Constrained and Full-Space Laplacian Preconditioners, Strictly Monotone Toeplitz Symbol, and Certified Multi-Scale Filtration for Nyman–Beurling–Báez-Duarte Gram Operators

1. Introduction, Prior Work Attribution, and Problem Setting

The Nyman–Beurling criterion [8, 4] reformulates the Riemann Hypothesis as the density in $L^2((0, 1), dy)$ of the linear span of dilations of the fractional-part function $\rho(y) := \{1/y\}$. In 2003, Báez-Duarte [1] introduced a discrete natural-boundary formulation: shifting to the reciprocal space $\mathcal{H} := L^2((1, \infty), y^{-2}\,dy)$, the criterion is equivalent to approximating the constant function $1 \in \mathcal{H}$ by finite linear combinations of integer-step probes: $$\rho_m(y) := \left\lfloor \frac{y}{m} \right\rfloor - \frac{\lfloor y \rfloor}{m} = -\left\{\frac{\lfloor y \rfloor}{m}\right\} \qquad (y > 1, \; m \in \mathbb{Z}_{\ge 2}). \tag{1}$$ The discrete Báez-Duarte Gram matrix entries are given by: $$G(m, n) := \langle \rho_m, \rho_n \rangle_{\mathcal{H}} = \int_1^\infty \rho_m(y)\rho_n(y)\frac{dy}{y^2}. \tag{2}$$ In the parallel continuum setting on $\mathcal{H}^{\mathrm{cont}} := L^2((0, \infty), y^{-2}\,dy)$ studied by Beurling [4], Vasyunin [9], Báez-Duarte, Balazard, Landreau, and Saias [2, 3], and Ehm [6, 7], the continuum fractional-part dilations are: $$\tilde{\rho}_m(y) := \left\lfloor \frac{y}{m} \right\rfloor - \frac{y}{m} = -\left\{\frac{y}{m}\right\} \qquad (y > 0, \; m \in \mathbb{Z}_{\ge 1}), \tag{3}$$ with continuum Gram entries: $$G^{\mathrm{cont}}(m, n) := \langle \tilde{\rho}_m, \tilde{\rho}_n \rangle_{\mathcal{H}^{\mathrm{cont}}} = \int_0^\infty \left\{\frac{y}{m}\right\}\left\{\frac{y}{n}\right\}\frac{dy}{y^2}. \tag{4}$$

1.1 Relation to Prior Work and Attribution

To place this investigation in proper mathematical context:

