Harmonic–Möbius Unitriangular Factorization, $15/\pi^2$ Dual Norm Asymptotics, and Wavelet Head-to-Tail Reconstruction in the Nyman–Beurling–Báez-Duarte Criterion

1. Introduction and Problem Setting

In Báez-Duarte's discrete natural-boundary formulation [1] of the Nyman–Beurling criterion [9, 3], the Riemann Hypothesis is equivalent to approximating the constant function $1$ in the anchored reciprocal Hilbert space $\mathcal{H} := L^2((1, \infty), y^{-2}\,dy)$ ($\langle f, g \rangle_{\mathcal{H}} := \int_1^\infty f(y)g(y)y^{-2}\,dy$, $\|1\|_{\mathcal{H}}^2 = 1$) by finite linear combinations of discrete fractional-part dilations: $$\rho_m(y) := \left\lfloor \frac{y}{m} \right\rfloor - \frac{\lfloor y \rfloor}{m} = -\left\{\frac{\lfloor y \rfloor}{m}\right\} \qquad (y > 1, \; m \in \mathbb{Z}_{\ge 2}). \tag{1}$$ For cumulative probe blocks $S_{\le N} := \{2, \dots, N\}$ ($N \ge 2$), let $\mathcal{V}_{S_{\le N}} := \operatorname{span}\{\rho_m\}_{m=2}^N \subset \mathcal{H}$ and define the least-squares distance and first-octave residual [16]: $$d_N^2 := \operatorname{dist}_{\mathcal{H}}^2\bigl(1,\, \mathcal{V}_{S_{\le N}}\bigr), \qquad \|r_1(N)\|_{\mathcal{H}}^2 := \operatorname{dist}_{\mathcal{H}}^2\bigl(\chi_1,\, \mathcal{V}_{S_{\le N}}\bigr) = \frac{1}{2} - \frac{1}{4} U_N, \tag{2}$$ where $\chi_1 := \mathbf{1}_{[1, 2)} \in \mathcal{H}$, $u_S(m) := 1/m$, and $U_N := u_{S_{\le N}}^{\mathsf T} G(S_{\le N})^{-1} u_{S_{\le N}}$. As shown by Báez-Duarte, Balazard, Landreau, and Saias [2], Burnol [5], Báez-Duarte [1], and Paper 31 v1 [16], the Riemann Hypothesis is equivalent both to $\lim_{N \to \infty} d_N^2 = 0$ and to $\lim_{N \to \infty} U_N = 2$.

A central structural feature of the discrete probes $\rho_m$ is that they lie inside the closed subspace of unit-cell step functions: $$\mathcal{S}_\infty := \overline{\bigcup_{M \ge 1} \mathcal{S}_M} \subsetneq \mathcal{H}, \qquad \mathcal{S}_M := \operatorname{span}\bigl\{\mathbf{1}_{[d, d+1)}\bigr\}_{d=1}^M \qquad (\dim \mathcal{S}_M = M). \tag{3}$$ Passing from the raw probes $\rho_m$ ($2 \le m \le N$) via telescoping differences $-m\rho_m = \sum_{n=1}^{m-1} f_n$ to the first-difference Haar wavelets $f_n(y) := n\lfloor y/n \rfloor - (n+1)\lfloor y/(n+1) \rfloor$ ($1 \le n \le N-1$, $\rho_1 \equiv 0$) and second differences $e_n := f_n - f_{n+1}$ ($1 \le n \le N-2$) [14, 16] pairs naturally with the Vasyunin–Balazard biorthogonal step functions $L_k \in \mathcal{S}_k$ ($\langle \rho_m, L_k \rangle_{\mathcal{H}} = \delta_{m, k}$ for $m, k \ge 2$, $L_1 := 2\chi_1$) [10, 4, 11, 12] and their harmonic cumulative sums: $$F_m^*(y) := \sum_{k=1}^m \frac{1}{k} L_k(y) = \sum_{d=1}^m d(d+1)\left(\frac{M_1(\lfloor m/d \rfloor)}{d} - \frac{M_1(\lfloor m/(d+1) \rfloor)}{d+1}\right)\mathbf{1}_{[d, d+1)}(y) \in \mathcal{S}_m, \tag{4}$$ where $M_1(x) := \sum_{k=1}^{\lfloor x \rfloor} \frac{\mu(k)}{k}$ [11, 13, 16, 18]. Balazard [11, §2 and Propositions 5–6] introduced the harmonic representers, proved their uniform boundedness and weak convergence in $\mathcal{H}$, and introduced the small-index / isolated-jump split. Ehm [7, Theorem 2.1 and §4] analyzed the associated Gram series, and Paper 9 v1 [18] established the analogous power-saving dual norm asymptotic $\|\mathcal{F}_N\|_{S_\sigma^*}^2 = \frac{\zeta(2\sigma+1)}{2\sigma\,\zeta(4\sigma+2)}N^{2\sigma-1} + O_\sigma(N^{\sigma-1/2})$ in the shifted setting $\sigma \in (1/2, 1)$, while leaving the critical endpoint $\sigma = 1/2$ ($\mathcal{H} = L^2((1, \infty), y^{-2}\,dy)$) open. In Paper 31 v1 [16, Theorem 3], we established the wavelet biorthogonality $\langle f_m, F_n^* \rangle_{\mathcal{H}} \equiv \delta_{m, n}$ for all $m, n \ge 1$ and observed numerically that $\|F_n^*\|_{\mathcal{H}}^2$ approaches $15/\pi^2 \approx 1.519818$. Meanwhile, in the forum discussion of Paper 29 v1 [14] and Paper 31 v1 [16], sylve_recursions [17] observed that on a single octave $2 \le M < N \le 2M$, integrating $f_m f_n$ up to the common cutoff $N$ gives $\int_1^N f_m f_n y^{-2}dy = \delta_{m, n} - \frac{1}{N}$ and introduced the coarse three-probe dyadic bridge $g_A = \rho_A - 3\rho_{2A} + 2\rho_{4A}$.

1.1 Relationship to Prior Work and Summary of Contributions

Building on the harmonic representers of Balazard [11], the shifted-endpoint analysis of Paper 9 v1 [18] and Paper 12 v1 [13], the primal-dual floor-quotient framework of Paper 16 v2 [15], and the wavelet biorthogonality and constraint-restoration framework of Paper 31 v1 [16], this paper establishes four results (we emphasize at the outset that no decay theorem for $d_N$ or resolution of the Riemann Hypothesis is obtained):

1. Endpoint $\frac{15}{\pi^2}$ Norm Limit, Exact Finite-Scale Squarefree Decomposition, and Fixed-Lag Covariance Oscillation vs. Cesàro Mean (Theorem 3): Settling the critical $\sigma = 1/2$ endpoint left open by Paper 9 v1 [18] and refining Balazard's small-index / isolated-jump split [11, Propositions 5–6], we prove the telescoped identity $\langle F_m^*, F_n^* \rangle_{\mathcal{H}} \equiv M_1(m)M_1(n) + \sum_{k=1}^{\min(m, n)}(M_1(\lfloor m/k \rfloor) - M_1(\lfloor m/(k+1) \rfloor))(M_1(\lfloor n/k \rfloor) - M_1(\lfloor n/(k+1) \rfloor))$. Splitting at $R(n) := \lfloor(\sqrt{4n+1}-1)/2\rfloor$ yields the exact finite-scale identity and universal lower bound: $$\|F_n^*\|_{\mathcal{H}}^2 \equiv \sum_{r=1}^{R(n)} \frac{\mu(r)^2}{r^2} + M_1(n)^2 + \sum_{k=1}^{\lfloor n/(R(n)+1) \rfloor}\bigl(M_1(\lfloor n/k \rfloor) - M_1(\lfloor n/(k+1) \rfloor)\bigr)^2 \;\ge\; M_1(n)^2 + \sum_{r=1}^{R(n)} \frac{\mu(r)^2}{r^2}, \tag{5}$$ proving unconditionally via the Prime Number Theorem that the diagonal norm converges to $\lim_{n \to \infty} \|F_n^*\|_{\mathcal{H}}^2 = \frac{\zeta(2)}{\zeta(4)} = \frac{15}{\pi^2}$. By contrast, for every fixed integer lag $h \ge 1$, the off-diagonal covariance $\langle F_n^*, F_{n+h}^* \rangle_{\mathcal{H}}$ does not converge as $n \to \infty$, but oscillates arithmetically according to $\sum_{r \ge h+1,\, \lfloor(n+h)/r\rfloor = \lfloor n/r \rfloor} \frac{\mu(r)^2}{r^2} + O_h((\log n)^{-2})$ (for instance, at $h = 1$, $\lim_{p \to \infty,\, p\text{ prime}}\langle F_{p-1}^*, F_p^* \rangle_{\mathcal{H}} = \frac{15}{\pi^2} - 1$ whereas $\lim_{a \to \infty}\langle F_{2^a-1}^*, F_{2^a}^* \rangle_{\mathcal{H}} = \frac{15}{\pi^2} - \frac{5}{4}$), while its Cesàro average converges to $\lim_{X \to \infty} \frac{1}{X}\sum_{n=1}^X \langle F_n^*, F_{n+h}^* \rangle_{\mathcal{H}} = \sum_{r=h+1}^\infty \frac{\mu(r)^2}{r^2}\bigl(1 - \frac{h}{r}\bigr)$.
2. Harmonic–Möbius Unitriangular Factorization & All-Band Quotient Stiffness Law (Theorem 2): We prove that on $[1, N)$, the arithmetic Haar wavelets $f_m\mathbf{1}_{[1, N)}$ and dual representers $F_n^*$ are intertwined with the canonical single-jump pair $(f_k^{(1)}, g_k^*)$ by the exact inverse upper-unitriangular floor-quotient matrices $(U_N)_{m, k} := H_{\lfloor k/m \rfloor} - H_{\lfloor k/(m+1) \rfloor}$ and $(W_N)_{k, n} := M_1(\lfloor n/k \rfloor) - M_1(\lfloor n/(k+1) \rfloor)$ ($U_N W_N \equiv I_{N-1}$, where $H_q := \sum_{j=1}^q 1/j$). Consequently, the restricted head Gram matrix $H_{\mathrm{head}}(N)_{m, n} := \int_1^N f_m f_n y^{-2}dy$ and the full dual Gram matrix $M_{F, \mathrm{full}}^*(N)_{m, n} := \langle F_m^*, F_n^* \rangle_{\mathcal{H}}$ are exact inverses for every $N \ge 2$: $$H_{\mathrm{head}}(N) \equiv U_N\left(I_{N-1} - \frac{1}{N}\mathbf{1}\mathbf{1}^{\mathsf T}\right)U_N^{\mathsf T}, \qquad M_{F, \mathrm{full}}^*(N) \equiv H_{\mathrm{head}}(N)^{-1} \equiv W_N^{\mathsf T}\left(I_{N-1} + \mathbf{1}\mathbf{1}^{\mathsf T}\right)W_N, \tag{6}$$ with $\det H_{\mathrm{head}}(N) \equiv \frac{1}{N}$ and $\det M_{F, \mathrm{full}}^*(N) \equiv N$. On any floor-quotient band $B_K(N-1) := (\lfloor(N-1)/(K+1)\rfloor, \lfloor(N-1)/K\rfloor]$ ($K \ge 1$), decomposing the row support of $(U_N)_{m, \bullet}$ into $K$ pairwise disjoint integer intervals $I_{m, q} := [qm, q(m+1)) \cap \mathbb{Z}$ of length $|I_{m, q}| = q$ yields the All-Band Harmonic Stiffness Law $\int_1^N f_m f_n y^{-2}dy \equiv H_K \delta_{m, n} - \frac{K^2}{N}$ and $\int_1^N e_m e_n y^{-2}dy \equiv H_K (L)_{m, n}$, together with a Causal Fine-to-Coarse Multi-Octave Schur Complement Bound.
3. Canonical Single-Jump Biorthogonality & Step-Space Parseval Identities (Theorem 1): We introduce the single-jump step basis $f_k^{(1)}(y) := k\,\mathbf{1}_{[k, k+1)}(y) - \mathbf{1}_{[k+1, N)}(y)$ and its dual $g_k^*(y) := \mathbf{1}_{[1, k)}(y) + (k+1)\,\mathbf{1}_{[k, k+1)}(y)$ ($1 \le k \le N-1$) in $\mathcal{S}_{N-1}$, proving $\langle f_j^{(1)}, g_k^* \rangle_{\mathcal{H}} \equiv \delta_{j, k}$, $\langle f_j^{(1)}, f_k^{(1)} \rangle_{\mathcal{H}} \equiv \delta_{j, k} - \frac{1}{N}$, and $\langle g_j^*, g_k^* \rangle_{\mathcal{H}} \equiv \delta_{j, k} + 1$, along with the Universal Single-Jump Parseval Identity on $\mathcal{S}_{N-1}$ and the Single-Octave Parseval and Variance Identities on $[M, N)$ ($N \le 2M$).
4. Wavelet Biorthogonal Interpolation, Head-from-Tail Reconstruction, & Harmonic-Coordinate Woodbury Expression (Theorem 4): As projection and Woodbury consequences of the wavelet biorthogonality of Paper 31 v1 [16] and the factorization (6), we show that every $h \in \mathcal{S}_{N-1}$ has a unique wavelet interpolant $\Phi_{h, N} := \sum_{n=1}^{N-1}\langle h, F_n^* \rangle_{\mathcal{H}} f_n \in \mathcal{V}_{S_{\le N}}$ agreeing with $h$ on $[1, N)$, satisfies $\Phi_{\chi_1, 2M} \equiv \Phi_{1, 2M} - D_2 \Phi_{1, M}$ on $(1, \infty)$, and yields an exact Head-from-Tail Energy Reconstruction Identity for every orthogonal residual $r \in \mathcal{S}_\infty \cap \mathcal{V}_{S_{\le N}}^\perp$. Finally, whereas Paper 16 v2 [15, Theorem 3] already reduced the $(M - N + 1)$-dimensional finite-support dual annihilator bound $\mathrm{LB}_{\mathrm{dual}}(N, M)$ to an $(N-1)$-dimensional primal least-squares problem independent of $M$, applying the Woodbury identity to (6) expresses that $(N-1)\times(N-1)$ reduction explicitly in the harmonic floor-quotient coordinates $U_{M+1}$.

