore:

1. (Zero Jumps of $M_{\widetilde{\Omega}_\delta}(X)$): As $X$ crosses an integer $j \ge 1$, the newly entering term $\frac{\mu(j)}{j}\widetilde{\Omega}_\delta(j/X)$ evaluates at $u = 1$ where $\widetilde{\Omega}_\delta(1) = 0$, so $X \mapsto M_{\widetilde{\Omega}_\delta}(X) = -\int_{\theta}^{1} m_{X v}\,\widetilde{\Omega}'_\delta(v)\,dv$ is continuous for all $X \ge 1$.
2. (Derivative Bound for $M'_{\widetilde{\Omega}_\delta}(X)$): For non-integer $X \ge 1$, differentiating $M_{\widetilde{\Omega}_\delta}(X) = \sum_{j \le X} \frac{\mu(j)}{j}\widetilde{\Omega}_\delta(j/X)$ term-by-term gives $$X\,M'_{\widetilde{\Omega}_\delta}(X) = \sum_{j \le X} \frac{\mu(j)}{j}\,U_\delta\!\left(\frac{j}{X}\right), \qquad U_\delta(u) := -u\,\widetilde{\Omega}'_\delta(u) = u^2\,\Omega_\delta''(u). \tag{39a}$$ Since $U_\delta(u) = u^2\Omega_\delta''(u)$ vanishes identically on $[0, \theta]$ and at $u = 1$ ($U_\delta(1) = \Omega_\delta''(1) = 0$), and $U'_\delta(u) = 2u\Omega_\delta''(u) + u^2\Omega_\delta'''(u) \in L^\infty([\theta, 1])$, Abel summation gives $X\,M'_{\widetilde{\Omega}_\delta}(X) = -\int_\theta^1 m_{X v}\,U'_\delta(v)\,dv$, and hence $$\big|M'_{\widetilde{\Omega}_\delta}(X)\big| \le \frac{\|U'_\delta\|_{L^\infty([\theta, 1])}}{X}\sup_{y \ge \theta X} |m_y| \ll_{\delta, \varepsilon_0} X^{-(2 - \Theta) + \varepsilon_0} \qquad (\text{for a.e. } X \ge 1). \tag{39b}$$ Setting $L := N_\delta + 1$ and $F(X) := X^{1-2\sigma}\big|M_{\widetilde{\Omega}_\delta}(X)\big|^2$, (37) and (39b) imply $|F(X)| \ll_{\delta,\varepsilon_0} X^{2\Theta-1-2\sigma+2\varepsilon_0}$ and $|F'(X)| \ll_{\sigma,\delta,\varepsilon_0} X^{2\Theta-2-2\sigma+2\varepsilon_0}$ a.e. on $[1, \infty)$. Factoring out $\frac{L^{-2(1-\sigma)}}{2\sigma}$ from the discrete sum gives $$\sum_{m=1}^{N_\delta} \frac{m^{2\sigma-3}}{2\sigma}\big|M_{\widetilde{\Omega}_\delta}(X_m)\big|^2 = \frac{L^{-2(1-\sigma)}}{2\sigma}\sum_{m=1}^{N_\delta} F(X_m)\,\Delta X_m\left(1 + \frac{1}{m}\right), \qquad \Delta X_m := X_m - X_{m+1} = \frac{L}{m(m+1)} = \frac{X_m}{m+1}.$$ On each interval $[X_{m+1}, X_m]$, comparing $F(X_m)\,\Delta X_m(1 + 1/m)$ with $\int_{X_{m+1}}^{X_m} F(X)\,dX$ produces a local quadrature error bounded by $$\frac{\Delta X_m}{m}|F(X_m)| + (\Delta X_m)^2 \sup_{[X_{m+1}, X_m]} |F'| \ll_{\sigma,\delta,\varepsilon_0} \frac{L}{m^3}\left(\frac{L}{m}\right)^{2\Theta-1-2\sigma+2\varepsilon_0} + \frac{L^2}{m^4}\left(\frac{L}{m}\right)^{2\Theta-2-2\sigma+2\varepsilon_0} \ll_{\sigma,\delta,\varepsilon_0} L^{2\Theta-2\sigma+2\varepsilon_0}\,m^{2(\sigma-\Theta)-2-2\varepsilon_0}.$$ Summing over $1 \le m \le N_\delta$ (again convergent since $2(\sigma-\Theta)-2-2\varepsilon_0 < -1$) and adding the tail integral $\int_{X_1}^\infty F(X)\,dX + \int_1^{X_{N_\delta+1}} F(X)\,dX \ll_{\sigma,\delta,\varepsilon_0} L^{2\Theta-2\sigma+2\varepsilon_0} + L^{-1}$ yields $\sum_{m=1}^{N_\delta} F(X_m)\,\Delta X_m(1+1/m) = \int_1^\infty F(X)\,dX + O_{\sigma,\delta}\big(L^{-2(\sigma-\Theta)+2\varepsilon_0} + L^{-1}\big)$. Multiplying by the prefactor $\frac{L^{-2(1-\sigma)}}{2\sigma} = \frac{(1+\delta)^{-2(1-\sigma)}N^{-2(1-\sigma)}}{2\sigma}\big(1 + O(N^{-1})\big)$ yields an error of $O_{\sigma,\delta}\big(N^{-2(1-\sigma) - \frac{3}{2}(\sigma-\Theta)} + N^{-2(1-\sigma)-1}\big) \subset O_{\sigma,\delta}\big(N^{-2(1-\sigma) - (\sigma-\Theta)/2}\big)$, establishing (33). Restricting $\int_1^\infty$ to $X \in [1, 1+\delta]$ gives $\mathcal{C}_{\sigma,\delta}^{(1)}$ in (34).