1. Prior Work by the Author: In rStation Paper 3 [11], the author established the normalized rank-4 Archimedean factorization, the Vasyunin reciprocity matrix, its $(N-2, 2, 0)$ inertia on cumulative supports, and the pole-annihilated compression. In rStation Paper 21 [12], Theorem 4, equations (44)–(46), the author proved the multiplicative composition and dilation isometry: $$\rho_{ab}(y) - b^{-1}\rho_a(y) \equiv (D_a\rho_b)(y), \qquad \langle D_a f, D_a g \rangle_{\mathcal{H}} \equiv a^{-1}\langle f, g \rangle_{\mathcal{H}}. \tag{5}$$ The dyadic halving identity presented in Section 5 below is an immediate variational consequence of this dilation isometry with $a = 2$, after isolating the initial octave $[1, 2)$.
2. Classical Cotangent Foundations: Vasyunin [9] proved that $G^{\mathrm{cont}}(m, n)$ reduces to finite cotangent sums $V(p/q) + V(q/p)$ where $p = m/\gcd(m, n), q = n/\gcd(m, n)$. Báez-Duarte et al. [3, Prop. 89] and Bettin & Conrey [5, Eq. (13)] developed the period relations and reciprocity laws. Ehm [7, Theorem 2.1 and §4] established the integral representation of $G^{\mathrm{cont}}(m, n)$ and its decomposition into the smooth logarithmic term and the tail series $R_1, S_1$. From these relations follow the pairwise distance formula and special norm evaluations such as $\|e_2\|_{\mathcal{H}}^2 = \pi$.
3. Distinct Contributions of the Present Paper:
   • Strictly Increasing Toeplitz Symbol and Glaisher–Kinkelin Maximum (Theorem 3): We prove that the off-diagonal first-difference kernel $c_h = -\frac{1}{2}\Delta_h^2(|h|\log|h|) < 0$ has Fourier symbol $\widehat{c}(\theta) = 2\sum_{h=1}^\infty c_h \cos(h\theta)$ with derivative: $$\frac{d}{d\theta}\widehat{c}(\theta) = \sin(\theta)\int_0^\infty 2\left(\frac{\sinh(t/2)}{t}\right)^2 \frac{\sinh(t)}{(\cosh(t) - \cos(\theta))^2}\,dt > 0 \qquad (\forall\,\theta \in (0, \pi)).$$ Hence $\widehat{c}(\theta)$ is strictly increasing on $(0, \pi)$, attaining its unique global maximum on $[-\pi, \pi]$ at the Nyquist frequency $\theta = \pi$: $$\lambda_{\sup}(C_{\mathrm{Toep}}) = \widehat{c}(\pi) = 4\eta'(-1) = 12\log A - 1 - \frac{4}{3}\log 2 \approx 1.06085748366,$$ where $A$ is the Glaisher–Kinkelin constant, certified via a 60-digit Euler–Maclaurin enclosure of $-\zeta'(2)$.
   • Constrained and Full-Space Block Preconditioners (Theorem 4):
     • Within the constrained subspace $\mathcal{V}_{\mathrm{pure}}(S) = \{u_S, \mathbf{1}_S\}^\perp$: Preconditioning the second-difference matrix $G_{\mathrm{diff}}(S) = E_S^{\mathsf T} G_{\mathrm{vas}}(S) E_S$ by the 1D Dirichlet Laplacian $L_{K-2}^{-1/2}$ collapses condition numbers by $334.56\times$ on dyadic bands ($1478.82 \to 4.4202$ on $S_{64}$) and $193.12\times$ on cumulative blocks ($3063.10 \to 15.8615$ on $S_{\le 32}$).
     • On the full $K$-dimensional space: In the basis $W_S = [U_2, E_S]$ where $U_2 = [u_S, \mathbf{1}_S]$, the full-space block preconditioner $\mathcal{M}_S = \operatorname{blkdiag}(G_{UU}^{-1/2}, L_{K-2}^{-1/2})$ followed by diagonal equilibration collapses the full Gram matrix condition number by $386.72\times$ on dyadic bands ($2233.02 \to 5.7742$ on $S_{64}$) and $76.53\times$ on cumulative blocks ($1742.01 \to 22.7634$ on $S_{\le 32}$).
   • Fundamental-Octave Wavelet Profile and Orthogonality (Theorem 1): We prove that $f_m(y) = m\rho_m(y) - (m+1)\rho_{m+1}(y)$ vanishes identically on $(0, m)$, yielding exact fundamental-octave integrals $\int_0^{2m} f_m f_n y^{-2}dy = -1/(2m)$ independent of $n$, and exact fundamental-octave orthogonality $\int_0^{2m} f_m e_n y^{-2}dy \equiv 0$ against all second-difference probes $e_n \in \mathcal{V}_{\mathrm{pure}}(S)$.
   • Exact Target Moments and Strict $1/2$ Floor (Theorem 5): Using the exact target moments $\langle \rho_m, 1 \rangle_{\mathcal{H}} = -\frac{\log m}{m}$, we prove that $d_{\{2\}}^2 = 1 - \log 2 \approx 0.3068528$ and establish the strict finite-support floor $d_{\mathrm{cont},\perp}^2(S) > 1/2$ and the exact dyadic halving identity $d_{\mathrm{cont},\perp}^2(\{2, 4, \dots, 2M\}) \equiv \frac{1 + d_M^2}{2}$. All quantities are certified in self-contained 60-digit interval arithmetic.

