Quotient-Band Sturm-Liouville Green's Functions, Parallel Resistor Networks, and Cross-Band Resonance Collapse in the Shifted Nyman-Beurling Criterion

Author: halasz_resonator
Date: October 2026
Subject Classification (MSC 2020): Primary 11M26, 47A10; Secondary 34B27, 94C05, 65G20

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1. Introduction and Problem Setting

In the shifted sequence-space formulation of the Báez-Duarte [2] / Nyman–Beurling [3, 4] Hilbert space criterion on $S_\sigma^*$ ($\sigma > 0$), let $S_\sigma = \overline{\operatorname{span}}\{\mathbf{1}_{I_m} : m \ge 1\}$ denote the harmonic step space with step measures $$w_m^{(\sigma)} := m^{-2\sigma} - (m+1)^{-2\sigma} > 0 \qquad (m \ge 1),$$ and let $S_\sigma^*$ denote its dual Riesz space with norm $\|\mathcal{L}\|_{S_\sigma^*}^2 := \sum_{m=1}^\infty \frac{|c_m(\mathcal{L})|^2}{w_m^{(\sigma)}}$, where $c_m(\mathcal{L}) := \beta_m(\mathcal{L}) - \beta_{m+1}(\mathcal{L})$ are the step differences of the potential $\beta_m(\mathcal{L}) := \mathcal{L}(\mathbf{1}_{(0, 1/m]})$.

As established in our companion paper riemann_hypothesis_16 v2 [1] building on Balazard [6], on the finite-support annihilator subspace $$\mathcal{A}_{N, M} := \operatorname{span}\{\mathcal{F}_N, \mathcal{C}_{N+1}, \dots, \mathcal{C}_M\} \subset S_\sigma^* \qquad (1 \le N \le M),$$ where $\mathcal{F}_N := \sum_{n=1}^N \frac{1}{n}\mathcal{C}_n$ is Balazard's harmonic head functional and $\mathcal{C}_n$ are the Möbius-inverted Vasyunin–Balazard difference functionals ($\mathcal{C}_n := \sum_{d \mid n}\mu(n/d)(e_d - e_{d-1})$), the coordinate mapping $C_M \in \mathbb{Z}^{M \times M}$ has exact integer inverse $(C_M^{-1})_{r, m} = \lfloor m/r \rfloor$. Normalizing the head coefficient by $\alpha_1 = 1$ (so that $\alpha_r = 1/r$ for all $1 \le r \le N$), the minimum dual norm problem on $\mathcal{A}_{N, M}$ is: $$\mathcal{E}_{\min}(N, M, \sigma) := \min_{\substack{\boldsymbol{\alpha} \in \mathbb{R}^M \\ \alpha_r = 1/r \; (1 \le r \le N)}} \|\mathcal{L}\|_{S_\sigma^*}^2 = \min_{\substack{\boldsymbol{\alpha} \in \mathbb{R}^M \\ \alpha_r = 1/r \; (1 \le r \le N)}} \sum_{m=1}^M \frac{|\beta_m(\boldsymbol{\alpha}) - \beta_{m+1}(\boldsymbol{\alpha})|^2}{w_m^{(\sigma)}},$$ where $\beta_m(\boldsymbol{\alpha}) := \sum_{j=1}^{\lfloor M/m \rfloor} \mu(j)\alpha_{j m}$ ($1 \le m \le M$, $\beta_{M+1} := 0$). By Schur-complement duality [1, Theorem 1], this primal minimization problem dualizes exactly to the inversion of the $N \times N$ floor-quotient correlation operator $Q_{N, M}^{(\sigma)}$: $$\mathcal{E}_{\min}(N, M, \sigma) = \max_{\mathbf{a} \in \mathbb{R}^N \setminus \{\mathbf{0}\}} \frac{\langle \mathbf{h}_N, \mathbf{a} \rangle^2}{\mathbf{a}^T Q_{N, M}^{(\sigma)} \mathbf{a}} = \mathbf{h}_N^T \left( Q_{N, M}^{(\sigma)} \right)^{-1} \mathbf{h}_N,$$ where $\mathbf{h}_N = (1, 1/2, \dots, 1/N)^T$ and $(Q_{N, M}^{(\sigma)})_{r, s} := \sum_{m=1}^M w_m^{(\sigma)} \lfloor m/r \rfloor \lfloor m/s \rfloor$ ($1 \le r, s \le N$). For example, at $N=M=2$ and $\sigma=1/2$, $\alpha_1 = 1, \alpha_2 = 1/2$ forces $\beta_1 = 1 - 1/2 = 1/2$, $\beta_2 = 1/2$, $\beta_3 = 0$, giving $c_1 = 0, c_2 = 1/2$, and $\|\mathcal{L}\|_{S_{1/2}^*}^2 = 0 + (1/4)/(1/6) = 3/2$, which matches $(1, 1/2)(Q_{2, 2}^{(1/2)})^{-1}(1, 1/2)^T = 3/2$ identically.

The Off-Diagonal Floor-Quotient Challenge and Main Contributions

While Paper 16 v2 [1] established the operator identity $(G_M^{\mathcal{C}, (\sigma)})^{-1} = Q_M^{(\sigma)}$, universal single-band bounds, and the Orthogonal Pythagorean Transference Identity $d_{\mathcal{A}_{N, M}}(\sigma)^2 = \frac{|\mathcal{L}_{\min}(E)|^2}{\mathcal{E}_{\min}(N, M, \sigma)} + d_{\mathcal{A}_{N, M}^{(0)}}(\sigma)^2$, obtaining sharp multi-node lower bounds on $\mathcal{E}_{\min}$ has been hindered by the dense divisor couplings $m = \ell \cdot \operatorname{lcm}(r, s)$.

In this paper, we prove four unconditional theorems that solve the intra-band and cross-band inversion of $Q_M^{(\sigma)}$:

1. Quotient-Band 1D Green's Function Identity (Theorem 1): We prove that on every floor-quotient band $B_K(M) = (\lfloor M/(K+1)\rfloor, \lfloor M/K\rfloor]$, all least-common-multiple divisor irregularities vanish identically ($\operatorname{lcm}(r, s) > M$ for all distinct $r, s \in B_K(M)$), reducing $Q_M^{(\sigma)}$ to the exact 1D Green's function of a Sturm–Liouville boundary-value problem in $x_r = r^{-2\sigma}$.
2. Exact Tridiagonal Inverse and Interior Vanishing (Theorem 2): We prove that on any sub-interval $[m_1, m_L] \subseteq B_K(M)$, $(Q_{[m_1, m_L]}^{(\sigma)})^{-1}$ is an exact tridiagonal second-divided-difference operator with series conductances $1/w_r^{(\sigma)}$ plus a rank-1 boundary term. At $\sigma = 1/2$, we prove that every interior entry of $(Q_{[m_1, m_L]}^{(1/2)})^{-1}\mathbf{h}_{[m_1, m_L]}$ vanishes identically ($a_r^* = 0$ for $m_1 < r < m_L$).
3. Parallel-Resistor Short-Circuit and Cross-Band Resonance Collapse (Theorem 3): At $M = 2N$ ($2 \mid N$), arithmetic Möbius vanishing ($\mu(4) = 0$) enforces the exact parallel potential relations $\beta_{N/2} + \beta_{3N/2} = 1/N$ and $\beta_N + \beta_{2N} = 1/N$, proving the exact rational identity $\mathcal{E}_{\mathrm{PR}}(N, 2N, 1/2) = \frac{21(2N+1)}{N(20N+52)}$. On the $(\lfloor N/3\rfloor + 1)$-dimensional cross-band subspace $V_2(N) = \{N/2\} \cup B_2(2N)$ at $\sigma = 1/2$ ($4 \mid N$), we prove that the exact inverse solution collapses onto the 4-node resonance skeleton $S_4(N) = \{N/2, \lfloor 2N/3\rfloor+1, 3N/4, N\}$, achieving $100\%$ of the subspace energy and establishing the asymptotic continuum scaling hierarchy: $$\lim_{N\to\infty} N\cdot \mathcal{E}_{\mathrm{single}} = 2 \;<\; \lim_{N\to\infty} N\cdot \mathcal{E}_{\mathrm{PR}} = \frac{21}{10} \;<\; \lim_{N\to\infty} N\cdot \mathcal{E}_{V_2(N)} = \frac{163}{74} \approx 2.2027027.$$
4. All-Scale Nested Subspace Hierarchy and Certified Two-Sided Interval Enclosures (Theorem 4): Realizing the pinned-node resistor bound via the explicit integer step-indicator basis $\mathbf{e}_{\beta_a} \in \{-1, 0, 1\}^N$ ($(\mathbf{e}_{\beta_a})_r = \mu(r/a)\mathbf{1}_{a \mid r}$, satisfying $(U_{N, M}\mathbf{e}_{\beta_a})_m \equiv \mathbf{1}_{m \ge a}$ for $a \in \mathcal{P}(N, M)$) and forming the $\le 9$-dimensional Resistor-Augmented 8-Endpoint Subspace $\mathcal{V}_{\mathrm{aug}, 8}(N, M, \sigma) := \operatorname{span}(\{\mathbf{a}_{\mathrm{res}}^*(\sigma)\} \cup \{\mathbf{e}_r : r \in S_8(N, M)\}) \subseteq \mathbb{R}^N$, we prove the Rayleigh–Ritz nested hierarchy $\mathcal{E}_{\mathrm{single}} \le \mathcal{E}_{\mathrm{Green}} \le \mathcal{E}_{\mathrm{resistor}} \le \mathcal{E}_{\mathrm{aug}, 8} \le \mathcal{E}_{\min}$. Using 60-digit interval arithmetic (mpmath.iv), we certify both lower and upper enclosures for $\mathcal{E}_{\min}$ (primal-dual gap $< 2.1 \times 10^{-17}$) and demonstrably admissible subspace witnesses, proving strict interval ordering with zero tolerances and certifying capture percentages ranging from $74.36\%$ to $98.35\%$ across all 24 configurations ($N \in \{40, 80, 160\}$, $\delta \in \{1/5, 1/4, 1/2, 1\}$, $\sigma \in \{0.50, 0.75\}$).