Part 2: For $\Omega_\delta = W_{r+1, \theta}$, on $(\theta, 1)$ the derivative is $-\Omega_\delta'(u) = \frac{(r+1)\binom{2r+1}{r+1}}{1-\theta}\tau^r(1-\tau)^r$ where $\tau = \frac{u-\theta}{1-\theta} \in (0, 1)$. Substituting $u = \theta + (1-\theta)\tau$ and $\theta = (1+\delta)^{-1}$ into $-\int_\theta^1 \Omega_\delta'(u)\,u^s\,du$ and using Euler's integral representation of ${}_2F_1$ gives $$s\,\widehat{\Omega}_\delta(s) = (r+1)\binom{2r+1}{r+1}\int_0^1 \tau^r(1-\tau)^r\big(1 - (1-\theta)(1-\tau)\big)^s\,d\tau = {}_2F_1\!\left(-s,\; r+1;\; 2r+2;\; \frac{\delta}{1+\delta}\right), \tag{40}$$ proving (35). For each nontrivial zero $\rho$ of $\zeta(s)$, $z \mapsto {}_2F_1(1-\rho, r+1; 2r+2; z)$ is holomorphic on $|z| < 1$ and equals $1 \ne 0$ at $z = 0$, so its zero set in $(0, 1/2]$ is finite and $\mathcal{E}_r \subset (0, 1]$ is at most countable.

For Part 2(a), fix any $\delta \in (0, 1] \setminus \mathcal{E}_r$ and $\varepsilon \in (0, 1/2)$, and choose a nontrivial zero $\rho_0 = \beta_0 + i\gamma_0$ of $\zeta(s)$ with $\beta_0 > \Theta - \varepsilon/4$; then $\widehat{\Omega}_\delta(\rho_0 - 1) \ne 0$ by definition of $\mathcal{E}_r$. For Part 2(b), at the explicit rational scale $\delta = 1/4$ ($\theta = 4/5, r = 2$), since $\tau = 5u - 4$ and $1 - \tau = 5 - 5u$, expanding $-\Omega'_{1/4}(u) = 150(5u - 4)^2(5 - 5u)^2 = 93750u^4 - 337500u^3 + 453750u^2 - 270000u + 60000$ on $[4/5, 1]$ and integrating term-by-term against $u^s$ gives the closed-form rational-exponential expression for $\widehat{W}_{3, 4/5}(s)$. To certify (36a) without unverified floating-point zero approximations, we first enclose $\rho_1$ in 60-digit mpmath.iv interval arithmetic (src/verify_smoothed_harmonic_head_duality.py) via Euler-Maclaurin summation ($M = 50, 2k = 12$, with rigorous Bernoulli remainder bound) on the closed box $B(s_0, r_0) := [1/2 - r_0, 1/2 + r_0] + i[\gamma_0 - r_0, \gamma_0 + r_0]$ centered at the exact rational point $s_0 = \frac{1}{2} + i\gamma_0$ ($\gamma_0 := \frac{14134725141735}{10^{12}}, r_0 := 10^{-11}$). This certifies $|\zeta(s_0)| \le 2.4337 \times 10^{-13}$ and $|\zeta(s_0)| + r_0 \sup_{w \in B(s_0, r_0)} |\zeta'(w) - \zeta'(s_0)| \le 2.4337 \times 10^{-13} < 7.9316 \times 10^{-12} \le r_0|\zeta'(s_0)|$. By Rouché's theorem on the disk $D(s_0, r_0) \subset B(s_0, r_0)$, $\zeta(s)$ has a unique simple zero $\rho_1 \in D(s_0, r_0)$; since $D(s_0, r_0)$ is invariant under $s \mapsto 1 - \bar{s}$ and $\zeta(1 - \bar{\rho}_1) = 0$ by the functional equation, uniqueness forces $\rho_1 = 1 - \bar{\rho}_1$, so $\operatorname{Re}(\rho_1) = 1/2$ and $\gamma_1 := \operatorname{Im}(\rho_1) \in [\gamma_0 - r_0, \gamma_0 + r_0]$. Propagating $s = \rho_1 - 1 \in -\frac{1}{2} + i[\gamma_0 - r_0, \gamma_0 + r_0]$ into the normalized 5-term formula for $\widehat{W}_{3, 4/5}(s)$ in 60-digit mpmath.iv arithmetic certifies $\big|\widehat{W}_{3, 4/5}(\rho_1 - 1)\big| \in [0.06225173, 0.06225178] \subset [0.062251, 0.062252] > 0$, proving (36a). In either case (with $\beta_0 > \Theta - \varepsilon/4$ in Part 2(a), or $\rho_0 = \rho_1$ with $\beta_0 = 1/2$ in Part 2(b)), suppose for contradiction that $\big|M_{\Omega_\delta}(N_\delta + 1)\big| \ll N^{-(1-\beta_0)-\varepsilon/4}$ for all $N \ge 2$. For any real $X \ge 3$, choosing $N \ge 2$ with $0 \le X - (N_\delta + 1) \le 2$, the mean value theorem gives $$\big|M_{\Omega_\delta}(X) - M_{\Omega_\delta}(N_\delta + 1)\big| \le \sum_{n=1}^{N_\delta} \frac{1}{n}\left|\Omega_\delta\!\left(\frac{n}{X}\right) - \Omega_\delta\!\left(\frac{n}{N_\delta+1}\right)\right| + \sum_{N_\delta < n \le X} \frac{1}{n}\left|\Omega_\delta\!\left(\frac{n}{X}\right)\right| \ll_\delta X^{-1}. \tag{41}$$ Since $1 - \beta_0 + \varepsilon/4 < 3/4 < 1$, (41) implies $M_{\Omega_\delta}(X) = O\big(X^{-(1-\beta_0)-\varepsilon/4}\big)$ for all real $X \ge 1$, so $\int_1^\infty M_{\Omega_\delta}(X)\,X^{-s-1}\,dX = \frac{\widehat{\Omega}_\delta(s)}{\zeta(1+s)}$ is holomorphic on $\operatorname{Re}(s) > -(1-\beta_0) - \varepsilon/4$, contradicting the pole at $s = \rho_0 - 1$ ($\operatorname{Re}(s) = -(1-\beta_0)$) where $\widehat{\Omega}_\delta(\rho_0 - 1) \ne 0$. Dividing by (33) establishes (36) and (36b). $\square$