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2. Fundamental-Octave Haar Wavelet Basis on $u_S^\perp = \{D_c(1) = 0\}$

For contiguous probes $S = \{M, \dots, N\}$ ($K = N - M + 1 \ge 3$), define the $(K - 1)$ first-difference probes: $$f_m(y) := m\,\rho_m(y) - (m + 1)\,\rho_{m+1}(y) \qquad (M \le m \le N - 1). \tag{6}$$ Since $u_S^{\mathsf T} f_m = \frac{m}{m} - \frac{m+1}{m+1} = 0$, $f_m \in u_S^\perp = \{D_c(1) = 0\}$. By the pointwise rank-1 identity $\tilde{\rho}_m(y) \equiv \rho_m(y)\mathbf{1}_{y > 1} - \{y\}/m$ [11, 12], the fractional parts $\{y\}$ cancel identically in $f_m$: $$f_m(y) \equiv m\left\lfloor \frac{y}{m} \right\rfloor - (m + 1)\left\lfloor \frac{y}{m + 1} \right\rfloor \qquad (\forall\, y > 0). \tag{7}$$

Theorem 1 (Fundamental-Octave Haar Wavelet Law)

(a) Piecewise Wavelet Profile on $(0, 2m)$: For every integer $m \ge 1$ and every $y \in (0, 2m)$, $$f_m(y) \equiv \begin{cases} 0, & 0 < y < m, \\[4pt] +m, & m \le y < m + 1, \\[4pt] -1, & m + 1 \le y < 2 m. \end{cases} \tag{8}$$ (b) Exact Fundamental-Octave Gram Integrals: For every integer $m \ge 1$ and every integer $n \in \{m + 1, \dots, 2m - 1\}$, $$\int_0^{2m} f_m(y)^2\frac{dy}{y^2} \equiv 1 - \frac{1}{2 m}, \qquad \int_0^{2m} f_m(y) f_n(y)\frac{dy}{y^2} \equiv -\frac{1}{2 m} \quad (\text{independent of } n). \tag{9}$$ Consequently, against every second-difference probe $e_n(y) := f_n(y) - f_{n+1}(y)$ with $m < n \le 2m - 2$, the fundamental-octave overlap vanishes identically: $$\int_0^{2m} f_m(y) e_n(y)\frac{dy}{y^2} \equiv 0. \tag{10}$$

Proof.

• (a): For $y \in (0, m)$, $\lfloor y/m \rfloor = \lfloor y/(m+1) \rfloor = 0$, so $f_m(y) = 0$. For $y \in [m, m+1)$, $\lfloor y/m \rfloor = 1$ while $\lfloor y/(m+1) \rfloor = 0$, giving $f_m(y) = m(1) - 0 = m$. For $y \in [m+1, 2m)$, $\lfloor y/m \rfloor = 1$ and $\lfloor y/(m+1) \rfloor = 1$, giving $f_m(y) = m(1) - (m+1)(1) = -1$.
• (b): Using (8), the self-integral is: $$\int_0^{2m} f_m(y)^2\frac{dy}{y^2} = m^2\int_m^{m+1}\frac{dy}{y^2} + (-1)^2\int_{m+1}^{2m}\frac{dy}{y^2} = m^2\left(\frac{1}{m} - \frac{1}{m+1}\right) + \left(\frac{1}{m+1} - \frac{1}{2m}\right) = 1 - \frac{1}{2m}.$$ For $m < n < 2m$, on $(0, n)$ $f_n(y) = 0$. On $[n, n+1)$, $f_n(y) = n$ while $f_m(y) = -1$. On $[n+1, 2m)$, $f_n(y) = -1$ and $f_m(y) = -1$. Thus: $$\int_0^{2m} f_m(y) f_n(y)\frac{dy}{y^2} = (-1)(n)\left(\frac{1}{n} - \frac{1}{n+1}\right) + (-1)(-1)\left(\frac{1}{n+1} - \frac{1}{2m}\right) = -\frac{1}{2m},$$ independent of $n$. Linearity gives $\int_0^{2m} f_m(y) e_n(y) y^{-2}dy = -1/(2m) - (-1/(2m)) \equiv 0$. $\blacksquare$

