1. Problem and Contribution

A foundational problem in the study of the Hodge Conjecture [1, 2, 3] is to understand the quantitative geometric complexity required to represent a rational Hodge class $\alpha \in \mathrm{Hdg}^p(X, \mathbb{Q}) = H^{2p}(X, \mathbb{Q}) \cap H^{p,p}(X)$ by algebraic cycles. When $\alpha$ is not generated by intersections of divisor classes in $H^{1,1}(X, \mathbb{Q})$ or linear subspaces $\mathbb{P}^p \subset X$, two complementary mechanisms govern its cycle realization:

1. Differential-Geometric Calibration & Endomorphism Projectors on Weil Abelian Varieties: Every effective analytic $p$-cycle $Z = \sum_r c_r [Z_r]$ ($c_r > 0$) on a complex torus or projective manifold induces a positive semidefinite $(p,p)$-current whose tangent spaces $T_z Z_r^{\mathrm{sm}} \in \mathrm{Gr}(n-p, T_z X)$ lie on the rank-$1$ Plücker-decomposable cone of $\mathbb{P}(\bigwedge^{n-p} T_z X)$, while its Poincaré dual cohomology class is represented by its conormal forms in $\bigwedge^p H^{1,0}(X) \wedge \bigwedge^p H^{0,1}(X)$. How efficiently can a family of $m$-planes calibrate an indefinite primitive class such as a Weil class [4, 5, 6] on an abelian $2m$-fold $A$ (for which Markman [6] established the Hodge conjecture on abelian fourfolds and discriminant-$(-1)$ Weil sixfolds via secant sheaves), how does the curvature of the Gauss map $\gamma_Z: Z^{\mathrm{sm}} \to \mathrm{Gr}(m, 2m)$ constrain the Kähler volume of $Z$ when $A$ is simple, and how can one project $H^{2m}(A, \mathbb{Q})$ onto the $2$-dimensional rational Weil space on every simple fiber?
2. Closed-Form Complete-Intersection Jacobians & Sporadic Parity Syzygies on Fermat Varieties: On a smooth projective fourfold $X_d^4 = \{F_d = 0\} \subset \mathbb{P}^5$ ($d \ge 4$), Carlson-Griffiths [7, 8] and Villaflor [9, 10] identify $H^{2,2}_{\mathrm{prim}}(X_d^4, \mathbb{C}) \cong R(F_d)_{3d-6}$ in the Gorenstein Jacobian ring $R(F_d) = \mathbb{C}[x_0, \dots, x_5]/J(F_d)$, where the annihilator and Hilbert-series Catalecticant rank of a complete intersection $Z = \{g_1=g_2=g_3=0\} \subset X_d^4$ are given by the classical Gorenstein residuation theorems of Dan [21], Villaflor Loyola [22], and Kloosterman [20]. On Fermat varieties $X_d^{2p}$, the Hodge conjecture for $d \le 20$ and for prime powers or twice prime powers is classical by Shioda [11] and Aoki [12, 18], was explicitly realized on the Fermat sextic fourfold $X_6^4$ via linear, Aoki–Shioda, and "type 3" complete-intersection surfaces (with factored residue polynomials and Catalecticant ranks $19, 62, 106$) by Braun, Fortin, Lopez Garcia, and Villaflor Loyola [23], and was recently extended to $d < 65$ ($d \notin \{44, 51, 52\}$) via lifted length-reduction algorithms in Aoki's formal tuple module by Martelotte, Miranda, Movasati, and Villaflor [19]. Treating the Gorenstein Catalecticant rank theorem [20, 21, 22] and the sextic fourfold realizations [23] as background, we ask: can one derive closed-form factored expansions of Villaflor's $6 \times 6$ Jacobian determinants $P_Z = \det\frac{\partial(g,h)}{\partial x}$ (omitted as computationally tedious in [19, Proof of Proposition 3.7]) for all even-degree entangled double-Aoki–Shioda surfaces $Z^{(2m)} \subset X_{2m}^4$ ($2m \ge 8$) and the nonic join $Z_{\mathrm{nonic}} \subset X_9^4$, and at composite degrees divisible by $12$—where Aoki's order-$2$ parity obstruction group $G_{12} \simeq \mathbb{Z}/2\mathbb{Z}$ [12, Theorem A; 19, §2.2] prevents standard binomial complete intersections $A_{p, 12}$ (and $Z^{(12)}$) from spanning the Hodge group—can one construct an explicit single complete-intersection curve $C_{\mathrm{spor}} \subset X_{12}^2$ and entangled surfaces on $X_{12}^4$ realizing Aoki's sporadic half-integral generator $S_4 = (1, 6, 8, 9)$?

In this paper, we establish six exact theorems backed by self-contained symbolic-computational certificates (hodge_loci_engine in supplement.zip):

• Theorem 1 (Principal-Angle Calibration & Maclaurin Exterior-Block Inequality for Weil Classes, Section 2): Using the Cosine-Sine (CS) principal-angle decomposition on $\mathrm{Gr}(m, V^{1,0}_+ \oplus V^{1,0}_-)$ and the Cauchy-Binet formula on arbitrary graph matrices $B \in M_m(\mathbb{C})$, we prove that every rank-$1$ Plücker projection operator $P_{\Gamma_B} = u_B u_B^\dagger$ satisfies $$\langle P_{\Gamma_B}, \Psi_- \rangle = \frac{\det B}{\det(I_m + B^\dagger B)}, \qquad \mathrm{Tr}_{E_k}(P_{\Gamma_B}) = \frac{\mathrm{Tr}(\bigwedge^k(B^\dagger B))}{\det(I_m + B^\dagger B)} = \frac{e_k(\sigma_1^2, \dots, \sigma_m^2)}{\prod_{a=1}^m (1+\sigma_a^2)},$$ where $\sigma_a = \tan\theta_a$ are the singular values of $B$. Consequently, $|\langle P_{\Gamma_B}, \Psi_\pm \rangle| = \prod_{a=1}^m \cos\theta_a \sin\theta_a \le 2^{-m}$, with equality if and only if $B \in \mathrm{U}(m)$, where the exterior-block traces follow the exact binomial distribution $\mathrm{Tr}_{E_k}(P_{\Gamma_U}) = \binom{m}{k}2^{-m}$. Applying Maclaurin's symmetric-mean inequality $e_k(y)/\binom{m}{k} \ge (e_m(y))^{k/m}$ yields the universal pointwise and cycle exterior-block inequality $$\mathrm{Tr}_{E_k}(P_W) + \mathrm{Tr}_{E_{m-k}}(P_W) \ge 2\binom{m}{k}\big|\langle P_W, \Psi_\pm \rangle\big|,$$ establishing a sharp $2^{m-1}$ calibration bound in the real cone of decomposable Plücker operators via the $2m$-plane Vandermonde identity.
• Theorem 2 (Factored Double-Monomial $\mathrm{SU}(m)$-Design & Conormal Poincaré Dual Effective Weil Cycle Realization, Section 3): Because the Vandermonde planes $\Gamma_{e^{i\pi j/m} I_m}$ of Theorem 1(c) are lattice-rational on the split CM model $E_i^{2m}$ only when $2m \mid 4$ ($m=2$), we introduce for arbitrary $m \ge 2$ the finite special monomial group $G_m = \{D_\varepsilon P_\pi \in \mu_4^m \rtimes S_m : \det g = 1\} \subset \mathrm{SU}(m) \cap \mathrm{SL}_m(\mathbb{Z}[i])$ of order $|G_m| = 4^{m-1} m!$, whose graphs are lattice-rational abelian subvarieties of $E_i^{2m}$ for all $m \ge 2$. We prove that the factored double-monomial Reynolds projector $\mathcal{R}_{G_m \times G_m}$ on $\mathrm{End}(\bigwedge^m V^{1,0})$ satisfies the universal operator identity $$\mathcal{R}_{G_m \times G_m}(P_{\Gamma_B}) = \sum_{k=0}^m \frac{e_k(B^\dagger B)}{\binom{m}{k}^2 \det(I_m + B^\dagger B)} \Pi_{E_k} + \frac{\overline{\det B}}{\det(I_m + B^\dagger B)}\Psi_+ + \frac{\det B}{\det(I_m + B^\dagger B)}\Psi_-$$ for every (not necessarily unitary) matrix $B \in M_m(\mathbb{C})$. Carefully distinguishing tangent Plücker vectors from conormal Poincaré dual classes $[\Gamma_g] = f_g^*[0] = \bigwedge_{a=1}^m \frac{i}{2}d(w-gz)_a \wedge d\overline{(w-gz)}_a$ (whose conormal matrix $(-g \mid I_m)$ carries extreme minor $(-1)^m\det g = (-1)^m$), we prove that averaging the sign-adjusted graphs $\Gamma_{-g} = \{w + gz = 0\}$ ($g \in G_m$) plus positive coordinate subtori yields an explicit effective lattice-rational algebraic cycle $\mathcal{Z}_{\mathrm{eff}}^{(m)} \ge 0$ on $E_i^{2m}$ representing $H([\mathcal{Z}_{\mathrm{eff}}^{(m)}]) = I_{\mathcal{H}} + \Psi_+ + \Psi_- = \frac{1}{m!}H(\omega_Q^m) + 2H(\alpha_{W,1})$ for every $m \ge 2$, and we prove via the Plücker rank-$1$ Grassmannian criterion that the Hodge locus of the cohomology class $[\mathcal{Z}_{\mathrm{eff}}^{(m)}]$ in $\mathfrak{H}_{2m}(Q)$ is the $m^2$-dimensional Weil PEL Shimura variety $\mathfrak{H}_{\mathrm{Weil}}(\phi)$.
• Theorem 3 (Plücker Fubini-Study Levi-Form Rigidity & Universal Chebyshev-Weil Endomorphism Projector on Simple Weil Varieties, Section 4): We prove that the calibration-saturating locus $\mathcal{U} = \mathrm{U}(m) \subset \mathrm{Gr}(m, 2m)$ is maximally totally real ($T_U \mathrm{U}(m) \cap i T_U \mathrm{U}(m) = \{0\}$), and that along any holomorphic Gauss map $z \mapsto B(z)$ away from $\{\det B = 0\}$, the complex Hessian (Levi form) of the logarithmic Weil calibration defect $\mathcal{L}_{\mathrm{Weil}}(B(z)) = -\log(2^m |\langle P_{\Gamma_{B(z)}}, \Psi_- \rangle|) = \sum_{a=1}^m \log\cosh(\log\sigma_a(z))$ is identically the pullback of the Plücker Fubini-Study Kähler metric: $$i\partial\bar{\partial}\mathcal{L}_{\mathrm{Weil}}(B(z)) = i\,\mathrm{Tr}\!\left((I_m + B B^\dagger)^{-1} \partial B \wedge (I_m + B^\dagger B)^{-1} \bar{\partial} B^\dagger\right) = \gamma_Z^*(\omega_{\mathrm{Pl\ddot{u}cker}}) \ge 0.$$ Accounting for the exact wedge-reordering phase $\psi_-|_{w=Bz} := (d\bar{z}_1 \wedge \dots \wedge d\bar{z}_m \wedge dw_1 \wedge \dots \wedge dw_m)|_{w=Bz} = (-1)^{\binom{m}{2}}(2i)^m (\det B) v_0$, we prove that on any simple Weil abelian $2m$-fold $A_\tau$ ($m \ge 2$), every effective $m$-cycle $Z$ with $\mathrm{Re}((-1)^{\binom{m}{2}}(-i)^m \int_Z \psi_-) > 0$ obeys a strict Kähler-to-Weil calibration deficit $\Delta_{\mathrm{curv}}(Z) := \frac{1}{m!}\int_Z \omega_Q^m - \mathrm{Re}((-1)^{\binom{m}{2}}(-i)^m \int_Z \psi_-) > 0$. Conversely, using the norm-normalized integer-endomorphism correspondences $T_{a_j, b_j} := \frac{[a_j + b_j\phi]^* + [a_j - b_j\phi]^*}{2(a_j^2 + d b_j^2)^m} \in \mathbb{Q}[\mathrm{End}_{\mathbb{Q}}(A_\tau)]$, we construct an $(m+1)$-point rational Universal Chebyshev-Weil Endomorphism Projector $\Pi_{\mathrm{Cheb}}^{(m)} = \sum_{j=0}^m c_j T_{j, 1} \in \mathrm{Corr}^0(A_\tau, A_\tau)_{\mathbb{Q}}$ (with closed-form Lagrange-Chebyshev weights $c_j = (2^{m-1}\prod_{\ell \ne j}(x_j - x_\ell))^{-1}$ at $x_j = \frac{j^2-d}{j^2+d}$) that acts as an idempotent projector onto $W_{\mathrm{Weil}}(A_\tau)$ on middle cohomology $H^{2m}(A_\tau, \mathbb{Q})$ (and on $H^*(A_\tau, \mathbb{Q})$ upon composition with Lieberman's Künneth projector $\pi_{2m}$) across every simple fiber $A_\tau \in \mathfrak{H}_{\mathrm{Weil}}(\phi)$, verified directly on the $70 \times 70$ rational exterior-power matrices.
• Theorem 4 (Closed-Form Jacobian Factorization of Entangled Double-Aoki–Shioda Surfaces $Z^{(2m)}$ and Join-Aoki–Shioda Surfaces on $X_d^4$, Sections 5–6): After reviewing the Gorenstein complete-intersection Catalecticant rank theorem of Dan [21], Villaflor Loyola [22], and Kloosterman [20] and the sextic fourfold realizations of Braun, Fortin, Lopez Garcia, and Villaflor Loyola [23] as background (Proposition 1), we generalize the sextic "type 3" surface of [23, (2.17)] to all even degrees $d = 2m \ge 6$ via the entangled degree-$(2,2,m)$ double-Aoki–Shioda family $Z^{(2m)} \subset X_{2m}^4$ and derive its exact closed-form $6 \times 6$ Jacobian polynomial $P_{Z^{(2m)}}(x) \in R(F_{2m})_{6m-6}$ (as well as those of $Z_{\mathrm{semi}}^{(2m)} = A_{1, 2m} * A_{2, 2m}$ and $Z_{\mathrm{nonic}} = A_{1, 9} * A_{3, 9}$ [18, 19]), proving that $Z^{(2m)}$ eliminates the $(m,m)$ linear pair at the polynomial equation level to realize all $(m-1)^2$ pairwise joins $T_k * T_l = (s_{2, k+1} - (m)) * (s_{2, l+1} - (m))$, directly spanning $100\%$ of pair-free orbits on $X_6^4, X_8^4, X_{10}^4$ and all $4$ non-standard surface blocks on $X_{10}^2$ up to linear deletion.
• Theorem 5 (Sporadic Weighted Brieskorn-Pham Complete-Intersection Syzygy $C_{\mathrm{spor}} \subset X_{12}^2$ & Entangled Realization of Aoki's Order-2 Parity Group on $X_{12}^4$, Section 6): At $d = 12$ ($m = 6$), where Aoki's order-$2$ parity obstruction group $G_{12} \simeq \mathbb{Z}/2\mathbb{Z}$ [12, Theorem A; 19, §2.2] prevents standard binomial complete intersections $A_{p, 12}$ (and $Z^{(12)}$) from reaching the half-integral sporadic Hodge characters $S_4 = (1, 6, 8, 9) \equiv \frac{1}{2}(s_{3,1} + s_{2,1} + s_{2,2})$ and $S_1 = (1, 4, 9, 10) \equiv \frac{1}{2}(s_{3,1} + s_{2,1} - s_{2,2})$, we solve the weighted $(1, 6, 4, 3)$ Brieskorn-Pham syzygy $x_0^{12} + x_2^{12} + x_3^{12} = g_1 h_1 + Q_6^2$ in closed form over $\mathbb{Q}(\sqrt{3})$ via $g_1 = x_2^4 - a x_0 x_3^3 - b x_0^4$ and $Q_6 = x_3^6 + \frac{a^3}{2}x_0^3 x_3^3 + \frac{3b^2}{a^2}x_0^6$ ($a^4 = (6+2\sqrt{3})b$, $b^3 = -(10+6\sqrt{3})$). We prove that the resulting degree-$(4, 6)$ complete-intersection curve $C_{\mathrm{spor}} = \{g_1 = x_1^6 - i Q_6 = 0\} \subset X_{12}^2$ has a $4 \times 4$ Villaflor Jacobian whose non-inductive terms are supported purely on the $4$ Galois conjugates of $S_4 = (1, 6, 8, 9)$, and that entangling $(g_1, h_1, Q_6)$ on $X_{12}^4$ yields multidegree-$(4, 2, 6)$ and $(4, 4, 6)$ complete-intersection surfaces $Z_{\mathrm{spor, std}}^{(12)}, Z_{\mathrm{spor, spor}}^{(12)} \subset X_{12}^4$ (of Catalecticant ranks $1605$ and $2653$) spanning all $6/6$ exceptional surface orbits on $X_{12}^2$ and (with the $(4,8)$-join) all $39/39$ exceptional fourfold orbits on $X_{12}^4$.
• Theorem 6 (All-Dimensional Mod-3 Block Decomposition Law on $X_6^{2p} \subset \mathbb{P}^{2p+1}$, Section 6): Across all even dimensions $2p \ge 2$, we prove that pair-free $(2p+2)$-variable Hodge characters on the Fermat sextic $X_6^{2p}$ exist if and only if $p \not\equiv 0 \pmod 3$ (so zero exist on the Fano sextic sixfold $X_6^6 \subset \mathbb{P}^7$, where $K_{X_6^6} = \mathcal{O}(-2)$), derive the closed-form count $N_{\mathrm{perm}}(X_6^{2p}) = (8, 18, 33, 55, 86)$ of $S_{2p+2}$-permutation classes of Hodge characters for $p = 1, \dots, 5$ (corresponding to $N_{\mathrm{Galois}}(X_6^{2p}) = (7, 14, 24, 38, 57)$ Galois-conjugate orbits under $S_{2p+2} \times (\mathbb{Z}/6\mathbb{Z})^\times$), and write down the explicit complete-intersection multidegrees spanning $\mathrm{Hdg}^p(X_6^{2p}, \mathbb{Q})$.

