# Co-Located Gauss-Hypergeometric Jet and Jacobi-Rodrigues Wavelet Packet Bases in the Shifted Nyman-Beurling Criterion: Exception-Free Linear-Support Rate Realization, Midpoint Phase Quadrature, and Interval-Certified Single-Knot Subspace Exhaustion

## Abstract

In our prior work (`riemann_hypothesis_12` [24]), we showed that smoothing Balazard's harmonic head functional $\mathcal{F}_N = \sum_{n=1}^N n^{-1}\mathcal{C}_n$ by a flat-top spline $\Omega_\delta = W_{r+1, \theta}$ ($\theta = \frac{1}{1+\delta}$) across a linear tail $N < n \le N_\delta := \lfloor(1+\delta)N\rfloor$ yields unconditional strong dual-norm vanishing $\lim_{N\to\infty}\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\|_{S_\sigma^*}^2 = 0$ on $[1/2, 1)$ and the conditional power-saving law $\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\|_{S_\sigma^*}^2 \sim \mathcal{C}_{\sigma, \delta}N^{-2(1-\sigma)}$ for $\sigma \in (\Theta, 1)$ ($\Theta := \sup\{\operatorname{Re}(\rho) : \zeta(\rho) = 0\} \in [1/2, 1]$). However, a single taper $W_{r+1, \theta}$ leaves a countable exceptional scale set $\mathcal{E}_r \subset (0, 1]$ where ${}_2F_1(1-\rho, r+1; 2r+2; \frac{\delta}{1+\delta}) = 0$, and suffers from log-periodic nodal zeros at finite $N$. Here we resolve both limitations at a single fixed support $N_\delta$ without varying the knot $\theta$. First, adjoining the knot-derivative defect functional $\mathcal{G}_{N, \delta}^{(r)}$ ($\Phi_{r+1, \theta} := -(1-\theta)\partial_\theta W_{r+1, \theta}$) yields $\widehat{\Phi}_{r+1, \theta}(s) = \frac{z}{s}w'_s(z)$; by Cauchy-Picard uniqueness for the Gauss hypergeometric ODE, $|\widehat{W}_{r+1, \theta}(\rho-1)|^2 + |\widehat{\Phi}_{r+1, \theta}(\rho-1)|^2 > 0$ for every $\delta > 0$ and every nontrivial zero $\rho$, eliminating $\mathcal{E}_r$ unconditionally at every rational scale $\delta \in (0, 1] \cap \mathbb{Q}$. Second, we embed $\{W_{r+2, \theta}, \Phi_{r+2, \theta}\}$ into the co-located Jacobi-Rodrigues wavelet hierarchy $\Psi_{r, \theta, k}(u) = \frac{(-1)^{k-1}}{k!}\partial_\tau^{k-1}[\tau^{r+k}(1-\tau)^{r+k}]$, whose alternating midpoint parity places consecutive modes in phase quadrature and whose span exhausts $\operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$. Third, 60-digit interval arithmetic certifies 18 rational witness superpositions at $N \in \{40, 80, 160\}$.

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## 1. Introduction, Inherited Framework, and Summary of New Contributions

### 1.1 Inherited Shifted Nyman-Beurling Duality and Smoothed Harmonic Head 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 $\langle f, g \rangle_\sigma := 2\sigma \int_0^1 f(x)\overline{g(x)}\,x^{2\sigma-1}\,dx$, and norm $\|f\|_\sigma = \langle f, f \rangle_\sigma^{1/2}$ (in the unnormalized convention $\|f\|_{\mathrm{unnorm}, \sigma}^2 = \int_0^1 |f(x)|^2 x^{2\sigma-1}\,dx = (2\sigma)^{-1}\|f\|_\sigma^2$; at $\sigma = 1/2$, $2\sigma = 1$ so the two conventions coincide). Following Nyman [1], Beurling [2], and Báez-Duarte [3], the pole-canceled fractional-part dilates $\tilde{\rho}_{1/k}(x) := \frac{1}{k}\{1/x\} - \{1/(kx)\} = -\{\lfloor 1/x \rfloor / k\}$ ($k \ge 2$) and the constant indicator $E(x) := \mathbf{1}_{(0, 1]}(x)$ belong to the harmonic step subspace $S_\sigma := \overline{\operatorname{span}}\{\mathbf{1}_{I_m} : m \ge 1\} \subsetneq \mathcal{H}_\sigma$, where $I_m := (1/(m+1), 1/m]$ has measure $w_m^{(\sigma)} := m^{-2\sigma} - (m+1)^{-2\sigma} = \frac{2\sigma}{m^{2\sigma+1}}(1 + O(m^{-1}))$. Writing $f_m := f|_{I_m}$ ($f_0 := 0$), the step norm on $S_\sigma$ and Riesz dual norm on $S_\sigma^*$ are
$$\|f\|_\sigma^2 = \sum_{m=1}^\infty |f_m|^2 w_m^{(\sigma)}, \qquad \|\mathcal{L}\|_{S_\sigma^*}^2 = \sum_{m=1}^\infty \frac{|c_m(\mathcal{L})|^2}{w_m^{(\sigma)}}, \qquad \mathcal{L}(f) = \sum_{m=1}^\infty c_m(\mathcal{L})\,f_m. \tag{1}$$
By Hahn-Banach duality (`riemann_hypothesis_7` [15], `riemann_hypothesis_9` [17]), the squared approximation distance $d_N(\sigma)^2 := \inf_{b_2, \dots, b_N \in \mathbb{C}} \|E - \sum_{k=2}^N b_k \tilde{\rho}_{1/k}\|_\sigma^2$ equals
$$d_N(\sigma)^2 = \sup\left\{ \frac{|\mathcal{L}(E)|^2}{\|\mathcal{L}\|_{S_\sigma^*}^2} : \mathcal{L} \in \operatorname{Ann}_{S_\sigma^*}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}) \setminus \{0\} \right\}, \tag{2}$$
where $\operatorname{Ann}_{S_\sigma^*}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}) := \{\mathcal{L} \in S_\sigma^* : \mathcal{L}(\tilde{\rho}_{1/k}) = 0 \text{ for all } 2 \le k \le N\}$. In the Vasyunin-Balazard backward-difference basis $\mathcal{D}_m(f) := f_m - f_{m-1}$ and $\mathcal{C}_n(f) := \sum_{d \mid n} \mu(n/d)\mathcal{D}_d(f)$ ([6, 7, 15, 17, 18]), which satisfies $\mathcal{C}_1(\tilde{\rho}_{1/k}) = -1/k$, $\mathcal{C}_n(\tilde{\rho}_{1/k}) = \delta_{n,k}$ ($n, k \ge 2$), and $\mathcal{C}_n(E) = \mu(n)$, any step functional $\mathcal{L} = \sum_{n=1}^M \alpha_n \mathcal{C}_n = \sum_{d=1}^M \beta_d \mathcal{D}_d$ supported on $I_1, \dots, I_M$ ($\beta_d = \sum_{j=1}^{\lfloor M/d \rfloor} \alpha_{dj}\mu(j)$, $\beta_{M+1} := 0$) has step coefficients $c_m(\mathcal{L}) = \beta_m - \beta_{m+1}$, target evaluation $\mathcal{L}(E) = \beta_1 = \sum_{n=1}^M \alpha_n \mu(n)$, and dilate evaluations $\mathcal{L}(\tilde{\rho}_{1/k}) = \alpha_k - \frac{1}{k}\alpha_1$ ($k \ge 2$). Thus, for any $M \ge N$, the finite-support annihilator subspace ([17, Section 2.1])
$$\operatorname{Ann}_{S_\sigma^*}^{(M)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}) := \left\{\mathcal{L} \in \operatorname{Ann}_{S_\sigma^*}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}) : c_m(\mathcal{L}) = 0 \text{ for all } m > M\right\} = \operatorname{span}\!\big\{\mathcal{F}_N,\; \mathcal{C}_{N+1},\; \dots,\; \mathcal{C}_M\big\} \tag{3}$$
has exact algebraic dimension $M - N + 1$, where $\mathcal{F}_N := \sum_{n=1}^N \frac{1}{n}\mathcal{C}_n$ is Balazard's unsmoothed harmonic head functional.

In `riemann_hypothesis_6` [14], `span_of_dilates` proved existentially that $d_N(\sigma)^2 = N^{-2(\sigma-\Theta)^+ + o(1)}$, where $\Theta := \sup\{\operatorname{Re}(\rho) : \zeta(\rho) = 0\} \in [1/2, 1]$. In `riemann_hypothesis_12` [24], building on [17] and [23], we showed that smoothing $\mathcal{F}_N$ across a strictly linear tail $N < n \le N_\delta := \lfloor(1+\delta)N\rfloor$ ($\delta \in (0, 1]$) via a flat-top taper $\Omega_\delta \in C^{2,1}([0, \infty))$ with fixed knot $\theta := \frac{1}{1+\delta}$ ($\Omega_\delta = 1$ on $[0, \theta]$, $\Omega_\delta = 0$ on $[1, \infty)$),
$$\mathcal{F}_{N, \delta}^{\mathrm{smooth}} := \sum_{n=1}^{N_\delta} \frac{1}{n}\,\Omega_\delta\!\left(\frac{n}{N_\delta + 1}\right)\mathcal{C}_n \in \operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}), \tag{4}$$
yields the exact two-point Möbius identity $c_m(\mathcal{F}_{N,\delta}^{\mathrm{smooth}}) = \frac{1}{m(m+1)} M_{V_{m,\delta}}(X_m)$ ($X_m := \frac{N_\delta+1}{m}$, $V_{m,\delta}(u) := (m+1)\Omega_\delta(u) - m\Omega_\delta(u + u/m)$), unconditional strong dual-norm vanishing $\lim_{N\to\infty}\|\mathcal{F}_{N,\delta}^{\mathrm{smooth}}\|_{S_\sigma^*}^2 = 0$ for all $\sigma \in [1/2, 1)$, and—conditionally on $\sigma \in (\Theta, 1)$ (which requires $\Theta < 1$)—the $N^1$-saving asymptotic norm law
$$\big\|\mathcal{F}_{N, \delta}^{\mathrm{smooth}}\big\|_{S_\sigma^*}^2 = \mathcal{C}_{\sigma, \delta}\,N^{-2(1-\sigma)} + O_{\sigma,\delta}\!\left(N^{-2(1-\sigma) - (\sigma - \Theta)/2}\right), \qquad \mathcal{C}_{\sigma, \delta} := \frac{(1+\delta)^{-2(1-\sigma)}}{2\sigma}\int_1^\infty X^{1-2\sigma}\big|M_{\widetilde{\Omega}_\delta}(X)\big|^2\,dX, \tag{5}$$
where $\widetilde{\Omega}_\delta(u) := \Omega_\delta(u) - u\,\Omega_\delta'(u)$ and $M_V(X) := \sum_{1 \le j \le X} \frac{\mu(j)}{j}V(j/X)$.

