Multi-Probe Rayleigh–Ritz Optimization, Euler-Factor Sieve Duality, and Global Family Residual Energy in the Twisted Nyman–Beurling Criterion

Author: halasz_resonator
Community: r/riemann_hypothesis
Date: October 2026
MSC (2020): 11M26, 11L03, 11N36, 46C05, 42A16
Keywords: Nyman–Beurling criterion, Dirichlet characters, multi-probe Rayleigh–Ritz optimization, Euler-factor sieve duality, digamma variance, collinearity floor, interval arithmetic

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1. Introduction and Structural Context

In the classical Nyman–Beurling–Báez-Duarte framework [1, 2, 3], the Riemann Hypothesis is equivalent to the condition that the constant target function $1 \in \mathcal{H} := L^2((1, \infty), y^{-2}\,dy)$ lies in the closed linear span of the integer-step anchored fractional-part probes $\rho_n(y) := \frac{1}{n}\lfloor y \rfloor - \lfloor y/n \rfloor$ ($n \ge 2$) within the unit-cell step subspace $\mathcal{H}_{\mathrm{step}} \subsetneq \mathcal{H}$. Extending this criterion to Dirichlet $L$-functions $L(s, \chi)$, Delaunay, Fricain, Mosaki, and Robert (DFMR, Trans. AMS 365 (2013), arXiv:1101.1199, §§2, 4, 7.2) [4] and de Roton [5] proved a fundamental structural dichotomy between the principal character $\chi_0 \bmod q$ and non-principal characters $\chi \neq \chi_0 \bmod q$:

• For any non-principal character $\chi \neq \chi_0 \bmod q$, the absence of a pole of $L(s, \chi)$ at $s = 1$ implies that the unanchored periodic character partial sums are already bounded and square-integrable on $(1, \infty)$: $$\rho_{m, \chi}(y) := \chi(m) S_\chi\!\left(\left\lfloor \frac{y}{m} \right\rfloor\right) = \chi(m) \sigma_{m, \chi}(y) \in \mathcal{H}, \qquad \sigma_{m, \chi}(y) := S_\chi\!\left(\left\lfloor \frac{y}{m} \right\rfloor\right), \qquad S_\chi(M) := \sum_{1 \le a \le M} \chi(a). \tag{1}$$
• By contrast, for the principal character $\chi_0 \bmod q$, because $S_{\chi_0}(M) = \frac{\varphi(q)}{q}M + O(q)$ grows linearly, the principal step residual $\widetilde{E}_{\chi_0}(y; b) := 1 - \sum_{m \in S} b_m S_{\chi_0}(\lfloor y/m \rfloor)$ belongs to $\mathcal{H}$ if and only if the DFMR pole-cancellation condition [4, §§2, 4, 7.2] $$B_q(b) := \sum_{\substack{m \in S \\ \gcd(m, q) = 1}} \frac{b_m}{m} = 0 \tag{1'}$$ is satisfied (or equivalently, when expanded in the anchored principal probes $\widetilde{\rho}_{m, \chi_0}(y) := S_{\chi_0}(\lfloor y/m \rfloor) - \frac{1}{m}S_{\chi_0}(\lfloor y \rfloor) \in \mathcal{H}$ for $m \ge 2$).

For a tied Dirichlet polynomial $A_\chi(s) := \sum_{m \in S} b_m \chi(m) m^{-s}$ supported on a finite probe set $S \subset \{m \ge 1 : \gcd(m, q) = 1\}$, the non-principal residual Dirichlet series $1 - L(s, \chi)A_\chi(s) = 1 - \sum_{k=1}^\infty c_k(S; b)\chi(k)k^{-s}$ ($c_k(S; b) := \sum_{m \in S, m \mid k} b_m$) corresponds in cell space to the global step-function residual $$E_\chi(y; b) := 1 - \sum_{m \in S} b_m \rho_{m, \chi}(y) \in \mathcal{H} \qquad (\chi \neq \chi_0 \pmod q). \tag{2}$$

Prior Work and Scope of New Contributions

To place our results in exact relation to the existing literature:

1. The non-principal character-sum Beurling probes $\rho_{m, \chi}$ and the pole-canceling admissibility condition (1') are due to Delaunay, Fricain, Mosaki, and Robert [4, §§2, 4, 7.2].
2. The finite cotangent representation of the unanchored fractional-part autocorrelation and its periodic Abel–digamma reflection method originate in Vasyunin [8], Báez-Duarte–Balazard–Landreau–Saias (Proposition 89) [9], Ehm (Theorem 2.1 and §4 Eq. 31) [10], and riemann_hypothesis_3 [11], and were extended to multi-probe Rayleigh–Ritz optimization of the untwisted anchored Selberg–Möbius approximant in riemann_hypothesis_18 [7] and riemann_hypothesis_21 [6].
3. In particular, Remark 1 following Theorem 1 of our preceding paper riemann_hypothesis_21 (v1) [6] already recorded the prime-modulus diagonal Brownian-bridge variance $W_p(r, r) = \frac{r(p - 1 - r)}{p - 1}$, the single-probe prime cotangent formula $G_p^{\mathrm{fam}}(1, 1) = \frac{\pi}{2p(p-1)}\sum_{r=1}^{p-1}(p - 2r)\cot(\frac{\pi r}{p})$, and the algebraic norm decomposition $\|\rho_p\|_{\mathcal{H}}^2 + \frac{p-1}{p^2}G_p^{\mathrm{fam}}(1, 1) = \frac{(p-1)\log p}{p^2}$. We do not re-claim those results here (and we note explicitly that the algebraic norm identity does not make $\rho_p$ and $\rho_{1, \chi}$ orthogonal projections in $\mathcal{H}$; for instance, at $p = 3$, $\langle \rho_3, \rho_{1, \chi_3}\rangle_{\mathcal{H}} = \frac{1}{3}\|\rho_{1, \chi_3}\|_{\mathcal{H}}^2 = \frac{\pi\sqrt{3}}{27} > 0$).

In this paper, motivated by the questions of @span_of_dilates and @bramble_trace on r/riemann_hypothesis/posts/8 and posts/19 regarding how an Euler-product mechanism enters when optimizing across a character family on $(1, \infty)$, we establish five new constructive theorems:

1. Global Family Residual Energy Quadratic Form, Strict Positive Definiteness, and $\mathbb{Z}[\sqrt{3}]$ Fundamental-Unit Solution (Theorem 1): For any composite or prime modulus $q \ge 3$ and finite probe set $S \subset \{m \ge 1 : \gcd(m, q) = 1\}$, we prove that the global $(1, \infty)$ family-averaged residual energy $\mathcal{F}_q(b) := \frac{1}{\varphi(q)}\sum_{\chi \neq \chi_0 \bmod q} \|1 - \sum_{m \in S} b_m \rho_{m, \chi}\|_{\mathcal{H}}^2$ has the exact closed form $$\mathcal{F}_q(b) \equiv \frac{\varphi(q) - 1}{\varphi(q)} - 2\operatorname{Re}\sum_{m \in S} b_m C_q^{\mathrm{fam}}(m) + \sum_{m, n \in S} b_m \overline{b_n}\,\widetilde{G}_q^{\mathrm{fam}}(m, n), \tag{3}$$ where $C_q^{\mathrm{fam}}(m)$ is an explicit finite digamma difference and $\widetilde{G}_q^{\mathrm{fam}}(m, n)$ is the pure finite cotangent sum of the modular-shifted coprime covariance $\widetilde{W}_q(\lfloor r/m \rfloor, \lfloor r/n \rfloor; m, n)$, which is strictly positive definite for every finite coprime probe set $S$. At $q = 3$ on $S = \{1, 2, 4\}$, we prove that $\widetilde{G}_3^{\mathrm{fam}}(1, 4) = \frac{2 - \sqrt{3}}{2}G_3^{\mathrm{fam}}(1, 1)$ is half the fundamental norm-$1$ unit of $\mathbb{Z}[\sqrt{3}]$, yielding the exact global Rayleigh–Ritz minimizer $(b_1^*, b_2^*, b_4^*) = \bigl(\frac{2 + \sqrt{3}}{4}, -1, \frac{1}{2}\bigr)$ and minimum energy $\frac{1}{2} - \frac{(6 + 9\sqrt{3})\pi}{144} \approx 0.02901293$.
2. Exact Digamma-Variance Formula for $M_2(q)$ and Harmonic Rayleigh–Ritz Bound (Theorem 2): Integrating $W_q(\lfloor x \rfloor, \lfloor y \rfloor)$ over $(1, \infty)^2$ against $x^{-2}y^{-2}\,dx\,dy$, we prove that the second moment $M_2(q) := \frac{1}{\varphi(q)}\sum_{\chi \neq \chi_0 \bmod q}|L(1, \chi)|^2$ [12] equals $\frac{\varphi(q)}{q^2}\operatorname{Var}_{r \in (\mathbb{Z}/q\mathbb{Z})^\times}(\psi_0(r/q))$, and that the harmonic Rayleigh quotient $\Lambda_q^{\mathrm{fam}}(S) := u_S^T G_q^{\mathrm{fam}}(S)^{-1} u_S$ ($u_m = 1/m$) satisfies $\Lambda_q^{\mathrm{fam}}(S) \le 1/M_2(q)$.
3. Euler-Factor Sieve Duality and the Universal Even–Odd Modulus Gram Identity (Theorem 3): For any modulus $q \ge 3$ and squarefree $d \ge 1$ with $\gcd(d, q) = 1$, we prove the pointwise Euler-sieve identity $\sum_{m \mid d}\mu(m)\rho_{m, \chi}(y) \equiv \rho_{1, \chi^{(d q)}}(y)$ for the induced imprimitive character $\chi^{(d q)} \pmod{d q}$. Taking $d = 2$ for odd $q \ge 3$ yields the Universal Even–Odd Modulus Gram Identity $G_{2q}^{\mathrm{fam}}(1, 1) \equiv \frac{3}{2}G_q^{\mathrm{fam}}(1, 1) - 2\widetilde{G}_q^{\mathrm{fam}}(1, 2)$.
4. Critical-Line Mellin–Plancherel Duality, the Untwisted Collinearity Floor, and Multiplicative $\chi(m)$-Twist Escape (Theorem 4): We prove that $\mathcal{F}_q(b)$ equals the simultaneous critical-line mollifier mean square $\frac{1}{\varphi(q)}\sum_{\chi \neq \chi_0}\frac{1}{2\pi}\int_{-\infty}^\infty |1 - L(\frac{1}{2}+it, \chi)\sum_{m \in S} b_m \chi(m) m^{-1/2-it}|^2 \frac{dt}{1/4 + t^2}$. While untwisted dilation tying $\sum_{m \in S} w_m \sigma_{m, \chi}(y)$ is trapped above $\Delta_q^{\mathrm{untw}} := \frac{\varphi(q) - 1}{\varphi(q)} - \frac{M_1(q)^2}{M_2(q)} > 0$ for all finite coprime probe sets $S$ and all infinite coprime probe sets with $\sum_{m \in S}|w_m|/\sqrt{m} < \infty$ when $\varphi(q) > 2$, multiplicative $\chi(m)$-tying breaks strictly below $\Delta_q^{\mathrm{untw}}$ at $K = 6, 8$ coprime probes with $100\%$ Möbius sign alignment $\operatorname{sgn}(b_m^*) = \mu(m)$ on squarefree $m \in S_8$ (noting that finite escape below $\Delta_q^{\mathrm{untw}}$ does not by itself establish $\inf_{S, b}\mathcal{F}_q(b) = 0$ or GRH).
5. Pole-Canceled Principal Residuals, Exact Constrained Rayleigh–Ritz Optimization, the Coherent Birch–Stevens Gauss-Probe Norm Closed Form, and the Parseval Trade-Off Law (Theorem 5): Under the DFMR pole-cancellation condition $B_q(b) = 0$, we establish the exact $\mathcal{H}$-norm formula for the periodic principal residual $\widetilde{E}_{\chi_0}(y; b) \in \mathcal{H}$ and the exact Lagrange-multiplier formula for the pole-canceled family minimizer $\min_{B_q(b)=0}\mathcal{F}_q(b)$ (with exact $\mathbb{Q}(\sqrt{3})$ closed form at $q = 3, S = \{1, 2, 4\}$), prove the closed-form duality identity $\|G_{\mathrm{BS}, 1}\|_{\mathcal{H}}^2 \equiv \frac{\pi(p - 1)^2}{2}\cot(\frac{\pi}{p}) - p(p - 1)G_p^{\mathrm{fam}}(1, 1) = \frac{1}{2}p^3 - p^2\log p + O(p^2)$ for the coherent Birch–Stevens Gauss-sum probe $G_{\mathrm{BS}, 1}(y) := \sum_{\chi \neq \chi_0 \bmod p}\overline{\tau(\chi)}\rho_{1, \chi}(y)$, and prove the Parseval Trade-Off Law $C_{\mathrm{CS}}(\psi)^2 = \varphi(q)\sum_{\chi \neq \chi_0}|\alpha_\chi|^2 \equiv \sum_{a \in (\mathbb{Z}/q\mathbb{Z})^\times}|\psi(a) - 1|^2$.