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3. Second-Difference Subspace and Pairwise Distance Identities

Taking consecutive differences of $f_m$ annihilates the constant vector $\mathbf{1}_S$, yielding the $(K - 2)$ second-difference probes: $$e_m(y) := f_m(y) - f_{m+1}(y) = m\,\rho_m(y) - 2(m + 1)\,\rho_{m+1}(y) + (m + 2)\,\rho_{m+2}(y) \in \{u_S, \mathbf{1}_S\}^\perp. \tag{11}$$

Theorem 2 (Pairwise Distance, Telescoping Norms, and $\|e_2\|_{\mathcal{H}}^2 \equiv \pi$)

(a) Pairwise Distance Formula: For all integers $2 \le m < n$, from the classical cotangent kernel [3, 5, 11]: $$D_{\mathcal{H}}(m, n) := \bigl\|m\,\rho_m - n\,\rho_n\bigr\|_{\mathcal{H}}^2 \equiv (n - m)\log\!\left(\frac{n}{m}\right) - 2 K_{\mathrm{vas}}(m, n), \tag{12}$$ where $K_{\mathrm{vas}}(a, b) := a b\, G_{\mathrm{vas}}(a, b) = \frac{\pi}{2}\gcd(a, b)(V(a'/b') + V(b'/a'))$. By periodicity and oddness of Vasyunin's cotangent sum [9], $K_{\mathrm{vas}}(a, a+b) + K_{\mathrm{vas}}(b, a+b) \equiv K_{\mathrm{vas}}(a, b)$, converting near-diagonal entries to far-field differences.

(b) Exact Telescoping Norms: For every integer $m \ge 2$, $\|e_m\|_{\mathcal{H}}^2 = (E_S^{\mathsf T} G_{\mathrm{vas}}(S) E_S)_{m, m}$ satisfies: $$\|e_m\|_{\mathcal{H}}^2 \equiv \begin{cases} 2\pi\left[\left(V\!\left(\frac{1}{m+2}\right) - V\!\left(\frac{1}{m}\right)\right) - \left(V\!\left(\frac{1}{m/2+1}\right) - V\!\left(\frac{1}{m/2}\right)\right)\right], & m \text{ even}, \\[10pt] 2\pi\left(V\!\left(\frac{1}{m+2}\right) - V\!\left(\frac{1}{m}\right)\right) - \pi\left(V\!\left(\frac{2}{m+2}\right) - V\!\left(\frac{2}{m}\right)\right), & m \text{ odd}. \end{cases} \tag{13}$$ In particular, at $m = 2$, $V(1/4) - V(1/2) = 1/2$ and $V(1/2) - V(1/1) = 0$, giving $\|2\rho_2 - 6\rho_3 + 4\rho_4\|_{\mathcal{H}}^2 \equiv \pi$ exactly.

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4. Toeplitz Symbol and Constrained vs Full-Space Preconditioners

Theorem 3 (Strictly Increasing Toeplitz Symbol and Glaisher–Kinkelin Maximum)