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2. Canonical Single-Jump Biorthogonality and Step-Space Parseval Identities

Let $N \ge 2$ and let $\mathcal{S}_{N-1} := \operatorname{span}\{\mathbf{1}_{[d, d+1)}\}_{d=1}^{N-1} \subset \mathcal{H}$ be the $(N-1)$-dimensional space of unit-cell step functions supported on $[1, N)$, equipped with the induced inner product $\langle g, h \rangle_{\mathcal{H}} = \sum_{d=1}^{N-1} \frac{g(d)h(d)}{d(d+1)}$.

Theorem 1 (Canonical Single-Jump Biorthogonal Bases and Parseval/Variance Identities)

(a) Canonical Single-Jump Biorthogonal Pair: For each integer $1 \le k \le N-1$, define the single-jump step wavelet $f_k^{(1)} \in \mathcal{S}_{N-1}$ and the cumulative dual step function $g_k^* \in \mathcal{S}_{N-1}$ by: $$f_k^{(1)}(y) := k\,\mathbf{1}_{[k, k+1)}(y) - \mathbf{1}_{[k+1, N)}(y), \qquad g_k^*(y) := \mathbf{1}_{[1, k)}(y) + (k+1)\,\mathbf{1}_{[k, k+1)}(y). \tag{7}$$ Then $\{f_k^{(1)}\}_{k=1}^{N-1}$ and $\{g_k^*\}_{k=1}^{N-1}$ are exact biorthogonal bases of $\mathcal{S}_{N-1}$ with rank-1-perturbed identity Gram matrices for all $1 \le j, k \le N-1$: $$\bigl\langle f_j^{(1)}, g_k^* \bigr\rangle_{\mathcal{H}} \equiv \delta_{j, k}, \qquad \bigl\langle 1, f_k^{(1)} \bigr\rangle_{\mathcal{H}} \equiv \frac{1}{N}, \qquad \bigl\langle 1, g_k^* \bigr\rangle_{\mathcal{H}} \equiv 1, \tag{8}$$ $$\bigl\langle f_j^{(1)}, f_k^{(1)} \bigr\rangle_{\mathcal{H}} \equiv \delta_{j, k} - \frac{1}{N}, \qquad \bigl\langle g_j^*, g_k^* \bigr\rangle_{\mathcal{H}} \equiv \delta_{j, k} + 1, \qquad \sum_{k=1}^{N-1} f_k^{(1)}(y) \equiv \mathbf{1}_{[1, N)}(y). \tag{9}$$

(b) Universal Single-Jump Parseval Identity on $\mathcal{S}_{N-1}$: Every step function $h \in \mathcal{S}_{N-1}$ expands uniquely as $h = \sum_{k=1}^{N-1} \langle h, f_k^{(1)} \rangle_{\mathcal{H}} g_k^* = \sum_{k=1}^{N-1} \langle h, g_k^* \rangle_{\mathcal{H}} f_k^{(1)}$ with $\langle h, f_k^{(1)} \rangle_{\mathcal{H}} = \frac{h(k)}{k+1} - \int_{k+1}^N h(y)y^{-2}dy$, and satisfies the exact Parseval identity: $$\int_1^N h(y)^2\frac{dy}{y^2} \;\equiv\; \left(\int_1^N h(y)\frac{dy}{y^2}\right)^2 + \sum_{k=1}^{N-1} \bigl\langle h, f_k^{(1)} \bigr\rangle_{\mathcal{H}}^2. \tag{10}$$

(c) Single-Octave Step-Space Parseval, Variance, and Two-Cell Localization Identities: For any integers $2 \le M < N \le 2M$ ($J := N - M \ge 1$), the restrictions $f_m\big|_{[1, N)}$ ($M \le m < N$) of the arithmetic Haar wavelets $f_m = m\rho_m - (m+1)\rho_{m+1}$ coincide identically with $f_m^{(1)}$, are supported on $[M, N)$, and form an invertible basis of $\mathcal{S}_{[M, N)} := \operatorname{span}\{\mathbf{1}_{[d, d+1)}\}_{d=M}^{N-1}$ satisfying $\sum_{m=M}^{N-1} f_m(y) \equiv M\,\mathbf{1}_{[M, N)}(y)$ on $[M, N)$, while each second difference $e_n = f_n - f_{n+1}$ ($M \le n \le N-2$) localizes on $[1, N)$ to two adjacent cells: $e_n(y)\mathbf{1}_{[1, N)}(y) \equiv n\,\mathbf{1}_{[n, n+1)}(y) - (n+2)\,\mathbf{1}_{[n+1, n+2)}(y)$. Consequently, for every unit-cell step function $g \in \mathcal{S}_\infty$, its local wavelet moments $\alpha_m^{[M, N)}(g) := \int_M^N g(y)f_m(y)y^{-2}dy$ ($M \le m < N$) and second-difference moments $\beta_n^{[M, N)}(g) := \int_M^N g(y)e_n(y)y^{-2}dy \equiv \frac{g(n) - g(n+1)}{n + 1}$ ($M \le n \le N-2$) satisfy: $$\sum_{m=M}^{N-1} \alpha_m^{[M, N)}(g) \;\equiv\; M\int_M^N g(y)\frac{dy}{y^2}, \qquad \int_M^N g(y)^2\frac{dy}{y^2} \;\equiv\; \sum_{m=M}^{N-1} \alpha_m^{[M, N)}(g)^2 + \frac{1}{M}\left(\sum_{m=M}^{N-1} \alpha_m^{[M, N)}(g)\right)^2, \tag{11}$$ $$\beta^{[M, N)}(g)^{\mathsf T} L_{J-1}^{-1} \beta^{[M, N)}(g) \;\equiv\; \sum_{m=M}^{N-1} \alpha_m^{[M, N)}(g)^2 - \frac{1}{J}\left(\sum_{m=M}^{N-1} \alpha_m^{[M, N)}(g)\right)^2 \;\equiv\; \int_M^N g(y)^2\frac{dy}{y^2} - \frac{M N}{N - M}\left(\int_M^N g(y)\frac{dy}{y^2}\right)^2, \tag{12}$$ where $L_{J-1} := D^{\mathsf T} D = \operatorname{tridiag}(-1, 2, -1) \in \mathbb{Z}^{(J-1)\times(J-1)}$ is the Dirichlet Laplacian and the right-hand side of (12) is the exact weighted variance $\operatorname{Var}_{L^2([M, N), y^{-2}dy)}(g)$.

Proof.