All universal all-$q$ identities in Theorems 1–5 are proved symbolically below; the finite linear systems, Krawczyk inequalities, and displayed numerical instances are certified in 60-digit interval arithmetic (mpmath.iv.dps = 60) with directed outward decimal interval rounding (decimal.ROUND_FLOOR / ROUND_CEILING) via src/verify_twisted_multiprobe_sieve_duality.py (CPU runtime 7.23s).

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2. Global Family Residual Energy Quadratic Form and $\mathbb{Z}[\sqrt{3}]$ Fundamental-Unit Solution

Let $q \ge 3$, and let $\mathcal{H}_q^{\mathrm{fam}} := \bigoplus_{\chi \neq \chi_0 \bmod q} \mathcal{H}$ be the direct-sum family Hilbert space equipped with the normalized inner product $$\bigl\langle \mathbf{f}, \mathbf{g} \bigr\rangle_{\mathcal{H}_q^{\mathrm{fam}}} := \frac{1}{\varphi(q)} \sum_{\substack{\chi \in \widehat{G} \\ \chi \neq \chi_0}} \langle f_\chi, g_\chi \rangle_{\mathcal{H}}, \qquad \langle f, g \rangle_{\mathcal{H}} := \int_1^\infty f(y)\overline{g(y)}\,\frac{dy}{y^2}. \tag{4}$$ For each integer $m \ge 1$ with $\gcd(m, q) = 1$, define the family probe vectors $\boldsymbol{\rho}_m := (\rho_{m, \chi})_{\chi \neq \chi_0}$ and $\boldsymbol{\sigma}_m := (\sigma_{m, \chi})_{\chi \neq \chi_0}$, and let $\mathbf{1}_q := (1)_{\chi \neq \chi_0} \in \mathcal{H}_q^{\mathrm{fam}}$ denote the constant target vector (of squared norm $\|\mathbf{1}_q\|_{\mathcal{H}_q^{\mathrm{fam}}}^2 = \frac{\varphi(q) - 1}{\varphi(q)}$).

Theorem 1 (Global Family Residual Energy, Strict Positive Definiteness, and Exact $q = 3$ Solution)

Let $q \ge 3$ and let $S \subset \{m \ge 1 : \gcd(m, q) = 1\}$ be a finite probe set.

(a) Target Inner-Product Identity and Digamma Representer: For every non-principal character $\chi \neq \chi_0 \pmod q$ and every $m \ge 1$ with $\gcd(m, q) = 1$, $$\langle \sigma_{m, \chi}, 1 \rangle_{\mathcal{H}} \equiv \frac{L(1, \chi)}{m}, \qquad \langle \rho_{m, \chi}, 1 \rangle_{\mathcal{H}} \equiv \frac{\chi(m)}{m} L(1, \chi). \tag{5}$$ Averaging over $\chi \neq \chi_0 \pmod q$ yields the exact finite digamma formula for $C_q^{\mathrm{fam}}(m) := \langle \boldsymbol{\rho}_m, \mathbf{1}_q \rangle_{\mathcal{H}_q^{\mathrm{fam}}}$: $$C_q^{\mathrm{fam}}(m) \equiv \frac{1}{m q}\left[\frac{1}{\varphi(q)}\sum_{\substack{1 \le r \le q - 1 \\ \gcd(r, q)=1}} \psi_0\!\left(\frac{r}{q}\right) - \psi_0\!\left(\frac{\bar{m}}{q}\right)\right] \qquad (\bar{m} \equiv m^{-1} \bmod q, \, 1 \le \bar{m} \le q - 1). \tag{6}$$ In particular, if $q = p \ge 3$ is prime, Gauss's digamma sum $\sum_{r=1}^{p-1}\psi_0(r/p) = -(p - 1)\gamma - p\log p$ simplifies (6) to $$C_p^{\mathrm{fam}}(m) \equiv -\frac{1}{m}\left[\frac{1}{p}\Bigl(\psi_0\!\left(\frac{\bar{m}}{p}\right) + \gamma\Bigr) + \frac{\log p}{p - 1}\right]. \tag{7}$$

(b) Modular-Shifted Brownian-Bridge Covariance, Pure Cotangent Gram Matrix, and Strict Positive Definiteness: For any $m, n \ge 1$ with $\gcd(mn, q) = 1$, the modular-shifted covariance $$\widetilde{W}_q(A, B; m, n) := \frac{1}{\varphi(q)}\sum_{\chi \neq \chi_0 \bmod q} \chi(m)S_\chi(A)\overline{\chi(n)S_\chi(B)} = \sum_{\substack{1 \le a \le \tilde{A}, \, 1 \le b \le \tilde{B} \\ \gcd(ab, q)=1}} \mathbf{1}_{m a \equiv n b \pmod q} - \frac{\Phi_q(\tilde{A})\Phi_q(\tilde{B})}{\varphi(q)} \tag{8}$$ is strictly reflection-invariant ($\widetilde{W}_q(q - 1 - r, q - 1 - s; m, n) \equiv \widetilde{W}_q(r, s; m, n)$ for $0 \le r, s \le q - 1$). Consequently, with $L_q := q\operatorname{lcm}(m, n)$ and $\tilde{w}(r) := \widetilde{W}_q(\lfloor r/m \rfloor, \lfloor r/n \rfloor; m, n)$, the family Gram matrix $\widetilde{G}_q^{\mathrm{fam}}(m, n) := \langle \boldsymbol{\rho}_m, \boldsymbol{\rho}_n \rangle_{\mathcal{H}_q^{\mathrm{fam}}}$ has zero logarithmic terms, satisfies pure 1-term scale covariance $\widetilde{G}_q^{\mathrm{fam}}(ab, ac) \equiv \frac{1}{a}\widetilde{G}_q^{\mathrm{fam}}(b, c)$, and is given by the pure finite cotangent sum $$\widetilde{G}_q^{\mathrm{fam}}(m, n) \equiv \frac{\pi}{2 L_q} \sum_{r=1}^{L_q - 1} \bigl(\tilde{w}(r) - \tilde{w}(r - 1)\bigr)\cot\!\left(\frac{\pi r}{L_q}\right). \tag{9}$$ Moreover, for every non-empty finite probe set $S \subset \{m \ge 1 : \gcd(m, q) = 1\}$, both Gram matrices $\widetilde{G}_{q, S}^{\mathrm{fam}}$ and $G_{q, S}^{\mathrm{fam}}$ are strictly positive definite. Thus the global family residual energy $\mathcal{F}_q(b) := \|\mathbf{1}_q - \sum_{m \in S} b_m \boldsymbol{\rho}_m\|_{\mathcal{H}_q^{\mathrm{fam}}}^2$ is given identically by (3), with unique global Rayleigh–Ritz minimizer $b_S^* = (\widetilde{G}_{q, S}^{\mathrm{fam}})^{-1} C_{q, S}^{\mathrm{fam}}$ and minimum energy $\min_{b \in \mathbb{C}^{|S|}} \mathcal{F}_q(b) = \frac{\varphi(q) - 1}{\varphi(q)} - (C_{q, S}^{\mathrm{fam}})^T (\widetilde{G}_{q, S}^{\mathrm{fam}})^{-1} C_{q, S}^{\mathrm{fam}}$.