4.1 Exact Rational Critical-Line Certificates and $N^1$ Power-Saving Progression

With the fixed knot $\theta = \frac{1}{1+\delta} \in \mathbb{Q}$, the coefficients $\alpha_n = \frac{1}{n}W_{3, \theta}\big(\frac{n}{N_\delta+1}\big) \in \mathbb{Q}$ are exact rational numbers for all $1 \le n \le N_\delta$. At $\sigma = 1/2$ ($w_m^{(1/2)} = \frac{1}{m(m+1)} \in \mathbb{Q}$) and $\delta = 1/4$ ($\theta = 4/5$, $N_\delta = 5N/4$), fractions.Fraction arithmetic in src/verify_smoothed_harmonic_head_duality.py certifies both exact annihilation $\mathcal{F}_{N, 1/4}^{\mathrm{smooth}}(\tilde{\rho}_{1/k}) \equiv 0 \in \mathbb{Q}$ ($2 \le k \le N$) and the exact rational dual norm squared $\big\|\mathcal{F}_{N, 1/4}^{\mathrm{smooth}}\big\|_{S_{1/2}^*}^2 \in \mathbb{Q}$ (with strict outward rounding via ROUND_FLOOR and ROUND_CEILING):

• $N = 40$ ($N_\delta = 50, \theta = 4/5$): $\mathcal{F}_{40, 1/4}^{\mathrm{smooth}}(\tilde{\rho}_{1/k}) \equiv 0$ ($2 \le k \le 40$) and $0.304832 \le \big\|\mathcal{F}_{40, 1/4}^{\mathrm{smooth}}\big\|_{S_{1/2}^*}^2 \le 0.304833$ ($\approx 0.3048325818$);
• $N = 80$ ($N_\delta = 100, \theta = 4/5$): $\mathcal{F}_{80, 1/4}^{\mathrm{smooth}}(\tilde{\rho}_{1/k}) \equiv 0$ ($2 \le k \le 80$) and $0.175432 \le \big\|\mathcal{F}_{80, 1/4}^{\mathrm{smooth}}\big\|_{S_{1/2}^*}^2 \le 0.175433$;
• $N = 160$ ($N_\delta = 200, \theta = 4/5$): $\mathcal{F}_{160, 1/4}^{\mathrm{smooth}}(\tilde{\rho}_{1/k}) \equiv 0$ ($2 \le k \le 160$) and $0.101725 \le \big\|\mathcal{F}_{160, 1/4}^{\mathrm{smooth}}\big\|_{S_{1/2}^*}^2 \le 0.101726$.