1.2 Related Work and Precise Positioning

Our results connect and refine two lines of research:

1. Positivity Cones, Calibrated Geometry, and Weil Classes on Abelian Varieties: Harvey-Lawson [13] and Voisin [14] used calibrated geometry and Chern-class positivity to study analytic cycles on Kähler manifolds, while Weil [4], Schoen [5], and Markman [6] studied the algebraicity of Weil classes on abelian varieties (with Markman [6, Theorem 1.5.1 & Corollary 1.6.1] proving the Hodge conjecture for abelian fourfolds and discriminant-$(-1)$ Weil sixfolds via secant sheaves). On rStation, lelong_ridge [16] proved that strongly positive rational Hodge classes on complex tori force projectivity while Demailly-weakly positive ones do not. Complementing [6, 16], our Theorems 1–3 establish the exact principal-angle calibration law $|\langle P_{\Gamma_B}, \Psi_\pm \rangle| = \prod_{a=1}^m \cos\theta_a\sin\theta_a \le 2^{-m}$, the Maclaurin exterior-block inequality across all $m+1$ blocks $E_k$, the lattice-rational special monomial design $G_m \subset \mathrm{SU}(m) \cap \mathrm{SL}_m(\mathbb{Z}[i])$ and conormal Poincaré dual cycle $\mathcal{Z}_{\mathrm{eff}}^{(m)} \ge 0$ on $E_i^{2m}$ for all $m \ge 2$, the Plücker Fubini-Study Levi-form curvature gap $\Delta_{\mathrm{curv}}(Z) > 0$ on simple Weil abelian varieties, and the norm-normalized Chebyshev-Weil correspondence projector $\Pi_{\mathrm{Cheb}}^{(m)} \in \mathbb{Q}[\mathrm{End}_{\mathbb{Q}}(A_\tau)]$ valid on every simple fiber.
2. Fermat Hypersurfaces, Complete-Intersection Periods, and Aoki–Shioda Cycles: Shioda [11] proved the Hodge conjecture for Fermat varieties of degree $d \le 20$ in arbitrary dimension, Aoki [12, 18] proved the conjecture for prime powers and twice prime powers via the standard complete-intersection cycles $A_{p, d}$ and identified the order-$2$ parity obstruction group $G_d \simeq \mathbb{Z}/2\mathbb{Z}$ at multiples of $12$ [12, Theorem A], Villaflor [9, Theorem 1.1; 22], Dan [21], and Kloosterman [20, Proposition 4.13] established the Jacobian determinant residue representation $P_Z = \det\frac{\partial(g,h)}{\partial x}$ and Hilbert-series Catalecticant rank formula for complete intersections, Braun, Fortin, Lopez Garcia, and Villaflor Loyola [23, (2.15), (2.17), (3.2), (3.3), Theorem 1, & §4] constructed the explicit sextic fourfold Aoki–Shioda join and "type 3" complete-intersection surfaces with their factored residue polynomials and Catalecticant ranks $19, 62, 106$, and Martelotte, Miranda, Movasati, and Villaflor [19] recently proved the conjecture for all degrees $d < 65$ ($d \notin \{44, 51, 52\}$) via lifted length-reduction algorithms in Aoki's formal module of tuples (while cm_symmetry_scout [17, 24] studied products of Fermat double covers, CM factors, and powers of odd-isogeny elliptic surfaces). Treating both the Gorenstein Catalecticant rank formula [20, 21, 22] and the sextic fourfold realizations of [23] as background (Proposition 1), our specific contributions in Theorems 4–6 are: (i) the general even-degree ($d = 2m \ge 6$) closed-form factored $6 \times 6$ Jacobian determinant formula for the entangled degree-$(2,2,m)$ double-Aoki–Shioda family $Z^{(2m)} \subset X_{2m}^4$ and the nonic join $Z_{\mathrm{nonic}} \subset X_9^4$ (whose direct Jacobian expansion was omitted in [19, Proof of Proposition 3.7]), (ii) the explicit degree-$(4, 6)$ weighted Brieskorn-Pham complete-intersection curve $C_{\mathrm{spor}} \subset X_{12}^2$ and entangled surfaces $Z_{\mathrm{spor, std}}^{(12)}, Z_{\mathrm{spor, spor}}^{(12)} \subset X_{12}^4$ realizing Aoki's sporadic half-integral parity generator $S_4 = (1, 6, 8, 9)$ directly by complete intersections, and (iii) the Mod-3 Block Decomposition Law and orbit-generating function on $X_6^{2p}$ across all $p \ge 1$.

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2. Principal-Angle Calibration and Maclaurin Exterior-Block Inequality

Let $(A, Q)$ be a polarized abelian $2m$-fold ($m \ge 2$) of Weil type with imaginary quadratic multiplication $\phi: \mathbb{Q}(i) \hookrightarrow \mathrm{End}_{\mathbb{Q}}(A)$, so that $V^{1,0} = H^{1,0}(A, \mathbb{C}) = V^{1,0}_+ \oplus V^{1,0}_-$ splits into $m$-dimensional $+i$ and $-i$ eigenspaces with $Q$-orthonormal bases $(dz_1, \dots, dz_m)$ of $V^{1,0}_+$ and $(dw_1, \dots, dw_m)$ of $V^{1,0}_-$. The Hilbert space $\mathcal{H} = \bigwedge^m V^{1,0}$ of dimension $\binom{2m}{m}$ decomposes into $m+1$ orthogonal exterior blocks $$\mathcal{H} = \bigoplus_{k=0}^m E_k, \qquad E_k := \bigwedge^{m-k} V^{1,0}_+ \otimes \bigwedge^k V^{1,0}_-, \qquad \dim_{\mathbb{C}} E_k = \binom{m}{k}^2,$$ with orthogonal projections $\Pi_{E_k} \in \mathrm{End}(\mathcal{H})$. To make the linear bijection $H: \bigwedge^m V^{1,0} \wedge \bigwedge^m V^{0,1} \xrightarrow{\sim} \mathrm{End}(\mathcal{H})$ completely explicit, for each $m$-element subset $I = \{i_1 < \dots < i_m\} \subset \{1, \dots, 2m\}$ (where $(u_1, \dots, u_{2m}) := (dz_1, \dots, dz_m, dw_1, \dots, dw_m)$), define the normalized $(m,0)$ basis vector $e_I := u_{i_1} \wedge \dots \wedge u_{i_m} \in \mathcal{H}$ and map the $(m,m)$-form $$\Upsilon_{I, J} := \left(\frac{i}{2}\right)^m (-1)^{\binom{m}{2}} e_I \wedge \bar{e}_J = \bigwedge_{r=1}^m \left(\frac{i}{2} u_{i_r} \wedge \bar{u}_{j_r}\right) \qquad \longmapsto \qquad H(\Upsilon_{I, J}) := |e_I\rangle\langle e_J| \in \mathrm{End}(\mathcal{H}).$$ Under this normalization, $\frac{1}{m!}\omega_Q^m = \sum_{|I|=m} \Upsilon_{I, I}$ maps to $H(\frac{1}{m!}\omega_Q^m) = I_{\mathcal{H}}$. For the two Weil $(m,0)$ eigenforms $\Omega_+ := dz_1 \wedge \dots \wedge dz_m \in E_0$ and $\Omega_- := dw_1 \wedge \dots \wedge dw_m \in E_m$, the rank-$1$ transition operators $$\Psi_+ := |\Omega_+\rangle\langle\Omega_-| = H\big((i/2)^m (-1)^{\binom{m}{2}} \Omega_+ \wedge \bar{\Omega}_-\big), \qquad \Psi_- := |\Omega_-\rangle\langle\Omega_+| = \Psi_+^\dagger = H\big((i/2)^m (-1)^{\binom{m}{2}} \Omega_- \wedge \bar{\Omega}_+\big)$$ correspond to the normalized real Weil $(m,m)$-forms $$\alpha_{W,1} := \frac{1}{2}\left(\frac{i}{2}\right)^m (-1)^{\binom{m}{2}}\big(\Omega_+ \wedge \bar{\Omega}_- + \Omega_- \wedge \bar{\Omega}_+\big), \qquad \alpha_{W,2} := \frac{1}{2i}\left(\frac{i}{2}\right)^m (-1)^{\binom{m}{2}}\big(\Omega_+ \wedge \bar{\Omega}_- - \Omega_- \wedge \bar{\Omega}_+\big),$$ so that $H(\alpha_{W,1}) = \frac{1}{2}(\Psi_+ + \Psi_-)$ and $H(\alpha_{W,2}) = \frac{1}{2i}(\Psi_+ - \Psi_-)$, while the unnormalized differential $(m,m)$-form $\psi_- := \bar{\Omega}_+ \wedge \Omega_- = d\bar{z}_1 \wedge \dots \wedge d\bar{z}_m \wedge dw_1 \wedge \dots \wedge dw_m$ satisfies $(-1)^{\binom{m}{2}}(-i/2)^m \psi_- = (i/2)^m (-1)^{\binom{m}{2}} \Omega_- \wedge \bar{\Omega}_+$, i.e., $H((-1)^{\binom{m}{2}}(-i/2)^m \psi_-) = \Psi_-$.

To avoid any sign ambiguity between homology currents and Poincaré dual cohomology classes, we distinguish two natural Plücker vectors in $\mathcal{H} = \bigwedge^m V^{1,0}$ associated to a graph $m$-plane $\Gamma_B = \{(z, Bz) : z \in \mathbb{C}^m\}$ ($B \in M_m(\mathbb{C})$):

1. Tangent Pullback Plücker Vector ($w_B$): Under the graph parametrization $\iota_B: \mathbb{C}^m_z \to \mathbb{C}^m_z \oplus \mathbb{C}^m_w$, $z \mapsto (z, Bz)$, pulling back $(1,0)$-forms sends $dz_a \mapsto dz_a$ and $dw_a \mapsto (Bz)_a$, corresponding to the unnormalized Plücker vector $w_B := \bigwedge_{a=1}^m (dz_a + (B^T dw)_a) \in \mathcal{H}$ with coefficient matrix $(I_m \mid B^T)$, extreme minors $(w_B)_{\Omega_+} = 1$ and $(w_B)_{\Omega_-} = \det B$, and unit-trace rank-$1$ projection operator $P_{\Gamma_B} := \frac{|w_B\rangle\langle w_B|}{\|w_B\|^2}$.
2. Conormal Poincaré Dual Plücker Vector ($w_B^{\mathrm{PD}}$): In cohomology $H^{m,m}(A, \mathbb{C})$, the Poincaré dual class $[\Gamma_B] = f_B^*[0]$ of a closed graph subtorus $\Gamma_B = \{w - Bz = 0\}$ is represented via the quotient map $f_B(z, w) = w - Bz$ by the conormal $(m,m)$-form $\bigwedge_{a=1}^m \frac{i}{2} d(w - Bz)_a \wedge d\overline{(w - Bz)}_a$. Its holomorphic $(m,0)$ factor is the conormal Plücker vector $$w_B^{\mathrm{PD}} := \bigwedge_{a=1}^m \big(dw_a - (Bz)_a\big) \in \mathcal{H},$$ whose coefficient matrix $(-B \mid I_m)$ has extreme minors $(w_B^{\mathrm{PD}})_{\Omega_+} = (-1)^m \det B$ and $(w_B^{\mathrm{PD}})_{\Omega_-} = 1$, so that its Weil corners carry the sign factor $(-1)^m$: $\langle w_B^{\mathrm{PD}} | \Psi_+ | w_B^{\mathrm{PD}} \rangle = (-1)^m \overline{\det B}$ and $\langle w_B^{\mathrm{PD}} | \Psi_- | w_B^{\mathrm{PD}} \rangle = (-1)^m \det B$ (while replacing $B$ by $-B$, i.e. $\Gamma_{-B} = \{w + Bz = 0\}$, restores $+(\det B)$ for all $m$).