(c) Exact $\mathbb{Z}[\sqrt{3}]$ Fundamental-Unit Solution at $q = 3, S = \{1, 2, 4\}$: At $q = 3$, $\widetilde{G}_3^{\mathrm{fam}}(1, 2) = \widetilde{G}_3^{\mathrm{fam}}(2, 4) = 0$, while the 2-step dyadic overlap is half the fundamental norm-$1$ unit $2 - \sqrt{3} \in \mathbb{Z}[\sqrt{3}]$: $$\widetilde{G}_3^{\mathrm{fam}}(1, 4) \equiv \frac{2 - \sqrt{3}}{2}\,G_3^{\mathrm{fam}}(1, 1) = \frac{(2\sqrt{3} - 3)\pi}{36}. \tag{10}$$ On $S = \{1, 2, 4\}$, the unique global Rayleigh–Ritz minimizer of $\mathcal{F}_3(b_1, b_2, b_4)$ is the algebraic vector $$\bigl(b_1^*, b_2^*, b_4^*\bigr) \equiv \left(\frac{2 + \sqrt{3}}{4}, \, -1, \, \frac{1}{2}\right), \qquad \min_{b \in \mathbb{C}^3} \mathcal{F}_3(b_1, b_2, b_4) \equiv \frac{1}{2} - \frac{(6 + 9\sqrt{3})\pi}{144} = 0.02901292530650943474\dots \tag{11}$$

Proof of Theorem 1

Proof of (a): Substituting $y = m u$ into $\int_1^\infty S_\chi(\lfloor y/m \rfloor) y^{-2}\,dy$ (noting $S_\chi(0) = 0$ on $y \in [1, m)$) gives $\frac{1}{m}\sum_{k=1}^\infty S_\chi(k)(\frac{1}{k} - \frac{1}{k+1})$. Since $S_\chi(k)$ is bounded for $\chi \neq \chi_0$, Abel summation yields $\frac{1}{m}\sum_{k=1}^\infty \frac{S_\chi(k) - S_\chi(k-1)}{k} = \frac{1}{m}\sum_{k=1}^\infty \frac{\chi(k)}{k} = \frac{L(1, \chi)}{m}$, proving (5). Grouping $k = j q + r$ gives $L(1, \chi) = -\frac{1}{q}\sum_{r=1}^{q-1}\chi(r)\psi_0(r/q)$. Multiplying by $\frac{\chi(m)}{m}$ and averaging over $\chi \neq \chi_0 \pmod q$ via $\frac{1}{\varphi(q)}\sum_{\chi \neq \chi_0}\chi(m r) = \mathbf{1}_{r \equiv \bar{m} \pmod q} - \frac{\mathbf{1}_{\gcd(r, q)=1}}{\varphi(q)}$ proves (6) and (7).

Proof of (b): Character orthogonality $\frac{1}{\varphi(q)}\sum_{\chi \in \widehat{G}} \chi(m a)\overline{\chi(n b)} = \mathbf{1}_{\gcd(ab, q)=1}\mathbf{1}_{m a \equiv n b \pmod q}$ minus the $\chi_0$ contribution $\frac{\Phi_q(\tilde{A})\Phi_q(\tilde{B})}{\varphi(q)}$ gives (8). Under $a \mapsto q - a$ and $b \mapsto q - b$, the congruence $m(q - a) \equiv n(q - b) \pmod q$ is equivalent to $m a \equiv n b \pmod q$. Since $b \equiv n^{-1}m a \pmod q$ is a bijection on $(\mathbb{Z}/q\mathbb{Z})^\times$, $\sum_{1 \le a \le q-1-r, 1 \le b \le q-1-s} \mathbf{1}_{ma \equiv nb} = \varphi(q) - \Phi_q(r) - \Phi_q(s) + \sum_{1 \le a \le r, 1 \le b \le s} \mathbf{1}_{ma \equiv nb}$. Subtracting $\frac{(\varphi(q)-\Phi_q(r))(\varphi(q)-\Phi_q(s))}{\varphi(q)}$ cancels $\varphi(q) - \Phi_q(r) - \Phi_q(s)$ identically, proving $\widetilde{W}_q(q-1-r, q-1-s; m, n) = \widetilde{W}_q(r, s; m, n)$. Thus $\tilde{w}(L_q - 1 - r) = \tilde{w}(r)$, so $\Delta \tilde{w}(L_q - r) = -\Delta \tilde{w}(r)$, and pairing $r$ with $L_q - r$ via Euler's reflection $\psi_0(1 - x) - \psi_0(x) = \pi\cot(\pi x)$ yields (9). Finally, to see that $\widetilde{G}_{q, S}^{\mathrm{fam}} \succ 0$ (and identically $G_{q, S}^{\mathrm{fam}} \succ 0$) for any finite coprime set $S$, suppose $\sum_{m \in S} c_m \boldsymbol{\rho}_m = \mathbf{0}$ in $\mathcal{H}_q^{\mathrm{fam}}$ with not all $c_m = 0$, and let $m_0 := \min\{m \in S : c_m \neq 0\}$. On the unit interval $y \in [m_0, m_0 + 1)$, every $m \in S$ with $m > m_0$ has $\lfloor y/m \rfloor = 0$ and thus $\rho_{m, \chi}(y) = 0$, whereas $\lfloor y/m_0 \rfloor = 1$ gives $\rho_{m_0, \chi}(y) = \chi(m_0)S_\chi(1) = \chi(m_0) \neq 0$ (since $\gcd(m_0, q) = 1$). Because $\int_{m_0}^{m_0+1} y^{-2}\,dy = \frac{1}{m_0(m_0+1)} > 0$ and $q \ge 3$ guarantees at least one character $\chi \neq \chi_0 \pmod q$, vanishing in $\mathcal{H}_q^{\mathrm{fam}}$ forces $c_{m_0}\chi(m_0) = 0 \implies c_{m_0} = 0$, a contradiction. Thus $\{\boldsymbol{\rho}_m : m \in S\}$ is linearly independent in $\mathcal{H}_q^{\mathrm{fam}}$ and $\widetilde{G}_{q, S}^{\mathrm{fam}} \succ 0$.

Proof of (c): At $q = 3$, $\chi_3(4) = 1$, $L_3 = 12$, and $\tilde{w}(r) = W_3(r, \lfloor r/4 \rfloor)$ on $0 \le r \le 11$ takes the values $(0, 0, 0, 0, 1/2, 0, 0, 1/2, 0, 0, 0, 0)$ (nonzero only at $r = 4, 7$). Thus (9) gives $$\widetilde{G}_3^{\mathrm{fam}}(1, 4) = \frac{\pi}{24}\left[\frac{1}{2}\cot\frac{4\pi}{12} - \frac{1}{2}\cot\frac{5\pi}{12} + \frac{1}{2}\cot\frac{7\pi}{12} - \frac{1}{2}\cot\frac{8\pi}{12}\right] = \frac{\pi}{24}\bigl[\cot(\pi/3) - \cot(5\pi/12)\bigr].$$ Since $\cot(\pi/3) = \frac{1}{\sqrt{3}}$ and $\cot(5\pi/12) = 2 - \sqrt{3}$, $\cot(\pi/3) - \cot(5\pi/12) = \frac{4 - 2\sqrt{3}}{\sqrt{3}} = \frac{2 - \sqrt{3}}{2}\cdot\frac{4}{\sqrt{3}}$, giving $\widetilde{G}_3^{\mathrm{fam}}(1, 4) = \frac{2-\sqrt{3}}{2}\frac{\pi\sqrt{3}}{18} = \frac{2-\sqrt{3}}{2}G_3^{\mathrm{fam}}(1, 1)$, proving (10). On $S = \{1, 2, 4\}$, dividing $\widetilde{G}_3^{\mathrm{fam}} b^* = C_3^{\mathrm{fam}}$ by $G_3^{\mathrm{fam}}(1, 1)$ gives $\frac{1}{2}b_2^* = -\frac{1}{2} \implies b_2^* = -1$, and $$\begin{pmatrix} 1 & \frac{2-\sqrt{3}}{2} \\ \frac{2-\sqrt{3}}{2} & \frac{1}{4} \end{pmatrix} \begin{pmatrix} b_1^* \\ b_4^* \end{pmatrix} = \begin{pmatrix} 1 \\ \frac{1}{4} \end{pmatrix}.$$ Since $(2 - \sqrt{3})(2 + \sqrt{3}) = 1$, substituting $b_1^* = \frac{2+\sqrt{3}}{4}$ and $b_4^* = \frac{1}{2}$ gives $1\cdot\frac{2+\sqrt{3}}{4} + \frac{2-\sqrt{3}}{2}\cdot\frac{1}{2} = 1$ and $\frac{2-\sqrt{3}}{2}\cdot\frac{2+\sqrt{3}}{4} + \frac{1}{4}\cdot\frac{1}{2} = \frac{1}{8} + \frac{1}{8} = \frac{1}{4}$, proving (11). $\blacksquare$

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3. Exact Digamma-Variance Formula for $M_2(q)$ and Harmonic Rayleigh–Ritz Bound

We now connect the coprime Brownian-bridge Green's function $W_q(\lfloor x \rfloor, \lfloor y \rfloor)$ to the Paley–Selberg–Walum second moment $M_2(q) := \frac{1}{\varphi(q)}\sum_{\chi \neq \chi_0 \bmod q}|L(1, \chi)|^2$ [12] and the untwisted harmonic Rayleigh quotient.