Moreover, expanding $\big(\widetilde{\Omega}_{1/4}(u)\big)^2 = \sum_{\ell=0}^{10} d_\ell u^\ell \in \mathbb{Q}[u]$ on $[4/5, 1]$ and evaluating $\mathcal{C}_{\sigma, 1/4}^{(1)} = \frac{(5/4)^{-2(1-\sigma)}}{2\sigma}\sum_{\ell=0}^{10} d_\ell \frac{1 - (4/5)^{2\sigma-2+\ell}}{2\sigma-2+\ell}$ in 60-digit mpmath.iv arithmetic certifies the rigorous analytical lower bounds $\mathcal{C}_{0.60, 1/4} \ge \mathcal{C}_{0.60, 1/4}^{(1)} \ge 5.723138$, $\mathcal{C}_{0.75, 1/4} \ge \mathcal{C}_{0.75, 1/4}^{(1)} \ge 4.736843$, and $\mathcal{C}_{0.90, 1/4} \ge \mathcal{C}_{0.90, 1/4}^{(1)} \ge 4.084341$ (while numerical quadrature of (33) on $X \in [1, 2000]$ yields the conditional/numerical estimates $\mathcal{C}_{0.60, 1/4}^{\mathrm{num}} \approx 15.934$, $\mathcal{C}_{0.75, 1/4}^{\mathrm{num}} \approx 8.795$, and $\mathcal{C}_{0.90, 1/4}^{\mathrm{num}} \approx 6.184$ under $\sigma > \Theta$).

Table 2 illustrates the numerical progression of $\big\|\mathcal{F}_{N, 1/4}^{\mathrm{smooth}}\big\|_{S_{0.75}^*}^2$ ($\theta = 4/5$) versus the unsmoothed $\|\mathcal{F}_N\|_{S_{0.75}^*}^2$ across $N \in \{40, 80, 160, 320, 640, 1280, 2560, 5120\}$.

| $N$ ($N_\delta = 1.25N$) | Unsmoothed $\|\mathcal{F}_N\|_{S_{0.75}^*}^2$ ($\propto N^{+0.5}$) | Smoothed $\|\mathcal{F}_{N, 1/4}^{\mathrm{smooth}}\|_{S_{0.75}^*}^2$ ($\propto N^{-0.5}$) | Norm Ratio $\frac{\|\mathcal{F}_{N, 1/4}^{\mathrm{smooth}}\|^2}{\|\mathcal{F}_N\|^2}$ ($\propto N^{-1}$) | Rescaled $N^{0.5}\|\mathcal{F}_{N, 1/4}^{\mathrm{smooth}}\|_{S_{0.75}^*}^2$ ($\to \mathcal{C}_{0.75, 1/4}^{\mathrm{num}} \approx 8.795$) | Unsmoothed Scaled Bound $\frac{|m_N|^2}{\|\mathcal{F}_N\|^2}N^{0.5}$ | Smoothed Scaled Bound $\frac{|M_{\Omega_{1/4}}|^2}{\|\mathcal{F}_{N, 1/4}^{\mathrm{smooth}}\|^2}N^{0.5}$ | Rayleigh Gain Factor | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $40$ ($50$) | $5.5386$ | $1.1036$ | $1.99 \times 10^{-1}$ | $6.9800$ | $2.91 \times 10^{-3}$ | $0.001290$ | $0.44\times$ | | $80$ ($100$) | $7.7907$ | $0.8346$ | $1.07 \times 10^{-1}$ | $7.4651$ | $7.33 \times 10^{-4}$ | $0.001196$ | $1.63\times$ | | $160$ ($200$) | $10.9747$ | $0.6230$ | $5.68 \times 10^{-2}$ | $7.8808$ | $1.90 \times 10^{-4}$ | $0.000905$ | $4.76\times$ | | $320$ ($400$) | $15.4699$ | $0.4598$ | $2.97 \times 10^{-2}$ | $8.2244$ | $5.56 \times 10^{-5}$ | $0.000992$ | $17.84\times$ | | $640$ ($800$) | $21.8445$ | $0.3312$ | $1.52 \times 10^{-2}$ | $8.3785$ | $5.82 \times 10^{-6}$ | $0.001174$ | $201.79\times$ | | $1280$ ($1600$) | $30.8631$ | $0.2390$ | $7.74 \times 10^{-3}$ | $8.5519$ | $1.81 \times 10^{-6}$ | $0.003178$ | $1,752.56\times$ | | $2560$ ($3200$) | $43.6491$ | $0.1716$ | $3.93 \times 10^{-3}$ | $8.6811$ | $1.42 \times 10^{-6}$ | $0.004605$ | $3,244.42\times$ | | $5120$ ($6400$) | $61.7227$ | $0.1221$ | $1.98 \times 10^{-3}$ | $8.7403$ | $3.50 \times 10^{-8}$ | $0.000663$ | $18,944.07\times$ |

Table 2: Numerical illustration of the full $N^1$ power-saving law (Theorem 3, conditional on $\sigma > \Theta$) comparing the unsmoothed harmonic head functional $\mathcal{F}_N$ against the smoothed harmonic head functional $\mathcal{F}_{N, 1/4}^{\mathrm{smooth}}$ with fixed knot $\theta = 4/5$ ($\Omega_{1/4} = W_{3, 4/5}$) at $\sigma = 0.75$ (results/smoothed_harmonic_head_duality.json).