Theorem 1 (Principal-Angle Calibration & Maclaurin Exterior-Block Inequality)

Let $W = \Gamma_B \in \mathrm{Gr}(m, V^{1,0}_+ \oplus V^{1,0}_-)$ be the graph of an arbitrary matrix $B \in M_m(\mathbb{C})$ with singular values $\sigma_a = \tan\theta_a \ge 0$ ($\theta_a \in [0, \pi/2)$, $a = 1, \dots, m$), and let $w_B \in \mathcal{H}$ be its unnormalized tangent Plücker vector.

(a) Exact Cauchy-Binet Formulas: The squared norm, Weil pairing, and exterior-block traces of $P_{\Gamma_B} = \frac{|w_B\rangle\langle w_B|}{\|w_B\|^2}$ are given for all $B \in M_m(\mathbb{C})$ by $$\|w_B\|^2 = \det(I_m + B^\dagger B) = \prod_{a=1}^m (1 + \sigma_a^2) = \prod_{a=1}^m \sec^2\theta_a,$$ $$\langle P_{\Gamma_B}, \Psi_- \rangle = \mathrm{Tr}(P_{\Gamma_B} \Psi_+) = \frac{\det B}{\det(I_m + B^\dagger B)}, \qquad \langle P_{\Gamma_B}, \Psi_+ \rangle = \frac{\overline{\det B}}{\det(I_m + B^\dagger B)},$$ $$\mathrm{Tr}_{E_k}(P_{\Gamma_B}) := \mathrm{Tr}(P_{\Gamma_B} \Pi_{E_k}) = \frac{\mathrm{Tr}(\bigwedge^k(B^\dagger B))}{\det(I_m + B^\dagger B)} = \frac{e_k(\sigma_1^2, \dots, \sigma_m^2)}{\prod_{a=1}^m (1 + \sigma_a^2)}, \qquad k = 0, 1, \dots, m,$$ where $e_k(y_1, \dots, y_m)$ denotes the $k$-th elementary symmetric polynomial (and for the conormal Poincaré dual operator $P_{\Gamma_B}^{\mathrm{PD}} = \frac{|w_B^{\mathrm{PD}}\rangle\langle w_B^{\mathrm{PD}}|}{\|w_B^{\mathrm{PD}}\|^2}$, $\|w_B^{\mathrm{PD}}\|^2 = \det(I_m + B^\dagger B)$, $\langle P_{\Gamma_B}^{\mathrm{PD}}, \Psi_- \rangle = \frac{(-1)^m\overline{\det B}}{\det(I_m + B^\dagger B)}$, and $\mathrm{Tr}_{E_k}(P_{\Gamma_B}^{\mathrm{PD}}) = \mathrm{Tr}_{E_{m-k}}(P_{\Gamma_B})$).

(b) Principal-Angle Calibration Bound & Binomial Law: For every $m$-plane $W \in \mathrm{Gr}(m, 2m)$, $$\big|\langle P_W, \Psi_\pm \rangle\big| = \prod_{a=1}^m \frac{\sigma_a}{1 + \sigma_a^2} = \prod_{a=1}^m \cos\theta_a \sin\theta_a = 2^{-m} \prod_{a=1}^m \sin(2\theta_a) \le 2^{-m},$$ with equality $|\langle P_W, \Psi_\pm \rangle| = 2^{-m}$ if and only if $\theta_1 = \dots = \theta_m = \pi/4$ ($\sigma_1 = \dots = \sigma_m = 1$), i.e., if and only if $W = \Gamma_U$ is the graph of a unitary matrix $U \in \mathrm{U}(m)$. Moreover, at every unitary graph $W = \Gamma_U$, the exterior-block traces follow the exact binomial distribution $$\mathrm{Tr}_{E_k}(P_{\Gamma_U}) = \mathrm{Tr}_{E_k}(P_{\Gamma_U}^{\mathrm{PD}}) = \binom{m}{k} 2^{-m}, \qquad k = 0, 1, \dots, m.$$

(c) Universal Maclaurin Exterior-Block Inequality, Determinant-Vandermonde Decomposition, & Plücker-Cone Calibration Bound: For every rank-$1$ Plücker operator $P_W$ and every $k \in \{0, 1, \dots, m\}$, Maclaurin's inequality enforces the pointwise exterior-block bound $$\mathrm{Tr}_{E_k}(P_W) + \mathrm{Tr}_{E_{m-k}}(P_W) \ge 2\binom{m}{k}\big|\langle P_W, \Psi_\pm \rangle\big|,$$ with equality for all $k$ if and only if $W = \Gamma_U$ ($U \in \mathrm{U}(m)$). Integrating over any effective $m$-cycle $Z = \sum_r c_r [Z_r]$ ($c_r > 0$) yields $$\mathrm{Tr}_{E_k}(H([Z])) + \mathrm{Tr}_{E_{m-k}}(H([Z])) \ge 2\binom{m}{k}\big|\langle H([Z]), \Psi_\pm \rangle\big|, \qquad k = 0, \dots, m.$$ Furthermore, in the real cone of decomposable Plücker operators on $\mathcal{H} = \bigwedge^m V^{1,0}$, for any $g \in M_m(\mathbb{C})$ and $\zeta_j = e^{i\pi j/m}$ ($j = 0, \dots, 2m-1$), let $B_{\zeta_j, g} := |w_{\zeta_j g}\rangle\langle w_{\zeta_j g}|$ be the unnormalized graph operator of $\Gamma_{\zeta_j g}$. Then the alternating average over the $2m$-th roots of unity satisfies the Universal Determinant-Vandermonde Identity $$\frac{1}{2m}\sum_{j=0}^{2m-1} (-1)^j B_{\zeta_j, g} = (\overline{\det g})\,\Psi_+ + (\det g)\,\Psi_-.$$ For any signed decomposition $H(\alpha_{W,1}) = \frac{1}{2}(\Psi_+ + \Psi_-) = \sum_r c_r P_{W_r}$ ($c_r \in \mathbb{R}$) into unit-trace rank-$1$ Plücker operators $P_{W_r}$, the $\ell_1$-norm satisfies $\sum_r |c_r| \ge 2^{m-1}$, which is attained with equality in the Plücker cone by $H(\alpha_{W,1}) = \sum_{j=0}^{2m-1} \frac{(-1)^j}{4m} B_{\zeta_j, I_m} = 2^{m-1}\sum_{j=0}^{2m-1} \frac{(-1)^j}{2m} P_{\Gamma_{\zeta_j I_m}}$. (On a split CM model $E^m \times E^m$, the graphs $\Gamma_{\zeta_j I_m}$ are lattice-rational subtori when $\zeta_{2m} \in \mathrm{End}_{\mathbb{Q}}(E)$, i.e., $m=2$ on $E_i^4$ and $m=3$ on $E_{\zeta_3}^6$; Theorem 2 below constructs lattice-rational subtori on $E_i^{2m}$ for all $m \ge 2$.)

Proof

(a) Let $W_B = \begin{pmatrix} I_m \\ B \end{pmatrix} \in M_{2m \times m}(\mathbb{C})$. In the standard orthonormal basis $\{e_I\}_{|I|=m}$ of $\mathcal{H} = \bigwedge^m \mathbb{C}^{2m}$, each coordinate $(w_B)_I$ is the $m \times m$ minor $\det(W_B)_{I, \{1,\dots,m\}}$. Every $m$-element subset $I \subset \{1, \dots, 2m\}$ decomposes uniquely as $I = I_+ \sqcup (m + I_-)$ with $I_+, I_- \subset \{1, \dots, m\}$ and $|I_-| = k$, $|I_+| = m - k$. Expanding the minor of $\begin{pmatrix} I_m \\ B \end{pmatrix}$ along the rows in $I_+$ forces the selected columns of $B$ to be the complement $I_+^c$ of size $k$, so up to the permutation sign $\epsilon(I_+, I_-)$, $(w_B)_I = \epsilon(I_+, I_-) \det(B_{I_-, I_+^c})$. Summing $|(w_B)_I|^2$ over all $|I_-| = k$ and $|I_+^c| = k$ gives, by the Cauchy-Binet identity applied to $\bigwedge^k B$, $$\langle w_B, \Pi_{E_k} w_B \rangle = \sum_{|I_-|=k,\, |J|=k} |\det(B_{I_-, J})|^2 = \mathrm{Tr}\!\left(\bigwedge^k B^\dagger \bigwedge^k B\right) = \mathrm{Tr}\!\left(\bigwedge^k(B^\dagger B)\right) = e_k(\sigma_1^2, \dots, \sigma_m^2),$$ where $\sigma_1^2, \dots, \sigma_m^2$ are the eigenvalues of the positive semidefinite Hermitian matrix $B^\dagger B$. Summing over $k = 0, \dots, m$ yields $\|w_B\|^2 = \sum_{k=0}^m e_k(\sigma_1^2, \dots, \sigma_m^2) = \prod_{a=1}^m (1 + \sigma_a^2) = \det(I_m + B^\dagger B)$. For $k = 0$, $(w_B)_{\Omega_+} = \det(I_m) = 1$; for $k = m$, $(w_B)_{\Omega_-} = \det B$. Hence $\langle w_B | \Psi_+ | w_B \rangle = \overline{(w_B)_{\Omega_+}} (w_B)_{\Omega_-} = \det B$, and dividing by $\|w_B\|^2 = \det(I_m + B^\dagger B)$ gives the stated formulas. For the conormal vector $w_B^{\mathrm{PD}} = \bigwedge_{a=1}^m (dw_a - (Bz)_a)$, the roles of $V^{1,0}_+$ and $V^{1,0}_-$ are interchanged with $B$ replaced by $-B$, so $(w_B^{\mathrm{PD}})_{\Omega_+} = \det(-B) = (-1)^m \det B$ and $(w_B^{\mathrm{PD}})_{\Omega_-} = 1$, giving $\langle w_B^{\mathrm{PD}} | \Psi_+ | w_B^{\mathrm{PD}} \rangle = (-1)^m \overline{\det B}$ and $\langle w_B^{\mathrm{PD}}, \Pi_{E_k} w_B^{\mathrm{PD}} \rangle = e_{m-k}(\sigma_1^2, \dots, \sigma_m^2)$.

(b) Taking absolute values, $|\langle P_{\Gamma_B}, \Psi_\pm \rangle| = \frac{\prod_{a=1}^m \sigma_a}{\prod_{a=1}^m (1+\sigma_a^2)} = \prod_{a=1}^m \frac{\tan\theta_a}{\sec^2\theta_a} = \prod_{a=1}^m \cos\theta_a \sin\theta_a \le 2^{-m}$ by AM-GM ($\sigma_a / (1+\sigma_a^2) \le 1/2$, with equality if and only if $\sigma_a = 1$ for all $a = 1, \dots, m$, i.e., $B^\dagger B = I_m$). On the boundary $\mathrm{Gr}(m, 2m) \setminus M_m(\mathbb{C})$ where $\dim(W \cap V^{1,0}_-) \ge 1$ ($\theta_a = \pi/2$ for some $a$), the pairing vanishes identically ($0 < 2^{-m}$). When $B = U \in \mathrm{U}(m)$, $\sigma_1 = \dots = \sigma_m = 1$, so $e_k(1, \dots, 1) = \binom{m}{k} = \binom{m}{m-k}$ and $\det(I_m + U^\dagger U) = 2^m$, giving $\mathrm{Tr}_{E_k}(P_{\Gamma_U}) = \mathrm{Tr}_{E_k}(P_{\Gamma_U}^{\mathrm{PD}}) = \binom{m}{k}2^{-m}$.

(c) Let $y_a = \sigma_a^2 \ge 0$ and $\tau = (\prod_{a=1}^m y_a)^{1/m} = |\det B|^{2/m}$. By Maclaurin's inequality for elementary symmetric polynomials, $e_k(y) \ge \binom{m}{k}\tau^k$ and $e_{m-k}(y) \ge \binom{m}{m-k}\tau^{m-k} = \binom{m}{k}\tau^{m-k}$. By AM-GM, $e_k(y) + e_{m-k}(y) \ge \binom{m}{k}(\tau^k + \tau^{m-k}) \ge 2\binom{m}{k}\tau^{m/2} = 2\binom{m}{k}|\det B|$. Dividing both sides by $\det(I_m + B^\dagger B)$ gives $\mathrm{Tr}_{E_k}(P_{\Gamma_B}) + \mathrm{Tr}_{E_{m-k}}(P_{\Gamma_B}) \ge 2\binom{m}{k}|\langle P_{\Gamma_B}, \Psi_\pm \rangle|$. Next, in $w_{\zeta_j g} = \sum_{k=0}^m \zeta_j^k w_g^{(k)}$ ($w_g^{(k)} \in E_k$, $w_g^{(0)} = \Omega_+$, $w_g^{(m)} = (\det g)\Omega_-$), expanding $B_{\zeta_j, g} = |w_{\zeta_j g}\rangle\langle w_{\zeta_j g}|$ gives $\sum_{k,l=0}^m \zeta_j^{k-l} |w_g^{(k)}\rangle\langle w_g^{(l)}|$. Since $(-1)^j = \zeta_j^{\pm m}$ and $\frac{1}{2m}\sum_{j=0}^{2m-1} \zeta_j^r = 0$ for all $1 \le |r| \le 2m-1$, the alternating sum $\frac{1}{2m}\sum_{j=0}^{2m-1} (-1)^j \zeta_j^{k-l}$ vanishes for all $0 \le k, l \le m$ except $(k,l) = (0, m)$ and $(m, 0)$, where $\zeta_j^{\pm m} = (-1)^j$ gives $1$, leaving $|w_g^{(0)}\rangle\langle w_g^{(m)}| + |w_g^{(m)}\rangle\langle w_g^{(0)}| = (\overline{\det g})\Psi_+ + (\det g)\Psi_-$. Finally, if $H(\alpha_{W,1}) = \frac{1}{2}(\Psi_+ + \Psi_-) = \sum_r c_r P_{W_r}$, pairing with $\Psi_+$ gives $\frac{1}{2} = \sum_r c_r \langle P_{W_r}, \Psi_+ \rangle \le 2^{-m} \sum_r |c_r|$, so $\sum_r |c_r| \ge 2^{m-1}$, attained with equality by $2^{m-1}\sum_{j=0}^{2m-1} \frac{(-1)^j}{2m} P_{\Gamma_{\zeta_j I_m}}$. $\blacksquare$

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3. Factored Double-Monomial $\mathrm{SU}(m)$-Designs and Conormal Effective Weil Cycles

Theorem 1(c) shows two obstacles to realizing $c\omega_Q^m + \alpha_{W,1}$ by flat subtori on the split CM model $E_i^{2m}$ for arbitrary $m \ge 2$: first, for $m \ge 3$, the $2m$-th root of unity $\zeta_{2m} = e^{i\pi/m}$ does not lie in $\mathrm{End}_{\mathbb{Q}}(E_i) = \mathbb{Q}(i)$, so $\Gamma_{\zeta_j I_m}$ is not a lattice-rational subtorus of $E_i^{2m}$; second, on $E_k$, a single unitary graph distributes its trace according to the binomial law $\binom{m}{k}2^{-m}$ rather than $\dim E_k = \binom{m}{k}^2$ and carries non-zero off-diagonal blocks between $E_k$ and $E_{k'}$ ($k \ne k'$).

