Quotient-Shell Prime-Separation Kernels, Unimodular Active-Core Inversion, and Sufficient Overflow Bounds in the Squarefree Complex

1. Introduction, Setup, and Conventions

Let $P \ge 2$ be a prime or $x \ge 2$ be a real additive cutoff, and let $p_{\max}(x) := \max\{p \le x : p \text{ prime}\}$ denote the largest prime at most $x$. Write $\mathcal{S}_P := \{m \in \mathbb{Z}_{\ge 1} : \mu(m) \ne 0,\; P^+(m) \le P\}$ for the finite Boolean lattice of $2^{\pi(P)}$ squarefree $P$-smooth integers ($P^+(1) := 0$, $P^-(1) := \infty$, where $P^\pm(n)$ are the largest and smallest prime factors of $n$), and $\mathcal{S}_{\le x} := \{m \in \mathbb{Z}_{\ge 1} : \mu(m) \ne 0,\; m \le x\}$ for the downward-closed divisor ideal of squarefree integers up to $x$. The full $P$-smooth simplicial complex $C_{\mathrm{smooth}, P}^{\mathrm{full}} := \bigoplus_{m \in \mathcal{S}_P} \mathbb{C} e_m \cong \bigotimes_{p \le P}(\mathbb{C} e_1 \oplus \mathbb{C} e_p)$ and the height-truncated squarefree complex $C_x := \bigoplus_{m \in \mathcal{S}_{\le x}} \mathbb{C} e_m$ [1, 2, 3, 4] are graded by prime-divisor degree $k = \omega(m)$ ($C_x = \bigoplus_{k \ge 0} C_x^k$) and equipped with the standard Euclidean inner product $\langle u, v \rangle := \sum_m \overline{u_m}\, v_m$ (conjugate-linear in the first slot) and coordinate projections $Q_y(\sum_m u_m e_m) := \sum_{m \le y} u_m e_m$. Given positive prime weights $\{a_p\} \subset (0, 1)$ with cumulative weight $S(t) := \sum_{p \le t} a_p^2$ and $a_m := \prod_{p \mid m} a_p$, the deletion boundary $d_x : C_x^k \to C_x^{k-1}$, coboundary $d_x^* : C_x^k \to C_x^{k+1}$, and Hodge Laplacian $L_x : C_x^k \to C_x^k$ act by [3, 4]: $$d_x(e_m) \;:=\; \sum_{p \mid m} (-1)^{I(p, m)} a_p\, e_{m/p}, \qquad d_x^*(e_n) \;:=\; \sum_{\substack{p \le x,\; p \nmid n \\ p n \le x}} (-1)^{I(p, p n)} a_p\, e_{p n}, \qquad L_x \;:=\; d_x d_x^* + d_x^* d_x,$$ where $I(p, m) := |\{q \mid m\text{ prime} : q < p\}|$, with cycle space $Z_x := \ker(d_x)$, boundary space $B_x := \operatorname{im}(d_x) = Z_{x/2}$ [3], coboundary space $B_x^* := \operatorname{im}(d_x^*) = Z_x^\perp$, harmonic space $H_x := \ker(L_x) = Z_x \cap \ker(d_x^*)$, and orthogonal projectors $P_{Z_x}, P_{B_x}, \Pi_x := P_{H_x}$. For any sequence $\beta = \{\beta_p\}$, we define the multiplicative cochain $\psi_x(\beta) := \sum_{m \in \mathcal{S}_{\le x}}\bigl(\prod_{p \mid m}\beta_p\bigr)e_m$ (and $\psi_P(\beta)$ on $C_{\mathrm{smooth}, P}^{\mathrm{full}}$), specializing to the signed Möbius cochain $\mu_x(\beta) := \psi_x(-\beta) = \sum_{m \le x\text{ sqfree}}\mu(m)\bigl(\prod_{p \mid m}\beta_p\bigr)e_m$ and unsigned cochain $\mathbf{1}_x(\beta) := \psi_x(+\beta) = \sum_{m \le x\text{ sqfree}}\bigl(\prod_{p \mid m}\beta_p\bigr)e_m$.

In the standard Euclidean metric $\langle e_m, e_n \rangle = \delta_{m, n}$, the unitary $U(1)$ degree-grading group $U(\theta)e_m := e^{i\theta\omega(m)}e_m$ ($\theta \in \mathbb{R}/2\pi\mathbb{Z}$) satisfies $U(\theta)d_x U(\theta)^{-1} = e^{-i\theta}d_x$ and $U(\theta)d_x^* U(\theta)^{-1} = e^{i\theta}d_x^*$ because $d_x$ lowers $\omega(m)$ by $1$ and $d_x^*$ raises $\omega(m)$ by $1$. Consequently, $U(\theta)$ commutes identically with $d_x d_x^*$, $d_x^* d_x$, $L_x$, $\Pi_x$, $P_{Z_x}$, and $Q_y$; since $U(\pi)e_m = \mu(m)e_m$ maps $\mu_x(a)$ to $\mathbf{1}_x(a)$, every degree-preserving Euclidean Hodge observable satisfies $\|\mathrm{Op}\,\mu_x(a)\|_2^2 \equiv \|\mathrm{Op}\,\mathbf{1}_x(a)\|_2^2$ and is blind to the Möbius sign $\mu(m)$.

Relation to Prior Work and Separation of Inherited Machinery vs. New Results: Several foundational components used below are established in the literature and our prior papers: the squarefree simplicial complex and its connection to Mertens sums go back to Björner [1] and Duval–Kook–Lee–Martin [2]; the compatible finite-cutoff Hodge/cycle framework and observation estimates were developed in [3]; the divisor-incidence $LDL^{\mathsf{T}}$ factorization and LCM-interior inverse of meet matrices on lower-closed divisor posets are classical [7, 8, Theorem 2.5]; the Euclidean odd-degree coboundary orthogonality and full-complex Euler-product cycle norm of $\mu_P(a)$ were established in [4, Theorems 3–4]; the noncancellation of Mellin poles between smooth terms and $\zeta(s+z)^{-1}$ is a classical mechanism also used in [9, Theorem 7.1]; and the vertical mean-value theorem for generalized Dirichlet polynomials is due to Montgomery and Vaughan [10, Corollary 2] (see also Broucke–Hilberdink [11, Section 4]). Building on and distinguishing that inherited baseline, the new contributions of the present paper address two structural and quantitative questions:

1. What is the exact operator difference between the finite-height Hodge Laplacian $L_x$ on $C_x$ (and $L_{>x}$ on $C_{>x}$) and the scalar Cartan identity $S(x) I$, and what sufficient conditions on the prime overlaps $\{\rho_p\}$ guarantee sub-scale decay of the Cholesky inverse overflow operator $\mathcal{M}_{x, \rho} := Q_x\mathcal{C}_{\mathrm{full}, \rho}^{-1}Q_{>x}$ on $C_y \subset C_x$ ($y \le x$), as opposed to the exponential divergence that occurs for non-zero constant overlaps $\rho_p \equiv \rho_0 \in \mathbb{D}\setminus\{0\}$?
2. On the non-vacuum quotient shells $\mathcal{S}_n(x) := \{k \in C_{x/2} : \lfloor x/k \rfloor = n\}$ of the conjugated boundary $d_{x, \rho}^{(w)}$, what is the exact Dirichlet-polynomial expansion of each active-core fiber $m = k_{\le n} \mid P_{\pi(n)}^\#$, what is the exact off-diagonal prime-separation kernel $W(d_1, d_2)$ across active cores, and what pointwise (fixed-$s$, zero-$t$-averaging) inversion, coercivity, and finite-$x$ support-preserving estimates does $W(d_1, d_2)$ enable beyond generic vertical Dirichlet-polynomial averaging [10, 11]?

We establish four theorems:

1. Exact Cartan Overflow Decomposition, $2 \times 2$ Block Duality, Contiguous-Suffix Parity Cancellation & Unitary Harmonic Isomorphism (Theorem 1): On $C_{\mathrm{smooth}, p_{\max}(x)}^{\mathrm{full}} = C_x \oplus C_{>x}$, the full boundary operator is upper-block-triangular $d_{\mathrm{full}} = \begin{pmatrix} d_x & \mathcal{T}_x^* \\ 0 & d_{>x} \end{pmatrix}$, where $\mathcal{T}_x := Q_{>x}(w_x \wedge \cdot)|_{C_x} : C_x \to C_{(x, x p_{\max}(x)]}$ is the coboundary overflow operator. Cartan's anticommutator implies the dual overflow identities $L_x \equiv S(x) I_{C_x} - \mathcal{T}_x^* \mathcal{T}_x$ and $L_{>x} \equiv S(x)I_{C_{>x}} - \mathcal{K}_x$ ($\mathcal{K}_x := \mathcal{T}_x \mathcal{T}_x^*$), where off-diagonal couplings $(L_x)_{m, n}$ ($m \ne n$) are nonzero if and only if $m = d q_1, n = d q_2$ with $\operatorname{lcm}(m, n) = d q_1 q_2 > x$, $L_x|_{C_{x/p_{\max}(x)}} \equiv S(x)I_{C_{x/p_{\max}(x)}}$, the contiguous suffix of overflow divisors $u/p_j \le x$ ($N_{\mathrm{ovf}}(u; x) := |\{p \mid u : u/p \le x\}|$) cancels in adjacent pairs via $(\mathcal{T}_x\psi_x(\gamma a))_u \equiv (-\gamma)^{\omega(u)-1}\mathbf{1}_{\{N_{\mathrm{ovf}}(u; x)\text{ odd}\}}\prod_{p \mid u} a_p$, and the scaled overflow map $\mathcal{U}_x := S(x)^{-1/2}\mathcal{T}_x|_{H_x} : H_x^k \stackrel{\sim}{\longrightarrow} H_{>x}^{k+1}$ is an exact unitary isomorphism from $H_x = \ker(L_x)$ onto $H_{>x} = \ker(L_{>x})$ on $C_{>x}^{\mathrm{ovf}}$.
2. All-Scale Divisor-Triangular Cholesky Factorization, Order-Ideal Approximation Residual & Sufficient Overflow Bounds vs. Constant-Overlap Divergence on $\mathbb{D}^{\pi(x)}$ (Theorem 2): Extending the meet-matrix $LDL^{\mathsf{T}}$ factorization [7, 8] to complex overlaps $\{\rho_p\}_{p \le x} \subset \mathbb{D}$ via the abelian Hasse-derivative group $\mathcal{U}_{x, \rho} = \exp(\sum_{p \le x}\rho_p\partial_p)$ and Möbius–Beltrami similarity $\mathcal{C}_{x, \rho} = \mathcal{J}_{x, \rho} D_{x, \rho} \mathcal{J}_{x, \rho}^{-1}$, we obtain the closed-form LCM-interior inverse $(G_{x, \rho}^{-1})_{m, n}$ [7, 8] disjoint in off-diagonal support from $L_x$, the Polydisc-Invariant Mertens Supertrace $\operatorname{STr}(G_{x, \rho}^{-1}) \equiv M(x)$, the overflow Schur complement $G_{x, \rho}^{-1} + \mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^* \equiv \Delta_x(\rho)^{-1} G_{x, -\rho}$, the Exact Order-Ideal Approximation Identity $\min_{v \in \mathcal{A}}\|v - G_{x, \rho}^{-1}\psi_x(\beta)\|_{G_{x, \rho}}^2 \equiv \sum_{k \in C_x \setminus \mathcal{A}}\prod_{p \mid k}\frac{|\beta_p - \overline{\rho}_p|^2}{1 - |\rho_p|^2}$ for every divisor-closed subspace $\mathcal{A} \subset C_x$, and two complementary overflow bounds (6f): (i) the Rankin–Euler-Product Majorant $0 \le (\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^*)_{m, m} \le C_\alpha(x; \rho)\bigl(\prod_{p \mid m}\frac{1}{1 + |\rho_p|^2(p^\alpha - 1)}\bigr)(m/x)^\alpha$ and $\operatorname{Tr}(P_{C_y}\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^* P_{C_y}) \le C_\alpha(x; \rho) x^{-\alpha}\bigl(y^\alpha + \frac{y^{\alpha+1}}{\alpha+1}\bigr)$ ($\forall \alpha \ge 0$), which gives a sufficient $O((m/x)^\alpha)$ decay bound whenever $\sum_p |\rho_p|^2 p^\alpha < \infty$ ($\sup_p |\rho_p| < 1$), together with a Direct Power-Law Tail Bound showing that whenever $\frac{|\rho_p|^2}{1 - |\rho_p|^2} \le p^{-2\sigma}$ ($\sigma > \frac{1}{2}$, e.g., at the Selberg–Helmert overlap $|\rho_p|^2 = \frac{1}{1+p^{2\sigma}}$), $(\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^*)_{m, m} \le \frac{2\sigma}{2\sigma - 1} K_\sigma(m/x)^{2\sigma - 1}$ and $\operatorname{Tr}(P_{C_y}\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^* P_{C_y}) = O(y^{2\sigma} x^{1 - 2\sigma})$ hold at the borderline exponent $\alpha = 2\sigma - 1$ even though $\sum_p p^{2\sigma - 1}|\rho_p|^2 = \infty$; and (ii) the Universal Constant-Overlap Lower Bound showing that for every constant $\rho_p \equiv \rho_0 \in \mathbb{D}\setminus\{0\}$ ($u_0 := \frac{|\rho_0|^2}{1 - |\rho_0|^2} > 0$) and all $x \ge 2$, $(\mathcal{M}_{x, \rho_0}\mathcal{M}_{x, \rho_0}^*)_{1, 1} \ge (1 + \min(u_0, 1))^{\pi(x)} - |C_x| \ge (1 + \min(u_0, 1))^{\pi(x)} - x \to \infty$.
3. Pythagorean Dual Overlap $\rho_p^\star = \frac{2a_p}{1+a_p^2}$, Transported vs. Untransported Parity Sectors & Complex Phase Dispersion (Theorem 3): Specializing the Euclidean coboundary identity $d_x^*(\mu_x^{2r}(a)) \equiv -\mu_x^{2r+1}(a)$ [4] under Cholesky transport at $\rho_p^\star := \frac{2a_p}{1 + a_p^2}$, $\mathcal{C}_{x, \rho^\star}^*\mu_x(a) \equiv \mathbf{1}_x(a)$ and $(\mathcal{C}_{x, \rho^\star}^{-1})^*\mathbf{1}_x(a) \equiv \mu_x(a)$ hold at every $x \ge 1$, while on $C_{\mathrm{smooth}, P}^{\mathrm{full}}$ the Möbius chain $\mu_P(a)$ is an exact eigenvector of $\mathcal{C}_{\rho^\star}$ with eigenvalue $\lambda_\mu(P) = \prod_{p \le P}\frac{1 - a_p^2}{1 + a_p^2}$ and zero transported odd-degree cycle energy, whereas $\mathbf{1}_P(a)$ satisfies the degree-$1$ Lagrange lower bound $\bigl\|P_{Z_{P, \rho^\star}} P_{\mathrm{odd}, \rho^\star}\mathbf{1}_P(a)\bigr\|_{G_{\rho^\star}}^2 \ge \frac{16\Lambda_+(P)^2}{S(P)}\sum_{p < q \le P}\frac{a_p^2 a_q^2(a_p^2 - a_q^2)^2}{(1+3a_p^2)^2(1+3a_q^2)^2} \ge 0$ (equal to $\frac{4}{325}$ at $P = 3, a_p = 1/p$), strictly positive iff $P \ge 3$ and $a_p$ is non-constant across $p \le P$.
4. Quotient-and-Core Shell Decomposition, Unimodular Complement-Divisor Active-Core Inversion, Prime-Separation Kernel $W(d_1, d_2)$, Pointwise Fixed-$s$ Coercivity & Support-Preserving Shell Bounds (Theorem 4): Partitioning $C_{x/2} = \{1\} \sqcup \bigsqcup_{n=2}^{\lfloor x/2 \rfloor}\mathcal{S}_n(x)$ into quotient shells $\mathcal{S}_n(x) := \{k \text{ sqfree} : \lfloor x/k \rfloor = n\}$ and active-core fibers $k = m v$ ($m = k_{\le n} \mid P_{\pi(n)}^\#$, $P^-(v) > n$) decouples $\mathcal{F}(n; m; \gamma) := \mathcal{E}(n; m; \gamma) - \mu(m)M_m(n; \gamma)$ from $v \in (\frac{x}{(n+1)m}, \frac{x}{nm}]$. We prove that on every active-core fiber $m \mid P_{\pi(n)}^\#$, $\mathcal{F}(n; m; \gamma)$ expands as $\mathcal{F}(n; m; \gamma) \equiv -\sum_{2 \le d \le n \text{ sqfree}, \gcd(d, m)=1} (-1)^{r_m(d)}\mu(d)\gamma(d)$ ($r_m(d) := \#\{p \mid m : p < P^+(d)\}$, $c_{n, m}(1) \equiv 0$), and across fibers $m \mid P_{\pi(n)}^\#$ with normalized weight $A_n m^{-1}$ ($A_n := \prod_{p \le n}\frac{p}{p+1}$), the quadratic form $A_n\mathcal{Q}_n(\gamma)$ has the exact $n$-independent off-diagonal prime-separation kernel $W(d_1, d_2) \equiv \prod_{p \le n}\bigl(\frac{p}{p+1} + \frac{1}{p+1}\sigma_p(d_1)\sigma_p(d_2)\bigr) \equiv \bigl(\prod_{p \mid \operatorname{lcm}(d_1, d_2)}\frac{p}{p+1}\bigr)\bigl(\prod_{\min(P^+(d_1), P^+(d_2)) < p < \max(P^+(d_1), P^+(d_2)), p \nmid d_1 d_2}\frac{p-1}{p+1}\bigr)$ ($\sigma_p(d) := \mathbf{1}_{\{p \nmid d\}}(-1)^{\mathbf{1}_{\{p < P^+(d)\}}}$). Beyond vertical $t$-averaging (where the Montgomery–Vaughan mean-value theorem [10, Corollary 2], [11, Section 4] yields the $\log n$-free uniform bound $\bigl|\frac{1}{T}\int_T^{2T} A_n\mathcal{Q}_n(p^{-\sigma - it})\,dt - \sum_{2 \le d \le n\text{ sqfree}}\frac{d^{1 - 2\sigma}}{\sigma_1(d)}\bigr| \le \frac{\pi}{T}\sum_{2 \le d \le n\text{ sqfree}}\frac{d^{2 - 2\sigma}}{\sigma_1(d)}$), we establish two pointwise (fixed-$s$, zero-$t$-averaging) and finite-support consequences of the active-core structure: (i) re-indexing $m = P_{\pi(n)}^\# / c$ by its complement divisor $c \in \mathcal{I}_n := \{2 \le c \le n\text{ sqfree}\}$ makes the interior fiber map $(\mathcal{B}_n^{\mathrm{int}})_{c, d} := \mathbf{1}_{\{d \mid c\}}(-1)^{\pi(P^+(d)) - 1 - \#\{p \mid c : p < P^+(d)\}}$ (supported strictly on divisibility pairs $d \mid c$ in $\mathcal{I}_n$) unimodular lower divisor-triangular with exact $\{-1, 0, +1\}$-valued inverse $((\mathcal{B}_n^{\mathrm{int}})^{-1})_{c, d} = \mathbf{1}_{\{d \mid c\}}(-1)^{\pi(P^+(c))}\mu(c)(-1)^{\mathbf{1}_{\{P^+(c) \nmid d\}}}$, decomposing $W_n \equiv W_n^{\mathrm{int}} + W_n^{\mathrm{ovf}} \succeq W_n^{\mathrm{int}} = \frac{1}{\sigma_1(P_{\pi(n)}^\#)}(\mathcal{B}_n^{\mathrm{int}})^T\operatorname{diag}(c)_{c \in \mathcal{I}_n}\mathcal{B}_n^{\mathrm{int}} \succ 0$ with closed-form GCD inverse $(W_n^{\mathrm{int}})^{-1}$ that vanishes identically on all coprime pairs $\gcd(d_1, d_2) = 1$ and yields the pointwise pairwise-coprime and Bertrand-prime coercivity bounds $A_n\mathcal{Q}_n(\gamma) \ge \frac{1}{\sigma_1(P_{\pi(n)}^\#)}\sum_{d \in \mathcal{C}}\frac{d}{\sigma_1(d)-d}|\gamma(d)|^2$ and $A_n\mathcal{Q}_n(\gamma) \ge \frac{p_{\max}(n)}{(p_{\max}(n)+1)^2}|\gamma_{p_{\max}(n)}|^2$ at every fixed $s \in \mathbb{C}$; and (ii) restricting to short active cores $m \le M_0 \le \frac{x}{n(n+1)}$ yields a support-preserving finite-$(x, n, M_0)$ shell identity with explicit Rankin truncation and rough-cofactor lattice remainders.

Scope Limit: These results establish finite-dimensional operator identities, sufficient sub-scale overflow bounds, order-ideal approximation residuals (which depend only on $\sigma = \operatorname{Re}(s)$ and do not distinguish zero locations), and an explicit quotient-shell prime-separation kernel $W(d_1, d_2)$ with unimodular complement-divisor inversion and pointwise coercivity bounds, while vertical $t$-averaging is controlled for $n = o(T)$ by the classical Montgomery–Vaughan mean-value theorem [10, 11]. Controlling long-range signed cancellation of $M_s(x)$ for $\sigma > 1/2$ remains open; thus no new zero-free region or proof of the Riemann Hypothesis is claimed.

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2. Exact Cartan Overflow Decomposition of the Finite-Height Hodge Laplacian

Let $w_x = \sum_{p \le x} a_p e_p = d_x^*(e_1)$, and decompose the full $p_{\max}(x)$-smooth complex as $C_{\mathrm{smooth}, p_{\max}(x)}^{\mathrm{full}} = C_x \oplus C_{>x}$ via the complementary coordinate projections $Q_x$ (onto $m \le x$) and $Q_{>x} := I - Q_x$ (onto $m > x$). Let $d_{>x} := Q_{>x} d_{\mathrm{full}}|_{C_{>x}}$ and $L_{>x} := d_{>x} d_{>x}^* + d_{>x}^* d_{>x}$ be the exterior complement boundary operator and Hodge Laplacian on $C_{>x}$.

Theorem 1 (Exact Cartan Overflow Decomposition, $2 \times 2$ Block Duality & Unitary Harmonic Isomorphism). (a) $2 \times 2$ Block-Triangular Decomposition & Dual Overflow Laplacians: Because $d_{\mathrm{full}}(C_x) \subseteq C_{x/2} \subset C_x$, the full boundary and coboundary operators on $C_{\mathrm{smooth}, p_{\max}(x)}^{\mathrm{full}} = C_x \oplus C_{>x}$ have the exact $2 \times 2$ block-triangular decomposition: $$d_{\mathrm{full}} \;\equiv\; \begin{pmatrix} d_x & \mathcal{T}_x^* \\ 0 & d_{>x} \end{pmatrix}, \qquad d_{\mathrm{full}}^* \;\equiv\; \begin{pmatrix} d_x^* & 0 \\ \mathcal{T}_x & d_{>x}^* \end{pmatrix}, \qquad \mathcal{T}_x \;:=\; Q_{>x}(w_x \wedge \cdot)\big|_{C_x} \;\equiv\; Q_{>x} d_{\mathrm{full}}^*\big|_{C_x}. \tag{1}$$ Consequently, the interior Laplacian $L_x = d_x d_x^* + d_x^* d_x$ on $C_x$ and the exterior Laplacian $L_{>x} = d_{>x} d_{>x}^* + d_{>x}^* d_{>x}$ on $C_{>x}$ satisfy the dual Cartan overflow identities: $$L_x \;\equiv\; S(x)\, I_{C_x} \;-\; \mathcal{T}_x^* \mathcal{T}_x, \qquad L_{>x} \;\equiv\; S(x)\, I_{C_{>x}} \;-\; \mathcal{K}_x, \qquad \mathcal{K}_x \;:=\; \mathcal{T}_x \mathcal{T}_x^*, \tag{1a}$$ where $\mathcal{K}_x = \mathcal{T}_x \mathcal{T}_x^*$ is supported strictly on the finite overflow boundary shell $C_{>x}^{\mathrm{ovf}} := \operatorname{span}\{e_u : u \in \mathcal{S}_{p_{\max}(x)},\; u/P^+(u) \le x < u\} \subset C_{(x, x p_{\max}(x)]}$ with closed-form entries $(\mathcal{K}_x)_{u, u} = \sum_{q \mid u, u/q \le x} a_q^2$ and $(\mathcal{K}_x)_{m q_1, m q_2} = (-1)^{I(q_1, m q_1) + I(q_2, m q_2)} a_{q_1} a_{q_2}$ ($m = \gcd(u, v) \le x$).

(b) Standard-Basis Diagonal, LCM-Overflow Off-Diagonal Entries & Contiguous-Suffix Parity Cancellation of $L_x$: In the standard basis $\{e_m\}_{m \le x\text{ sqfree}}$, the matrix entries $(L_x)_{m, n} = \langle e_m, L_x e_n \rangle$ are given in closed form by: $$(L_x)_{m, m} \;\equiv\; S(x/m) \;+\; \sum_{\substack{p \mid m \\ p > x/m}} a_p^2 \;=\; S(x) \;-\; \sum_{\substack{p \le x,\; p \nmid m \\ m p > x}} a_p^2, \tag{2}$$ $$(L_x)_{m, n} \;\equiv\; (-1)^{I(q_1, m) + I(q_2, n)}\, a_{q_1} a_{q_2}\, \mathbf{1}_{\{m = d q_1,\, n = d q_2,\, \operatorname{lcm}(m, n) = d q_1 q_2 > x\}} \qquad (m \ne n), \tag{3}$$ where $d = \gcd(m, n)$. In particular, $(L_x)_{m, n} \equiv 0$ for all $m \ne n$ with $\operatorname{lcm}(m, n) \le x$, and every monomial $e_m$ whose prime factors contain all primes in $(x/m, x]$ satisfies $\mathcal{T}_x(e_m) = 0$ and is an exact eigenvector $L_x(e_m) = S(x) e_m$. Moreover, for any squarefree overflow integer $u = p_1 \cdots p_k > x$ ($p_1 < \dots < p_k \le x$), the divisors $p_j \mid u$ with $u/p_j \le x$ form a contiguous suffix of length $N_{\mathrm{ovf}}(u; x) := |\{p \mid u : u/p \le x\}|$ whose alternating signs $\sum_{j = k - N_{\mathrm{ovf}}(u; x) + 1}^k (-1)^{j-1} = -\mu(u)\mathbf{1}_{\{N_{\mathrm{ovf}}(u; x)\text{ odd}\}}$ cancel in adjacent pairs, so for any constant degree fugacity $\gamma \in \mathbb{C}$: $$\bigl(\mathcal{T}_x\psi_x(\gamma a)\bigr)_u \;\equiv\; (-\gamma)^{\omega(u)-1}\,\mathbf{1}_{\{N_{\mathrm{ovf}}(u; x)\text{ odd}\}}\prod_{p \mid u} a_p, \quad \bigl(L_x\mu_x(a)\bigr)_m \;\equiv\; \Lambda_x(m)\,\bigl(\mu_x(a)\bigr)_m, \quad \bigl\langle \mu_x(a), \, L_x\,\mu_x(a) \bigr\rangle \;\equiv\; S(x)\,\bigl\|\mu_x(a)\bigr\|_2^2 \;-\; \sum_{\substack{u > x\text{ sqfree}, \, P^+(u) \le x \\ N_{\mathrm{ovf}}(u; x)\text{ odd}}} \prod_{p \mid u} a_p^2 \quad (\forall x \ge 2), \tag{3'}$$ where $\Lambda_x(m) := S(x) - \sum_{p \le x, p \nmid m, mp > x, N_{\mathrm{ovf}}(mp; x)\text{ odd}} (-1)^{|\{q \mid m : q > p\}|} a_p^2$ satisfies $\bigl(L_x\mathbf{1}_x(a)\bigr)_m \equiv \Lambda_x(m)\bigl(\mathbf{1}_x(a)\bigr)_m$ and $\Lambda_x(m) \equiv S(x)$ whenever $\omega(m) \le 1$ or $m \le x/p_{\max}(x)$, while $\mathcal{T}_x\mu_x(a)$ is entrywise non-negative on $C_{>x}^{\mathrm{ovf}}$ with $(\mathcal{T}_x\mathbf{1}_x(a))_u \equiv -\mu(u)(\mathcal{T}_x\mu_x(a))_u$.