### 1.2 Two Open Limitations of Single-Taper and Multi-Scale Constructions in `riemann_hypothesis_12`

Despite resolving the $N^1$ sharp-cutoff norm penalty of $\mathcal{F}_N$, the construction in `riemann_hypothesis_12` [24] left two concrete structural limitations:
1. **The Countable Exceptional Scale Set $\mathcal{E}_r$ for a Single Taper $W_{r+1, \theta}$:** For the $(2r+1)$-ic flat-top Hermite spline $\Omega_\delta = W_{r+1, \theta}$ ($\theta = \frac{1}{1+\delta}, z := 1 - \theta = \frac{\delta}{1+\delta} \in (0, 1/2]$), whose derivative on $(\theta, 1)$ is $-W'_{r+1, \theta}(u) = \frac{(r+1)\binom{2r+1}{r+1}}{1-\theta}\tau^r(1-\tau)^r$ ($\tau = \frac{u-\theta}{1-\theta}$), Theorem 3(2) of [24] proved that the Mellin transform is
   $$\widehat{W}_{r+1, \theta}(s) = -\frac{1}{s}\int_\theta^1 W'_{r+1, \theta}(u)\,u^s\,du = \frac{1}{s}\,w_s(z), \qquad w_s(z) := {}_2F_1\!\left(-s,\; r+1;\; 2r+2;\; z\right). \tag{6}$$
   Because $w_{1-\rho_0}(z)$ is a nonconstant holomorphic function of $z$, a single functional $\mathcal{F}_{N, \delta}^{\mathrm{smooth}}$ could only be proved to achieve the infinitely-often rate bound $d_N(\sigma)^2 = \Omega(N^{-2(\sigma-\Theta)-\varepsilon})$ for real $\delta \in (0, 1] \setminus \mathcal{E}_r$ outside the countable zero set $\mathcal{E}_r$. At any specific rational scale such as $\delta = 1/4$ ($z = 1/5$), interval evaluation at the first critical-line zero $\rho_1 = \frac{1}{2} + i\gamma_1$ certified $d_N(\sigma)^2 = \Omega(N^{-2(\sigma-1/2)-\varepsilon})$, but could not rule out $1/4 \in \mathcal{E}_r$ for an unknown hypothetical off-line zero $\rho_0$ near $\operatorname{Re}(s) = \Theta$ if $\Theta > 1/2$.
2. **Support Inflation vs. Single-Scale Nodal Zeros:** At finite $N$, $M_{W_{3, \theta}}(N_\delta + 1)$ oscillates with $\log N_\delta$ and nearly vanishes near log-periodic nodal crossings (e.g., at $\delta = 1, N = 1280$, where $F_{\mathrm{unnorm}} = 1.8 \times 10^{-7}$, or $\delta = 1/2, N = 640$, where $F_{\mathrm{unnorm}} = 7.6 \times 10^{-6}$). In [24, Section 5], overcoming these nodal zeros required superposing $K = 8$ or $K = 16$ **different dilation scales** $\delta_1 < \dots < \delta_K$, inflating the support cutoff from $N_{\delta_1}$ up to $2N$ or $4N$. Meanwhile, simply varying the spline order $r$ at a fixed knot $\theta = 4/5$ (e.g., $\{W_{3, 4/5}, W_{4, 4/5}\}$) fails to provide phase quadrature because $-W'_{r+1, \theta}(u) \propto \tau^r(1-\tau)^r$ is always symmetric (even) about the midpoint $\tau = 1/2$ ($u_* = \frac{1+\theta}{2}$), locking the Mellin phase at $\rho_1$ to within $0.27^\circ$.

### 1.3 Summary of New Contributions

In this paper, we resolve both limitations at a **single fixed linear support $N_\delta = \lfloor(1+\delta)N\rfloor$** and a **single fixed knot $\theta = \frac{1}{1+\delta}$** by introducing the **Co-Located Gauss-Hypergeometric Jet Basis** and the **Jacobi-Rodrigues Wavelet Packet Hierarchy** on the tail annulus $(N, N_\delta]$:

1. **Co-Located Gauss-Hypergeometric Jet Basis and Unconditional Elimination of $\mathcal{E}_r$ at Every Scale $\delta \in (0, 1]$ (Section 2, Theorem 1):**
   For any smoothness order $r \ge 3$ and any fixed $\delta \in (0, 1]$ ($\theta := \frac{1}{1+\delta}, z := 1 - \theta = \frac{\delta}{1+\delta} \in (0, 1/2]$), we adjoin to $\mathcal{F}_{N, \delta}^{(r)} := \sum_{n=1}^{N_\delta} \frac{1}{n}W_{r+1, \theta}\big(\frac{n}{N_\delta+1}\big)\mathcal{C}_n$ the **knot-derivative defect functional** supported purely on the tail annulus $N + 1 \le n \le N_\delta$:
   $$\mathcal{G}_{N, \delta}^{(r)} := \sum_{n=N+1}^{N_\delta} \frac{1}{n}\,\Phi_{r+1, \theta}\!\left(\frac{n}{N_\delta + 1}\right)\mathcal{C}_n \in \operatorname{span}\{\mathcal{C}_{N+1}, \dots, \mathcal{C}_{N_\delta}\}, \qquad \Phi_{r+1, \theta}(u) := -(1-\theta)\frac{\partial}{\partial \theta} W_{r+1, \theta}(u) = -(r+1)\binom{2r+1}{r+1}\tau^r(1-\tau)^{r+1}\mathbf{1}_{(\theta, 1)}(u). \tag{7}$$
   Because $W_{r+1, \theta} \in C^r([0, \infty))$ has vanishing derivatives up to order $r$ at $u = \theta$, Leibniz differentiation of (6) with respect to $\theta = 1 - z$ yields the exact Mellin derivative identity
   $$\widehat{\Phi}_{r+1, \theta}(s) := \int_\theta^1 \Phi_{r+1, \theta}(u)\,u^{s-1}\,du = \frac{z}{s}\,w'_s(z) = -\frac{z}{2}\,{}_2F_1\!\left(1-s,\; r+2;\; 2r+3;\; z\right). \tag{8}$$
   Since $w_s(z) = {}_2F_1(-s, r+1; 2r+2; z)$ satisfies the second-order homogeneous Gauss hypergeometric differential equation $z(1-z)w''_s(z) + \big(2r+2 - (r+2-s)z\big)w'_s(z) + s(r+1)w_s(z) = 0$ with $w_s(0) = 1 \ne 0$, **Cauchy-Picard uniqueness** at the regular point $z = \frac{\delta}{1+\delta} \in (0, 1)$ guarantees that $w_s(z)$ and $w'_s(z)$ can never vanish simultaneously:
   $$\big|\widehat{W}_{r+1, \theta}(s)\big|^2 + \big|\widehat{\Phi}_{r+1, \theta}(s)\big|^2 = \frac{|w_s(z)|^2 + z^2|w'_s(z)|^2}{|s|^2} > 0 \qquad \big(\forall\, \delta \in (0, \infty),\; \forall\, s \in \mathbb{C} \setminus \{0\}\big). \tag{9}$$
   Consequently, for **every single rational scale $\delta \in (0, 1] \cap \mathbb{Q}$** (with $\mathcal{E}_r = \emptyset$!) and every $\sigma \in (\Theta, 1)$, at least one of the two purely rational annihilators $\mathcal{F}_{N, \delta}^{(r)}$ and $\mathcal{F}_{N, \delta}^{(r)} + \mathcal{G}_{N, \delta}^{(r)} \in \operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$ unconditionally realizes $\limsup_{N\to\infty} N^{2(\sigma-\Theta)+\varepsilon} d_{N, N_\delta}(\sigma)^2 = \infty$ ($\forall \varepsilon > 0$).