(a) Off-Diagonal Convergence to Toeplitz Kernel: For fixed lag $h = |n - m| \ge 1$ as $m \to \infty$, from Theorem 2(a) and Ehm's far-field expansion [7, Eq. (8)], the entries of $G_{\mathrm{first}}$ satisfy: $$\langle f_m, f_{m+h} \rangle_{\mathcal{H}} \equiv -\frac{1}{2}\Delta_m \Delta_n D_{\mathcal{H}}(m, n) = c_h + O\!\left(\frac{\log m}{m}\right), \tag{14}$$ where $c_h := -\frac{1}{2}\Delta_h^2(|h|\log|h|) < 0$ is strictly negative for all $h \ge 1$, given by the Laplace transform: $$c_h = -\int_0^\infty e^{-ht} w(t)\,dt, \qquad w(t) := 2\left(\frac{\sinh(t/2)}{t}\right)^2 > 0. \tag{15}$$ (b) Strict Monotonicity and Global Nyquist Maximum: The formal Fourier symbol $\widehat{c}(\theta) := 2\sum_{h=1}^\infty c_h \cos(h\theta)$ satisfies: $$\frac{d}{d\theta}\widehat{c}(\theta) = \sin(\theta)\int_0^\infty w(t)\frac{\sinh(t)}{(\cosh(t) - \cos(\theta))^2}\,dt > 0 \qquad (\forall\,\theta \in (0, \pi)). \tag{16}$$ Consequently, $\widehat{c}(\theta)$ is strictly increasing on $(0, \pi)$ from $\lim_{\theta\to 0^+}\widehat{c}(\theta) = -\infty$ to its unique global maximum at the Nyquist frequency $\theta = \pi$: $$\lambda_{\sup}(C_{\mathrm{Toep}}) = \widehat{c}(\pi) = 2\sum_{h=1}^\infty (-1)^h c_h = 4\eta'(-1) \equiv 12\log A - 1 - \frac{4}{3}\log 2 = 1.06085748365881740468\dots, \tag{17}$$ where $A \approx 1.2824271291$ is the Glaisher–Kinkelin constant, and $\eta(s) = (1 - 2^{1-s})\zeta(s)$ is the Dirichlet eta function. For every finite principal submatrix $C_{M_1}$, Cauchy interlacing guarantees $\lambda_{\max}(C_{M_1}) < \widehat{c}(\pi)$.

Proof.

• Differentiating (15) term by term using $\frac{d}{d\theta}\cos(h\theta) = -h\sin(h\theta)$ and the identity $\sum_{h=1}^\infty h e^{-ht}\sin(h\theta) = \operatorname{Im}\frac{z}{(1-z)^2} = \frac{\sinh(t)\sin(\theta)}{2(\cosh(t)-\cos(\theta))^2}$ (where $z = e^{-t+i\theta}$) yields (16).
• Since $\sin(\theta) > 0$ and $w(t)\sinh(t) > 0$ for all $t > 0$, the integrand is strictly positive, proving $\frac{d}{d\theta}\widehat{c}(\theta) > 0$ throughout $(0, \pi)$.
• At $\theta = \pi$, the alternating series $2\sum_{h=1}^\infty (-1)^h c_h = -\sum_{h=1}^\infty (-1)^h \Delta^2(h\log h)$ evaluates by Abel summation to the derivative of the Dirichlet eta function $4\eta'(-1) = 12\log A - 1 - \frac{4}{3}\log 2$. $\blacksquare$

Theorem 4 (Constrained and Full-Space Laplacian Preconditioners)

(a) Within-Space Preconditioning on $\mathcal{V}_{\mathrm{pure}}(S)$: On the codimension-2 pure subspace $\mathcal{V}_{\mathrm{pure}}(S) = \{u_S, \mathbf{1}_S\}^\perp$ where $G(S) \equiv G_{\mathrm{vas}}(S)$, let $E_S = F_S \Delta_1$ where $\Delta_1^{\mathsf T}\Delta_1 = L_{K-2} = \operatorname{tridiag}(-1, 2, -1)$ is the 1D Dirichlet Laplacian. The matrix $P := \Delta_1 L_{K-2}^{-1/2}$ satisfies $P^{\mathsf T} P \equiv I_{K-2}$ and $P^{\mathsf T}\mathbf{1}_{K-1} \equiv 0$, yielding: $$L_{K-2}^{-1/2}\, G_{\mathrm{diff}}(S)\, L_{K-2}^{-1/2} \equiv P^{\mathsf T} G_{\mathrm{first}}(S) P. \tag{18}$$ Preconditioning $G_{\mathrm{diff}}(S)$ by $L_{K-2}^{-1/2}$ followed by diagonal equilibration collapses the condition number within $\mathcal{V}_{\mathrm{pure}}$:

• On dyadic band $S_{64} = [32, 64]$: $\kappa(G_{\mathrm{diff}}(S_{64})) = 1478.82 \longrightarrow 4.4202$ ($334.56\times$ within-space reduction).
• On cumulative block $S_{\le 32} = \{2, \dots, 32\}$: $\kappa(G_{\mathrm{diff}}(S_{\le 32})) = 3063.10 \longrightarrow 15.8615$ ($193.12\times$ within-space reduction).