(c) Overflow Intertwining, Three-Block Hodge Split & Unitary Harmonic Isomorphism $H_x^k \cong H_{>x}^{k+1}$: The overflow operator satisfies the exact intertwining identities: $$\mathcal{T}_x d_x^* \;\equiv\; -\,d_{>x}^* \mathcal{T}_x, \qquad d_{>x} \mathcal{T}_x \;\equiv\; -\,\mathcal{T}_x d_x, \qquad d_x \mathcal{T}_x^* \;\equiv\; -\,\mathcal{T}_x^* d_{>x}. \tag{3a}$$ For all $x \ge 2$, $0 \preceq L_x \preceq S(x) I_{C_x}$, $L_x|_{C_{x/p_{\max}(x)}} \equiv S(x) I_{C_{x/p_{\max}(x)}}$ (since $m \le x/p_{\max}(x) \implies mp \le m p_{\max}(x) \le x$ for all primes $p \le x$, whereas at $x = 10, p_{\max}(10) = 7, a_p = 1/p$, $e_1 \in C_{10/7}$ has $(L_{10})_{1, 1} = S(10)$ while $e_2 \notin C_{10/7}$ has $(L_{10})_{2, 2} = S(5) = \frac{361}{900} \ne S(10)$), the non-zero spectra of $\mathcal{T}_x^* \mathcal{T}_x$ on $C_x$ and $\mathcal{K}_x = \mathcal{T}_x \mathcal{T}_x^*$ on $C_{>x}^{\mathrm{ovf}}$ are identical, and all three orthogonal Hodge subspaces $B_x = Z_{x/2}$, $H_x = \ker(L_x)$, and $B_x^* = d_x^*(Z_{x/2})$ are invariant under $\mathcal{T}_x^* \mathcal{T}_x$, with $0 \preceq \mathcal{T}_x^* \mathcal{T}_x|_{Z_{x/2}} \equiv S(x)I_{Z_{x/2}} - d_x d_x^*|_{Z_{x/2}} \preceq (S(x) - a_2^2) I_{Z_{x/2}}$. Finally, $\Phi \in H_x = \ker(L_x) \iff \mathcal{T}_x^* \mathcal{T}_x \Phi = S(x)\Phi$ and $\Psi \in H_{>x} = \ker(L_{>x}) \iff \mathcal{K}_x \Psi = S(x)\Psi$, so the scaled overflow map $$\mathcal{U}_x \;:=\; \frac{1}{\sqrt{S(x)}}\,\mathcal{T}_x\big|_{H_x} \;:\; H_x^k \;\stackrel{\sim}{\longrightarrow}\; H_{>x}^{k+1} \;\subset\; C_{>x}^{\mathrm{ovf}}, \qquad \mathcal{U}_x^{-1} \;=\; \mathcal{U}_x^* \;=\; \frac{1}{\sqrt{S(x)}}\,\mathcal{T}_x^*\big|_{H_{>x}^{k+1}}, \tag{3b}$$ is an exact unitary isomorphism with harmonic reconstruction $d_{\mathrm{full}}(\mathcal{T}_x \Phi) \equiv S(x)\,\Phi$ ($\forall \Phi \in H_x$).

Proof.

• (a): For any $m \le x$ and $p \mid m$, $m/p \le x/2 < x$, so $Q_{>x} d_{\mathrm{full}} Q_x = 0$, giving the $2 \times 2$ block decomposition (1). Substituting (1) into $d_{\mathrm{full}}d_{\mathrm{full}}^* + d_{\mathrm{full}}^* d_{\mathrm{full}} = S(x)I$ gives $(d_x d_x^* + d_x^* d_x) + \mathcal{T}_x^* \mathcal{T}_x = S(x)I_{C_x}$ in the $(C_x, C_x)$ block and $(d_{>x} d_{>x}^* + d_{>x}^* d_{>x}) + \mathcal{T}_x \mathcal{T}_x^* = S(x)I_{C_{>x}}$ in the $(C_{>x}, C_{>x})$ block, proving (1a). Since $\mathcal{T}_x^*(e_u) = \sum_{q \mid u, u/q \le x} (-1)^{I(q, u)} a_q e_{u/q}$, taking inner products $\langle \mathcal{T}_x^* e_u, \mathcal{T}_x^* e_v \rangle_{C_x}$ gives the closed-form entries of $\mathcal{K}_x$.
• (b): Expanding $\|\mathcal{T}_x e_m\|^2 = \sum_{p \le x, p \nmid m, mp > x} a_p^2$ gives (2), and $\langle \mathcal{T}_x e_m, \mathcal{T}_x e_n \rangle$ ($m \ne n$) is nonzero iff $m q_2 = n q_1 > x$ ($m = d q_1, n = d q_2$), where $(-1)^{I(q_2, d q_1 q_2) + I(q_1, d q_1 q_2)} = -(-1)^{I(q_1, d q_1) + I(q_2, d q_2)}$ gives (3). Since $u/p_1 > \dots > u/p_k$, $\{j : u/p_j \le x\} = \{k - N_{\mathrm{ovf}}(u; x) + 1, \dots, k\}$, so $(\mathcal{T}_x\psi_x(\gamma a))_u = \gamma^{k-1}(\prod_{p \mid u} a_p)\sum_{j=k-N_{\mathrm{ovf}}(u; x)+1}^k (-1)^{j-1} = (-\gamma)^{k-1}\mathbf{1}_{\{N_{\mathrm{ovf}}(u; x)\text{ odd}\}}\prod_{p \mid u} a_p$, and applying $\mathcal{T}_x^*$ with $\mu(m)(-1)^{I(p, mp)} = (-1)^{|\{q \mid m : q > p\}|}$ yields $\Lambda_x(m)$ and (3').
• (c): The off-diagonal blocks of $d_{\mathrm{full}}^2 = 0$, $(d_{\mathrm{full}}^*)^2 = 0$, and $d_{\mathrm{full}}d_{\mathrm{full}}^* + d_{\mathrm{full}}^* d_{\mathrm{full}} = S(x)I$ yield (3a). By singular value decomposition of $\mathcal{T}_x$, $\mathcal{T}_x^* \mathcal{T}_x$ and $\mathcal{K}_x = \mathcal{T}_x \mathcal{T}_x^*$ share the same non-zero eigenvalues, and the eigenvalue-$S(x)$ eigenspaces $H_x = \ker(S(x)I - \mathcal{T}_x^* \mathcal{T}_x)$ and $H_{>x} = \ker(S(x)I - \mathcal{T}_x \mathcal{T}_x^*)$ are mapped unitarily onto each other by $\mathcal{U}_x = S(x)^{-1/2}\mathcal{T}_x$. $\blacksquare$

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3. All-Scale Divisor-Triangular Cholesky Factorization on the Complex Polydisc

For any sequence of real or complex prime overlap parameters $\{\rho_p\}_{p \ge 2} \subset \mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$, define the Hermitian Multiplicative Overlap / GCD Metric $G_{x, \rho}$ on $C_x$ by: $$(G_{x, \rho})_{m, n} \;:=\; \langle e_m,\, G_{x, \rho} e_n \rangle \;=\; \left(\prod_{p \mid (m \setminus n)} \overline{\rho_p}\right)\left(\prod_{p \mid (n \setminus m)} \rho_p\right) \qquad (m, n \le x\text{ sqfree}), \tag{4}$$ which reduces for real $\rho_p \in (-1, 1)$ to $\prod_{p \mid (m \triangle n)} \rho_p = \frac{\rho(m)\rho(n)}{\rho(\gcd(m, n))^2}$ where $m \triangle n = \frac{\operatorname{lcm}(m, n)}{\gcd(m, n)}$—a diagonally scaled meet matrix on the lower-closed divisor poset $\mathcal{S}_{\le x}$ [7, 8].

Theorem 2 (All-Scale Divisor-Triangular Cholesky Factorization, LCM-Interior Inverse & Overflow Schur Complement). (a) Abelian Hasse-Exponential Factorization, Square-Root-Free $LDL^*$ Decomposition & LCM-Interior Inverse: Let $\partial_p(e_m) := \mathbf{1}_{\{p \mid m\}} e_{m/p}$ ($p \le x$) be the commuting 2-nilpotent Hasse prime-divisor derivatives ($\partial_p\partial_q = \partial_q\partial_p$, $\partial_p^2 = 0$), which generate the abelian unitriangular divisor-zeta group $\mathcal{U}_{x, \rho} := \exp\bigl(\sum_{p \le x}\rho_p\partial_p\bigr) = \prod_{p \le x}(I_{C_x} + \rho_p\partial_p)$ satisfying $(\mathcal{U}_{x, \rho})_{d, m} = \mathbf{1}_{\{d \mid m\}}\prod_{p \mid (m/d)}\rho_p$, $\mathcal{U}_{x, \rho^{(1)}}\mathcal{U}_{x, \rho^{(2)}} \equiv \mathcal{U}_{x, \rho^{(1)}+\rho^{(2)}}$, and $\mathcal{U}_{x, \rho}^{-1} \equiv \mathcal{U}_{x, -\rho} \equiv U(\pi)\mathcal{U}_{x, \rho}U(\pi)^{-1}$. Setting $D_{x, \rho} := \operatorname{diag}\bigl(\prod_{p \mid m}\sqrt{1 - |\rho_p|^2}\bigr)_{m \le x\text{ sqfree}}$, define the divisor-triangular Cholesky operator $\mathcal{C}_{x, \rho} := D_{x, \rho}\mathcal{U}_{x, \rho} : C_x \to C_x$ by: $$\mathcal{C}_{x, \rho}(e_m) \;:=\; \sum_{d \mid m} \left(\prod_{p \mid d} \sqrt{1 - |\rho_p|^2}\right)\left(\prod_{p \mid (m/d)} \rho_p\right) e_d. \tag{5}$$ Then for every additive cutoff $x \ge 1$ and every $\rho \in \mathbb{D}^{\pi(x)}$, $\mathcal{C}_{x, \rho}$ preserves $C_x$, satisfies the Cholesky and square-root-free unitriangular $LDL^*$ factorizations $G_{x, \rho} \equiv \mathcal{C}_{x, \rho}^* \mathcal{C}_{x, \rho} \equiv \mathcal{U}_{x, \rho}^* D_{x, \rho}^2 \mathcal{U}_{x, \rho}$ [7, 8], intertwines every squarefree Hasse divisor derivative $\partial_k = \prod_{p \mid k}\partial_p$ ($k \le x$) via $\partial_k \mathcal{C}_{x, \rho} \equiv \bigl(\prod_{p \mid k}\sqrt{1 - |\rho_p|^2}\bigr)\mathcal{C}_{x, \rho} \partial_k$, and has exact divisor-triangular inverse $\mathcal{C}_{x, \rho}^{-1} \equiv \mathcal{U}_{x, -\rho} D_{x, \rho}^{-1}$ on $C_x$; moreover, for $\rho \in (\mathbb{D}\setminus\{0\})^{\pi(x)}$ with Beltrami–Poincaré half-angle parameter $\alpha_p(\rho_p) := \frac{\overline{\rho_p}}{1 + \sqrt{1 - |\rho_p|^2}} = \frac{1 - \sqrt{1 - |\rho_p|^2}}{\rho_p} \in \mathbb{D}$ ($\alpha_\rho(k) := \prod_{p \mid k}\alpha_p(\rho_p)$), the divisor-poset Möbius–Beltrami similarity $(\mathcal{J}_{x, \rho})_{k, d} := \mathbf{1}_{\{k \mid d\}}\mu(k)\alpha_\rho(k)$ with inverse $(\mathcal{J}_{x, \rho}^{-1})_{d, m} \equiv \mathbf{1}_{\{d \mid m\}}\frac{\mu(d)}{\alpha_\rho(m)}$ diagonalizes $\mathcal{C}_{x, \rho} \equiv \mathcal{J}_{x, \rho} D_{x, \rho} \mathcal{J}_{x, \rho}^{-1} \equiv \sum_{d \le x\text{ sqfree}}\bigl(\prod_{p \mid d}\sqrt{1 - |\rho_p|^2}\bigr)\bigl|\Phi_d^{(\rho)}\bigr\rangle\bigl\langle \widetilde{\Psi}_{x, d}^{(\rho)}\bigr|$ at every finite cutoff $x \ge 1$ across the biorthogonal right/left eigenbases $\Phi_d^{(\rho)} = \sum_{k \mid d}\mu(k)\alpha_\rho(k)e_k$ and $\widetilde{\Psi}_{x, d}^{(\rho)} = \mu(d)\sum_{d \mid m \le x}\overline{\alpha_\rho(m)}^{-1} e_m$ ($\langle \widetilde{\Psi}_{x, d}^{(\rho)}, \Phi_{d'}^{(\rho)} \rangle \equiv \delta_{d, d'}$), yielding closed-form complex powers $\mathcal{C}_{x, \rho}^s = \mathcal{J}_{x, \rho} D_{x, \rho}^s \mathcal{J}_{x, \rho}^{-1}$ ($s \in \mathbb{C}$) and infinitesimal generator $\log\mathcal{C}_{x, \rho} \equiv \log D_{x, \rho} - \sum_{q \le x}\frac{\log(1 - |\rho_q|^2)}{2(1 - \sqrt{1 - |\rho_q|^2})}\rho_q\partial_q$: $$\mathcal{C}_{x, \rho}^{-1}(e_m) \;\equiv\; \left(\prod_{p \mid m} \frac{1}{\sqrt{1 - |\rho_p|^2}}\right) \sum_{d \mid m} \mu(m/d)\left(\prod_{p \mid (m/d)} \rho_p\right) e_d, \quad \bigl(\mathcal{C}_{x, \rho}^s\bigr)_{k, m} \;\equiv\; \mathbf{1}_{\{k \mid m\}}\left(\prod_{p \mid k}\bigl(1 - |\rho_p|^2\bigr)^{s/2}\right)\prod_{q \mid (m/k)}\left(\rho_q\,\frac{1 - (1 - |\rho_q|^2)^{s/2}}{1 - \sqrt{1 - |\rho_q|^2}}\right), \quad \det(G_{x, \rho}) \;\equiv\; \prod_{p \le x} \bigl(1 - |\rho_p|^2\bigr)^{N_p(x)}, \tag{6}$$ where $N_p(x) := |\{m \le x\text{ sqfree} : p \mid m\}|$. Consequently, as in classical meet-matrix inversion [7, 8], $G_{x, \rho}^{-1} \equiv \mathcal{C}_{x, \rho}^{-1}(\mathcal{C}_{x, \rho}^*)^{-1}$ factors through its diagonal entries at $\ell := \operatorname{lcm}(m, n)$, and for any multiplicative function $\beta(m) = \prod_{p \mid m}\beta_p$ the diagonal sum collapses via $k = m r \le x$ into a single multiplicative sum: $$(G_{x, \rho}^{-1})_{m, n} \;\equiv\; \mathbf{1}_{\{\ell \le x\}}\,(G_{x, -\rho})_{m, n}\,(G_{x, \rho}^{-1})_{\ell, \ell}, \qquad (G_{x, \rho}^{-1})_{\ell, \ell} \;\equiv\; \prod_{p \mid \ell}\frac{1}{1 - |\rho_p|^2}\sum_{\substack{r \le x/\ell\text{ sqfree} \\ \gcd(r, \ell) = 1}} \prod_{p \mid r} \frac{|\rho_p|^2}{1 - |\rho_p|^2}, \qquad \sum_{m \le x\text{ sqfree}}\beta(m)\,(G_{x, \rho}^{-1})_{m, m} \;\equiv\; \sum_{k \le x\text{ sqfree}}\prod_{p \mid k}\frac{\beta_p + |\rho_p|^2}{1 - |\rho_p|^2}, \tag{6a}$$ so $\beta_p = z \in \mathbb{C}$ gives the degree-graded trace polynomial $\sum_{d \ge 0} z^d\operatorname{Tr}(G_{x, \rho}^{-1}|_{C_x^d}) \equiv \sum_{k \le x\text{ sqfree}}\prod_{p \mid k}\frac{z+|\rho_p|^2}{1-|\rho_p|^2}$ unifying Selberg's denominator $(G_{x, \rho}^{-1})_{1, 1}$ ($z = 0$), $\operatorname{Tr}(G_{x, \rho}^{-1})$ ($z = 1$), and the Polydisc-Invariant Mertens Supertrace Identity $\operatorname{STr}(G_{x, \rho}^{-1}) := \operatorname{Tr}(U(\pi)G_{x, \rho}^{-1}) \equiv \sum_{k \le x}\mu(k) = M(x)$ ($z = -1$, $\forall \rho \in \mathbb{D}^{\pi(x)}$, whose $\beta_p = -1 + \varepsilon w_p$ derivative yields $\sum_{m \le x\text{ sqfree}}\mu(m)(\sum_{p \mid m} w_p)(G_{x, \rho}^{-1})_{m, m} \equiv \sum_{k \le x\text{ sqfree}}\mu(k)\sum_{p \mid k}\frac{w_p}{1-|\rho_p|^2}$), while $\beta_p = -|\rho_p|^2$ gives $\sum_{m \le x\text{ sqfree}}\mu(m)(\prod_{p \mid m}|\rho_p|^2)(G_{x, \rho}^{-1})_{m, m} \equiv 1$. Moreover, on $C_{\mathrm{smooth}, x}^{\mathrm{full}} = C_x \oplus C_{>x}$ where $\mathcal{C}_{\mathrm{full}, \rho}^{-1} = \begin{pmatrix} \mathcal{C}_{x, \rho}^{-1} & \mathcal{M}_{x, \rho} \\ 0 & \mathcal{C}_{>x, \rho}^{-1} \end{pmatrix}$ has upper-right block $(\mathcal{M}_{x, \rho})_{m, d} \equiv \mathbf{1}_{\{m \mid d\}}\bigl(\prod_{p \mid d}\frac{1}{\sqrt{1 - |\rho_p|^2}}\bigr)\mu(d/m)\prod_{q \mid (d/m)}\rho_q$ ($m \le x < d$) and $G_{\mathrm{full}, \rho}^{-1} \equiv \Delta_x(\rho)^{-1} G_{\mathrm{full}, -\rho}$ ($\Delta_x(\rho) := \prod_{p \le x}(1 - |\rho_p|^2)$), the global inverse satisfies the positive-semidefinite overflow Schur complement and Dual-Sign Parity Reciprocity Identity ($U(\pi)e_m = \mu(m)e_m$): $$\bigl(G_{\mathrm{full}, \rho}^{-1}\bigr)_{C_x, C_x} \;\equiv\; G_{x, \rho}^{-1} + \mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^* \;\equiv\; \frac{1}{\Delta_x(\rho)}\, G_{x, -\rho} \;\equiv\; \frac{1}{\Delta_x(\rho)}\, U(\pi)\,G_{x, \rho}\,U(\pi)^{-1}, \qquad \operatorname{STr}\bigl(\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^*\bigr) \;\equiv\; \left(\frac{1}{\Delta_x(\rho)} - 1\right) M(x), \tag{6b}$$ with $P_{C_x^k}(G_{x, \rho}^{-1} + \mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^*)P_{C_x^k} \equiv \Delta_x(\rho)^{-1}P_{C_x^k} G_{x, \rho} P_{C_x^k}$ for every degree $k \ge 0$. In particular, under the independent prime Bernoulli product measure $\mathbb{P}_{x, \rho}(\mathbf{d} = d) := \Delta_x(\rho)\prod_{p \mid d}\frac{|\rho_p|^2}{1-|\rho_p|^2}$ ($\mathbb{P}_{x, \rho}(p \mid \mathbf{d}) = |\rho_p|^2$) on $\mathcal{S}_{P(x)} = \{d\text{ sqfree} : P^+(d) \le x\}$, every vector $w = \sum_{m \le x\text{ sqfree}} f_m(\prod_{p \mid m}\overline{\rho_p})e_m \in C_x$ satisfies the Divisor-Sum $L^2(\mathbb{P}_{x, \rho})$ Isometry $\|w\|_{G_{x, \rho}}^2 \equiv \mathbb{E}_{x, \rho}\bigl[\bigl|\sum_{m \mid \mathbf{d}, m \le x} f_m\bigr|^2\bigr]$ (projectively invariant under adding primes $q > x$) and $(G_{x, \rho}^{-1})_{\ell, \ell} \equiv \Delta_x(\rho)^{-1}\mathbb{P}_{x, \rho}(\mathbf{d} \le x \mid \ell \mid \mathbf{d})$ ($\forall \ell \le x\text{ sqfree}$), so for any multiplicative twist $\psi_x(\gamma\overline{\rho}) := \sum_{m \le x\text{ sqfree}}(\prod_{p \mid m}\gamma_p\overline{\rho_p})e_m$ ($\gamma_p \in \mathbb{C}$), $(\mathcal{M}_{x, -\rho}^*\psi_x(\gamma\overline{\rho}))_d \equiv \bigl(\prod_{p \mid d}\frac{\overline{\rho_p}}{\sqrt{1 - |\rho_p|^2}}\bigr)\sum_{m \mid d, m \le x}\prod_{p \mid m}\gamma_p$ ($d > x$), and evaluating (6b) at $-\rho$ yields the Universal Finite-$x$ GCD Norm Decomposition (unifying Dress–Iwaniec–Tenenbaum's $M(d, x) = \sum_{m \mid d, m \le x}\mu(m)$ at $\gamma_p = -1$ and Hooley's truncated divisor function $\tau_{\le x}(d) = \sum_{m \mid d, m \le x} 1$ at $\gamma_p = +1$): $$\bigl\|\psi_x(\gamma \overline{\rho})\bigr\|_{G_{x, \rho}}^2 \;\equiv\; \Delta_x(\rho)\left[\sum_{m \le x\text{ sqfree}}\prod_{p \mid m}\frac{|1 + \gamma_p|^2 |\rho_p|^2}{1 - |\rho_p|^2} \;+\; \sum_{\substack{d > x\text{ sqfree} \\ P^+(d) \le x}}\left(\prod_{p \mid d}\frac{|\rho_p|^2}{1 - |\rho_p|^2}\right)\left|\sum_{\substack{m \mid d \\ m \le x}}\prod_{p \mid m}\gamma_p\right|^2\right], \tag{6c}$$ which at $\gamma_p = -1$ ($\psi_x(-\overline{\rho}) = \mu_x(\overline{\rho})$) collapses the interior sum to $1$ ($m = 1$) and, combined with $w_x^{\mathrm{Sel}} := \frac{G_{x, \rho}^{-1}e_1}{(G_{x, \rho}^{-1})_{1, 1}}$ ($(w_x^{\mathrm{Sel}})_m \equiv \frac{(G_{x, \rho}^{-1})_{m, m}}{(G_{x, \rho}^{-1})_{1, 1}}(\mu_x(\overline{\rho}))_m = \frac{\mathbb{P}_{x, \rho}(\mathbf{d} \le x \mid m \mid \mathbf{d})}{\mathbb{P}_{x, \rho}(\mathbf{d} \le x)}(\mu_x(\overline{\rho}))_m$, satisfying $0 < \frac{(G_{x, \rho}^{-1})_{m, m}}{(G_{x, \rho}^{-1})_{1, 1}} \le 1$ by Harris–FKG monotonicity and $\sum_{m \mid d, m \le x}\mu(m)(G_{x, \rho}^{-1})_{m, m} \equiv \sum_{k \le x\text{ sqfree}}\mu(\gcd(d, k))\prod_{q \mid (k \setminus d)}\frac{|\rho_q|^2}{1-|\rho_q|^2}$), gives the Three-Tier Selberg–Cholesky Variational Hierarchy ($\min_{w \in C_x, w_1 = 1}\|w\|_{G_{x, \rho}}^2 \equiv \mathbb{P}_{x, \rho}(\mathbf{d} = 1 \mid \mathbf{d} \le x)$): $$\prod_{p \le x}\bigl(1 - |\rho_p|^2\bigr) \;\le\; \min_{\substack{w \in C_x \\ w_1 = 1}}\|w\|_{G_{x, \rho}}^2 \;=\; \left(\sum_{r \le x\text{ sqfree}}\prod_{p \mid r}\frac{|\rho_p|^2}{1 - |\rho_p|^2}\right)^{-1} \;\le\; \bigl\|\mu_x(\overline{\rho})\bigr\|_{G_{x, \rho}}^2 \;\equiv\; \Delta_x(\rho)\left[1 \;+\; \sum_{\substack{d > x\text{ sqfree} \\ P^+(d) \le x}}\left(\prod_{p \mid d}\frac{|\rho_p|^2}{1 - |\rho_p|^2}\right)\left(\sum_{\substack{m \mid d \\ m \le x}}\mu(m)\right)^2\right], \tag{6d}$$ with exact gap $\|\mu_x(\overline{\rho})\|_{G_{x, \rho}}^2 - (G_{x, \rho}^{-1})_{1, 1}^{-1} \equiv \|\mathcal{C}_{x, \rho}(\mu_x(\overline{\rho}) - w_x^{\mathrm{Sel}})\|_2^2 \ge 0$, where for every overflow integer $d > x$ ($P^+(d) \le x$), pairing $m'$ and $p^* m'$ at $p^* = P^\pm(d)$ expresses $M(d, x)$ and Hooley's strip count directly through the Cartan overflow operator $\mathcal{T}_x$ via $(-1)^{I(P^+(d), m' P^+(d))} = \mu(m')$ and $(-1)^{I(P^-(d), m' P^-(d))} = +1$ as $M(d, x) \equiv \sum_{m' \mid (d/P^+(d)), \, x/P^+(d) < m' \le x}\mu(m') \equiv a_{P^+(d)}^{-1}\sum_{m' \mid (d/P^+(d)), \, m' \le x}(\mathcal{T}_x)_{m' P^+(d), \, m'}$ and $\tau_{\le x}(d) - 2\tau_{\le x/P^-(d)}(d/P^-(d)) \equiv a_{P^-(d)}^{-1}\sum_{m' \mid (d/P^-(d)), \, m' \le x}(\mathcal{T}_x)_{m' P^-(d), \, m'}$ (and whenever $d/(P^-(d)P_2^-(d)) \le x$, which holds automatically for all $\omega(d) \in \{2, 3\}$, $M(d, x) \equiv (-1)^{\omega(d)-1}\bigl(N_{\mathrm{ovf}}(d; x) - (\omega(d) - 1)\bigr)$ and $\tau_{\le x}(d) \equiv 2^{\omega(d)} - 1 - (\omega(d) - N_{\mathrm{ovf}}(d; x))$ are linear functions of the Cartan overflow count $N_{\mathrm{ovf}}(d; x)$ of (3')). Moreover, at the Selberg–Helmert overlap $\rho_p^{\mathrm{Sel}} := \frac{a_p}{\sqrt{1 + a_p^2}}$ (where $\frac{(\rho_p^{\mathrm{Sel}})^2}{1 - (\rho_p^{\mathrm{Sel}})^2} \equiv a_p^2$), $(\mathcal{C}_{x, \rho^{\mathrm{Sel}}}^{-1})^* e_1 \equiv \mu_x(a)$ for all $x \ge 1$, so for every subspace $V_x \subseteq C_x$ with $P_{V_x}\mu_x(a) \ne 0$, the subspace-constrained Selberg minimum over $V_{x, \rho^{\mathrm{Sel}}} := \mathcal{C}_{x, \rho^{\mathrm{Sel}}}^{-1}(V_x)$ is the exact reciprocal $\min_{w \in V_{x, \rho^{\mathrm{Sel}}}, w_1 = 1}\|w\|_{G_{x, \rho^{\mathrm{Sel}}}}^2 \equiv \|P_{V_x}\mu_x(a)\|_2^{-2}$. Together with Theorem 1(b), the off-diagonal supports of $L_x$ (and hence of $\mathcal{T}_x^* \mathcal{T}_x = S(x)I_{C_x} - L_x$) and $G_{x, \rho}^{-1}$ are completely disjoint on $C_x \times C_x$ across the entire polydisc $\mathbb{D}^{\pi(x)}$ ($(L_x)_{m, n}(G_{x, \rho}^{-1})_{m, n} \equiv 0$ for all $m \ne n$), so $\operatorname{Tr}(L_x G_{x, \rho}^{-1}) \equiv \sum_{m \le x\text{ sqfree}} (L_x)_{m, m}(G_{x, \rho}^{-1})_{m, m}$ and for any multiplicative weight $\beta(m) = \prod_{q \mid m}\beta_q$, since $(L_x)_{m, m} = S(x) - \|\mathcal{T}_x e_m\|_2^2$, $\sum_{m \le x\text{ sqfree}}\beta(m)(L_x)_{m, m}(G_{x, \rho}^{-1})_{m, m} \equiv S(x)\sum_{k \le x\text{ sqfree}}\prod_{p \mid k}\frac{\beta_p + |\rho_p|^2}{1 - |\rho_p|^2} - \sum_{p \le x}|a_p|^2\sum_{x/p < k \le x\text{ sqfree}, p \nmid k}\bigl(\prod_{q \mid k}\frac{1}{1-|\rho_q|^2}\bigr)\sum_{r \mid k, r < kp/x}\beta(k/r)\prod_{s \mid r}|\rho_s|^2$ (for example, equal to $\frac{1}{4}(1 + \frac{41}{9}) - 0 = \frac{25}{18}$ at $x = 2, a_2 = \frac{1}{2}, \rho_2 = \frac{4}{5}, \beta \equiv 1$). Furthermore, for every divisor-closed approximant subspace $\mathcal{A} \subset C_x$ ($m \in \mathcal{A}, d \mid m \implies d \in \mathcal{A}$, encompassing the sub-scale complex $C_y$, the $h$-smooth complex $C_{x, h\text{-smooth}} := \operatorname{span}\{e_m \in C_x : P^+(m) \le h\}$, and the degree-$K$ sieved complex $C_x^{(\le K)} := \operatorname{span}\{e_m \in C_x : \omega(m) \le K\}$) and every multiplicative cochain $\psi_x(\beta) \in C_x$, the unique $G_{x, \rho}$-best approximant in $\mathcal{A}$ to $G_{x, \rho}^{-1}\psi_x(\beta)$ is $v_{\mathcal{A}}^* = G_{\mathcal{A}, \rho}^{-1}(P_{\mathcal{A}}\psi_x(\beta))$ ($G_{\mathcal{A}, \rho} := P_{\mathcal{A}}G_{x, \rho}P_{\mathcal{A}}$), with exact irreducible order-ideal approximation error: $$\min_{v \in \mathcal{A}}\bigl\|v - G_{x, \rho}^{-1}\psi_x(\beta)\bigr\|_{G_{x, \rho}}^2 \;\equiv\; \bigl\langle \psi_x(\beta), \, \bigl(G_{x, \rho}^{-1} - P_{\mathcal{A}}G_{\mathcal{A}, \rho}^{-1}P_{\mathcal{A}}\bigr)\psi_x(\beta)\bigr\rangle \;\equiv\; \sum_{k \in C_x \setminus \mathcal{A}}\prod_{p \mid k}\frac{|\beta_p - \overline{\rho_p}|^2}{1 - |\rho_p|^2}, \tag{6e}$$ while for every $\alpha \ge 0$ and all $m \in C_x$, the overflow Schur complement $\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^* = \Delta_x(\rho)^{-1}G_{x, -\rho} - G_{x, \rho}^{-1}$ satisfies the prime-factor-cancelling Rankin–Euler-product overflow estimate and phase-transition dichotomy ($C_\alpha(x; \rho) := \prod_{p \le x}\bigl(1 + \frac{|\rho_p|^2 p^\alpha}{1 - |\rho_p|^2}\bigr)$): $$0 \;\le\; \bigl(\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^*\bigr)_{m, m} \;\le\; C_\alpha(x; \rho)\left(\prod_{p \mid m}\frac{1}{1 + |\rho_p|^2(p^\alpha - 1)}\right)\left(\frac{m}{x}\right)^\alpha, \qquad \bigl\|P_{C_y}\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^* P_{C_y}\bigr\|_{\mathrm{op}} \;\le\; \operatorname{Tr}\bigl(P_{C_y}\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^* P_{C_y}\bigr) \;\le\; C_\alpha(x; \rho)\sum_{m \in C_y}\left(\prod_{p \mid m}\frac{1}{1 + |\rho_p|^2(p^\alpha - 1)}\right)\left(\frac{m}{x}\right)^\alpha \;\le\; C_\alpha(x; \rho)\,\frac{y^\alpha + \frac{y^{\alpha + 1}}{\alpha + 1}}{x^\alpha} \quad (\forall 1 \le y \le x), \tag{6f}$$ where: (i) in the weighted $\ell^2$ summable overlap regime $\sup_p|\rho_p| < 1$ and $\sum_p |\rho_p|^2 p^\alpha < \infty$ (a sufficient Rankin condition satisfied for $|\rho_p| \asymp p^{-\sigma}$ with $\sigma > \frac{1}{2}$ and $0 \le \alpha < 2\sigma - 1$), $C_\alpha(\infty; \rho) := \prod_p\bigl(1 + \frac{|\rho_p|^2 p^\alpha}{1 - |\rho_p|^2}\bigr) < \infty$ is finite and independent of $x$, yielding uniform $O((m/x)^\alpha)$ diagonal decay and $\|P_{C_y}\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^* P_{C_y}\|_{\mathrm{op}} \le C_\alpha(\infty; \rho) x^{-\alpha}\bigl(y^\alpha + \frac{y^{\alpha+1}}{\alpha+1}\bigr) \to 0$ as $x \to \infty$ whenever $\alpha > 0$ and $1 \le y = o(x^{\alpha/(\alpha+1)})$ (at $x = 5, y = 1, \alpha = 1, \rho_2 = \rho_3 = \frac{4}{5}, \rho_5 = 0$, $(\mathcal{M}_{5, \rho}\mathcal{M}_{5, \rho}^*)_{1, 1} = \frac{256}{81} \approx 3.1605$ is bounded by the discrete-sum bound $\frac{C_1(5; \rho)}{5} = \frac{779}{135} \approx 5.7704$ and Euler–Maclaurin bound $\frac{3 C_1(5; \rho)}{10} = \frac{2337}{270} \approx 8.6556$, whereas omitting the $y^\alpha$ endpoint from $\sum_{m \le y} m^\alpha$ would falsely give $\frac{779}{270} < \frac{256}{81}$), while even at the critical exponent $\alpha = 2\sigma - 1$ where $C_{2\sigma - 1}(\infty; \rho) = \infty$ (for instance, at the $\sigma = 1$ Selberg–Helmert overlap $|\rho_p|^2 = \frac{1}{1+p^2}$ where $\frac{|\rho_p|^2}{1-|\rho_p|^2} = p^{-2}$ and $\sum_p p|\rho_p|^2 = \infty$), bounding the exterior Dirichlet tail directly yields $(\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^*)_{m, m} \le \frac{2\sigma K_\sigma}{2\sigma - 1}\bigl(\frac{m}{x}\bigr)^{2\sigma - 1}$ and $\operatorname{Tr}(P_{C_y}\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^* P_{C_y}) \le \frac{2\sigma K_\sigma}{2\sigma - 1}\frac{y^{2\sigma-1} + \frac{y^{2\sigma}}{2\sigma}}{x^{2\sigma-1}}$ ($K_\sigma := \prod_p(1 + p^{-2\sigma}) = \frac{\zeta(2\sigma)}{\zeta(4\sigma)}$, giving $(\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^*)_{m, m} \le \frac{30}{\pi^2}\frac{m}{x}$ and $\operatorname{Tr}(P_{C_y}\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^* P_{C_y}) \le \frac{15}{\pi^2}\frac{\lfloor y \rfloor(\lfloor y \rfloor + 1)}{x}$ at $\sigma = 1$); whereas (ii) in the non-summable constant-overlap regime $\rho_p \equiv \rho_0 \in \mathbb{D}\setminus\{0\}$ ($u_0 := \frac{|\rho_0|^2}{1 - |\rho_0|^2} > 0$), $C_\alpha(x; \rho_0)$ grows super-exponentially and for every $0 < |\rho_0| < 1$ and $x \ge 2$ the vacuum compression blows up exponentially as $(\mathcal{M}_{x, \rho_0}\mathcal{M}_{x, \rho_0}^*)_{1, 1} = \sum_{d > x\text{ sqfree}, P^+(d)\le x} u_0^{\omega(d)} \ge \bigl(1 + \min(u_0, 1)\bigr)^{\pi(x)} - |C_x| \ge \bigl(1 + \min(u_0, 1)\bigr)^{\pi(x)} - x \to \infty$ (equal to $(1 - |\rho_0|^2)^{-\pi(x)} - |C_x|$ when $|\rho_0|^2 \le \frac{1}{2}$ and to $2^{\pi(x)} - |C_x|$ when $|\rho_0|^2 \ge \frac{1}{2}$).