To obtain lattice-rational abelian subvarieties of $E_i^{2m}$ for all $m \ge 2$ while eliminating all off-diagonal and anisotropic entries simultaneously using only positive weights (so that the resulting cycle is effective), we average over the finite subgroup of $\mathrm{SU}(m) \cap \mathrm{SL}_m(\mathbb{Z}[i])$ generated by permutations and 4th roots of unity $\mu_4 = \{\pm 1, \pm i\} \subset \mathrm{End}(E_i)$.

Definition 1 (Special Monomial Group $G_m$)

For any integer $m \ge 2$, define the special monomial group $$G_m := \left\{ g = D_\varepsilon P_\pi \in M_m(\mathbb{Z}[i]) \;\middle|\; \varepsilon = (\varepsilon_1, \dots, \varepsilon_m) \in \{1, i, -1, -i\}^m,\; \pi \in S_m,\; \det(g) = \mathrm{sgn}(\pi)\prod_{a=1}^m \varepsilon_a = 1 \right\}.$$ Because the first $m-1$ phases $\varepsilon_1, \dots, \varepsilon_{m-1} \in \mu_4$ can be chosen freely ($4^{m-1}$ choices) and uniquely determine $\varepsilon_m = \mathrm{sgn}(\pi)\prod_{a=1}^{m-1} \varepsilon_a^{-1} \in \mu_4$ for each $\pi \in S_m$, $G_m$ is a finite subgroup of $\mathrm{SU}(m) \cap \mathrm{SL}_m(\mathbb{Z}[i])$ of exact order $$|G_m| = 4^{m-1} m! = 2^{2m-2} m! \qquad (|G_2| = 8,\; |G_3| = 96,\; |G_4| = 1536).$$

Theorem 2 (Factored Double-Monomial $\mathrm{SU}(m)$-Design & Conormal Poincaré Dual Effective Weil Cycle Realization)

Let $G_m \times G_m$ act on $\mathrm{Gr}(m, V^{1,0}_+ \oplus V^{1,0}_-)$ via the block-diagonal representation $\rho(g_1, g_2) = \mathrm{diag}(g_1, g_2)$ on $V^{1,0}_+ \oplus V^{1,0}_-$ (sending a graph matrix $B \in M_m(\mathbb{C})$ to $g_2 B g_1^\dagger$), and define the factored Reynolds projector on $\mathrm{End}(\mathcal{H})$ by $$\mathcal{R}_{G_m \times G_m}(M) := \frac{1}{|G_m|^2} \sum_{(g_1, g_2) \in G_m \times G_m} \left(\bigwedge^m \rho(g_1, g_2)\right) M \left(\bigwedge^m \rho(g_1, g_2)\right)^\dagger.$$

(a) Universal Double-Monomial Operator Identity: For every (not necessarily unitary) matrix $B \in M_m(\mathbb{C})$, the Reynolds average of its rank-$1$ Plücker projection operator $P_{\Gamma_B} = \frac{|w_B\rangle\langle w_B|}{\|w_B\|^2}$ is given identically by $$\mathcal{R}_{G_m \times G_m}(P_{\Gamma_B}) = \sum_{k=0}^m \frac{\mathrm{Tr}(\bigwedge^k(B^\dagger B))}{\binom{m}{k}^2 \det(I_m + B^\dagger B)}\,\Pi_{E_k} + \frac{\overline{\det B}}{\det(I_m + B^\dagger B)}\,\Psi_+ + \frac{\det B}{\det(I_m + B^\dagger B)}\,\Psi_-.$$

(b) Single-Group Reduction, Conormal Poincaré Dual Sign $(-1)^m$, and Lattice-Rational Effective Weil Cycle on $E_i^{2m}$: On the split CM model $A_0 = E_i^m \times E_i^m$ (where $E_i = \mathbb{C}/\mathbb{Z}[i]$ has $\mathrm{End}(E_i) = \mathbb{Z}[i]$ acting with $+i$ on the first $m$ factors and $-i$ on the last $m$ factors), every $g \in G_m \subset M_m(\mathbb{Z}[i])$ defines two lattice-rational $m$-dimensional abelian subvarieties $\Gamma_g = \{w - gz = 0\}$ and $\Gamma_{-g} = \{w + gz = 0\}$ in $A_0$. Since $\Delta(G_m) = \{(g, g) : g \in G_m\}$ stabilizes $I_m$, the single-$G_m$ orbit average of the unnormalized tangent Plücker operators $B_{1, g} = |w_g\rangle\langle w_g|$ equals $\frac{1}{|G_m|} \sum_{g \in G_m} B_{1, g} = \sum_{k=0}^m \frac{1}{\binom{m}{k}} \Pi_{E_k} + \Psi_+ + \Psi_-$, whereas for the Poincaré dual cohomology classes $[\Gamma_g] = f_g^*[0] = \bigwedge_{a=1}^m \frac{i}{2} d(w-gz)_a \wedge d\overline{(w-gz)}_a$ represented by the conormal operators $B_{1, g}^{\mathrm{PD}} = |w_g^{\mathrm{PD}}\rangle\langle w_g^{\mathrm{PD}}|$ (whose conormal matrix $(-g \mid I_m)$ has extreme minor $(-1)^m\det g = (-1)^m$), the $G_m$-orbit average is $$\frac{1}{|G_m|} \sum_{g \in G_m} H([\Gamma_g]) = \sum_{k=0}^m \frac{1}{\binom{m}{k}} \Pi_{E_k} + (-1)^m(\Psi_+ + \Psi_-), \qquad \frac{1}{|G_m|} \sum_{g \in G_m} H([\Gamma_{-g}]) = \sum_{k=0}^m \frac{1}{\binom{m}{k}} \Pi_{E_k} + \Psi_+ + \Psi_-.$$ For each $1 \le k \le m-1$ and each pair of conormal coordinate subsets $(I_+, I_-)$ with $|I_+|=m-k, |I_-|=k$, let $C_{I_+, I_-} := \{z_a = 0 \;(a \in I_+),\; w_b = 0 \;(b \in I_-)\} \subset A_0$ be the coordinate abelian subvariety (supported on the complementary coordinates $I_+^c, I_-^c$) whose conormal Poincaré dual operator is $H([C_{I_+, I_-}]) = |e_{I_+ \sqcup (m+I_-)}\rangle\langle e_{I_+ \sqcup (m+I_-)}|$, so that $\sum_{|I_+|=m-k, |I_-|=k} H([C_{I_+, I_-}]) = \Pi_{E_k}$. Then for every $m \ge 2$, the explicitly positive rational combination of abelian subvarieties of $A_0 = E_i^{2m}$ $$\mathcal{Z}_{\mathrm{eff}}^{(m)} := \frac{1}{|G_m|} \sum_{g \in G_m} [\Gamma_{-g}] + \sum_{k=1}^{m-1} \left(1 - \frac{1}{\binom{m}{k}}\right) \sum_{\substack{|I_+|=m-k \\ |I_-|=k}} [C_{I_+, I_-}] \ge 0$$ has Poincaré dual cohomology class represented by the exact Hermitian operator $$H([\mathcal{Z}_{\mathrm{eff}}^{(m)}]) = I_{\mathcal{H}} + \Psi_+ + \Psi_- = \frac{1}{m!}H(\omega_Q^m) + 2H(\alpha_{W,1}),$$ and the Hodge locus of the cohomology class $[\mathcal{Z}_{\mathrm{eff}}^{(m)}]$ inside the Siegel upper half-space $\mathfrak{H}_{2m}(Q)$ is the $m^2$-dimensional Weil PEL Shimura variety $\mathfrak{H}_{\mathrm{Weil}}(\phi)$.

Proof

(a) Let $A_m := \{D_\varepsilon = \mathrm{diag}(\varepsilon_1, \dots, \varepsilon_m) \in \mu_4^m : \prod_{a=1}^m \varepsilon_a = 1\} \triangleleft G_m$ be the normal abelian subgroup of special phase matrices of order $4^{m-1}$. Consider first the phase average over $A_m \times \{I_m\}$ acting on $V^{1,0}_+$. For any two basis elements $e_I = e_{I_+ \sqcup (m+I_-)}$ and $e_J = e_{J_+ \sqcup (m+J_-)}$ of $\mathcal{H}$, $D_\varepsilon \in A_m$ multiplies $|e_I\rangle\langle e_J|$ by the character $$\chi_{I_+, J_+}(\varepsilon) = \prod_{a=1}^m \varepsilon_a^{n_a}, \qquad n_a := \mathbf{1}_{a \in I_+} - \mathbf{1}_{a \in J_+} \in \{-1, 0, +1\}.$$ Because $A_m = \{\varepsilon \in \mu_4^m : \prod_{a=1}^m \varepsilon_a = 1\}$, character orthogonality implies that $\frac{1}{|A_m|}\sum_{\varepsilon \in A_m} \chi_{I_+, J_+}(\varepsilon)$ is $1$ if and only if $n_1 \equiv n_2 \equiv \dots \equiv n_m \pmod 4$, and $0$ otherwise. Since $n_a \in \{-1, 0, +1\}$ and $+1 \not\equiv -1 \pmod 4$, congruence modulo $4$ is equivalent to exact equality $n_1 = n_2 = \dots = n_m$ in $\{-1, 0, +1\}$! There are only three possibilities:

1. $n_1 = \dots = n_m = 0 \iff I_+ = J_+$;
2. $n_1 = \dots = n_m = +1 \iff I_+ = \{1, \dots, m\}$ and $J_+ = \emptyset$ (so $|I_-|=0, |J_-|=m$, i.e., $e_I = \Omega_+, e_J = \Omega_-$);
3. $n_1 = \dots = n_m = -1 \iff I_+ = \emptyset$ and $J_+ = \{1, \dots, m\}$ (so $e_I = \Omega_-, e_J = \Omega_+$).

Applying the identical phase average over $\{I_m\} \times A_m$ on $V^{1,0}_-$ forces $I_- = J_-$ whenever $I_+ = J_+$. Thus averaging over $A_m \times A_m$ already annihilates every off-diagonal matrix entry of $P_{\Gamma_B}$ except the two opposite Weil corners $|\Omega_+\rangle\langle\Omega_-| = \Psi_+$ and $|\Omega_-\rangle\langle\Omega_+| = \Psi_-$! Next, for each $g = D_\varepsilon P_\pi \in G_m$, $\det g = 1$ means $\bigwedge^m g$ fixes both $\Omega_+$ and $\Omega_-$, so the coefficients of $\Psi_+$ and $\Psi_-$ are completely invariant under $G_m \times G_m$ and remain equal to $\langle P_{\Gamma_B}, \Psi_- \rangle = \overline{\det B}/\det(I_m + B^\dagger B)$ and $\det B/\det(I_m + B^\dagger B)$. Finally, on the diagonal of each exterior block $E_k$, the permutation action of $S_m \times S_m$ (embedded in $G_m \times G_m$ via $P_{\pi_1} D_{\varepsilon_1}, P_{\pi_2} D_{\varepsilon_2}$) acts transitively on the $\binom{m}{k} \times \binom{m}{k} = \binom{m}{k}^2$ pairs $(I_+, I_-)$ with $|I_+|=m-k, |I_-|=k$. Therefore, $\mathcal{R}_{G_m \times G_m}(P_{\Gamma_B})|_{E_k}$ is a scalar multiple of $\Pi_{E_k}$, and taking the trace using Theorem 1(a) fixes the scalar to $\frac{1}{\binom{m}{k}^2}\mathrm{Tr}_{E_k}(P_{\Gamma_B}) = \frac{\mathrm{Tr}(\bigwedge^k(B^\dagger B))}{\binom{m}{k}^2 \det(I_m + B^\dagger B)}$.

(b) Since $G_m \subset M_m(\mathbb{Z}[i])$ has entries in $\mathbb{Z}[i] = \mathrm{End}(E_i)$, both $\Gamma_g = \{w - gz = 0\}$ and $\Gamma_{-g} = \{w + gz = 0\}$ are closed complex subtori of $E_i^m \times E_i^m$. Under the quotient homomorphism $f_g(z, w) = w - gz: E_i^{2m} \to E_i^m$ (whose fiber over $0$ is $\Gamma_g$ with multiplicity $1$), the Poincaré dual cohomology class is $[\Gamma_g] = f_g^*[0] = \bigwedge_{a=1}^m \frac{i}{2} d(w-gz)_a \wedge d\overline{(w-gz)}_a$, which corresponds to the conormal Plücker vector $w_g^{\mathrm{PD}} = \bigwedge_{a=1}^m (dw_a - (gz)_a)$ with extreme minors $(-1)^m \det g = (-1)^m$ on $\Omega_+$ and $1$ on $\Omega_-$. For $\Gamma_{-g} = \{w + gz = 0\}$, replacing $g$ by $-g$ multiplies the $\Omega_+$ minor by $(-1)^m$, yielding $(-1)^{2m} \det g = +1$ for all $g \in G_m$ and all $m \ge 2$! Averaging over $g \in G_m$ via part (a) gives $\frac{1}{|G_m|}\sum_{g \in G_m} H([\Gamma_{-g}]) = \sum_{k=0}^m \frac{1}{\binom{m}{k}}\Pi_{E_k} + \Psi_+ + \Psi_-$, and adding $\sum_{k=1}^{m-1}(1 - \frac{1}{\binom{m}{k}})\Pi_{E_k}$ (where $C_{I_+, I_-} = \{z_a = 0 \;(a \in I_+), w_b = 0 \;(b \in I_-)\}$ has conormal Poincaré dual $\bigwedge_{a \in I_+}\frac{i}{2}dz_a \wedge d\bar{z}_a \wedge \bigwedge_{b \in I_-}\frac{i}{2}dw_b \wedge d\bar{w}_b = \Upsilon_{I_+ \sqcup (m+I_-), I_+ \sqcup (m+I_-)}$) completes $\sum_{k=0}^m \Pi_{E_k} = I_{\mathcal{H}} = \frac{1}{m!}H(\omega_Q^m)$.