• (a): We verify $\langle f_j^{(1)}, g_k^* \rangle_{\mathcal{H}} = \delta_{j, k}$ across all three index orderings of $(j, k)$:
  1. Case $j > k$: $g_k^*$ is supported on $[1, k+1)$ whereas $f_j^{(1)}$ is supported on $[j, N) \subset [k+1, N)$, so their supports are disjoint ($f_j^{(1)} g_k^* \equiv 0$) and $\langle f_j^{(1)}, g_k^* \rangle_{\mathcal{H}} = 0$.
  2. Case $j < k$: Since $j + 1 \le k$, on $[1, k+1)$ where $g_k^*$ is supported we have $f_j^{(1)}(y) = j\,\mathbf{1}_{[j, j+1)}(y) - \mathbf{1}_{[j+1, k)}(y) - \mathbf{1}_{[k, k+1)}(y)$, while $g_k^*(y) = \mathbf{1}_{[1, k)}(y) + (k+1)\,\mathbf{1}_{[k, k+1)}(y)$. Integrating against $y^{-2}\,dy$ yields the exact cancellation: $$\bigl\langle f_j^{(1)}, g_k^* \bigr\rangle_{\mathcal{H}} = \frac{j \cdot 1}{j(j+1)} - \left(\frac{1}{j+1} - \frac{1}{k}\right) - \frac{1 \cdot (k+1)}{k(k+1)} = \frac{1}{j+1} - \frac{1}{j+1} + \frac{1}{k} - \frac{1}{k} = 0.$$
  3. Case $j = k$: $f_k^{(1)}$ and $g_k^*$ overlap only on $[k, k+1)$, where $\langle f_k^{(1)}, g_k^* \rangle_{\mathcal{H}} = \frac{k(k+1)}{k(k+1)} = 1$. Direct integration gives $\langle 1, f_k^{(1)} \rangle_{\mathcal{H}} = \frac{k}{k(k+1)} - (\frac{1}{k+1} - \frac{1}{N}) = \frac{1}{N}$ and $\langle 1, g_k^* \rangle_{\mathcal{H}} = (1 - \frac{1}{k}) + \frac{k+1}{k(k+1)} = 1$. For $j < k$, $f_j^{(1)} = -1$ on $[k, N) = \operatorname{supp}(f_k^{(1)})$, so $\langle f_j^{(1)}, f_k^{(1)} \rangle_{\mathcal{H}} = -\langle 1, f_k^{(1)} \rangle_{\mathcal{H}} = -\frac{1}{N}$, while $\|f_k^{(1)}\|_{\mathcal{H}}^2 = \frac{k^2}{k(k+1)} + (\frac{1}{k+1} - \frac{1}{N}) = 1 - \frac{1}{N}$. Similarly, for $j < k$, $g_k^* = 1$ on $[1, j+1) = \operatorname{supp}(g_j^*)$, so $\langle g_j^*, g_k^* \rangle_{\mathcal{H}} = \langle g_j^*, 1 \rangle_{\mathcal{H}} = 1$, while $\|g_k^*\|_{\mathcal{H}}^2 = (1 - \frac{1}{k}) + \frac{(k+1)^2}{k(k+1)} = 2$, establishing (8)–(9).
• (b): Since $\langle f_j^{(1)}, g_k^* \rangle_{\mathcal{H}} = \delta_{j, k}$ and $\dim \mathcal{S}_{N-1} = N-1$, any $h \in \mathcal{S}_{N-1}$ equals $\sum_{k=1}^{N-1} \langle h, f_k^{(1)} \rangle_{\mathcal{H}} g_k^*$. Taking the squared norm with $\langle g_j^*, g_k^* \rangle_{\mathcal{H}} = \delta_{j, k} + 1$ and using $\sum_{k=1}^{N-1} f_k^{(1)} = \mathbf{1}_{[1, N)}$ gives $\|h\|_{\mathcal{H}}^2 = \sum_{k=1}^{N-1}\langle h, f_k^{(1)} \rangle_{\mathcal{H}}^2 + (\sum_{k=1}^{N-1}\langle h, f_k^{(1)} \rangle_{\mathcal{H}})^2$, proving (10).
• (c): When $2 \le M \le m < N \le 2M$, we have $2m \ge 2M \ge N$, so on $[1, N)$ the floor quotients $\lfloor y/m \rfloor$ and $\lfloor y/(m+1) \rfloor$ jump only once (at $y = m$ and $y = m+1$ respectively), giving $f_m\big|_{[1, N)} \equiv f_m^{(1)}$ and $e_n\big|_{[1, N)} = f_n^{(1)} - f_{n+1}^{(1)} = n\mathbf{1}_{[n, n+1)} - (n+2)\mathbf{1}_{[n+1, n+2)}$, so $\beta_n^{[M, N)}(g) = \frac{n g(n)}{n(n+1)} - \frac{(n+2)g(n+1)}{(n+1)(n+2)} = \frac{g(n) - g(n+1)}{n+1}$. Since $\operatorname{supp}(f_m^{(1)}) = [m, N) \subset [M, N)$, their Gram matrix on $[M, N)$ is $I_J - \frac{1}{N}\mathbf{1}_J\mathbf{1}_J^{\mathsf T}$, whose exact inverse is $I_J + \frac{1}{M}\mathbf{1}_J\mathbf{1}_J^{\mathsf T}$ (since $(1 - J/N)(1 + J/M) = \frac{M}{N}\frac{N}{M} = 1$). On $[M, N)$, $\sum_{m=M}^{N-1} f_m(y) = M\lfloor y/M \rfloor - N\lfloor y/N \rfloor = M\,\mathbf{1}_{[M, N)}(y)$, proving (11). Since $D \in \mathbb{R}^{J \times (J-1)}$ has columns $e_i - e_{i+1}$ spanning $\mathbf{1}_J^\perp$, the projection matrix onto $\operatorname{col}(D)$ is $D(D^{\mathsf T} D)^{-1} D^{\mathsf T} = D L_{J-1}^{-1} D^{\mathsf T} = I_J - \frac{1}{J}\mathbf{1}_J\mathbf{1}_J^{\mathsf T}$. Applying this to $\beta^{[M, N)}(g) = D^{\mathsf T}\alpha^{[M, N)}(g)$ and subtracting $\frac{1}{M} + \frac{1}{J} = \frac{N}{M(N-M)}$ times $(\sum \alpha_m)^2 = M^2(\int_M^N g y^{-2}dy)^2$ proves (12). $\blacksquare$

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3. Harmonic–Möbius Unitriangular Factorization and All-Band Stiffness Law

We now extend Theorem 1 across all scales $1 \le m, n \le N-1$ by decomposing the multi-jump wavelets $f_m\big|_{[1, N)}$ into the single-jump basis $\{f_k^{(1)}\}_{k=1}^{N-1}$.

Theorem 2 (Harmonic–Möbius Unitriangular Factorization and All-Band Quotient Stiffness Law)

(a) Exact Intertwining by Inverse Harmonic–Möbius Unitriangular Matrices: For any $N \ge 2$, define the $(N-1)\times(N-1)$ upper-unitriangular rational matrices $U_N$ and $W_N$ ($1 \le m, k, n \le N-1$) by: $$(U_N)_{m, k} := H_{\lfloor k/m \rfloor} - H_{\lfloor k/(m+1) \rfloor}, \qquad (W_N)_{k, n} := M_1\bigl(\lfloor n/k \rfloor\bigr) - M_1\bigl(\lfloor n/(k+1) \rfloor\bigr), \tag{13}$$ where $H_q := \sum_{j=1}^q \frac{1}{j}$ ($H_0 := 0$) and $M_1(q) := \sum_{j=1}^q \frac{\mu(j)}{j}$ ($M_1(0) := 0$). Then $U_N$ and $W_N$ are exact inverses ($U_N W_N \equiv W_N U_N \equiv I_{N-1}$), and for every $1 \le m, n \le N-1$ on $y > 1$: $$f_m(y)\mathbf{1}_{[1, N)}(y) \;\equiv\; \sum_{k=m}^{N-1} (U_N)_{m, k}\, f_k^{(1)}(y), \qquad F_n^*(y) \;\equiv\; \sum_{k=1}^n (W_N)_{k, n}\, g_k^*(y), \qquad g_k^*(y) \;\equiv\; \sum_{m=1}^k (U_N)_{m, k}\, F_m^*(y). \tag{14}$$ Moreover, the row sums of $U_N$ and column sums of $W_N$ are given in closed form by: $$r_m(N) := \sum_{k=1}^{N-1} (U_N)_{m, k} \equiv N\bigl(H_{\lfloor(N-1)/m\rfloor} - H_{\lfloor(N-1)/(m+1)\rfloor}\bigr) - f_m(N-1), \qquad \sum_{k=1}^{N-1} (W_N)_{k, n} \equiv M_1(n). \tag{15}$$

(b) Exact Cholesky-Type Factorization of $H_{\mathrm{head}}(N)$ and $M_{F, \mathrm{full}}^*(N)$: Let $H_{\mathrm{head}}(N)_{m, n} := \int_1^N f_m(y)f_n(y)y^{-2}dy$ and $M_{F, \mathrm{full}}^*(N)_{m, n} := \langle F_m^*, F_n^* \rangle_{\mathcal{H}}$ ($1 \le m, n \le N-1$). Then $H_{\mathrm{head}}(N)$ and $M_{F, \mathrm{full}}^*(N)$ are exact inverses of one another on $\mathbb{Q}^{(N-1)\times(N-1)}$ and admit the exact factorizations: $$H_{\mathrm{head}}(N) \;\equiv\; U_N\left(I_{N-1} - \frac{1}{N}\mathbf{1}\mathbf{1}^{\mathsf T}\right)U_N^{\mathsf T}, \qquad M_{F, \mathrm{full}}^*(N) \;\equiv\; H_{\mathrm{head}}(N)^{-1} \;\equiv\; W_N^{\mathsf T}\left(I_{N-1} + \mathbf{1}\mathbf{1}^{\mathsf T}\right)W_N, \tag{16}$$ with exact determinants $\det H_{\mathrm{head}}(N) \equiv \frac{1}{N}$ and $\det M_{F, \mathrm{full}}^*(N) \equiv N$.

(c) All-Band Harmonic Quotient Stiffness Law: For each integer $K \ge 1$, let $B_K(N-1) := \bigl(\lfloor\frac{N-1}{K+1}\rfloor,\, \lfloor\frac{N-1}{K}\rfloor\bigr]$ be the $K$-th floor-quotient band. For any indices $m, n > \sqrt{N-1}$ such that $\{m, m+1\} \subset B_K(N-1)$ and $\{n, n+1\} \subset B_K(N-1)$: $$\int_1^N f_m(y)f_n(y)\frac{dy}{y^2} \;\equiv\; H_K\,\delta_{m, n} - \frac{K^2}{N}, \qquad \int_1^N f_m(y)\frac{dy}{y^2} \;\equiv\; \frac{K}{N}. \tag{17}$$ Consequently, on any sub-interval $\{m_1, \dots, m_1 + J - 1\} \subset B_K(N-1)$ with $m_1 > \sqrt{N-1}$ and $m_1 + J \le \lfloor(N-1)/K\rfloor$, the second-difference wavelets $e_m = f_m - f_{m+1}$ ($m_1 \le m \le m_1 + J - 2$) satisfy $\int_1^N e_m e_n y^{-2}dy \equiv H_K (L_{J-1})_{m, n}$, and the full Gram matrix $G_e := (\langle e_m, e_n \rangle_{\mathcal{H}})$ obeys the all-band operator lower bound: $$G_e \;\succeq\; H_K\, L_{J-1} + N\,\mathbf{b}_e \mathbf{b}_e^{\mathsf T} \;\succeq\; H_K\, L_{J-1}, \qquad L_{J-1}^{-1/2} G_e L_{J-1}^{-1/2} \;\succeq\; H_K\, I_{J-1}, \tag{18}$$ where $H_1 = 1, H_2 = \frac{3}{2}, H_3 = \frac{11}{6}, \dots$ and $(\mathbf{b}_e)_m := \langle e_m, 1 \rangle_{\mathcal{H}} = \log\frac{(m+1)^2}{m(m+2)}$.