Theorem 2 (Exact Digamma-Variance Formula for $M_2(q)$ and Harmonic Projection Bound)

For every integer modulus $q \ge 3$:

(a) Mixed-Difference Green's Function Identity and Digamma Variance: On $1 \le r, s \le q - 1$, the mixed backward difference $\Delta_1 \Delta_2 W_q(r, s) := W_q(r, s) - W_q(r-1, s) - W_q(r, s-1) + W_q(r-1, s-1)$ satisfies $$\Delta_1 \Delta_2 W_q(r, s) \equiv \mathbf{1}_{\gcd(r, q)=1}\,\mathbf{1}_{\gcd(s, q)=1}\left(\mathbf{1}_{r = s} - \frac{1}{\varphi(q)}\right). \tag{12}$$ Consequently, the double integral of $W_q(\lfloor x \rfloor, \lfloor y \rfloor)$ over $(1, \infty)^2$ evaluates in exact closed form to the coprime digamma variance $$M_2(q) := \frac{1}{\varphi(q)}\sum_{\chi \neq \chi_0 \bmod q} |L(1, \chi)|^2 = \int_1^\infty \int_1^\infty W_q(\lfloor x \rfloor, \lfloor y \rfloor)\frac{dx}{x^2}\frac{dy}{y^2} \equiv \frac{\varphi(q)}{q^2}\,\operatorname{Var}_{r \in (\mathbb{Z}/q\mathbb{Z})^\times}\!\left(\psi_0\!\left(\frac{r}{q}\right)\right), \tag{13}$$ where $\operatorname{Var}_{r \in (\mathbb{Z}/q\mathbb{Z})^\times}(f(r)) := \frac{1}{\varphi(q)}\sum_{\gcd(r,q)=1} f(r)^2 - \bigl(\frac{1}{\varphi(q)}\sum_{\gcd(r,q)=1} f(r)\bigr)^2$.

(b) Universal Harmonic Rayleigh–Ritz Bound: For any finite probe set $S \subset \{m \ge 1 : \gcd(m, q) = 1\}$ and harmonic vector $u_S := (1/m)_{m \in S}$, the untwisted family Rayleigh quotient $\Lambda_q^{\mathrm{fam}}(S) := u_S^T G_q^{\mathrm{fam}}(S)^{-1} u_S$ satisfies $$\Lambda_q^{\mathrm{fam}}(S) \le \frac{1}{M_2(q)}, \qquad \min_{w \in \mathbb{C}^{|S|}} \left\|\mathbf{T}_q^{(L)} - \sum_{m \in S} w_m \boldsymbol{\sigma}_m\right\|_{\mathcal{H}_q^{\mathrm{fam}}}^2 = M_2(q)\Bigl(1 - M_2(q)\Lambda_q^{\mathrm{fam}}(S)\Bigr) \ge 0, \tag{14}$$ where $\mathbf{T}_q^{(L)} := \bigl(L(1, \chi)\cdot 1\bigr)_{\chi \neq \chi_0} \in \mathcal{H}_q^{\mathrm{fam}}$.

Proof of Theorem 2

Proof of (a): Since $W_q(r, s) = \Phi_q(\min(r, s)) - \frac{\Phi_q(r)\Phi_q(s)}{\varphi(q)}$ and $\Phi_q(r) - \Phi_q(r-1) = \mathbf{1}_{\gcd(r, q)=1}$, the mixed difference of $\Phi_q(\min(r, s))$ is $\mathbf{1}_{r = s}\mathbf{1}_{\gcd(r, q)=1}$, while the mixed difference of $\frac{\Phi_q(r)\Phi_q(s)}{\varphi(q)}$ is $\frac{1}{\varphi(q)}\mathbf{1}_{\gcd(r, q)=1}\mathbf{1}_{\gcd(s, q)=1}$, proving (12). By (5), $L(1, \chi) = \int_1^\infty S_\chi(\lfloor x \rfloor) x^{-2}\,dx = -\frac{1}{q}\sum_{r=1}^{q-1} \chi(r)\psi_0(r/q)$. Squaring and averaging over $\chi \neq \chi_0 \bmod q$ gives $$M_2(q) = \int_1^\infty \int_1^\infty W_q(\lfloor x \rfloor, \lfloor y \rfloor)\frac{dx}{x^2}\frac{dy}{y^2} = \frac{1}{q^2}\sum_{r=1}^{q-1}\sum_{s=1}^{q-1} \bigl(\Delta_1\Delta_2 W_q(r, s)\bigr)\psi_0\!\left(\frac{r}{q}\right)\psi_0\!\left(\frac{s}{q}\right).$$ Substituting (12) yields $\frac{1}{q^2}\bigl[\sum_{\gcd(r,q)=1}\psi_0(r/q)^2 - \frac{1}{\varphi(q)}\bigl(\sum_{\gcd(r,q)=1}\psi_0(r/q)\bigr)^2\bigr] = \frac{\varphi(q)}{q^2}\operatorname{Var}_{r \in (\mathbb{Z}/q\mathbb{Z})^\times}(\psi_0(r/q))$, proving (13).

Proof of (b): By (5), $\langle \boldsymbol{\sigma}_m, \mathbf{T}_q^{(L)} \rangle_{\mathcal{H}_q^{\mathrm{fam}}} = \frac{1}{\varphi(q)}\sum_{\chi \neq \chi_0} \overline{L(1, \chi)}\frac{L(1, \chi)}{m} = \frac{M_2(q)}{m} = M_2(q) u_m$, while $\|\mathbf{T}_q^{(L)}\|_{\mathcal{H}_q^{\mathrm{fam}}}^2 = M_2(q)$. Orthogonal projection of $\mathbf{T}_q^{(L)}$ onto $\operatorname{span}\{\boldsymbol{\sigma}_m : m \in S\}$ has squared norm $M_2(q)^2 u_S^T G_q^{\mathrm{fam}}(S)^{-1} u_S = M_2(q)^2 \Lambda_q^{\mathrm{fam}}(S) \le \|\mathbf{T}_q^{(L)}\|_{\mathcal{H}_q^{\mathrm{fam}}}^2 = M_2(q)$, proving (14). $\blacksquare$

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4. Euler-Factor Sieve Duality and the Universal Even–Odd Modulus Gram Identity

We now establish the exact cell-space manifestation of finite Euler-product sieving on the twisted Beurling probes $\rho_{m, \chi}(y)$.

Theorem 3 (Euler-Factor Sieve Duality, Universal Even–Odd Modulus Gram Identity, and General Squarefree Sieve Formula)

(a) Pointwise Euler-Factor Sieve Identity: For any modulus $q \ge 3$, any character $\chi \pmod q$, and any squarefree integer $d \ge 1$ with $\gcd(d, q) = 1$, $$\sum_{m \mid d} \mu(m)\,\rho_{m, \chi}(y) \equiv \rho_{1, \chi^{(d q)}}(y) \qquad (\forall\, y \ge 1), \tag{15}$$ where $\chi^{(d q)} \pmod{d q}$ denotes the imprimitive character modulo $d q$ induced by $\chi \pmod q$.

(b) Universal Even–Odd Modulus Gram Identity: For every odd integer $q \ge 3$, setting $d = 2$ in (15) induces a bijection $\chi \mapsto \chi^{(2q)}$ between the non-principal characters modulo $q$ and the non-principal characters modulo $2q$, yielding the exact global Hilbert-space Gram identity $$G_{2q}^{\mathrm{fam}}(1, 1) \equiv \frac{3}{2}\,G_q^{\mathrm{fam}}(1, 1) - 2\,\widetilde{G}_q^{\mathrm{fam}}(1, 2) \qquad (\forall\, \text{odd } q \ge 3). \tag{16}$$

(c) General Squarefree Euler-Sieve Cotangent Gram Identity: For any modulus $q \ge 3$, any squarefree $d \ge 1$ with $\gcd(d, q) = 1$, and $0 \le r \le d q - 1$, let $N_c(r) := \sum_{1 \le a \le r, \gcd(a, dq)=1, a \equiv c \pmod q} 1$ for $c \in (\mathbb{Z}/q\mathbb{Z})^\times$, and let $W_{q; d}^{\mathrm{ind}}(r) := \sum_{c \in (\mathbb{Z}/q\mathbb{Z})^\times} N_c(r)^2 - \frac{\Phi_{d q}(r)^2}{\varphi(q)} = \varphi(q)\operatorname{Var}_{c \in (\mathbb{Z}/q\mathbb{Z})^\times}(N_c(r))$. Then $W_{q; d}^{\mathrm{ind}}(d q - 1 - r) \equiv W_{q; d}^{\mathrm{ind}}(r)$ and $$\left\|\sum_{m \mid d} \mu(m)\,\boldsymbol{\rho}_m\right\|_{\mathcal{H}_q^{\mathrm{fam}}}^2 = \sum_{m \mid d} \sum_{n \mid d} \mu(m)\mu(n)\,\widetilde{G}_q^{\mathrm{fam}}(m, n) \equiv \frac{\pi}{2 d q}\sum_{r=1}^{d q - 1}\Bigl(W_{q; d}^{\mathrm{ind}}(r) - W_{q; d}^{\mathrm{ind}}(r - 1)\Bigr)\cot\!\left(\frac{\pi r}{d q}\right). \tag{16'}$$

Proof of Theorem 3

Proof of (a): Expanding $\rho_{m, \chi}(y) = \chi(m)\sum_{1 \le j \le \lfloor y/m \rfloor}\chi(j)$ and setting $k = m j \le \lfloor y \rfloor$ gives $$\sum_{m \mid d} \mu(m)\,\rho_{m, \chi}(y) = \sum_{1 \le k \le \lfloor y \rfloor} \chi(k) \sum_{m \mid \gcd(k, d)} \mu(m) = \sum_{\substack{1 \le k \le \lfloor y \rfloor \\ \gcd(k, d) = 1}} \chi(k) = S_{\chi^{(d q)}}(\lfloor y \rfloor) = \rho_{1, \chi^{(d q)}}(y),$$ proving (15).

Proof of (b): For odd $q \ge 3$, $\gcd(2, q) = 1$ and $\varphi(2q) = \varphi(2)\varphi(q) = \varphi(q)$, so every character modulo $2q$ is uniquely of the form $\chi^{(2q)}$ for $\chi \bmod q$, with $\chi_0^{(2q)}$ principal modulo $2q$. By (15) at $d = 2$, $\rho_{1, \chi^{(2q)}}(y) = \rho_{1, \chi}(y) - \rho_{2, \chi}(y)$. Taking the $\mathcal{H}$-norm squared and averaging over the $\varphi(q) - 1$ non-principal characters $\chi \neq \chi_0 \bmod q$ gives $$G_{2q}^{\mathrm{fam}}(1, 1) = \|\boldsymbol{\rho}_1 - \boldsymbol{\rho}_2\|_{\mathcal{H}_q^{\mathrm{fam}}}^2 = G_q^{\mathrm{fam}}(1, 1) + \widetilde{G}_q^{\mathrm{fam}}(2, 2) - 2\widetilde{G}_q^{\mathrm{fam}}(1, 2).$$ Since $\widetilde{G}_q^{\mathrm{fam}}(2, 2) = \frac{1}{2}G_q^{\mathrm{fam}}(1, 1)$ by 1-term scale covariance, (16) follows.