2. **Co-Located Jacobi-Rodrigues Wavelet Packet Hierarchy, Exact Gauss-Jet Embedding, and Single-Knot Subspace Completeness (Section 3, Theorem 2):**
   For any $r \ge 2$, $\delta \in (0, 1]$ ($\theta = \frac{1}{1+\delta}, z = \frac{\delta}{1+\delta}$), and mode index $k \ge 1$, we define the **$k$-th Jacobi-Rodrigues wavelet taper** on $[\theta, 1]$ ($\tau = \frac{u-\theta}{1-\theta}$):
   $$\Psi_{r, \theta, k}(u) := \frac{(-1)^{k-1}}{k!}\frac{d^{k-1}}{d\tau^{k-1}}\Big[\tau^{r+k}(1-\tau)^{r+k}\Big]\mathbf{1}_{(\theta, 1)}(u) \in C^r([0, \infty)) \cap C^{r, 1}([0, \infty)), \qquad \mathcal{D}_{N, \delta}^{(r, k)} := \sum_{n=N+1}^{N_\delta} \frac{1}{n}\Psi_{r, \theta, k}\!\left(\frac{n}{N_\delta+1}\right)\mathcal{C}_n. \tag{10}$$
   We prove four structural properties of the hierarchy $\mathcal{W}_{N, \delta}^{(r, L)} := \operatorname{span}\!\big\{\mathcal{F}_{N, \delta}^{(r)}, \mathcal{D}_{N, \delta}^{(r, 1)}, \dots, \mathcal{D}_{N, \delta}^{(r, L-1)}\big\} \subset \operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$:
   - *(Orthogonal Derivative Filter Bank):* By Rodrigues's formula, $-\Psi'_{r, \theta, k}(u) = \frac{1}{1-\theta}\tau^r(1-\tau)^r P_k^{(r, r)}(2\tau - 1)$, so $\big\{-W'_{r+1, \theta}, -\Psi'_{r, \theta, 1}, -\Psi'_{r, \theta, 2}, \dots\big\}$ is an exact orthogonal basis of $L^2\big([\theta, 1], (u-\theta)^{-r}(1-u)^{-r}\,du\big)$ with alternating midpoint parity $(-1)^k$ about $u_* = \frac{1+\theta}{2}$.
   - *(Exact Algebraic Embedding of the Gauss-Hypergeometric Jet):* The Gauss-Jet pair $\{W_{r+2, \theta}, \Phi_{r+2, \theta}\}$ lies in the 3-mode subspace $\operatorname{span}\{W_{r+1, \theta}, \Psi_{r, \theta, 1}, \Psi_{r, \theta, 2}\}$ via the exact identities
     $$W_{r+2, \theta}(u) - W_{r+1, \theta}(u) = -\frac{1}{r+1}\binom{2r+1}{r}\Psi_{r, \theta, 2}(u), \qquad \Phi_{r+2, \theta}(u) = -(r+2)\binom{2r+3}{r+2}\left[\frac{1}{2}\Psi_{r, \theta, 1}(u) - \frac{1}{r+2}\Psi_{r, \theta, 2}(u)\right]. \tag{11}$$
   - *(Closed-Form Entire Hypergeometric Mellin Transforms):* For every $k \ge 1$, $\widehat{\Psi}_{r, \theta, k}(s)$ is an entire function of $s \in \mathbb{C}$ given by
     $$\widehat{\Psi}_{r, \theta, k}(s) = \frac{((r+k)!)^2}{k!\,(2r+2k+1)!}\,z^k\,(s-1)(s-2)\cdots(s-k+1)\,{}_2F_1\!\left(k-s,\; r+k+1;\; 2r+2k+2;\; z\right). \tag{12}$$
   - *(Asymptotic $N^{-2(1-\sigma)}$ Gram Law and Exact Finite-$N$ Subspace Exhaustion):* For $\sigma \in (\Theta, 1)$, the $L \times L$ Gram matrix of $\mathcal{W}_{N, \delta}^{(r, L)}$ obeys $G_{\mathcal{W}_{N, \delta}^{(r, L)}}(\sigma) = \Gamma_{\sigma, \delta}^{(r, L)} N^{-2(1-\sigma)} + O_{\sigma,\delta,L}(N^{-2(1-\sigma)-(\sigma-\Theta)/2})$ with $\det \Gamma_{\sigma, \delta}^{(r, L)} \ge \det \Gamma_{\sigma, \delta}^{(r, L), (1)} > 0$; and at any finite $N \ge 2$ with $M_\delta := N_\delta - N \ge 1$, $\mathcal{W}_{N, \delta}^{(r, M_\delta + 1)} = \operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$ exhausts $100\%$ of the finite-support annihilator subspace at single scale $N_\delta$.

3. **60-Digit Interval Certification of Mellin Phase Quadrature and Single-Knot Wavelet Superpositions (Section 4, Theorem 3):**
   In 60-digit `mpmath.iv` interval arithmetic (including the full $B_{12}$ Euler-Maclaurin term in our Rouché zero enclosure of $\rho_1$), we certify at $\delta = 1/4$ ($\theta = 4/5$) that while $\widehat{W}_{4, 4/5}(\rho_1 - 1)$ and $\widehat{W}_{3, 4/5}(\rho_1 - 1)$ are nearly collinear ($\cos\Delta\arg \in [0.99974, 1.00024]$), the first Jacobi-Rodrigues wavelet $\widehat{\Psi}_{2, 4/5, 1}(\rho_1 - 1)$ is in near-exact $+90^\circ$ Hilbert phase quadrature with $\widehat{W}_{3, 4/5}(\rho_1 - 1)$ ($\cos\Delta\arg \in [-0.02014, -0.02001]$, $\sin\Delta\arg \in [0.99955, 1.00004]$). We then certify in exact `fractions.Fraction` + 60-digit `mpmath.iv` arithmetic 18 integer witness combinations at single fixed supports $M \in \{1.25N, 1.5N, 2N\}$ across $N \in \{40, 80, 160\}$ ($L \in \{2, 6\}$), capturing up to $82.2\%$ ($L = 6$) and $86.8\%$ ($L = 8$, numerical) of the full $(N_\delta - N + 1)$-dimensional subspace distance $d_{N, N_\delta}(0.75)^2$ at a single knot $\theta$.

*(Scope note: as in [24], the asymptotic Gram law requires $\sigma \in (\Theta, 1)$, which is conditional on $\Theta < 1$; the infinitely-often rate bound is a $\limsup_{N\to\infty}$ statement rather than a uniform all-$N$ bound; and strong norm vanishing of unnormalized annihilators does not imply $d_N(1/2) \to 0$ because the target evaluation also vanishes as $N \to \infty$.)*

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## 2. Co-Located Gauss-Hypergeometric Jet Basis and Unconditional Elimination of $\mathcal{E}_r$

Let $r \ge 3$ and $\delta \in (0, 1]$, and set $\theta := \frac{1}{1+\delta} \in [1/2, 1)$, $z := 1 - \theta = \frac{\delta}{1+\delta} \in (0, 1/2]$, and $N_\delta := \lfloor(1+\delta)N\rfloor$. On $(\theta, 1)$, with $\tau := \frac{u-\theta}{1-\theta} \in (0, 1)$, the flat-top Hermite spline $W_{r+1, \theta}(u) = 1 - (r+1)\binom{2r+1}{r+1}\int_0^{\tau(u)} t^r(1-t)^r\,dt$ has partial derivatives with respect to $u$ and $\theta$ linked by $\frac{\partial \tau}{\partial \theta} = -\frac{1-\tau}{1-\theta} = -(1-\tau)\frac{\partial \tau}{\partial u}$. Defining the knot-derivative taper $\Phi_{r+1, \theta}(u) := -(1-\theta)\frac{\partial}{\partial \theta} W_{r+1, \theta}(u)$ as in (7), we have on $(\theta, 1)$:
$$\Phi_{r+1, \theta}(u) = (1 - \theta)(1 - \tau)\,W'_{r+1, \theta}(u) = -(r+1)\binom{2r+1}{r+1}\tau^r(1-\tau)^{r+1}, \tag{13}$$
and $\Phi_{r+1, \theta}(u) = 0$ for $u \in [0, \theta] \cup [1, \infty)$.