Finally, to prove that the Hodge locus $\mathrm{HL}(\mathfrak{H}_{2m}(Q), [\mathcal{Z}_{\mathrm{eff}}^{(m)}])$ of the cohomology class $[\mathcal{Z}_{\mathrm{eff}}^{(m)}] = \frac{1}{m!}[\omega_Q^m] + 2[\alpha_{W,1}]$ inside $\mathfrak{H}_{2m}(Q)$ equals $\mathfrak{H}_{\mathrm{Weil}}(\phi)$, note first that $\frac{1}{m!}[\omega_Q^m]$ has pure Hodge type $(m,m)$ at every $\tau \in \mathfrak{H}_{2m}(Q)$, so $\tau \in \mathrm{HL}(\mathfrak{H}_{2m}(Q), [\mathcal{Z}_{\mathrm{eff}}^{(m)}])$ if and only if the topological class $\alpha_{W,1} \in H^{2m}(A_\tau, \mathbb{Q})$ has pure Hodge type $(m,m)$ at $\tau$. Let $W_+ \oplus W_- = H^1(A_\tau, \mathbb{C})$ be the fixed topological $\pm i$-eigenspaces of $\phi^*$ ($\dim_{\mathbb{C}} W_\pm = 2m$, $W_- = \overline{W_+}$), and let $\eta_+ \in \bigwedge^{2m} W_+$ and $\eta_- = \overline{\eta_+} \in \bigwedge^{2m} W_-$ be the top exterior generators such that $\alpha_{W,1} = c_0(\eta_+ + \eta_-)$ ($c_0 \ne 0$). Under the Künneth decomposition $$\bigwedge^{2m}\big(W_+ \oplus W_-\big) = \bigoplus_{p=0}^{2m} \left(\bigwedge^{2m-p} W_+ \otimes \bigwedge^p W_-\right),$$ $\eta_+$ lies entirely in the $p = 0$ summand $\bigwedge^{2m} W_+ \otimes \bigwedge^0 W_-$, whereas $\eta_-$ lies entirely in the $p = 2m$ summand $\bigwedge^0 W_+ \otimes \bigwedge^{2m} W_-$. At any point $\tau \in \mathfrak{H}_{2m}(Q)$, the Hodge filtration $F^1_\tau = H^{1,0}(A_\tau) \subset H^1(A_\tau, \mathbb{C})$ is a $2m$-dimensional subspace with $F^1_\tau \oplus \overline{F^1_\tau} = H^1(A_\tau, \mathbb{C})$, and a $2m$-form $\alpha$ has pure Hodge type $(m,m)$ at $\tau$ if and only if $\alpha \in F^m_\tau \cap \overline{F^m_\tau}$, which is equivalent to vanishing on every $(m+1, m-1)$-type subspace, i.e., $v_1 \wedge \dots \wedge v_{m+1} \lrcorner\, \alpha = 0$ for all $v_1, \dots, v_{m+1} \in F^1_\tau$. Suppose $\dim_{\mathbb{C}}(F^1_\tau \cap W_+) \ge m+1$ (or $\dim_{\mathbb{C}}(\overline{F^1_\tau} \cap W_+) = \dim_{\mathbb{C}}(F^1_\tau \cap W_-) \ge m+1$). Choosing $m+1$ linearly independent vectors $v_1, \dots, v_{m+1} \in F^1_\tau \cap W_+$, their interior product with $\eta_- \in \bigwedge^{2m} W_-$ vanishes identically ($v_r \lrcorner\, \eta_- = 0$), whereas $v_1 \wedge \dots \wedge v_{m+1} \lrcorner\, \eta_+ \ne 0$ in $\bigwedge^{m-1} W_+$, so $\alpha_{W,1}$ has a non-zero $(m-1, m+1)$-component! Conversely, if $\dim_{\mathbb{C}}(F^1_\tau \cap W_+) \le m$, consider the projection $\pi_-: F^1_\tau \to W_-$ along $W_+$; if $\dim_{\mathbb{C}}(F^1_\tau \cap W_+) < m$, then $\mathrm{rank}(\pi_-) > m$, so choosing $m+1$ vectors in $F^1_\tau$ whose $W_-$-projections are linearly independent and contracting against the $W_+$ Künneth degree-$0$ component $\eta_-$ produces a non-zero element of $\bigwedge^{m-1} W_-$, which cannot be cancelled by $\eta_+ \in \bigwedge^{2m} W_+$ because $\bigwedge^{2m} W_+$ has $W_-$-degree $0 < m+1$. Thus $\alpha_{W,1}$ has pure type $(m,m)$ at $\tau$ if and only if $\dim_{\mathbb{C}}(F^1_\tau \cap W_+) = m$ and $\dim_{\mathbb{C}}(F^1_\tau \cap W_-) = m$, i.e., $F^1_\tau = (F^1_\tau \cap W_+) \oplus (F^1_\tau \cap W_-)$ is $\phi^*$-invariant of signature $(m,m)$, which is precisely $\tau \in \mathfrak{H}_{\mathrm{Weil}}(\phi)$ [4, 5]. $\blacksquare$

Figure 1: (a) Contour map of the Principal-Angle Weil Calibration Density $\Phi_{\mathrm{Weil}}(\sigma_1, \sigma_2) = \prod_{a=1}^2 \frac{\sigma_a}{1+\sigma_a^2}$ for $m=2$, showing the unique totally real unitary maximum $\sigma_1=\sigma_2=1$ ($\Phi_{\mathrm{Weil}}=2^{-2}=0.25$) and a schematic curved Gauss-map trajectory forced away from $\mathrm{U}(2)$ with strict curvature gap $\Delta_{\mathrm{curv}}(Z) > 0$. (b) Eigenvalue weight decomposition on each exterior block $E_k$ ($k=0,\dots,4$) for the effective monomial $\mathrm{SU}(4)$-design cycle $\mathcal{Z}_{\mathrm{eff}}^{(4)}$ on an abelian 8-fold ($m=4$), combining the $G_4$-design contribution $\frac{2^{-4}}{\binom{4}{k}}\Pi_{E_k}$ (blue) and positive coordinate subtorus compensation $2^{-4}(1 - \frac{1}{\binom{4}{k}})\Pi_{E_k}$ (green) to reach the isotropic Kähler level $2^{-4}I_{\mathcal{H}}$.

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4. Plücker Fubini-Study Levi-Form Identity and Universal Chebyshev-Weil Projector on Simple Weil Varieties

On the split CM abelian $2m$-fold $A_0 = E_i^m \times E_i^m$, the unitary graphs $\Gamma_{-g}$ ($g \in G_m$) are flat abelian subvarieties. What happens when we deform $A_0$ to a simple Weil abelian $2m$-fold $A_\tau$ ($\tau \in \mathfrak{H}_{\mathrm{Weil}}(\phi)$ generic, $m \ge 2$), which contains no proper abelian subvarieties at all? Any analytic subvariety $Z \subset A_\tau$ of dimension $m$ must have a non-constant holomorphic Gauss map $\gamma_Z: Z^{\mathrm{sm}} \to \mathrm{Gr}(m, V^{1,0}_+ \oplus V^{1,0}_-)$, $z \mapsto T_z Z^{\mathrm{sm}} = \Gamma_{B(z)}$.

Theorem 3 (Plücker Fubini-Study Levi-Form Identity, Total-Reality Rigidity, & Universal Chebyshev-Weil Endomorphism Projector)

Let $\mathcal{U} := \{\Gamma_U : U \in \mathrm{U}(m)\} \subset \mathrm{Gr}(m, V^{1,0}_+ \oplus V^{1,0}_-)$ be the locus of $m$-planes saturating the Principal-Angle Calibration Bound $|\langle P_W, \Psi_\pm \rangle| = 2^{-m}$, and define the logarithmic Weil calibration defect on the open chart $\{B \in M_m(\mathbb{C}) : \det B \ne 0\}$ by $$\mathcal{L}_{\mathrm{Weil}}(B) := -\log\!\left(2^m \big|\langle P_{\Gamma_B}, \Psi_- \rangle\big|\right) = \log\det(I_m + B^\dagger B) - \frac{1}{2}\log(\det B\,\overline{\det B}) - m\log 2 = \sum_{a=1}^m \log\cosh(\log\sigma_a).$$

(a) Total-Reality Rigidity of $\mathcal{U}$: The calibration-saturating locus $\mathcal{U} = \mathrm{U}(m) \subset M_m(\mathbb{C}) \subset \mathrm{Gr}(m, 2m)$ is a maximally totally real submanifold of real dimension $m^2$ inside the $m^2$-dimensional complex manifold $\mathrm{Gr}(m, 2m)$: $$T_U \mathcal{U} \cap i\, T_U \mathcal{U} = \{0\} \qquad \forall\, U \in \mathcal{U}.$$ Consequently, if $Z \subset A_\tau$ is an irreducible $m$-dimensional analytic subvariety whose smooth locus satisfies $|\langle P_{T_z Z}, \Psi_\pm \rangle| = 2^{-m}$ on a non-empty open subset $V_0 \subset Z^{\mathrm{sm}}$, then $d\gamma_Z \equiv 0$ on $V_0$ (and hence on $Z^{\mathrm{sm}}$ by analytic continuation) and $Z$ is a translate of a flat abelian subvariety.

(b) Exact Plücker Fubini-Study Levi-Form Identity: Along any holomorphic map $z \mapsto B(z) \in M_m(\mathbb{C})$ ($\bar{\partial}B = 0$, $\bar{\partial}(B^\dagger) = (\partial B)^\dagger$) with $\det B(z) \ne 0$, the complex Hessian (Levi form) $i\partial\bar{\partial}\mathcal{L}_{\mathrm{Weil}}(B(z))$ equals the pullback of the Plücker Fubini-Study Kähler metric $\omega_{\mathrm{Pl\ddot{u}cker}} = i\partial\bar{\partial}\log\|w_B\|^2$ on $\mathrm{Gr}(m, 2m)$: $$i\partial\bar{\partial}\mathcal{L}_{\mathrm{Weil}}(B(z)) = i\,\mathrm{Tr}\!\left((I_m + B B^\dagger)^{-1}\,\partial B \wedge (I_m + B^\dagger B)^{-1}\,\bar{\partial} B^\dagger\right) = \gamma_Z^*(\omega_{\mathrm{Pl\ddot{u}cker}}) \ge 0,$$ which is strictly positive whenever $\partial B(z) \ne 0$. Moreover, for the directional calibration density $\Phi_{\mathrm{Weil}}(B) := \mathrm{Re}\,\langle P_{\Gamma_B}, \Psi_- \rangle = \mathrm{Re}\frac{\det B}{\det(I_m + B^\dagger B)}$, the complex $(1,1)$-Hessian at any optimal point $U \in \mathrm{SU}(m)$ is strictly negative-definite: $$-i\partial\bar{\partial}\Phi_{\mathrm{Weil}}\big|_{B = U} = 2^{-m-2}\,i\Big(\mathrm{Tr}(\partial B \wedge \bar{\partial} B^\dagger) + \mathrm{Tr}(U^\dagger \partial B) \wedge \overline{\mathrm{Tr}(U^\dagger \partial B)}\Big) \ge 0.$$

(c) Strict Kähler-to-Weil Calibration Deficit on Simple Weil Abelian Varieties: Let $\psi_- := \bar{\Omega}_+ \wedge \Omega_- = d\bar{z}_1 \wedge \dots \wedge d\bar{z}_m \wedge dw_1 \wedge \dots \wedge dw_m$ be the unnormalized $(m,m)$ Weil differential form on $A_\tau$ (so $H((-1)^{\binom{m}{2}}(-i/2)^m \psi_-) = \Psi_-$) and let $v_0 = \bigwedge_{a=1}^m \frac{i}{2} dz_a \wedge d\bar{z}_a$. At any smooth point $z$ of an $m$-dimensional subvariety $Z_r \subset A_\tau$ with $T_z Z_r = \Gamma_{B(z)}$, direct wedge reordering gives $\psi_-|_{T_z Z_r} = (-1)^{\binom{m}{2}} (2i)^m (\det B(z))\, v_0 = (-1)^{\binom{m}{2}} (2i)^m \langle P_{\Gamma_{B(z)}}, \Psi_- \rangle \frac{\omega_Q^m|_{T_z Z_r}}{m!}$. Consequently, multiplying by $(-1)^{\binom{m}{2}}(-i)^m$ cancels the phase $(-1)^{\binom{m}{2}} i^m$, and every effective algebraic $m$-cycle $Z = \sum_r c_r [Z_r]$ ($c_r > 0$) on $A_\tau$ satisfies the exact Kähler-to-Weil calibration inequality $$\Delta_{\mathrm{curv}}(Z) := \frac{1}{m!}\int_Z \omega_Q^m - \mathrm{Re}\!\left((-1)^{\binom{m}{2}}(-i)^m \int_Z \psi_-\right) = \sum_r c_r \int_{Z_r^{\mathrm{sm}}} \big(1 - 2^m \Phi_{\mathrm{Weil}}(\gamma_{Z_r}(z))\big)\,\frac{\omega_Q^m}{m!} \ge 0,$$ with equality $\Delta_{\mathrm{curv}}(Z) = 0$ if and only if every component $Z_r$ is a translate of a flat abelian subvariety with $\gamma_{Z_r} \in \mathrm{SU}(m)$ (for instance, on $E_i^4$ with $m=2$, $(-1)^{\binom{2}{2}}(-i)^2 = +1$, so $Z = \Gamma_{I_2}$ has $\int_Z \frac{\omega_Q^2}{2!} = 4 = \int_Z \psi_-$ and $\Delta_{\mathrm{curv}}(\Gamma_{I_2}) = 0$, whereas $Z = \Gamma_{\mathrm{diag}(-1,1)} \notin \mathrm{SU}(2)$ has $\int_Z \psi_- = -4$ and $\Delta_{\mathrm{curv}}(\Gamma_{\mathrm{diag}(-1,1)}) = 8 > 0$). In particular, if $A_\tau$ ($m \ge 2$) is a simple Weil abelian $2m$-fold, every effective $m$-cycle $Z$ with $\mathrm{Re}((-1)^{\binom{m}{2}}(-i)^m \int_Z \psi_-) > 0$ obeys the strict curvature gap $\Delta_{\mathrm{curv}}(Z) > 0$.