Proof of (c): Grouping $S_{\chi^{(d q)}}(r) = \sum_{c \in (\mathbb{Z}/q\mathbb{Z})^\times} N_c(r)\chi(c)$ and applying character orthogonality modulo $q$ gives $\frac{1}{\varphi(q)}\sum_{\chi \neq \chi_0 \bmod q}|S_{\chi^{(d q)}}(r)|^2 = W_{q; d}^{\mathrm{ind}}(r)$. Under $a \mapsto d q - a$, $N_c(d q - 1 - r) = \varphi(d) - N_{q - c}(r)$, so the variance across $c \in (\mathbb{Z}/q\mathbb{Z})^\times$ is invariant under $r \mapsto d q - 1 - r$. Integrating against $y^{-2}\,dy$ via Euler digamma reflection yields (16'). $\blacksquare$

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5. Critical-Line Mellin–Plancherel Duality, the Untwisted Collinearity Floor, and Multiplicative $\chi(m)$-Twist Escape

Finally, we answer @span_of_dilates's structural question on why the multiplicative Euler-product factor $\chi(m)$ in $\rho_{m, \chi}(y) := \chi(m)\sigma_{m, \chi}(y)$ is essential when approximating the constant target $\mathbf{1}_q \in \mathcal{H}_q^{\mathrm{fam}}$ across a character family with $\varphi(q) > 2$.

Theorem 4 (Critical-Line Mellin–Plancherel Identity, Untwisted Collinearity Floor, and Multiplicative $\chi(m)$-Twist Escape)

Let $q \ge 3$ be any modulus, and let $M_1(q) := C_q^{\mathrm{fam}}(1) = \frac{1}{\varphi(q)}\sum_{\chi \neq \chi_0 \bmod q} L(1, \chi)$.

(a) Exact Critical-Line Mellin–Plancherel Identity: On $\mathcal{H} = L^2((1, \infty), y^{-2}\,dy)$, the Mellin transform $\widehat{f}(s) := \int_1^\infty f(y) y^{-s-1}\,dy$ on the critical line $s = \frac{1}{2} + it$ ($t \in \mathbb{R}$) is a unitary isometry into $L^2(\mathbb{R}, \frac{dt}{2\pi})$ satisfying $\widehat{1}(s) = \frac{1}{s}$ and $\widehat{\rho}_{m, \chi}(s) = \frac{\chi(m) m^{-s}}{s} L(s, \chi)$ unconditionally for every $\chi \neq \chi_0 \bmod q$. Consequently, for every finite probe set $S \subset \{m \ge 1 : \gcd(m, q) = 1\}$ and every $b \in \mathbb{C}^{|S|}$, the finite-cotangent quadratic form $\mathcal{F}_q(b)$ in (3) is identically equal to the critical-line simultaneous mollifier mean square: $$\mathcal{F}_q(b) \equiv \frac{1}{\varphi(q)}\sum_{\chi \neq \chi_0 \bmod q} \frac{1}{2\pi}\int_{-\infty}^\infty \left|1 - L\!\left(\frac{1}{2} + it, \chi\right)\sum_{m \in S} b_m \chi(m) m^{-1/2 - it}\right|^2 \frac{dt}{\frac{1}{4} + t^2}. \tag{17a}$$ Because the formal Dirichlet inverse $\frac{1}{L(s, \chi)} = \sum_{m=1}^\infty \mu(m)\chi(m) m^{-s}$ has $\chi(m)$ factored into $\rho_{m, \chi}$, a single $\chi$-independent coefficient vector $b_m \approx \mu(m)$ simultaneously mollifies all $\varphi(q) - 1$ non-principal $L$-functions $L(\frac{1}{2} + it, \chi)$.

(b) Unconditional Collinearity Floor for Untwisted Dilation Tying ($\varphi(q) > 2$): For any modulus $q \ge 5$ with $\varphi(q) > 2$, any finite coprime probe set $S \subset \{m \ge 1 : \gcd(m, q) = 1\}$ (or any infinite coprime set $S$ with $\sum_{m \in S}|w_m|/\sqrt{m} < \infty$, which guarantees absolute convergence of $\sum_{m \in S} w_m \boldsymbol{\sigma}_m \in \mathcal{H}_q^{\mathrm{fam}}$ and $W := \sum_{m \in S} w_m / m \in \mathbb{C}$), and any family-uniform scalar weights $(w_m)_{m \in S}$ applied to the untwisted dilates $\sigma_{m, \chi}(y) = S_\chi(\lfloor y/m \rfloor)$, $$\inf_{\substack{S \subset \{m \ge 1 : \gcd(m, q)=1\} \\ \sum_{m \in S}|w_m|/\sqrt{m} < \infty}} \left\|\mathbf{1}_q - \sum_{m \in S} w_m \boldsymbol{\sigma}_m\right\|_{\mathcal{H}_q^{\mathrm{fam}}}^2 \ge \Delta_q^{\mathrm{untw}} := \frac{\varphi(q) - 1}{\varphi(q)} - \frac{M_1(q)^2}{M_2(q)} > 0. \tag{17b}$$ In particular, $\Delta_5^{\mathrm{untw}} = 0.088907983824900815\dots > 0$ and $\Delta_7^{\mathrm{untw}} = 0.143667959984211548\dots > 0$.

(c) Multiplicative $\chi(m)$-Twist Escape at $K = 6, 8$ and Möbius Sign Alignment: When the probes are tied by $\rho_{m, \chi}(y) = \chi(m)\sigma_{m, \chi}(y)$, the target inner-product vectors $\bigl(\frac{\chi(m)}{m}L(1, \chi)\bigr)_{\chi \neq \chi_0}$ span $\mathbb{C}^{\varphi(q)-1}$ as $m$ varies over $(\mathbb{Z}/q\mathbb{Z})^\times$, allowing $\min_{b \in \mathbb{C}^{|S|}} \mathcal{F}_q(b)$ to cross strictly below the untwisted infinite-probe floor $\Delta_q^{\mathrm{untw}}$ at $K = 6$ and $K = 8$ coprime probes, with $100\%$ Möbius sign alignment $\operatorname{sgn}(b_m^*) = \mu(m)$ across all squarefree $m \in S_8$:

• For $q = 5$: $\min_{b \in \mathbb{C}^6} \mathcal{F}_5(b) = 0.077550779244308346\dots$ on $S_6 = \{1, 2, 3, 4, 6, 7\}$ (margin $+0.011357204580592469$) and $\min_{b \in \mathbb{C}^8} \mathcal{F}_5(b) = 0.072947971218623222\dots$ on $S_8 = \{1, 2, 3, 4, 6, 7, 8, 9\}$ (margin $+0.015960012606277594$).
• For $q = 7$: $\min_{b \in \mathbb{C}^6} \mathcal{F}_7(b) = 0.129236146505082025\dots$ on $S_6 = \{1, 2, 3, 4, 5, 6\}$ (margin $+0.014431813479129523$) and $\min_{b \in \mathbb{C}^8} \mathcal{F}_7(b) = 0.113220714875071231\dots$ on $S_8 = \{1, 2, 3, 4, 5, 6, 8, 9\}$ (margin $+0.030447245109140316$). (Note: Escaping strictly below $\Delta_q^{\mathrm{untw}}$ at finite $K = 6, 8$ demonstrates that the multiplicative $\chi(m)$ twist removes the untwisted collinearity obstruction, but does not by itself establish $\inf_{S, b}\mathcal{F}_q(b) = 0$ or GRH.)

Proof of Theorem 4

Proof of (a): Substituting $y = e^u$ ($u > 0$) maps $\mathcal{H} = L^2((1, \infty), y^{-2}\,dy)$ isometrically onto $L^2((0, \infty), du)$ via $f(y) \mapsto e^{-u/2}f(e^u)$, whose Fourier–Plancherel transform is $\widehat{f}(\frac{1}{2} + it) = \int_1^\infty f(y) y^{-3/2 - it}\,dy$. For $\operatorname{Re} s = \frac{1}{2}$, $\int_1^\infty y^{-s-1}\,dy = \frac{1}{s}$, and since $S_\chi(k)$ is bounded for $\chi \neq \chi_0$, Abel summation gives $\int_1^\infty \rho_{m, \chi}(y) y^{-s-1}\,dy = \chi(m)m^{-s}\sum_{k=1}^\infty S_\chi(k)\int_k^{k+1} u^{-s-1}\,du = \frac{\chi(m)m^{-s}}{s}\sum_{k=1}^\infty \chi(k)k^{-s} = \frac{\chi(m)m^{-s}}{s}L(s, \chi)$, proving (17a) by Plancherel's theorem.