(d) Causal Fine-to-Coarse Multi-Octave and Coarse-Bridge Schur Complement Bounds: Partition the wavelet indices into dyadic octaves $\mathcal{B}_k := \{f_m : 2^k \le m < 2^{k+1}\}$ ($J_k = 2^k$) and $\mathcal{E}_k := \{e_m : 2^k \le m \le 2^{k+1}-2\}$, and let $\mathcal{F}_{\ge 2^{k+1}} := \overline{\operatorname{span}}\{f_n : n \ge 2^{k+1}\} \subset \mathcal{H}$ be the closed subspace spanned by all finer octaves. Because every $f_n \in \mathcal{F}_{\ge 2^{k+1}}$ vanishes identically on $[1, 2^{k+1})$, the orthogonal projection $P_{\mathcal{F}_{\ge 2^{k+1}}^\perp}$ leaves restrictions to $[1, 2^{k+1})$ invariant, so the Schur complements conditioned on any finite or infinite family of finer octaves satisfy: $$\operatorname{Schur}\bigl(G_{\mathcal{B}_k} \;\big|\; \mathcal{F}_{\ge 2^{k+1}}\bigr) \;\succeq\; I_{2^k} - \frac{1}{2^{k+1}}\mathbf{1}\mathbf{1}^{\mathsf T} \;\succeq\; \frac{1}{2}\,I_{2^k}, \qquad \operatorname{Schur}\bigl(G_{\mathcal{E}_k} \;\big|\; \mathcal{F}_{\ge 2^{k+1}}\bigr) \;\succeq\; L_{2^k - 1}. \tag{19}$$ Moreover, for every integer $A \ge 2$, the coarse dyadic bridge $g_A := \rho_A - 3\rho_{2A} + 2\rho_{4A} = \frac{1}{A}\sum_{m=A}^{2A-1} f_m - \frac{1}{2A}\sum_{m=2A}^{4A-1} f_m$ ($\|g_A\|_{\mathcal{H}}^2 = \frac{\pi}{4A}$) [17] conditioned on the entire infinite union of internal second-difference octaves $\mathcal{B}_{\ge A}^{\mathrm{int}} := \overline{\operatorname{span}}\bigcup_{\ell=0}^\infty \mathcal{B}_{2^\ell A}^{\mathrm{int}}$ ($\mathcal{B}_B^{\mathrm{int}} := \{e_n\}_{n=B}^{2B-2}$) satisfies $(\mathbf{w}_A)_n := \int_{2A}^{4A} e_n(y)y^{-2}dy = u_{A, n} - u_{A, n+1}$ ($u_{A, m} := \int_{2A}^{4A} f_m(y)y^{-2}dy$, $\sum_{m=A}^{2A-1} u_{A, m} \equiv \frac{1}{12}$), $4A\,\mathbf{w}_A^{\mathsf T} L_{A-1}^{-1}\mathbf{w}_A \equiv \frac{1}{18}$ if $3 \mid A$ and $\frac{A-1}{18A}$ if $3 \nmid A$, and the all-scale Schur complement lower bound: $$\operatorname{Schur}\!\left(\|g_A\|_{\mathcal{H}}^2 \;\Big|\; \bigcup_{\ell=0}^\infty \mathcal{B}_{2^\ell A}^{\mathrm{int}}\right) \;\ge\; \frac{1}{2A} + \frac{\frac{1}{9A}}{1 + 4A\,\mathbf{w}_A^{\mathsf T} L_{A-1}^{-1}\mathbf{w}_A} \;\equiv\; \begin{cases} \dfrac{23}{38\,A} = \dfrac{46}{19\pi}\,\|g_A\|_{\mathcal{H}}^2, & 3 \mid A, \\[8pt] \dfrac{23A - 1}{(38A - 2)A} > \dfrac{46}{19\pi}\,\|g_A\|_{\mathcal{H}}^2, & 3 \nmid A. \end{cases} \tag{19a}$$

Proof.