(b) Finite-$x$ Cholesky–Mertens Coordinates, Adjoint Affine Cochain Calculus, Single-Sum Intertwined Boundary & Hodge-Block Schur Complement: For any multiplicative cochain $\psi_x(\gamma a) = \sum_{m \le x\text{ sqfree}} (\prod_{p \mid m} \gamma_p a_p) e_m$ ($\gamma_p \in \mathbb{C}$) and any squarefree $d \le x$: $$\langle e_d,\, \mathcal{C}_{x, \rho}\bigl(\psi_x(\gamma a)\bigr) \rangle \;\equiv\; \left(\prod_{p \mid d}\sqrt{1 - |\rho_p|^2}\,\gamma_p a_p\right) \sum_{\substack{n \le x/d\text{ sqfree} \\ \gcd(n, d) = 1}} \prod_{q \mid n} (\rho_q \gamma_q a_q). \tag{7}$$ Because every divisor $d \mid m$ of $m \le x$ satisfies $d \le x$, $\mathcal{U}_{x, \rho}^* = \exp(\sum_{p \le x}\overline{\rho_p}\partial_p^*)$ acts on multiplicative cochains by pure prime-weight translation $\mathcal{U}_{x, \rho}^*\psi_x(\beta) \equiv \psi_x(\beta + \overline{\rho})$, so $\mathcal{C}_{x, \rho}^* = \mathcal{U}_{x, \rho}^* D_{x, \rho}$ and $(\mathcal{C}_{x, \rho}^{-1})^* = D_{x, \rho}^{-1}\mathcal{U}_{x, -\rho}^*$ act by exact prime-wise affine transformations at every finite additive cutoff $x \ge 1$ without truncation overflow: $$\mathcal{C}_{x, \rho}^*\bigl(\psi_x(\beta)\bigr) \;\equiv\; \psi_x\!\left(\overline{\rho} + \sqrt{1 - |\rho|^2}\,\beta\right), \quad \bigl(\mathcal{C}_{x, \rho}^{-1}\bigr)^*\bigl(\psi_x(\beta)\bigr) \;\equiv\; \psi_x\!\left(\frac{\beta - \overline{\rho}}{\sqrt{1 - |\rho|^2}}\right), \quad \bigl(G_{x, \rho}^{-1}\psi_x(\beta)\bigr)_m \;\equiv\; \left(\prod_{p \mid m}\frac{\beta_p - \overline{\rho_p}}{1 - |\rho_p|^2}\right)\sum_{\substack{r \le x/m\text{ sqfree} \\ \gcd(r, m) = 1}}\mu(r)\prod_{q \mid r}\frac{\rho_q(\beta_q - \overline{\rho_q})}{1 - |\rho_q|^2} \quad (\forall x \ge 1), \tag{7a}$$ which diagonalizes $\langle \psi_x(\beta^{(1)}), G_{x, \rho}^{-1}\psi_x(\beta^{(2)}) \rangle = \langle \psi_x(\beta^{(1)}-\overline{\rho}), D_{x, \rho}^{-2}\psi_x(\beta^{(2)}-\overline{\rho}) \rangle$ at every finite $x \ge 1$ and shows that in the Unimodular Hasse Metric $\mathcal{W}_{x, \rho} := \mathcal{U}_{x, \rho}\mathcal{U}_{x, \rho}^*$ ($\det \mathcal{W}_{x, \rho} \equiv 1$ for all $\rho \in \mathbb{C}^{\pi(x)}$, $(\mathcal{W}_{x, \rho})_{m, n} \equiv \mathbf{1}_{\{\ell \le x\}}(G_{x, \rho})_{m, n}\sum_{r \le x/\ell\text{ sqfree}, \gcd(r, \ell)=1}\prod_{q \mid r}|\rho_q|^2$, $\mathcal{W}_{x, \rho}\mu_x(\overline{\rho}) \equiv e_1$) and in the dual lower-Cholesky LCM-interior metric $\breve{G}_{x, \rho} := \mathcal{C}_{x, \rho}\mathcal{C}_{x, \rho}^* = D_{x, \rho}\mathcal{W}_{x, \rho} D_{x, \rho}$ (which satisfies $(L_x)_{m, n}(\breve{G}_{x, \rho})_{m, n} \equiv 0$ for all $m \ne n$, $\sum_{m \le x\text{ sqfree}}\beta(m)(\breve{G}_{x, \rho})_{m, m} \equiv \sum_{k \le x\text{ sqfree}}\prod_{p \mid k}(\beta_p(1-|\rho_p|^2)+|\rho_p|^2)$, and $(\mathcal{C}_{x, \rho}^* w)_1 \equiv w_1$), the untwisted Möbius chain $\mu_x(a)$ is the unique global variational minimizer achieving $\min_{w \in C_x, w_1 = 1}\|w\|_{\breve{G}_{x, \rho^{\mathrm{Sel}}}}^2 = 1$ at $\rho_p^{\mathrm{Sel}} = \frac{a_p}{\sqrt{1 + a_p^2}}$, with exact Pythagoras excess and phase-dispersion identity: $$\bigl\langle \psi_x(\beta^{(1)}), \, G_{x, \rho}^{-1} \psi_x(\beta^{(2)}) \bigr\rangle \;\equiv\; \sum_{m \le x\text{ sqfree}} \prod_{p \mid m} \frac{\bigl(\overline{\beta_p^{(1)}} - \rho_p\bigr)\bigl(\beta_p^{(2)} - \overline{\rho_p}\bigr)}{1 - |\rho_p|^2}, \qquad \bigl\|\mu_x(a p^{-it})\bigr\|_{\breve{G}_{x, \rho^{\mathrm{Sel}}}}^2 - 1 \;=\; \bigl\|\mu_x(a p^{-it}) - \mu_x(a)\bigr\|_{\breve{G}_{x, \rho^{\mathrm{Sel}}}}^2 \;\equiv\; \sum_{2 \le m \le x\text{ sqfree}} \prod_{p \mid m} \frac{4\sin^2\!\bigl(\frac{t\log p}{2}\bigr)\,a_p^2}{1 + a_p^2} \qquad (\forall x \ge 1), \tag{7b}$$ so at $t = 0$, $\mathcal{C}_{x, \rho^{\mathrm{Sel}}}^* \mu_x(a) \equiv e_1 = z_2 \in Z_x^0$ and $\breve{G}_{x, \rho^{\mathrm{Sel}}}\mu_x(a) = \mathcal{C}_{x, \rho^{\mathrm{Sel}}}e_1 \equiv e_1$ ($\langle \mu_x(a), u \rangle_{\breve{G}_{x, \rho^{\mathrm{Sel}}}} \equiv u_1$ for all $u \in C_x$), giving identically zero coexact and $k \ge 1$ cycle energy at every finite cutoff $x \ge 2$. On the full $P$-smooth complex $C_{\mathrm{smooth}, P}^{\mathrm{full}}$, $\mathcal{C}_\rho = \bigotimes_{p \le P} \begin{pmatrix} 1 & \rho_p \\ 0 & \sqrt{1 - |\rho_p|^2} \end{pmatrix}$ maps $\psi_P(\gamma a)$ to the rescaled multiplicative cochain: $$\mathcal{C}_\rho\bigl(\psi_P(\gamma a)\bigr) \;\equiv\; \left[\prod_{p \le P} (1 + \rho_p \gamma_p a_p)\right] \psi_P\bigl(\widetilde{\alpha}(\gamma; \rho)\bigr), \qquad \widetilde{\alpha}_p(\gamma; \rho) \;:=\; \frac{\sqrt{1 - |\rho_p|^2}\,\gamma_p a_p}{1 + \rho_p \gamma_p a_p}, \tag{8}$$ while for any $\gamma \in \mathbb{C}^{\pi(P)}$ and $\rho \in \mathbb{D}^{\pi(P)}$, $\bigl(\mathcal{U}_{P, -\rho}^* d_P \psi_P(\gamma\overline{\rho})\bigr)_m \equiv \bigl(\prod_{t \mid m}\overline{\rho_t}\bigr)\sum_{p \le P}(-1)^{\mathbf{1}_{\{p \mid m\}}} a_p\gamma_p\overline{\rho_p}\bigl(\prod_{q \mid m, q < p}(-1-\gamma_q)\bigr)\bigl(\prod_{r \mid m, r > p}(\gamma_r-1)\bigr)$ ($\forall m \in \mathcal{S}_P$, and at any finite cutoff $x \ge 2$ and $m \le x/2$, all internal terms $p \mid m$ and external terms $p \le x/m$ retain this exact product without truncation while $p \in (x/m, x]$ sums $\mu(m/n)(-1)^{I(p, np)}\gamma(n)$ over $n \mid m, n \le x/p$, with $\bigl((\mathcal{C}_{x, \rho}^{-1})^* d_x u\bigr)_1 \equiv (d_x u)_1 = \sum_{p \le x} a_p u_p$ for all $u \in C_x$ and $\|d_x u\|_{G_{x, \rho}}^2 \equiv |(d_x u)_1|^2$, $\langle d_x u, G_{x, \rho}^{-1} d_x u \rangle \equiv (G_{x, \rho}^{-1})_{1, 1}|(d_x u)_1|^2$ for all $u \in C_x^1$), so for $\beta = \overline{\rho}$ ($\gamma_p = \pm 1$), $\bigl(\mathcal{U}_{P, \mp\overline{\beta}}^* d_P \psi_P(\beta)\bigr)_m$ equals $\sum_{p \le P} a_p\beta_p$ at $m = 1$ and localizes for $m > 1$ to the largest prime factor $P^+(m)$ (at $-\overline{\beta}$, giving $-(-2)^{\omega(m)-1}\beta(m)(a_{P^+(m)}\beta_{P^+(m)} + 2\sum_{r > P^+(m)} a_r\beta_r)$) or smallest prime factor $P^-(m)$ (at $+\overline{\beta}$, giving $2^{\omega(m)-1}\beta(m)(a_{P^-(m)}\beta_{P^-(m)} + 2\sum_{q < P^-(m)} a_q\beta_q)$), and by (6b) the signed and unsigned GCD-metric boundary norms collapse into the prime-order-reversal dual $O(\pi(P))$ polynomial sums: $$\bigl\|d_P\,\mu_P(\beta)\bigr\|_{G_{P, \overline{\beta}}}^2 \;\equiv\; \Delta_P(\beta)\left|\sum_{p \le P} a_p\beta_p\right|^2 \;+\; \sum_{p \le P} |\beta_p|^2\,\left|a_p\beta_p + 2\sum_{p < r \le P} a_r\beta_r\right|^2 \left(\prod_{q < p}\bigl(1 + 3|\beta_q|^2\bigr)\right)\left(\prod_{p < s \le P}\bigl(1 - |\beta_s|^2\bigr)\right), \tag{8'a}$$ $$\bigl\|d_P\,\mathbf{1}_P(\beta)\bigr\|_{G_{P, \overline{\beta}}}^2 \;\equiv\; \Delta_P(\beta)\left|\sum_{p \le P} a_p\beta_p\right|^2 \;+\; \sum_{p \le P} |\beta_p|^2\,\left|a_p\beta_p + 2\sum_{q < p} a_q\beta_q\right|^2 \left(\prod_{q < p}\bigl(1 - |\beta_q|^2\bigr)\right)\left(\prod_{p < s \le P}\bigl(1 + 3|\beta_s|^2\bigr)\right), \tag{8'b}$$ and for the coboundary $d_x^* : C_{x/2} \to C_x$ at every finite additive cutoff $x \ge 2$ and any twist $\gamma \in \mathbb{C}^{\pi(x)}$, $\bigl(\mathcal{U}_{x, -\rho}^* d_x^*\psi_{x/2}(\gamma\overline{\rho})\bigr)_m$ vanishes at $m = 1$ and equals $\bigl(\prod_{t \mid m}\overline{\rho_t}\bigr)\sum_{p \mid m}\frac{a_p}{\overline{\rho_p}}\bigl(\prod_{q \mid m, q < p}(-1-\gamma_q)\bigr)\bigl(\prod_{r \mid m, r > p}(\gamma_r-1)\bigr)$ for all $2 \le m \le x$ without truncation error (specializing at $\gamma_p = +1$ and $\gamma_p = -1$ to $(-2)^{\omega(m)-1}\bigl(\prod_{t \mid m}\overline{\rho_t}\bigr)\frac{a_{P^+(m)}}{\overline{\rho_{P^+(m)}}}$ and $(-2)^{\omega(m)-1}\bigl(\prod_{t \mid m}\overline{\rho_t}\bigr)\frac{a_{P^-(m)}}{\overline{\rho_{P^-(m)}}}$, so $\bigl\langle d_x^*\mathbf{1}_{x/2}(\overline{\rho}), G_{x, \rho}^{-1} d_x^*\mu_{x/2}(\overline{\rho})\bigr\rangle \equiv \sum_{2 \le m \le x\text{ sqfree}} 4^{\omega(m)-1}\bigl(\prod_{t \mid m}\frac{|\rho_t|^2}{1-|\rho_t|^2}\bigr)\frac{\overline{a_{P^+(m)}}a_{P^-(m)}}{\rho_{P^+(m)}\overline{\rho_{P^-(m)}}}$), yielding at every $x \ge 2$ (where $|\beta_p| = |a_p|$ equates both diagonal terms to $\frac{1}{4}\prod_{p \mid m}\frac{4|a_p|^2}{1-|a_p|^2}$, while for all degrees $k, \ell \ge 0$, $\bigl\langle \mu_x^k(\overline{\rho}), G_{x, \rho}^{-1}\mu_x^\ell(\overline{\rho})\bigr\rangle \equiv \sum_{m \le x\text{ sqfree}}\binom{\omega(m)}{k}\binom{\omega(m)}{\ell}\prod_{p \mid m}\frac{|\rho_p|^2}{1-|\rho_p|^2}$ and $(-1)^{k+\ell}\bigl\langle \mu_x^k(\overline{\rho}), G_{x, \rho}\mu_x^\ell(\overline{\rho})\bigr\rangle \equiv \mathbb{E}_{x, \rho}\bigl[\tau_{\le x}^{(k)}(\mathbf{d})\tau_{\le x}^{(\ell)}(\mathbf{d})\bigr] \equiv \Delta_x(\rho)\bigl\langle \mu_x^k(\overline{\rho}), G_{x, \rho}^{-1}\mu_x^\ell(\overline{\rho})\bigr\rangle + \mathbb{E}_{x, \rho}\bigl[\tau_{\le x}^{(k)}(\mathbf{d})\tau_{\le x}^{(\ell)}(\mathbf{d})\mathbf{1}_{\{\mathbf{d}>x\}}\bigr]$ with $\tau_{\le x}^{(k)}(d) := |\{m \mid d : m \le x, \omega(m)=k\}|$, so $\bigl\|\mu_x^{\mathrm{even/odd}}(\overline{\rho})\bigr\|_{G_{x, \rho}}^2 \equiv \mathbb{E}_{x, \rho}\bigl[\bigl(\frac{\tau_{\le x}(\mathbf{d}) \pm M(\mathbf{d}, x)}{2}\bigr)^2\bigr]$ and $\bigl\langle \mu_x^{\mathrm{even}}(\overline{\rho}), G_{x, \rho}\mu_x^{\mathrm{odd}}(\overline{\rho})\bigr\rangle \equiv -\mathbb{E}_{x, \rho}\bigl[\frac{\tau_{\le x}(\mathbf{d})^2 - M(\mathbf{d}, x)^2}{4}\bigr]$) and on $C_{\mathrm{smooth}, P}^{\mathrm{full}}$: $$\bigl\langle d_x^*\psi_{x/2}(\beta), \, G_{x, \pm\overline{\beta}}^{-1}\,d_x^*\psi_{x/2}(\beta)\bigr\rangle \;\equiv\; \sum_{2 \le m \le x\text{ sqfree}}\frac{|a_{P^\pm(m)}|^2}{1 - |\beta_{P^\pm(m)}|^2}\prod_{\substack{q \mid m \\ q \lessgtr P^\pm(m)}}\frac{4|\beta_q|^2}{1 - |\beta_q|^2}, \quad \bigl\langle d_x^*\mu_{x/2}(a e^{-i\theta}), \, G_{x, a e^{i\theta}}^{-1}\,d_x^*\mu_{x/2}(a e^{-i\theta})\bigr\rangle \;\equiv\; \frac{1}{4}\Bigl(\bigl\langle \mu_x(a e^{-i\theta}), \, G_{x, a e^{i\theta}}^{-1}\,\mu_x(a e^{-i\theta})\bigr\rangle - 1\Bigr), \quad \bigl\|d_P^*\mu_P(a e^{-i\theta})\bigr\|_{G_{P, a e^{i\theta}}}^2 \;\equiv\; \frac{1}{4}\left[\prod_{p \le P}\bigl(1 + 3|a_p|^2\bigr) - \prod_{p \le P}\bigl(1 - |a_p|^2\bigr)\right]. \tag{8'c}$$ Moreover, because $\{m \le x\text{ sqfree}\}$ is a downward-closed order ideal under divisibility, every divisor-decreasing operator $A_{\mathrm{full}}$ satisfies $A_{\mathrm{full}} Q_x = Q_x A_{\mathrm{full}} Q_x = A_x Q_x$ and $Q_x A_{\mathrm{full}}^* = A_x^* Q_x$, so every normal-ordered product $B_x^* A_x \equiv Q_x B_{\mathrm{full}}^* A_{\mathrm{full}} Q_x$ has zero truncation overflow at all $x \ge 1$ (in particular, $d_x^* d_x \equiv Q_x d_{\mathrm{full}}^* d_{\mathrm{full}} Q_x$, $(\breve{G}_{x, \rho})^{-1} \equiv Q_x(\breve{G}_{\mathrm{full}, \rho})^{-1}Q_x$, $d_{x, \rho}^* d_{x, \rho} \equiv Q_x d_{\mathrm{full}, \rho}^* d_{\mathrm{full}, \rho} Q_x$, and $\widetilde{d}_{x, \rho}^*\widetilde{d}_{x, \rho} \equiv Q_x \widetilde{d}_{\mathrm{full}, \rho}^*\widetilde{d}_{\mathrm{full}, \rho} Q_x$). Writing $d_x = \sum_{p \le x} a_p\widehat{\varepsilon}_p$ with signed Jordan–Wigner components $\widehat{\varepsilon}_p(e_m) := \mathbf{1}_{\{p \mid m\}}(-1)^{I(p, m)}e_{m/p}$, the Hasse derivatives $\partial_q$ satisfy the Chiral Exchange Law $\partial_q\widehat{\varepsilon}_p \equiv +\widehat{\varepsilon}_p\partial_q$ ($p \le q$) and $\partial_q\widehat{\varepsilon}_p \equiv -\widehat{\varepsilon}_p\partial_q$ ($p > q$), so $[d_x, \partial_q](e_m) \equiv 2\,\mathbf{1}_{\{q \mid m\}}\sum_{p \mid m, \, p > q}(-1)^{I(p, m/q)} a_p e_{m/(qp)}$, $(\operatorname{ad}_{\partial_q})^2(d_x) \equiv 0$, $\mathcal{U}_{x, \rho}^{-1}\widehat{\varepsilon}_p\mathcal{U}_{x, \rho} \equiv \widehat{\varepsilon}_p\prod_{q < p}(I_{C_x} + 2\rho_q\partial_q)$, and $[d_x, \log\mathcal{C}_{x, \rho}](e_m) \equiv \frac{1}{2}\sum_{p \mid m}(-1)^{I(p, m/p)}a_p\log(1-|\rho_p|^2)e_{m/p} - \sum_{q < p, \, qp \mid m}(-1)^{I(p, m/(qp))}a_p\frac{\log(1-|\rho_q|^2)}{1-\sqrt{1-|\rho_q|^2}}\rho_q e_{m/(qp)}$ is supported strictly on $\omega(m/d) \in \{1, 2\}$. In the biorthogonal Möbius–Beltrami eigenbases $\{\Phi_d^{(\rho)}, \widetilde{\Psi}_{x, d}^{(\rho)}\}_{d \le x\text{ sqfree}}$ of (6) (where $\widetilde{\Psi}_{x, 1}^{(\rho)} = \mathbf{1}_x(\overline{\alpha(\rho)}^{-1})$ is the unique eigenvalue-$1$ fixed point of $\mathcal{C}_{x, \rho}^*$ and $\lim_{\operatorname{Re}(s)\to+\infty}\mathcal{C}_{x, \rho}^s \equiv |e_1\rangle\langle\mathbf{1}_x(\overline{\alpha(\rho)}^{-1})|$), $J_q(\alpha_q)^{-1} Z_q J_q(\alpha_q) \equiv Z_q + 2\varepsilon_q$ is independent of $\alpha_q$, so $\bigl(\mathcal{J}_{x, \rho}^{-1} d_x \mathcal{J}_{x, \rho}\bigr)_{k, d} \equiv -\mathbf{1}_{\{k \mid d, \, k < d\}}(-1)^{I(p_*(d/k), k)} 2^{\omega(d/k)-1} a_{p_*(d/k)}\alpha_{p_*(d/k)}(\rho_{p_*(d/k)})$ ($p_*(n) := P^+(n)$) and $d_x^*\widetilde{\Psi}_{x, k}^{(\rho)} \equiv -\sum_{k \mid d \le x\text{ sqfree}, \, d > k}(-1)^{I(p_*(d/k), k)} 2^{\omega(d/k)-1}\overline{a_{p_*(d/k)}\alpha_{p_*(d/k)}(\rho_{p_*(d/k)})}\,\widetilde{\Psi}_{x, d}^{(\rho)}$, and for every additive cutoff $x \ge 2$ and every complex power $s \in \mathbb{C}$, $d_{x, \rho}^{(s)} := \mathcal{C}_{x, \rho}^{-s} d_x \mathcal{C}_{x, \rho}^s \equiv \sum_{p \le x} a_p(1 - |\rho_p|^2)^{s/2}\widehat{\varepsilon}_p\prod_{q < p}\bigl(I_{C_x} + 2\rho_q\frac{1 - (1 - |\rho_q|^2)^{s/2}}{1 - \sqrt{1 - |\rho_q|^2}}\partial_q\bigr) : C_x \to C_{x/2}$ acts on $e_m$ by $\sum_{d \mid m, d < m}(-1)^{I(p_*(m/d), d)} a_{p_*(m/d)}(1 - |\rho_{p_*(m/d)}|^2)^{s/2}\bigl(\prod_{q \mid (m/d), q < p_*(m/d)} 2\rho_q\frac{1 - (1 - |\rho_q|^2)^{s/2}}{1 - \sqrt{1 - |\rho_q|^2}}\bigr)e_d$, specializing at $s = +1$ (the Cholesky-intertwined boundary $d_{x, \rho} := \mathcal{C}_{x, \rho}^{-1} d_x \mathcal{C}_{x, \rho}$) and $s = -1$ (Witten's deformed boundary $\widetilde{d}_{x, \rho} := \mathcal{C}_{x, \rho} d_x \mathcal{C}_{x, \rho}^{-1}$) to the dual single-sum divisor formulas: $$d_{x, \rho}(e_m) \;\equiv\; \sum_{\substack{d \mid m \\ d < m}} (-1)^{I(p_*(m/d),\, d)}\, a_{p_*(m/d)}\sqrt{1 - |\rho_{p_*(m/d)}|^2} \left(\prod_{\substack{q \mid (m/d) \\ q < p_*(m/d)}} 2\rho_q\right) e_d, \qquad \widetilde{d}_{x, \rho}(e_m) \;\equiv\; -\sum_{\substack{d \mid m \\ d < m}} (-1)^{I(p_*(m/d),\, d)}\, \mu(m/d)\,\frac{a_{p_*(m/d)}}{\sqrt{1 - |\rho_{p_*(m/d)}|^2}} \left(\prod_{\substack{q \mid (m/d) \\ q < p_*(m/d)}} \frac{2\rho_q}{\sqrt{1 - |\rho_q|^2}}\right) e_d. \tag{8a}$$ Moreover, $\widetilde{d}_{x, \rho}$ determines the $G_{x, \rho}$-orthogonal projector onto the undeformed cycle space $Z_x = \ker(d_x)$ via $\Pi_{Z_x}^{(G_{x, \rho})} = I_{C_x} - G_{x, \rho}^{-1} d_x^*(d_x G_{x, \rho}^{-1} d_x^*|_{Z_{x/2}})^{-1} d_x \equiv \mathcal{C}_{x, \rho}^{-1} P_{\ker(\widetilde{d}_{x, \rho})} \mathcal{C}_{x, \rho}$, where at every finite $x \ge 2$ the cycle Green's operator satisfies $d_x G_{x, \rho}^{-1} d_x^* + (d_x\mathcal{M}_{x, \rho})(d_x\mathcal{M}_{x, \rho})^* \equiv \Delta_x(\rho)^{-1} d_x G_{x, -\rho} d_x^*$, has LCM-interior entries ($\ell := \operatorname{lcm}(n_1, n_2)$, $A_n(m) := \sum_{p \mid (m/n)}(-1)^{I(p, np)}\frac{a_p}{\overline{\rho_p}}$) disjoint in off-diagonal support from $L_x$, and for any multiplicative weight $\beta(n) = \prod_{q \mid n}\beta_q$ satisfies $\sum_{n \le x/2\text{ sqfree}}\beta(n)(d_x G_{x, \rho}^{-1} d_x^*)_{n, n} \equiv \sum_{2 \le m \le x\text{ sqfree}}\prod_{t \mid m}\frac{1}{1-|\rho_t|^2}\bigl[\sum_{p \mid m}|a_p|^2\prod_{q \mid m, q \ne p}(\beta_q+|\rho_q|^2) + 2\sum_{p < q \mid m}\operatorname{Re}(a_p\overline{\rho_p}\overline{a_q}\rho_q)\bigl(\prod_{s \mid m, p < s < q}(|\rho_s|^2-\beta_s)\bigr)\bigl(\prod_{r \mid m, r \notin [p,q]}(\beta_r+|\rho_r|^2)\bigr)\bigr]$ (where at $\beta_q = z \in \mathbb{C}$ the two bracketed terms give the exact diagonal and off-diagonal parts of $\sum_{k \ge 0} z^k\operatorname{Tr}_{C_x^{k+1}}(d_x^* d_x G_{x, \rho}^{-1})$, the latter cancelling the off-diagonal part of $\sum_{k \ge 0} z^k\operatorname{Tr}_{C_x^{k+1}}(d_x d_x^* G_{x, \rho}^{-1})$ identically at every $x \ge 2$ since $(d_x^* d_x + d_x d_x^*)_{n_1, n_2} = -(L_x)_{n_1, n_2} = 0$ on $\operatorname{supp}(G_{x, \rho}^{-1})$), which at $\beta_p = \mp |\rho_p|^2$ localizes to extreme prime factors $P^-(m) < P^+(m)$ in the signed ($\mu(n)$) case ($\sum_{p \le x}\frac{|a_p|^2}{1-|\rho_p|^2} + 2\sum_{m \le x\text{ sqfree}, \omega(m)\ge 2}\frac{\operatorname{Re}(a_{P^-(m)}\overline{\rho_{P^-(m)}}\overline{a_{P^+(m)}}\rho_{P^+(m)})}{(1-|\rho_{P^-(m)}|^2)(1-|\rho_{P^+(m)}|^2)}\prod_{s \mid m, P^-(m) < s < P^+(m)}\frac{2|\rho_s|^2}{1-|\rho_s|^2}$) versus adjacent prime factors $(p_i, p_{i+1})$ in the unsigned ($+1$) case ($\sum_{2 \le m \le x\text{ sqfree}} 2^{\omega(m)-1}(\prod_{p \mid m}\frac{|\rho_p|^2}{1-|\rho_p|^2})\bigl[\frac{|\xi_{P^-(m)}|^2+|\xi_{P^+(m)}|^2}{2} + \frac{1}{2}\sum_{i=1}^{\omega(m)-1}|\xi_{p_i}+\xi_{p_{i+1}}|^2\bigr]$ with $\xi_p := a_p/\rho_p$, reducing at $\rho_p = a_p$ to $\frac{1}{2}\sum_{2 \le m \le x\text{ sqfree}}\prod_{p \mid m}\frac{2a_p^2}{1-a_p^2}$ and $\frac{1}{2}\sum_{2 \le m \le x\text{ sqfree}}(2\omega(m)-1)\prod_{p \mid m}\frac{2a_p^2}{1-a_p^2}$), and at $\beta_p = 0$ ($n_1 = n_2 = 1$) gives via $\mathcal{U}_{x, \rho}^*\mu_x(\overline{\rho}) \equiv \mathcal{C}_{x, \rho}^*\mu_x\bigl(\frac{\overline{\rho}}{\sqrt{1-|\rho|^2}}\bigr) \equiv e_1$ the All-Scale Hasse & Witten Deformed Coboundary Actions $\bigl(\mathcal{U}_{x, -\rho}^* d_x^* \mathcal{U}_{x, \rho}^*\mu_x(\overline{\rho})\bigr)_m \equiv -\bigl(\sum_{p \mid m}\frac{a_p}{\overline{\rho_p}}\bigr)(\mu_x(\overline{\rho}))_m$ and $\bigl(\widetilde{d}_{x, \rho}^*\mu_x\bigl(\frac{\overline{\rho}}{\sqrt{1-|\rho|^2}}\bigr)\bigr)_m \equiv -\bigl(\sum_{p \mid m}\frac{a_p}{\overline{\rho_p}}\bigr)\bigl(\mu_x\bigl(\frac{\overline{\rho}}{\sqrt{1-|\rho|^2}}\bigr)\bigr)_m$ ($\forall m \le x\text{ sqfree}$, reducing at $\rho_p^{\mathrm{Sel}}$ to $-\bigl(\sum_{p \mid m}\sqrt{1+a_p^2}\bigr)(\mu_x(a))_m$ and at $\rho_p = \overline{a_p}$ to the exact prime-divisor number operator $-\omega(m)$ on $\mu_x(a)$ and $\mu_x\bigl(\frac{a}{\sqrt{1-|a|^2}}\bigr)$) together with the Homological Selberg Sieve ($\min_{v \in C_x, (d_x v)_1 = 1}\|v\|_{G_{x, \rho}}^2 = (d_x G_{x, \rho}^{-1} d_x^*)_{1, 1}^{-1} \equiv \|\widetilde{d}_{x, \rho}^*\mu_x(\frac{\overline{\rho}}{\sqrt{1-|\rho|^2}})\|_2^{-2}$, achieved uniquely at $\mathcal{C}_{x, \rho} v_x^{\mathrm{hom}} \equiv \|\widetilde{d}_{x, \rho}^*\mu_x(\frac{\overline{\rho}}{\sqrt{1-|\rho|^2}})\|_2^{-2}\,\widetilde{d}_{x, \rho}^*\mu_x(\frac{\overline{\rho}}{\sqrt{1-|\rho|^2}})$), while on $C_{\mathrm{smooth}, P}^{\mathrm{full}} = Z_P \oplus B_P^*$ (with orthonormal bases $Z_{\mathrm{mat}}, W_{\mathrm{mat}}$ satisfying $d_P^* Z_{\mathrm{mat}} = \sqrt{S(P)}W_{\mathrm{mat}}$ and Hodge Gram blocks $G_{Z, \rho}, G_{B^*, \rho}, G_{Z B^*, \rho}$) the Hodge-block Schur complement and deformed traces satisfy: $$\bigl(d_x G_{x, \rho}^{-1} d_x^*\bigr)_{n_1, n_2} \;\equiv\; \mathbf{1}_{\{\ell \le x\}}(G_{x, -\rho})_{n_1, n_2}\prod_{q \mid \ell}\frac{1}{1 - |\rho_q|^2}\sum_{\substack{k \le x/\ell\text{ sqfree} \\ \gcd(k, \ell) = 1}}\left(\prod_{r \mid k}\frac{|\rho_r|^2}{1 - |\rho_r|^2}\right)\overline{A_{n_1}(\ell k)}\,A_{n_2}(\ell k), \quad G_{Z, \rho} - G_{Z B^*, \rho} G_{B^*, \rho}^{-1} G_{Z B^*, \rho}^* \;\equiv\; \Delta_P(\rho)\,G_{Z, -\rho}^{-1}, \quad Z_{\mathrm{mat}}^*\bigl(d_P G_{P, \rho}^{-1} d_P^*\bigr)Z_{\mathrm{mat}} \;\equiv\; \frac{S(P)}{\Delta_P(\rho)}\,G_{B^*, -\rho}, \tag{8b}$$ $$\operatorname{Tr}\!\left(d_{P, \rho} d_{P, \rho}^*\right) \;\equiv\; 2^{\pi(P)-1} \sum_{q \le P} |a_q|^2\bigl(1 - |\rho_q|^2\bigr) \prod_{p < q} \bigl(1 + 2|\rho_p|^2\bigr), \quad \operatorname{Tr}\!\left(\widetilde{d}_{P, \rho}\widetilde{d}_{P, \rho}^*\right) \;\equiv\; 2^{\pi(P)-1} \sum_{q \le P} \frac{|a_q|^2}{1 - |\rho_q|^2} \prod_{p < q} \frac{1 + |\rho_p|^2}{1 - |\rho_p|^2}, \quad \sum_{k=0}^{\pi(P)-1} z^k\operatorname{Tr}_{C_P^k}\!\left(d_P G_{P, \rho}^{-1} d_P^*\right) \;\equiv\; \frac{(1+z)^{\pi(P)-1}S(P) + 2\sum_{p < q \le P}\operatorname{Re}(a_p\overline{\rho_p}\overline{a_q}\rho_q)(1-z)^{d(p,q)}(1+z)^{\pi(P)-2-d(p,q)}}{\Delta_P(\rho)}, \tag{8c}$$ where $d(p, q) := |\{s\text{ prime} : p < s < q\}|$ (so $z = 1$ isolates nearest-neighbor hops $\operatorname{Tr}(d_P^* d_P G_{P, \rho}^{-1}) = \frac{2^{\pi(P)-1}}{\Delta_P(\rho)}\bigl[S(P) + \sum_{p < P}\operatorname{Re}(a_p\overline{\rho_p}\overline{a_{p^+}}\rho_{p^+})\bigr]$ and $z = -1$ isolates the endpoint bridge $\operatorname{STr}(d_P G_{P, \rho}^{-1} d_P^*) = \frac{2^{\pi(P)-1}}{\Delta_P(\rho)}\operatorname{Re}(a_2\overline{\rho_2}\overline{a_P}\rho_P)$), while at every cutoff $x \ge 2$, $\rho \in \mathbb{D}^{\pi(x)}$, and $s \in \mathbb{C}$ (setting $A_p(s) := a_p(1-|\rho_p|^2)^{s/2}$, $c_q(s) := \rho_q\frac{1-(1-|\rho_q|^2)^{s/2}}{1-\sqrt{1-|\rho_q|^2}}$, $\Lambda_p(s) := \frac{A_p(s)}{2 c_p(s)}$, $g := \gcd(m_1, m_2)$, and $p_i^* := P^+(m_i/g)$ with $p_i^* = 0$ if $m_i = g$ and $p_1^* \le p_2^*$), for any signed multiplicative cochain $\mu_x(\beta) = \psi_x(-\beta) = \sum_{m \le x\text{ sqfree}}\mu(m)\bigl(\prod_{p \mid m}\beta_p\bigr)e_m$ and $z$-smooth coprime Mertens sum $\Sigma_z(Y; k; \gamma) := \sum_{u \le Y\text{ sqfree}, \, P^+(u) \le z, \, \gcd(u, k) = 1}\mu(u)\prod_{q \mid u}\gamma_q$ ($\gamma_q := 2 c_q(s)\beta_q$), every coordinate $k = p_1\cdots p_{\omega(k)} \in C_{x/2}$ ($p_1 < \dots < p_{\omega(k)}$) satisfies the All-Scale Buchstab–Beltrami Smooth-Mertens & Uniform-Scale Stieltjes Representation $\bigl(d_{x, \rho}^{(s)}\mu_x(\beta)\bigr)_k \equiv -\mu_x(\beta)_k\sum_{p \le x/k, \, p \nmid k}(-1)^{I(p, k)}A_p(s)\beta_p\Sigma_{p-1}\bigl(\frac{x}{k p}; k; \gamma\bigr) \equiv -\mu_x(\beta)_k\sum_{p \le x/k, \, p \nmid k}(-1)^{I(p, k)}\Lambda_p(s)\bigl[\Sigma_{p-1}\bigl(\frac{x}{k}; k; \gamma\bigr) - \Sigma_p\bigl(\frac{x}{k}; k; \gamma\bigr)\bigr]$ (where on $C_{\mathrm{smooth}, P}^{\mathrm{full}}$ at $\beta_p = \frac{1}{2 c_p(s)}$, $\Sigma_{p-1}(\infty; k; \mathbf{1}) = \prod_{q < p, \, q \nmid k}(1 - 1) = \mathbf{1}_{\{p = \operatorname{mex}(k)\}}$ localizes for every $k \in \mathcal{S}_P$ to the minimum excluded prime factor $\operatorname{mex}(k) := \min\{p \le P : p \nmid k\}$ as $\bigl(d_{P, \rho}^{(s)}\mu_P(\frac{1}{2 c(s)})\bigr)_k \equiv -\mathbf{1}_{\{k < P^\#\}}(-1)^{\pi(\operatorname{mex}(k))-1}\Lambda_{\operatorname{mex}(k)}(s)\mu_P(\frac{1}{2 c(s)})_k$, yielding $\bigl\|d_{P, \rho}^{(s)}\mu_P(\frac{1}{2 c(s)})\bigr\|_2^2 \equiv \sum_{q \le P}|\Lambda_q(s)|^2\bigl(\prod_{p < q} w_p^{-1}\bigr)\bigl(\prod_{q < r \le P}(1 + w_r^{-1})\bigr)$ with $w_q := 4|c_q(s)|^2$) and the Chiral Buchstab–Mertens Doubling Identity $\sum_{p \le x/k, \, p \nmid k}(-1)^{I(p, k)}\gamma_p\Sigma_{p-1}\bigl(\frac{x}{k p}; k; \gamma\bigr) \equiv \mathcal{E}_x(k; \gamma) - \mu(k)M_k\bigl(\frac{x}{k}; \gamma\bigr)$ with $M_k(Y; \gamma) := \Sigma_Y(Y; k; \gamma)$ and $\mathcal{E}(Y; k; \gamma) := 1 - 2\sum_{j=1}^{\omega(k)}(-1)^{j-1}\Sigma_{p_j - 1}(Y; k; \gamma)$, $\mathcal{E}_x(k; \gamma) := \mathcal{E}\bigl(\frac{x}{k}; k; \gamma\bigr)$ (which along the Selberg–Beltrami locus $\Lambda_p(s) \equiv \lambda$—in particular at $s = 1$, $\rho_p = \rho_p^{\mathrm{Sel}}(a_p)$, $\lambda = \frac{1}{2}$—evaluates $\bigl(d_{x, \rho}^{(s)}\mu_x(\beta)\bigr)_1 \equiv -\lambda(1 - M(x; \gamma))$ at $k = 1$ for every $\beta \in \mathbb{C}^{\pi(x)}$ and equates the non-vacuum coexact inner product for any $\beta^{(1)}, \beta^{(2)} \in \mathbb{C}^{\pi(x)}$, $\gamma^{(i)} := 2 c(s)\beta^{(i)}$, to $|\lambda|^{-2}\sum_{1 < k \le x/2\text{ sqfree}}\overline{(d_{x, \rho}^{(s)}\mu_x(\beta^{(1)}))_k}\,(d_{x, \rho}^{(s)}\mu_x(\beta^{(2)}))_k \equiv \sum_{1 < d \le x\text{ sqfree}}\bigl(\prod_{q \mid d}\overline{\beta_q^{(1)}}\beta_q^{(2)}\bigr)\overline{M_d(\frac{x}{d}; \gamma^{(1)})}\,M_d(\frac{x}{d}; \gamma^{(2)}) - \mathcal{D}_x(\beta^{(1)}, \beta^{(2)})$ with single-sum chiral GCD defect $\mathcal{D}_x(\beta^{(1)}, \beta^{(2)}) \equiv \sum_{1 < k \le x\text{ sqfree}}\bigl(\prod_{q \mid k}\overline{\beta_q^{(1)}}\beta_q^{(2)}\bigr)\bigl[\mu(k)\overline{\mathcal{E}_x(k; \gamma^{(1)})}M_k(\frac{x}{k}; \gamma^{(2)}) + \mu(k)\overline{M_k(\frac{x}{k}; \gamma^{(1)})}\mathcal{E}_x(k; \gamma^{(2)}) - \overline{\mathcal{E}_x(k; \gamma^{(1)})}\mathcal{E}_x(k; \gamma^{(2)})\bigr]$), and the normal-ordered coexact operator $\bigl(d_{x, \rho}^{(s)*} d_{x, \rho}^{(s)}\bigr)_{m_1, m_2} \equiv \bigl(\prod_{q \mid (m_1\setminus g), q < p_1^*} 2\overline{c_q(s)}\bigr)\bigl(\prod_{q \mid (m_2\setminus g), q < p_2^*} 2 c_q(s)\bigr)\bigl[\mathbf{1}_{\{p_1^* > 0\}}(-1)^{|\{r \mid g : p_1^* < r < p_2^*\}|}\overline{A_{p_1^*}(s)}A_{p_2^*}(s)\prod_{q \mid g, q < p_1^*}(1+4|c_q(s)|^2) + (2\overline{c_{p_1^*}(s)})^{\mathbf{1}_{\{p_1^* > 0\}}}\sum_{p \mid g, \, p_1^* < p < p_2^*}(-1)^{|\{r \mid g : p < r < p_2^*\}|}\overline{A_p(s)}(2 c_p(s))A_{p_2^*}(s)\prod_{q \mid g, q < p}(1+4|c_q(s)|^2) + (2\overline{c_{p_1^*}(s)})^{\mathbf{1}_{\{p_1^* > 0\}}}(2 c_{p_2^*}(s))^{\mathbf{1}_{\{p_2^* > 0\}}}\sum_{p \mid g, \, p > p_2^*}|A_p(s)|^2\prod_{q \mid g, q < p}(1+4|c_q(s)|^2)\bigr]$ has zero truncation overflow (giving $\|d_{x, \rho}^{(s)} e_m\|_2^2 \equiv \sum_{p \mid m}|a_p|^2(1 - |\rho_p|^2)^{\operatorname{Re}(s)}\prod_{q \mid m, q < p}(1 + 4|c_q(s)|^2)$ on the diagonal $m_1 = m_2 = m$, specializing at $s = +1$ and $s = -1$ to $\|d_{x, \rho} e_m\|_2^2 \equiv \sum_{p \mid m}|a_p|^2(1 - |\rho_p|^2)\prod_{q \mid m, q < p}(1 + 4|\rho_q|^2)$ and $\|\widetilde{d}_{x, \rho} e_m\|_2^2 \equiv \sum_{p \mid m}\frac{|a_p|^2}{1 - |\rho_p|^2}\prod_{q \mid m, q < p}\frac{1 + 3|\rho_q|^2}{1 - |\rho_q|^2}$ for all $m \le x\text{ sqfree}$), so summing $\beta(m)\|d_{P, \rho} e_m\|_2^2$ and $\beta(m)\|\widetilde{d}_{P, \rho} e_m\|_2^2$ over $m \in \mathcal{S}_P$ gives $\sum_{q \le P}\beta_q|a_q|^2(1-|\rho_q|^2)\bigl(\prod_{p < q}(1+\beta_p(1+4|\rho_p|^2))\bigr)\bigl(\prod_{q < r \le P}(1+\beta_r)\bigr)$ and $\sum_{q \le P}\frac{\beta_q|a_q|^2}{1-|\rho_q|^2}\bigl(\prod_{p < q}(1+\beta_p\frac{1+3|\rho_p|^2}{1-|\rho_p|^2})\bigr)\bigl(\prod_{q < r \le P}(1+\beta_r)\bigr)$ (collapsing to single monomials at $q = P$ when $\beta_p = -1$, where $\operatorname{STr}(d_{P, \rho} d_{P, \rho}^*) \equiv \mu(P^\#)\operatorname{STr}(d_{P, \rho}^* d_{P, \rho}) \equiv |a_P|^2(1 - |\rho_P|^2)\prod_{p < P}(4|\rho_p|^2)$ and $\operatorname{STr}(\widetilde{d}_{P, \rho}\widetilde{d}_{P, \rho}^*) \equiv \mu(P^\#)\operatorname{STr}(\widetilde{d}_{P, \rho}^*\widetilde{d}_{P, \rho}) \equiv \frac{|a_P|^2}{\Delta_P(\rho)}\prod_{p < P}(4|\rho_p|^2)$, and dually at $q = 2$ when $\beta_p = -\frac{1}{1+4|\rho_p|^2}$ or $-\frac{1-|\rho_p|^2}{1+3|\rho_p|^2}$, giving $-\frac{|a_2|^2(1-|\rho_2|^2)}{1+4|\rho_2|^2}\prod_{2 < r \le P}\frac{4|\rho_r|^2}{1+4|\rho_r|^2}$ and $-\frac{|a_2|^2}{1+3|\rho_2|^2}\prod_{2 < r \le P}\frac{4|\rho_r|^2}{1+3|\rho_r|^2}$), and telescoping both globally on $C_{\mathrm{smooth}, P}^{\mathrm{full}}$ and individually at every squarefree integer $m \le x$ at every finite cutoff $x \ge 2$ at $|\rho_p^{\mathrm{Sel}}|^2 = \frac{|a_p|^2}{1+|a_p|^2}$ and $|\rho_p| = |a_p|$ into: $$\bigl\|d_{x, \rho^{\mathrm{Sel}}} e_m\bigr\|_2^2 \;\equiv\; \frac{1}{4}\left[\prod_{p \mid m}\frac{1 + 5|a_p|^2}{1 + |a_p|^2} - 1\right], \quad \bigl\|\widetilde{d}_{x, a} e_m\bigr\|_2^2 \;\equiv\; \frac{1}{4}\left[\prod_{p \mid m}\frac{1 + 3|a_p|^2}{1 - |a_p|^2} - 1\right], \quad \operatorname{Tr}\!\left(d_{P, \rho^{\mathrm{Sel}}} d_{P, \rho^{\mathrm{Sel}}}^*\right) \;\equiv\; 2^{\pi(P)-2}\left[\prod_{p \le P}\frac{1 + 3|a_p|^2}{1 + |a_p|^2} - 1\right], \quad \operatorname{Tr}\!\left(\widetilde{d}_{P, a}\widetilde{d}_{P, a}^*\right) \;\equiv\; 2^{\pi(P)-2}\left[\prod_{p \le P}\frac{1 + |a_p|^2}{1 - |a_p|^2} - 1\right]. \tag{8d}$$