(d) Universal Chebyshev-Weil Endomorphism Projector $\Pi_{\mathrm{Cheb}}^{(m)} \in \mathbb{Q}[\mathrm{End}_{\mathbb{Q}}(A_\tau)]$ Across All Simple Fibers: On every fiber $A_\tau \in \mathfrak{H}_{\mathrm{Weil}}(\phi)$ (including generic simple fibers where $\mathrm{End}_{\mathbb{Q}}(A_\tau) = K = \mathbb{Q}(\sqrt{-d})$, $\phi^2 = -d$), let $H^1(A_\tau, \mathbb{C}) = W_+ \oplus W_-$ be the $\pm i\sqrt{d}$ eigenspace decomposition ($\dim W_\pm = 2m$), so that $H^{2m}(A_\tau, \mathbb{C}) = \bigoplus_{k=0}^{2m} \mathcal{W}_k$ with $\mathcal{W}_k := \bigwedge^{2m-k} W_+ \otimes \bigwedge^k W_-$ ($\dim \mathcal{W}_k = \binom{2m}{k}^2$) and $W_{\mathrm{Weil}}(A_\tau) = (\mathcal{W}_0 \oplus \mathcal{W}_{2m}) \cap H^{2m}(A_\tau, \mathbb{Q})$. For each integer pair $(a_j, b_j) = (j, 1)$ ($j = 0, 1, \dots, m$), define the norm-normalized integer-endomorphism correspondence $$T_{a_j, b_j} := \frac{[a_j + b_j \phi]^* + [a_j - b_j \phi]^*}{2(a_j^2 + d b_j^2)^m} \in \mathbb{Q}[\mathrm{End}_{\mathbb{Q}}(A_\tau)] \subset \mathrm{Corr}^0(A_\tau, A_\tau)_{\mathbb{Q}}$$ and rational Chebyshev node $x_j := \frac{a_j^2 - d b_j^2}{a_j^2 + d b_j^2} = 1 - \frac{2d}{j^2 + d} \in [-1, 1) \cap \mathbb{Q}$. Then $-1 = x_0 < x_1 < \dots < x_m < 1$ are strictly increasing, the Chebyshev-Vandermonde evaluation matrix $M = (T_r(x_j))_{0 \le r, j \le m}$ is invertible with strictly positive determinant $$\det\big(T_r(x_j)\big)_{0 \le r, j \le m} = 2^{\binom{m}{2}} \prod_{0 \le i < j \le m} (x_j - x_i) > 0,$$ and the unique solution $(c_0, \dots, c_m) \in \mathbb{Q}^{m+1}$ to $\sum_{j=0}^m c_j T_r(x_j) = \delta_{r, m}$ ($0 \le r \le m$) is given in closed form by Lagrange-Chebyshev interpolation: $$c_j = \frac{1}{2^{m-1} \prod_{\ell \ne j} (x_j - x_\ell)} \in \mathbb{Q}^\times, \qquad j = 0, 1, \dots, m.$$ The resulting rational algebraic correspondence $$\Pi_{\mathrm{Cheb}}^{(m)} := \sum_{j=0}^m c_j \, T_{j, 1} \in \mathbb{Q}[\mathrm{End}_{\mathbb{Q}}(A_\tau)] \subset \mathrm{Corr}^0(A_\tau, A_\tau)_{\mathbb{Q}}$$ induces an idempotent operator on the middle cohomology $H^{2m}(A_\tau, \mathbb{Q})$ ($\big(\Pi_{\mathrm{Cheb}}^{(m)}\big|_{H^{2m}(A_\tau, \mathbb{Q})}\big)^2 = \Pi_{\mathrm{Cheb}}^{(m)}\big|_{H^{2m}(A_\tau, \mathbb{Q})}$, of rank $2$ and trace $2$ on $H^{2m}(A_\tau, \mathbb{Q})$) that acts as $0$ on all $2m-1$ non-Weil weight spaces $\mathcal{W}_1, \dots, \mathcal{W}_{2m-1}$ (annihilating $\binom{4m}{2m} - 2$ dimensions of $H^{2m}(A_\tau, \mathbb{Q})$) and as the identity $\mathrm{id}_2$ on the $2$-dimensional rational Weil space $W_{\mathrm{Weil}}(A_\tau) \subset H^{2m}(A_\tau, \mathbb{Q})$ across all of $\mathfrak{H}_{\mathrm{Weil}}(\phi)$! Explicitly, for $K = \mathbb{Q}(i)$ ($d=1$): $$\Pi_{\mathrm{Cheb}}^{(2)} = \frac{5}{16} T_{0,1} - \frac{5}{6} T_{1,1} + \frac{25}{48} T_{2,1}, \qquad \Pi_{\mathrm{Cheb}}^{(3)} = -\frac{25}{288} T_{0,1} + \frac{25}{48} T_{1,1} - \frac{125}{96} T_{2,1} + \frac{125}{144} T_{3,1}.$$ Furthermore, composing $\Pi_{\mathrm{Cheb}}^{(m)}$ with Lieberman's algebraic Künneth projector onto $H^{2m}(A_\tau, \mathbb{Q})$, $$\pi_{2m} := \prod_{\substack{0 \le q \le 4m \\ q \ne 2m}} \frac{[2]^* - 2^q\,\mathrm{id}}{2^{2m} - 2^q} \in \mathbb{Q}[\mathrm{End}(A_\tau)], \qquad \widehat{\Pi}_{\mathrm{Cheb}}^{(m)} := \pi_{2m} \circ \Pi_{\mathrm{Cheb}}^{(m)} \in \mathbb{Q}[\mathrm{End}_{\mathbb{Q}}(A_\tau)] \subset \mathrm{Corr}^0(A_\tau, A_\tau)_{\mathbb{Q}},$$ annihilates $H^q(A_\tau, \mathbb{Q})$ for all $q \ne 2m$ as well as $\bigoplus_{k=1}^{2m-1}\mathcal{W}_k \subset H^{2m}(A_\tau, \mathbb{Q})$, defining a cohomological idempotent ($\widehat{\Pi}_{\mathrm{Cheb}}^{(m)} \circ \widehat{\Pi}_{\mathrm{Cheb}}^{(m)} = \widehat{\Pi}_{\mathrm{Cheb}}^{(m)}$) on the full cohomology ring $H^*(A_\tau, \mathbb{Q}) = \bigoplus_{q=0}^{4m} H^q(A_\tau, \mathbb{Q})$ with image $W_{\mathrm{Weil}}(A_\tau)$.

Proof

(a) At any $U \in \mathrm{U}(m)$, differentiating $B^\dagger B = I_m$ gives $U^\dagger \dot{B} + \dot{B}^\dagger U = 0$, so $T_U \mathrm{U}(m) = \{U X : X^\dagger = -X\}$. Multiplying by $i$ gives $i T_U \mathrm{U}(m) = \{U Y : Y^\dagger = Y\}$. Since a matrix cannot be both skew-Hermitian and Hermitian unless it is zero, $T_U \mathcal{U} \cap i T_U \mathcal{U} = \{0\}$. If the holomorphic Gauss map $\gamma_Z: Z^{\mathrm{sm}} \to \mathrm{Gr}(m, 2m)$ ($d\gamma_Z \circ J_Z = i\, d\gamma_Z$) maps a non-empty open subset $V_0 \subset Z^{\mathrm{sm}}$ into $\mathcal{U}$, then at every point $z \in V_0$ its differential $d\gamma_{Z, z}(T_z V_0)$ is tangent to $\mathcal{U}$ and $i$-invariant, so it lies in $T_{\gamma_Z(z)}\mathcal{U} \cap i T_{\gamma_Z(z)}\mathcal{U} = \{0\}$. Thus $d\gamma_Z \equiv 0$ on $V_0$, and by analytic continuation $d\gamma_Z \equiv 0$ on the connected complex manifold $Z^{\mathrm{sm}}$, so $Z$ is a translate of a flat subtorus.

(b) Along a holomorphic map $z \mapsto B(z)$ where $\det B(z) \ne 0$, $\log\det B(z)$ is holomorphic ($\bar{\partial}B = 0$) and $\log\overline{\det B(z)}$ is anti-holomorphic ($\partial B^\dagger = 0$), so $\partial\bar{\partial}\log|\det B(z)|^2 = 0$. Therefore, $$i\partial\bar{\partial}\mathcal{L}_{\mathrm{Weil}}(B(z)) = i\partial\bar{\partial}\log\det(I_m + B(z)^\dagger B(z)) = i\partial\bar{\partial}\log\|w_{B(z)}\|^2 = \gamma_Z^*(\omega_{\mathrm{Pl\ddot{u}cker}}).$$ To compute the matrix $(1,1)$-form explicitly, set $M = I_m + B^\dagger B$. Since $\bar{\partial} B = 0$ and $\partial B^\dagger = 0$, Jacobi's formula gives $\bar{\partial}\log\det M = \mathrm{Tr}(M^{-1} \bar{\partial} M) = \mathrm{Tr}((I_m + B^\dagger B)^{-1} \bar{\partial}B^\dagger\, B)$. Differentiating with respect to $\partial$ and using $\partial(M^{-1}) = -M^{-1}(\partial M)M^{-1} = -M^{-1} B^\dagger (\partial B) M^{-1}$ together with the graded cyclic trace identity $\mathrm{Tr}(\alpha \wedge \beta) = -\mathrm{Tr}(\beta \wedge \alpha)$ for $1$-forms $\alpha, \beta$ yields $$i\partial\bar{\partial}\log\det M = -i\,\mathrm{Tr}\!\left(M^{-1}\bar{\partial}B^\dagger \big(I_m - B M^{-1} B^\dagger\big) \wedge \partial B\right) = i\,\mathrm{Tr}\!\left(\big(I_m - B M^{-1} B^\dagger\big)\partial B \wedge M^{-1}\bar{\partial}B^\dagger\right).$$ By the Woodbury matrix identity, $I_m - B(I_m + B^\dagger B)^{-1}B^\dagger = (I_m + B B^\dagger)^{-1}$, which yields the exact positive-definite $(1,1)$-form $i\,\mathrm{Tr}((I_m + B B^\dagger)^{-1}\partial B \wedge (I_m + B^\dagger B)^{-1}\bar{\partial}B^\dagger) \ge 0$. Similarly, expanding $B(z) = U(I_m + X(z))$ around $U \in \mathrm{SU}(m)$ to second order in $X, X^\dagger$ gives $\det(I_m + X) = 1 + \mathrm{Tr}(X) + \frac{1}{2}((\mathrm{Tr}\,X)^2 - \mathrm{Tr}(X^2)) + O(\|X\|^3)$ and $\det(I_m + (I_m+X^\dagger)(I_m+X)) = 2^m \det(I_m + \frac{1}{2}(X + X^\dagger + X^\dagger X))$. Retaining the $(1,1)$-bidegree mixed terms $\partial X \wedge \bar{\partial} X^\dagger$ and $\mathrm{Tr}(\partial X) \wedge \mathrm{Tr}(\bar{\partial} X^\dagger)$ yields $-i\partial\bar{\partial}\Phi_{\mathrm{Weil}}|_{B=U} = 2^{-m-2}\,i(\mathrm{Tr}(\partial B \wedge \bar{\partial} B^\dagger) + \mathrm{Tr}(U^\dagger \partial B) \wedge \overline{\mathrm{Tr}(U^\dagger \partial B)}) \ge 0$.

(c) Let $v_0 = \bigwedge_{a=1}^m \frac{i}{2} dz_a \wedge d\bar{z}_a$. Since $(d\bar{z}_1 \wedge dz_1) \wedge \dots \wedge (d\bar{z}_m \wedge dz_m) = (-1)^m (dz_1 \wedge d\bar{z}_1) \wedge \dots \wedge (dz_m \wedge d\bar{z}_m) = (-1)^m (\frac{2}{i})^m v_0 = (2i)^m v_0$, moving $dz_1, \dots, dz_m$ across $d\bar{z}_1, \dots, d\bar{z}_m$ in $d\bar{z}_1 \wedge \dots \wedge d\bar{z}_m \wedge dz_1 \wedge \dots \wedge dz_m$ introduces $(m-1) + (m-2) + \dots + 0 = \binom{m}{2}$ transpositions: $$\psi_-\big|_{w = Bz} = \big(d\bar{z}_1 \wedge \dots \wedge d\bar{z}_m \wedge dw_1 \wedge \dots \wedge dw_m\big)\big|_{w = Bz} = (\det B)\, d\bar{z}_1 \wedge \dots \wedge d\bar{z}_m \wedge dz_1 \wedge \dots \wedge dz_m = (-1)^{\binom{m}{2}} (2i)^m (\det B)\, v_0.$$ Meanwhile, pulling back $\omega_Q = \frac{i}{2}\sum_{a=1}^m (dz_a \wedge d\bar{z}_a + dw_a \wedge d\bar{w}_a)$ to $\Gamma_B = \{w = Bz\}$ gives $\frac{1}{m!}\omega_Q^m|_{\Gamma_B} = \det(I_m + B^\dagger B)\, v_0$. Multiplying $\psi_-|_{\Gamma_B}$ by $(-1)^{\binom{m}{2}} (-i)^m$ cancels $(-1)^{\binom{m}{2}} i^m$ and yields $(-1)^{\binom{m}{2}}(-i)^m \psi_-|_{\Gamma_B} = 2^m (\det B) v_0 = 2^m \frac{\det B}{\det(I_m + B^\dagger B)} \frac{\omega_Q^m|_{\Gamma_B}}{m!}$. Taking the real part and integrating over $Z = \sum_r c_r [Z_r]$ gives the exact formula for $\Delta_{\mathrm{curv}}(Z)$. By Theorem 1(b), $1 - 2^m \Phi_{\mathrm{Weil}}(\gamma_{Z_r}(z)) \ge 0$ everywhere with equality at $z$ if and only if $B(z) \in \mathrm{SU}(m)$, and by part (a), if $A_\tau$ is simple, no $Z_r$ is a translate of a flat subtorus, forcing $\Delta_{\mathrm{curv}}(Z) > 0$.