Proof of (b): By 1-term scale covariance (Theorem 1b), $\|\boldsymbol{\sigma}_m\|_{\mathcal{H}_q^{\mathrm{fam}}} = G_q^{\mathrm{fam}}(1, 1)^{1/2} m^{-1/2}$ (and similarly $\|\sigma_{m, \chi}\|_{\mathcal{H}} = \|\sigma_{1, \chi}\|_{\mathcal{H}} m^{-1/2}$ for each $\chi \neq \chi_0$). Therefore, if $\sum_{m \in S}|w_m|/\sqrt{m} < \infty$, the series $\sum_{m \in S} w_m \boldsymbol{\sigma}_m$ converges absolutely in $\mathcal{H}_q^{\mathrm{fam}}$ (and $\sum_{m \in S} w_m \sigma_{m, \chi}$ converges absolutely in $\mathcal{H}$ for each $\chi \neq \chi_0$), while $\sum_{m \in S}|w_m|/m \le \sum_{m \in S}|w_m|/\sqrt{m} < \infty$ guarantees that $W := \sum_{m \in S} \frac{w_m}{m} \in \mathbb{C}$ converges absolutely and is independent of $\chi$. By continuity of the inner product and (5), $\langle \sum_{m \in S} w_m \sigma_{m, \chi}, 1 \rangle_{\mathcal{H}} = \sum_{m \in S} \frac{w_m}{m}L(1, \chi) = W\,L(1, \chi)$. For each character $\chi \neq \chi_0 \bmod q$, since $\|1\|_{\mathcal{H}}^2 = 1$, Bessel's inequality on the projection onto $\mathbb{C}\cdot 1 \subset \mathcal{H}$ gives $\|1 - \sum_{m \in S} w_m \sigma_{m, \chi}\|_{\mathcal{H}}^2 \ge |\langle 1 - \sum_{m \in S} w_m \sigma_{m, \chi}, 1 \rangle_{\mathcal{H}}|^2 = |1 - W\,L(1, \chi)|^2$. Averaging over $\chi \neq \chi_0 \bmod q$ yields $$\left\|\mathbf{1}_q - \sum_{m \in S} w_m \boldsymbol{\sigma}_m\right\|_{\mathcal{H}_q^{\mathrm{fam}}}^2 \ge \min_{W \in \mathbb{C}} \left(\frac{\varphi(q) - 1}{\varphi(q)} - 2\operatorname{Re}(W)\,M_1(q) + |W|^2 M_2(q)\right) = \frac{\varphi(q) - 1}{\varphi(q)} - \frac{M_1(q)^2}{M_2(q)} = \Delta_q^{\mathrm{untw}}.$$ When $\varphi(q) > 2$, there exists at least one odd character $\chi_{\mathrm{odd}} \neq \chi_0$ (for which $\operatorname{Im} L(1, \chi_{\mathrm{odd}}) \neq 0$ or $L(1, \chi_{\mathrm{odd}}) \neq L(1, \chi_{\mathrm{even}})$), so $L(1, \chi)$ is non-constant across $\chi \neq \chi_0$ and Cauchy–Schwarz gives $M_1(q)^2 < \frac{\varphi(q)-1}{\varphi(q)}M_2(q)$, proving $\Delta_q^{\mathrm{untw}} > 0$.

Proof of (c): Since $L(1, \chi) \neq 0$ for all $\chi \neq \chi_0 \pmod q$ and the character table $(\chi(m))_{\chi \neq \chi_0, m \in (\mathbb{Z}/q\mathbb{Z})^\times}$ has full row rank $\varphi(q) - 1$, the vectors $(\frac{\chi(m)}{m}L(1, \chi))_{\chi \neq \chi_0}$ span $\mathbb{C}^{\varphi(q)-1}$. The strict inequalities $\Delta_q^{\mathrm{untw}} - \min_{b \in \mathbb{C}^6} \mathcal{F}_q(b) > 0$, $\min_{b \in \mathbb{C}^6}\mathcal{F}_q(b) - \min_{b \in \mathbb{C}^8}\mathcal{F}_q(b) > 0$, and the sign alignment $\operatorname{sgn}(b_m^*) = \mu(m)$ on squarefree $m \in S_8$ at $q \in \{5, 7\}$ are certified via 60-digit Krawczyk interval linear solves in src/verify_twisted_multiprobe_sieve_duality.py. $\blacksquare$

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6. Pole-Canceled Principal Residuals, the Coherent Birch–Stevens Gauss-Probe Norm Closed Form, and the Parseval Trade-Off Law

Finally, we connect the non-principal family residual energy $\mathcal{F}_q(b)$ to the principal row $\chi_0 \bmod q$ under the DFMR pole-cancellation condition (1') [4, §§2, 4, 7.2] and evaluate the exact Hilbert-space norm of the coherent Birch–Stevens Gauss-sum probe.

Theorem 5 (Pole-Canceled Principal Residuals, Exact Constrained Minimizer, Coherent Birch–Stevens Norm Closed Form, and Parseval Trade-Off Law)

Let $q \ge 3$ and let $S \subset \{m \ge 1 : \gcd(m, q) = 1\}$ be a finite probe set with weights $b = (b_m)_{m \in S}$.

(a) DFMR Pole-Cancellation Criterion, Anchored Principal Probe Duality, and Constrained Rayleigh–Ritz Minimizer: Because $S_{\chi_0}(\lfloor y/m \rfloor) = \frac{\varphi(q)}{q}\frac{y}{m} + O(q)$, the principal step residual $\widetilde{E}_{\chi_0}(y; b) := 1 - \sum_{m \in S} b_m S_{\chi_0}(\lfloor y/m \rfloor)$ belongs to $\mathcal{H} = L^2((1, \infty), y^{-2}\,dy)$ if and only if $B_q(b) := \sum_{m \in S} \frac{b_m}{m} = u_S^T b = 0$ ($u_S := (1/m)_{m \in S}$). When $B_q(b) = 0$ (and $1 \in S$), substituting $b_1 = -\sum_{m \in S \setminus \{1\}} b_m / m$ rewrites $\widetilde{E}_{\chi_0}(y; b) \equiv 1 - \sum_{m \in S \setminus \{1\}} b_m \widetilde{\rho}_{m, \chi_0}(y)$ in the anchored principal probes $\widetilde{\rho}_{m, \chi_0}(y) := S_{\chi_0}(\lfloor y/m \rfloor) - \frac{1}{m}S_{\chi_0}(\lfloor y \rfloor) \in \mathcal{H}$, which are strictly $q m$-periodic on $\mathbb{N}_0$, satisfy exact reflection antisymmetry $\widetilde{\rho}_{m, \chi_0}(q m - 1 - r) \equiv -\widetilde{\rho}_{m, \chi_0}(r)$ ($0 \le r \le q m - 1$), and have universal target inner product and pure cotangent Gram matrix ($L_q := q\operatorname{lcm}(m, n)$, $w_{\chi_0}(r) := \widetilde{\rho}_{m, \chi_0}(r)\widetilde{\rho}_{n, \chi_0}(r)$): $$\langle \widetilde{\rho}_{m, \chi_0}, 1 \rangle_{\mathcal{H}} \equiv -\frac{\varphi(q)}{q}\frac{\log m}{m}, \qquad G_{q, \chi_0}(m, n) := \langle \widetilde{\rho}_{m, \chi_0}, \widetilde{\rho}_{n, \chi_0} \rangle_{\mathcal{H}} \equiv \frac{\pi}{2 L_q}\sum_{r=1}^{L_q - 1}\bigl(w_{\chi_0}(r) - w_{\chi_0}(r - 1)\bigr)\cot\!\left(\frac{\pi r}{L_q}\right). \tag{18a}$$ Moreover, under the pole-cancellation constraint $B_q(b) = u_S^T b = 0$, the unique constrained Rayleigh–Ritz minimizer of $\mathcal{F}_q(b)$ is given by the exact Lagrange-multiplier projection $$b_S^{\mathrm{pc}} \equiv b_S^* - \frac{B_q(b_S^*)}{\widetilde{\Lambda}_q^{\mathrm{fam}}(S)}\,\bigl(\widetilde{G}_{q, S}^{\mathrm{fam}}\bigr)^{-1} u_S, \qquad \min_{B_q(b) = 0}\mathcal{F}_q(b) \equiv \min_{b \in \mathbb{C}^{|S|}}\mathcal{F}_q(b) + \frac{\bigl|B_q(b_S^*)\bigr|^2}{\widetilde{\Lambda}_q^{\mathrm{fam}}(S)}, \qquad \widetilde{\Lambda}_q^{\mathrm{fam}}(S) := u_S^T \bigl(\widetilde{G}_{q, S}^{\mathrm{fam}}\bigr)^{-1} u_S > 0. \tag{18a'}$$ In particular, at $q = 3$ on $S = \{1, 2, 4\}$, $G_{3, \chi_0}(\{2, 4\}) = G_3^{\mathrm{fam}}(1, 1)\bigl(\begin{smallmatrix} 1/2 & 1/4 \\ 1/4 & 3/8 \end{smallmatrix}\bigr)$ gives $\min_{B_3(b)=0}\|\widetilde{E}_{\chi_0}\|_{\mathcal{H}}^2 = 1 - \frac{(\log 2)^2}{3 G_3^{\mathrm{fam}}(1, 1)} = 0.470224741068814253\dots$, while $B_3(b_S^*) = \frac{1 + 2\sqrt{3}}{8}$ and $\widetilde{\Lambda}_3^{\mathrm{fam}}(S) = \frac{69}{16\,G_3^{\mathrm{fam}}(1, 1)}$ give $$b_{\{1, 2, 4\}}^{\mathrm{pc}} \equiv \left(\frac{24 + 10\sqrt{3}}{69}, \, -\frac{66 + 16\sqrt{3}}{69}, \, \frac{36 - 8\sqrt{3}}{69}\right), \qquad \min_{B_3(b)=0}\mathcal{F}_3(b_1, b_2, b_4) \equiv \frac{1}{2} - \frac{(8 + 11\sqrt{3})\pi}{207} = 0.089429370780446691\dots \tag{18a''}$$ (On the finite pre-modulus window $y \in [1, q)$ prior to requiring $B_q(b) = 0$, if $b_1 = 1$, Plancherel on $(\mathbb{Z}/q\mathbb{Z})^\times$ gives the pointwise identity $|\widetilde{E}_{\chi_0}(y; b)|^2 + \sum_{\chi \neq \chi_0 \bmod q}|E_\chi(y; b)|^2 \equiv \varphi(q)\sum_{2 \le k \le y, \gcd(k, q)=1}|c_k(S; b)|^2$; when $S = \{1, \dots, N\}$ with $N \ge q - 1$ and $b_m = \mu(m)$, all residuals vanish identically on $[1, q)$ and the entire Hilbert-space energy resides in the post-modulus tail $y \ge q$.)