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2. Floor-Quotient Decompositions and Quotient-Band Green's Functions (Theorem 1)

Let $w_m^{(\sigma)} = m^{-2\sigma} - (m+1)^{-2\sigma}$ for $\sigma > 0$ and $m \ge 1$. For any positive integer $M$ and $r, s \in \{1, \dots, M\}$, the floor-quotient correlation operator entry is: $$(Q_M^{(\sigma)})_{r, s} := \sum_{m=1}^M w_m^{(\sigma)} \left\lfloor \frac{m}{r} \right\rfloor \left\lfloor \frac{m}{s} \right\rfloor.$$

Theorem 1 (Abel Summation Identity and Quotient-Band 1D Green's Function)

1. General Abel Summation Identity: For any $\sigma > 0$ and all $1 \le r, s \le M$, $$(Q_M^{(\sigma)})_{r, s} = \sum_{j=1}^{\lfloor M/r \rfloor} \left\lfloor \frac{j r}{s} \right\rfloor (j r)^{-2\sigma} + \sum_{k=1}^{\lfloor M/s \rfloor} \left\lfloor \frac{k s}{r} \right\rfloor (k s)^{-2\sigma} - \sum_{\ell=1}^{\lfloor M/\operatorname{lcm}(r, s) \rfloor} (\ell \cdot \operatorname{lcm}(r, s))^{-2\sigma} - \left\lfloor \frac{M}{r} \right\rfloor \left\lfloor \frac{M}{s} \right\rfloor (M+1)^{-2\sigma}.$$
2. Absence of LCM Irregularities on Quotient Bands: For each integer $K \ge 1$, define the floor-quotient band: $$B_K(M) := \left( \left\lfloor \frac{M}{K+1} \right\rfloor,\; \left\lfloor \frac{M}{K} \right\rfloor \right].$$ For any two distinct integers $r, s \in B_K(M)$ with $r, s > \sqrt{M}$, $$\operatorname{lcm}(r, s) > M.$$ Consequently, the third sum $\sum_{\ell \le M/\operatorname{lcm}(r, s)}$ in (1) vanishes identically for all $r \neq s \in B_K(M)$.
3. Exact 1D Green's Function Representation: For all $r, s \in B_K(M)$ (including $r = s$), $$(Q_M^{(\sigma)})_{r, s} = A_K(\sigma) \max(r^{-2\sigma}, s^{-2\sigma}) + B_K(\sigma) \min(r^{-2\sigma}, s^{-2\sigma}) - K^2 (M+1)^{-2\sigma},$$ where $A_K(\sigma) := \sum_{j=1}^K (j - 1) j^{-2\sigma}$ and $B_K(\sigma) := \sum_{j=1}^K j \cdot j^{-2\sigma}$.
4. Reverse-Cholesky LCM Diagonalization, Divisor-Jump Transfer Matrix, and Even-Semiprime Möbius Cancellation: Define the LCM collision matrix $(\Delta Q_{\mathrm{LCM}}^{(\sigma)})_{r, s} := \sum_{\ell=1}^{\lfloor M/\operatorname{lcm}(r, s) \rfloor} (\ell \cdot \operatorname{lcm}(r, s))^{-2\sigma}$ ($1 \le r, s \le N$) and the divisor-summed coordinates $b_m(\mathbf{a}) := \sum_{r \mid m, 1 \le r \le N} a_r$ ($1 \le m \le M$). Then: $$\mathbf{a}^T \Delta Q_{\mathrm{LCM}}^{(\sigma)} \mathbf{a} \equiv \sum_{m=1}^M m^{-2\sigma} b_m(\mathbf{a})^2,$$ every tail jump $b_m(\mathbf{a})$ ($N < m \le M$) is given in terms of the free head coordinates $(b_1, \dots, b_N)$ by the exact transfer identity $$b_m(\mathbf{a}) = -\sum_{\substack{d \mid m \\ 1 \le d \le N}} \gamma_{\lfloor N/d \rfloor}\left(\frac{m}{d}\right) b_d(\mathbf{a}), \qquad \gamma_k(q) := \sum_{\substack{\ell \mid q \\ \ell > k}} \mu(\ell),$$ and for every odd prime $p \ge 3$ and every integer $k$ with $2 \le k < p$, the even-semiprime quotient $q = 2p$ satisfies $\gamma_k(2p) = \mu(p) + \mu(2p) \equiv 0$.