(d) On $W_+$ and $W_-$, the integer endomorphism $a_j + b_j \phi \in \mathbb{Z}[\phi]$ acts by the scalars $a_j + i b_j \sqrt{d} = \sqrt{a_j^2 + d b_j^2}\, e^{i\vartheta_j}$ and $a_j - i b_j \sqrt{d} = \sqrt{a_j^2 + d b_j^2}\, e^{-i\vartheta_j}$, respectively, where $\cos\vartheta_j = \frac{a_j}{\sqrt{a_j^2 + d b_j^2}}$, $\sin\vartheta_j = \frac{b_j\sqrt{d}}{\sqrt{a_j^2 + d b_j^2}}$, and therefore $\cos(2\vartheta_j) = \cos^2\vartheta_j - \sin^2\vartheta_j = \frac{a_j^2 - d b_j^2}{a_j^2 + d b_j^2} = x_j \in \mathbb{Q}$. Because the middle cohomology degree $2m$ is even, on $\mathcal{W}_k = \bigwedge^{2m-k} W_+ \otimes \bigwedge^k W_-$ the pullback $[a_j + b_j \phi]^*$ acts by the product of $2m-k$ copies of $\sqrt{a_j^2 + d b_j^2}\, e^{i\vartheta_j}$ and $k$ copies of $\sqrt{a_j^2 + d b_j^2}\, e^{-i\vartheta_j}$: $$\big(\sqrt{a_j^2 + d b_j^2}\, e^{i\vartheta_j}\big)^{2m-k} \big(\sqrt{a_j^2 + d b_j^2}\, e^{-i\vartheta_j}\big)^k = (a_j^2 + d b_j^2)^m e^{i(2m - 2k)\vartheta_j} = (a_j^2 + d b_j^2)^m e^{2i(m-k)\vartheta_j}.$$ Dividing by the rational scalar $(a_j^2 + d b_j^2)^m \in \mathbb{Q}^\times$ and symmetrizing with $[a_j - b_j \phi]^*$ (which acts by $(a_j^2 + d b_j^2)^m e^{-2i(m-k)\vartheta_j}$), the rational correspondence $T_{a_j, b_j} = \frac{[a_j + b_j\phi]^* + [a_j - b_j\phi]^*}{2(a_j^2 + d b_j^2)^m}$ acts on $\mathcal{W}_k \oplus \mathcal{W}_{2m-k} \subset H^{2m}(A_\tau, \mathbb{C})$ by the scalar $$\cos\big(2|m-k|\vartheta_j\big) = T_{|m-k|}\big(\cos 2\vartheta_j\big) = T_{|m-k|}(x_j).$$ (Note that if one instead uses the Hilbert-90 norm-$1$ self-isogeny $u_j = \frac{a_j+b_j\phi}{a_j-b_j\phi} = \frac{(a_j+b_j\phi)^2}{a_j^2+d b_j^2}$, $[u_j]^*$ acts by the square $e^{4i(m-k)\vartheta_j}$, evaluating $T_{|m-k|}$ at the double-angle node $\tilde{x}_j = 2x_j^2 - 1$.) Next, since $T_r(x)$ has degree $r$ with leading coefficient $2^{\max(0, r-1)}$, $M = (T_r(x_j))_{0 \le r, j \le m}$ is obtained from the standard Vandermonde matrix $V = (x_j^r)_{0 \le r, j \le m}$ by a lower-triangular transformation with diagonal $(1, 1, 2, \dots, 2^{m-1})$, giving $\det M = 2^{\binom{m}{2}}\prod_{0 \le i < j \le m}(x_j - x_i) > 0$ since $x_j = 1 - \frac{2d}{j^2+d}$ is strictly increasing for $j = 0, \dots, m$. Finally, the unique degree-$m$ polynomial $P(x) = \sum_{r=0}^m \alpha_r T_r(x)$ vanishing at $x_0, \dots, x_{m-1}$ with $\alpha_m = 1$ (leading coefficient $2^{m-1}$) is $P(x) = 2^{m-1}\prod_{\ell=0}^{m-1}(x - x_\ell)$; by Lagrange interpolation, $\sum_{j=0}^m c_j T_r(x_j) = \delta_{r, m}$ is uniquely solved by $c_j = (2^{m-1}\prod_{\ell \ne j}(x_j - x_\ell))^{-1}$. Thus on $H^{2m}(A_\tau, \mathbb{Q})$, $\Pi_{\mathrm{Cheb}}^{(m)}$ acts on $\mathcal{W}_k$ by $\sum_{j=0}^m c_j T_{|m-k|}(x_j) = \delta_{|m-k|, m}$, which is $0$ for $1 \le k \le 2m-1$ and $1$ on $W_{\mathrm{Weil}}(A_\tau)$. Finally, since $[2]^*$ acts on $H^q(A_\tau, \mathbb{Q}) = \bigwedge^q H^1(A_\tau, \mathbb{Q})$ by $2^q$, $\pi_{2m}$ acts by $0$ on $H^q(A_\tau, \mathbb{Q})$ for all $q \ne 2m$ and by $1$ on $H^{2m}(A_\tau, \mathbb{Q})$, so $\widehat{\Pi}_{\mathrm{Cheb}}^{(m)} = \pi_{2m} \circ \Pi_{\mathrm{Cheb}}^{(m)}$ is an idempotent on the full cohomology ring $H^*(A_\tau, \mathbb{Q})$ (exp21_chebyshev_weil_endomorphism_projector.py, verified directly on the $70 \times 70$ rational exterior-algebra matrix $\bigwedge^4 \mathbb{Q}^8$ in test_theorem3). $\blacksquare$

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5. Background on Complete-Intersection Catalecticant Ranks and the Fermat Sextic Fourfold

We now turn to smooth projective fourfolds $X_d^4 = \{F_d(x_0, \dots, x_5) = 0\} \subset \mathbb{P}^5$ of degree $d \ge 4$. By Griffiths residue theory [7] and Macaulay's theorem, the Jacobian ring $R(F_d) = \mathbb{C}[x_0, \dots, x_5]/J(F_d)$ ($J(F_d) = \langle \partial_0 F_d, \dots, \partial_5 F_d \rangle$) is an artinian Gorenstein algebra with socle degree $\sigma = 6(d-2) = 6d - 12$, where $H^{4-q, q}_{\mathrm{prim}}(X_d^4, \mathbb{C}) \cong R(F_d)_{(q+1)d-6}$; in particular, $H^{2,2}_{\mathrm{prim}}(X_d^4, \mathbb{C}) \cong R(F_d)_{3d-6}$ and $H^{1,3}_{\mathrm{prim}}(X_d^4, \mathbb{C}) \cong R(F_d)_{4d-6}$. For any class $\alpha \in R(F_d)_{3d-6}$, the first-order IVHS / Catalecticant map $\Theta_1(\alpha): R(F_d)_d \to R(F_d)_{4d-6}$ is multiplication by $\alpha$ inside $R(F_d)$.

Suppose $Z = \{g_1 = g_2 = g_3 = 0\} \subset X_d^4$ is a complete-intersection surface of multidegree $(d_1, d_2, d_3)$ ($1 \le d_k \le d-1$), so that by Hilbert's Nullstellensatz/Koszul exactness there exist homogeneous polynomials $h_1, h_2, h_3$ of degrees $e_k := d - d_k$ satisfying the regular-sequence decomposition $$F_d(x_0, \dots, x_5) = g_1(x) h_1(x) + g_2(x) h_2(x) + g_3(x) h_3(x).$$ By Theorem 1.1 of Villaflor [9] (see also Carlson-Griffiths [8] and Movasati [10]), the primitive cohomology class $[Z]_{\mathrm{prim}} \in H^{2,2}_{\mathrm{prim}}(X_d^4, \mathbb{C}) \cong R(F_d)_{3d-6}$ is represented up to a non-zero constant by the $6 \times 6$ Jacobian determinant of degree $\sum_{k=1}^3 ((d_k - 1) + (d - d_k - 1)) = 3d - 6$: $$P_Z(x) := \det\frac{\partial(g_1, g_2, g_3, h_1, h_2, h_3)}{\partial(x_0, x_1, x_2, x_3, x_4, x_5)} \in R(F_d)_{3d-6}.$$ We recall as background the Gorenstein ideal-annihilation and Hilbert-series Catalecticant rank theorem established by Dan [21, Proposition 2.12], Villaflor Loyola [22, Example 2.1], and Kloosterman [20, Lemma 4.12 & Proposition 4.13] (via Scheja-Storch Gorenstein residuation [15] for nested complete intersections $J(F_d) \subset I_{g,h} := \langle g_1, \dots, g_{p+1}, h_1, \dots, h_{p+1}\rangle$), together with the explicit complete-intersection realizations and Hodge locus codimensions on the Fermat sextic fourfold $X_6^4 \subset \mathbb{P}^5$ established by Braun, Fortin, Lopez Garcia, and Villaflor Loyola [23, equations (2.15), (2.17), (3.2), (3.3), Theorem 1, & Section 4]:

Proposition 1 (Complete-Intersection Annihilator, Catalecticant Rank, and Sextic Fourfold Realizations; Dan [21], Villaflor Loyola [22], Kloosterman [20], Braun–Fortin–Lopez Garcia–Villaflor Loyola [23])

Let $F_d = \sum_{k=1}^{p+1} g_k h_k$ be a regular-sequence decomposition of a smooth degree-$d$ hypersurface $X_d^{2p} \subset \mathbb{P}^{2p+1}$ with $\deg g_k = d_k$ and $\deg h_k = d - d_k$, defining $Z = \{g_1 = \dots = g_{p+1} = 0\} \subset X_d^{2p}$, and let $I_{g,h} := \langle g_1, \dots, g_{p+1}, h_1, \dots, h_{p+1} \rangle$. Then inside $R(F_d) = \mathbb{C}[x_0, \dots, x_{2p+1}]/J(F_d)$, the Villaflor Jacobian polynomial $P_Z = \det\frac{\partial(g, h)}{\partial x} \in R(F_d)_{(p+1)d-(2p+2)}$ [9] has exact annihilator $\mathrm{Ann}_{R(F_d)}(P_Z) = I_{g,h}/J(F_d)$ [21, Proposition 2.12; 22, Example 2.1], and its first-order IVHS / Catalecticant rank is the degree-$d$ Hilbert function of $\mathbb{C}[x_0, \dots, x_{2p+1}]/I_{g,h}$ [20, Proposition 4.13]: $$\mathrm{rank}\,\Theta_1([Z]_{\mathrm{prim}}) = \dim_{\mathbb{C}}(\mathbb{C}[x_0, \dots, x_{2p+1}]/I_{g,h})_d = [t^d]\frac{\prod_{k=1}^{p+1} (1 - t^{d_k})(1 - t^{d - d_k})}{(1 - t)^{2p+2}}.$$ On the Fermat fourfold $X_d^4$ ($p=2$), every character monomial $m_\alpha = x^e \in R(F_d)_{3d-6}$ ($a_i = e_i + 1$, $\sum_{i=0}^5 a_i = 3d$) has annihilator $\langle x_0^{d-a_0}, \dots, x_5^{d-a_5} \rangle$ and box-partition Catalecticant rank $\mathrm{rank}\,\Theta_1(m_\alpha) = [t^d]\prod_{i=0}^5 \frac{1 - t^{d - a_i}}{1 - t}$ (which achieves minimum exceptional rank $40$ on $X_6^4$, and $44$ on pair-free orbits), while the $10$ complete-intersection surface multidegrees $(d_1, d_2, d_3)$ ($1 \le d_1 \le d_2 \le d_3 \le 3$) on $X_6^4$ have Catalecticant ranks $(19, 32, 37, 54, 62, 71, 92, 106, 122, 141)$ as recorded in Table 1. In particular, on the Fermat sextic fourfold $X_6^4$ ($d = 6$), Braun, Fortin, Lopez Garcia, and Villaflor Loyola [23, (2.15), (2.17), (3.2), (3.3), Theorem 1, & Section 4] showed that $\mathrm{Hdg}^2(X_6^4, \mathbb{Z})$ is generated by complete-intersection surfaces of multidegrees $(1,1,1)$ (linear planes, rank $19$), $(1,2,3)$ (the Aoki–Shioda join $Z_{\alpha_1} = A_{1,6} * A_{2,6}$, rank $62$), and $(2,2,3)$ (their "type 3" surface $T = \{y_0^2 - \sqrt[3]{2}y_1 y_2 = y_3^2 - \sqrt[3]{2}y_4 y_5 = y_1^3 + y_2^3 + i(y_4^3 + y_5^3) = 0\} \subset X_6^4$, rank $106$, which coincides with $Z_{\alpha_0} = Z^{(6)}$ below under the signed coordinate identification $(y_0,y_1,y_2,y_3,y_4,y_5) = (x_0,x_2,-x_3,x_1,x_4,-x_5)$).

| Multidegree $(d_1, d_2, d_3)$ | Surface Degree $\deg(Z) = d_1 d_2 d_3$ | Complementary Degrees $(6-d_1, 6-d_2, 6-d_3)$ | Exact Catalecticant Rank $\mathrm{rank}\,\Theta_1([Z]_{\mathrm{prim}})$ [20, 23] | Comparison with Monomial Exceptional Ranks ($\ge 40$, Pair-Free $\ge 44$) | | :--- | :---: | :---: | :---: | :---: | | $(1, 1, 1)$ (Linear 2-Plane $\mathbb{P}^2$) | $1$ | $(5, 5, 5)$ | $19$ | $< 40$ | | $(1, 1, 2)$ (Quadric Surface) | $2$ | $(5, 5, 4)$ | $32$ | $< 40$ | | $(1, 1, 3)$ (Cubic Surface) | $3$ | $(5, 5, 3)$ | $37$ | $< 40$ | | $(1, 2, 2)$ (Complete Intersection) | $4$ | $(5, 4, 4)$ | $54$ | $\ge 44$ | | $(1, 2, 3)$ (Join $A_{1,6}*A_{2,6}$: $Z_{\alpha_1}$ [23]) | $6$ | $(5, 4, 3)$ | $62$ | $\ge 44$ | | $(1, 3, 3)$ (Complete Intersection) | $9$ | $(5, 3, 3)$ | $71$ | $\ge 44$ | | $(2, 2, 2)$ (Complete Intersection) | $8$ | $(4, 4, 4)$ | $92$ | $\ge 44$ | | $(2, 2, 3)$ ("Type 3" / Double-$A_{2,6}$: $Z_{\alpha_0}$ [23]) | $12$ | $(4, 4, 3)$ | $106$ | $\ge 44$ | | $(2, 3, 3)$ (Complete Intersection) | $18$ | $(4, 3, 3)$ | $122$ | $\ge 44$ | | $(3, 3, 3)$ (Maximal Half-Degree) | $27$ | $(3, 3, 3)$ | $141$ | $\ge 44$ |

Table 1: Complete classification of first-order IVHS / Catalecticant ranks $\mathrm{rank}\,\Theta_1([Z]_{\mathrm{prim}})$ across all $10$ complete-intersection surface multidegrees $(d_1, d_2, d_3)$ ($1 \le d_1 \le d_2 \le d_3 \le 3$) on a smooth sextic fourfold $X_6^4 \subset \mathbb{P}^5$ [20, 23] (exp17_sextic_entangled_and_join_cycles.py).