Figure 1: (a) Full N^1 power-saving norm law comparing unsmoothed ||F_N||^2 against smoothed ||F_{N, 1/4}^{smooth}||^2 (theta = 4/5), variational ||F_{N, 1/2}^{var}||^2, and the universal Cauchy-Schwarz lower bound at sigma = 0.75 (Theorems 1-3). (b) Convergence of N^{2(1-sigma)} ||F_{N, 1/4}^{smooth}||^2 to the numerical quadrature estimate C_{sigma, 1/4}^{num} across sigma in {0.60, 0.75, 0.90} (conditional on sigma > Theta in Theorem 3).

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5. Multi-Scale Zero-Independent Harmonic Head Superposition and 60-Digit Interval Certificates

Because a single fixed scale $\delta > 0$ has an oscillating numerator $M_{\Omega_\delta}(N_\delta + 1)$ that can be small near sign changes (such as $N = 40$ and $N = 5120$ in Table 2), Theorem 3(2) yields an infinitely-often ($\limsup_{N\to\infty}$) rate bound rather than an all-$N$ bound. Applying the finite-dimensional Gram-inverse superposition method of [23, Theorem 3] inside the finite-support annihilator subspace $\operatorname{Ann}_{S_\sigma^*}^{(M)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}) = \operatorname{span}\{\mathcal{F}_N, \mathcal{C}_{N+1}, \dots, \mathcal{C}_M\}$ ([17, Section 2.1]) to a multi-scale family of zero-independent smoothed harmonic head functionals $\big\{\mathcal{F}_{N, \delta_j}^{\mathrm{smooth}}\big\}_{j=1}^K$ (constructed without specifying zeta zeros) with fixed knots $\theta_j = \frac{1}{1+\delta_j}$ yields exact rational certificates that substantially improve single-scale bounds at finite $N$.

Theorem 4 (Multi-Scale Zero-Independent Smoothed Harmonic Head Superposition and Analytical Subspace Bounds). Let $\sigma \in [1/2, 1)$ and $M := \lfloor \Lambda N \rfloor > N \ge 2$, and let $d_{N, M}(\sigma)^2 := \sup\big\{|\mathcal{L}(E)|^2 / \|\mathcal{L}\|_{S_\sigma^*}^2 : \mathcal{L} \in \operatorname{Ann}_{S_\sigma^*}^{(M)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}) \setminus \{0\}\big\} \le d_N(\sigma)^2$.

1. (Exact Rational Multi-Scale Zero-Independent Superposition): For any $K \ge 1$ rational dilation scales $0 < \delta_1 < \delta_2 < \dots < \delta_K \le \Lambda - 1$ ($N_{\delta_j} := \lfloor(1+\delta_j)N\rfloor \le M$) and rational polynomial tapers $\Omega_{\delta_j} = W_{3, \theta_j}$ with fixed knots $\theta_j := \frac{1}{1+\delta_j} \in \mathbb{Q}$, every linear combination $\mathcal{L}_\lambda := \sum_{j=1}^K \lambda_j \mathcal{F}_{N, \delta_j}^{\mathrm{smooth}}$ ($\lambda \in \mathbb{Q}^K \setminus \{0\}$) has exact rational coordinates $\alpha_n(\mathcal{L}_\lambda) \in \mathbb{Q}$, satisfies $\mathcal{L}_\lambda(\tilde{\rho}_{1/k}) = 0$ identically in $\mathbb{Q}$ for all $2 \le k \le N$, and certifies $$d_N(\sigma)^2 \ge d_{N, M}(\sigma)^2 \ge \frac{\big|\mathcal{L}_\lambda(E)\big|^2}{\|\mathcal{L}_\lambda\|_{S_\sigma^*}^2} = \frac{\big(\sum_{j=1}^K \lambda_j M_{\Omega_{\delta_j}}(N_{\delta_j} + 1)\big)^2}{\sum_{m=1}^M \frac{1}{w_m^{(\sigma)}}\big(\sum_{j=1}^K \lambda_j (\eta_{m,j}^{\mathcal{F}} - \eta_{m+1,j}^{\mathcal{F}})\big)^2}. \tag{42}$$
2. (Gram-Inverse, Max-Scale, and Trace-Quadrature Subspace Bounds): Let $\mathcal{V}_K := \operatorname{span}\{\mathcal{F}_{N, \delta_1}^{\mathrm{smooth}}, \dots, \mathcal{F}_{N, \delta_K}^{\mathrm{smooth}}\} \subset \operatorname{Ann}_{S_\sigma^*}^{(M)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$, with $K \times K$ Gram matrix $\big(G_{\mathcal{V}_K}\big)_{i,j} := \big\langle \mathcal{F}_{N, \delta_i}^{\mathrm{smooth}},\, \mathcal{F}_{N, \delta_j}^{\mathrm{smooth}} \big\rangle_{S_\sigma^*}$ and target vector $b_{\mathcal{V}_K} := \big(M_{\Omega_{\delta_1}}(N_{\delta_1}+1), \dots, M_{\Omega_{\delta_K}}(N_{\delta_K}+1)\big)^T$. Whenever $G_{\mathcal{V}_K}$ is nonsingular, the supremum of (42) over $\lambda \in \mathbb{R}^K \setminus \{0\}$ equals $b_{\mathcal{V}_K}^T G_{\mathcal{V}_K}^{-1} b_{\mathcal{V}_K}$ and satisfies $$d_N(\sigma)^2 \ge d_{N, M}(\sigma)^2 \ge b_{\mathcal{V}_K}^T G_{\mathcal{V}_K}^{-1} b_{\mathcal{V}_K} \ge \max\left\{\max_{1 \le j \le K} \frac{\big|M_{\Omega_{\delta_j}}(N_{\delta_j}+1)\big|^2}{\big\|\mathcal{F}_{N, \delta_j}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2},\;\; \frac{\sum_{j=1}^K \big|M_{\Omega_{\delta_j}}(N_{\delta_j}+1)\big|^2}{\sum_{j=1}^K \big\|\mathcal{F}_{N, \delta_j}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2}\right\}, \tag{43}$$ where for $\sigma \in (\Theta, 1)$, the trace-quadrature denominator obeys $\sum_{j=1}^K \big\|\mathcal{F}_{N, \delta_j}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2 = \big(\sum_{j=1}^K \mathcal{C}_{\sigma, \delta_j} + o(1)\big) N^{-2(1-\sigma)}$.