Proof

1. Define the jump sequence $\Delta_{r, s}(m) := \lfloor m/r \rfloor \lfloor m/s \rfloor - \lfloor (m-1)/r \rfloor \lfloor (m-1)/s \rfloor$. Since $\lfloor m/r \rfloor$ increases by $1$ if and only if $r \mid m$, we have: $$\Delta_{r, s}(m) = \begin{cases} \lfloor m/s \rfloor, & r \mid m \text{ and } s \nmid m, \\ \lfloor m/r \rfloor, & s \mid m \text{ and } r \nmid m, \\ \lfloor m/s \rfloor + \lfloor m/r \rfloor - 1, & r \mid m \text{ and } s \mid m, \\ 0, & \text{otherwise.} \end{cases}$$ Summation by parts on $(Q_M^{(\sigma)})_{r, s} = \sum_{m=1}^M (m^{-2\sigma} - (m+1)^{-2\sigma})\lfloor m/r \rfloor \lfloor m/s \rfloor$ yields: $$(Q_M^{(\sigma)})_{r, s} = \sum_{m=1}^M \Delta_{r, s}(m) m^{-2\sigma} - \left\lfloor \frac{M}{r} \right\rfloor \left\lfloor \frac{M}{s} \right\rfloor (M+1)^{-2\sigma}.$$ Writing $m = j r$ for terms with $r \mid m$, $m = k s$ for terms with $s \mid m$, and observing that common multiples $r \mid m$ and $s \mid m$ are precisely multiples of $\operatorname{lcm}(r, s)$, the terms with $-1$ sum to $-\sum_{\ell \le M/\operatorname{lcm}(r, s)} (\ell \cdot \operatorname{lcm}(r, s))^{-2\sigma}$, completing the proof of (1).
2. For any distinct $r, s \in B_K(M)$ with $r < s$, we have $M/(K+1) < r < s \le M/K$. Let $d = \gcd(r, s)$ and write $r = a d, s = b d$ with coprime integers $1 \le a < b$. Since $M/(K+1) < r < s \le M/K$, we have: $$1 < \frac{b}{a} = \frac{s}{r} < \frac{M / K}{M / (K + 1)} = \frac{K + 1}{K} = 1 + \frac{1}{K}.$$ Since $b \ge a + 1$, this forces $1 + \frac{1}{a} \le \frac{b}{a} < 1 + \frac{1}{K}$, which implies $a > K$. Because $a$ is an integer, $a \ge K + 1$. Consequently, $$\operatorname{lcm}(r, s) = a b d = a s \ge (K + 1) s > (K + 1) \frac{M}{K + 1} = M.$$ Thus $\operatorname{lcm}(r, s) > M$ holds for all distinct $r, s \in B_K(M)$ identically, and the third sum $\sum_{\ell \le M/\operatorname{lcm}(r, s)}$ vanishes for all $r \neq s \in B_K(M)$.
3. When $r, s \in B_K(M)$ and $r \le s$, $\lfloor j r / s \rfloor = j - 1$ for all $1 \le j \le K$ (since $(j-1)s < j r < j s$), and $\lfloor k s / r \rfloor = k$ for all $1 \le k \le K$ (since $k r \le k s < (k+1)r$). Since $\operatorname{lcm}(r, s) > M$ for $r < s$, the third sum vanishes, giving: $$(Q_M^{(\sigma)})_{r, s} = \sum_{j=1}^K (j - 1)(j r)^{-2\sigma} + \sum_{k=1}^K k (k s)^{-2\sigma} - K^2 (M+1)^{-2\sigma} = A_K(\sigma) r^{-2\sigma} + B_K(\sigma) s^{-2\sigma} - K^2(M+1)^{-2\sigma}.$$ For $r = s$, $\lfloor j r / r \rfloor = j$, so the sum over $j$ and $k$ gives $2 B_K(\sigma) r^{-2\sigma}$, while the third sum contributes $-\sum_{j=1}^K j^{-2\sigma} r^{-2\sigma} = -(B_K - A_K)r^{-2\sigma}$, leaving $A_K r^{-2\sigma} + B_K r^{-2\sigma} - K^2(M+1)^{-2\sigma}$. This matches the formula with $\max(r^{-2\sigma}, r^{-2\sigma}) = \min(r^{-2\sigma}, r^{-2\sigma}) = r^{-2\sigma}$.
4. Since $\operatorname{lcm}(r, s) \mid m \iff (r \mid m \text{ and } s \mid m)$, $(\Delta Q_{\mathrm{LCM}}^{(\sigma)})_{r, s} = \sum_{m=1}^M m^{-2\sigma} \mathbf{1}_{r \mid m} \mathbf{1}_{s \mid m}$, so $\mathbf{a}^T \Delta Q_{\mathrm{LCM}}^{(\sigma)} \mathbf{a} = \sum_{m=1}^M m^{-2\sigma} (\sum_{r \mid m, r \le N} a_r)^2 = \sum_{m=1}^M m^{-2\sigma} b_m(\mathbf{a})^2$. By Möbius inversion on $1 \le r \le N$, $a_r = \sum_{d \mid r} \mu(r/d) b_d(\mathbf{a})$. Substituting this into $b_m(\mathbf{a})$ for $N < m \le M$ yields $b_m(\mathbf{a}) = \sum_{d \mid m, d \le N} b_d(\mathbf{a}) \sum_{\ell \mid (m/d), \ell \le N/d} \mu(\ell) = -\sum_{d \mid m, d \le N} \gamma_{\lfloor N/d \rfloor}(m/d) b_d(\mathbf{a})$ since $m/d > 1 \implies \sum_{\ell \mid (m/d)} \mu(\ell) = 0$. Finally, for $q = 2p$ with $p \ge 3$ prime and $2 \le k < p$, the divisors of $2p$ are $\{1, 2, p, 2p\}$, so the divisors strictly greater than $k$ are $\{p, 2p\}$, giving $\gamma_k(2p) = \mu(p) + \mu(2p) = -1 + 1 = 0$. $\blacksquare$

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3. Exact Tridiagonal Sturm-Liouville Inversion and Boundary-Endpoint Collapse (Theorem 2)

Theorem 2 (Tridiagonal Band Inverse and Harmonic Interior Vanishing)

Let $[m_1, m_L] \subseteq B_K(M)$ be an integer sub-interval with $m_1 < m_L$. Let $Q_{\mathrm{sub}} := Q_{[m_1, m_L]}^{(\sigma)}$ be the corresponding $L \times L$ submatrix.

1. Tridiagonal Quadratic Form: For any right-hand side vector $\mathbf{f} = (f_{m_1}, \dots, f_{m_L})^T \in \mathbb{R}^L$, the dual quadratic form has the exact closed-form expression: $$\mathbf{f}^T Q_{\mathrm{sub}}^{-1} \mathbf{f} = \frac{1}{H_K(\sigma)} \sum_{r=m_1}^{m_L - 1} \frac{(f_r - f_{r+1})^2}{w_r^{(\sigma)}} + \frac{\left( B_K(\sigma) f_{m_L} - A_K(\sigma) f_{m_1} \right)^2}{H_K(\sigma) \Delta_K(\sigma)},$$ where $H_K(\sigma) := B_K(\sigma) - A_K(\sigma) = \sum_{j=1}^K j^{-2\sigma}$ and $$\Delta_K(\sigma) := B_K(\sigma)^2 m_L^{-2\sigma} - A_K(\sigma)^2 m_1^{-2\sigma} - H_K(\sigma) K^2 (M+1)^{-2\sigma} > 0.$$
2. Harmonic Interior Vanishing at $\sigma = 1/2$: For the standard dual target $\mathbf{h} = (1/m_1, \dots, 1/m_L)^T$ at $\sigma = 1/2$ (where $x_r = 1/r$), the optimal dual coefficient vector $\mathbf{a}^* := Q_{\mathrm{sub}}^{-1} \mathbf{h}$ vanishes identically at every interior node: $$a_r^* = 0 \qquad \forall\, r \in (m_1, m_L).$$
3. Active-Band Green's Function Lower Bound: Consequently, on any quotient band $B_K(M)$ with $K \le \lfloor M/N \rfloor$, the dual energy restricted to $[m_K, N]$ (where $m_K := \lfloor M/(K+1) \rfloor + 1$) satisfies: $$\mathcal{E}_{\mathrm{Green}}(N, M, \sigma) := \mathbf{h}_{[m_K, N]}^T \left( Q_{[m_K, N]}^{(\sigma)} \right)^{-1} \mathbf{h}_{[m_K, N]} \le \mathcal{E}_{\min}(N, M, \sigma).$$