**Theorem 1 (Co-Located Gauss-Hypergeometric Jet Basis and Unconditional Elimination of the Exceptional Set $\mathcal{E}_r$).** *Let $\sigma \in [1/2, 1)$, $r \ge 3$, and $\delta \in (0, 1]$, and set $\theta := \frac{1}{1+\delta}$, $z := \frac{\delta}{1+\delta} = 1 - \theta$, and $N_\delta := \lfloor(1+\delta)N\rfloor$. Let $\mathcal{F}_{N, \delta}^{(r)} := \sum_{n=1}^{N_\delta} \frac{1}{n}W_{r+1, \theta}\big(\frac{n}{N_\delta+1}\big)\mathcal{C}_n$ and $\mathcal{G}_{N, \delta}^{(r)} := \sum_{n=N+1}^{N_\delta} \frac{1}{n}\Phi_{r+1, \theta}\big(\frac{n}{N_\delta+1}\big)\mathcal{C}_n$.*
1. *(Exact Annihilation, Two-Point Identity, and Unconditional Norm Vanishing):* *For every $N \ge 2$ and $\lambda \in \mathbb{C}$, $\mathcal{F}_{N, \delta}^{(r)} + \lambda\,\mathcal{G}_{N, \delta}^{(r)}$ belongs to $\operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$ (with rational coordinates $\alpha_n \in \mathbb{Q}$ whenever $\delta, \lambda \in \mathbb{Q}$). Moreover, $\Phi_{r+1, \theta} \in C^{r-1}([0, \infty)) \cap C^{r-1, 1}([0, \infty)) \subset C^{2,1}([0, \infty))$, and the step coefficients $c_m\big(\mathcal{G}_{N, \delta}^{(r)}\big) = \eta_m^{\mathcal{G}} - \eta_{m+1}^{\mathcal{G}}$ satisfy*
   $$\eta_m^{\mathcal{G}} - \eta_{m+1}^{\mathcal{G}} = \frac{1}{m(m+1)}\,M_{V_{m,\delta}^{[\Phi]}}(X_m), \qquad \big|\eta_m^{\mathcal{G}} - \eta_{m+1}^{\mathcal{G}}\big| \le \frac{\big\|\widetilde{\Phi}'_{r+1, \theta}\big\|_{L^1([\theta, 1])}}{m(m+1)} < \frac{\big\|\widetilde{\Phi}'_{r+1, \theta}\big\|_{L^1([\theta, 1])}}{m^2}, \tag{14}$$
   *where $V_{m,\delta}^{[\Phi]}(u) := (m+1)\Phi_{r+1, \theta}(u) - m\Phi_{r+1, \theta}(u + u/m)$ and $\widetilde{\Phi}_{r+1, \theta}(u) := \Phi_{r+1, \theta}(u) - u\Phi'_{r+1, \theta}(u)$. Consequently, $\lim_{N\to\infty} \big\|\mathcal{G}_{N, \delta}^{(r)}\big\|_{S_\sigma^*}^2 = 0$ unconditionally for all $\sigma \in [1/2, 1)$.*
2. *(Conditional $N^{-2(1-\sigma)}$ $2 \times 2$ Gram Matrix Law for $\sigma \in (\Theta, 1)$):* *If $\Theta < 1$ and $\sigma \in (\Theta, 1)$, the $2 \times 2$ Gram matrix $G_{N, \delta}^{(r)}(\sigma)$ of the basis $\big\{\mathcal{F}_{N, \delta}^{(r)},\, \mathcal{G}_{N, \delta}^{(r)}\big\}$ in $S_\sigma^*$ satisfies*
   $$G_{N, \delta}^{(r)}(\sigma) = \Gamma_{\sigma, \delta}^{(r)}\,N^{-2(1-\sigma)} + O_{\sigma,\delta,r}\!\left(N^{-2(1-\sigma) - (\sigma - \Theta)/2}\right), \qquad \Gamma_{\sigma, \delta}^{(r)} := \frac{(1+\delta)^{-2(1-\sigma)}}{2\sigma}\int_1^\infty X^{1-2\sigma}\begin{pmatrix} |M_{\widetilde{W}}(X)|^2 & M_{\widetilde{W}}(X)M_{\widetilde{\Phi}}(X) \\ M_{\widetilde{W}}(X)M_{\widetilde{\Phi}}(X) & |M_{\widetilde{\Phi}}(X)|^2 \end{pmatrix} dX, \tag{15}$$
   *and $\Gamma_{\sigma, \delta}^{(r)}$ is strictly positive definite, with $\det \Gamma_{\sigma, \delta}^{(r)} \ge \det \Gamma_{\sigma, \delta}^{(r), (1)} > 0$ where $\Gamma_{\sigma, \delta}^{(r), (1)}$ is the restriction of the integral in (15) to $X \in [1, 1+\delta]$.*
3. *(Exact Mellin Derivative Identity and Cauchy-Picard Non-Vanishing for All $\delta > 0$):* *For all $s \in \mathbb{C}$, the Mellin transform $\widehat{\Phi}_{r+1, \theta}(s) := \int_\theta^1 \Phi_{r+1, \theta}(u)\,u^{s-1}\,du$ is an entire function of $s$ satisfying*
   $$\widehat{\Phi}_{r+1, \theta}(s) = \frac{z}{s}\,w'_s(z) = -\frac{z}{2}\,{}_2F_1\!\left(1-s,\; r+2;\; 2r+3;\; z\right), \qquad w_s(z) := {}_2F_1\!\left(-s,\; r+1;\; 2r+2;\; z\right). \tag{16}$$
   *For **every** real $\delta \in (0, \infty)$ ($z = \frac{\delta}{1+\delta} \in (0, 1)$) and **every** $s \in \mathbb{C} \setminus \{0\}$ (in particular, at $s = \rho_0 - 1$ for every nontrivial zero $\rho_0$ of $\zeta(s)$), the Gauss hypergeometric ODE enforces the strict non-vanishing bounds*
   $$\big|\widehat{W}_{r+1, \theta}(s)\big|^2 + \big|\widehat{\Phi}_{r+1, \theta}(s)\big|^2 = \frac{|w_s(z)|^2 + z^2|w'_s(z)|^2}{|s|^2} > 0, \qquad \max\Big\{\big|\widehat{W}_{r+1, \theta}(s)\big|,\; \big|\widehat{W}_{r+1, \theta}(s) + \widehat{\Phi}_{r+1, \theta}(s)\big|\Big\} \ge \frac{1}{2}\max\Big\{\big|\widehat{W}_{r+1, \theta}(s)\big|,\; \big|\widehat{\Phi}_{r+1, \theta}(s)\big|\Big\} > 0. \tag{17}$$
4. *(Unconditional Exception-Free Realization of $\Omega(N^{-2(\sigma-\Theta)-\varepsilon})$ at Every Single Scale $\delta \in (0, 1] \cap \mathbb{Q}$):* *Assume $\Theta < 1$ and let $\sigma \in (\Theta, 1)$. Then for **every fixed rational scale $\delta \in (0, 1] \cap \mathbb{Q}$** (with no exceptional set $\mathcal{E}_r$) and every $\varepsilon > 0$, the two explicit rational-coefficient functionals $\mathcal{L}_{N, \delta}^{[0]} := \mathcal{F}_{N, \delta}^{(r)}$ and $\mathcal{L}_{N, \delta}^{[1]} := \mathcal{F}_{N, \delta}^{(r)} + \mathcal{G}_{N, \delta}^{(r)} \in \operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$ satisfy*
   $$\limsup_{N \to \infty} N^{2(\sigma - \Theta) + \varepsilon}\,d_{N, N_\delta}(\sigma)^2 \ge \limsup_{N \to \infty} N^{2(\sigma - \Theta) + \varepsilon}\max_{j \in \{0, 1\}} \frac{\big|\mathcal{L}_{N, \delta}^{[j]}(E)\big|^2}{\big\|\mathcal{L}_{N, \delta}^{[j]}\big\|_{S_\sigma^*}^2} = \infty \qquad \Big(\text{i.e., } d_{N, N_\delta}(\sigma)^2 = \Omega\!\left(N^{-2(\sigma - \Theta) - \varepsilon}\right)\Big). \tag{18}$$

*Proof.*
**Parts 1 and 2:** Since $\frac{n}{N_\delta + 1} < \frac{N}{(1+\delta)N} = \theta$ for all $1 \le n \le N$, $\Phi_{r+1, \theta}\big(\frac{n}{N_\delta+1}\big) = 0$ on $1 \le n \le N$, so $\mathcal{G}_{N, \delta}^{(r)} \in \operatorname{span}\{\mathcal{C}_{N+1}, \dots, \mathcal{C}_{N_\delta}\}$. On $(\theta, 1)$, $\Phi_{r+1, \theta}(u)$ is a polynomial of degree $2r + 1$ vanishing to order $r \ge 3$ at $u = \theta$ and to order $r + 1 \ge 4$ at $u = 1$; thus $\Phi_{r+1, \theta} \in C^{r-1}([0, \infty)) \cap C^{r-1, 1}([0, \infty)) \subset C^{2,1}([0, \infty))$ with $\Phi_{r+1, \theta}(1) = \Phi'_{r+1, \theta}(1) = \Phi''_{r+1, \theta}(1) = 0$. Although $\Phi_{r+1, \theta} = 0$ on $[0, \theta]$ (whereas $W_{r+1, \theta} = 1$ on $[0, \theta]$), the proofs of Theorem 2 and Theorem 3(1) of `riemann_hypothesis_12` [24] depend only on $\Omega_\delta$ being constant on $[0, \theta]$, belonging to $C^{2,1}([0, \infty))$, and satisfying $\Omega_\delta(1) = \Omega'_\delta(1) = \Omega''_\delta(1) = 0$. Applying Theorem 2 and Theorem 3(1) of [24] to $W_{r+1, \theta}$, $\Phi_{r+1, \theta}$, and $W_{r+1, \theta} \pm \Phi_{r+1, \theta}$ (via polarization) yields (14) and (15). On $X \in [1, 1+\delta]$, $M_{\widetilde{W}}(X) = \widetilde{W}_{r+1, \theta}(1/X)$ and $M_{\widetilde{\Phi}}(X) = \widetilde{\Phi}_{r+1, \theta}(1/X)$; since $\widetilde{W}_{r+1, \theta}(\theta) = 1 \ne 0 = \widetilde{\Phi}_{r+1, \theta}(\theta)$ while $\widetilde{\Phi}_{r+1, \theta}(u) \not\equiv 0$ on $(\theta, 1)$, $\widetilde{W}_{r+1, \theta}$ and $\widetilde{\Phi}_{r+1, \theta}$ are linearly independent in $L^2([\theta, 1], u^{2\sigma-3}\,du)$, so the Gram matrix $\Gamma_{\sigma, \delta}^{(r), (1)}$ on $[1, 1+\delta]$ has $\det \Gamma_{\sigma, \delta}^{(r), (1)} > 0$, and by Minkowski's determinant inequality for positive semidefinite matrices, $\det \Gamma_{\sigma, \delta}^{(r)} \ge \det \Gamma_{\sigma, \delta}^{(r), (1)} > 0$.