Proof.

• (a) Step 1 (Abelian Hasse Factorization, $LDL^*$, Determinant & Möbius–Beltrami Spectral Diagonalization): On $C_{\mathrm{smooth}, P}^{\mathrm{full}} = \bigotimes_{p \le P}(\mathbb{C} e_1 \oplus \mathbb{C} e_p)$, the unsigned prime deletion operators $\partial_p = (\bigotimes_{q < p} I_2) \otimes \varepsilon_p \otimes (\bigotimes_{q > p} I_2)$ ($\varepsilon_p = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$) commute pairwise and satisfy $\partial_p^2 = 0$, so $\mathcal{U}_\rho := \exp(\sum_{p \le P}\rho_p\partial_p) = \prod_{p \le P}(I + \rho_p\partial_p)$ is an abelian group representation ($\mathcal{U}_{\rho^{(1)}}\mathcal{U}_{\rho^{(2)}} = \mathcal{U}_{\rho^{(1)}+\rho^{(2)}}$, $\mathcal{U}_\rho^{-1} = \mathcal{U}_{-\rho}$). On each prime factor, $G_p(\rho_p) = \begin{pmatrix} 1 & \rho_p \\ \overline{\rho_p} & 1 \end{pmatrix} = C_p(\rho_p)^* C_p(\rho_p)$ where $C_p(\rho_p) = \begin{pmatrix} 1 & \rho_p \\ 0 & \sqrt{1 - |\rho_p|^2} \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-|\rho_p|^2} \end{pmatrix}\begin{pmatrix} 1 & \rho_p \\ 0 & 1 \end{pmatrix}$, so $G_{\mathrm{full}, \rho} = \mathcal{C}_{\mathrm{full}, \rho}^*\mathcal{C}_{\mathrm{full}, \rho} = \mathcal{U}_{\mathrm{full}, \rho}^* D_{\mathrm{full}, \rho}^2 \mathcal{U}_{\mathrm{full}, \rho}$. Because $\mathcal{S}_{\le x}$ is downward-closed under divisibility ($d \mid m \le x \implies d \le x$), $\mathcal{U}_{\mathrm{full}, \rho}$ and $\mathcal{C}_{\mathrm{full}, \rho}$ are upper-block-triangular on $C_x \oplus C_{>x}$ ($Q_{>x}\mathcal{C}_{\mathrm{full}, \rho} Q_x = 0$), so $(G_{\mathrm{full}, \rho})_{C_x, C_x} = (Q_x\mathcal{C}_{\mathrm{full}, \rho}^* Q_x)(Q_x\mathcal{C}_{\mathrm{full}, \rho} Q_x) = \mathcal{C}_{x, \rho}^*\mathcal{C}_{x, \rho} = \mathcal{U}_{x, \rho}^* D_{x, \rho}^2 \mathcal{U}_{x, \rho} = G_{x, \rho}$ and $\det(G_{x, \rho}) = \det(D_{x, \rho}^2) = \prod_{m \le x\text{ sqfree}}\prod_{p \mid m}(1 - |\rho_p|^2) = \prod_{p \le x}(1 - |\rho_p|^2)^{N_p(x)}$. For $\rho_p \ne 0$, the $2\times 2$ factor $C_p(\rho_p)$ has right eigenvectors $e_1$ (eigenvalue $1$) and $e_1 - \alpha_p(\rho_p)e_p$ (eigenvalue $\sqrt{1 - |\rho_p|^2}$ where $\alpha_p(\rho_p) = \frac{1 - \sqrt{1-|\rho_p|^2}}{\rho_p}$), so tensoring over $p \mid d$ gives the right eigenvectors $\Phi_d^{(\rho)} = \sum_{k \mid d}\mu(k)\alpha_\rho(k)e_k \in C_d \subseteq C_x$ of $\mathcal{C}_{x, \rho}$ for every $d \le x\text{ sqfree}$. By Möbius inversion on the divisor poset $\mathcal{S}_{\le x}$, $(\mathcal{J}_{x, \rho}^{-1})_{d, m} = \mathbf{1}_{\{d \mid m\}}\frac{\mu(d)}{\alpha_\rho(m)}$ satisfies $(\mathcal{J}_{x, \rho}\mathcal{J}_{x, \rho}^{-1})_{k, m} = \mathbf{1}_{\{k \mid m\}}\frac{\alpha_\rho(k)}{\alpha_\rho(m)}\sum_{k \mid d \mid m}\mu(k)\mu(d) = \delta_{k, m}$, giving $\mathcal{C}_{x, \rho} = \mathcal{J}_{x, \rho} D_{x, \rho}\mathcal{J}_{x, \rho}^{-1}$ and $(\mathcal{C}_{x, \rho}^s)_{k, m} = (\mathcal{J}_{x, \rho} D_{x, \rho}^s \mathcal{J}_{x, \rho}^{-1})_{k, m} = \mathbf{1}_{\{k \mid m\}}\frac{\mu(k)\alpha_\rho(k)}{\alpha_\rho(m)}\sum_{j \mid (m/k)}\mu(k j)(1 - |\rho_{k j}|^2)^{s/2} = \mathbf{1}_{\{k \mid m\}}(\prod_{p \mid k}(1 - |\rho_p|^2)^{s/2})\prod_{q \mid (m/k)}\frac{1 - (1 - |\rho_q|^2)^{s/2}}{\alpha_q(\rho_q)}$, which proves (6); differentiating at $s = 0$ gives $\log\mathcal{C}_{x, \rho}$.
• (a) Step 2 (LCM-Interior Inverse, Mertens Supertrace, Overflow Schur Complement, Order-Ideal Obstruction & Rankin Overflow Phase-Transition): Expanding $(G_{x, \rho}^{-1})_{m, n} = \sum_{k \le x\text{ sqfree}}(\mathcal{C}_{x, \rho}^{-1})_{m, k}\overline{(\mathcal{C}_{x, \rho}^{-1})_{n, k}}$, the summand is nonzero iff $m \mid k$ and $n \mid k$, i.e., $\ell := \operatorname{lcm}(m, n) \mid k$, so writing $k = \ell r$ ($r \le x/\ell\text{ sqfree}$, $\gcd(r, \ell) = 1$) factors out $\mu(\ell/m)\mu(\ell/n)(\prod_{p \mid (\ell/m)}\rho_p)(\prod_{p \mid (\ell/n)}\overline{\rho_p}) = (G_{x, -\rho})_{m, n}$ times $(G_{x, \rho}^{-1})_{\ell, \ell}$, proving the first two identities of (6a) [7, 8]. For any multiplicative $\beta(m)$, swapping sums over $m \le x$ and $k = m r \le x$ gives $\sum_{m \le x}\beta(m)(G_{x, \rho}^{-1})_{m, m} = \sum_{k \le x\text{ sqfree}}(\prod_{p \mid k}(1 - |\rho_p|^2)^{-1})\sum_{m \mid k}\beta(m)\prod_{q \mid (k/m)}|\rho_q|^2 = \sum_{k \le x\text{ sqfree}}\prod_{p \mid k}\frac{\beta_p + |\rho_p|^2}{1 - |\rho_p|^2}$, which at $\beta_p = -1$ cancels $\frac{-1 + |\rho_p|^2}{1 - |\rho_p|^2} \equiv -1$ identically across all $\rho \in \mathbb{D}^{\pi(x)}$ to give $\operatorname{STr}(G_{x, \rho}^{-1}) = \sum_{k \le x}\mu(k) = M(x)$. Since $C_p(\rho_p)^{-1}(C_p(\rho_p)^*)^{-1} = (1 - |\rho_p|^2)^{-1}\begin{pmatrix} 1 & -\rho_p \\ -\overline{\rho_p} & 1 \end{pmatrix} = (1 - |\rho_p|^2)^{-1} G_p(-\rho_p)$, on $C_{\mathrm{smooth}, x}^{\mathrm{full}} = C_x \oplus C_{>x}$ we have $G_{\mathrm{full}, \rho}^{-1} = \Delta_x(\rho)^{-1} G_{\mathrm{full}, -\rho}$ while $\mathcal{C}_{\mathrm{full}, \rho}^{-1} = \begin{pmatrix} \mathcal{C}_{x, \rho}^{-1} & \mathcal{M}_{x, \rho} \\ 0 & \mathcal{C}_{>x, \rho}^{-1} \end{pmatrix}$, whose $(C_x, C_x)$ block in $\mathcal{C}_{\mathrm{full}, \rho}^{-1}(\mathcal{C}_{\mathrm{full}, \rho}^*)^{-1}$ yields (6b). Comparing $(G_{x, \rho}^{-1})_{m, n} \propto \mathbf{1}_{\{\operatorname{lcm}(m, n) \le x\}}$ with $(L_x)_{m, n} \propto \mathbf{1}_{\{\operatorname{lcm}(m, n) > x\}}$ from (3) proves $(L_x)_{m, n}(G_{x, \rho}^{-1})_{m, n} \equiv 0$ for all $m \ne n$. Under $\mathbb{P}_{x, \rho}(\mathbf{d} = d) = \Delta_x(\rho)\prod_{p \mid d}\frac{|\rho_p|^2}{1 - |\rho_p|^2}$, $\mathbb{P}_{x, \rho}(m_1 \mid \mathbf{d}\text{ and }m_2 \mid \mathbf{d}) = \prod_{p \mid \operatorname{lcm}(m_1, m_2)}|\rho_p|^2 = (\prod_{p \mid m_1}\rho_p)(\prod_{p \mid m_2}\overline{\rho_p})(G_{x, \rho})_{m_1, m_2}$, proving $\|w\|_{G_{x, \rho}}^2 = \mathbb{E}_{x, \rho}[|\sum_{m \mid \mathbf{d}, m \le x} f_m|^2]$, and since $\mu(d/m)\prod_{q \mid (d/m)}(-\overline{\rho_q}) = \prod_{q \mid (d/m)}\overline{\rho_q}$, $(\mathcal{M}_{x, -\rho}^*\psi_x(\gamma\overline{\rho}))_d = (\prod_{p \mid d}\frac{\overline{\rho_p}}{\sqrt{1-|\rho_p|^2}})\sum_{m \mid d, m \le x}\prod_{p \mid m}\gamma_p$ ($d > x$), which combined with (6b) at $-\rho$ proves (6c) and (6d). For any divisor-closed subspace $\mathcal{A} \subset C_x$, $m \in \mathcal{A}$ and $d \mid m \implies d \in \mathcal{A}$, so $\mathcal{C}_{x, \rho}(\mathcal{A}) \subset \mathcal{A}$ and $P_{\mathcal{A}}G_{x, \rho}P_{\mathcal{A}} = \mathcal{C}_{\mathcal{A}, \rho}^*\mathcal{C}_{\mathcal{A}, \rho} = G_{\mathcal{A}, \rho}$; hence for any $v \in \mathcal{A}$, $\|v - G_{x, \rho}^{-1}\psi_x(\beta)\|_{G_{x, \rho}}^2 = \|\mathcal{C}_{\mathcal{A}, \rho}v - (\mathcal{C}_{\mathcal{A}, \rho}^{-1})^* P_{\mathcal{A}}\psi_x(\beta)\|_{\ell^2(\mathcal{A})}^2 + \|P_{C_x\setminus\mathcal{A}}(\mathcal{C}_{x, \rho}^{-1})^*\psi_x(\beta)\|_2^2$, which is minimized at $v_{\mathcal{A}}^* = G_{\mathcal{A}, \rho}^{-1}(P_{\mathcal{A}}\psi_x(\beta))$ with residual given by (7a) as (6e). Finally, by (6a) and (6b), $(\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^*)_{m, m} = \Delta_x(\rho)^{-1} - (G_{x, \rho}^{-1})_{m, m} = (\prod_{p \mid m}\frac{1}{1-|\rho_p|^2})\sum_{r > x/m\text{ sqfree}, P^+(r)\le x, \gcd(r, m)=1}\prod_{q \mid r}\frac{|\rho_q|^2}{1-|\rho_q|^2}$; bounding $\mathbf{1}_{\{r > x/m\}} \le (rm/x)^\alpha$ ($\alpha \ge 0$) and extending the sum to all squarefree $r$ with $P^+(r) \le x$ and $\gcd(r, m) = 1$ gives $(\frac{m}{x})^\alpha(\prod_{p \mid m}\frac{1}{1-|\rho_p|^2})\prod_{q \le x, q \nmid m}\bigl(1 + \frac{|\rho_q|^2 q^\alpha}{1-|\rho_q|^2}\bigr) = C_\alpha(x; \rho)(\prod_{p \mid m}\frac{1}{1+|\rho_p|^2(p^\alpha-1)})(\frac{m}{x})^\alpha$. Since $u \mapsto u^\alpha$ is non-decreasing for $\alpha \ge 0$, $\sum_{m \in C_y} m^\alpha \le \sum_{m=1}^{\lfloor y \rfloor} m^\alpha \le \int_0^{\lfloor y \rfloor - 1} (u+1)^\alpha\,du + \lfloor y \rfloor^\alpha \le \frac{y^{\alpha+1}}{\alpha+1} + y^\alpha$, which establishes (6f). When $\frac{|\rho_p|^2}{1 - |\rho_p|^2} \le p^{-2\sigma}$ ($\sigma > \frac{1}{2}$), bounding $\prod_{p \mid m}(1 + p^{-2\sigma}) \le K_\sigma$ and $\sum_{r > \lfloor x/m \rfloor} r^{-2\sigma} \le (\lfloor x/m \rfloor + 1)^{-2\sigma} + \int_{\lfloor x/m \rfloor + 1}^\infty u^{-2\sigma}\,du \le (m/x)^{2\sigma} + \frac{1}{2\sigma - 1}(m/x)^{2\sigma - 1} \le \frac{2\sigma}{2\sigma - 1}(m/x)^{2\sigma - 1}$ directly gives $(\mathcal{M}_{x, \rho}\mathcal{M}_{x, \rho}^*)_{m, m} \le \frac{2\sigma K_\sigma}{2\sigma - 1}(m/x)^{2\sigma - 1}$ even at $\alpha = 2\sigma - 1$ where $C_{2\sigma - 1}(\infty; \rho) = \infty$; conversely, for constant $\rho_p \equiv \rho_0 \in \mathbb{D}\setminus\{0\}$ ($u_0 = \frac{|\rho_0|^2}{1 - |\rho_0|^2} > 0$), $u_0 \mapsto \sum_{d > x\text{ sqfree}, P^+(d)\le x} u_0^{\omega(d)}$ is non-decreasing on $(0, \infty)$, so replacing $u_0$ by $u_* := \min(u_0, 1) \in (0, 1]$ (where $u_*^{\omega(r)} \le 1$ for all $r \in C_x$) yields $(\mathcal{M}_{x, \rho_0}\mathcal{M}_{x, \rho_0}^*)_{1, 1} \ge \sum_{d > x\text{ sqfree}, P^+(d)\le x} u_*^{\omega(d)} = (1 + u_*)^{\pi(x)} - \sum_{r \in C_x} u_*^{\omega(r)} \ge (1 + \min(u_0, 1))^{\pi(x)} - |C_x| \ge (1 + \min(u_0, 1))^{\pi(x)} - x \to \infty$ for all $0 < |\rho_0| < 1$ and $x \ge 2$.
• (b) Step 3 (Adjoint Affine Cochain Calculus, Divisor-Poset Wick Theorem & Conjugated Boundary $d_{x, \rho}^{(s)}$): Expanding $\langle e_d, \mathcal{C}_{x, \rho}\psi_x(\gamma a) \rangle$ over $m = d n \le x$ gives (7) and (8). Since $(\mathcal{U}_{x, \rho}^*\psi_x(\beta))_m = \sum_{d \mid m}\beta(d)\prod_{q \mid (m/d)}\overline{\rho_q} = \prod_{p \mid m}(\beta_p + \overline{\rho_p}) = \psi_x(\beta + \overline{\rho})_m$, $\mathcal{C}_{x, \rho}^* = \mathcal{U}_{x, \rho}^* D_{x, \rho}$ and $(\mathcal{C}_{x, \rho}^{-1})^* = D_{x, \rho}^{-1}\mathcal{U}_{x, -\rho}^*$ act by (7a) without truncation error at every $x \ge 1$, diagonalizing $\langle \psi_x(\beta^{(1)}), G_{x, \rho}^{-1}\psi_x(\beta^{(2)}) \rangle = \langle (\mathcal{C}_{x, \rho}^{-1})^*\psi_x(\beta^{(1)}), (\mathcal{C}_{x, \rho}^{-1})^*\psi_x(\beta^{(2)}) \rangle$ to give (7b). Because $\mathcal{S}_{\le x}$ is downward-closed, every divisor-lowering operator $A_{\mathrm{full}}$ satisfies $A_{\mathrm{full}} Q_x = Q_x A_{\mathrm{full}} Q_x = A_x Q_x$ and $Q_x A_{\mathrm{full}}^* = A_x^* Q_x$, so $B_x^* A_x = Q_x B_{\mathrm{full}}^* Q_x A_{\mathrm{full}} Q_x \equiv Q_x B_{\mathrm{full}}^* A_{\mathrm{full}} Q_x$ has zero truncation overflow. Writing $d_x = \sum_{p \le x} a_p\widehat{\varepsilon}_p$ where $\widehat{\varepsilon}_p(e_m) = \mathbf{1}_{\{p \mid m\}}(-1)^{I(p, m)}e_{m/p}$, removing $q$ from $m$ leaves $I(p, m/q) = I(p, m)$ for $p < q$ and $I(p, m/q) = I(p, m) - 1$ for $p > q$, so $\partial_q\widehat{\varepsilon}_p = +\widehat{\varepsilon}_p\partial_q$ ($p \le q$) and $\partial_q\widehat{\varepsilon}_p = -\widehat{\varepsilon}_p\partial_q$ ($p > q$). Consequently, $[d_x, \partial_q] = 2\sum_{p > q} a_p\widehat{\varepsilon}_p\partial_q$ and $(\operatorname{ad}_{\partial_q})^2(d_x) = 0$ (since $\partial_q^2 = 0$), and conjugating $\widehat{\varepsilon}_p$ by $\mathcal{C}_{x, \rho}^s = D_{x, \rho}^s \prod_{q \le x}(I_{C_x} + c_q(s)\partial_q)$ cancels all factors $q \ge p$ while doubling all factors $q < p$ into $(I_{C_x} + c_q(s)\partial_q)^2 = I_{C_x} + 2 c_q(s)\partial_q$, yielding $d_{x, \rho}^{(s)} = \sum_{p \le x} A_p(s)\widehat{\varepsilon}_p\prod_{q < p}(I_{C_x} + 2 c_q(s)\partial_q)$ and (8a) (each proper divisor $d \mid m$ arising uniquely at $p = P^+(m/d)$).
• (b) Step 4 (Cycle Green's Matrix, Buchstab–Beltrami Smooth-Mertens Representation, Chiral Doubling & Coexact GCD Gram Matrix): Expanding $(d_x G_{x, \rho}^{-1} d_x^*)_{n_1, n_2} = \langle (\mathcal{C}_{x, \rho}^{-1})^* d_x^* e_{n_1}, (\mathcal{C}_{x, \rho}^{-1})^* d_x^* e_{n_2} \rangle$ over common multiples $m = \ell k \le x$ ($\ell = \operatorname{lcm}(n_1, n_2)$) gives the LCM-interior formula in (8b), while on $C_{\mathrm{smooth}, P}^{\mathrm{full}} = Z_P \oplus B_P^*$, $(G_{\mathrm{Hodge}, \rho}^{-1})_{Z, Z} = \Delta_P(\rho)^{-1} G_{Z, -\rho}$ and $Z_{\mathrm{mat}}^*(d_P G_{P, \rho}^{-1} d_P^*)Z_{\mathrm{mat}} = S(P)W_{\mathrm{mat}}^* G_{P, \rho}^{-1} W_{\mathrm{mat}} = S(P)\Delta_P(\rho)^{-1} G_{B^*, -\rho}$ by (6b), and summing over $\omega(m) = k$ yields the Krawtchouk trace polynomial in (8c). For $\mu_x(\beta)_m = \mu(m)\prod_{p \mid m}\beta_p$ and $k \in C_{x/2}$, writing $m = k p u \le x$ where $p = P^+(m/k) \nmid k$ and $u = (m/k)/p \le \frac{x}{k p}$ is $(p-1)$-smooth and coprime to $k$ gives $(d_{x, \rho}^{(s)}\mu_x(\beta))_k = -\mu_x(\beta)_k\sum_{p \le x/k, p \nmid k}(-1)^{I(p, k)}A_p(s)\beta_p\Sigma_{p-1}(\frac{x}{kp}; k; \gamma)$, and since $\gamma_p\Sigma_{p-1}(\frac{x}{kp}; k; \gamma) = \Sigma_{p-1}(\frac{x}{k}; k; \gamma) - \Sigma_p(\frac{x}{k}; k; \gamma)$ is the exact contribution of multiples of $p$ to $\Sigma_p(\frac{x}{k}; k; \gamma)$, both the uniform-scale Stieltjes representation and (since $(-1)^{I(p, k)} = (-1)^j$ on $p \in (p_j, p_{j+1})$ and $\Sigma_{p_j}(\frac{x}{k}; k; \gamma) = \Sigma_{p_j - 1}(\frac{x}{k}; k; \gamma)$ for $p_j \mid k$) the Chiral Buchstab–Mertens Doubling Identity follow immediately. Contracting $(d_{x, \rho}^{(s)*} d_{x, \rho}^{(s)})_{m_1, m_2} = \sum_{d \mid g}\overline{(d_{x, \rho}^{(s)})_{d, m_1}}(d_{x, \rho}^{(s)})_{d, m_2}$ ($g = \gcd(m_1, m_2)$, $p_1^* \le p_2^*$) over $d = g$ (where $P^+(m_i/g) = p_i^*$) and $d < g$ grouped by $p = P^+(g/d) \mid g$ (which for $p_1^* < p < p_2^*$ gives $P^+(m_1/d) = p$ with weight $\overline{A_p(s)}$ and $p < P^+(m_2/d) = p_2^*$ with interior factor $2 c_p(s)$, while for $p > p_2^*$ gives $P^+(m_1/d) = P^+(m_2/d) = p$ with weight $|A_p(s)|^2$, and for $p < p_1^*$ sums over $g/(dp) \mid \prod_{q \mid g, q < p} q$ to $\prod_{q \mid g, q < p}(1 + 4|c_q(s)|^2)$) establishes the $O(\omega(g))$ coexact GCD Gram formula and (8d). $\blacksquare$

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4. Pythagorean Dual Overlap, Möbius Eigenvectors & Complex Phase Dispersion

Equip $(C_x, G_{x, \rho})$ with the Cholesky-intertwined boundary operator $d_{x, \rho} = \mathcal{C}_{x, \rho}^{-1} d_x \mathcal{C}_{x, \rho}$ of (8a), whose $G_{x, \rho}$-adjoint is $d_{x, \rho}^{*_\rho} \equiv \mathcal{C}_{x, \rho}^{-1} d_x^* \mathcal{C}_{x, \rho}$ and whose $G_{x, \rho}$-orthogonal cycle projector is $P_{Z_{x, \rho}} \equiv \mathcal{C}_{x, \rho}^{-1} P_{Z_x} \mathcal{C}_{x, \rho}$.

Theorem 3 (Pythagorean Dual Overlap $\rho_p^\star = \frac{2a_p}{1+a_p^2}$, Exact Möbius Eigenvectors & Complex Phase Dispersion). Set the prime overlap parameter to the Pythagorean rational $\rho_p^\star := \frac{2 a_p}{1 + a_p^2} \in (0, 1)$, so that $\sqrt{1 - (\rho_p^\star)^2} \equiv \frac{1 - a_p^2}{1 + a_p^2}$ is purely rational in $a_p$. Then on $C_{\mathrm{smooth}, P}^{\mathrm{full}}$: (a) All-Scale Degree-Graded Coboundary Identity, Boundary Energy Formula, All-Scale Pythagorean Parity-Flip & Coboundary-Equivalence, Exact Möbius Eigenvector & Co-Twisted Complex Pythagorean Family: At every finite additive cutoff $x \ge 1$ and every degree $k \ge 0$, since $m \le x \implies m/p_j \le m/2 \le x/2$ and $\sum_{j=1}^{\omega(m)}(-1)^{j-1} = \frac{1 - \mu(m)}{2}$, the degree-$k$ slice $\mu_x^k(a) := P_{C_x^k}\mu_x(a) = (-1)^k\mathbf{1}_x^k(a)$ satisfies $d_x^*(\mu_x^{2r}(a)) \equiv -\mu_x^{2r+1}(a)$ and $d_x^*(\mu_x^{2r+1}(a)) \equiv 0$ ($\forall r \ge 0$), so every odd-degree slice $\mu_x^{2r+1}(a)$ is individually an exact coboundary in $B_x^{*, 2r+1} \subset Z_x^\perp$, the total odd part $\mu_x^{\mathrm{odd}}(a) = -\mathbf{1}_x^{\mathrm{odd}}(a)$ is determined by the exact projection $P_{B_x}\mu_x(a) \in B_x^{\mathrm{even}} = Z_{x/2}^{\mathrm{even}}$ ($B_x = \operatorname{im}(d_x) = \ker(d_x^*)^\perp$), and by (3') the degree-$k$ boundary energy at every $x \ge 2$ is given by a purely positive monomial sum: $$\mathbf{1}_x^{\mathrm{odd}}(a) \;=\; -\,\mu_x^{\mathrm{odd}}(a) \;\equiv\; d_x^*\bigl(\mu_{x/2}^{\mathrm{even}}(a)\bigr) \;\equiv\; d_x^*\bigl(P_{B_x}\mu_x(a)\bigr) \;\in\; B_x^* \;=\; Z_x^\perp, \qquad S(x)\bigl\|P_{B_x}\mu_x(a)\bigr\|_2^2 - \bigl\|\mu_x^{\mathrm{odd}}(a)\bigr\|_2^2 \;\equiv\; \bigl\|\mathcal{T}_x P_{B_x}\mu_x(a)\bigr\|_2^2 \;\ge\; 0, \tag{9a}$$ $$\bigl\|d_x \mu_x^k(a)\bigr\|_2^2 \;=\; \bigl\|d_x \mathbf{1}_x^k(a)\bigr\|_2^2 \;\equiv\; S(x)\,\bigl\|\mu_x^k(a)\bigr\|_2^2 \;-\; \mathbf{1}_{\{k\text{ even}\}}\,\bigl\|\mu_x^{k+1}(a)\bigr\|_2^2 \;-\; \sum_{\substack{u > x\text{ sqfree}, \, P^+(u) \le x \\ \omega(u) = k+1, \, N_{\mathrm{ovf}}(u; x)\text{ odd}}} \prod_{p \mid u} a_p^2 \qquad (\forall k \ge 0, \; \forall x \ge 2), \tag{9b}$$ and more generally, for any frequency $t \in \mathbb{R}$, $\bigl\|d_x \mu_x^k(a p^{-it})\bigr\|_2^2 \equiv S(x)\bigl\|\mu_x^k(a)\bigr\|_2^2 - \sum_{u \le x\text{ sqfree}, \omega(u)=k+1} a_u^2\bigl|\sum_{j=1}^{k+1}(-1)^j p_j^{it}\bigr|^2 - \sum_{u > x\text{ sqfree}, P^+(u)\le x, \omega(u)=k+1} a_u^2\bigl|\sum_{j=k+2-N_{\mathrm{ovf}}(u; x)}^{k+1}(-1)^j p_j^{it}\bigr|^2$ (partitioning at $k = 1$ the prime-pair Dirichlet variance into $\|d_x^*\mu_x^1(a p^{-it})\|_2^2 = 4\sum_{p < q \le x, pq \le x} a_p^2 a_q^2\sin^2(\frac{t\log(q/p)}{2})$ and $\|\mathcal{T}_x\mu_x^1(a p^{-it})\|_2^2 = 4\sum_{p < q \le x, pq > x} a_p^2 a_q^2\sin^2(\frac{t\log(q/p)}{2})$), so $P_{Z_x}\mu_x^{\mathrm{odd}}(a) \equiv P_{Z_x}\mathbf{1}_x^{\mathrm{odd}}(a) \equiv 0$, $P_{Z_x}\mu_x(a) \equiv P_{Z_x}\mathbf{1}_x(a)$, and $\Pi_x\mu_x(a) \equiv \Pi_x\mathbf{1}_x(a)$ hold as vector identities at every $x \ge 1$ (with $P_{B_x}\mu_x(a) \equiv e_1$ and $\mathcal{T}_x P_{B_x}\mu_x(a) \equiv 0$ for all $2 \le x < 30$), and on $C_{\mathrm{smooth}, P}^{\mathrm{full}}$ (where $C_{>x} = \{0\}$), $\bigl\|P_{Z_P}\mu_P^{2r+1}(a)\bigr\|_2^2 \equiv 0$ and $\bigl\|P_{Z_P}\mu_P^{2r}(a)\bigr\|_2^2 \equiv \frac{1}{S(P)}\bigl\|\mu_P^{2r+1}(a)\bigr\|_2^2$ for every $r \ge 0$. Furthermore, at every finite additive cutoff $x \ge 1$, because $\rho_p^\star - \sqrt{1 - (\rho_p^\star)^2}\,a_p = \frac{2a_p}{1+a_p^2} - \frac{1-a_p^2}{1+a_p^2}\,a_p \equiv +a_p$, the lower-triangular Cholesky adjoint $\mathcal{C}_{x, \rho^\star}^*$ and its inverse $(\mathcal{C}_{x, \rho^\star}^{-1})^*$ act by (7a) without truncation overflow as exact all-scale parity-flipping and coboundary-equivalence maps: $$\mathcal{C}_{x, \rho^\star}^*\bigl(\mu_x(a)\bigr) \;\equiv\; \mathbf{1}_x(a), \quad \bigl(\mathcal{C}_{x, \rho^\star}^{-1}\bigr)^*\bigl(\mathbf{1}_x(a)\bigr) \;\equiv\; \mu_x(a), \quad \bigl(\mathcal{C}_{x, \rho^\star}^* - I_{C_x}\bigr)\mu_x(a) \;\equiv\; 2\,d_x^*\bigl(\mu_{x/2}^{\mathrm{even}}(a)\bigr) \;\in\; B_x^* \;=\; Z_x^\perp, \quad \widetilde{d}_{x, \rho^\star}^*\bigl(\mu_{x/2}(a)\bigr) \;\equiv\; \frac{1}{2}\left[\mu_x(a) - \mu_x\!\left(\frac{a(3+a^2)}{1-a^2}\right)\right] \quad (\forall x \ge 2), \tag{9c}$$ with $(\mathcal{C}_{x, \rho^\star}^{-1})^*\mu_x(a) \equiv \mu_x\bigl(\frac{a(3+a^2)}{1-a^2}\bigr)$, $\mathcal{C}_{x, \rho^\star}^*\mathbf{1}_x(a) \equiv \mathbf{1}_x\bigl(\frac{a(3-a^2)}{1+a^2}\bigr)$, $\bigl(G_{x, \rho^\star}^{-1}\mathbf{1}_x(a)\bigr)_m = \bigl(\mathcal{C}_{x, \rho^\star}^{-1}\mu_x(a)\bigr)_m \equiv (\mu_x(a))_m\bigl(\prod_{p \mid m}\frac{1+a_p^2}{1-a_p^2}\bigr)\sum_{r \le x/m\text{ sqfree}, \gcd(r, m)=1}\prod_{q \mid r}\frac{2a_q^2}{1-a_q^2}$ (preserving the exact sign $\mu(m)$ at every coordinate $m \le x$), $\bigl\langle \mathbf{1}_x(a), G_{x, \rho^\star}^{-1}\mu_x(a)\bigr\rangle \equiv \sum_{m \le x\text{ sqfree}}\prod_{p \mid m}\frac{a_p^2(3+a_p^2)}{1-a_p^2} > 0$, $P_{Z_x}\mathcal{C}_{x, \rho^\star}^*\mu_x(a) \equiv P_{Z_x}(\mathcal{C}_{x, \rho^\star}^{-1})^*\mathbf{1}_x(a) \equiv P_{Z_x}\mu_x(a)$, $\Pi_x\mathcal{C}_{x, \rho^\star}^*\mu_x(a) \equiv \Pi_x\mu_x(a)$, and $\|\mu_x(a)\|_{\breve{G}_{x, \rho^\star}}^2 = \langle \mathbf{1}_x(a), G_{x, \rho^\star}^{-1}\mathbf{1}_x(a)\rangle = \langle \mu_x(a), G_{x, -\rho^\star}^{-1}\mu_x(a)\rangle \equiv \|\mu_x(a)\|_2^2 = \sum_{m \le x\text{ sqfree}} a_m^2$ at every $x \ge 1$. Dually, on $C_{\mathrm{smooth}, P}^{\mathrm{full}}$, for $\mu_P(a)$ ($\gamma_p = -1$), $1 - \rho_p^\star a_p \equiv \frac{1 - a_p^2}{1 + a_p^2} \equiv \sqrt{1 - (\rho_p^\star)^2}$, so $\widetilde{\alpha}_p(-1; \rho^\star) \equiv -a_p$ for all $p \le P$, $\mu_P(a)$ is an exact eigenvector of $\mathcal{C}_{\rho^\star}$, and combining $\mathcal{C}_{\rho^\star}\mu_P(a) \equiv \lambda_\mu(P)\mu_P(a)$ with $\mathcal{C}_{\rho^\star}^*\mu_P(a) \equiv \mathbf{1}_P(a)$ makes the full GCD Gram matrix $G_{P, \rho^\star} = \mathcal{C}_{\rho^\star}^* \mathcal{C}_{\rho^\star}$ and its inverse $G_{P, \rho^\star}^{-1}$ exact scalar-rescaled parity flips between $\mu_P(a)$ and $\mathbf{1}_P(a)$: $$\mathcal{C}_{\rho^\star}\bigl(\mu_P(a)\bigr) \;\equiv\; \lambda_\mu(P)\,\mu_P(a), \qquad G_{P, \rho^\star}\bigl(\mu_P(a)\bigr) \;\equiv\; \lambda_\mu(P)\,\mathbf{1}_P(a), \qquad G_{P, \rho^\star}^{-1}\bigl(\mathbf{1}_P(a)\bigr) \;\equiv\; \lambda_\mu(P)^{-1}\,\mu_P(a), \qquad \lambda_\mu(P) \;:=\; \prod_{p \le P} \frac{1 - a_p^2}{1 + a_p^2}. \tag{9}$$ Because $\mathcal{C}_\rho$ is upper divisor-triangular rather than degree-diagonal, it does not commute with the Euclidean parity projectors $P_{\mathrm{even}}, P_{\mathrm{odd}}$; instead, the $G_\rho$-orthogonal parity decomposition that commutes with $P_{Z_{P, \rho}} = \mathcal{C}_\rho^{-1} P_{Z_P}\mathcal{C}_\rho$ is given by the Cholesky-transported parity projectors $P_{\mathrm{even/odd}, \rho} := \mathcal{C}_\rho^{-1} P_{\mathrm{even/odd}} \mathcal{C}_\rho$ (satisfying $\|P_{Z_{P, \rho}} P_{\mathrm{even/odd}, \rho} w\|_{G_\rho}^2 \equiv \|P_{Z_P} P_{\mathrm{even/odd}}(\mathcal{C}_\rho w)\|_2^2$). Consequently, by (9a)–(9b) and [4, Theorem 3], for the transported parity projectors at $\rho^\star$, $\bigl\|P_{Z_{P, \rho^\star}} P_{\mathrm{odd}, \rho^\star}\mu_P(a)\bigr\|_{G_{\rho^\star}}^2 = \bigl\|P_{Z_P} P_{\mathrm{odd}}(\mathcal{C}_{\rho^\star}\mu_P(a))\bigr\|_2^2 \equiv 0$ identically, and $\bigl\|P_{Z_{P, \rho^\star}} P_{\mathrm{even}, \rho^\star}\mu_P(a)\bigr\|_{G_{\rho^\star}}^2 = \bigl\|P_{Z_{P, \rho^\star}}\mu_P(a)\bigr\|_{G_{\rho^\star}}^2 \equiv \frac{\lambda_\mu(P)^2}{S(P)}\|\mu_P^{\mathrm{odd}}(a)\|_2^2 \equiv \lambda_\mu(P)^2\,\frac{\prod_{p \le P}(1+a_p^2) - \prod_{p \le P}(1-a_p^2)}{2 S(P)}$. More generally, for any complex Dirichlet twist $\alpha_p = a_p e^{-i\theta_p}$ ($a_p \in (0, 1)$, $\theta_p \in \mathbb{R}$), at the co-twisted complex Pythagorean overlap $\rho_p^\star(\alpha_p) := \frac{2\overline{\alpha_p}}{1 + a_p^2} \in \mathbb{D}$, $\overline{\rho_p^\star(\alpha_p)} - \frac{1-a_p^2}{1+a_p^2}\alpha_p \equiv +\alpha_p$ and $\rho_p^\star(\alpha_p)(-\alpha_p) = -\frac{2a_p^2}{1+a_p^2}$ cancel $e^{\pm i\theta_p}$ identically, so $\mathcal{C}_{x, \rho^\star(\alpha)}^*\bigl(\mu_x(\alpha)\bigr) \equiv \mathbf{1}_x(\alpha)$ and $\bigl(\mathcal{C}_{x, \rho^\star(\alpha)}^{-1}\bigr)^*\bigl(\mathbf{1}_x(\alpha)\bigr) \equiv \mu_x(\alpha)$ at every finite cutoff $x \ge 1$, while $\mathcal{C}_{\rho^\star(\alpha)}\bigl(\mu_P(\alpha)\bigr) \equiv \lambda_\mu(P)\,\mu_P(\alpha)$, $G_{P, \rho^\star(\alpha)}\bigl(\mu_P(\alpha)\bigr) \equiv \lambda_\mu(P)\,\mathbf{1}_P(\alpha)$, and $\bigl\|P_{Z_P(\overline{\alpha})} P_{\mathrm{odd}}\bigl(\mathcal{C}_{\rho^\star(\alpha)}\mu_P(\alpha)\bigr)\bigr\|_2^2 \equiv 0$ for all $\theta \in \mathbb{R}^{\pi(P)}$.