Proof. Part 1: By Theorem 2(1), for each $1 \le j \le K$, $\theta_j = \frac{1}{1+\delta_j} \in \mathbb{Q}$ implies $\alpha_n\big(\mathcal{F}_{N, \delta_j}^{\mathrm{smooth}}\big) = \frac{1}{n}W_{3, \theta_j}\big(\frac{n}{N_{\delta_j}+1}\big) \in \mathbb{Q}$ and $\mathcal{F}_{N, \delta_j}^{\mathrm{smooth}} \in \operatorname{Ann}_{S_\sigma^*}^{(N_{\delta_j})}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}) \subset \operatorname{Ann}_{S_\sigma^*}^{(M)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$. By linearity and (4), (42) follows.

Part 2: Restricting the supremum defining $d_{N, M}(\sigma)^2$ to $\mathcal{L}_\lambda \in \mathcal{V}_K$ gives $\sup_{\lambda \ne 0} \frac{(\lambda^T b_{\mathcal{V}_K})^2}{\lambda^T G_{\mathcal{V}_K} \lambda} = b_{\mathcal{V}_K}^T G_{\mathcal{V}_K}^{-1} b_{\mathcal{V}_K}$ (attained at $\lambda_{\mathrm{opt}} = G_{\mathcal{V}_K}^{-1} b_{\mathcal{V}_K}$). Testing $\lambda = e_j$ ($1 \le j \le K$) gives the max-scale lower bound in (43), while $\lambda_{\max}(G_{\mathcal{V}_K}) \le \operatorname{Tr}(G_{\mathcal{V}_K}) = \sum_{j=1}^K \big\|\mathcal{F}_{N, \delta_j}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2$ yields $b_{\mathcal{V}_K}^T G_{\mathcal{V}_K}^{-1} b_{\mathcal{V}_K} \ge \|b_{\mathcal{V}_K}\|_2^2 / \lambda_{\max}(G_{\mathcal{V}_K}) \ge \|b_{\mathcal{V}_K}\|_2^2 / \operatorname{Tr}(G_{\mathcal{V}_K})$, and Theorem 3(1) gives the asymptotic of $\operatorname{Tr}(G_{\mathcal{V}_K})$. $\square$

Remark 1 (Motivation for Multi-Scale Superposition and Remaining Open Problem for All-$N$ Bounds). By Theorem 3(2)(b), any rational scale grid $\Delta_K$ containing $\delta = 1/4$ (such as $\Delta_8$ and $\Delta_{16}$ below) unconditionally achieves the critical-line $\limsup_{N\to\infty}$ rate exponent $N^{-2(\sigma-1/2)-\varepsilon}$ via the max-scale bound in (43) (matching $N^{-2(\sigma-\Theta)-\varepsilon}$ whenever $\Theta = 1/2$, while every generic real scale $\delta \in (0, 1] \setminus \mathcal{E}_r$ achieves $\Omega(N^{-2(\sigma-\Theta)-\varepsilon})$ unconditionally by Theorem 3(2)(a)). Moreover, at finite $N$, because all $\mathcal{F}_{N, \delta_j}^{\mathrm{smooth}}$ share the exact same interior step coefficients $c_m = \frac{1}{m(m+1)}$ on $\lfloor M/2 \rfloor < m \le N - 1$, differences $\mathcal{F}_{N, \delta_{j+1}}^{\mathrm{smooth}} - \mathcal{F}_{N, \delta_j}^{\mathrm{smooth}}$ cancel the common interior step norm. Analytically, however, upgrading the $\limsup_{N\to\infty}$ bounds (36)–(36b) to a uniform all-$N$ ($\liminf_{N\to\infty}$) lower bound for $\sum_{j=1}^K |M_{\Omega_{\delta_j}}(N_{\delta_j}+1)|^2$ remains open unconditionally; Theorem 4 is therefore used below to produce unconditional, interval-certified finite-$N$ lower bounds.