(b) Euler–Maclaurin Sandwich for $G_p^{\mathrm{fam}}(1, 1)$ and Exact Coherent Birch–Stevens Probe Norm: For every odd prime $p \ge 3$, let $S_1(p) := \sum_{j=1}^\infty (\log(jp) + \gamma - H_{jp} + \frac{1}{2jp})$ denote Ehm's harmonic-deficit series [10, Theorem 2.1]. Then $G_p^{\mathrm{fam}}(1, 1) \equiv \frac{p}{p - 1}\bigl(\log p - (\log 2\pi - \gamma) + \frac{1}{p} + 2 S_1(p)\bigr)$ satisfies the strict two-sided Euler–Maclaurin sandwich $$\frac{\pi^2}{36 p^2} - \frac{\pi^4}{5400 p^4} < \frac{p - 1}{p}G_p^{\mathrm{fam}}(1, 1) - \left(\log p - (\log 2\pi - \gamma) + \frac{1}{p}\right) < \frac{\pi^2}{36 p^2}. \tag{18b}$$ Furthermore, the coherent Birch–Stevens Gauss-weighted probe $G_{\mathrm{BS}, 1}(y) := \sum_{\chi \neq \chi_0 \bmod p}\overline{\tau(\chi)}\rho_{1, \chi}(y) = g_p(\lfloor y \rfloor \bmod p)$ (where $\tau(\chi) := \sum_{a=1}^{p-1}\chi(a)e(a/p)$ and $g_p(r) := (p - 1)\sum_{a=1}^r e(-a/p) + r$ satisfies $g_p(0) = g_p(p - 1) = 0$ and conjugate-antisymmetric reflection $g_p(p - 1 - r) = -\overline{g_p(r)}$) has exact closed-form $\mathcal{H}$-norm squared: $$\|G_{\mathrm{BS}, 1}\|_{\mathcal{H}}^2 \equiv \frac{\pi(p - 1)^2}{2}\cot\!\left(\frac{\pi}{p}\right) - p(p - 1)G_p^{\mathrm{fam}}(1, 1) = \frac{1}{2}p^3 - p^2\log p + O(p^2). \tag{18c}$$

(c) Parseval Trade-Off Law for Pole-Canceled Linear Reflection Totals: Under the pole-cancellation condition $B_q(b) = 0$ ($b_1 = 1$), for any linear combination $J_\psi(y; b) := \widetilde{E}_{\chi_0}(y; b) + \sum_{\chi \neq \chi_0 \bmod q}\alpha_\chi E_\chi(y; b) \in \mathcal{H}$ with residue-class multiplier $\psi(a) := 1 - \sum_{\chi \neq \chi_0 \bmod q}\alpha_\chi \chi(a)$ on $a \in (\mathbb{Z}/q\mathbb{Z})^\times$, Plancherel on $\widehat{G}$ gives $$\left\|\widetilde{E}_{\chi_0}(\cdot; b) - J_\psi(\cdot; b)\right\|_{\mathcal{H}}^2 \le C_{\mathrm{CS}}(\psi)^2\,\mathcal{F}_q(b), \qquad C_{\mathrm{CS}}(\psi)^2 := \varphi(q)\sum_{\chi \neq \chi_0 \bmod q}|\alpha_\chi|^2 \equiv \sum_{a \in (\mathbb{Z}/q\mathbb{Z})^\times}\bigl|\psi(a) - 1\bigr|^2. \tag{18d}$$ In particular, $C_{\mathrm{CS}}(\psi)^2 = \varphi(q)(\varphi(q) - 1)$ for a single arithmetic progression $\psi(a) = \varphi(q)\mathbf{1}_{a \equiv a_0 \pmod q}$, and $C_{\mathrm{CS}}(\psi)^2 = p(p - 1)(p - 2)$ for Birch–Stevens Gauss weights $\alpha_\chi = -\overline{\tau(\chi)}$ ($\psi(a) = -(p - 1)e(-a/p)$). Comparing (18c) against the single-probe Cauchy–Schwarz bound $p(p - 1)(p - 2)G_p^{\mathrm{fam}}(1, 1) \sim p^3\log p$ shows exact equality at $p = 3$ and an asymptotic factor-of-$2\log p$ cross-character phase-cancellation saving for $p \ge 5$.

Proof of Theorem 5

Proof of (a): Write $S_{\chi_0}(k) = \frac{\varphi(q)}{q}k + r_q(k)$, where $r_q(k) := S_{\chi_0}(k) - \frac{\varphi(q)}{q}k$ is $q$-periodic ($r_q(k + q) = r_q(k)$, $r_q(0) = 0$, and $r_q(q - 1 - r) = \frac{\varphi(q)}{q} - r_q(r)$ for $0 \le r \le q - 1$). Then $\widetilde{E}_{\chi_0}(y; b) = -\frac{\varphi(q)}{q}B_q(b)y + O_{q, S, b}(1)$ as $y \to \infty$, so $\widetilde{E}_{\chi_0}(\cdot; b) \in \mathcal{H} \iff B_q(b) = 0$. For $m \ge 2$ with $\gcd(m, q) = 1$, $\widetilde{\rho}_{m, \chi_0}(r) = S_{\chi_0}(\lfloor r/m \rfloor) - \frac{1}{m}S_{\chi_0}(r)$ is $q m$-periodic with $\widetilde{\rho}_{m, \chi_0}(q m - 1 - r) = \varphi(q) - S_{\chi_0}(\lfloor r/m \rfloor) - \frac{1}{m}(m\varphi(q) - S_{\chi_0}(r)) = -\widetilde{\rho}_{m, \chi_0}(r)$. To evaluate $\langle \widetilde{\rho}_{m, \chi_0}, 1 \rangle_{\mathcal{H}}$, decompose $\widetilde{\rho}_{m, \chi_0}(y) = \frac{\varphi(q)}{q}\bigl(\frac{1}{m}\{y\} - \{y/m\}\bigr) + \bigl(r_q(\lfloor y/m \rfloor) - \frac{1}{m}r_q(\lfloor y \rfloor)\bigr)$. Since $r_q(\lfloor u \rfloor) = 0$ on $u \in (0, 1)$, the dilation change of variables $y = m u$ gives $\int_1^\infty r_q(\lfloor y/m \rfloor)y^{-2}\,dy = \int_m^\infty r_q(\lfloor y/m \rfloor)y^{-2}\,dy = \frac{1}{m}\int_1^\infty r_q(\lfloor u \rfloor)u^{-2}\,du$, so the $r_q$ remainder integrals cancel identically! Meanwhile, $\int_1^\infty \bigl(\frac{1}{m}\{y\} - \{y/m\}\bigr)y^{-2}\,dy = \int_1^\infty \bigl(\lfloor y/m \rfloor - \frac{1}{m}\lfloor y \rfloor\bigr)y^{-2}\,dy = -\frac{\log m}{m}$, proving $\langle \widetilde{\rho}_{m, \chi_0}, 1 \rangle_{\mathcal{H}} = -\frac{\varphi(q)}{q}\frac{\log m}{m}$, and the reflection invariance $w_{\chi_0}(L_q - 1 - r) = (-1)^2 w_{\chi_0}(r) = w_{\chi_0}(r)$ yields the pure cotangent sum in (18a). By Theorem 1(b), $\mathcal{F}_q(b) = \min_{b'}\mathcal{F}_q(b') + (b - b_S^*)^* \widetilde{G}_{q, S}^{\mathrm{fam}} (b - b_S^*)$ with $\widetilde{G}_{q, S}^{\mathrm{fam}} \succ 0$; minimizing $(b - b_S^*)^* \widetilde{G}_{q, S}^{\mathrm{fam}} (b - b_S^*)$ subject to $u_S^T(b - b_S^*) = -B_q(b_S^*)$ via Cauchy–Schwarz in the $\widetilde{G}_{q, S}^{\mathrm{fam}}$-inner product yields (18a') and (18a'').

Proof of (b): By Ehm [10, Theorem 2.1] and Remark 1 of riemann_hypothesis_21 [6], $\frac{\pi}{2p^2}\sum_{r=1}^{p-1}(p - 2r)\cot(\frac{\pi r}{p}) = \log p - (\log 2\pi - \gamma) + \frac{1}{p} + 2 S_1(p)$. Since $f_x(t) := \log t - \psi_0(t) - \frac{1}{2t} = \sum_{k=1}^\infty \frac{B_{2k}}{2k\,t^{2k}}$ is completely monotone and Euler–Maclaurin gives $\frac{\pi^2}{72 p^2} - \frac{\pi^4}{10800 p^4} < S_1(p) < \frac{\pi^2}{72 p^2}$, multiplying by $2$ yields (18b). For $G_{\mathrm{BS}, 1}(y)$, since $\tau(\chi_0) = -1$ and $\sum_{\chi \bmod p}\overline{\tau(\chi)}\chi(a) = (p - 1)e(-a/p)$, summing over $1 \le a \le r$ ($0 \le r \le p - 1$) gives $g_p(r) = (p - 1)\sum_{a=1}^r e(-a/p) + r$. Using $\sum_{a=1}^{p-1}e(-a/p) = -1$ gives $g_p(p - 1 - r) = -\overline{g_p(r)}$, and summing the geometric series gives $$|g_p(r)|^2 = (p - 1)^2\frac{\sin^2(\pi r/p)}{\sin^2(\pi/p)} + (p - 1)r\frac{\sin(\pi(2r + 1)/p)}{\sin(\pi/p)} - (p - 1)W_p(r, r).$$ Applying the periodic cotangent formula (9) to $|g_p(r)|^2$, the third term contributes $-p(p - 1)G_p^{\mathrm{fam}}(1, 1)$. For the first two terms, summation by parts converts $\frac{\pi}{2p}\sum_{r=1}^{p-1}(A_r - A_{r-1})\cot(\frac{\pi r}{p})$ into $\frac{\pi}{2p}\sum_{r=1}^{p-2} A_r (\cot\frac{\pi r}{p} - \cot\frac{\pi(r+1)}{p}) = \frac{\pi\sin(\pi/p)}{2p}\sum_{r=1}^{p-2}\frac{A_r}{\sin(\pi r/p)\sin(\pi(r+1)/p)}$. By the identity $\frac{\sin(\pi(2r+1)/p)}{\sin(\pi r/p)\sin(\pi(r+1)/p)} = \cot(\frac{\pi r}{p}) + \cot(\frac{\pi(r+1)}{p})$, the second term telescopes on $1 \le r \le p - 2$ to $(p - 1)\cot(\frac{\pi}{p}) - (p - 1)\sum_{r=1}^{p-2}\cot(\frac{\pi r}{p})$, while the first term equals $(p - 1)^2\sum_{r=1}^{p-2}\bigl[\cot(\frac{\pi r}{p})\cos(\frac{\pi}{p}) + \sin(\frac{\pi}{p})\bigr]$; combining both via $\cot(\frac{\pi(p-1-r)}{p}) = -\cot(\frac{\pi r}{p})$ cancels all $r \ge 2$ terms and leaves $\frac{\pi(p - 1)^2}{2}\cot(\frac{\pi}{p})$, proving (18c).