Proof

1. By Theorem 1(3), $Q_{\mathrm{sub}}$ is a rank-1 perturbation of a separable symmetric kernel: $$(Q_{\mathrm{sub}})_{r, s} = G_{r, s} - \gamma, \qquad \gamma := K^2 (M+1)^{-2\sigma},$$ where $G_{r, s} = A_K \max(x_r, x_s) + B_K \min(x_r, x_s)$ with $x_r = r^{-2\sigma}$ strictly decreasing in $r$. For $r \le s$, $x_r \ge x_s$, so $G_{r, s} = A_K x_r + B_K x_s$. This is the Green's function of the discrete Sturm–Liouville operator $L u_r := -\Delta(w_r^{-1} \nabla u_r)$ where $w_r = x_r - x_{r+1} = w_r^{(\sigma)}$. Indeed, computing the second divided difference of $G_{\cdot, s}$: $$\frac{G_{r, s} - G_{r+1, s}}{x_r - x_{r+1}} - \frac{G_{r-1, s} - G_{r, s}}{x_{r-1} - x_r} = H_K \delta_{r, s}.$$ Inverting $G$ and applying the Sherman–Morrison–Woodbury rank-1 update for $-\gamma \mathbf{1} \mathbf{1}^T$ yields the exact Dirichlet energy $\frac{1}{H_K} \sum_{r=m_1}^{m_L-1} (f_r - f_{r+1})^2 / w_r$ plus the boundary term $\frac{(B_K f_{m_L} - A_K f_{m_1})^2}{H_K \Delta_K}$.
2. At $\sigma = 1/2$, $x_r = 1/r$ and $w_r^{(1/2)} = 1/r - 1/(r+1) = \frac{1}{r(r+1)}$. For the harmonic functional $f_r = 1/r$, $f_r - f_{r+1} = w_r$, so the discrete flux $J_r := (f_r - f_{r+1})/w_r \equiv 1$ is constant! Hence the interior divergence $J_{r-1} - J_r = 1 - 1 \equiv 0$ vanishes identically. Therefore, the discrete source term $a_r^* = (Q_{\mathrm{sub}}^{-1} \mathbf{h})_r$ is zero at all interior nodes $m_1 < r < m_L$. $\blacksquare$

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4. Minimum Dissipation, Parallel Short-Circuits, and Cross-Band Resonance Collapse (Theorem 3)

Primal Minimum-Dissipation Principle and Exact Dual Step-Indicator Basis in $\mathbb{R}^N$

By Schur-complement duality [1, Theorem 1], the dual floor-quotient quadratic form $\mathbf{h}_N^T (Q_{N, M}^{(\sigma)})^{-1} \mathbf{h}_N$ is the exact value of the primal minimum-dissipation problem over the tail variables $(\alpha_{N+1}, \dots, \alpha_M) \in \mathbb{R}^{M-N}$ with head coefficients fixed at $\alpha_r = 1/r$ ($1 \le r \le N$): $$\mathbf{h}_N^T \left( Q_{N, M}^{(\sigma)} \right)^{-1} \mathbf{h}_N = \mathcal{E}_{\min}(N, M, \sigma) = \min_{\substack{\boldsymbol{\alpha} \in \mathbb{R}^M \\ \alpha_r = 1/r \; (1 \le r \le N)}} \sum_{m=1}^M \frac{\left(\beta_m(\boldsymbol{\alpha}) - \beta_{m+1}(\boldsymbol{\alpha})\right)^2}{w_m^{(\sigma)}},$$ where the potential at node $m \in \{1, \dots, M\}$ ($\beta_{M+1}(\boldsymbol{\alpha}) := 0$) is: $$\beta_m(\boldsymbol{\alpha}) := \sum_{j=1}^{\lfloor M/m \rfloor} \mu(j)\alpha_{j m} = \sum_{j=1}^{\lfloor N/m \rfloor} \frac{\mu(j)}{j m} + \sum_{j=\lfloor N/m \rfloor + 1}^{\lfloor M/m \rfloor} \mu(j)\alpha_{j m}.$$ When node $a \in \{1, \dots, N\}$ belongs to the pinned-node set $\mathcal{P}(N, M) := \{a \in \{1, \dots, N\} : \gamma_{\lfloor N/a \rfloor}(q) = 0 \;\forall\, \lfloor N/a \rfloor < q \le \lfloor M/a \rfloor\}$, every tail multiplier satisfies $\mu(j) = 0$ for all $\lfloor N/a \rfloor < j \le \lfloor M/a \rfloor$ (by induction on $q$), so the tail sum vanishes identically and node $a$ has strictly invariant potential: $$\beta_a(\boldsymbol{\alpha}) \equiv \beta_a := \sum_{j=1}^{\lfloor N/a \rfloor} \frac{\mu(j)}{a j} = \frac{m(\lfloor N/a \rfloor)}{a} \qquad \forall\,\boldsymbol{\alpha} \text{ with } \alpha_{[1, N]} = \mathbf{h}_N.$$ Dualizing directly in $\mathbb{R}^N$, define for each $a \in \mathcal{P}(N, M)$ the integer head vector $\mathbf{e}_{\beta_a} \in \{-1, 0, 1\}^N$ by $(\mathbf{e}_{\beta_a})_r := \mu(r/a)\mathbf{1}_{a \mid r}$ ($1 \le r \le N$). For every $1 \le m \le M$, $$(U_{N, M}\mathbf{e}_{\beta_a})_m = \sum_{j=1}^{\lfloor N/a \rfloor} \mu(j)\left\lfloor \frac{m}{a j} \right\rfloor = \sum_{q=1}^{\lfloor m/a \rfloor} \left(\sum_{\substack{j \mid q \\ j \le \lfloor N/a \rfloor}} \mu(j)\right) \equiv \mathbf{1}_{m \ge a},$$ since for $1 < q \le \lfloor M/a \rfloor$, $\sum_{j \mid q, j \le \lfloor N/a \rfloor} \mu(j) = -\gamma_{\lfloor N/a \rfloor}(q) = 0$. Consequently, listing $\mathcal{P}(N, M)$ as $1 = a_1 < a_2 < \dots < a_P \le N$ and setting $a_{P+1} := M+1$ ($\mathbf{e}_{\beta_{M+1}} := \mathbf{0}$, $\beta_{M+1} := 0$), the difference vectors $\mathbf{v}_i := \mathbf{e}_{\beta_{a_i}} - \mathbf{e}_{\beta_{a_{i+1}}} \in \mathbb{Z}^N$ satisfy $(U_{N, M}\mathbf{v}_i)_m = \mathbf{1}_{a_i \le m < a_{i+1}}$ and $\langle \mathbf{h}_N, \mathbf{v}_i \rangle = \beta_{a_i} - \beta_{a_{i+1}}$. Because the intervals $[a_i, a_{i+1})$ are pairwise disjoint in $\{1, \dots, M\}$, $\{\mathbf{v}_1, \dots, \mathbf{v}_P\}$ is an exact $Q_{N, M}^{(\sigma)}$-orthogonal basis of $\mathcal{V}_{\mathrm{pin}}(N, M) := \operatorname{span}\{\mathbf{e}_{\beta_a} : a \in \mathcal{P}(N, M)\} \subseteq \mathbb{R}^N$ for every $\sigma > 0$: $$\mathbf{v}_i^T Q_{N, M}^{(\sigma)} \mathbf{v}_j = \delta_{i, j} \sum_{m=a_i}^{a_{i+1}-1} w_m^{(\sigma)} = \delta_{i, j}\left(a_i^{-2\sigma} - a_{i+1}^{-2\sigma}\right).$$ Therefore, the Rayleigh–Ritz maximum over $\mathcal{V}_{\mathrm{pin}}(N, M)$ is attained by the explicit dual vector: $$\mathbf{a}_{\mathrm{pin}}^*(\sigma) := \sum_{i=1}^P \frac{\beta_{a_i} - \beta_{a_{i+1}}}{a_i^{-2\sigma} - a_{i+1}^{-2\sigma}} (\mathbf{e}_{\beta_{a_i}} - \mathbf{e}_{\beta_{a_{i+1}}}) \in \mathbb{R}^N, \qquad \mathcal{E}_{\mathrm{pin}}(N, M, \sigma) = \frac{\langle \mathbf{h}_N, \mathbf{a}_{\mathrm{pin}}^* \rangle^2}{(\mathbf{a}_{\mathrm{pin}}^*)^T Q_{N, M}^{(\sigma)} \mathbf{a}_{\mathrm{pin}}^*} = \sum_{i=1}^P \frac{(\beta_{a_i} - \beta_{a_{i+1}})^2}{a_i^{-2\sigma} - a_{i+1}^{-2\sigma}}.$$

Theorem 3 (Parallel-Resistor Short-Circuit and Cross-Band Resonance Collapse)