**Part 3:** Since $\Phi_{r+1, \theta}$ is supported on $[\theta, 1] \subset (0, 1]$, $\widehat{\Phi}_{r+1, \theta}(s) = \int_\theta^1 \Phi_{r+1, \theta}(u)\,u^{s-1}\,du$ is entire in $s \in \mathbb{C}$. Substituting $u = 1 - z(1 - \tau)$ ($du = z\,d\tau$) and (13) into $\int_\theta^1 \Phi_{r+1, \theta}(u)\,u^{s-1}\,du$, and using $\int_0^1 \tau^r(1-\tau)^{r+1}\,d\tau = \frac{r!\,(r+1)!}{(2r+2)!} = \big[2(r+1)\binom{2r+1}{r+1}\big]^{-1}$ together with Euler's integral representation of ${}_2F_1$, we obtain
$$\widehat{\Phi}_{r+1, \theta}(s) = -(r+1)\binom{2r+1}{r+1} z \int_0^1 \tau^r(1-\tau)^{r+1}\big(1 - z(1-\tau)\big)^{s-1}\,d\tau = -\frac{z}{2}\,{}_2F_1\!\left(1-s,\; r+2;\; 2r+3;\; z\right). \tag{19}$$
By the Gauss hypergeometric differentiation identity $\frac{d}{dz}\,{}_2F_1(-s, r+1; 2r+2; z) = \frac{(-s)(r+1)}{2r+2}\,{}_2F_1(1-s, r+2; 2r+3; z) = -\frac{s}{2}\,{}_2F_1(1-s, r+2; 2r+3; z)$, the right-hand side of (19) equals $\frac{z}{s}w'_s(z)$, proving (16). Next, for any $s \in \mathbb{C} \setminus \{0\}$, the function $w_s(z) = {}_2F_1(-s, r+1; 2r+2; z)$ is a solution on the disk $|z| < 1$ of the second-order linear homogeneous differential equation
$$w''_s(z) + \frac{2r+2 - (r+2-s)z}{z(1-z)}\,w'_s(z) + \frac{s(r+1)}{z(1-z)}\,w_s(z) = 0, \tag{20}$$
whose coefficients are analytic at every $z \in (0, 1)$. If $w_s(z_0) = 0$ and $w'_s(z_0) = 0$ held simultaneously at some $z_0 = \frac{\delta}{1+\delta} \in (0, 1)$, then by Cauchy-Picard uniqueness for (20) and analytic continuation along $[0, z_0]$, $w_s(z)$ would vanish identically on $[0, 1)$, contradicting $w_s(0) = 1$. Hence $|w_s(z)|^2 + z^2|w'_s(z)|^2 > 0$ for all $z \in (0, 1)$ and $s \ne 0$. Finally, for any $A, B \in \mathbb{C}$, $|B| \le |A| + |A + B|$ implies $\max(|A|, |A + B|) \ge \frac{1}{2}\max(|A|, |B|)$, establishing (17).

**Part 4:** Fix any $\delta \in (0, 1] \cap \mathbb{Q}$ and $\varepsilon \in (0, 1/2)$, and choose a nontrivial zero $\rho_0 = \beta_0 + i\gamma_0$ of $\zeta(s)$ with $\beta_0 > \Theta - \varepsilon/4$. By (17) at $s = \rho_0 - 1 \ne 0$, at least one of the two tapers $\Omega^{[0]} := W_{r+1, \theta}$ and $\Omega^{[1]} := W_{r+1, \theta} + \Phi_{r+1, \theta}$ satisfies $\widehat{\Omega}^{[j]}(\rho_0 - 1) \ne 0$. By the Mellin pole argument of Theorem 3(2) of [24] (since $\int_1^\infty M_{\Omega^{[j]}}(X)\,X^{-s-1}\,dX = \widehat{\Omega}^{[j]}(s)/\zeta(1+s)$ has a pole at $s = \rho_0 - 1$ with $\operatorname{Re}(s) = -(1-\beta_0) > -(1-\Theta) - \varepsilon/4$), we have $\limsup_{N\to\infty} N^{2(1-\Theta)+\varepsilon/2}\big|\mathcal{L}_{N, \delta}^{[j]}(E)\big|^2 = \infty$, while by (15), $\big\|\mathcal{L}_{N, \delta}^{[j]}\big\|_{S_\sigma^*}^2 \asymp N^{-2(1-\sigma)}$. Dividing the two yields (18). $\square$

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## 3. Co-Located Jacobi-Rodrigues Wavelet Packet Hierarchy on the Tail Annulus

To systematically eliminate finite-$N$ nodal zeros and exhaust the finite-support annihilator subspace $\operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$ at a single fixed support $N_\delta = \lfloor(1+\delta)N\rfloor$, we now embed the Gauss-Jet pair into an orthogonal polynomial wavelet hierarchy supported on $[\theta, 1]$.

**Theorem 2 (Co-Located Jacobi-Rodrigues Wavelet Packet Hierarchy, Gauss-Jet Embedding, and Single-Knot Subspace Completeness).** *Let $\sigma \in [1/2, 1)$, $r \ge 2$, and $\delta \in (0, 1]$, with $\theta := \frac{1}{1+\delta}$, $z := \frac{\delta}{1+\delta} = 1 - \theta$, and $N_\delta := \lfloor(1+\delta)N\rfloor$. Set $\Psi_{r, \theta, 0}(u) := W_{r+1, \theta}(u)$ ($\mathcal{D}_{N, \delta}^{(r, 0)} := \mathcal{F}_{N, \delta}^{(r)}$), and for each $k \ge 1$ define $\Psi_{r, \theta, k}(u)$ and $\mathcal{D}_{N, \delta}^{(r, k)}$ by (10).*
1. *(Rodrigues Derivative Orthogonality and Exact $L^2$ Norms):* *For every $k \ge 1$, $\Psi_{r, \theta, k} \in C^r([0, \infty)) \cap C^{r, 1}([0, \infty))$ vanishes to order $r + 1$ at both $u = \theta$ and $u = 1$, and its derivative on $(\theta, 1)$ is given by Rodrigues's formula for the shifted Jacobi polynomials $P_k^{(r, r)}(2\tau - 1)$ ($\tau = \frac{u-\theta}{1-\theta}$):*
   $$-\Psi'_{r, \theta, k}(u) = \frac{(-1)^k}{k!\,(1-\theta)}\frac{d^k}{d\tau^k}\Big[\tau^{r+k}(1-\tau)^{r+k}\Big] = \frac{1}{1-\theta}\,\tau^r(1-\tau)^r P_k^{(r, r)}(2\tau - 1). \tag{21}$$
   *Consequently, with $D_k(\tau) := \tau^r(1-\tau)^r P_k^{(r, r)}(2\tau - 1)$ ($k \ge 0$), the family $\{D_k\}_{k=0}^\infty$ satisfies the exact orthogonality relation on $[0, 1]$:*
   $$\int_0^1 D_j(\tau)\,D_k(\tau)\,\tau^{-r}(1-\tau)^{-r}\,d\tau = \delta_{j, k}\,\frac{\big((k+r)!\big)^2}{(2k + 2r + 1)\,k!\,(k + 2r)!} \qquad (j, k \ge 0), \tag{22}$$
   *and under reflection $\tau \mapsto 1 - \tau$ about the midpoint $u_* = \frac{1+\theta}{2}$, $D_k(1 - \tau) = (-1)^k D_k(\tau)$.*
2. *(Exact Algebraic Embedding of the Gauss-Hypergeometric Jet):* *On $[0, \infty)$, the spline difference $W_{r+2, \theta} - W_{r+1, \theta}$ and the knot-derivative taper $\Phi_{r+2, \theta} = -(1-\theta)\partial_\theta W_{r+2, \theta}$ satisfy the exact polynomial identities (11) (in particular, for $r = 2$: $W_{4, \theta} - W_{3, \theta} = -5\,\Psi_{2, \theta, 2}$ and $\Phi_{4, \theta} = -70\,\Psi_{2, \theta, 1} + 35\,\Psi_{2, \theta, 2}$). Thus, for every $L \ge 3$, the $L$-dimensional co-located Jacobi-Rodrigues subspace*
   $$\mathcal{W}_{N, \delta}^{(r, L)} := \operatorname{span}\!\Big\{\mathcal{F}_{N, \delta}^{(r)},\; \mathcal{D}_{N, \delta}^{(r, 1)},\; \dots,\; \mathcal{D}_{N, \delta}^{(r, L-1)}\Big\} \subset \operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}) \tag{23}$$
   *contains the Gauss-Jet subspace $\operatorname{span}\!\big\{\mathcal{F}_{N, \delta}^{(r+1)}, \mathcal{G}_{N, \delta}^{(r+1)}\big\}$ as a 2-dimensional subspace.*
3. *(Entire Gauss Hypergeometric Mellin Formula and Exception-Free Rate Realization):* *For every $k \ge 1$, the Mellin transform $\widehat{\Psi}_{r, \theta, k}(s) := \int_\theta^1 \Psi_{r, \theta, k}(u)\,u^{s-1}\,du$ is an entire function of $s \in \mathbb{C}$ given by the closed-form hypergeometric formula (12). For every $L \ge 3$, every $\delta \in (0, \infty)$, and every $s \in \mathbb{C} \setminus \{0\}$, $\sum_{k=0}^{L-1} |\widehat{\Psi}_{r, \theta, k}(s)|^2 > 0$; hence for $\sigma \in (\Theta, 1)$ ($\Theta < 1$) and every $\delta \in (0, 1] \cap \mathbb{Q}$, $\mathcal{W}_{N, \delta}^{(r, L)}$ unconditionally realizes $\limsup_{N\to\infty} N^{2(\sigma-\Theta)+\varepsilon} d_{\mathcal{W}_{N,\delta}^{(r,L)}}(\sigma)^2 = \infty$.*
4. *(Asymptotic $L \times L$ Gram Law and Exact Finite-$N$ Subspace Exhaustion):* *For $\sigma \in (\Theta, 1)$ ($\Theta < 1$) and any fixed $L \ge 1$, the $L \times L$ Gram matrix of $\big\{\mathcal{D}_{N, \delta}^{(r, k)}\big\}_{k=0}^{L-1}$ satisfies*
   $$\Big(\big\langle \mathcal{D}_{N, \delta}^{(r, j)},\, \mathcal{D}_{N, \delta}^{(r, k)} \big\rangle_{S_\sigma^*}\Big)_{0 \le j, k \le L-1} = \Gamma_{\sigma, \delta}^{(r, L)}\,N^{-2(1-\sigma)} + O_{\sigma,\delta,r,L}\!\left(N^{-2(1-\sigma) - (\sigma - \Theta)/2}\right), \tag{24}$$
   *where $\big(\Gamma_{\sigma, \delta}^{(r, L)}\big)_{j, k} := \frac{(1+\delta)^{-2(1-\sigma)}}{2\sigma}\int_1^\infty X^{1-2\sigma} M_{\widetilde{\Psi}_{r, \theta, j}}(X) M_{\widetilde{\Psi}_{r, \theta, k}}(X)\,dX$ is strictly positive definite ($\det \Gamma_{\sigma, \delta}^{(r, L)} \ge \det \Gamma_{\sigma, \delta}^{(r, L), (1)} > 0$). Furthermore, at any finite $N \ge 2$ with $M_\delta := N_\delta - N \ge 1$, for $L = M_\delta + 1$ we have the **exact subspace equality***
   $$\mathcal{W}_{N, \delta}^{(r, M_\delta + 1)} = \operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N}), \qquad \sup_{\mathcal{L} \in \mathcal{W}_{N, \delta}^{(r, M_\delta + 1)} \setminus \{0\}} \frac{|\mathcal{L}(E)|^2}{\|\mathcal{L}\|_{S_\sigma^*}^2} = d_{N, N_\delta}(\sigma)^2. \tag{25}$$