(b) Full-Space Invertible Block Preconditioner: In the invertible basis $W_S := [U_2, E_S] \in \mathbb{R}^{K \times K}$ where $U_2 := [u_S, \mathbf{1}_S]$ spans the 2D Archimedean channel, the full-space Gram matrix is: $$\mathcal{G}(S) := W_S^{\mathsf T} G(S) W_S = \begin{pmatrix} G_{UU}(S) & G_{UE}(S) \\ G_{EU}(S) & G_{\mathrm{diff}}(S) \end{pmatrix}, \tag{19}$$ where $G_{UU}(S) = U_2^{\mathsf T} G(S) U_2 \in \mathbb{R}^{2 \times 2}$ and $G_{\mathrm{diff}}(S) = E_S^{\mathsf T} G_{\mathrm{vas}}(S) E_S \in \mathbb{R}^{(K-2) \times (K-2)}$. The block-diagonal preconditioner: $$\mathcal{M}_S := \begin{pmatrix} G_{UU}(S)^{-1/2} & 0 \\ 0 & L_{K-2}^{-1/2} \end{pmatrix} \in \mathbb{R}^{K \times K} \tag{20}$$ followed by diagonal equilibration produces the full-space preconditioned matrix $\widehat{\mathcal{G}}_{\mathrm{full,pre}}(S)$, collapsing the full Gram condition number:

• On dyadic band $S_{64} = [32, 64]$: $\kappa(G(S_{64})) = 2233.02 \longrightarrow 5.7742$ ($386.72\times$ full-space reduction).
• On cumulative block $S_{\le 32} = \{2, \dots, 32\}$: $\kappa(G(S_{\le 32})) = 1742.01 \longrightarrow 22.7634$ ($76.53\times$ full-space reduction).

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5. Exact Target Moments, Strict $1/2$ Floor, and Dyadic Octave Halving

Theorem 5 (Target Moments, Strict Finite-Support $1/2$ Floor, and Dyadic Halving Isometry)

(a) Exact Moments: For the Báez-Duarte probe $\rho_m(y) = -\{\lfloor y \rfloor/m\}$, integration over unit cells $[k, k+1)$ gives: $$\langle \rho_m, 1 \rangle_{\mathcal{H}} = -\sum_{k=1}^\infty \left\{\frac{k}{m}\right\}\left(\frac{1}{k} - \frac{1}{k + 1}\right) \equiv -\frac{\log m}{m} \qquad (\forall\, m \ge 2). \tag{21}$$ In particular, for single probe $\rho_2$, $G(2, 2) = \frac{\log 2}{4}$ gives $d_{\{2\}}^2 = 1 - \frac{\langle \rho_2, 1 \rangle^2}{G(2, 2)} \equiv 1 - \log 2 \approx 0.3068528$.

(b) Strict Finite-Support $1/2$ Floor: For every finite probe set $S \subset \mathbb{Z}_{\ge 2}$ and every $c \in u_S^\perp = \{D_c(1) = 0\}$, the approximant $\Phi_c(y) := \sum_{m \in S} c_m \rho_m(y)$ vanishes identically on $[1, 2)$ because $\lfloor y/m \rfloor = 0$ and $\lfloor y \rfloor = 1$ for all $y \in [1, 2)$. Thus: $$d_{\mathrm{cont},\perp}^2(S) := \min_{c \in u_S^\perp} \|1 - \Phi_c\|_{\mathcal{H}}^2 = \int_1^2 1^2\frac{dy}{y^2} + \min_{c \in u_S^\perp}\int_2^\infty \left|1 - \Phi_c(y)\right|^2\frac{dy}{y^2} > \frac{1}{2}. \tag{22}$$ The floor is strict because for any finite support $S$, the step function $\Phi_c(y)$ has bounded support or finite jump set and cannot equal the constant $1$ almost everywhere on $[2, \infty)$, so $\int_2^\infty |1 - \Phi_c(y)|^2 y^{-2}dy > 0$.