• (a): Let $\Phi_m(y) := m\lfloor y/m \rfloor\mathbf{1}_{[1, N)}(y)$ and $\Phi_k^{(1)}(y) := \sum_{j=k}^{N-1} f_j^{(1)}(y)$. Summing $f_j^{(1)}(y) = j\mathbf{1}_{[j, j+1)}(y) - \mathbf{1}_{[j+1, N)}(y)$ over $k \le j \le N-1$ gives $\Phi_k^{(1)}(y) = \sum_{d=k}^{N-1}(d - (d - k))\mathbf{1}_{[d, d+1)}(y) = k\,\mathbf{1}_{[k, N)}(y)$. Since $m\lfloor y/m \rfloor = \sum_{q=1}^{\lfloor(N-1)/m\rfloor} m\,\mathbf{1}_{[qm, N)}(y) = \sum_{q=1}^{\lfloor(N-1)/m\rfloor} \frac{1}{q}\Phi_{qm}^{(1)}(y)$ on $[1, N)$, taking differences $f_m = \Phi_m - \Phi_{m+1}$ and $\Phi_k^{(1)} = \sum_{j=k}^{N-1} f_j^{(1)}$ shows that the coefficient of $f_k^{(1)}$ in $f_m\mathbf{1}_{[1, N)}$ is $\sum_{q \ge 1 : qm \le k}\frac{1}{q} - \sum_{q \ge 1 : q(m+1) \le k}\frac{1}{q} = H_{\lfloor k/m \rfloor} - H_{\lfloor k/(m+1) \rfloor} = (U_N)_{m, k}$. Conversely, by (4), on $[1, N)$ we have $F_n^*(y) = \sum_{d=1}^n d(d+1)\bigl(\frac{M_1(\lfloor n/d \rfloor)}{d} - \frac{M_1(\lfloor n/(d+1) \rfloor)}{d+1}\bigr)\mathbf{1}_{[d, d+1)}(y)$. Writing $g_k^*(y) - g_{k-1}^*(y) = (k+1)\mathbf{1}_{[k, k+1)}(y) - k\mathbf{1}_{[k-1, k)}(y)$ and summing by parts yields $F_n^* = \sum_{k=1}^n (M_1(\lfloor n/k \rfloor) - M_1(\lfloor n/(k+1) \rfloor))g_k^* = \sum_{k=1}^n (W_N)_{k, n} g_k^*$. Since $\langle f_m\mathbf{1}_{[1, N)}, F_n^* \rangle_{\mathcal{H}} = \langle f_m, F_n^* \rangle_{\mathcal{H}} = \delta_{m, n}$ and $\langle f_j^{(1)}, g_k^* \rangle_{\mathcal{H}} = \delta_{j, k}$, we have $\delta_{m, n} = \sum_{k=1}^{N-1} (U_N)_{m, k}(W_N)_{k, n}$, proving $U_N W_N = I_{N-1}$ and (14)–(15).
• (b): Combining (14) with $\langle f_j^{(1)}, f_k^{(1)} \rangle_{\mathcal{H}} = \delta_{j, k} - \frac{1}{N}$ and $\langle g_j^*, g_k^* \rangle_{\mathcal{H}} = \delta_{j, k} + 1$ from Theorem 1(a) gives $H_{\mathrm{head}}(N) = U_N(I_{N-1} - \frac{1}{N}\mathbf{1}\mathbf{1}^{\mathsf T})U_N^{\mathsf T}$ and $M_{F, \mathrm{full}}^*(N) = W_N^{\mathsf T}(I_{N-1} + \mathbf{1}\mathbf{1}^{\mathsf T})W_N$. Since $U_N W_N = I_{N-1}$ and $(I_{N-1} - \frac{1}{N}\mathbf{1}\mathbf{1}^{\mathsf T})(I_{N-1} + \mathbf{1}\mathbf{1}^{\mathsf T}) = I_{N-1} + (1 - 1 - \frac{N-1}{N})\mathbf{1}\mathbf{1}^{\mathsf T} + \frac{1}{N}\mathbf{1}\mathbf{1}^{\mathsf T} = I_{N-1}$, they are exact inverses, and $\det U_N = \det W_N = 1$ gives $\det H_{\mathrm{head}}(N) = \det(I_{N-1} - \frac{1}{N}\mathbf{1}\mathbf{1}^{\mathsf T}) = 1 - \frac{N-1}{N} = \frac{1}{N}$.
• (c): Let $m \in B_K(N-1)$ such that $m+1 \in B_K(N-1)$, so $\lfloor(N-1)/m\rfloor = \lfloor(N-1)/(m+1)\rfloor = K$. For any $1 \le k \le N-1$, we have $\lfloor k/m \rfloor \le K$ and $\lfloor k/(m+1) \rfloor \le K$. The difference $(U_N)_{m, k} = H_{\lfloor k/m \rfloor} - H_{\lfloor k/(m+1) \rfloor}$ is nonzero if and only if $\lfloor k/m \rfloor > \lfloor k/(m+1) \rfloor$. Since $k/m - k/(m+1) = \frac{k}{m(m+1)} \le \frac{N-1}{m(m+1)} < 1$ for all $m > \sqrt{N-1}$, the two floor quotients differ by at most $1$. Thus $\lfloor k/m \rfloor - \lfloor k/(m+1) \rfloor = 1$ occurs if and only if there exists an integer $q \in \{1, \dots, K\}$ such that $$qm \le k < q(m+1) \iff k \in I_{m, q} := [qm, q(m+1)) \cap \mathbb{Z}.$$ For each $q \in \{1, \dots, K\}$, the interval $I_{m, q}$ contains exactly $q(m+1) - qm = q$ consecutive integers, and on $I_{m, q}$ we have $\lfloor k/m \rfloor = q$ and $\lfloor k/(m+1) \rfloor = q - 1$, so $(U_N)_{m, k} = H_q - H_{q-1} = \frac{1}{q}$. For $k \notin \bigcup_{q=1}^K I_{m, q}$, $\lfloor k/m \rfloor = \lfloor k/(m+1) \rfloor$, so $(U_N)_{m, k} = 0$. Since $q(m+1) \le (q+1)m$ for all $q \le m$, the $K$ intervals $I_{m, 1}, \dots, I_{m, K}$ are pairwise disjoint subsets of $\{1, \dots, N-1\}$. Consequently, the row sum and squared row norm of row $m$ are given by: $$r_m(N) = \sum_{k=1}^{N-1} (U_N)_{m, k} = \sum_{q=1}^K \sum_{k \in I_{m, q}} \frac{1}{q} = \sum_{q=1}^K q \cdot \frac{1}{q} = K,$$ $$(U_N U_N^{\mathsf T})_{m, m} = \sum_{k=1}^{N-1} (U_N)_{m, k}^2 = \sum_{q=1}^K \sum_{k \in I_{m, q}} \frac{1}{q^2} = \sum_{q=1}^K q \cdot \frac{1}{q^2} = \sum_{q=1}^K \frac{1}{q} = H_K.$$ Now suppose $m \ne n$ both satisfy $\{m, m+1\}, \{n, n+1\} \subset B_K(N-1)$ with $m, n > \sqrt{N-1}$. We claim that the support sets $\operatorname{supp}((U_N)_{m, \bullet}) = \bigcup_{q=1}^K I_{m, q}$ and $\operatorname{supp}((U_N)_{n, \bullet}) = \bigcup_{p=1}^K I_{n, p}$ are completely disjoint. Suppose for contradiction that there exists $k \in I_{m, q} \cap I_{n, p}$ for some $q, p \in \{1, \dots, K\}$.
  • If $p = q$, then $qm \le k < qm + q$ and $qn \le k < qn + q$. Since $m \ne n$ are integers, $|m - n| \ge 1 \implies |qm - qn| \ge q$, so the half-open intervals $[qm, qm+q)$ and $[qn, qn+q)$ of length $q$ with integer endpoints are disjoint, a contradiction.
  • If $p \ne q$, without loss of generality assume $1 \le p < q \le K$. Then $k \in I_{m, q} \cap I_{n, p}$ requires $qm \le k < q(m+1)$ and $pn \le k < p(n+1)$. In particular, $qm < p(n+1)$ and $pn < q(m+1)$, which implies $\frac{m}{n+1} < \frac{p}{q} < \frac{m+1}{n}$. However, both $m, n \in B_K(N-1)$ means $\frac{N-1}{K+1} < m, n \le \frac{N-1}{K}$, so $\frac{m}{n} > \frac{K}{K+1}$. Since $m > \sqrt{N-1} \ge K$, we have $\frac{m}{n+1} = \frac{m}{n}\frac{n}{n+1} > \frac{K}{K+1}\frac{\sqrt{N-1}}{\sqrt{N-1}+1} \ge \frac{K-1}{K} \ge \frac{p}{q}$ (since $p \le q - 1 \le K - 1$), contradicting $\frac{m}{n+1} < \frac{p}{q}$. Thus $I_{m, q} \cap I_{n, p} = \emptyset$ for all $1 \le q, p \le K$, which proves $(U_N U_N^{\mathsf T})_{m, n} = 0$ for all $m \ne n$. Substituting into (16) yields $H_{\mathrm{head}}(N)_{m, n} = (U_N U_N^{\mathsf T})_{m, n} - \frac{1}{N}r_m(N)r_n(N) = H_K \delta_{m, n} - \frac{K^2}{N}$, and $\int_1^N f_m y^{-2}dy = \langle f_m\mathbf{1}_{[1, N)}, \mathbf{1}_{[1, N)} \rangle_{\mathcal{H}} = \sum_{k=1}^{N-1}(U_N)_{m, k}\langle f_k^{(1)}, 1 \rangle_{\mathcal{H}} = \frac{r_m(N)}{N} = \frac{K}{N}$, proving (17). Passing to the second differences $e_m = f_m - f_{m+1}$, since $r_m(N) = K$ is constant across $m$, the rank-1 component $(K/N)\mathbf{1}$ is annihilated identically: $\int_1^N e_m y^{-2}dy = \frac{K}{N} - \frac{K}{N} = 0$, and $\int_1^N e_m e_n y^{-2}dy = H_K(L_{J-1})_{m, n}$. Since $(\mathbf{b}_e)_m = \int_1^\infty e_m y^{-2}dy = \int_N^\infty e_m y^{-2}dy$, the full Gram matrix splits as $G_e = H_{\mathrm{head}, e} + T_{\mathrm{tail}, e} = H_K L_{J-1} + \int_N^\infty e(y)e(y)^{\mathsf T}y^{-2}dy$. Applying Cauchy–Schwarz against the constant tail functional $\mathbf{1}_{[N, \infty)}$ ($\|\mathbf{1}_{[N, \infty)}\|_{\mathcal{H}}^2 = 1/N$) gives $\int_N^\infty e(y)e(y)^{\mathsf T}y^{-2}dy \succeq \frac{\mathbf{b}_e \mathbf{b}_e^{\mathsf T}}{\|\mathbf{1}_{[N, \infty)}\|_{\mathcal{H}}^2} = N \mathbf{b}_e \mathbf{b}_e^{\mathsf T} \succeq 0$, establishing (18).
• (d): For any vector $c \in \mathbb{R}^{2^k}$, $\Phi := \sum_{m=2^k}^{2^{k+1}-1} c_m f_m$ satisfies $\operatorname{dist}_{\mathcal{H}}^2(\Phi, \mathcal{F}_{\ge 2^{k+1}}) = \inf_{\Psi \in \mathcal{F}_{\ge 2^{k+1}}} \bigl(\int_1^{2^{k+1}}(\Phi - \Psi)^2 y^{-2}dy + \int_{2^{k+1}}^\infty(\Phi - \Psi)^2 y^{-2}dy\bigr) \ge \int_1^{2^{k+1}} \Phi(y)^2 y^{-2}dy = c^{\mathsf T}(I_{2^k} - 2^{-(k+1)}\mathbf{1}\mathbf{1}^{\mathsf T})c$ because every $\Psi \in \mathcal{F}_{\ge 2^{k+1}}$ vanishes identically on $[1, 2^{k+1})$, proving (19). Similarly, decomposing any $\Psi \in \mathcal{B}_{\ge A}^{\mathrm{int}}$ as $\Psi_0 + \Psi_1 + \Psi_{\ge 2}$ with $\Psi_\ell \in \operatorname{span}(\mathcal{B}_{2^\ell A}^{\mathrm{int}})$ ($\ell \in \{0, 1\}$) and $\Psi_{\ge 2} \in \overline{\operatorname{span}}\bigcup_{\ell=2}^\infty \mathcal{B}_{2^\ell A}^{\mathrm{int}}$, $\Psi_{\ge 2}$ vanishes on $[1, 4A)$. On $[1, 2A)$, $\Psi_1 \equiv 0$ and $g_A\big|_{[1, 2A)} = \mathbf{1}_{[A, 2A)} \perp \Psi_0\big|_{[1, 2A)}$, giving $\int_1^{2A}(g_A - \Psi_0)^2 y^{-2}dy = \frac{1}{2A} + c_0^{\mathsf T} L_{A-1} c_0$. On $[2A, 4A)$, minimizing over $\Psi_1 \in \operatorname{span}(\mathcal{B}_{2A}^{\mathrm{int}})$ leaves $4A(\int_{2A}^{4A}(g_A - \Psi_0)y^{-2}dy)^2 = 4A(-\frac{1}{6A} - \mathbf{w}_A^{\mathsf T} c_0)^2$. Since $\sum_{m=A}^{2A-1} f_m\big|_{[2A, 4A)} = A\mathbf{1}_{[3A, 4A)}$, we have $\sum_{m=A}^{2A-1} u_{A, m} = \frac{1}{12}$, while $u_{A, m} = \frac{1}{4A}$ for $3(m+1) \le 4A$, $0$ for $3m \ge 4A$, and $\frac{4A-3m}{12A}$ if $3m < 4A < 3(m+1)$, giving $4A\mathbf{w}_A^{\mathsf T} L_{A-1}^{-1}\mathbf{w}_A = 4A(\sum u_{A, m}^2 - \frac{1}{A}(\frac{1}{12})^2) = \frac{1}{18}$ if $3 \mid A$ and $\frac{A-1}{18A}$ if $3 \nmid A$. Minimizing over $c_0$ yields (19a). $\blacksquare$

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4. Telescoped Möbius Quotient Formula and $15/\pi^2$ Dual Norm Asymptotics

Using the exact factorization $M_{F, \mathrm{full}}^*(N) = W_N^{\mathsf T}(I_{N-1} + \mathbf{1}\mathbf{1}^{\mathsf T})W_N$ from Theorem 2(b), we obtain an explicit finite-scale squarefree decomposition and unconditional asymptotic limit for Balazard's harmonic representers $F_n^*$.

Theorem 3 (Telescoped Möbius Quotient Formula, Squarefree Decomposition, and $15/\pi^2$ Limit)

(a) Telescoped Möbius Quotient Formula: For all integers $m, n \ge 1$, the inner product of the harmonic representers $F_m^*, F_n^* \in \mathcal{H}$ is given identically by: $$\langle F_m^*, F_n^* \rangle_{\mathcal{H}} \;\equiv\; M_1(m)\,M_1(n) + \sum_{k=1}^{\min(m, n)} \Bigl(M_1\bigl(\lfloor m/k \rfloor\bigr) - M_1\bigl(\lfloor m/(k+1) \rfloor\bigr)\Bigr)\Bigl(M_1\bigl(\lfloor n/k \rfloor\bigr) - M_1\bigl(\lfloor n/(k+1) \rfloor\bigr)\Bigr). \tag{20}$$

(b) Exact Finite-Scale Squarefree Decomposition and Universal Lower Bound: For every integer $n \ge 2$, let $R(n) := \bigl\lfloor\frac{\sqrt{4n+1}-1}{2}\bigr\rfloor \ge 1$ be the largest integer satisfying $R(n)(R(n)+1) \le n$. Then: $$\sum_{k=\lfloor n/(R(n)+1) \rfloor + 1}^n \Bigl(M_1\bigl(\lfloor n/k \rfloor\bigr) - M_1\bigl(\lfloor n/(k+1) \rfloor\bigr)\Bigr)^2 \;\equiv\; \sum_{r=1}^{R(n)} \frac{\mu(r)^2}{r^2}, \tag{21}$$ and consequently $\|F_n^*\|_{\mathcal{H}}^2$ decomposes identically as: $$\|F_n^*\|_{\mathcal{H}}^2 \;\equiv\; \sum_{r=1}^{R(n)} \frac{\mu(r)^2}{r^2} + M_1(n)^2 + \sum_{k=1}^{\lfloor n/(R(n)+1) \rfloor} \Bigl(M_1\bigl(\lfloor n/k \rfloor\bigr) - M_1\bigl(\lfloor n/(k+1) \rfloor\bigr)\Bigr)^2 \;\ge\; M_1(n)^2 + \sum_{r=1}^{R(n)} \frac{\mu(r)^2}{r^2}. \tag{22}$$ Similarly, for any lag $h \ge 0$ and any integer $R \ge 1$ such that $\lfloor n/r \rfloor > \lfloor(n+h)/(r+1)\rfloor$ for all $1 \le r \le R$, the tail covariance sum over $\lfloor(n+h)/(R+1)\rfloor < k \le n$ equals $\sum_{1 \le r \le R : \lfloor(n+h)/r\rfloor = \lfloor n/r \rfloor} \frac{\mu(r)^2}{r^2}$ identically.