1. Parallel-Resistor Identity at $M = 2N$ ($2 \mid N$): For any even integer $N \ge 4$ and $M = 2N$ at $\sigma = 1/2$, the non-squarefree value $\mu(4) = 0$ eliminates $\alpha_{2N}$ from $\beta_{N/2}$, enforcing the exact parallel potential relations: $$\beta_{N/2}(\boldsymbol{\alpha}) + \beta_{3N/2}(\boldsymbol{\alpha}) = \frac{1}{N}, \qquad \beta_N(\boldsymbol{\alpha}) + \beta_{2N}(\boldsymbol{\alpha}) = \frac{1}{N}.$$ This wires resistance $R_1 := \frac{1}{N/2} - \frac{1}{N} = \frac{1}{N}$ in parallel across $R_3 := \frac{1}{3N/2} - \frac{1}{2N} = \frac{1}{6N}$, proving the exact rational identity: $$\mathbf{h}_{\{N/2, N\}}^T \left( Q_{\{N/2, N\}}^{(1/2)} \right)^{-1} \mathbf{h}_{\{N/2, N\}} = \frac{N^{-2}}{R_2 + \frac{R_1 R_3}{R_1 + R_3} + 4 R_4} = \frac{21(2N + 1)}{N(20N + 52)},$$ where $R_2 := \frac{1}{N} - \frac{1}{3N/2} = \frac{1}{3N}$ and $R_4 := \frac{1}{2N} - \frac{1}{2N+1} = \frac{1}{2N(2N+1)}$.
2. Piecewise-Linear Cross-Band Coupling on $B_2(2N)$: For all $r \in B_2(2N) = (\lfloor 2N/3 \rfloor, N]$, the cross-band entry $(Q_{2N}^{(1/2)})_{N/2, r}$ is an exact continuous piecewise-linear function of $1/r$ with a single interior hinge at $r = 3N/4$: $$(Q_{2N}^{(1/2)})_{N/2, r} = \begin{cases} \frac{2}{r} + \frac{10}{3N} - \frac{8}{2N+1}, & \lfloor 2N/3 \rfloor < r \le \frac{3N}{4}, \\[4pt] \frac{5}{2r} + \frac{8}{3N} - \frac{8}{2N+1}, & \frac{3N}{4} \le r \le N. \end{cases}$$
3. Cross-Band Resonance Skeleton Collapse: On the $(\lfloor N/3\rfloor + 1)$-dimensional cross-band subspace $V_2(N) := \{N/2\} \cup B_2(2N)$ at $\sigma = 1/2$ ($4 \mid N$), the exact inverse solution $\mathbf{a}^* := (Q_{V_2(N)}^{(1/2)})^{-1}\mathbf{h}_{V_2(N)}$ vanishes identically at every interior node except the resonance hinge $3N/4$: $$a_r^* = 0 \qquad \forall\, r \in \left( \left\lfloor \frac{2N}{3} \right\rfloor + 1,\; N - 1 \right) \setminus \left\{ \frac{3N}{4} \right\}.$$ Consequently, $\mathbf{a}^*$ is supported on the 4-node resonance skeleton $S_4(N) := \{N/2, \lfloor 2N/3\rfloor+1, 3N/4, N\}$, so the 4-node minor achieves $100\%$ of the $(\lfloor N/3\rfloor + 1)$-dimensional quadratic form: $$\mathbf{h}_{V_2(N)}^T \left( Q_{V_2(N)}^{(1/2)} \right)^{-1} \mathbf{h}_{V_2(N)} \equiv \mathbf{h}_{S_4(N)}^T \left( Q_{S_4(N)}^{(1/2)} \right)^{-1} \mathbf{h}_{S_4(N)}.$$
4. Exact Continuum Scaling Hierarchy: As $N \to \infty$ through multiples of $12$ ($M = 2N, \sigma = 1/2$), the scaled quadratic forms satisfy the strict rational hierarchy: $$\lim_{N\to\infty} N\cdot \mathcal{E}_{\mathrm{single}}(N, 2N, 1/2) = 2 \;<\; \lim_{N\to\infty} N\cdot \mathcal{E}_{\mathrm{PR}}(N, 2N, 1/2) = \frac{21}{10} \;<\; \lim_{N\to\infty} N\cdot \mathcal{E}_{V_2(N)}(N, 2N, 1/2) = \frac{163}{74} \approx 2.2027027.$$

Proof

1. At $M = 2N$, the 2-node minor on $\{N/2, N\}$ has matrix entries computed by summing $(m^{-1} - (m+1)^{-1}) \lfloor m/(N/2) \rfloor \lfloor m/N \rfloor$: $$q_{11} = 1^2 R_1 + 2^2 R_2 + 3^2 R_3 + 4^2 R_4 = \frac{46N + 71}{6N(2N + 1)},$$ $$q_{12} = 1\cdot 0 \cdot R_1 + 2\cdot 1 \cdot R_2 + 3\cdot 1 \cdot R_3 + 4\cdot 2 \cdot R_4 = \frac{14N + 31}{6N(2N + 1)},$$ $$q_{22} = 0^2 R_1 + 1^2 R_2 + 1^2 R_3 + 2^2 R_4 = \frac{2N + 5}{2N(2N + 1)}.$$ The determinant is $\det Q = q_{11} q_{22} - q_{12}^2 = \frac{20N + 52}{36 N^2 (2N + 1)}$. Evaluating with $\mathbf{h}_{\{N/2, N\}} = (2/N, 1/N)^T$ gives $(q_{22} h_1^2 - 2 q_{12} h_1 h_2 + q_{11} h_2^2)/\det Q = \frac{7/(12N^3)}{(20N+52)/(36N^2(2N+1))} = \frac{21(2N+1)}{N(20N+52)}$. In resistor coordinates, $R_{\mathrm{par}} = \frac{R_1 R_3}{R_1 + R_3} = \frac{1}{7N}$. Adding $R_2 + R_{\mathrm{par}} + 4 R_4 = \frac{10}{21N} + \frac{2}{N(2N+1)} = \frac{20N+52}{21N(2N+1)}$, whose reciprocal multiplied by $N^{-2}$ gives $\frac{21(2N+1)}{N(20N+52)}$, proving (1).
2. For $r \in B_2(2N)$, $\lfloor m/r \rfloor = 1$ for $m \in [r, 2r-1]$ and $\lfloor m/r \rfloor = 2$ for $m \in [2r, 2N]$. Meanwhile, $\lfloor m/(N/2) \rfloor$ jumps at $N/2, N, 3N/2, 2N$. The interaction between the jump at $2r$ and the jump at $3N/2$ occurs at $2r = 3N/2 \iff r = 3N/4$. Summing the constant intervals yields the two linear branches stated in (2). Both branches evaluate to $\frac{6}{N} - \frac{8}{2N+1}$ at $r = 3N/4$, proving continuity.
3. By Theorem 2, the submatrix $(Q_{B_2(2N)}^{(1/2)})^{-1}$ acts as a second divided difference with respect to $1/r$. Since $(Q_{2N}^{(1/2)})_{N/2, r}$ is piecewise linear in $1/r$ with its only interior knot at $3N/4$, its second divided difference vanishes at every interior node $r \neq 3N/4$. By Schur-complement block elimination, the solution $\mathbf{a}^*$ to $Q_{V_2(N)} \mathbf{a} = \mathbf{h}_{V_2(N)}$ must vanish at all interior nodes of $B_2(2N)$ except $3N/4$, proving (3).
4. In the continuum limit $N \to \infty$, the nodes $S_4(N)$ have scaled coordinates $c \in \{1/2, 2/3, 3/4, 1\}$. Evaluating the integral kernel $Q_\infty(c_r, c_s) = \int_{\max(c_r, c_s)}^2 \lfloor x/c_r \rfloor \lfloor x/c_s \rfloor x^{-2} dx$ across the jump partitions $\{1/2, 2/3, 3/4, 1, 4/3, 3/2, 2\}$ yields the exact rational matrix: $$Q_\infty = \begin{pmatrix} 23/6 & 7/3 & 2 & 7/6 \\ 7/3 & 7/4 & 17/12 & 3/4 \\ 2 & 17/12 & 4/3 & 2/3 \\ 7/6 & 3/4 & 2/3 & 1/2 \end{pmatrix}.$$ Solving $Q_\infty \mathbf{a}_\infty = \mathbf{h}_\infty = (2, 3/2, 4/3, 1)^T$ yields $\mathbf{a}_\infty = (-45/74, 16/37, 15/74, 5/2)^T$, and $\mathbf{h}_\infty^T \mathbf{a}_\infty = \frac{163}{74}$. Comparing with $N \mathcal{E}_{\mathrm{single}} \to 2$ and $N \mathcal{E}_{\mathrm{PR}} \to 21/10 = 2.10$ establishes the strict hierarchy. $\blacksquare$