(b) Unsigned Gauge-Symmetry Breaking, Transported vs. Untransported Parity Slices, Complex Two-Recurrence Formula & Fixed-Metric Phase Dispersion: Under the parity involution $U(\pi)e_m = \mu(m)e_m$, $U(\pi)\mathcal{C}_\rho U(\pi)^{-1} \equiv \mathcal{C}_{-\rho}$ and $U(\pi)G_{x, \rho}U(\pi)^{-1} \equiv G_{x, -\rho}$. For any complex sequence $\{\beta_p\}_{p \le P} \subset \mathbb{D}$, define the two complex scalar 1-step recurrences ($R_{2^-} = R_{2^-}^{\mathrm{alt}} := 0$): $$R_p(\beta) \;:=\; \frac{1 - |\beta_p|^2}{1 + |\beta_p|^2}\, R_{p^-}(\beta) + \frac{a_p \beta_p}{1 + |\beta_p|^2}, \qquad R_p^{\mathrm{alt}}(\beta) \;:=\; \frac{1 + |\beta_p|^2}{1 - |\beta_p|^2}\, R_{p^-}^{\mathrm{alt}}(\beta) + \frac{a_p \beta_p}{1 - |\beta_p|^2}. \tag{10}$$ Then the total and parity-difference cycle energies of $\psi_P(\beta) = \psi_P^{\mathrm{even}}(\beta) + \psi_P^{\mathrm{odd}}(\beta)$ on $C_{\mathrm{smooth}, P}^{\mathrm{full}}$ are given in closed form by: $$\|P_{Z_P}\psi_P(\beta)\|^2 \;\equiv\; \left[\prod_{q \le P}(1 + |\beta_q|^2)\right]\left(1 \;-\; \frac{1}{S(P)}\sum_{p \le P}\frac{a_p^2 |\beta_p|^2 + 2 a_p\operatorname{Re}\bigl(\overline{\beta_p}\,R_{p^-}(\beta)\bigr)}{1 + |\beta_p|^2}\right), \tag{10a}$$ $$\|P_{Z_P}\psi_P^{\mathrm{even}}(\beta)\|^2 - \|P_{Z_P}\psi_P^{\mathrm{odd}}(\beta)\|^2 \;\equiv\; \left[\prod_{q \le P}(1 - |\beta_q|^2)\right]\left(1 \;+\; \frac{1}{S(P)}\sum_{p \le P}\frac{a_p^2 |\beta_p|^2 + 2 a_p\operatorname{Re}\bigl(\overline{\beta_p}\,R_{p^-}^{\mathrm{alt}}(\beta)\bigr)}{1 - |\beta_p|^2}\right). \tag{10b}$$ In particular: (i) for the unsigned chain $\mathbf{1}_P(a)$ ($\gamma_p = +1$), $\mathcal{C}_{\rho^\star}(\mathbf{1}_P(a)) \equiv \Lambda_+(P)\,\psi_P(\widetilde{\gamma}^{(+)} a)$ with $\Lambda_+(P) := \prod_{p \le P}\frac{1 + 3a_p^2}{1 + a_p^2}$ and $\widetilde{\gamma}_p^{(+)} := \frac{1 - a_p^2}{1 + 3a_p^2} \ne 1$, so the transported odd-degree cycle energy satisfies the exact degree-$1$ Lagrange lower bound $\bigl\|P_{Z_{P, \rho^\star}} P_{\mathrm{odd}, \rho^\star}\mathbf{1}_P(a)\bigr\|_{G_{\rho^\star}}^2 = \bigl\|P_{Z_P} P_{\mathrm{odd}}(\mathcal{C}_{\rho^\star}\mathbf{1}_P(a))\bigr\|_2^2 \ge \bigl\|P_{Z_P^1} P_{C_P^1}(\mathcal{C}_{\rho^\star}\mathbf{1}_P(a))\bigr\|_2^2 \equiv \frac{16\Lambda_+(P)^2}{S(P)}\sum_{p < q \le P}\frac{a_p^2 a_q^2(a_p^2 - a_q^2)^2}{(1+3a_p^2)^2(1+3a_q^2)^2} \ge 0$ (equal to $\frac{4}{325}$ at $P = 3, a_2 = \frac{1}{2}, a_3 = \frac{1}{3}$), which is strictly positive if and only if $P \ge 3$ and $a_p$ is not constant across $p \le P$ (vanishing identically when $P = 2$ or $a_2 = \dots = a_P = a \in (0, 1)$, where $\widetilde{\gamma}_p^{(+)} = g$ is constant so $\psi_P^{2r+1}(g a) = g^{2r+1}\mathbf{1}_P^{2r+1}(a) \in \operatorname{im}(d_P^*) = Z_P^\perp$ for all $r \ge 0$), whereas for the untransported Euclidean parity slices $\mu_P^{\mathrm{odd}}(a) = -\mathbf{1}_P^{\mathrm{odd}}(a) = \frac{1}{2}(\mu_P(a) - \mathbf{1}_P(a))$ and $\mu_P^{\mathrm{even}}(a) = \mathbf{1}_P^{\mathrm{even}}(a) = \frac{1}{2}(\mathbf{1}_P(a) + \mu_P(a))$, the signed and unsigned chains have identical slice norms $\bigl\|P_{Z_{P, \rho^\star}}\mu_P^{\mathrm{odd}}(a)\bigr\|_{G_{\rho^\star}}^2 \equiv \bigl\|P_{Z_{P, \rho^\star}}\mathbf{1}_P^{\mathrm{odd}}(a)\bigr\|_{G_{\rho^\star}}^2 \equiv \frac{1}{4}\bigl\|P_{Z_P}(\Lambda_+(P)\psi_P(\widetilde{\gamma}^{(+)}a) - \lambda_\mu(P)\mu_P(a))\bigr\|_2^2 > 0$ for all $P \ge 2$ (equal to $\frac{4}{25}$ at $P = 2, a_2 = \frac{1}{2}$ and $\frac{18074636}{38130625}$ at $P = 5, a_p = \frac{1}{p}$) and $\bigl\|P_{Z_{P, \rho^\star}}\mu_P^{\mathrm{even}}(a)\bigr\|_{G_{\rho^\star}}^2 \equiv \bigl\|P_{Z_{P, \rho^\star}}\mathbf{1}_P^{\mathrm{even}}(a)\bigr\|_{G_{\rho^\star}}^2 \equiv \frac{1}{4}\bigl\|P_{Z_P}(\Lambda_+(P)\psi_P(\widetilde{\gamma}^{(+)}a) + \lambda_\mu(P)\mu_P(a))\bigr\|_2^2$ (equal to $1$ at $P = 2, a_2 = \frac{1}{2}$), so in the untransported grading the whole-vector cycle-energy gap $\|P_{Z_{P, \rho^\star}}\mathbf{1}_P(a)\|_{G_{\rho^\star}}^2 - \|P_{Z_{P, \rho^\star}}\mu_P(a)\|_{G_{\rho^\star}}^2 = -4\operatorname{Re}\langle P_{Z_P}\mathcal{C}_{\rho^\star}\mu_P^{\mathrm{even}}(a), P_{Z_P}\mathcal{C}_{\rho^\star}\mu_P^{\mathrm{odd}}(a) \rangle_2$ arises from the $G_{\rho^\star}$-non-orthogonality of $P_{\mathrm{even}}$ and $P_{\mathrm{odd}}$; and (ii) under the fixed real Pythagorean metric $G_{P, \rho^\star(a)}$, any complex-twisted Möbius chain $\mu_P(a e^{-i\theta})$ satisfies the exact phase-dispersion Euler product $$\bigl\|\mu_P(a e^{-i\theta})\bigr\|_{G_{P, \rho^\star(a)}}^2 \;\equiv\; \left(\prod_{p \le P}\frac{(1 - a_p^2)^2}{1 + a_p^2}\right)\prod_{p \le P}\left(1 + \frac{8 a_p^2}{(1 - a_p^2)^2}\sin^2\!\left(\frac{\theta_p}{2}\right)\right) \tag{10c}$$ and maps under $\mathcal{C}_{\rho^\star(a)}$ to $\bigl[\prod_{p \le P}\bigl(1 - \frac{2a_p^2}{1+a_p^2}e^{-i\theta_p}\bigr)\bigr]\psi_P\bigl(\widetilde{\gamma}(\theta)a\bigr)$ with $\widetilde{\gamma}_p(\theta_p) := -\frac{(1 - a_p^2)e^{-i\theta_p}}{1 + a_p^2 - 2a_p^2 e^{-i\theta_p}}$, whose transported parity-sector cycle energies $\bigl\|P_{Z_P} P_{\mathrm{even/odd}}(\mathcal{C}_{\rho^\star(a)}\mu_P(a e^{-i\theta}))\bigr\|_2^2$ are given exactly by (10a)–(10b) with $\beta_p = \widetilde{\gamma}_p(\theta_p)a_p$.

Proof.

• (a): For any $m = p_1\cdots p_{k+1} \le x$ ($p_1 < \dots < p_{k+1}$), $m/p_j \le m/2 \le x/2$, so $(d_x^*\mu_x^k(a p^{-it}))_m = (-1)^k a_m m^{-it}\sum_{j=1}^{k+1}(-1)^{j-1} p_j^{it}$ while for $u > x$ ($P^+(u) \le x$, $\omega(u) = k+1$) only the $N_{\mathrm{ovf}}(u; x)$ largest prime factors satisfy $u/p_j \le x$, giving $(\mathcal{T}_x\mu_x^k(a p^{-it}))_u = (-1)^k a_u u^{-it}\sum_{j=k+2-N_{\mathrm{ovf}}(u; x)}^{k+1}(-1)^{j-1} p_j^{it}$. Substituting these into the degree-$k$ block of Theorem 1(a) ($\|d_x u\|_2^2 = S(x)\|u\|_2^2 - \|d_x^* u\|_2^2 - \|\mathcal{T}_x u\|_2^2$) yields the general frequency-$t$ identity and (at $t = 0$, where $\sum_{j=1}^{k+1}(-1)^{j-1} = \mathbf{1}_{\{k\text{ even}\}}$) proves $d_x^*\mu_x^{2r}(a) \equiv -\mu_x^{2r+1}(a)$, $d_x^*\mu_x^{2r+1}(a) \equiv 0$, (9a), and (9b). Substituting $\beta_p = -\alpha_p$ ($\psi_x(-\alpha) = \mu_x(\alpha)$) and $\rho_p^\star(\alpha_p) = \frac{2\overline{\alpha_p}}{1+a_p^2}$ into (7a) gives $\overline{\rho_p^\star(\alpha_p)} - \sqrt{1-|\rho_p^\star(\alpha_p)|^2}\alpha_p = \frac{2\alpha_p}{1+a_p^2} - \frac{1-a_p^2}{1+a_p^2}\alpha_p \equiv +\alpha_p$, so $\mathcal{C}_{x, \rho^\star(\alpha)}^*\mu_x(\alpha) \equiv \mathbf{1}_x(\alpha)$ and $(\mathcal{C}_{x, \rho^\star(\alpha)}^{-1})^*\mathbf{1}_x(\alpha) \equiv \mu_x(\alpha)$ at every $x \ge 1$; combining with $\mathbf{1}_x(a) - \mu_x(a) = 2\mathbf{1}_x^{\mathrm{odd}}(a) \equiv 2 d_x^*\mu_{x/2}^{\mathrm{even}}(a)$ and $d_x^*\mathcal{C}_{x/2, \rho^\star}^*\mu_{x/2}(a) = d_x^*\mathbf{1}_{x/2}(a) = \frac{1}{2}(\mathbf{1}_x(a) - \mu_x(a))$ proves (9c). Substituting $\rho_p^\star(\alpha_p) = \frac{2\overline{\alpha_p}}{1+a_p^2}$ and $\gamma_p a_p = -\alpha_p$ into (8) gives $1 + \rho_p^\star(\alpha_p)(-\alpha_p) = 1 - \frac{2a_p^2}{1+a_p^2} = \frac{1-a_p^2}{1+a_p^2} = \sqrt{1 - |\rho_p^\star(\alpha_p)|^2}$, so $\widetilde{\alpha}_p = -\alpha_p$, yielding (9); since $\mathcal{C}_\rho P_{Z_{P, \rho}} P_{\mathrm{even/odd}, \rho} w = P_{Z_P} P_{\mathrm{even/odd}}\mathcal{C}_\rho w$, the transported parity identities follow immediately from (9) and [4, Theorem 3].
• (b): On $C_{\mathrm{smooth}, P}^{\mathrm{full}}$, $P_{Z_P} = I - \frac{1}{S(P)}d_{\mathrm{full}}^* d_{\mathrm{full}}$, so $\langle P_{Z_P}\psi_P(\beta), \psi_P(\pm\beta) \rangle = \prod_{q \le P}(1 \pm |\beta_q|^2) - \frac{1}{S(P)}\langle d_{\mathrm{full}}\psi_P(\beta), d_{\mathrm{full}}\psi_P(\pm\beta) \rangle$. Expanding $d_{\mathrm{full}}\psi_P(\beta) = \sum_{p \le P} a_p\beta_p(\bigotimes_{q < p}(e_1 - \beta_q e_q))\otimes e_1 \otimes (\bigotimes_{r > p}(e_1 + \beta_r e_r))$ and taking inner products over diagonal ($p = p'$) and ordered off-diagonal ($p' < p$) prime pairs factors out $\prod_{r > p}(1 \pm |\beta_r|^2)$ and satisfies the one-step linear recurrence (10) (generalizing [4, Theorem 3] to complex weights $\beta_p \in \mathbb{D}$), establishing (10a)–(10b). On $C_P^1$, $P_{Z_P^1} = I_{C_P^1} - S(P)^{-1}|w_P\rangle\langle w_P|$ and $\widetilde{\gamma}_p^{(+)} - \widetilde{\gamma}_q^{(+)} = \frac{4(a_q^2 - a_p^2)}{(1+3a_p^2)(1+3a_q^2)}$, so by Lagrange's identity $\|P_{Z_P^1} P_{C_P^1}(\mathcal{C}_{\rho^\star}\mathbf{1}_P(a))\|_2^2 = \frac{\Lambda_+(P)^2}{S(P)}\sum_{p < q \le P} a_p^2 a_q^2(\widetilde{\gamma}_p^{(+)} - \widetilde{\gamma}_q^{(+)})^2 = \frac{16\Lambda_+(P)^2}{S(P)}\sum_{p < q \le P}\frac{a_p^2 a_q^2(a_p^2 - a_q^2)^2}{(1+3a_p^2)^2(1+3a_q^2)^2}$, which is $> 0$ whenever $P \ge 3$ and $a_p$ are not all equal; conversely, if $P = 2$ or $a_2 = \dots = a_P = a$, then $\widetilde{\gamma}_p^{(+)} = g$ is constant so $\psi_P^{2r+1}(g a) = g^{2r+1}\mathbf{1}_P^{2r+1}(a) \in \operatorname{im}(d_P^*) = Z_P^\perp$ for every $r \ge 0$. For the untransported slices, $\mu_P^{\mathrm{even}}(a) = \mathbf{1}_P^{\mathrm{even}}(a) = \frac{1}{2}(\mathbf{1}_P(a) + \mu_P(a))$ and $\mu_P^{\mathrm{odd}}(a) = -\mathbf{1}_P^{\mathrm{odd}}(a) = \frac{1}{2}(\mu_P(a) - \mathbf{1}_P(a))$ give $\mathcal{C}_{\rho^\star}\mu_P^{\mathrm{even/odd}}(a) = \pm\frac{1}{2}(\Lambda_+(P)\psi_P(\widetilde{\gamma}^{(+)}a) \pm \lambda_\mu(P)\mu_P(a))$ and the polarization identity $\|P_{Z_{P, \rho^\star}}\mathbf{1}_P(a)\|_{G_{\rho^\star}}^2 - \|P_{Z_{P, \rho^\star}}\mu_P(a)\|_{G_{\rho^\star}}^2 = -4\operatorname{Re}\langle P_{Z_P}\mathcal{C}_{\rho^\star}\mu_P^{\mathrm{even}}(a), P_{Z_P}\mathcal{C}_{\rho^\star}\mu_P^{\mathrm{odd}}(a) \rangle_2$. For $\mu_P(a e^{-i\theta})$ under $G_{P, \rho^\star(a)} = \bigotimes_{p \le P}\begin{pmatrix} 1 & \rho_p^\star \\ \rho_p^\star & 1 \end{pmatrix}$, each prime factor contributes $\begin{pmatrix} 1 & -a_p e^{i\theta_p} \end{pmatrix}\begin{pmatrix} 1 & \rho_p^\star \\ \rho_p^\star & 1 \end{pmatrix}\begin{pmatrix} 1 \\ -a_p e^{-i\theta_p} \end{pmatrix} = 1 + a_p^2 - \frac{4a_p^2}{1+a_p^2}\cos\theta_p = \frac{(1-a_p^2)^2 + 8a_p^2\sin^2(\theta_p/2)}{1+a_p^2}$, proving (10c), while (8) with $\gamma_p = -e^{-i\theta_p}$ gives $\widetilde{\gamma}_p(\theta_p)$. $\blacksquare$

Theorem 4 (Quotient-and-Core Shell Decomposition, Vertical $t$-Averaged Mean Values, Off-Diagonal Prime-Separation Kernel & Order-Ideal Approximation Residuals). (a) Quotient-and-Core Shell Decomposition, Unit-Modulus Dirichlet Expansion, Vertical $t$-Averaged Mean-Value & Off-Diagonal Kernel Theorem: For every cutoff $x \ge 2$, every complex variable $s = \sigma + it \in \mathbb{C}$, and every overlap sequence $\rho \in \mathbb{D}^{\pi(x)}$, the partial Dirichlet series $M_s(x) := \sum_{n \le x}\mu(n)n^{-s}$ of $\zeta(s)^{-1}$ is represented identically across the polydisc $\mathbb{D}^{\pi(x)}$ by the $G_{x, \rho}^{-1}$-bilinear form, the weighted diagonal trace, and (along any Selberg–Beltrami locus $\Lambda_p(w) = \frac{A_p(w)}{2 c_p(w)} \equiv \lambda \ne 0$, such as $w = 1, \rho_p = \rho_p^{\mathrm{Sel}}(a_p), \lambda = \frac{1}{2}$) the coexact vacuum coordinate $\mathcal{V}_{x, \rho}^{(w)}(s) := \bigl\langle e_1, d_{x, \rho}^{(w)}\mu_x\bigl(\frac{p^{-s}}{2 c(w)}\bigr)\bigr\rangle$: $$M_s(x) \;\equiv\; \Bigl\langle \psi_x(\overline{\rho} - 1), \, G_{x, \rho}^{-1}\,\psi_x\bigl(\overline{\rho} + (1 - |\rho|^2)p^{-s}\bigr)\Bigr\rangle \;\equiv\; \sum_{m \le x\text{ sqfree}}\mu(m)\left[\prod_{p \mid m}\Bigl(|\rho_p|^2 + (1 - |\rho_p|^2)p^{-s}\Bigr)\right](G_{x, \rho}^{-1})_{m, m}, \qquad \mathcal{V}_{x, \rho}^{(w)}(s) \;\equiv\; -\lambda\bigl(1 - M_s(x)\bigr). \tag{11a}$$ Moreover, for every non-vacuum coordinate $k = p_1\cdots p_r \in C_{x/2}$, let $n := \lfloor x/k \rfloor \in [2, \lfloor x/2 \rfloor]$ be its integer quotient and $m := k_{\le x/k} = k_{\le n} := \prod_{p \mid k, \, p \le n} p \mid P_{\pi(n)}^\#$ ($P_{\pi(n)}^\# := \prod_{p \le n} p$) be its $n$-smooth active core, so $k = m v$ with $n$-rough cofactor $v := k_{>n}$ satisfying $P^-(v) > n$ and $\frac{x}{(n+1)m} < v \le \frac{x}{nm}$. Retaining the scale $Y = \frac{x}{k}$ (or equivalently $n = \lfloor x/k \rfloor$) in the two-variable Buchstab functional $\mathcal{E}(Y; m; \gamma) = 1 - 2\sum_{j=1}^{\omega(m)}(-1)^{j-1}\Sigma_{p_j - 1}(Y; m; \gamma)$ of Theorem 2(b), all inactive prime factors $p \mid v$ decouple completely via $\bigl(d_{x, \rho}^{(w)}\mu_x(\beta)\bigr)_k \equiv -\lambda\,\mu_x(\beta)_k\,\mathcal{F}(n; m; \gamma)$ where $\mathcal{F}(n; m; \gamma) := \mathcal{E}(n; m; \gamma) - \mu(m)M_m(n; \gamma)$ (for example, at $x = 180, k = 35, n = 5, m = k_{\le 5} = 5, a_p = \frac{1}{2}, \rho_p = \frac{4}{5}, w = 1, \beta_p = \frac{5}{8}, \gamma_p = 1$, $\Sigma_4(5; 5; \mathbf{1}) = -1$ gives $\mathcal{E}(5; 5; \mathbf{1}) = 3$, $M_5(5; \mathbf{1}) = -1$, and $(d_{180, \rho}^{(1)}\mu_{180}(\beta))_{35} = -\frac{75}{512}$). Furthermore, $\Sigma_{p_j - 1}(Y; m; \gamma) \equiv \Sigma_{p_j - 1}(Y; m_{< p_j}; \gamma)$ depends only on the strict prefix $m_{< p_j} = p_1\cdots p_{j-1}$ and is constant across every contiguous prime run dividing $m$, so every contiguous prime run above the primorial prefix $P_s^\#$ of even length cancels identically in $\mathcal{E}(Y; m; \gamma)$, $\mathcal{E}(Y; 2m'; \gamma) \equiv -\mathcal{E}(Y; m'; \gamma_{\mathrm{odd}})$ ($2 \nmid m'$, $\gamma_{\mathrm{odd}}(d) := \mathbf{1}_{\{2 \nmid d\}}\gamma(d)$), $\mathcal{F}(n; P_{\pi(n)}^\#; \gamma) \equiv 0$, and $\mathcal{E}(Y; m; \gamma) \equiv \mu(m)$ holds identically in $(Y, \gamma)$ whenever every non-primorial maximal prime run of $m$ has even length (including all primorials $P_r^\# \in \{1, 2, 6, 30, \dots\}$ and all even-run cores $15, 35, 70, \dots$). Partitioning $C_{x/2} = \{1\} \sqcup \bigsqcup_{n=2}^{\lfloor x/2 \rfloor}\mathcal{S}_n(x)$ into quotient shells $\mathcal{S}_n(x) := \{k \text{ sqfree} : \lfloor x/k \rfloor = n\}$ yields the Exact Quotient-and-Core Norm Decomposition: $$\frac{1}{|\lambda|^2}\bigl\|d_{x, \rho}^{(w)}\,\mu_x(\beta)\bigr\|_2^2 \;\equiv\; \bigl|1 - M(x; \gamma)\bigr|^2 \;+\; \sum_{n = 2}^{\lfloor x/2 \rfloor}\sum_{\substack{m \mid P_{\pi(n)}^\# \\ m < P_{\pi(n)}^\#}} \bigl|\mathcal{F}(n; m; \gamma)\bigr|^2\left(\prod_{p \mid m}|\beta_p|^2\right)\sum_{\substack{\frac{x}{(n+1)m} < v \le \frac{x}{nm} \\ v\text{ sqfree},\; P^-(v) > n}}\prod_{q \mid v}|\beta_q|^2. \tag{11a'}$$ Consequently, letting $\Theta := \sup\{\operatorname{Re}(\rho_\zeta) : \zeta(\rho_\zeta) = 0\} \in [\frac{1}{2}, 1]$:

1. (Unit-Modulus Dirichlet Expansion, Unimodular Complement-Divisor Inversion, Closed-Form GCD Inverse $(W_n^{\mathrm{int}})^{-1}$, Pointwise Fixed-$s$ Coercivity, Support-Preserving Short-Core Bounds & Vertical $t$-Averaging): For every $n \ge 2$ and every active core $m \mid P_{\pi(n)}^\#$, the fiber amplitude $\mathcal{F}(n; m; \gamma) = \mathcal{E}(n; m; \gamma) - \mu(m)M_m(n; \gamma)$ expands on $\mathcal{I}_n := \{2 \le d \le n\text{ sqfree}\}$ ($|\mathcal{I}_n| = Q(n) - 1$) as $$\mathcal{F}(n; m; \gamma) \;\equiv\; -\sum_{\substack{d \in \mathcal{I}_n \\ \gcd(d, m) = 1}} (-1)^{r_m(d)}\,\mu(d)\,\gamma(d), \qquad r_m(d) \;:=\; \#\bigl\{p \mid m : p < P^+(d)\bigr\},$$ in which the principal frequency $d = 1$ cancels identically ($c_{n, m}(1) = 1 - (1 - \mu(m)) - \mu(m) \equiv 0$) on every fiber $m$, and every coprime squarefree frequency $d \in \mathcal{I}_n$ has unit-modulus sign $c_{n, m}(d) = -(-1)^{r_m(d)} \in \{\pm 1\}$. Writing $A_n := \prod_{p \le n}\frac{p}{p+1} = \frac{P_{\pi(n)}^\#}{\sigma_1(P_{\pi(n)}^\#)}$ and $\mathcal{Q}_n(\gamma) := \sum_{m \mid P_{\pi(n)}^\#, m < P_{\pi(n)}^\#} m^{-1}|\mathcal{F}(n; m; \gamma)|^2$, at $\rho_p \equiv \frac{1}{2}, w = 1$ ($\beta_p = \gamma_p = p^{-s}$, $\Lambda_p(1) = \frac{\sqrt{3}}{2}a_p \equiv \lambda$), the fixed-$s$ shell asymptotic is $c_n(\sigma, t) := \lim_{x \to \infty}(|\lambda|^2 x^{1 - 2\sigma})^{-1}\sum_{k \in \mathcal{S}_n(x)}|(d_{x, 1/2}^{(1)}\mu_x(p^{-s}))_k|^2 = \frac{6}{\pi^2} A_n(\int_n^{n+1}u^{2\sigma - 2}\,du)\mathcal{Q}_n(p^{-s})$, and summing over all active cores $m \mid P_{\pi(n)}^\#$ with normalized weight $A_n m^{-1}$ yields the $n$-independent off-diagonal prime-separation kernel $W(d_1, d_2)$ ($\sigma_p(d) := \mathbf{1}_{\{p \nmid d\}}(-1)^{\mathbf{1}_{\{p < P^+(d)\}}} \in \{-1, 0, +1\}$): $$A_n\,\mathcal{Q}_n(\gamma) \;\equiv\; \sum_{d_1, d_2 \in \mathcal{I}_n} W(d_1, d_2)\,\mu(d_1)\gamma(d_1)\,\overline{\mu(d_2)\gamma(d_2)}, \qquad W(d_1, d_2) \;\equiv\; \prod_{p \le n}\left(\frac{p}{p+1} + \frac{1}{p+1}\sigma_p(d_1)\sigma_p(d_2)\right) \;\equiv\; \left(\prod_{p \mid \operatorname{lcm}(d_1, d_2)}\frac{p}{p+1}\right)\left(\prod_{\substack{\min(P^+(d_1), P^+(d_2)) < p < \max(P^+(d_1), P^+(d_2)) \\ p \nmid d_1 d_2}}\frac{p-1}{p+1}\right). \tag{11a''}$$ This kernel structure yields three consequences:
   • (i) Unimodular Complement-Divisor Active-Core Inversion & Closed-Form Coprime-Vanishing Inverse $(W_n^{\mathrm{int}})^{-1}$: Re-indexing each active core $m = P_{\pi(n)}^\# / c$ by its complement divisor $c \in \mathcal{S}_{p_{\max}(n)} \setminus \{1\}$ converts the coprimality condition $\gcd(d, m) = 1$ into exact divisibility $d \mid c$ with fiber weight $A_n m^{-1} = \frac{c}{\sigma_1(P_{\pi(n)}^\#)}$ and $r_{P_{\pi(n)}^\#/c}(d) = \pi(P^+(d)) - 1 - \#\{p \mid c : p < P^+(d)\}$. Because $\mathcal{I}_n = \{2 \le c \le n\text{ sqfree}\}$ is a downward-closed order ideal under divisibility, the restriction $\mathcal{B}_n^{\mathrm{int}}$ to the $Q(n) - 1$ interior complement cores $c \in \mathcal{I}_n$ (rows indexed by $c \in \mathcal{I}_n$, columns indexed by $d \in \mathcal{I}_n$) is unimodular lower divisor-triangular ($\det\mathcal{B}_n^{\mathrm{int}} = \pm 1$, supported strictly on divisibility pairs $d \mid c$, so $(\mathcal{B}_n^{\mathrm{int}})_{c, d} = 0$ whenever $d \nmid c$) with exact $\{-1, 0, +1\}$-valued inverse $(\mathcal{B}_n^{\mathrm{int}})^{-1}$ (also indexed by row $c \in \mathcal{I}_n$, column $d \in \mathcal{I}_n$): $$\bigl(\mathcal{B}_n^{\mathrm{int}}\bigr)_{c, d} \;:=\; \mathbf{1}_{\{d \mid c\}}\,(-1)^{\pi(P^+(d)) - 1 - \#\{p \mid c \,:\, p < P^+(d)\}}, \qquad \bigl((\mathcal{B}_n^{\mathrm{int}})^{-1}\bigr)_{c, d} \;\equiv\; \mathbf{1}_{\{d \mid c\}}\,(-1)^{\pi(P^+(c))}\,\mu(c)\,(-1)^{\mathbf{1}_{\{P^+(c) \nmid d\}}} \qquad (c, d \in \mathcal{I}_n).$$ (For example, at $n = 3$, $\mathcal{I}_3 = \{2, 3\}$ and $P_2^\# = 6$; since $2 \nmid 3$, the $(c, d) = (3, 2)$ entry is $\mathbf{1}_{\{2 \mid 3\}} = 0$ while at $(3, 3)$, $P^+(3) = 3$, $\pi(3) - 1 = 1$, and $\#\{p \mid 3 : p < 3\} = 0$, so $\mathcal{B}_3^{\mathrm{int}} = (\mathcal{B}_3^{\mathrm{int}})^{-1} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ is diagonal. At $n = 6$, $\mathcal{I}_6 = \{2, 3, 5, 6\}$ and $P_3^\# = 30$; the only off-diagonal divisibility pairs $d \mid c$ ($d < c$) in $\mathcal{I}_6$ are $2 \mid 6$ and $3 \mid 6$ on row $c = 6$, giving $\mathcal{B}_6^{\mathrm{int}} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 \end{pmatrix}$ and $(\mathcal{B}_6^{\mathrm{int}})^{-1} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -1 & 1 & 0 & 1 \end{pmatrix}$.) Splitting $c \in \mathcal{S}_{p_{\max}(n)}\setminus\{1\}$ into interior cores $c \in \mathcal{I}_n$ and overflow cores $c > n$ decomposes $W_n = \bigl(W(d_1, d_2)\bigr)_{d_1, d_2 \in \mathcal{I}_n} \equiv W_n^{\mathrm{int}} + W_n^{\mathrm{ovf}} \succeq W_n^{\mathrm{int}} := \frac{1}{\sigma_1(P_{\pi(n)}^\#)}(\mathcal{B}_n^{\mathrm{int}})^T\operatorname{diag}(c)_{c \in \mathcal{I}_n}\mathcal{B}_n^{\mathrm{int}} \succ 0$, whose exact inverse $(W_n^{\mathrm{int}})^{-1} = \sigma_1(P_{\pi(n)}^\#)(\mathcal{B}_n^{\mathrm{int}})^{-1}\operatorname{diag}(c^{-1})_{c \in \mathcal{I}_n}((\mathcal{B}_n^{\mathrm{int}})^{-1})^T$ is given in closed form for all $d_1, d_2 \in \mathcal{I}_n$ ($g := \gcd(d_1, d_2)$, $q_i := P^+(d_i)$, $q_- := \min(q_1, q_2)$) by $$\bigl((W_n^{\mathrm{int}})^{-1}\bigr)_{d_1, d_2} \;\equiv\; \sigma_1\bigl(P_{\pi(n)}^\#\bigr)\,(-1)^{\pi(q_1) + \pi(q_2)}\,\mu(d_1)\mu(d_2)\begin{cases} 0, & \text{if } g = 1, \\[4pt] \dfrac{\sigma_1(g)}{g} - 1, & \text{if } g > 1\text{ and } (q_1 = q_2\text{ or } q_- \nmid g), \\[6pt] \dfrac{q_- - 1}{q_-}\,\dfrac{\sigma_1(g/q_-)}{g/q_-} - 1, & \text{if } q_1 \ne q_2\text{ and } q_- \mid g, \end{cases}$$ which vanishes identically on all coprime pairs $\gcd(d_1, d_2) = 1$ (e.g., $(W_3^{\mathrm{int}})^{-1} = \operatorname{diag}(6, 4)$ on $\mathcal{I}_3 = \{2, 3\}$ and $(W_6^{\mathrm{int}})^{-1} = \begin{pmatrix} 36 & 0 & 0 & -36 \\ 0 & 24 & 0 & -24 \\ 0 & 0 & 72/5 & 0 \\ -36 & -24 & 0 & 72 \end{pmatrix}$ on $\mathcal{I}_6 = \{2, 3, 5, 6\}$)! Consequently, for every complex sequence $\gamma$ (at any fixed $s \in \mathbb{C}$, with zero $t$-averaging) and any pairwise coprime subset $\mathcal{C} \subset \mathcal{I}_n$ (such as the primes $\{p \le n\}$), Cauchy–Schwarz on $\operatorname{span}(\mathcal{C})$ gives the pointwise lower bound $A_n\mathcal{Q}_n(\gamma) \ge \frac{1}{\sigma_1(P_{\pi(n)}^\#)}\sum_{d \in \mathcal{C}}\frac{d}{\sigma_1(d) - d}|\gamma(d)|^2 \ge \frac{1}{\sigma_1(P_{\pi(n)}^\#)}\sum_{p \le n} p\,|\gamma_p|^2$; furthermore, for every Bertrand prime $q \in (n/2, n]$ ($R_{>q} := \prod_{q < r \le n} r$) and every $m' \mid P_{\pi(q)-1}^\#$, pairing $m_1 = m' R_{>q}$ and $m_2 = q m' R_{>q}$ gives the exact martingale difference $\mathcal{F}(n; m' R_{>q}; \gamma) - \mathcal{F}(n; q m' R_{>q}; \gamma) \equiv \mu(m')\gamma_q$, yielding the $\sigma_1(P_{\pi(n)}^\#)$-free pointwise coercivity bound $A_n\mathcal{Q}_n(\gamma) \ge \sum_{n/2 < q \le n\text{ prime}}\frac{q}{(q+1)^2}\bigl(\prod_{q < r \le n}\frac{1}{r+1}\bigr)|\gamma_q|^2 \ge \frac{p_{\max}(n)}{(p_{\max}(n) + 1)^2}\,\bigl|\gamma_{p_{\max}(n)}\bigr|^2$.
   • (ii) Support-Preserving Short-Core Shell Decomposition: For any finite cutoff $x \ge 2$, quotient $n \ge 2$, and short-core threshold $1 \le M_0 \le \frac{x}{n(n+1)}$, the short-core kernel $W_{n, \le M_0}(d_1, d_2) := A_n\sum_{m \mid P_{\pi(n)}^\#, m \le M_0, \gcd(m, d_1 d_2)=1} m^{-1}(-1)^{r_m(d_1)+r_m(d_2)}$ satisfies $|W(d_1, d_2) - W_{n, \le M_0}(d_1, d_2)| \le M_0^{-\alpha}\frac{d_1}{\sigma_1(d_1)}\frac{d_2}{\sigma_1(d_2)}\bigl(\prod_{p \mid \gcd(d_1, d_2)}\frac{p+1}{p}\bigr)\prod_{p \le n, p \nmid d_1 d_2}\frac{1+p^{\alpha-1}}{1+p^{-1}}$ ($\forall \alpha \in (0, 1]$), while the finite-$Y$ $n$-rough cofactor count $N_n(Y) := \#\{v \le Y\text{ sqfree} : P^-(v) > n\}$ satisfies $|N_n(Y) - \frac{6}{\pi^2}A_n Y| \le (2^{\pi(n)} + 1)\sqrt{Y}$ ($\forall Y \ge 1$).
   • (iii) Vertical $t$-Averaged Mean Value & Comparison with the Montgomery–Vaughan Baseline [10, 11]: Under vertical $t$-averaging, $A_n\lim_{T \to \infty}\frac{1}{T}\int_T^{2T}\mathcal{Q}_n(p^{-\sigma - it})\,dt \equiv \sum_{d \in \mathcal{I}_n}\frac{d^{1 - 2\sigma}}{\sigma_1(d)}$ and $\sum_{n=2}^N \lim_{T \to \infty}\frac{1}{T}\int_T^{2T} c_n(\frac{1}{2}, t)\,dt \equiv \frac{6}{\pi^2}\sum_{2 \le d \le N\text{ sqfree}}\frac{\log((N+1)/d)}{\sigma_1(d)}$. Moreover, applying the Montgomery–Vaughan mean-value theorem [10, Corollary 2], [11, Section 4] directly to each fiber polynomial $\mathcal{F}(n; m; p^{-\sigma - it})$ (whose frequencies $\log d$ have spacing $\delta_d \ge \log(1 + 1/d) > \frac{1}{d + 1/2}$) and averaging with weight $A_n m^{-1}$ yields the $\log n$-free uniform bound $\bigl|\frac{1}{T}\int_T^{2T} A_n\mathcal{Q}_n(p^{-\sigma - it})\,dt - \sum_{d \in \mathcal{I}_n}\frac{d^{1 - 2\sigma}}{\sigma_1(d)}\bigr| \le \frac{\pi}{T}\sum_{d \in \mathcal{I}_n}\frac{d^{2 - 2\sigma}}{\sigma_1(d)}$ ($\forall T > 0, n \ge 2$, valid for all $n = o(T)$ at $\sigma = \frac{1}{2}$; by comparison, direct termwise integration of $W(d_1, d_2)(d_2/d_1)^{it}$ for $\sigma \ge \frac{1}{2}$ gives the elementary harmonic bound $\frac{2 H_{n-1}}{T}\sum_{d \in \mathcal{I}_n}\frac{d^{2 - 2\sigma}}{\sigma_1(d)}$). Finally, for $\sigma < \Theta$, $\bigl\|d_{x, \rho}^{(w)}\mu_x\bigl(\frac{p^{-\sigma}}{2 c(w)}\bigr)\bigr\|_2 = \Omega(x^{\Theta - \sigma - \varepsilon})$, and for $\sigma_0 \in [\frac{1}{2}, 1)$, $\Theta \le \sigma_0 \iff \mathcal{V}_{x, \rho}^{(w)}(\sigma) + \lambda(1 - \zeta(\sigma)^{-1}) = O_{\sigma, \varepsilon}(x^{\sigma_0 - \sigma + \varepsilon})$ ($\forall \sigma > \sigma_0$).
2. (Every-Coordinate Mellin Pole Separation & Zeta-Zero Divergence): For every squarefree integer $k \ge 1$ and $\gamma_p = 2 c_p(w)\beta_p = p^{-s}$, the Mellin transform of $(d_{x, \rho}^{(w)}\mu_x(\beta))_k$ satisfies $\int_k^\infty \frac{(d_{x, \rho}^{(w)}\mu_x(\beta))_k}{-\lambda\,\mu_x(\beta)_k} x^{-z-1}\,dx \equiv \frac{k^{-z}}{z}\bigl[\mathcal{E}_\infty(k; p^{-(s+z)}) - \frac{\mu(k)}{\zeta(s+z)\prod_{p \mid k}(1 - p^{-(s+z)})}\bigr]$, where $\mathcal{E}_\infty(k; \alpha) := 1 - 2\sum_{j=1}^{\omega(k)}(-1)^{j-1}\prod_{q < p_j, q \nmid k}(1 - \alpha_q)$ is an entire finite Dirichlet polynomial in $z$. By the classical pole-noncancellation mechanism (compare [9, Theorem 7.1]), at $s = \rho_\zeta = \beta_\zeta + i\gamma_\zeta$ (a nontrivial zero of $\zeta$ of multiplicity $m_\zeta \ge 1$), the smooth Buchstab term $\frac{k^{-z}}{z}\mathcal{E}_\infty(k; p^{-(\rho_\zeta + z)})$ has at most a simple pole ($O(z^{-1})$) at $z = 0$ (and for $s \ne \rho_\zeta$ is holomorphic at $z = \rho_\zeta - s \ne 0$) and therefore cannot cancel the order-$(m_\zeta + 1) \ge 2$ pole of $-\frac{\mu(k)k^{-z}}{z\zeta(\rho_\zeta + z)\prod_{p \mid k}(1 - p^{-(\rho_\zeta + z)})}$ at $z = 0$, forcing every coordinate $k \ge 1$ simultaneously to diverge logarithmically with amplification factor $\prod_{p \mid k}|1 - p^{-\rho_\zeta}|^{-1}$: $$\limsup_{x \to \infty}\frac{\left|\bigl(d_{x, \rho}^{(w)}\,\mu_x(\beta)\bigr)_k\right|}{(\log x)^{m_\zeta}} \;\ge\; \frac{|\lambda|\,\bigl|\mu_x(\beta)_k\bigr|}{\bigl|\zeta^{(m_\zeta)}(\rho_\zeta)\bigr|\prod_{p \mid k}\bigl|1 - p^{-\rho_\zeta}\bigr|} \;>\; 0 \quad (\forall k \ge 1\text{ sqfree}), \qquad \limsup_{x \to \infty} \frac{\left|\Bigl\langle \psi_x(\overline{\rho} - 1), \, G_{x, \rho}^{-1}\,w_{\rho_\zeta}\Bigr\rangle\right|}{(\log x)^{m_\zeta}} \;\ge\; \frac{1}{\bigl|\zeta^{(m_\zeta)}(\rho_\zeta)\bigr|} \;>\; 0. \tag{11b}$$
3. (Order-Ideal Approximation Residual & Its $t$-Independence): For the reciprocal-zeta cochain $w_s := \psi_x\bigl(\overline{\rho} + (1 - |\rho|^2)p^{-s}\bigr)$ of (11a), every divisor-closed approximant class $\mathcal{A} \subset C_x$ incurs by (6e) the exact $G_{x, \rho}$-approximation residual $\min_{v \in \mathcal{A}}\|v - G_{x, \rho}^{-1}w_s\|_{G_{x, \rho}}^2 \equiv \sum_{k \in C_x \setminus \mathcal{A}} k^{-2\sigma}\prod_{p \mid k}(1 - |\rho_p|^2)$, which depends only on $\sigma = \operatorname{Re}(s)$ and is independent of $t = \operatorname{Im}(s)$ (reducing at $\rho_p = 0$ to the Euclidean tail norm of $\mu_x(p^{-s})$); in particular, at every point $s = \frac{1}{2} + it$ on the critical line (whether or not $s$ is a zero of $\zeta$) and any overlap locus with $\Delta_\infty(\rho) := \prod_p(1 - |\rho_p|^2) > 0$, every sub-scale approximant $v \in C_y$ ($1 \le y < x$) incurs the logarithmic tail error $\min_{v \in C_y}\|v - G_{x, \rho}^{-1}w_{1/2 + it}\|_{G_{x, \rho}}^2 \ge \Delta_\infty(\rho)\sum_{y < k \le x\text{ sqfree}} k^{-1} = \frac{6\Delta_\infty(\rho)}{\pi^2}\log(x/y) + O(y^{-1/2})$.

(b) Asymptotic Critical-Weight Pythagorean GCD Invariants, Rational Zeta Ratios & Validated Rouché / Euler–Maclaurin Enclosures: For $a_p = p^{-\sigma}$ with $\sigma > 1/2$, as $P \to \infty$, the Möbius Cholesky eigenvalue $\lambda_\mu(\sigma)$, total $G_{\rho^\star}$-norm squared, and transported even-degree $G_{\rho^\star}$-cycle energy converge unconditionally to: $$\lambda_\mu(\sigma) \;=\; \frac{\zeta(4\sigma)}{\zeta(2\sigma)^2}, \qquad \|\mu(p^{-\sigma})\|_{G_{\rho^\star}}^2 \;=\; \frac{\zeta(4\sigma)}{\zeta(2\sigma)^3}, \qquad \bigl\|P_{Z_{\infty, \rho^\star}} P_{\mathrm{even}, \rho^\star}\mu(p^{-\sigma})\bigr\|_{G_{\rho^\star}}^2 \;=\; \bigl\|P_{Z_{\infty, \rho^\star}}\mu(p^{-\sigma})\bigr\|_{G_{\rho^\star}}^2 \;=\; \frac{\zeta(4\sigma)\bigl(\zeta(2\sigma)^2 - \zeta(4\sigma)\bigr)}{2\, P(2\sigma)\, \zeta(2\sigma)^5}, \tag{11c}$$ while the unsigned $G_{\rho^\star}$-norm squared admits the $O(p^{-4\sigma})$-convergent zeta-renormalized Euler product and $\mathbb{Z}[\sqrt{2}]$ Pell–Lucas prime-zeta expansion ($T_m := (3 + 2\sqrt{2})^m + (3 - 2\sqrt{2})^m$, $T_0 = 2, T_1 = 6, T_{m+1} = 6 T_m - T_{m-1}$): $$\|\mathbf{1}(p^{-\sigma})\|_{G_{\rho^\star}}^2 \;=\; \prod_{p \ge 2}\frac{1 + 6 p^{-2\sigma} + p^{-4\sigma}}{1 + p^{-2\sigma}} \;\equiv\; \zeta(2\sigma)^5\,\zeta(4\sigma)\prod_{p \ge 2}\Bigl(1 - 20 p^{-4\sigma} + 64 p^{-6\sigma} - 90 p^{-8\sigma} + 64 p^{-10\sigma} - 20 p^{-12\sigma} + p^{-16\sigma}\Bigr), \tag{11d}$$ $$\log\left(\frac{1 + 6 p^{-2\sigma} + p^{-4\sigma}}{1 + p^{-2\sigma}}\right) \;\equiv\; \sum_{m=1}^\infty \frac{(-1)^{m-1}(T_m - 1)}{m}\, p^{-2m\sigma} \qquad \bigl(\forall p^{2\sigma} > 3 + 2\sqrt{2}\bigr), \tag{11e}$$ and at every positive integer $\sigma = m \in \mathbb{Z}_{\ge 1}$, $\lambda_\mu(m) = \frac{2(2m)!^2}{(4m)!}\frac{|B_{4m}|}{B_{2m}^2} \in \mathbb{Q}$ ($\lambda_\mu(1) = \frac{2}{5}$, $\lambda_\mu(2) = \frac{6}{7}$). Moreover, for any $\sigma > 1/2$ and prime cutoff $P_{\max} \ge 2$, the infinite-prime phase-dispersion tail in (10c) satisfies $1 \le \prod_{p > P_{\max}}\bigl(1 + \frac{8 p^{-2\sigma}\sin^2(\theta_p/2)}{(1 - p^{-2\sigma})^2}\bigr) \le \exp\!\left(\frac{8 P_{\max}^{1 - 2\sigma}}{(2\sigma - 1)(1 - P_{\max}^{-2\sigma})^2}\right)$, and for $\operatorname{Re}(s) = \sigma_0 > 0$ ($s \ne 1$), defining the explicit upper majorant $\overline{R}_{N, M}(s) := \frac{2|B_{2M+2}|}{(2M+2)!}|(s)_{2M+2}|\frac{N^{-\sigma_0 - 2M - 1}}{\sigma_0 + 2M + 1}$ for the $(N, M)$ even-Bernoulli Euler–Maclaurin remainder $R_{N, M}(s) := -\frac{(s)_{2M+2}}{(2M+2)!}\int_N^\infty \bigl(B_{2M+2}(\{u\}) - B_{2M+2}\bigr) u^{-s-2M-2}\,du$, $\zeta(s)$ and its derivatives satisfy $|R_{N, M}(s)| \le \overline{R}_{N, M}(s)$, $|R'_{N, M}(s)| \le \overline{R}_{N, M}(s)\,D_{N, M}(s)$, and $|R''_{N, M}(s)| \le \overline{R}_{N, M}(s)\bigl(D_{N, M}(s)^2 + (\sigma_0 + 2M + 1)^{-2} + \sum_{j=0}^{2M+1}|s+j|^{-2}\bigr)$ where $D_{N, M}(s) := \log N + (\sigma_0 + 2M + 1)^{-1} + \sum_{j=0}^{2M+1}|s+j|^{-1}$.

Proof.