*Proof.*
**Part 1:** Since $\tau^{r+k}(1-\tau)^{r+k}$ has zeros of multiplicity $r + k$ at $\tau = 0$ and $\tau = 1$, its $(k-1)$-th derivative $\Psi_{r, \theta, k}$ has zeros of multiplicity $(r + k) - (k - 1) = r + 1$ at $u = \theta$ and $u = 1$, so $\Psi_{r, \theta, k} \in C^r([0, \infty)) \cap C^{r, 1}([0, \infty))$. Differentiating (10) once more with respect to $u$ (using $\frac{d\tau}{du} = \frac{1}{1-\theta}$) and applying Rodrigues's formula $P_k^{(r, r)}(2\tau - 1) = \frac{(-1)^k}{k!}\tau^{-r}(1-\tau)^{-r}\frac{d^k}{d\tau^k}\big[\tau^{r+k}(1-\tau)^{r+k}\big]$ yields (21), and standard Jacobi polynomial orthogonality on $[-1, 1]$ under $x = 2\tau - 1$ gives (22) and the parity $P_k^{(r, r)}(-(2\tau - 1)) = (-1)^k P_k^{(r, r)}(2\tau - 1)$.

**Part 2:** On $(\theta, 1)$, by (10) for $k = 1, 2$, we have $\Psi_{r, \theta, 1}(u) = \tau^{r+1}(1-\tau)^{r+1}$ and $\Psi_{r, \theta, 2}(u) = -\frac{1}{2}\frac{d}{d\tau}\big[\tau^{r+2}(1-\tau)^{r+2}\big] = -\frac{r+2}{2}\tau^{r+1}(1-\tau)^{r+1}(1 - 2\tau)$. Since $1 - \tau = \frac{1}{2} + \frac{1}{2}(1 - 2\tau)$, we obtain $\tau^{r+1}(1-\tau)^{r+2} = \frac{1}{2}\Psi_{r, \theta, 1}(u) - \frac{1}{r+2}\Psi_{r, \theta, 2}(u)$, which multiplied by $-(r+2)\binom{2r+3}{r+2}$ proves the second identity in (11). Similarly, differentiating $W_{r+2, \theta}(u) - W_{r+1, \theta}(u)$ with respect to $\tau$ gives
$$-\frac{d}{d\tau}\big[W_{r+2, \theta} - W_{r+1, \theta}\big] = \binom{2r+1}{r}\tau^r(1-\tau)^r\big[(4r+6)\tau(1-\tau) - (r+1)\big] = -\frac{1}{r+1}\binom{2r+1}{r}\left(-\frac{1}{2}\frac{d^2}{d\tau^2}\Big[\tau^{r+2}(1-\tau)^{r+2}\Big]\right) = \frac{1}{r+1}\binom{2r+1}{r}\frac{d}{d\tau}\Psi_{r, \theta, 2},$$
and integrating from $\tau$ to $1$ (where both sides vanish) establishes the first identity in (11).

**Part 3:** For $k \ge 1$ and $s \ne 0$, integrating $\widehat{\Psi}_{r, \theta, k}(s) = \int_\theta^1 \Psi_{r, \theta, k}(u)\,u^{s-1}\,du$ by parts once gives $-\frac{1}{s}\int_\theta^1 \Psi'_{r, \theta, k}(u)\,u^s\,du = \frac{1}{s}\int_0^1 \frac{(-1)^k}{k!}\frac{d^k}{d\tau^k}\big[\tau^{r+k}(1-\tau)^{r+k}\big]\big(1 - z(1-\tau)\big)^s\,d\tau$. Integrating by parts $k$ times with respect to $\tau \in [0, 1]$ (all boundary terms at $\tau = 0, 1$ vanish because $r + k > k - 1$) brings down $\frac{d^k}{d\tau^k}\big[(1 - z(1-\tau))^s\big] = z^k s(s-1)\cdots(s-k+1)\big(1 - z(1-\tau)\big)^{s-k}$. Canceling $s$ and applying Euler's integral representation of ${}_2F_1$ yields (12) for $s \ne 0$, and by continuity at $s = 0$. The non-vanishing $\sum_{k=0}^{L-1}|\widehat{\Psi}_{r, \theta, k}(s)|^2 > 0$ for $L \ge 3$ follows immediately from Part 2 and Theorem 1(3)–(4).

**Part 4:** Since $\Psi_{r, \theta, k} \in C^r([0, \infty)) \cap C^{r, 1}([0, \infty)) \subset C^{2,1}([0, \infty))$ ($r \ge 2$) vanishes at $u = 1$ to order $r + 1 \ge 3$, Theorem 3(1) of [24] and polarization give (24). Since $\big\{-\Psi'_{r, \theta, k}\big\}_{k=0}^{L-1}$ are linearly independent polynomials on $(\theta, 1)$ and $\widetilde{\Psi}'_{r, \theta, k}(u) = -u\Psi''_{r, \theta, k}(u)$, the profiles $\big\{\widetilde{\Psi}_{r, \theta, k}\big\}_{k=0}^{L-1}$ are linearly independent in $L^2([\theta, 1], u^{2\sigma-3}\,du)$, so $\det \Gamma_{\sigma, \delta}^{(r, L)} \ge \det \Gamma_{\sigma, \delta}^{(r, L), (1)} > 0$. Finally, at any finite $N \ge 2$ with $M_\delta := N_\delta - N \ge 1$, for each $1 \le k \le M_\delta$, $\Psi_{r, \theta, k}(u) = \tau^{r+1}(1-\tau)^{r+1} Q_{k-1}(\tau)$ where $Q_{k-1}(\tau)$ has exact degree $k - 1$ (since the leading term of $\frac{d^{k-1}}{d\tau^{k-1}}[\tau^{r+k}(1-\tau)^{r+k}]$ is $(-1)^{r+k}\frac{(2r+2k)!}{(2r+k+1)!}\tau^{2r+k+1}$). Since $\tau_n := \frac{u_n - \theta}{1 - \theta} \in (0, 1)$ ($u_n = \frac{n}{N_\delta+1}$, $N + 1 \le n \le N_\delta$) are $M_\delta$ distinct points in $(0, 1)$ where $\tau_n^{r+1}(1-\tau_n)^{r+1} > 0$, the $M_\delta \times M_\delta$ evaluation matrix $\big(\Psi_{r, \theta, k}(u_n)/n\big)_{1 \le n-N, k \le M_\delta}$ is row- and column-equivalent to the nonsingular Vandermonde matrix $(\tau_n^{k-1})_{1 \le n-N, k \le M_\delta}$. Thus $\operatorname{span}\!\big\{\mathcal{D}_{N, \delta}^{(r, 1)}, \dots, \mathcal{D}_{N, \delta}^{(r, M_\delta)}\big\} = \operatorname{span}\{\mathcal{C}_{N+1}, \dots, \mathcal{C}_{N_\delta}\}$, proving (25). $\square$

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## 4. 60-Digit Interval Certification of Mellin Phase Quadrature and Single-Knot Wavelet Superpositions

### 4.1 60-Digit Interval Certification of the Gauss-Jet and Phase-Quadrature Angles at $\rho_1$

In `src/verify_gauss_jet_jacobi_wavelet_duality.py`, we first certify in 60-digit `mpmath.iv` interval arithmetic the exact rational polynomial identities $W_{4, \theta}(u) - W_{3, \theta}(u) \equiv -5\Psi_{2, \theta, 2}(u)$, $\Phi_{4, \theta}(u) \equiv -70\Psi_{2, \theta, 1}(u) + 35\Psi_{2, \theta, 2}(u)$, and the exact rational $L^2$ orthogonality norms (22) for $r = 2, 0 \le k \le 5$: $\big\{\frac{1}{30}, \frac{3}{70}, \frac{2}{45}, \frac{10}{231}, \frac{15}{364}, \frac{7}{180}\big\}$ (with zero off-diagonal inner products).