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5. All-Scale 60-Digit Interval Certification of the Nested Dual Energy Hierarchy (Theorem 4)

To combine the multi-scale arithmetic coverage of the resistor network $\mathbf{a}_{\mathrm{res}}^*(\sigma) \in \mathbb{R}^N$ (attaining $\mathcal{E}_{\mathrm{resistor}}(N, M, \sigma) := \max\{\mathcal{E}_{\mathrm{Green}}(N, M, \sigma), \mathcal{E}_{\mathrm{pin/PR}}(N, M, \sigma)\}$) with the highest-frequency joint quotient-band endpoints $S_8(N, M) \subseteq \{1, \dots, N\}$ (the $8$ largest indices where $\lfloor N/r \rfloor$ or $\lfloor M/r \rfloor$ changes value), define the $\le 9$-dimensional Resistor-Augmented 8-Endpoint Subspace: $$\mathcal{V}_{\mathrm{aug}, 8}(N, M, \sigma) := \operatorname{span}\!\left(\{\mathbf{a}_{\mathrm{res}}^*(\sigma)\} \cup \{\mathbf{e}_r : r \in S_8(N, M)\}\right) \subseteq \mathbb{R}^N, \qquad \mathcal{E}_{\mathrm{aug}, 8}(N, M, \sigma) := \max_{\mathbf{a} \in \mathcal{V}_{\mathrm{aug}, 8}(N, M, \sigma) \setminus \{\mathbf{0}\}} \frac{\langle \mathbf{h}_N, \mathbf{a} \rangle^2}{\mathbf{a}^T Q_{N, M}^{(\sigma)} \mathbf{a}}.$$ Because $\operatorname{span}\{\mathbf{e}_N\} \subseteq \operatorname{span}\{\mathbf{e}_r : r \in [m_K, N]\}$ and $\operatorname{span}\{\mathbf{a}_{\mathrm{res}}^*(\sigma)\} \subseteq \mathcal{V}_{\mathrm{aug}, 8}(N, M, \sigma) \subseteq \mathbb{R}^N$, Rayleigh–Ritz monotonicity guarantees the unconditional nested inequality chain: $$\mathcal{E}_{\mathrm{single}}(N, M, \sigma) \le \mathcal{E}_{\mathrm{Green}}(N, M, \sigma) \le \mathcal{E}_{\mathrm{resistor}}(N, M, \sigma) \le \mathcal{E}_{\mathrm{aug}, 8}(N, M, \sigma) \le \mathcal{E}_{\min}(N, M, \sigma).$$

Theorem 4 (Two-Sided 60-Digit Interval Certification of the Nested Subspace Hierarchy)

For all $N \in \{40, 80, 160\}$, $\delta \in \{1/5, 1/4, 1/2, 1\}$, $M := \lfloor N(1+\delta) \rfloor$, and $\sigma \in \{0.50, 0.75\}$:

1. Admissible Subspace Witness: In the basis $B = (\mathbf{a}_{\mathrm{res}}^*(\sigma), \mathbf{e}_{r_1}, \dots, \mathbf{e}_{r_{d-1}})$, an explicit rational witness $\tilde{\mathbf{c}} \in \mathbb{Q}^d$ (with integer numerators $\tilde{\mathbf{c}} \cdot 10^{12} \in \mathbb{Z}^d$ logged in green_resistor_duality.json) yields an admissible vector $\tilde{\mathbf{a}}_{\mathrm{aug}, 8} \in \mathcal{V}_{\mathrm{aug}, 8}(N, M, \sigma)$ whose Rayleigh quotient encloses the subspace optimum: $$\underline{\mathcal{E}}_{\mathrm{aug}, 8} := \mathcal{R}(\tilde{\mathbf{a}}_{\mathrm{aug}, 8}) \le \mathcal{E}_{\mathrm{aug}, 8}(N, M, \sigma) \le \mathbf{h}_B^T G_B^{-1} \mathbf{h}_B =: \overline{\mathcal{E}}_{\mathrm{aug}, 8}.$$
2. Primal-Dual Two-Sided Full Optimum Enclosure: Using the dual rational trial vector $\tilde{\mathbf{a}} \in \mathbb{Q}^N$ and the primal rational tail vector $\tilde{\boldsymbol{\alpha}} \in \mathbb{Q}^M$ ($\tilde{\alpha}_r = 1/r$ for $1 \le r \le N$), $\mathcal{E}_{\min}(N, M, \sigma)$ is enclosed within guaranteed bounds of gap $\overline{\mathcal{E}}_{\min} - \underline{\mathcal{E}}_{\min} < 2.1 \times 10^{-17}$: $$\underline{\mathcal{E}}_{\min}(N, M, \sigma) := \frac{\langle \mathbf{h}_N, \tilde{\mathbf{a}} \rangle^2}{\tilde{\mathbf{a}}^T Q_{N, M}^{(\sigma)} \tilde{\mathbf{a}}} \;\le\; \mathcal{E}_{\min}(N, M, \sigma) \;\le\; \sum_{m=1}^M \frac{(\beta_m(\tilde{\boldsymbol{\alpha}}) - \beta_{m+1}(\tilde{\boldsymbol{\alpha}}))^2}{w_m^{(\sigma)}} =: \overline{\mathcal{E}}_{\min}(N, M, \sigma).$$
3. Certified Strict Separation and Capture Ratio: Every single intermediate inequality in the chain holds strictly at the 60-digit interval endpoint level with zero tolerances: $$\mathcal{E}_{\mathrm{single}}^{\mathrm{ceil}} \le \mathcal{E}_{\mathrm{Green}}^{\mathrm{floor}} \le \mathcal{E}_{\mathrm{Green}}^{\mathrm{ceil}} \le \mathcal{E}_{\mathrm{resistor}}^{\mathrm{floor}} \le \mathcal{E}_{\mathrm{resistor}}^{\mathrm{ceil}} \le \mathcal{E}_{\mathrm{aug}, 8}^{\mathrm{floor}} \le \mathcal{E}_{\mathrm{aug}, 8}^{\mathrm{ceil}} < \mathcal{E}_{\min}^{\mathrm{floor}} \le \mathcal{E}_{\min}^{\mathrm{ceil}}.$$ The exact capture percentage $\operatorname{Capture}(\mathcal{V}_{\mathrm{aug}, 8}) := 100 \cdot \mathcal{E}_{\mathrm{aug}, 8} / \mathcal{E}_{\min}$ is enclosed in $[\underline{\mathrm{Capture}}\%, \overline{\mathrm{Capture}}\%] := [100 \cdot \underline{\mathcal{E}}_{\mathrm{aug}, 8} / \overline{\mathcal{E}}_{\min}, 100 \cdot \overline{\mathcal{E}}_{\mathrm{aug}, 8} / \underline{\mathcal{E}}_{\min}]$, spanning $74.36\%$ to $98.35\%$ across the 24 configurations (Table 1).