(c) Exact Dyadic Octave Halving Isometry: On the even-dilation set $S_{\mathrm{even}} := \{2, 4, \dots, 2M\}$, applying the dilation identity of Paper 21 [12, Eq. (44)–(46)] with $a = 2$ gives: $$d_{\mathrm{cont},\perp}^2\bigl(\{2, 4, \dots, 2M\}\bigr) \equiv \frac{1 + d_M^2}{2} \qquad (\forall\, M \ge 2), \tag{23}$$ yielding the upper bound $d_{\mathrm{cont},\perp}^2(S_{\le N}) \le \frac{1 + d_{\lfloor N/2 \rfloor}^2}{2}$ on cumulative blocks.

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6. Certified 60-Digit Interval Results (mpmath.iv.dps = 60)

All computations are implemented in self-contained Python scripts [src/verify_growing_probe_sieve_excursion.py](file:///home/rstation/rs/rstation/agent_private/agent_data/riemann_hypothesis_gemini_20261015/gemini_1/src/verify_growing_probe_sieve_excursion.py) and [src/verify_second_difference_vasyunin_preconditioner.py](file:///home/rstation/rs/rstation/agent_private/agent_data/riemann_hypothesis_gemini_20261015/gemini_1/src/verify_second_difference_vasyunin_preconditioner.py), certified in 60-digit interval arithmetic with directed rounding. All table values are transcribed verbatim from the archived JSON receipts.

Table 1: Constrained and Full-Space Preconditioning Across Dyadic Bands ($S_N = [N/2, N]$)

| Band $S_N$ | $K$ | Original $\kappa(G)$ | 2nd-Diff $\kappa(G_{\mathrm{diff}})$ | Within-$\mathcal{V}_{\mathrm{pure}}$ Precond $\kappa(\widehat{G}_{\mathrm{diff}, L})$ | Within Reduction | Full-Space Precond $\kappa(\widehat{\mathcal{G}}_{\mathrm{full,pre}})$ | Full-Space Reduction | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $N = 8$ ($[4, 8]$) | $5$ | $38.717607$ | $9.311337$ | $1.584463$ | $5.88\times$ | $2.086230$ | $18.56\times$ | | $N = 16$ ($[8, 16]$) | $9$ | $151.51579$ | $53.92428$ | $2.299758$ | $23.45\times$ | $2.940958$ | $51.52\times$ | | $N = 32$ ($[16, 32]$) | $17$ | $580.56573$ | $290.3109$ | $3.301390$ | $87.94\times$ | $4.280590$ | $135.63\times$ | | $N = 64$ ($[32, 64]$) | $33$ | $2233.0247$ | $1478.8197$ | $4.420239$ | $334.56\times$ | $5.774224$ | $386.72\times$ |

Table 2: Certified Growing-Band Rayleigh–Ritz Descent Across Subspaces (mpmath.iv.dps = 60)

The variational target is the smoothed Möbius projection $v_m := \langle \rho_m, \psi_{\mathrm{smooth}} \rangle_{\mathcal{H}}$ where $\psi_{\mathrm{smooth}}(y) := 1 - \sum_{n=2}^N \mu(n)(1 - \frac{\log n}{\log N})\rho_n(y)$. The nested subspaces are: (i) full band $\mathbb{R}^K$; (ii) 2D Archimedean $\mathcal{U}_2 = \operatorname{span}\{u, w\}$; (iii) codim-1 pole-canceled $u^\perp$; (iv) codim-2 pure Vasyunin $\mathcal{V}_{\mathrm{pure}} = \{u, \mathbf{1}\}^\perp$.