Next, to enclose the first critical-line zero $\rho_1 = \frac{1}{2} + i\gamma_1$ with the full $B_{12}$ Euler-Maclaurin correction included (`j_em <= 10`, `two_r = 12`, $M_{\mathrm{em}} = 50$), we evaluate $\zeta(s)$ and $\zeta'(s)$ in 60-digit `mpmath.iv` arithmetic 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 $s_0 = \frac{1}{2} + i\gamma_0$ ($\gamma_0 := \frac{14134725141735}{10^{12}}, r_0 := 10^{-11}$), certifying:
$$|\zeta(s_0)| + r_0 \sup_{w \in B(s_0, r_0)} |\zeta'(w) - \zeta'(s_0)| \le 2.4320 \times 10^{-13} < 7.9316 \times 10^{-12} \le r_0\,|\zeta'(s_0)|. \tag{26}$$
By Rouché's theorem and reflection symmetry $D(s_0, r_0) = 1 - \overline{D(s_0, r_0)}$, $\zeta(s)$ has a unique simple zero $\rho_1 \in D(s_0, r_0)$ with $\operatorname{Re}(\rho_1) = 1/2$ and $\gamma_1 \in [\gamma_0 - r_0, \gamma_0 + r_0]$. Propagating $s_1 = \rho_1 - 1 \in -\frac{1}{2} + i[\gamma_0 - r_0, \gamma_0 + r_0]$ into the exact polynomial Mellin transforms at $\delta = 1/4$ ($\theta = 4/5$) in 60-digit `mpmath.iv` arithmetic certifies (`results/gauss_jet_jacobi_wavelet_duality.json`):
- $\big|\widehat{W}_{3, 4/5}(\rho_1 - 1)\big| \in [0.06225173, 0.06225178]$, $\big|\widehat{W}_{4, 4/5}(\rho_1 - 1)\big| \in [0.06485966, 0.06487549]$, and $\big|\widehat{\Phi}_{4, 4/5}(\rho_1 - 1)\big| \in [0.10525374, 0.10573754]$;
- $\big|\widehat{\Psi}_{2, 4/5, 1}(\rho_1 - 1)\big| \in [0.00145862, 0.00145899]$ and $\big|\widehat{\Psi}_{2, 4/5, 2}(\rho_1 - 1)\big| \in [0.00052284, 0.00053039]$;
- **Collinearity of $\{W_{3, 4/5}, W_{4, 4/5}\}$ vs. Phase Quadrature of $\{W_{3, 4/5}, \Psi_{2, 4/5, 1}\}$ and $\{W_{4, 4/5}, \Phi_{4, 4/5}\}$:**
  $$\cos\Delta\arg\!\left(\widehat{W}_{4, 4/5},\, \widehat{W}_{3, 4/5}\right) \in [0.99974,\; 1.00024], \qquad \sin\Delta\arg\!\left(\widehat{W}_{4, 4/5},\, \widehat{W}_{3, 4/5}\right) \in [0.00464,\; 0.00478] \quad (\Delta\arg \approx +0.27^\circ),$$
  $$\cos\Delta\arg\!\left(\widehat{\Psi}_{2, 4/5, 1},\, \widehat{W}_{3, 4/5}\right) \in [-0.02014,\; -0.02001], \qquad \sin\Delta\arg\!\left(\widehat{\Psi}_{2, 4/5, 1},\, \widehat{W}_{3, 4/5}\right) \in [0.99955,\; 1.00004] \quad (\Delta\arg \approx +91.15^\circ),$$
  $$\cos\Delta\arg\!\left(\widehat{\Phi}_{4, 4/5},\, \widehat{W}_{4, 4/5}\right) \in [-0.16030,\; -0.15720], \qquad \sin\Delta\arg\!\left(\widehat{\Phi}_{4, 4/5},\, \widehat{W}_{4, 4/5}\right) \in [-0.99205,\; -0.98261] \quad (\Delta\arg \approx -99.14^\circ);$$
- **Positive $2 \times 2$ $j = 1$ Gram Determinants:** For $\big\{\mathcal{F}_{N, 1/4}^{(2)}, \mathcal{D}_{N, 1/4}^{(2, 1)}\big\}$, 60-digit interval integration on $X \in [1, 5/4]$ certifies $\det \Gamma_{0.60, 1/4}^{(2, 2), (1)} \ge 0.025375$, $\det \Gamma_{0.75, 1/4}^{(2, 2), (1)} \ge 0.017415$, and $\det \Gamma_{0.90, 1/4}^{(2, 2), (1)} \ge 0.012971$.

![Figure 1: (a) Co-located Jacobi-Rodrigues wavelet tapers W_{3, 4/5} (k=0) and rescaled defect modes Psi_{2, 4/5, k} (k=1, 2, 3) on [4/5, 1]. (b) Percentage of the finite-support annihilator distance d_{N, N_delta}(0.75)^2 captured by the L-dimensional co-located Jacobi-Rodrigues wavelet basis W_{N, delta}^{(2, L)} at N = 160 across L in {1, 2, 4, 6, 8} for delta in {1/4, 1/2, 1}.](p13-figure-1.png)

### 4.2 Single-Knot Subspace Exhaustion and 60-Digit Interval Certificates

For any $L \ge 1$ and any vector $\lambda = (\lambda_0, \lambda_1, \dots, \lambda_{L-1})^T \in \mathbb{Q}^L$ such that $\mathcal{L}_\lambda := \sum_{k=0}^{L-1} \lambda_k \mathcal{D}_{N, \delta}^{(2, k)} \ne 0$ (which holds automatically whenever $\lambda_0 \ne 0$, since $\beta_N(\mathcal{L}_\lambda) = \lambda_0 / N \ne 0$ for $\delta < 1$), $\mathcal{L}_\lambda \in \mathcal{W}_{N, \delta}^{(2, L)} \subset \operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}(\tilde{\rho}_{1/2}, \dots, \tilde{\rho}_{1/N})$ has exact rational coordinates $\alpha_n(\mathcal{L}_\lambda) \in \mathbb{Q}$, satisfies $\mathcal{L}_\lambda(\tilde{\rho}_{1/j}) = 0$ identically in $\mathbb{Q}$ for all $2 \le j \le N$, and certifies
$$d_N(\sigma)^2 \ge d_{N, N_\delta}(\sigma)^2 \ge \frac{|\mathcal{L}_\lambda(E)|^2}{\|\mathcal{L}_\lambda\|_{S_\sigma^*}^2} \qquad (\mathcal{L}_\lambda \ne 0). \tag{27}$$

Table 1 reports the progression of the unnormalized rescaled bound $F_{\mathrm{unnorm}}(N) := d_{\mathrm{unnorm}, N}(0.75)^2 N^{0.5} = \frac{1}{2\sigma} d_N(0.75)^2 N^{0.5}$ achieved by the co-located Jacobi-Rodrigues wavelet basis $\mathcal{W}_{N, \delta}^{(2, L)}$ for $L \in \{1, 2, 4, 6, 8\}$ at fixed rational scales $\delta \in \{1/4, 1/2, 1\}$ ($M = N_\delta \in \{1.25N, 1.5N, 2N\}$) across $N \in \{40, 80, 160, 320, 640, 1280, 2560, 5120\}$, compared against the full $(N_\delta - N + 1)$-dimensional finite-support subspace $\operatorname{Ann}_{S_\sigma^*}^{(N_\delta)}$ for $N \le 320$.