(c) Unconditional Asymptotic Limit and Off-Diagonal Almost-Periodicity: Unconditionally as $n \to \infty$: $$\lim_{n \to \infty} \|F_n^*\|_{\mathcal{H}}^2 \;=\; \sum_{r=1}^\infty \frac{\mu(r)^2}{r^2} \;=\; \frac{\zeta(2)}{\zeta(4)} \;=\; \frac{15}{\pi^2} \;=\; 1.51981775463506657\dots, \tag{23}$$ with quantitative rate $\|F_n^*\|_{\mathcal{H}}^2 = \frac{15}{\pi^2} + O\bigl((\log n)^{-2}\bigr)$, and for every fixed integer lag $h \ge 1$: $$\langle F_n^*, F_{n+h}^* \rangle_{\mathcal{H}} \;=\; \sum_{\substack{r=h+1 \\ \lfloor(n+h)/r\rfloor = \lfloor n/r \rfloor}}^\infty \frac{\mu(r)^2}{r^2} + O_h\!\left(\frac{1}{(\log n)^2}\right), \qquad \lim_{X \to \infty} \frac{1}{X}\sum_{n=1}^X \langle F_n^*, F_{n+h}^* \rangle_{\mathcal{H}} \;=\; \sum_{r=h+1}^\infty \frac{\mu(r)^2}{r^2}\left(1 - \frac{h}{r}\right). \tag{24}$$ In particular, for $h = 1$, $\langle F_n^*, F_{n+1}^* \rangle_{\mathcal{H}} = \frac{15}{\pi^2} - \prod_{p \mid (n+1)}\bigl(1 + \frac{1}{p^2}\bigr) + O\bigl((\log n)^{-2}\bigr)$.

Proof.

• (a): Choose $N > \max(m, n)$ in Theorem 2(b). By (16) and (15), the $(m, n)$ entry of $M_{F, \mathrm{full}}^*(N) = W_N^{\mathsf T} W_N + (W_N^{\mathsf T}\mathbf{1})(W_N^{\mathsf T}\mathbf{1})^{\mathsf T}$ is $\sum_{k=1}^{\min(m, n)}(W_N)_{k, m}(W_N)_{k, n} + M_1(m)M_1(n)$, which is (20) (independently verified by summing the algebraic identity $d(d+1)(\frac{a_d}{d} - \frac{a_{d+1}}{d+1})(\frac{b_d}{d} - \frac{b_{d+1}}{d+1}) = (a_d - a_{d+1})(b_d - b_{d+1}) + (\frac{a_d b_d}{d} - \frac{a_{d+1}b_{d+1}}{d+1})$ over $d \ge 1$).
• (b): Since $R(n)(R(n)+1) \le n$, for every $1 \le r \le R(n)$ we have $\frac{n}{r} - \frac{n}{r+1} = \frac{n}{r(r+1)} \ge 1$, so the integers $k_r := \lfloor n/r \rfloor$ ($1 \le r \le R(n)+1$) are strictly decreasing: $n = k_1 > k_2 > \dots > k_{R(n)+1} = \lfloor n/(R(n)+1)\rfloor$. For $\lfloor n/(R(n)+1) \rfloor < k \le n$, $\lfloor n/k \rfloor$ is constant on each interval $(k_{r+1}, k_r]$ where it equals $r$. Thus $\lfloor n/k \rfloor \ne \lfloor n/(k+1) \rfloor$ occurs inside $(\lfloor n/(R(n)+1)\rfloor, n]$ if and only if $k = k_r$ for some $1 \le r \le R(n)$, at which point $\lfloor n/k_r \rfloor = r$ and $\lfloor n/(k_r+1) \rfloor = r - 1$, so $M_1(\lfloor n/k_r \rfloor) - M_1(\lfloor n/(k_r+1) \rfloor) = M_1(r) - M_1(r-1) = \frac{\mu(r)}{r}$. Squaring and summing over $1 \le r \le R(n)$ proves (21) and (22). For $h \ge 1$, when $\lfloor n/r \rfloor > \lfloor(n+h)/(r+1)\rfloor$ for $1 \le r \le R$, the jump points $k_r = \lfloor n/r \rfloor$ of $\lfloor n/k \rfloor$ coincide with a jump of $\lfloor(n+h)/k\rfloor$ (necessarily from $r$ to $r-1$) if and only if $\lfloor(n+h)/k_r\rfloor = r \iff k_r \ge \lfloor(n+h)/r\rfloor \iff \lfloor(n+h)/r\rfloor = \lfloor n/r \rfloor$.
• (c): For $1 \le k \le \lfloor n/(R(n)+1) \rfloor \le \sqrt{n}$, the trivial harmonic bound gives $|M_1(\lfloor n/k \rfloor) - M_1(\lfloor n/(k+1) \rfloor)| \le \sum_{\lfloor n/(k+1)\rfloor < j \le \lfloor n/k \rfloor}\frac{1}{j} \le \frac{1}{k} + \frac{k}{n} \le \frac{2}{k}$, providing the $n$-independent summable majorant $\frac{4}{k^2}$. Since $\lim_{x \to \infty} M_1(x) = 0$ by the Prime Number Theorem, each fixed-$k$ summand tends to $0$ as $n \to \infty$, so dominated convergence immediately yields $\lim_{n \to \infty}\|F_n^*\|_{\mathcal{H}}^2 = \sum_{r=1}^\infty \frac{\mu(r)^2}{r^2} = \frac{\zeta(2)}{\zeta(4)} = \frac{15}{\pi^2}$. Quantitatively, by de la Vallée Poussin's error term $\sup_{x \ge Y}|M_1(x)| \ll (\log(2+Y))^{-2}$, splitting the head sum at $K_0 := \lfloor(\log n)^2\rfloor$ bounds $1 \le k \le K_0$ by $\ll K_0(\log n)^{-4} = (\log n)^{-2}$ and $K_0 < k \le \lfloor n/(R(n)+1)\rfloor$ by $\ll \sum_{k > K_0} k^{-2} \ll K_0^{-1} \asymp (\log n)^{-2}$, while $\sum_{r > R(n)}\frac{\mu(r)^2}{r^2} \ll R(n)^{-1} \ll n^{-1/2}$, proving (23). For the off-diagonal covariance at fixed lag $h \ge 1$, we apply Theorem 3(a) and split the sum at $k_{\mathrm{split}} := \lfloor(n+h)/(R(n)+1)\rfloor$. In the tail range $k_{\mathrm{split}} < k \le n$, Theorem 3(b) shows that the product of differences is nonzero if and only if $k = \lfloor n/r \rfloor = \lfloor(n+h)/r\rfloor$ for some $1 \le r \le R(n)$, in which case the product equals $\frac{\mu(r)^2}{r^2}$. For $1 \le r \le h$, the interval $(n/r, (n+h)/r]$ has length $h/r \ge 1$ and contains at least one integer, so $\lfloor(n+h)/r\rfloor > \lfloor n/r \rfloor$; hence $\lfloor(n+h)/r\rfloor = \lfloor n/r \rfloor$ requires $r \ge h+1$. Bounding the head sum $1 \le k \le k_{\mathrm{split}}$ by Cauchy–Schwarz and the tail over $r > R(n)$ by $O(R(n)^{-1}) = O(n^{-1/2})$ gives the arithmetic asymptotic formula in (24). In particular, for $h = 1$, $\lfloor(n+1)/r\rfloor = \lfloor n/r \rfloor \iff r \nmid (n+1)$, so $\sum_{r \ge 2 : r \nmid (n+1)}\frac{\mu(r)^2}{r^2} = \sum_{r=1}^\infty \frac{\mu(r)^2}{r^2} - \sum_{r \mid (n+1)}\frac{\mu(r)^2}{r^2} = \frac{15}{\pi^2} - \prod_{p \mid (n+1)}(1 + \frac{1}{p^2})$. Along $n + 1 = p$ (odd primes), this equals $\frac{15}{\pi^2} - (1 + \frac{1}{p^2}) \to \frac{15}{\pi^2} - 1$, whereas along $n + 1 = 2^a$ ($a \ge 2$), it equals $\frac{15}{\pi^2} - (1 + \frac{1}{4}) = \frac{15}{\pi^2} - \frac{5}{4}$, demonstrating persistent oscillation. Finally, for each fixed $r \ge h+1$, the condition $\lfloor(n+h)/r\rfloor = \lfloor n/r \rfloor$ is equivalent to $n \bmod r \in \{0, 1, \dots, r - h - 1\}$, which has asymptotic density $\frac{r-h}{r} = 1 - \frac{h}{r}$. Since $\sum_{r=h+1}^\infty \frac{\mu(r)^2}{r^2} < \infty$, the dominated convergence theorem for Cesàro averages allows interchanging the limit with the summation over $r$: $$\lim_{X \to \infty} \frac{1}{X}\sum_{n=1}^X \langle F_n^*, F_{n+h}^* \rangle_{\mathcal{H}} = \sum_{r=h+1}^\infty \frac{\mu(r)^2}{r^2} \lim_{X \to \infty}\frac{1}{X}\sum_{n=1}^X \mathbf{1}_{\lfloor(n+h)/r\rfloor = \lfloor n/r \rfloor} = \sum_{r=h+1}^\infty \frac{\mu(r)^2}{r^2}\left(1 - \frac{h}{r}\right),$$ completing the proof of (24). $\blacksquare$

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5. Wavelet Biorthogonal Interpolation, Head-from-Tail Reconstruction, and Woodbury Annihilator Reduction

Theorem 4 (Wavelet Interpolation, Head-from-Tail Reconstruction, and $(N-1)\times(N-1)$ Woodbury Reduction)