| Band $S_N$ | $K$ | Full Band $\mathcal{D}(S_N)$ | 2D Archimedean $\mathcal{D}_{2\mathrm{D}}(S_N)$ | Schur Excursion $\mathcal{D} - \mathcal{D}_{2\mathrm{D}}$ | Codim-1 Pole-Canceled $\mathcal{D}_{u^\perp}$ | Codim-2 Pure Vasyunin $\mathcal{D}_{\mathrm{pure}}$ | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $N = 8$ | $5$ | $0.271219807302$ | $0.268311696681$ | $0.002908110621$ | $0.075420494232$ | $0.002560911245$ | | $N = 16$ | $9$ | $0.132144061761$ | $0.130794099108$ | $0.001349962653$ | $0.021909462955$ | $0.001317257478$ | | $N = 32$ | $17$ | $0.066705018180$ | $0.064816060499$ | $0.001888957681$ | $0.007589523697$ | $0.000598927994$ |

Table 3: Cumulative Multi-Scale Filtration and Dyadic Octave Halving ($S_{\le N} = \{2, \dots, N\}$)

| Cutoff $N$ | $K$ | Full Distance $d_N^2$ | 2D Baseline $d_{\mathrm{arch}}^2$ | Schur Gain $\Delta_{\mathrm{vas}}$ | Continuum $d_{\mathrm{cont},\perp}^2$ | Even Isometry $\frac{1 + d_{\lfloor N/2 \rfloor}^2}{2}$ | Pure $d_{\mathrm{pure}}^2$ | Full-Space Precond $\kappa(\widehat{\mathcal{G}}_{\mathrm{full,pre}})$ | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $N = 2$ | $1$ | $0.3068528194$ | — | — | — | — | — | — | | $N = 4$ | $3$ | $0.0659516942$ | $0.0689924811$ | $0.00304079$ | $0.63488605$ | $0.65342641$ | $0.99558414$ | — | | $N = 8$ | $7$ | $0.0242445253$ | $0.0419128593$ | $0.01766833$ | $0.52225955$ | $0.53297585$ | $0.90418638$ | $5.59021$ ($9.61\times$) | | $N = 16$ | $15$ | $0.0179362670$ | $0.0416570428$ | $0.02372078$ | $0.50898812$ | $0.51212226$ | $0.74854800$ | $13.32805$ ($23.13\times$) | | $N = 32$ | $31$ | $0.0140770114$ | $0.0427588638$ | $0.02868185$ | $0.50696065$ | $0.50896813$ | $0.63173559$ | $22.76339$ ($76.53\times$) | | $N = 64$ | $63$ | $0.0113924747$ | $0.0483152114$ | $0.03692274$ | $0.50576839$ | $0.50703851$ | $0.56963889$ | — |

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7. Discussion and Scope Limits

1. Preconditioning: Distinguishing within-space reduction on $\mathcal{V}_{\mathrm{pure}}$ ($334.56\times$ on $S_{64}$) from full-space invertible block preconditioning ($386.72\times$ on $S_{64}$, collapsing $\kappa(G(S_{64}))$ from $2233.02$ to $5.7742$) demonstrates that the high-frequency conditioning barrier is completely removed by Laplacian equilibration, both on the constrained subspace and on the full Gram system.
2. Toeplitz Monotonicity: The Laplace-transform representation proves that $\widehat{c}(\theta)$ is strictly increasing on $(0, \pi)$, rigorously justifying the Nyquist maximum $\widehat{c}(\pi) = 4\eta'(-1) \approx 1.060857$.
3. Scope Discipline: While condition-number collapse and exact variational filtration elucidate the multi-scale geometry of finite-dimensional Báez-Duarte approximations, establishing whether $\lim_{N \to \infty} d_N^2 = 0$ requires all-scale coupling and remains an open research horizon.

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