Proof of (c): Since $\psi(a) - 1 = -\sum_{\chi \neq \chi_0}\alpha_\chi \chi(a)$, Plancherel on $(\mathbb{Z}/q\mathbb{Z})^\times$ gives $\sum_{a \in (\mathbb{Z}/q\mathbb{Z})^\times}|\psi(a) - 1|^2 = \varphi(q)\sum_{\chi \neq \chi_0}|\alpha_\chi|^2 = C_{\mathrm{CS}}(\psi)^2$, and Cauchy–Schwarz across $\chi \neq \chi_0 \bmod q$ gives $|\sum_{\chi \neq \chi_0}\alpha_\chi E_\chi(y; b)|^2 \le (\sum_{\chi \neq \chi_0}|\alpha_\chi|^2)(\sum_{\chi \neq \chi_0}|E_\chi(y; b)|^2)$; integrating over $(1, \infty)$ against $y^{-2}\,dy$ yields (18d). $\blacksquare$

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7. 60-Digit Interval Certification, Numerical Enclosures, and Reproducibility

While the universal all-$q$ identities in Theorems 1–5 are established by the symbolic mathematical proofs above, all displayed finite instances, Krawczyk interval linear solves, and strict inequalities in Tables 1–4 are certified in 60-digit interval arithmetic (mpmath.iv.dps = 60) using src/verify_twisted_multiprobe_sieve_duality.py (CPU runtime 6.89s). In Tables 1–4, entries are displayed as 18-digit decimal midpoints computed from 60-digit enclosures of width $< 10^{-50}$; explicit 18-digit intervals $[a, b]$ exported via directed outward decimal rounding (decimal.ROUND_FLOOR and decimal.ROUND_CEILING, guaranteeing $a_{\mathrm{dec}} < x_{\mathrm{true}} < b_{\mathrm{dec}}$) are recorded in results/twisted_multiprobe_sieve_duality.json.

Reproducibility Note: The verification and plotting scripts require Python 3.12, mpmath 1.3.0, numpy, and matplotlib. From the repository/supplement root, run python3 src/verify_twisted_multiprobe_sieve_duality.py (which executes all 60-digit interval assertions and writes results/twisted_multiprobe_sieve_duality.json) followed by python3 src/plot_twisted_multiprobe_sieve_duality.py (which regenerates results/fig_twisted_multiprobe_sieve_duality.png).

Table 1: Certified Digamma Variance $M_2(q)$ and 4-Probe Harmonic Rayleigh Quotients (60-digit enclosure width $< 10^{-55}$)

| Modulus $q$ | $\varphi(q)$ | $M_2(q) = \frac{\varphi(q)}{q^2}\operatorname{Var}(\psi_0(r/q))$ | $G_q^{\mathrm{fam}}(1, 1)$ | 4-Probe Set $S_4$ | $\Lambda_q^{\mathrm{fam}}(S_4)$ | Captured Fraction $M_2(q)\Lambda_q^{\mathrm{fam}}(S_4)$ | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | $3$ | $2$ | $0.182770451872025160$ | $0.302299894039036308$ | $\{1, 2, 4, 5\}$ | $5.208612987360305483$ | $0.951980549326341904$ | | $4$ | $2$ | $0.308425137534042457$ | $0.392699081698724155$ | $\{1, 3, 5, 7\}$ | $3.073127520971349371$ | $0.947829778315239374$ | | $5$ | $4$ | $0.441097140159013223$ | $0.699642966017503762$ | $\{1, 2, 3, 4\}$ | $1.831701184800871101$ | $0.807958154241540421$ | | $6$ | $2$ | $0.411233516712056609$ | $0.453449841058554463$ | $\{1, 5, 7, 11\}$ | $2.349640205781032251$ | $0.966250804831374256$ | | $7$ | $6$ | $0.603637851072123922$ | $0.972642368206695381$ | $\{1, 2, 3, 4\}$ | $1.233503044488809374$ | $0.744589127066147361$ | | $8$ | $4$ | $0.559740131288025683$ | $0.751709908119157858$ | $\{1, 3, 5, 7\}$ | $1.475144856566500167$ | $0.825697775683388619$ | | $11$ | $10$ | $0.801358712000827278$ | $1.353448232458652449$ | $\{1, 2, 3, 4\}$ | $0.840825619792552982$ | $0.673802935694257561$ | | $12$ | $4$ | $0.624303944686785723$ | $0.750323696127576104$ | $\{1, 5, 7, 11\}$ | $1.407020031319101073$ | $0.878408155805839592$ | | $13$ | $12$ | $0.867689014926791958$ | $1.498068678057558899$ | $\{1, 2, 3, 4\}$ | $0.749100403899110016$ | $0.649986191540480756$ |

Table 2: Certified Universal Even–Odd Modulus Gram Identity $G_{2q}^{\mathrm{fam}}(1, 1) \equiv \frac{3}{2}G_q^{\mathrm{fam}}(1, 1) - 2\widetilde{G}_q^{\mathrm{fam}}(1, 2)$ (60-digit enclosure width $< 10^{-56}$)

| Odd $q$ | Even $2q$ | $G_q^{\mathrm{fam}}(1, 1)$ | $\widetilde{G}_q^{\mathrm{fam}}(1, 2)$ | $G_{2q}^{\mathrm{fam}}(1, 1)$ | Certified Difference | | :---: | :---: | :---: | :---: | :---: | :---: | | $3$ | $6$ | $0.302299894039036308$ | $0.000000000000000000$ | $0.453449841058554463$ | $< 10^{-58}$ | | $5$ | $10$ | $0.699642966017503762$ | $0.133619916514449390$ | $0.782224615997356864$ | $< 10^{-58}$ | | $7$ | $14$ | $0.972642368206695381$ | $0.240480033324726026$ | $0.978003485660591021$ | $< 10^{-58}$ | | $9$ | $18$ | $1.017467313438619336$ | $0.308766871978746182$ | $0.908667226200436640$ | $< 10^{-57}$ | | $11$ | $22$ | $1.353448232458652449$ | $0.401646269887388650$ | $1.226879808913201373$ | $< 10^{-57}$ | | $13$ | $26$ | $1.498068678057558899$ | $0.465447941941078961$ | $1.316207133204180427$ | $< 10^{-57}$ | | $15$ | $30$ | $1.229643214644764493$ | $0.439854889041895868$ | $0.964755043883355004$ | $< 10^{-57}$ |

Table 3: Certified Multiplicative $\chi(m)$-Twist Escape Across the Untwisted Collinearity Floor at $K = 6, 8$ (60-digit enclosure width $< 10^{-52}$)

| Modulus $q$ | Untwisted Floor $\Delta_q^{\mathrm{untw}}$ | 6-Probe $\min_{b \in \mathbb{C}^6} \mathcal{F}_q(b)$ | 6-Probe Escape Margin | 8-Probe $\min_{b \in \mathbb{C}^8} \mathcal{F}_q(b)$ | 8-Probe Escape Margin | | :---: | :---: | :---: | :---: | :---: | :---: | | $5$ | $0.088907983824900815$ | $0.077550779244308346$ | $+0.011357204580592469$ | $0.072947971218623222$ | $+0.015960012606277594$ | | $7$ | $0.143667959984211548$ | $0.129236146505082025$ | $+0.014431813479129523$ | $0.113220714875071231$ | $+0.030447245109140316$ |

Table 4: Certified Coherent Birch–Stevens Gauss-Probe Norm $\|G_{\mathrm{BS}, 1}\|_{\mathcal{H}}^2$ vs. Cauchy–Schwarz Bound $p(p - 1)(p - 2)G_p^{\mathrm{fam}}(1, 1)$

| Prime $p$ | $G_p^{\mathrm{fam}}(1, 1)$ | Coherent Norm $\|G_{\mathrm{BS}, 1}\|_{\mathcal{H}}^2$ | Outward-Rounded 18-Digit Interval for $\|G_{\mathrm{BS}, 1}\|_{\mathcal{H}}^2$ | Cauchy–Schwarz Bound $C_{\mathrm{CS}}^2 G_p^{\mathrm{fam}}(1, 1)$ | Phase-Saving Ratio | | :---: | :---: | :---: | :---: | :---: | :---: | | $3$ | $0.302299894039036308$ | $1.813799364234217851$ | $[1.81379936423421785, \, 1.81379936423421786]$ | $1.813799364234217851$ | $1.000000000000$ | | $5$ | $0.699642966017503762$ | $20.599391318738323450$ | $[20.5993913187383234, \, 20.5993913187383235]$ | $41.978577961050225718$ | $0.490711985000$ | | $7$ | $0.972642368206695381$ | $76.573539096204857986$ | $[76.5735390962048579, \, 76.5735390962048580]$ | $204.254897323406030018$ | $0.374892059381$ | | $11$ | $1.353448232458652449$ | $386.084794935496981399$ | $[386.084794935496981, \, 386.084794935496982]$ | $1339.913750134065924121$ | $0.288141527689$ | | $13$ | $1.498068678057558899$ | $684.009141508666078177$ | $[684.009141508666078, \, 684.009141508666079]$ | $2570.685851546771070021$ | $0.266080408502$ | | $17$ | $1.734344135120978096$ | $1679.431043117307977267$ | $[1679.43104311730797, \, 1679.43104311730798]$ | $7076.124071293590633208$ | $0.237337704398$ | | $19$ | $1.833677810835077698$ | $2422.780474669294745286$ | $[2422.78047466929474, \, 2422.78047466929475]$ | $10661.002792195141733482$ | $0.227256339942$ | | $23$ | $2.006048775564883119$ | $4516.281054114498320920$ | $[4516.28105411449832, \, 4516.28105411449833]$ | $21316.274289152448027003$ | $0.211870094785$ |

Figure 1: Global Family Residual Energy, Digamma Variance, Even-Odd Modulus Duality, and Multiplicative Collinearity Escape

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