(a) Exact Wavelet Interpolation on $\mathcal{S}_{N-1}$ and Global Residual Identities: For every $N \ge 2$ and every step function $h \in \mathcal{S}_{N-1}$, define its wavelet interpolant $\Phi_{h, N} := \sum_{n=1}^{N-1} \langle h, F_n^* \rangle_{\mathcal{H}}\, f_n \in \mathcal{V}_{S_{\le N}}$. Then $\Phi_{h, N}(y) \equiv h(y)$ identically on $[1, N)$. In particular, for $h = \mathbf{1}_{[1, N)}$ and $h = \chi_1 = \mathbf{1}_{[1, 2)}$: $$\Phi_{1, N} := \sum_{n=1}^{N-1} M_1(n)\,f_n = -\sum_{m=2}^{N-1}\mu(m)\rho_m - N M_1(N-1)\rho_N, \qquad \Phi_{\chi_1, N} := \sum_{n=1}^{N-1}\left(M_1(n) - \frac{1}{2}M_1(\lfloor n/2 \rfloor)\right)f_n. \tag{25}$$ For the global constant function $1 \in \mathcal{H}$ on $(1, \infty)$ and the first-octave indicator $\chi_1 \in \mathcal{H}$, define the global residuals $R_{1, N} := 1 - \Phi_{1, N}$ and $R_{\chi_1, N} := \chi_1 - \Phi_{\chi_1, N}$. Both vanish identically on $[1, N)$ (so $\operatorname{supp}(R_{1, N}) \subset [N, \infty)$ and $\operatorname{supp}(R_{\chi_1, N}) \subset [N, \infty)$). For every integer $d \ge 1$, the dilation operator $(D_d g)(y) := g(y/d)\mathbf{1}_{y \ge d}$ acts on the Haar wavelets by $(D_d f_k)(y) \equiv \frac{1}{d}\sum_{j=0}^{d-1} f_{dk+j}(y)$ on $(1, \infty)$, so for every even cutoff $N = 2M$ ($M \ge 2$): $$\Phi_{\chi_1, 2M}(y) \;\equiv\; \Phi_{1, 2M}(y) - \bigl(D_2 \Phi_{1, M}\bigr)(y), \qquad R_{\chi_1, 2M}(y) \;\equiv\; R_{1, 2M}(y) - \bigl(D_2 R_{1, M}\bigr)(y) \qquad (\forall\, y > 1). \tag{26}$$ Moreover, writing $T_{\mathrm{tail}}(N)_{m, n} := \int_N^\infty f_m(y)f_n(y)y^{-2}dy = G_{\mathrm{first}}^{\mathrm{full}}(N)_{m, n} - H_{\mathrm{head}}(N)_{m, n} \succ 0$ and $\mathbf{a}_{\mathrm{tail}}(N)_m := \int_N^\infty f_m(y)y^{-2}dy = \log\frac{m+1}{m} - \frac{r_m(N)}{N}$, the distances $d_N^2$ and $\|r_1(N)\|_{\mathcal{H}}^2$ satisfy the exact Pythagorean identities: $$d_N^2 \;\equiv\; \|R_{1, N}\|_{\mathcal{H}}^2 - \mathbf{s}_1(N)^{\mathsf T} G_{\mathrm{first}}^{\mathrm{full}}(N)^{-1} \mathbf{s}_1(N), \qquad \|r_1(N)\|_{\mathcal{H}}^2 \;\equiv\; \|R_{\chi_1, N}\|_{\mathcal{H}}^2 - \mathbf{t}_{\chi_1}(N)^{\mathsf T} G_{\mathrm{first}}^{\mathrm{full}}(N)^{-1} \mathbf{t}_{\chi_1}(N), \tag{27}$$ where $\|R_{1, N}\|_{\mathcal{H}}^2 = \frac{1}{N} - 2\mathbf{m}^{\mathsf T}\mathbf{a}_{\mathrm{tail}}(N) + \mathbf{m}^{\mathsf T} T_{\mathrm{tail}}(N)\mathbf{m}$, $\|R_{\chi_1, N}\|_{\mathcal{H}}^2 = \mathbf{m}_1^{\mathsf T} T_{\mathrm{tail}}(N)\mathbf{m}_1$, $\mathbf{s}_1(N) := \mathbf{a}_{\mathrm{tail}}(N) - T_{\mathrm{tail}}(N)\mathbf{m}$, and $\mathbf{t}_{\chi_1}(N) := T_{\mathrm{tail}}(N)\mathbf{m}_1$ (with equality $d_2^2 = \|R_{1, 2}\|_{\mathcal{H}}^2 = 1 - \log 2$ at $N = 2$ and strict inequality for $N \ge 3$).

(b) Head-from-Tail Residual Reconstruction and Variational Energy Ratio: Let $\mathcal{S}_{[N, \infty)} := \{g \in \mathcal{S}_\infty : \operatorname{supp}(g) \subset [N, \infty)\}$. The extension map $\mathcal{E}_N : \mathcal{S}_{[N, \infty)} \to \mathcal{S}_\infty \cap \mathcal{V}_{S_{\le N}}^\perp$ defined by: $$\bigl(\mathcal{E}_N r_{\mathrm{tail}}\bigr)(y) \;:=\; r_{\mathrm{tail}}(y) - \sum_{n=1}^{N-1}\left(\int_N^\infty r_{\mathrm{tail}}(t)\,f_n(t)\frac{dt}{t^2}\right) F_n^*(y) \tag{28}$$ is a linear bijection with inverse $r \mapsto r\,\mathbf{1}_{[N, \infty)}$. Consequently, for every orthogonal residual $r \in \mathcal{S}_\infty \cap \mathcal{V}_{S_{\le N}}^\perp$ (including $r_{\mathrm{full}, N}$ and $r_1(N)$), its head restriction $r\,\mathbf{1}_{[1, N)}$ is uniquely determined by its tail $r\,\mathbf{1}_{[N, \infty)}$, with head energy: $$E_{\mathrm{head}}(r; N) := \int_1^N r(y)^2\frac{dy}{y^2} \;\equiv\; \boldsymbol{\tau}(r; N)^{\mathsf T} M_{F, \mathrm{full}}^*(N)\,\boldsymbol{\tau}(r; N), \qquad \tau_n(r; N) := -\int_N^\infty r(y)f_n(y)\frac{dy}{y^2}, \tag{29}$$ and the supremum of the head-to-tail energy ratio over all nonzero $r \in \mathcal{S}_\infty \cap \mathcal{V}_{S_{\le N}}^\perp$ is given by: $$\sup_{0 \ne r \in \mathcal{S}_\infty \cap \mathcal{V}_{S_{\le N}}^\perp} \frac{\int_1^N r(y)^2\frac{dy}{y^2}}{\int_N^\infty r(y)^2\frac{dy}{y^2}} \;\equiv\; \lambda_{\max}\!\Bigl(M_{F, \mathrm{full}}^*(N)\,T_{\mathrm{tail}}(N)\Bigr) \;=\; \lambda_{\max}\!\Bigl(H_{\mathrm{head}}(N)^{-1} T_{\mathrm{tail}}(N)\Bigr). \tag{30}$$

(c) Dimension-Reduced $(N-1)\times(N-1)$ Woodbury Formula for Dual Annihilator Bounds: For any integers $M \ge N \ge 2$, let $G_{\mathcal{D}^*}(N, M) := \bigl(\langle F_m^*, F_n^* \rangle_{\mathcal{H}}\bigr)_{N \le m, n \le M} \in \mathbb{Q}^{(M-N+1)\times(M-N+1)}$ and $\mathrm{LB}_{\mathrm{dual}}(N, M) := \mathbf{m}_{N, M}^{\mathsf T} G_{\mathcal{D}^*}(N, M)^{-1} \mathbf{m}_{N, M}$. Partitioning $W_{M+1} = \begin{pmatrix} W_{AA} & W_{AB} \\ 0 & W_{BB} \end{pmatrix}$ onto $A = \{1, \dots, N-1\}$ and $B = \{N, \dots, M\}$, we have $G_{\mathcal{D}^*}(N, M) \equiv K_{\mathcal{D}^*}(N, M) + \mathbf{m}_{N, M}\mathbf{m}_{N, M}^{\mathsf T}$ where $K_{\mathcal{D}^*}(N, M) := W_{BB}^{\mathsf T} W_{BB} + W_{AB}^{\mathsf T} W_{AB} \succ 0$ ($W_{BB} = I_{M-N+1}$ whenever $M \le 2N-1$), and $\mathrm{LB}_{\mathrm{dual}}(N, M)$ reduces to the fixed-dimension $(N-1)\times(N-1)$ system: $$\mathrm{LB}_{\mathrm{dual}}(N, M) \;\equiv\; \frac{\mathcal{Q}_{\mathrm{dual}}(N, M)}{1 + \mathcal{Q}_{\mathrm{dual}}(N, M)}, \qquad \mathcal{Q}_{\mathrm{dual}}(N, M) \;\equiv\; M - \mathbf{r}_A(M+1)^{\mathsf T}\,\Bigl(\bigl(U_{M+1} U_{M+1}^{\mathsf T}\bigr)_{AA}\Bigr)^{-1}\,\mathbf{r}_A(M+1), \tag{31}$$ where $\bigl((U_{M+1} U_{M+1}^{\mathsf T})_{AA}\bigr)_{m, n} = \sum_{k=1}^M (H_{\lfloor k/m \rfloor} - H_{\lfloor k/(m+1) \rfloor})(H_{\lfloor k/n \rfloor} - H_{\lfloor k/(n+1) \rfloor})$ ($1 \le m, n \le N-1$) and $r_m(M+1) = (M+1)(H_{\lfloor M/m \rfloor} - H_{\lfloor M/(m+1) \rfloor}) - f_m(M)$.

Proof.