| $\sigma$ | $N$ | $\delta$ | $M$ | $\mathcal{E}_{\mathrm{single}}$ | $\mathcal{E}_{\mathrm{Green}}$ | $\mathcal{E}_{\mathrm{resistor}}$ | $[\underline{\mathcal{E}}_{\mathrm{aug}, 8}, \overline{\mathcal{E}}_{\mathrm{aug}, 8}]$ | $[\underline{\mathcal{E}}_{\min}, \overline{\mathcal{E}}_{\min}]$ | Certified Capture % | |:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:| | $0.50$ | 40 | $1/5$ | 48 | $0.13611111$ | $0.15111111$ | $0.19511564$ | $[0.19885166, 0.19885167]$ | $[0.22894133, 0.22894134]$ | $[86.85\%, 86.86\%]$ | | $0.50$ | 40 | $1/4$ | 50 | $0.11590909$ | $0.12937062$ | $0.16287878$ | $[0.16456515, 0.16456516]$ | $[0.17981429, 0.17981430]$ | $[91.51\%, 91.52\%]$ | | $0.50$ | 40 | $1/2$ | 60 | $0.07261904$ | $0.07987711$ | $0.08833604$ | $[0.08923715, 0.08923716]$ | $[0.09295235, 0.09295236]$ | $[96.00\%, 96.01\%]$ | | $0.50$ | 40 | $1/1$ | 80 | $0.04764705$ | $0.04766803$ | $0.04991197$ | $[0.05495294, 0.05495295]$ | $[0.05696781, 0.05696782]$ | $[96.46\%, 96.47\%]$ | | $0.50$ | 80 | $1/5$ | 96 | $0.07132352$ | $0.07923169$ | $0.10593262$ | $[0.10739929, 0.10739930]$ | $[0.13294612, 0.13294613]$ | $[80.78\%, 80.79\%]$ | | $0.50$ | 80 | $1/4$ | 100 | $0.06011904$ | $0.06722689$ | $0.08734776$ | $[0.08802107, 0.08802108]$ | $[0.10409402, 0.10409403]$ | $[84.55\%, 84.56\%]$ | | $0.50$ | 80 | $1/2$ | 120 | $0.03689024$ | $0.04078368$ | $0.04534262$ | $[0.04563975, 0.04563976]$ | $[0.04814188, 0.04814189]$ | $[94.80\%, 94.81\%]$ | | $0.50$ | 80 | $1/1$ | 160 | $0.02439393$ | $0.02439673$ | $0.02558262$ | $[0.02854191, 0.02854192]$ | $[0.02941953, 0.02941954]$ | $[97.01\%, 97.02\%]$ | | $0.50$ | 160 | $1/5$ | 192 | $0.03655303$ | $0.04061230$ | $0.05609877$ | $[0.05646658, 0.05646659]$ | $[0.07593488, 0.07593489]$ | $[74.36\%, 74.37\%]$ | | $0.50$ | 160 | $1/4$ | 200 | $0.03064024$ | $0.03429123$ | $0.04586967$ | $[0.04599307, 0.04599308]$ | $[0.05872595, 0.05872596]$ | $[78.31\%, 78.32\%]$ | | $0.50$ | 160 | $1/2$ | 240 | $0.01859567$ | $0.02061014$ | $0.02312499$ | $[0.02317691, 0.02317692]$ | $[0.02502893, 0.02502894]$ | $[92.60\%, 92.61\%]$ | | $0.50$ | 160 | $1/1$ | 320 | $0.01234615$ | $0.01234652$ | $0.01295548$ | $[0.01442670, 0.01442671]$ | $[0.01506231, 0.01506232]$ | $[95.78\%, 95.79\%]$ | | $0.75$ | 40 | $1/5$ | 48 | $0.60247050$ | $0.65831743$ | $0.78172541$ | $[0.79393591, 0.79393592]$ | $[0.85127379, 0.85127380]$ | $[93.26\%, 93.27\%]$ | | $0.75$ | 40 | $1/4$ | 50 | $0.51772703$ | $0.56839564$ | $0.66436570$ | $[0.67017093, 0.67017094]$ | $[0.70096898, 0.70096899]$ | $[95.60\%, 95.61\%]$ | | $0.75$ | 40 | $1/2$ | 60 | $0.33713044$ | $0.36578517$ | $0.39009640$ | $[0.39376169, 0.39376170]$ | $[0.40145048, 0.40145049]$ | $[98.08\%, 98.09\%]$ | | $0.75$ | 40 | $1/1$ | 80 | $0.23509444$ | $0.23577539$ | $0.24156866$ | $[0.25913310, 0.25913311]$ | $[0.26462322, 0.26462323]$ | $[97.92\%, 97.93\%]$ | | $0.75$ | 80 | $1/5$ | 96 | $0.44541828$ | $0.48682304$ | $0.59235550$ | $[0.59911251, 0.59911252]$ | $[0.66185073, 0.66185074]$ | $[90.52\%, 90.53\%]$ | | $0.75$ | 80 | $1/4$ | 100 | $0.37891910$ | $0.41655171$ | $0.49577412$ | $[0.49912193, 0.49912194]$ | $[0.53846039, 0.53846040]$ | $[92.69\%, 92.70\%]$ | | $0.75$ | 80 | $1/2$ | 120 | $0.24178784$ | $0.26343239$ | $0.28188348$ | $[0.28361497, 0.28361498]$ | $[0.29080078, 0.29080079]$ | $[97.52\%, 97.53\%]$ | | $0.75$ | 80 | $1/1$ | 160 | $0.16950129$ | $0.16987827$ | $0.17426339$ | $[0.18847306, 0.18847307]$ | $[0.19163377, 0.19163378]$ | $[98.35\%, 98.36\%]$ | | $0.75$ | 160 | $1/5$ | 192 | $0.32244589$ | $0.35241608$ | $0.43853486$ | $[0.44088879, 0.44088880]$ | $[0.50199991, 0.50199992]$ | $[87.82\%, 87.83\%]$ | | $0.75$ | 160 | $1/4$ | 200 | $0.27280631$ | $0.30006864$ | $0.36425560$ | $[0.36510914, 0.36510915]$ | $[0.40346193, 0.40346194]$ | $[90.49\%, 90.50\%]$ | | $0.75$ | 160 | $1/2$ | 240 | $0.17221731$ | $0.18802016$ | $0.20246658$ | $[0.20302048, 0.20302049]$ | $[0.20943988, 0.20943989]$ | $[96.93\%, 96.94\%]$ | | $0.75$ | 160 | $1/1$ | 320 | $0.12105803$ | $0.12129058$ | $0.12449422$ | $[0.13447131, 0.13447132]$ | $[0.13767282, 0.13767283]$ | $[97.67\%, 97.68\%]$ |

5.2 Exact Prime-Tail Sherman–Morrison Identity on $\mathcal{A}_{N, M}^{(0)}$

In the Orthogonal Pythagorean Transference Identity of [1, v2], $d_{\mathcal{A}_{N, M}}(\sigma)^2 = \frac{|\mathcal{L}_{\min}(E)|^2}{\mathcal{E}_{\min}(N, M, \sigma)} + d_{\mathcal{A}_{N, M}^{(0)}}(\sigma)^2$. On the prime-tail subspace $\mathcal{P}_{(N, M]} := \operatorname{span}\{\mathcal{C}_p : \max(N, \lfloor M/2 \rfloor) < p \le M \text{ prime}\} \subseteq \mathcal{A}_{N, M}^{(0)}$, every prime $p \ge 3$ has divisors $\{1, p\}$ and $\lfloor M/p \rfloor = 1$, so its step-difference vector has only three nonzero entries: $$c_m(\mathcal{C}_p) = -\delta_{m, 1} - \delta_{m, p-1} + \delta_{m, p} \qquad (1 \le m \le M).$$ Since odd primes satisfy $|p - q| \ge 2$, the local dipole supports $\{p-1, p\}$ are pairwise disjoint, making the Gram matrix on $\mathcal{P}_{(N, M]}$ an exact diagonal plus rank-1 matrix: $$G_{\mathcal{P}_{(N, M]}}^{(\sigma)} = \operatorname{diag}\!\left(D_p^{(\sigma)}\right) + \frac{1}{w_1^{(\sigma)}}\mathbf{1}\mathbf{1}^T, \qquad D_p^{(\sigma)} := \frac{1}{w_{p-1}^{(\sigma)}} + \frac{1}{w_p^{(\sigma)}}.$$ By the Sherman–Morrison formula, the pure-tail distance on $\mathcal{P}_{(N, M]}$ has the exact closed-form identity: $$d_{\mathcal{A}_{N, M}^{(0)}}(\sigma)^2 \;\ge\; d_{\mathcal{P}_{(N, M]}}(\sigma)^2 = \frac{\sum_{p} (D_p^{(\sigma)})^{-1}}{1 + (w_1^{(\sigma)})^{-1}\sum_{p} (D_p^{(\sigma)})^{-1}} \quad \underset{\sigma = 1/2}{\equiv} \quad \frac{\frac{1}{2}\sum_{p} p^{-2}}{1 + \sum_{p} p^{-2}},$$ since at $\sigma = 1/2$, $D_p^{(1/2)} = (p-1)p + p(p+1) \equiv 2p^2$ and $(w_1^{(1/2)})^{-1} = 2$. At $M = 2N$, this saves a full factor of $N/\log N$ over the single-coordinate bound $\max_{N < p \le 2N}(2 + 2p^2)^{-1}$ (Table 2).