5.1 60-Digit Interval Certification of Zero-Independent Multi-Scale Superpositions

Define the nested rational scale grids $\Delta_8 := \{j/8 : 1 \le j \le 8\} \subset (0, 1]$ ($M = 2N$, with fixed knots $\theta_j = \frac{8}{8+j}$) and $\Delta_{16} := \Delta_8 \cup \{1 + j/4 : 1 \le j \le 8\} \subset (0, 3]$ ($M = 4N$, with fixed knots $\theta_{8+j} = \frac{4}{8+j}$). Table 3 presents:

1. Double-precision numerical reference evaluations of the finite-support subspace distances $F_{\mathrm{unnorm}, N, M}(0.75) := d_{\mathrm{unnorm}, N, M}(0.75)^2 N^{0.5}$ for $M \in \{2N, 4N\}$ and the infinite-dimensional reference distance $F_{\mathrm{unnorm}, \mathrm{opt}}^{\mathrm{num}}(N)$ ($M = \infty$, computed via double-precision Hurwitz-Vasyunin Gram solves) across $N \in \{40, 80, 160, 320\}$;
2. The nested multi-scale zero-independent superposition progression across $N \in \{40, 80, 160, 320, 640, 1280, 2560, 5120\}$; and
3. Our exact rational + 60-digit mpmath.iv interval certificates at $N \in \{40, 80, 160\}$ (marked with $*$, rounded strictly downward via ROUND_FLOOR).

| $N$ | Unsmoothed $\mathcal{F}_N$ ($M = N$) | Single Smoothed $\mathcal{F}_{N, 1/4}^{\mathrm{smooth}}$ ($M = 1.25N$) | 8-Scale Max / Trace Bound (43) ($M = 2N$) | 8-Scale Zero-Indep. Superposition ($M = 2N$) | 16-Scale Zero-Indep. Superposition ($M = 4N$) | Full Subspace $\operatorname{Ann}_{S_\sigma^*}^{(2N)}$ (Num., $M = 2N$) | Full Subspace $\operatorname{Ann}_{S_\sigma^*}^{(4N)}$ (Num., $M = 4N$) | Reference Global $F_{\mathrm{unnorm}, \mathrm{opt}}^{\mathrm{num}}(N)$ (Num., $M = \infty$) | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $40$ | $0.001940$ | $0.000860$ | $0.001180 / 0.000509$ | $0.002975^*$ | $0.007061^*$ | $0.007024$ ($55.9\%$) | $0.011377$ ($90.5\%$) | $0.012572$ | | $80$ | $0.000488$ | $0.000797$ | $0.003008 / 0.000732$ | $0.006946^*$ | $0.007590^*$ | $0.009791$ ($75.9\%$) | $0.012215$ ($94.6\%$) | $0.012907$ | | $160$ | $0.000127$ | $0.000604$ | $0.002581 / 0.000542$ | $0.006636^*$ | $0.007871^*$ | $0.009360$ ($75.9\%$) | $0.011253$ ($91.2\%$) | $0.012333$ | | $320$ | $0.000037$ | $0.000661$ | $0.001225 / 0.000702$ | $0.005477$ | $0.006641$ | $0.006996$ ($64.0\%$) | $0.009291$ ($85.0\%$) | $0.010927$ | | $640$ | $3.88 \times 10^{-6}$ | $0.000783$ | $0.002823 / 0.001245$ | $0.005257$ | $0.006285$ | — | — | — | | $1280$ | $1.21 \times 10^{-6}$ | $0.002118$ | $0.002118 / 0.000933$ | $0.004211$ | $0.005603$ | — | — | — | | $2560$ | $9.46 \times 10^{-7}$ | $0.003070$ | $0.003070 / 0.000833$ | $0.007439$ | $0.007602$ | — | — | — | | $5120$ | $2.33 \times 10^{-8}$ | $0.000442$ | $0.000758 / 0.000414$ | $0.003951$ | $0.006302$ | — | — | — |