• (a): Substituting $\beta_p^{(1)} = \overline{\rho_p} - 1$ and $\beta_p^{(2)} = \overline{\rho_p} + (1 - |\rho_p|^2)p^{-s}$ into (7b) gives $\frac{(\overline{\beta_p^{(1)}} - \rho_p)(\beta_p^{(2)} - \overline{\rho_p})}{1 - |\rho_p|^2} = -p^{-s}$, while setting $\beta_p = -|\rho_p|^2 - (1 - |\rho_p|^2)p^{-s}$ in (6a) gives $\frac{\beta_p + |\rho_p|^2}{1 - |\rho_p|^2} = -p^{-s}$, and taking $k = 1$ in the Chiral Buchstab–Mertens Doubling Identity with $\gamma_p = 2 c_p(w)\frac{p^{-s}}{2 c_p(w)} = p^{-s}$ gives $\mathcal{V}_{x, \rho}^{(w)}(s) = -\lambda(1 - M_s(x))$, establishing (11a). For any $k = p_1\cdots p_r \in C_{x/2}$ with $n = \lfloor x/k \rfloor$ and $m = k_{\le n} = p_1\cdots p_{j_*}$, every $j > j_*$ has $p_j > n \ge u$ for all $u \le x/k$, so all $u \le x/k$ are automatically $(p_j - 1)$-smooth and coprime to $v = k_{>n}$; hence $\Sigma_{p_j - 1}(x/k; k; \gamma) \equiv \Sigma_{p_j - 1}(n; m; \gamma)$ for all $j \le j_*$ and $\Sigma_{p_j - 1}(x/k; k; \gamma) \equiv M_m(n; \gamma)$ for all $j > j_*$, and summing $-2\sum_{j=j_*+1}^r(-1)^{j-1} - (-1)^r = -\mu(m)$ proves $\mathcal{E}(x/k; k; \gamma) - \mu(k)M_k(x/k; \gamma) \equiv \mathcal{F}(n; m; \gamma)$. Since $P^+(u) \le p_j - 1 \implies \gcd(u, p_j\cdots p_{j_*}) = 1$, $\Sigma_{p_j - 1}(Y; m; \gamma) \equiv \Sigma_{p_j - 1}(Y; m_{< p_j}; \gamma)$; if $p_{i_j}$ and $p_{i_{j+1}}$ are consecutive primes ($i_{j+1} = i_j + 1$), the only prime in $(p_{i_j}-1, p_{i_{j+1}}-1]$ is $p_{i_j} \mid m$, which is excluded by $\gcd(u, m) = 1$, so $\Sigma_{p_{i_{j+1}}-1}(Y; m; \gamma) \equiv \Sigma_{p_{i_j}-1}(Y; m; \gamma)$ is constant across every contiguous prime run of $m$ and cancels in pairs on every even-length run. Grouping $k = mv \in \mathcal{S}_n(x)$ by $n = \lfloor x/k \rfloor$ and $m = k_{\le n} \mid P_{\pi(n)}^\#$ (where $\mathcal{F}(n; P_{\pi(n)}^\#; \gamma) = \mu(P_{\pi(n)}^\#)(1 - 1) = 0$) gives (11a'). Expanding $\mathcal{F}(n; m; \gamma) = \sum_{d \le n\text{ sqfree}, \gcd(d, m)=1} c_{n, m}(d)\mu(d)\gamma(d)$ for $m = p_1\cdots p_{\omega(m)} \mid P_{\pi(n)}^\#$, the $d = 1$ frequency appears in every $\Sigma_{p_j - 1}$ ($p_1 - 1 \ge 1$) and in $M_m(n; \gamma)$, giving $c_{n, m}(1) = 1 - 2\sum_{j=1}^{\omega(m)}(-1)^{j-1} - \mu(m) = 1 - (1 - \mu(m)) - \mu(m) \equiv 0$; for $d \in \mathcal{I}_n$ with $\gcd(d, m) = 1$, $P^+(d) \le p_j - 1 \iff j \ge r_m(d) + 1$ where $r_m(d) = \#\{p \mid m : p < P^+(d)\}$, so $c_{n, m}(d) = -2\sum_{j=r_m(d)+1}^{\omega(m)}(-1)^{j-1} - \mu(m) = -((-1)^{r_m(d)} - \mu(m)) - \mu(m) \equiv -(-1)^{r_m(d)} \in \{\pm 1\}$. Writing $\sigma_p(d) = \mathbf{1}_{\{p \nmid d\}}(-1)^{\mathbf{1}_{\{p < P^+(d)\}}}$, we have $\mathbf{1}_{\{\gcd(d, m)=1\}}(-1)^{r_m(d)} = \prod_{p \mid m}\sigma_p(d)$, so for all $d_1, d_2 \in \mathcal{I}_n$, $A_n\sum_{m \mid P_{\pi(n)}^\#} m^{-1}\mathbf{1}_{\{\gcd(d_1 d_2, m)=1\}}(-1)^{r_m(d_1)+r_m(d_2)} = \prod_{p \le n}\bigl(\frac{p}{p+1} + \frac{1}{p+1}\sigma_p(d_1)\sigma_p(d_2)\bigr) = W(d_1, d_2)$, proving (11a''). Under the complement substitution $c = P_{\pi(n)}^\# / m$, $\gcd(d, m) = 1 \iff d \mid c$ and $r_m(d) = \pi(P^+(d)) - 1 - \#\{p \mid c : p < P^+(d)\}$, so $(\mathcal{B}_n^{\mathrm{int}})_{c, d} = \mathbf{1}_{\{d \mid c\}}(-1)^{\pi(P^+(d)) - 1 - \#\{p \mid c : p < P^+(d)\}}$ is lower divisor-triangular on the downward-closed ideal $\mathcal{I}_n$ with diagonal entries $(-1)^{\pi(P^+(d)) - \omega(d)}$. For any $c \mid k$ in $\mathcal{I}_n$, setting $q = P^+(k)$ gives $\sum_{c \mid d \mid k}\bigl((\mathcal{B}_n^{\mathrm{int}})^{-1}\bigr)_{k, d}(\mathcal{B}_n^{\mathrm{int}})_{d, c} = (-1)^{\pi(q)+\pi(P^+(c))-1}\mu(k)\sum_{c \mid d \mid k}(-1)^{\mathbf{1}_{\{q \nmid d\}} - \#\{p \mid d : p < P^+(c)\}}$; if $c < k$, choosing $p_0 \mid (k/c)$ flips $(-1)^{\mathbf{1}_{\{q \nmid d\}}}$ when $p_0 = q$ and flips $(-1)^{-\#\{p \mid d : p < P^+(c)\}}$ when $p_0 < P^+(c)$ while leaving both signs invariant when $P^+(c) < p_0 < q$, so pairing $d$ and $d \triangle \{p_0\}$ cancels the sum to $0$ and proves $(\mathcal{B}_n^{\mathrm{int}})^{-1}\mathcal{B}_n^{\mathrm{int}} = I_{\mathcal{I}_n}$. Consequently, $\bigl((W_n^{\mathrm{int}})^{-1}\bigr)_{d_1, d_2} = \sigma_1(P_{\pi(n)}^\#)\sum_{c \mid g} c^{-1}\bigl((\mathcal{B}_n^{\mathrm{int}})^{-1}\bigr)_{d_1, c}\bigl((\mathcal{B}_n^{\mathrm{int}})^{-1}\bigr)_{d_2, c}$ ($g = \gcd(d_1, d_2)$, $c \ge 2$) vanishes when $g = 1$ and evaluates via $\sum_{c \mid g, c \ge 2} c^{-1} = \frac{\sigma_1(g)}{g} - 1$ (or $\sum_{c \mid (g/q_-)} c^{-1} - q_-^{-1}\sum_{c' \mid (g/q_-)} (c')^{-1} - 1 = \frac{q_- - 1}{q_-}\frac{\sigma_1(g/q_-)}{g/q_-} - 1$ when $q_1 \ne q_2$ and $q_- \mid g$) to the stated closed form, whose restriction to any pairwise coprime subset $\mathcal{C} \subset \mathcal{I}_n$ is diagonal with entries $\sigma_1(P_{\pi(n)}^\#)\frac{\sigma_1(d)-d}{d}$, giving $A_n\mathcal{Q}_n(\gamma) \ge \xi^* W_n^{\mathrm{int}}\xi \ge \frac{1}{\sigma_1(P_{\pi(n)}^\#)}\sum_{d \in \mathcal{C}}\frac{d}{\sigma_1(d)-d}|\gamma(d)|^2$. Moreover, for any prime $q \in (n/2, n]$ and $m' \mid P_{\pi(q)-1}^\#$, the only multiple of $q$ in $\mathcal{I}_n$ is $d = q$ itself (since $2q > n$) and every $d \in \mathcal{I}_n \setminus \{q\}$ coprime to $m' R_{>q}$ is $(q-1)$-smooth ($P^+(d) < q$), so $r_{q m' R_{>q}}(d) = r_{m' R_{>q}}(d)$ for all $d \ne q$, cancelling all $d \ne q$ in $\mathcal{F}(n; m' R_{>q}; \gamma) - \mathcal{F}(n; q m' R_{>q}; \gamma) = -(-1)^{\omega(m')}\mu(q)\gamma_q = \mu(m')\gamma_q$, and applying $(m_1^{-1} + m_2^{-1})^{-1} = \frac{q}{(q+1)m' R_{>q}}$ across the disjoint pairs $(m' R_{>q}, q m' R_{>q})$ yields $A_n\mathcal{Q}_n(\gamma) \ge \sum_{n/2 < q \le n\text{ prime}}\frac{q}{(q+1)^2}\bigl(\prod_{q < r \le n}\frac{1}{r+1}\bigr)|\gamma_q|^2$. Bounding $\mathbf{1}_{\{m > M_0\}} \le (m/M_0)^\alpha$ and $\mu(v)^2\mathbf{1}_{\{\gcd(v, P_{\pi(n)}^\#)=1\}} = \sum_{a^2 \mid v, \gcd(a, P_{\pi(n)}^\#)=1}\mu(a)\sum_{b \mid \gcd(v, P_{\pi(n)}^\#)}\mu(b)$ gives (ii). For (iii), applying [10, Corollary 2] (see also [11, Section 4], where the local frequency spacing $\delta_d = \min(\log\frac{d}{d-1}, \log\frac{d+1}{d}) = \log(1 + \frac{1}{d}) > \frac{1}{d + 1/2}$ has $\delta_d^{-1} < d + \frac{1}{2} < \frac{3}{2}d$ for $d \ge 2$) to each fiber polynomial $\mathcal{F}(n; m; p^{-\sigma - it})$ on $[T, 2T]$ and averaging over $m \mid P_{\pi(n)}^\#$ with weight $A_n m^{-1}$ yields $\bigl|\frac{1}{T}\int_T^{2T} A_n\mathcal{Q}_n(p^{-\sigma - it})\,dt - \sum_{d \in \mathcal{I}_n}\frac{d^{1 - 2\sigma}}{\sigma_1(d)}\bigr| \le \frac{\pi}{T}\sum_{d \in \mathcal{I}_n}\frac{d^{2 - 2\sigma}}{\sigma_1(d)}$; by comparison, for $\sigma \ge \frac{1}{2}$, integrating $W(d_1, d_2)(d_2/d_1)^{it}$ directly using $|\frac{1}{T}\int_T^{2T}(d_2/d_1)^{it}\,dt| \le \frac{2\max(d_1, d_2)}{T|d_2 - d_1|}$, $W(d_1, d_2) \le \min\bigl(\frac{d_1}{\sigma_1(d_1)}, \frac{d_2}{\sigma_1(d_2)}\bigr)$, and $d_1^{-\sigma}d_2^{-\sigma}\max(d_1, d_2) = (\max(d_1, d_2))^{1 - 2\sigma}\bigl(\frac{\min(d_1, d_2)}{\max(d_1, d_2)}\bigr)^{-\sigma} \le \max(d_1, d_2)^{2 - 2\sigma}$ gives the elementary harmonic bound $\frac{2 H_{n-1}}{T}\sum_{d \in \mathcal{I}_n}\frac{d^{2 - 2\sigma}}{\sigma_1(d)}$. Claim 2 follows by substituting $x = k Y$ ($Y \ge 1$) and comparing the simple pole of $\frac{k^{-z}}{z}\mathcal{E}_\infty(k; p^{-(\rho_\zeta + z)})$ at $z = 0$ with the order-$(m_\zeta + 1) \ge 2$ pole of $-\frac{\mu(k)k^{-z}}{z\zeta(\rho_\zeta + z)\prod_{p \mid k}(1 - p^{-(\rho_\zeta + z)})}$ [9, Theorem 7.1], and Claim 3 follows from (6e).
• (b): Since $\prod_{p \ge 2}(1 - p^{-2\sigma}) = \zeta(2\sigma)^{-1}$ and $\prod_{p \ge 2}(1 + p^{-2\sigma}) = \zeta(2\sigma)/\zeta(4\sigma)$ converge absolutely for $\sigma > 1/2$, taking $P \to \infty$ in Theorem 3(a) and [4, Theorem 4] yields (11c). Multiplying $\frac{1 + 6x + x^2}{1 + x}$ ($x = p^{-2\sigma}$) by $(1 - x)^5(1 - x^2) = (1 - x)^6(1 + x)$ cancels $1 + x$ and gives $(1 - x)^6(1 + 6x + x^2) \equiv 1 - 20 x^2 + 64 x^3 - 90 x^4 + 64 x^5 - 20 x^6 + x^8$, so factoring out $\zeta(2\sigma)^5\zeta(4\sigma) = \prod_{p \ge 2}\frac{1}{(1-x)^5(1-x^2)}$ gives (11d), and expanding $\log(1 + (3\pm 2\sqrt{2})x) - \log(1+x)$ gives (11e). Euler's formula $\zeta(2k) = \frac{(-1)^{k-1}(2\pi)^{2k} B_{2k}}{2(2k)!}$ cancels $\pi^{4m}$ at $\sigma = m \in \mathbb{Z}_{\ge 1}$. For $p > P_{\max}$, $\log\bigl(1 + \frac{8 p^{-2\sigma}\sin^2(\theta_p/2)}{(1 - p^{-2\sigma})^2}\bigr) \le \frac{8 p^{-2\sigma}}{(1 - P_{\max}^{-2\sigma})^2}$ and $\sum_{p > P_{\max}} p^{-2\sigma} \le \int_{P_{\max}}^\infty u^{-2\sigma}\,du = \frac{P_{\max}^{1 - 2\sigma}}{2\sigma - 1}$. In the even-index Euler–Maclaurin summation formula truncated after $B_{2M}$, integrating by parts once more with antiderivative $\frac{B_{2M+2}(\{u\}) - B_{2M+2}}{(2M+2)!}$ of $\frac{B_{2M+1}(\{u\})}{(2M+1)!}$ yields $R_{N, M}(s)$; since $\max_{u \in \mathbb{R}}|B_{2M+2}(\{u\})| = |B_{2M+2}|$, $|B_{2M+2}(\{u\}) - B_{2M+2}| \le 2|B_{2M+2}|$ for all $u \in \mathbb{R}$, and differentiating $(s)_{2M+2} u^{-s-2M-2}$ under the integral sign with respect to $s$ once and twice and bounding $\int_N^\infty (\log u)^r u^{-\sigma_0 - 2M - 2}\,du$ by the upper majorant $\overline{R}_{N, M}(s)$ yields the stated bounds for $|R_{N, M}(s)|$, $|R'_{N, M}(s)|$, and $|R''_{N, M}(s)|$. Finally, to certify the existence and simplicity of $\rho_j \in D(s_{j, 0}, r_0) \subset B_{r_0}(s_{j, 0})$ ($r_0 = 10^{-40}$) around each approximate zero $s_{j, 0} = \frac{1}{2} + i t_{j, 0}$, Taylor's theorem on the convex box $B_{r_0}(s_{j, 0})$ gives $|\zeta(s) - (\zeta(s_{j, 0}) + \zeta'(s_{j, 0})(s - s_{j, 0}))| \le \frac{1}{2} r_0^2 \sup_{w \in B_{r_0}(s_{j, 0})} |\zeta''(w)|$ on $|s - s_{j, 0}| = r_0$; verifying $|\zeta(s_{j, 0})| + \frac{1}{2} r_0^2 \sup_{B_{r_0}(s_{j, 0})}|\zeta''| < r_0|\zeta'(s_{j, 0})|$ via rigorous interval arithmetic implies by Rouché's theorem that $\zeta(s)$ has a unique simple zero $\rho_j \in D(s_{j, 0}, r_0)$, and evaluating $|\zeta'(B_{r_0}(s_{j, 0}))|$ encloses $|\zeta'(\rho_j)|$ across the entire certified root neighborhood. $\blacksquare$

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5. Verification Tables and Rigorous Interval Enclosures

All finite rational identities (including the Cartan overflow decomposition, harmonic reconstruction $d_{\mathrm{full}}(\mathcal{T}_x\Phi) = S(x)\Phi$ and $\|\mathcal{T}_x\Phi\|_2^2 = S(x)\|\Phi\|_2^2$ on $H_x^1$, divisor-triangular Cholesky factorization, order-ideal approximation residual (6e), and the discrete-sum, Euler–Maclaurin, direct Dirichlet-tail, and universal constant-overlap bounds in (6f)) were verified with zero rational error (fractions.Fraction) on the full complex $C_x \subset C_{\mathrm{smooth}, p_{\max}(x)}^{\mathrm{full}}$ ($x \in \{10, 14, 15, 22, 30, 40\}$, Table 1) in src/verify_gcd_cholesky_overflow_laplacian.py. The complex polydisc identities, full SVD unitary harmonic isomorphism, quotient-and-core shell decomposition (11a'), and contiguous prime-run telescoping were verified in complex double precision ($\le 4.02 \times 10^{-14}$) on the $11$-smooth completion $C_{x, 11} \subset C_{\mathrm{smooth}, 11}^{\mathrm{full}}$ ($x \in \{11, 13, 15, 17, 21, 30\}$) in src/verify_complex_cholesky_overflow_gram.py, and the unit-modulus Dirichlet expansion, unimodular complement-divisor active-core matrix $\mathcal{B}_n^{\mathrm{int}}$ and its $\{-1, 0, +1\}$-valued inverse, closed-form coprime-vanishing inverse $(W_n^{\mathrm{int}})^{-1}$, pointwise fixed-$s$ coercivity bounds, vertical $t$-averaged shell mean-value identity, and off-diagonal prime-separation kernel $W(d_1, d_2)$ in (11a'') were verified with zero rational error across all $n \in \{2, \dots, 15\}$ (together with floating-point evaluation of the closed-form trigonometric antiderivatives over $[T, 2T]$ at $n = 10, \sigma = 1/2, T \in \{10, 50, 200, 1000\}$ against the Montgomery–Vaughan [10] and elementary harmonic bounds) in src/verify_vertical_active_core_mean_values.py (Table 5). All asymptotic constants and nontrivial-zero invariants were enclosed via rigorous Euler–Maclaurin interval arithmetic (mpmath.iv.dps = 60–70 working precision via zeta_iv_rigorous and _zeta_and_deriv_iv_rigorous with zero calls to mp.zeta), and every exported (mid, rad) pair in results/gcd_cholesky_overflow_laplacian.json and results/complex_cholesky_overflow_gram.json is verified in exact rational arithmetic (certify_iv_pair) to satisfy $\operatorname{Fraction}(\texttt{mid}) - \operatorname{Fraction}(\texttt{rad}) \le a_{\mathrm{exact}} \le b_{\mathrm{exact}} \le \operatorname{Fraction}(\texttt{mid}) + \operatorname{Fraction}(\texttt{rad})$, enclosing both the internal interval width and the decimal rounding of the exported midpoint string: in gcd_cholesky_overflow_laplacian.json (50-digit midpoints), the four rational-zeta columns of Table 2 have certified radius $\le 4.68 \times 10^{-51}$, the $K_{\mathrm{pell}} = 500$ unsigned norm $\|\mathbf{1}(p^{-\sigma})\|_{G_{\rho^\star}}^2$ has certified radius $1.07 \times 10^{-27}$ at $\sigma = 0.6$, $2.62 \times 10^{-39}$ at $\sigma = 0.75$, and $\le 5.58 \times 10^{-50}$ for $\sigma \ge 1.0$, and the $K_p = 2000$ infinite unsigned recurrence at $\sigma = 1$ (bottom row of Table 3) has certified radius $\le 7.48 \times 10^{-8}$; in complex_cholesky_overflow_gram.json (40-digit midpoints), the root-derived quantities of Table 4 have certified radius $\le 1.26 \times 10^{-37}$ ($\rho_1$), $\le 8.02 \times 10^{-38}$ ($\rho_2$), and $\le 8.43 \times 10^{-38}$ ($\rho_3$), while the $P_{\max} = 7919$ phase-dispersion tail in (10c) at $\gamma_1$ has certified radii $13.4407$ ($\sigma = 0.75$), $8.32 \times 10^{-3}$ ($\sigma = 1.0$), and $1.04 \times 10^{-7}$ ($\sigma = 1.5$) (in Tables 2 and 4 below, midpoints are abbreviated to 18 and 20 decimal places for display, which introduces an additional $\pm 5 \times 10^{-19}$ and $\pm 5 \times 10^{-21}$ display rounding relative to the full 50- and 40-digit certified midpoints in the JSON files).

Table 1: Exact Rational Cartan Overflow Laplacian, Harmonic Reconstruction & GCD Cholesky Verification on Full $C_x$

| Cutoff $x$ | $\dim C_x$ | Off-Diagonal LCM-Overflow Pairs $(L_x)_{m, n} \ne 0$ | Harmonic Reconstruction $d_{\mathrm{full}}(\mathcal{T}_x\Phi) \equiv S(x)\Phi$ on $H_x^1$ | Cholesky, Order-Ideal (6e) & Overflow Bounds (6f) | Inverse Disjointness $(L_x)_{m, n}(G_{x, \rho}^{-1})_{m, n} \equiv 0$ | |:---:|:---:|:---:|:---:|:---:|:---:| | 10 | 7 | 5 | Exact (0 rational error) | Exact at $x=10$ (0 rational error) | Exact at $x=10$ (0 rational error) | | 14 | 10 | 15 | Exact (0 rational error) | Exact at $x=15$ (0 rational error) | Exact at $x=15$ (0 rational error) | | 22 | 15 | 33 | Exact (0 rational error) | Exact at $x=22$ (0 rational error) | Exact at $x=22$ (0 rational error) | | 30 | 19 | 50 | Exact (0 rational error) | Exact at $x=30$ (0 rational error) | Exact at $x=30$ (0 rational error) | | 40 | 26 | 90 | Exact (0 rational error) | Verified (SVD iso. $\le 5.6 \times 10^{-16}$ on $C_{x,11}$) | Exact (0 rational error) |

Table 2: Certified Interval Enclosures for Pythagorean GCD Metric Invariants ($a_p = p^{-\sigma}$; 18-Digit Display, Full 50-Digit JSON Midpoints: Cols. 2–6 Radius $\le 4.68 \times 10^{-51}$, Col. 7 Radius Listed Below)

| $\sigma$ | Cholesky Eigenvalue $\lambda_\mu(\sigma) = \frac{\zeta(4\sigma)}{\zeta(2\sigma)^2}$ | Möbius Norm $\|\mu(p^{-\sigma})\|_{G_{\rho^\star}}^2$ | Transported Even Cycle $\|P_{Z_{\infty, \rho^\star}} P_{\mathrm{even}, \rho^\star}\mu\|_{G_{\rho^\star}}^2$ | Transported Odd Cycle $\|P_{Z_{\infty, \rho^\star}} P_{\mathrm{odd}, \rho^\star}\mu\|_{G_{\rho^\star}}^2$ | Cycle Share $\frac{\|P_Z \mu\|^2}{\|\mu\|^2}$ | Unsigned Norm $\|\mathbf{1}(p^{-\sigma})\|_{G_{\rho^\star}}^2$ (50-Digit JSON Radius) | |:---:|:---:|:---:|:---:|:---:|:---:|:---:| | 0.60 | 0.044244609625049993 | 0.007912717033235905 | 0.002488083839240829 | 0 (exact) | 0.314441149454743851 | 207.2720545745806870 ($1.07 \times 10^{-27}$) | | 0.75 | 0.176138329414212231 | 0.067424587168472033 | 0.032692427583082417 | 0 (exact) | 0.484873974851261789 | 17.04352930786598468 ($2.62 \times 10^{-39}$) | | 1.00 | 0.4 ($2/5$, JSON rad $4.80 \times 10^{-52}$) | 0.243170840741610651 | 0.161308277260838748 | 0 (exact) | 0.663353701327381922 | 4.735464761658951740 ($5.58 \times 10^{-50}$) | | 1.50 | 0.704072487320784478 | 0.585723093033397288 | 0.495905700254270295 | 0 (exact) | 0.846655537663757942 | 1.971113090590836351 ($1.90 \times 10^{-50}$) | | 2.00 | 0.857142857142857143 ($6/7$) | 0.791947202504220143 | 0.734710358545453469 | 0 (exact) | 0.927726439618982467 | 1.391927015192705165 ($1.35 \times 10^{-50}$) |

Table 3: Exact Rational Finite-$P$ Pythagorean Gauge-Symmetry Breaking at $a_p = 1/p$ ($\sigma = 1$, Transported Parity Projectors $P_{\mathrm{even/odd}, \rho^\star}$; $P = \infty$ Unsigned Cycle Columns Certified to Radius $\le 7.48 \times 10^{-8}$)

| Prime Cutoff $P$ ($\pi(P)$) | $\lambda_\mu(P)$ | $\|\mu_P\|_{G_{\rho^\star}}^2$ | $\|P_{Z_{P, \rho^\star}} P_{\mathrm{even}, \rho^\star}\mu_P\|_{G_{\rho^\star}}^2$ | $\|P_{Z_{P, \rho^\star}} P_{\mathrm{odd}, \rho^\star}\mu_P\|_{G_{\rho^\star}}^2$ | $\|\mathbf{1}_P\|_{G_{\rho^\star}}^2$ | $\|P_{Z_{P, \rho^\star}} P_{\mathrm{even}, \rho^\star}\mathbf{1}_P\|_{G_{\rho^\star}}^2$ | $\|P_{Z_{P, \rho^\star}} P_{\mathrm{odd}, \rho^\star}\mathbf{1}_P\|_{G_{\rho^\star}}^2$ | |:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:| | 5 (3) | 0.44307692 | 0.28356923 | 0.19686097 | 0 (exact) | 3.69827009 | 3.27549124 | 0.02915536 | | 7 (4) | 0.42535385 | 0.26667082 | 0.18177266 | 0 (exact) | 4.06960659 | 3.54529725 | 0.04177693 | | 13 (6) | 0.41345870 | 0.25554991 | 0.17201086 | 0 (exact) | 4.36142482 | 3.75188154 | 0.05306777 | | 29 (10) | 0.40583155 | 0.24850829 | 0.16590142 | 0 (exact) | 4.56740633 | 3.89569948 | 0.06133688 | | 97 (25) | 0.40145755 | 0.24450119 | 0.16245003 | 0 (exact) | 4.69260522 | 3.98242508 | 0.06643407 | | $\infty$ | 0.40000000 ($2/5$) | 0.24317084 | 0.16130828 | 0 (exact) | 4.73546476 | 4.01201568 | 0.06818086 |

(Note: For the untransported Euclidean parity slices $\mu_P^{\mathrm{odd}}(a) = -\mathbf{1}_P^{\mathrm{odd}}(a)$, both chains share the positive untransported odd-slice energy $\|P_{Z_{P, \rho^\star}}\mu_P^{\mathrm{odd}}(a)\|_{G_{\rho^\star}}^2 = \|P_{Z_{P, \rho^\star}}\mathbf{1}_P^{\mathrm{odd}}(a)\|_{G_{\rho^\star}}^2 = \frac{4}{25} = 0.16$ at $P = 2, a_2 = \frac{1}{2}$ and $\frac{18074636}{38130625} \approx 0.47401887$ at $P = 5, a_p = \frac{1}{p}$, while the transported odd-cycle energy of $\mathbf{1}_P(a)$ at $P = 3, a_2 = \frac{1}{2}, a_3 = \frac{1}{3}$ is $\frac{4}{325}$.)

Table 4: Validated Rouché Root Disks ($r_0 = 10^{-40}$), Multi-Shell Lower-Bound Constants & Non-Vacuum Zeta-Zero Pole Constants (Theorem 4(a); 20-Digit Display, Full 40-Digit JSON Midpoints Certified to Radius $\le 1.26 \times 10^{-37}$)

| Nontrivial Zero $\rho_j = \frac{1}{2} + i\gamma_j \in D(s_{j, 0}, 10^{-40})$ | $\bigl|\zeta'(\rho_j)\bigr|$ | Vacuum Pole Const. $\frac{1}{\bigl|\zeta'(\rho_j)\bigr|}$ | $k=2$ Pole Amp. $\bigl|1 - 2^{-\rho_j}\bigr|^{-1}$ | Shell Invariant $\mathcal{Q}_3(p^{-\rho_j})$ | Multi-Shell Bound $C_{\mathrm{shell}, 4}(\rho_j)$ | 40-Digit JSON Radius | |:---:|:---:|:---:|:---:|:---:|:---:|:---:| | $\rho_1 = \frac{1}{2} + i\,14.134725141734693790\dots$ | $0.79316043335650611601$ | $1.26077897729742727049$ | $0.59579338258662571919$ | $1.86187477088084623557$ | $0.37126211861541268764$ | $\le 1.26 \times 10^{-37}$ | | $\rho_2 = \frac{1}{2} + i\,21.022039638771554993\dots$ | $1.13683910682797482214$ | $0.87963194966983031716$ | $0.69089862210483629953$ | $0.65981170550777400674$ | $0.18461493695912972089$ | $\le 8.02 \times 10^{-38}$ | | $\rho_3 = \frac{1}{2} + i\,25.010857580145688763\dots$ | $1.37172128721612993731$ | $0.72901106756859630456$ | $0.83953497836379650659$ | $0.55080636726727463500$ | $0.16768941969289160211$ | $\le 8.43 \times 10^{-38}$ |

Table 5: Exact Rational Quotient-Shell Invariants, Pointwise ($t = 0$) Coercivity Bounds & Vertical Mean Values (Theorem 4(a)(1), (11a''))

| Quotient Shell $n$ ($\pi(n)$, $\|\mathcal{I}_n\|$) | Exact Vertical Mean at $\sigma = \frac{1}{2}$: $\sum_{d \in \mathcal{I}_n}\frac{1}{\sigma_1(d)}$ | Exact Vertical Mean at $\sigma = 1$: $\sum_{d \in \mathcal{I}_n}\frac{1}{d\,\sigma_1(d)}$ | Pointwise Fixed-$s$ ($s = 1$): $A_n\mathcal{Q}_n(p^{-1})$ | Interior Lower Bound $\xi^* W_n^{\mathrm{int}}\xi$ ($s = 1$) | Unimodular $\mathcal{B}_n^{\mathrm{int}}$, $(W_n^{\mathrm{int}})^{-1}$ & $W_n$ Residual | |:---:|:---:|:---:|:---:|:---:|:---:| | 2 (1, 1) | $\frac{1}{3} \approx 0.333333$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | 0 (exact) | | 3 (2, 2) | $\frac{7}{12} \approx 0.583333$ | $\frac{1}{4}$ | $\frac{5}{12}$ | $\frac{5}{72}$ | 0 (exact) | | 4 (2, 2) | $\frac{7}{12} \approx 0.583333$ | $\frac{1}{4}$ | $\frac{5}{12}$ | $\frac{5}{72}$ | 0 (exact) | | 5 (3, 3) | $\frac{3}{4} \approx 0.750000$ | $\frac{17}{60}$ | $\frac{53}{90}$ | $\frac{31}{2160}$ | 0 (exact) | | 6 (3, 4) | $\frac{5}{6} \approx 0.833333$ | $\frac{107}{360}$ | $\frac{157}{360}$ | $\frac{37}{720}$ | 0 (exact) | | 10 (4, 6) | $\frac{73}{72} \approx 1.013889$ | $\frac{101}{315}$ | $\frac{1207}{2520}$ | $\frac{127}{13440}$ | 0 (exact) | | 15 (6, 10) | $\frac{631}{504} \approx 1.251984$ | $\frac{48931}{144144}$ | $\frac{1063723}{2162160}$ | $\frac{403867}{2905943040}$ | 0 (exact) |

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References

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