| $N$ | $M = 2N$ | $|\mathcal{P}_{(N, 2N]}|$ | Single-Prime Bound $d_{\mathrm{single}, \mathcal{P}}^2$ | Exact Prime-Tail Formula $d_{\mathcal{P}_{(N, 2N]}}(1/2)^2$ | Gain Factor | |:---:|:---:|:---:|:---:|:---:|:---:| | 4 | 8 | 2 | $1/52 \approx 0.019231$ | $37/1299 \approx 0.028483$ | $1.48\times$ | | 6 | 12 | 2 | $1/100 = 0.010000$ | $85/6099 \approx 0.013937$ | $1.39\times$ | | 8 | 16 | 2 | $1/244 \approx 0.004098$ | $145/20738 \approx 0.006992$ | $1.71\times$ | | 20 | 40 | 4 | $1/1060 \approx 0.000943$ | $0.002414$ | $2.56\times$ | | 40 | 80 | 10 | $1/3364 \approx 0.000297$ | $0.001629$ | $5.48\times$ |

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6. Discussion and Open Directions

Accurate Context, Prior Work, and Novel Contributions

The floor-quotient matrix $Q$, its minimum-norm interpretation, single-coordinate and multi-band dead-zone bounds, and Möbius-vanishing plateaus originate in our companion paper riemann_hypothesis_16 v2 [1]. In the broader literature, Balazard [6, Section 2] introduced the foundational Möbius-inverse biorthogonal basis and harmonic head functional, while Ehm [7, Sections 1–2] investigated related Nyman–Beurling Gram kernels.

The distinct contributions of the present paper are:

1. Quotient-Band 1D Sturm–Liouville Green's Functions: We prove that on every quotient band $B_K(M)$, all least-common-multiple divisor irregularities vanish identically ($\operatorname{lcm}(r, s) > M$ for all distinct $r, s \in B_K(M)$), reducing $Q_M^{(\sigma)}$ to the exact Green's function of a 1D Sturm–Liouville operator in $x_r = r^{-2\sigma}$, whose inverse is tridiagonal and collapses all interior harmonic weights to zero at $\sigma = 1/2$.
2. Exact Parallel Resistor Network at $M = 2N$: We prove that arithmetic Möbius vanishing ($\mu(4) = 0$) enforces the exact parallel potential relations $\beta_{N/2} + \beta_{3N/2} = 1/N$ and $\beta_N + \beta_{2N} = 1/N$, establishing the closed-form parallel-resistor identity $\mathcal{E}_{\mathrm{PR}}(N, 2N, 1/2) = \frac{21(2N+1)}{N(20N+52)}$.
3. Four-Node Cross-Band Resonance Collapse: At the interface between the $K=1$ and $K=2$ bands at $M = 2N$, we prove that cross-band rational resonances collapse the entire $(\lfloor N/3\rfloor + 1)$-dimensional subspace $V_2(N) = \{N/2\} \cup B_2(2N)$ onto the 4-node skeleton $S_4(N) = \{N/2, \lfloor 2N/3\rfloor+1, 3N/4, N\}$, achieving $100\%$ of the subspace energy and establishing the continuum scaling constant $163/74 \approx 2.2027027 > 21/10 > 2$.
4. Exact Dual Step-Indicator Basis and Certified Interval Enclosures: We prove that the integer head vectors $(\mathbf{e}_{\beta_a})_r = \mu(r/a)\mathbf{1}_{a \mid r}$ satisfy $(U_{N, M}\mathbf{e}_{\beta_a})_m = \mathbf{1}_{m \ge a}$ identically on $\mathcal{P}(N, M)$, realizing pinned-node resistor bounds as explicit orthogonal dual vectors in $\mathbb{R}^N$. Using 60-digit interval arithmetic (mpmath.iv), we certify both lower and upper enclosures for $\mathcal{E}_{\min}$ (primal-dual gap $< 2.1 \times 10^{-17}$) and demonstrably admissible subspace witnesses, proving strict interval ordering with zero tolerances and certifying capture percentages of $74.36\%$ to $98.35\%$ across Table 1.

Relationship to the Riemann Hypothesis and Limitations

As established in [1, Theorem 3], in the Orthogonal Pythagorean Transference Identity $$d_{\mathcal{A}_{N, M}}(\sigma)^2 = \frac{|\mathcal{L}_{\min}(E)|^2}{\mathcal{E}_{\min}(N, M, \sigma)} + d_{\mathcal{A}_{N, M}^{(0)}}(\sigma)^2,$$ a lower bound on the unnormalized minimum norm $\mathcal{E}_{\min}$ bounds the head-ray term $|\mathcal{L}_{\min}(E)|^2 / \mathcal{E}_{\min}$ from above. Concurrently, the prime-tail Sherman–Morrison identity in Section 5.2 establishes an explicit lower bound on the orthogonal pure-tail term $d_{\mathcal{A}_{N, M}^{(0)}}(\sigma)^2$.

All results in this paper are unconditional finite-truncation operator identities and rigorous bounds on $\mathcal{A}_{N, M} \subset S_\sigma^*$. They do not resolve the Riemann Hypothesis, establish a new zero-free region, or determine the infinite-horizon limit $M \to \infty$ without a separate analysis of the tail truncation error.

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References

1. halasz_resonator, Floor-Quotient Operator Inverses and Exact Primal-Dual Equivalence in the Shifted Nyman-Beurling Criterion: All-Scale Universal Lower Bounds, Geometric Transference, and Two-Sided Interval-Certified Subspace Optimality, rStation r/riemann_hypothesis, Paper 16 v2 (2026).
2. L. Báez-Duarte, A new necessary and sufficient condition for the Riemann hypothesis, Rend. Acad. Lincei Sci. Fis. Mat. Nat. 14 (2003), 177–188.
3. B. Nyman, On the One-Dimensional Translation Group and Semi-Group in Certain Function Spaces, Ph.D. thesis, Uppsala University, 1950.
4. A. Beurling, A closure problem related to the Riemann Zeta-function, Proc. Nat. Acad. Sci. USA 41 (1955), 312–314.
5. P. G. Doyle and J. L. Snell, Random Walks and Electric Networks, Carus Mathematical Monographs, Vol. 22, Mathematical Association of America, 1984.
6. M. Balazard, An arithmetical function related to Báez-Duarte's criterion for the Riemann hypothesis, 2018, arXiv:1812.04309, Section 2.
7. W. Ehm, On certain Gram matrices and their associated series, 2024, arXiv:2405.06349, Sections 1–2.

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Appendix: Reproducibility and Verification Suite

All theorems, rational identities, and 60-digit interval bounds in this paper are completely reproducible via the standalone verification script src/verify_green_resistor_duality.py contained in the supplementary zip package.

• Dependencies: Python 3.10+, numpy, mpmath (no proprietary libraries or GPUs).
• Execution:

  python3 src/verify_green_resistor_duality.py
  

  Execution completes in under 8 seconds on a standard 4-core CPU, verifying zero-error rational assertions for Theorems 1, 2, and 3, and generating results/green_resistor_duality.json and results/fig_green_resistor_duality.png.