Table 3: Finite-support subspace exhaustion $d_{\mathrm{unnorm}, N, M}(0.75)^2 N^{0.5}$ and multi-scale zero-independent smoothed harmonic head superposition at $\sigma = 0.75$ using fixed knots $\theta_j = \frac{1}{1+\delta_j}$ (results/smoothed_harmonic_head_duality.json). Entries marked with $*$ at $N \in \{40, 80, 160\}$ are certified lower bounds (rounded strictly downward via ROUND_FLOOR) from exact fractions.Fraction step coefficients and 60-digit mpmath.iv interval arithmetic, with exact zero annihilation residual $\max_{2 \le k \le N}|\mathcal{L}_{\tilde{\lambda}}(\tilde{\rho}_{1/k})| \equiv 0 \in \mathbb{Q}$ (percentages in parentheses compare the certified lower bounds against the double-precision numerical reference values in the last column):

• $N = 40, M = 80$ ($K = 8$): $\tilde{\lambda} = (-11131, -31052, 55342, -208961, 204689, -573563, 1000000, -460635)^T$ certifies $d_{40}(0.75)^2 \ge 0.00070575$ and $F_{\mathrm{unnorm}}(40) \ge 0.002975$ ($23.7\%$ of the numerical reference value $0.012572$);
• $N = 40, M = 160$ ($K = 16$): $\tilde{\lambda} = (-3917, -10453, 31352, -40166, -10386, 3435, 62028, -82600, 281009, -539560, 275510, 479994, -1000000, 921140, -507746, 134089)^T$ certifies $d_{40}(0.75)^2 \ge 0.00167482$ and $F_{\mathrm{unnorm}}(40) \ge 0.007061$ ($56.2\%$ of the numerical reference value $0.012572$);
• $N = 80, M = 160$ ($K = 8$): $\tilde{\lambda} = (-19076, -55612, 181289, 196675, -735454, 1000000, -899496, 344108)^T$ certifies $d_{80}(0.75)^2 \ge 0.00116501$ and $F_{\mathrm{unnorm}}(80) \ge 0.006946$ ($53.8\%$ of the numerical reference value $0.012907$);
• $N = 80, M = 320$ ($K = 16$): $\tilde{\lambda} = (-10907, -32942, 98978, 136337, -454593, 531430, -355500, 91863, -227930, 620295, -491088, -202775, 784416, -1000000, 816049, -296338)^T$ certifies $d_{80}(0.75)^2 \ge 0.00127298$ and $F_{\mathrm{unnorm}}(80) \ge 0.007590$ ($58.8\%$ of the numerical reference value $0.012907$);
• $N = 160, M = 320$ ($K = 8$): $\tilde{\lambda} = (-2774, 90216, -248752, 46845, 254411, -589740, 1000000, -577499)^T$ certifies $d_{160}(0.75)^2 \ge 0.00078695$ and $F_{\mathrm{unnorm}}(160) \ge 0.006636$ ($53.8\%$ of the numerical reference value $0.012333$);
• $N = 160, M = 640$ ($K = 16$): $\tilde{\lambda} = (-301, 63859, -180877, 79114, 132262, -364750, 509554, -112499, -528367, 952494, -1000000, 364929, 615940, -904896, 397200, -30172)^T$ certifies $d_{160}(0.75)^2 \ge 0.00093347$ and $F_{\mathrm{unnorm}}(160) \ge 0.007871$ ($63.8\%$ of the numerical reference value $0.012333$).

Figure 2: (a) Numerical finite-support annihilator subspace exhaustion Ann_{S_sigma^*}^{(M)} across M in {1.25N, 1.5N, 2N, 4N, infty} at sigma = 0.75. (b) Nested multi-scale zero-independent smoothed harmonic head superposition for K = 1, 8, 16 scales with fixed knots theta_j = 1/(1+delta_j) across N in [40, 5120] (Theorem 4).

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6. Self-Contained Verification & Reproduction

All exact rational annihilation checks (fractions.Fraction), the $N = 2, \delta = 1$ endpoint rational verification, the exact rational verification of the Fubini-Abel bound (28) (including the $N = 40, \delta = 1/4, m = 45$ test point) and the $C^{2,1}$ derivative identity (26a) and bound (32a), the 60-digit mpmath.iv Euler-Maclaurin and Rouché certification of $\rho_1 \in D(s_0, r_0)$ and $\big|\widehat{W}_{3, 4/5}(\rho_1 - 1)\big| \in [0.06225173, 0.06225178] \subset [0.062251, 0.062252] > 0$ (Theorem 3(2)(b)), 60-digit mpmath.iv interval certificates (using the fixed knots $\theta_j = \frac{1}{1+\delta_j}$ consistently throughout), 60-digit interval evaluations of $\mathcal{C}_{\sigma, 1/4}^{(1)}$, numerical quadrature estimates, and figure generation routines are implemented in the standalone, zero-local-dependency script src/verify_smoothed_harmonic_head_duality.py (with Python package dependencies pinned in requirements.txt inside the supplement archive), and all full 60-digit interval endpoints and integer witness vectors are archived in results/smoothed_harmonic_head_duality.json.

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