• (a): Since $\{f_n\mathbf{1}_{[1, N)}\}_{n=1}^{N-1}$ and $\{F_n^*\}_{n=1}^{N-1}$ are biorthogonal bases of $\mathcal{S}_{N-1}$, any $h \in \mathcal{S}_{N-1}$ satisfies $h = \sum_{n=1}^{N-1}\langle h, F_n^* \rangle_{\mathcal{H}} f_n\mathbf{1}_{[1, N)} = \Phi_{h, N}\mathbf{1}_{[1, N)}$, so $R_{h, N} = h - \Phi_{h, N}$ vanishes on $[1, N)$. For $y > 1$, $(D_d f_k)(y) = k\lfloor y/(dk) \rfloor - (k+1)\lfloor y/(d(k+1)) \rfloor = \frac{1}{d}\sum_{j=0}^{d-1}((dk+j)\lfloor y/(dk+j) \rfloor - (dk+j+1)\lfloor y/(dk+j+1) \rfloor) = \frac{1}{d}\sum_{j=0}^{d-1} f_{dk+j}(y)$. Applying $D_2$ to $\Phi_{1, M} = \sum_{k=1}^{M-1} M_1(k) f_k$ gives $D_2 \Phi_{1, M} = \sum_{k=1}^{M-1}\frac{1}{2}M_1(k)(f_{2k} + f_{2k+1}) = \sum_{n=1}^{2M-1}\frac{1}{2}M_1(\lfloor n/2 \rfloor)f_n$, proving (26). Since $1 - R_{1, N} = \Phi_{1, N} \in \mathcal{V}_{S_{\le N}}$, we have $\operatorname{dist}_{\mathcal{H}}^2(1, \mathcal{V}_{S_{\le N}}) = \operatorname{dist}_{\mathcal{H}}^2(R_{1, N}, \mathcal{V}_{S_{\le N}})$, which expands to (27).
• (b): For any $r \in \mathcal{S}_\infty \cap \mathcal{V}_{S_{\le N}}^\perp$, we have $r\mathbf{1}_{[1, N)} \in \mathcal{S}_{N-1}$ and $\langle r\mathbf{1}_{[1, N)}, f_n \rangle_{\mathcal{H}} = -\langle r\mathbf{1}_{[N, \infty)}, f_n \rangle_{\mathcal{H}} = \tau_n(r; N)$. Expanding $r\mathbf{1}_{[1, N)}$ in the basis $\{F_n^*\}_{n=1}^{N-1}$ gives $r\mathbf{1}_{[1, N)} = \sum_{n=1}^{N-1} \tau_n(r; N) F_n^*$, proving (28) and (29). Since $r_{\mathrm{tail}} = r\mathbf{1}_{[N, \infty)}$ ranges freely over $\mathcal{S}_{[N, \infty)}$ and $\operatorname{span}\{f_n\mathbf{1}_{[N, \infty)}\}_{n=1}^{N-1} \subset \mathcal{S}_{[N, \infty)}$ has Gram matrix $T_{\mathrm{tail}}(N)$, maximizing $\frac{\boldsymbol{\tau}^{\mathsf T} M_{F, \mathrm{full}}^*(N)\boldsymbol{\tau}}{\|r_{\mathrm{tail}}\|_{\mathcal{H}}^2}$ over $r_{\mathrm{tail}} \in \mathcal{S}_{[N, \infty)}$ yields $\lambda_{\max}(T_{\mathrm{tail}}(N)^{1/2} M_{F, \mathrm{full}}^*(N) T_{\mathrm{tail}}(N)^{1/2}) = \lambda_{\max}(M_{F, \mathrm{full}}^*(N) T_{\mathrm{tail}}(N))$, proving (30).
• (c): Restricting $M_{F, \mathrm{full}}^*(M+1) = W_{M+1}^{\mathsf T}(I_M + \mathbf{1}_M\mathbf{1}_M^{\mathsf T})W_{M+1}$ to its bottom-right block $B \times B$ gives $G_{\mathcal{D}^*}(N, M) = K_{\mathcal{D}^*}(N, M) + \mathbf{m}_{N, M}\mathbf{m}_{N, M}^{\mathsf T}$, so Sherman–Morrison gives $\mathrm{LB}_{\mathrm{dual}}(N, M) = \frac{\mathcal{Q}_{\mathrm{dual}}}{1 + \mathcal{Q}_{\mathrm{dual}}}$ with $\mathcal{Q}_{\mathrm{dual}} := \mathbf{m}_{N, M}^{\mathsf T} K_{\mathcal{D}^*}(N, M)^{-1}\mathbf{m}_{N, M}$. Since $W_{M+1} U_{M+1} = I_M$, we have $W_{BB}^{-1} = U_{BB}$ and $Z_{AB} := W_{AB} U_{BB} = -W_{AA} U_{AB}$, so $K_{\mathcal{D}^*}(N, M)^{-1} = U_{BB}(I_B + Z_{AB}^{\mathsf T} Z_{AB})^{-1} U_{BB}^{\mathsf T}$ and $U_{BB}^{\mathsf T}\mathbf{m}_{N, M} = \mathbf{1}_B + Z_{AB}^{\mathsf T}\mathbf{1}_A$. Applying the Woodbury identity $(I_B + Z_{AB}^{\mathsf T} Z_{AB})^{-1} = I_B - Z_{AB}^{\mathsf T}(I_A + Z_{AB} Z_{AB}^{\mathsf T})^{-1} Z_{AB}$ yields $\mathcal{Q}_{\mathrm{dual}}(N, M) = M - (\mathbf{1}_A - Z_{AB}\mathbf{1}_B)^{\mathsf T}(I_A + Z_{AB} Z_{AB}^{\mathsf T})^{-1}(\mathbf{1}_A - Z_{AB}\mathbf{1}_B)$. Substituting $\mathbf{1}_A - Z_{AB}\mathbf{1}_B = W_{AA}(U_{AA}\mathbf{1}_A + U_{AB}\mathbf{1}_B) = W_{AA}\mathbf{r}_A(M+1)$ and $I_A + Z_{AB} Z_{AB}^{\mathsf T} = W_{AA}(U_{AA} U_{AA}^{\mathsf T} + U_{AB} U_{AB}^{\mathsf T})W_{AA}^{\mathsf T} = W_{AA}(U_{M+1} U_{M+1}^{\mathsf T})_{AA} W_{AA}^{\mathsf T}$ cancels $W_{AA}$ identically, establishing (31). $\blacksquare$

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6. Certified Numerical and Exact Rational Tables (mpmath.iv.dps = 60)

All finite numerical checks and exact rational identities are reproduced by the self-contained script src/verify_harmonic_mobius_factorization.py (results/harmonic_mobius_factorization.json).

Table 1: Exact Dual Norm $\|F_n^*\|_{\mathcal{H}}^2$, Squarefree Decomposition, and Convergence to $15/\pi^2 \approx 1.51981775$

| Index $n$ | Cutoff $R(n)$ | Exact Rational $\|F_n^*\|_{\mathcal{H}}^2$ | Decimal $\|F_n^*\|_{\mathcal{H}}^2$ | Squarefree Sum $\sum_{r=1}^{R(n)}\frac{\mu(r)^2}{r^2}$ | Bound $M_1(n)^2 + \sum_{r=1}^{R(n)}\frac{\mu(r)^2}{r^2}$ | Head Remainder | Difference $\|F_n^*\|_{\mathcal{H}}^2 - \frac{15}{\pi^2}$ | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $n = 2$ | $1$ | $3/2$ | $1.50000000$ | $1.00000000$ | $1.25000000$ | $0.50000000$ | $-0.01981775$ | | $n = 3$ | $1$ | $31/18$ | $1.72222222$ | $1.00000000$ | $1.02777778$ | $0.72222222$ | $+0.20240447$ | | $n = 4$ | $1$ | $25/18$ | $1.38888889$ | $1.00000000$ | $1.02777778$ | $0.38888889$ | $-0.13092887$ | | $n = 8$ | $2$ | $3411/2450$ | $1.39224490$ | $1.25000000$ | $1.25009070$ | $0.14224490$ | $-0.12757286$ | | $n = 16$ | $3$ | $70656309/50100050$ | $1.41030416$ | $1.36111111$ | $1.36480037$ | $0.04919305$ | $-0.10951359$ | | $n = 32$ | $5$ | — | $1.47936222$ | $1.40111111$ | $1.40504656$ | $0.07825110$ | $-0.04045554$ | | $n = 64$ | $7$ | — | $1.48538682$ | $1.44929705$ | $1.44955252$ | $0.03608976$ | $-0.03443094$ | | $n = 128$ | $10$ | — | $1.48401061$ | $1.45929705$ | $1.45929738$ | $0.02471356$ | $-0.03580714$ | | $n = 200$ | $13$ | — | $1.52441937$ | $1.47347867$ | $1.47442517$ | $0.05094070$ | $+0.00460162$ |

Table 2: Head $[1, N)$ vs. Tail $[N, \infty)$ Energy Split and Wavelet Interpolant Norms ($S_{\le N} = \{2, \dots, N\}$)

| Cutoff $N$ | Distance $d_N^2$ | Head $E_{\mathrm{head}}(r_{\mathrm{full}})$ | Tail $E_{\mathrm{tail}}(r_{\mathrm{full}})$ | Tail Share ($r_{\mathrm{full}}$) | Interpolant $\|R_{1, N}\|_{\mathcal{H}}^2$ | Octave $\|r_1(N)\|_{\mathcal{H}}^2$ | Tail Share ($r_1$) | Interpolant $\|R_{\chi_1, N}\|_{\mathcal{H}}^2$ | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $N = 2$ | $0.30685282$ | $0$ | $0.30685282$ | $100.000\%$ | $0.30685282$ | $0.13932624$ | $72.135\%$ | $0.19314718$ | | $N = 3$ | $0.09574778$ | $0.00183451$ | $0.09391328$ | $98.084\%$ | $0.09861029$ | $0.13774430$ | $71.690\%$ | $0.19314718$ | | $N = 4$ | $0.06595169$ | $0.00143985$ | $0.06451184$ | $97.817\%$ | $0.06761422$ | $0.09244028$ | $75.743\%$ | $0.12605619$ | | $N = 8$ | $0.02424453$ | $0.00477832$ | $0.01946621$ | $80.291\%$ | $0.03687741$ | $0.04675150$ | $61.468\%$ | $0.09417114$ | | $N = 16$ | $0.01793627$ | $0.00575646$ | $0.01217981$ | $67.906\%$ | $0.03717749$ | $0.03137248$ | $54.080\%$ | $0.08013985$ | | $N = 32$ | $0.01407701$ | $0.00669565$ | $0.00738136$ | $52.436\%$ | $0.03921085$ | $0.02627010$ | $42.508\%$ | $0.08647429$ | | $N = 64$ | $0.01139247$ | $0.00634818$ | $0.00504430$ | $44.277\%$ | $0.03628948$ | $0.02170346$ | $36.638\%$ | $0.08142694$ |

Table 3: Dimension-Reduced $(N-1)\times(N-1)$ Woodbury Annihilator Bounds $\mathrm{LB}_{\mathrm{dual}}(N, M)$

| Cutoff $N$ | Woodbury Size $N-1$ | $\mathrm{LB}_{\mathrm{dual}}(N, 2N-1)$ (Direct Size $N$) | $\mathrm{LB}_{\mathrm{dual}}(N, 2N)$ (Direct Size $N+1$) | $\mathrm{LB}_{\mathrm{dual}}(N, 4N)$ (Direct Size $3N+1$) | Full Distance $d_N^2$ | | :---: | :---: | :---: | :---: | :---: | :---: | | $N = 2$ | $1 \times 1$ | $0.16666667$ ($2 \times 2$) | $0.21666667$ ($3 \times 3$) | $0.25436508$ ($7 \times 7$) | $0.30685282$ | | $N = 3$ | $2 \times 2$ | $0.03947130$ ($3 \times 3$) | $0.06328083$ ($4 \times 4$) | $0.07631830$ ($10 \times 10$) | $0.09574778$ | | $N = 4$ | $3 \times 3$ | $0.03731411$ ($4 \times 4$) | $0.03847585$ ($5 \times 5$) | $0.05238213$ ($13 \times 13$) | $0.06595169$ | | $N = 8$ | $7 \times 7$ | $0.01247744$ ($8 \times 8$) | $0.01426724$ ($9 \times 9$) | $0.01947397$ ($25 \times 25$) | $0.02424453$ | | $N = 16$ | $15 \times 15$ | $0.01157172$ ($16 \times 16$) | $0.01194771$ ($17 \times 17$) | $0.01505304$ ($49 \times 49$) | $0.01793627$ |

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7. Discussion and Scope Limits

1. Exact Head–Dual Duality: Theorem 2 shows that the restricted first-difference wavelet Gram matrix $H_{\mathrm{head}}(N)$ on $[1, N)$ and the full dual harmonic representer Gram matrix $M_{F, \mathrm{full}}^*(N)$ are exact matrix inverses factored explicitly by the harmonic and Möbius floor-quotient matrices $U_N$ and $W_N$.
2. Asymptotic Resolution of $\|F_n^*\|_{\mathcal{H}}^2$: Theorem 3 resolves the asymptotic value of $\|F_n^*\|_{\mathcal{H}}^2$ unconditionally, proving $\lim_{n \to \infty}\|F_n^*\|_{\mathcal{H}}^2 = \frac{15}{\pi^2}$ and supplying an elementary all-scale squarefree lower bound.
3. Open Problems and Scope: While Theorem 4 reduces the $(M-N+1)\times(M-N+1)$ dual variational bound $\mathrm{LB}_{\mathrm{dual}}(N, M)$ to an $(N-1)\times(N-1)$ system and characterizes the head-to-tail energy ratio via $\lambda_{\max}(H_{\mathrm{head}}(N)^{-1} T_{\mathrm{tail}}(N))$, we emphasize that establishing uniform decay of $d_N^2 \to 0$ as $N \to \infty$ is equivalent to the Riemann Hypothesis and remains open; no unconditional decay theorem for $d_N$ or resolution of the Riemann Hypothesis is obtained in this paper.

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