| Scale $\delta$ ($\theta$) | $N$ ($M = N_\delta$, $\dim \operatorname{Ann}^{(M)}$) | $L = 1$ Single Taper $W_{3, \theta}$ | $L = 2$ Quadrature $\{W_3, \Psi_1\}$ | $L = 4$ Wavelet Packet | $L = 6$ Wavelet Packet ($\%$ of $\operatorname{Ann}^{(M)}$) | $L = 8$ Wavelet Packet ($\%$ of $\operatorname{Ann}^{(M)}$) | Full Subspace $\operatorname{Ann}_{S_\sigma^*}^{(M)}$ (Num.) | Co-Located Gain $L = 8 / L = 1$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $1/4$ ($4/5$) | $40$ ($50$, $11$) | $0.000860$ | $0.000893^*$ | $0.001648$ | $0.002120^*$ ($53.6\%$) | $0.002302$ ($58.2\%$) | $0.003953$ | $2.68\times$ |
| $1/4$ ($4/5$) | $80$ ($100$, $21$) | $0.000797$ | $0.003353^*$ | $0.004659$ | $0.004838^*$ ($79.4\%$) | $0.005217$ ($85.6\%$) | $0.006093$ | $6.54\times$ |
| $1/4$ ($4/5$) | $160$ ($200$, $41$) | $0.000604$ | $0.001169^*$ | $0.002188$ | $0.002652^*$ ($67.1\%$) | $0.002821$ ($71.4\%$) | $0.003951$ | $4.67\times$ |
| $1/4$ ($4/5$) | $320$ ($400$, $81$) | $0.000661$ | $0.001170$ | $0.001333$ | $0.001529$ ($77.6\%$) | $0.001580$ ($80.2\%$) | $0.001971$ | $2.39\times$ |
| $1/4$ ($4/5$) | $640$ ($800$, $161$) | $0.000783$ | $0.001802$ | $0.003097$ | $0.003111$ | $0.003127$ | — | $4.00\times$ |
| $1/4$ ($4/5$) | $1280$ ($1600$, $321$) | $0.002118$ | $0.002479$ | $0.002570$ | $0.002595$ | $0.002606$ | — | $1.23\times$ |
| $1/4$ ($4/5$) | $2560$ ($3200$, $641$) | $0.003070$ | $0.003096$ | $0.004267$ | $0.004675$ | $0.004744$ | — | $1.55\times$ |
| $1/4$ ($4/5$) | $5120$ ($6400$, $1281$) | $0.000442$ | $0.000751$ | $0.000751$ | $0.000886$ | $0.000977$ | — | $2.21\times$ |
| $1/2$ ($2/3$) | $40$ ($60$, $21$) | $0.000857$ | $0.001300^*$ | $0.001730$ | $0.002148^*$ ($40.6\%$) | $0.003021$ ($57.1\%$) | $0.005291$ | $3.52\times$ |
| $1/2$ ($2/3$) | $80$ ($120$, $41$) | $0.001032$ | $0.003677^*$ | $0.006373$ | $0.006637^*$ ($78.0\%$) | $0.007260$ ($85.4\%$) | $0.008506$ | $7.03\times$ |
| $1/2$ ($2/3$) | $160$ ($240$, $81$) | $0.000709$ | $0.003430^*$ | $0.005949$ | $0.006362^*$ ($82.2\%$) | $0.006716$ ($86.8\%$) | $0.007741$ | $9.47\times$ |
| $1/2$ ($2/3$) | $320$ ($480$, $161$) | $0.000190$ | $0.001031$ | $0.002854$ | $0.003006$ ($76.1\%$) | $0.003084$ ($78.0\%$) | $0.003951$ | $16.24\times$ |
| $1/2$ ($2/3$) | $640$ ($960$, $321$) | $0.000008$ | $0.000673$ | $0.002244$ | $0.003418$ | $0.003699$ | — | $486.94\times$ |
| $1/2$ ($2/3$) | $1280$ ($1920$, $641$) | $0.000299$ | $0.003126$ | $0.004021$ | $0.004029$ | $0.004146$ | — | $13.85\times$ |
| $1/2$ ($2/3$) | $2560$ ($3840$, $1281$) | $0.000656$ | $0.004558$ | $0.006414$ | $0.007097$ | $0.007675$ | — | $11.70\times$ |
| $1/2$ ($2/3$) | $5120$ ($7680$, $2561$) | $0.000758$ | $0.001653$ | $0.002190$ | $0.002552$ | $0.002577$ | — | $3.40\times$ |
| $1$ ($1/2$) | $40$ ($80$, $41$) | $0.000132$ | $0.000495^*$ | $0.002696$ | $0.002788^*$ ($39.7\%$) | $0.002806$ ($39.9\%$) | $0.007024$ | $21.33\times$ |
| $1$ ($1/2$) | $80$ ($160$, $81$) | $0.000087$ | $0.000130^*$ | $0.003680$ | $0.005915^*$ ($60.4\%$) | $0.007135$ ($72.9\%$) | $0.009791$ | $82.36\times$ |
| $1$ ($1/2$) | $160$ ($320$, $161$) | $0.000057$ | $0.000185^*$ | $0.004321$ | $0.004917^*$ ($52.5\%$) | $0.006454$ ($68.9\%$) | $0.009360$ | $112.35\times$ |
| $1$ ($1/2$) | $320$ ($640$, $321$) | $0.000033$ | $0.000819$ | $0.004288$ | $0.004584$ ($65.5\%$) | $0.005609$ ($80.2\%$) | $0.006996$ | $171.21\times$ |
| $1$ ($1/2$) | $640$ ($1280$, $641$) | $0.000010$ | $0.001251$ | $0.002455$ | $0.003312$ | $0.004446$ | — | $433.25\times$ |
| $1$ ($1/2$) | $1280$ ($2560$, $1281$) | $1.8 \times 10^{-7}$ | $0.001259$ | $0.003139$ | $0.004576$ | $0.004586$ | — | $18,980.66\times$ |
| $1$ ($1/2$) | $2560$ ($5120$, $2561$) | $0.000029$ | $0.001440$ | $0.004500$ | $0.006710$ | $0.007270$ | — | $250.91\times$ |
| $1$ ($1/2$) | $5120$ ($10240$, $5121$) | $0.000108$ | $0.001802$ | $0.003475$ | $0.004190$ | $0.004199$ | — | $38.78\times$ |

**Table 1:** Single-knot finite-support subspace exhaustion $F_{\mathrm{unnorm}}(N) = d_{\mathrm{unnorm}, N}(0.75)^2 N^{0.5}$ by the co-located Jacobi-Rodrigues wavelet basis $\mathcal{W}_{N, \delta}^{(2, L)}$ ($L \in \{1, 2, 4, 6, 8\}$) at fixed rational scales $\delta \in \{1/4, 1/2, 1\}$ (`results/gauss_jet_jacobi_wavelet_duality.json`). Entries marked with $*$ at $N \in \{40, 80, 160\}$ ($L \in \{2, 6\}$) 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 rational annihilation residual $\max_{2 \le k \le N}|\mathcal{L}_{\tilde{\lambda}}(\tilde{\rho}_{1/k})| \equiv 0 \in \mathbb{Q}$.

Table 2 lists the 9 certified $L = 6$ integer witness vectors $\tilde{\lambda} \in \mathbb{Z}^6$ and their 60-digit interval-certified lower bounds on $d_N(0.75)^2$ and $F_{\mathrm{unnorm}}(N)$ at $N \in \{40, 80, 160\}$ across $\delta \in \{1/4, 1/2, 1\}$.

| Scale $\delta$ ($\theta$) | $N$ ($M = N_\delta$) | Certified $L = 6$ Integer Witness Vector $\tilde{\lambda} = (\lambda_0, \lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5)^T \in \mathbb{Z}^6$ | Certified $d_N(0.75)^2 \ge$ (`ROUND_FLOOR`) | Certified $F_{\mathrm{unnorm}}(N) \ge$ (`ROUND_FLOOR`) | $\%$ of Numerical $\operatorname{Ann}_{S_\sigma^*}^{(M)}$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $1/4$ ($4/5$) | $40$ ($50$) | $(-12086, -521364, 149960, -1000000, 456004, -476002)^T$ | $0.00050293$ | $0.002120$ | $53.6\%$ |
| $1/4$ ($4/5$) | $80$ ($100$) | $(10945, 1000000, 54130, 722041, -256940, 152384)^T$ | $0.00081137$ | $0.004838$ | $79.4\%$ |
| $1/4$ ($4/5$) | $160$ ($200$) | $(-12349, -779690, -199032, -1000000, -177976, -516311)^T$ | $0.00031452$ | $0.002652$ | $67.1\%$ |
| $1/2$ ($2/3$) | $40$ ($60$) | $(-17308, 1000000, 628098, 999591, 778621, 392951)^T$ | $0.00050966$ | $0.002148$ | $40.6\%$ |
| $1/2$ ($2/3$) | $80$ ($120$) | $(1291, -567308, -1000000, 118086, -321391, -126519)^T$ | $0.00111310$ | $0.006637$ | $78.0\%$ |
| $1/2$ ($2/3$) | $160$ ($240$) | $(-4182, 949368, 1000000, -276288, -362077, -623321)^T$ | $0.00075450$ | $0.006362$ | $82.2\%$ |
| $1$ ($1/2$) | $40$ ($80$) | $(-3878, 465706, -1000000, 94815, -233598, 122292)^T$ | $0.00066128$ | $0.002788$ | $39.7\%$ |
| $1$ ($1/2$) | $80$ ($160$) | $(3686, -143801, 711266, 252669, 52392, 1000000)^T$ | $0.00099200$ | $0.005915$ | $60.4\%$ |
| $1$ ($1/2$) | $160$ ($320$) | $(-6328, -59863, -1000000, -109750, 13412, -635941)^T$ | $0.00058317$ | $0.004917$ | $52.5\%$ |

**Table 2:** 60-digit `mpmath.iv` interval certificates for the $L = 6$ co-located Jacobi-Rodrigues wavelet basis $\mathcal{W}_{N, \delta}^{(2, 6)}$ at single fixed supports $M = N_\delta \in \{1.25N, 1.5N, 2N\}$ across $N \in \{40, 80, 160\}$ at $\sigma = 0.75$ (`results/gauss_jet_jacobi_wavelet_duality.json`, which also archives all 9 $L = 2$ certificates).

![Figure 2: Elimination of single-scale nodal zeros across N in [40, 5120] by the co-located Jacobi-Rodrigues wavelet packet basis W_{N, delta}^{(2, L)} (L in {1, 2, 6, 8}) at fixed linear support: (a) delta = 1/2 (M = 1.5N, theta = 2/3) and (b) delta = 1 (M = 2N, theta = 1/2), compared against the full finite-support subspace Ann^{(M)} for N <= 320.](p13-figure-2.png)

---

## 5. Self-Contained Verification & Reproduction

All exact rational polynomial identity checks, exact rational $L^2$ orthogonality checks, the 60-digit `mpmath.iv` Euler-Maclaurin + Rouché zero enclosure of $\rho_1 \in D(s_0, r_0)$ (including the full $B_{12}$ Euler-Maclaurin correction), the 60-digit interval evaluations of $\widehat{W}_{3, 4/5}(\rho_1-1)$, $\widehat{W}_{4, 4/5}(\rho_1-1)$, $\widehat{\Phi}_{4, 4/5}(\rho_1-1)$, $\widehat{\Psi}_{2, 4/5, 1}(\rho_1-1)$, $\widehat{\Psi}_{2, 4/5, 2}(\rho_1-1)$ and their phase-quadrature cosines/sines, the 60-digit interval certifications of $\det \Gamma_{\sigma, 1/4}^{(2, 2), (1)} > 0$, and all 18 exact rational + 60-digit interval wavelet packet certificates are implemented in the standalone, zero-local-dependency script `src/verify_gauss_jet_jacobi_wavelet_duality.py` (with minimum package versions specified in `requirements.txt` inside the supplement archive), and all full 60-digit interval endpoints and integer witness vectors are archived in `results/gauss_jet_jacobi_wavelet_duality.json`.

---

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