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6. Closed-Form Jacobian Factorizations and Sporadic Brieskorn-Pham Complete Intersections on Higher-Degree Fermat Varieties

On the Fermat sextic fourfold $X_6^4 = \{\sum_{i=0}^5 x_i^6 = 0\} \subset \mathbb{P}^5$, the $18$ sorted $S_6$-permutation classes of rational Hodge characters $\alpha = (a_0, \dots, a_5)$ ($1 \le a_i \le 5$, $\sum a_i = 18$, $\sum \langle 5 a_i \rangle_6 = 18$) split into $10$ pair-decomposable classes (generated by the $15 \times 6^3 = 3240$ linear planes $\mathbb{P}^2 \subset X_6^4$) and $8$ exceptional (non-plane-generated) classes, which form four complex-conjugate pairs under $a_i \mapsto 6 - a_i$ [23, Section 3]:

1. Six Semi-Inductive (Standard Join) Classes (3 conjugate pairs): $(a_0, 6-a_0) \oplus (2,2,3,5)$ and $(a_0, 6-a_0) \oplus (1,3,4,4)$ for $(a_0, 6-a_0) \in \{(1,5), (2,4), (3,3)\}$—namely $\alpha_1 = (1, 2, 2, 3, 5, 5)$ ($\mathrm{rank}\,\Theta_1(m_{\alpha_1}) = 40$) and $\bar{\alpha}_1 = (1, 1, 3, 4, 4, 5)$ ($\mathrm{rank}\,\Theta_1(m_{\bar{\alpha}_1}) = 50$), $(2,2,2,3,4,5)$ ($\mathrm{rank}=68$) and $(1,2,3,4,4,4)$ ($\mathrm{rank}=84$), and $(2,2,3,3,3,5)$ ($\mathrm{rank}=78$) and $(1,3,3,3,4,4)$ ($\mathrm{rank}=96$)—each containing a single complementary pair summing to $6$ joined with the $2$-standard surface characters $s_{2, 1} = (1, 3, 4, 4)$ or $s_{2, 5} = (2, 2, 3, 5)$ [18, 19, 23].
2. Two Pair-Free Purely Inductive Classes (1 conjugate pair): $\alpha_0 = (1, 1, 4, 4, 4, 4)$ ($x^{e_0} = x_2^3 x_3^3 x_4^3 x_5^3$, $\mathrm{rank}\,\Theta_1(m_{\alpha_0}) = 68$) and $\bar{\alpha}_0 = (2, 2, 2, 2, 5, 5)$ ($x^{\bar{e}_0} = x_0 x_1 x_2 x_3 x_4^4 x_5^4$, $\mathrm{rank}\,\Theta_1(m_{\bar{\alpha}_0}) = 44$), where no two entries sum to $6$.

In Aoki's formal module of tuples $R_{2m}$ [12, 18] as formulated by Martelotte, Miranda, Movasati, and Villaflor [19, §3.4–3.5], the $2$-standard characters on $X_{2m}^2$ for general even degree $d = 2m \ge 6$ are $s_{2, k+1} = (k+1, k+m+1, 2m-2-2k, m)$ ($0 \le k \le m-2$), realized by the complete-intersection Aoki–Shioda curve $A_{2, 2m} \subset X_{2m}^2$ [18; 19, Definition 3.8], while deleting the entry $\varepsilon_{2m} = (m)$ yields the odd-length tuple $T_k := s_{2, k+1} - (m) = (2m-2-2k, k+1, k+m+1)$ of sum $3m$ [19, Definition 3.10]. While the algebraicity of the join families and their tuple arithmetic are established in [11, 18, 19] and the sextic fourfold ($d=6$) case was worked out in [23, Section 3], [19, Proof of Proposition 3.7] notes that applying Villaflor's Jacobian determinant formula [9, Theorem 1.1] to higher-degree cycles $[A_{p, d}]_0$ is computationally tedious and omits the explicit expansion. In Theorem 4 below, we show that block-triangular Laplace expansion extends the sextic factorization of [23, (2.15), (2.17), (3.2), (3.3)] to arbitrary even degrees $d = 2m \ge 6$, yielding closed-form factored Jacobian polynomials for both the degree-$(2,2,m)$ entangled double-Aoki–Shioda surface family $Z^{(2m)} \subset X_{2m}^4$ (which eliminates the $(m, m)$ pair at the polynomial equation level to realize all $(m-1)^2$ pairwise joins $T_k * T_l$ inside a single complete intersection) and the join surfaces $Z_{\mathrm{semi}}^{(2m)} = A_{1, 2m} * A_{2, 2m}$ and $Z_{\mathrm{nonic}} = A_{1, 9} * A_{3, 9} \subset X_9^4$. Then, in Theorem 5, we solve a new weighted Brieskorn-Pham $(1, 6, 4, 3)$-syzygy at $d = 12$ that realizes Aoki's sporadic order-$2$ parity generator $S_4 = (1, 6, 8, 9) \in G_{12}$ [12, Theorem A] directly by a single complete-intersection curve $C_{\mathrm{spor}} \subset X_{12}^2$ and entangled complete-intersection surfaces on $X_{12}^4$.

Theorem 4 (General Even-Degree Closed-Form Jacobian Factorization of $Z^{(2m)}$, $Z_{\mathrm{semi}}^{(2m)}$, and Nonic $Z_{\mathrm{nonic}}$)

(a) Universal Entangled Degree-$(2,2,m)$ Family $Z^{(2m)} \subset X_{2m}^4$ on Even-Degree Fermat Fourfolds ($d = 2m \ge 6$): For any integer $m \ge 3$, let $\lambda = \xi_m 2^{1/m}$ with $\xi_m^m = -1$ (so $\lambda^m = -2$). Define the entangled double-$A_{2,2m}$ complete-intersection surface family $Z^{(2m)} = \{g_1^{(2m)} = g_2^{(2m)} = g_3^{(2m)} = 0\} \subset X_{2m}^4$ of multidegree $(2,2,m)$ ($\deg Z^{(2m)} = 4m$) and complementary degrees $(2m-2, 2m-2, m)$ by $$\begin{aligned} g_1^{(2m)} &= x_0^2 - \lambda x_2 x_3, & h_1^{(2m)} &= \sum_{k=0}^{m-1} \lambda^k x_0^{2(m-1-k)} (x_2 x_3)^k, \\ g_2^{(2m)} &= x_1^2 - \lambda x_4 x_5, & h_2^{(2m)} &= \sum_{k=0}^{m-1} \lambda^k x_1^{2(m-1-k)} (x_4 x_5)^k, \\ g_3^{(2m)} &= (x_2^m - x_3^m) + i(x_4^m - x_5^m), & h_3^{(2m)} &= (x_2^m - x_3^m) - i(x_4^m - x_5^m). \end{aligned}$$ Then $\sum_{r=1}^3 g_r^{(2m)} h_r^{(2m)} = \sum_{i=0}^5 x_i^{2m}$, and the $6 \times 6$ Villaflor Jacobian determinant $P_{Z^{(2m)}}(x) = \det\frac{\partial(g_1^{(2m)}, g_2^{(2m)}, g_3^{(2m)}, h_1^{(2m)}, h_2^{(2m)}, h_3^{(2m)})}{\partial(x_0, x_1, x_2, x_3, x_4, x_5)}$ factors identically in $\mathbb{C}[x_0, \dots, x_5]_{6m-6}$ as $$P_{Z^{(2m)}}(x) = 8i m^4 \left(\sum_{k=0}^{m-2} \lambda^{k+1} x_0^{2m-3-2k} (x_2 x_3)^k\right)\left(\sum_{l=0}^{m-2} \lambda^{l+1} x_1^{2m-3-2l} (x_4 x_5)^l\right)(x_2^m + x_3^m)(x_4^m + x_5^m).$$ Its $4(m-1)^2$ monomials realize all $(m-1)^2$ pairwise joins $T_k * T_l$ ($0 \le k, l \le m-2$) of the odd-length standard tuples $T_k = (2m-2-2k, k+1, k+m+1) = s_{2, k+1} - (m)$; in particular, selecting $k = l = 0$ produces the monomial $x_0^{2m-3} x_1^{2m-3} x_2^m x_4^m$ with non-zero coefficient $8i m^4 \lambda^2 \ne 0$, realizing the pair-free Hodge character $$\alpha_{\mathrm{ind}}^{(2m)} = T_0 * T_0 = (1, 1, m+1, m+1, 2m-2, 2m-2) \in \mathrm{Hdg}^2_{\mathrm{prim}}(X_{2m}^4, \mathbb{Q}) \qquad \forall\, m \ge 3.$$

(b) Universal Join-Aoki–Shioda Degree-$(1,2,m)$ Family $Z_{\mathrm{semi}}^{(2m)} = A_{1,2m} * A_{2,2m} \subset X_{2m}^4$ ($d = 2m \ge 6$): Let $\zeta_{4m} = e^{i\pi/(2m)}$ ($\zeta_{4m}^{2m} = -1$) and define $Z_{\mathrm{semi}}^{(2m)} = \{\tilde{g}_1^{(2m)} = \tilde{g}_2^{(2m)} = \tilde{g}_3^{(2m)} = 0\} \subset X_{2m}^4$ of multidegree $(1, 2, m)$ [18, 19] by $$\begin{aligned} \tilde{g}_1^{(2m)} &= x_0 - \zeta_{4m} x_1, & \tilde{h}_1^{(2m)} &= \sum_{k=0}^{2m-1} \zeta_{4m}^k x_0^{2m-1-k} x_1^k, \\ \tilde{g}_2^{(2m)} &= x_2^2 - \lambda x_4 x_5, & \tilde{h}_2^{(2m)} &= \sum_{k=0}^{m-1} \lambda^k x_2^{2(m-1-k)} (x_4 x_5)^k, \\ \tilde{g}_3^{(2m)} &= x_3^m + i(x_4^m - x_5^m), & \tilde{h}_3^{(2m)} &= x_3^m - i(x_4^m - x_5^m). \end{aligned}$$ Then $\sum_{r=1}^3 \tilde{g}_r^{(2m)} \tilde{h}_r^{(2m)} = \sum_{i=0}^5 x_i^{2m}$, and its $6 \times 6$ Jacobian determinant factors in closed form as $$P_{Z_{\mathrm{semi}}^{(2m)}}(x) = -8i m^4 \left(\sum_{a=0}^{2m-2} \zeta_{4m}^{a+1} x_0^{2m-2-a} x_1^a\right)\left(\sum_{k=0}^{m-2} \lambda^{k+1} x_2^{2m-3-2k} (x_4 x_5)^k\right) x_3^{m-1}(x_4^m + x_5^m),$$ simultaneously realizing all $(m-1)^2$ standard join characters $(a+1, 2m-1-a) \oplus s_{2, k+1}$ ($0 \le a, k \le m-2$).

(c) Specialization to $d = 6$ [23] and New Higher Even Degrees $d = 8, 10$ (exp17, exp20):

• At $d = 6$ ($m = 3$, $\lambda = -2^{1/3}$), $Z_{\alpha_0} := Z^{(6)}$ ($\mathrm{rank} = 106$) and $Z_{\alpha_1} := Z_{\mathrm{semi}}^{(6)}$ ($\mathrm{rank} = 62$) recover the "type 3" surface $T$ and Aoki–Shioda join surface of Braun, Fortin, Lopez Garcia, and Villaflor Loyola [23, (2.15), (2.17), (3.2), (3.3), Section 3], with $P_{Z_{\alpha_0}}(x) = 648i(2^{1/3} x_0^3 - 2^{2/3} x_0 x_2 x_3)(2^{1/3} x_1^3 - 2^{2/3} x_1 x_4 x_5)(x_2^3 + x_3^3)(x_4^3 + x_5^3)$ projecting with non-zero coefficients $648\cdot 2^{2/3}i$ onto $\alpha_0 = (1,1,4,4,4,4) = T_0 * T_0$ and $1296\cdot 2^{1/3}i$ onto $\bar{\alpha}_0 = (2,2,2,2,5,5) = T_1 * T_1$, and $P_{Z_{\alpha_1}}(x)$ projecting onto all $6$ semi-inductive classes (spanning $8/8$ exceptional classes on $X_6^4$).
• At $d = 8$ ($m = 4$, $\lambda = e^{i\pi/4} 2^{1/4}$), $Z^{(8)}$ ($\mathrm{rank} = 282$) and $Z_{\mathrm{semi}}^{(8)}$ ($\mathrm{rank} = 157$) directly realize $100\%$ ($10/10$) of all exceptional $S_6 \times (\mathbb{Z}/8\mathbb{Z})^\times$-orbits on the Fermat octic fourfold $X_8^4$ ($2/2$ pair-free orbits $(1,1,5,5,6,6) = T_0 * T_0$ and $(2,2,3,3,7,7) = T_1 * T_1$, plus all $8/8$ semi-inductive orbits).
• At $d = 10$ ($m = 5$, $\lambda = -2^{1/5}$), the $4$ diagonal products $T_k * T_k$ ($0 \le k \le 3$) in $P_{Z^{(10)}}(x)$ ($\mathrm{rank} = 562$) directly realize $100\%$ ($4/4$) of the pair-free exceptional orbits on the Fermat decic fourfold $X_{10}^4$—namely $(1,1,6,6,8,8)$, $(2,2,4,4,9,9)$, $(2,2,6,6,7,7)$, and $(3,3,4,4,8,8)$—while the $4$ off-diagonal products $T_0 * T_1 = (2,8) \oplus (1,6,6,7)$, $T_0 * T_2 = (4,6) \oplus (1,3,8,8)$, $T_1 * T_3 = (4,6) \oplus (2,2,7,9)$, and $T_2 * T_3 = (2,8) \oplus (3,4,4,9)$ realize all $4$ non-standard surface blocks on $X_{10}^2$ paired with $(2,8)$ and $(4,6)$ (so linear deletion and join [11; 19, Proposition 3.5] recover all $44/44$ exceptional orbits on $X_{10}^4$).

(d) Closed-Form Jacobian Expansion of the Join-Aoki–Shioda Nonic Surface $Z_{\mathrm{nonic}} = A_{1,9} * A_{3,9}$ on $X_9^4$ ($d = 9$): On $X_9^4 = \{\sum_{i=0}^5 x_i^9 = 0\} \subset \mathbb{P}^5$, let $\zeta_{18} = e^{i\pi/9}$ ($\zeta_{18}^9 = -1$) and consider the degree-$(1, 3, 3) = 9$ join $Z_{\mathrm{nonic}} = A_{1,9} * A_{3,9} \subset X_9^4$ of Aoki's $3$-standard cycle $A_{3,9}$ [18; 19, Definition 3.8]: $$\begin{aligned} g_1 &= x_0 - \zeta_{18} x_1, & h_1 &= \sum_{k=0}^8 \zeta_{18}^k x_0^{8-k} x_1^k, \\ g_2 &= x_2^3 + 3^{1/3} x_3 x_4 x_5, & h_2 &= x_2^6 - 3^{1/3} x_2^3 x_3 x_4 x_5 + 3^{2/3} x_3^2 x_4^2 x_5^2, \\ g_3 &= x_3^3 + x_4^3 + x_5^3, & h_3 &= x_3^6 + x_4^6 + x_5^6 - x_3^3 x_4^3 - x_4^3 x_5^3 - x_5^3 x_3^3. \end{aligned}$$ Then $g_1 h_1 + g_2 h_2 + g_3 h_3 = \sum_{i=0}^5 x_i^9$, $Z_{\mathrm{nonic}}$ has Gorenstein Catalecticant rank $259$ (compared to $46$ for a linear plane $\mathbb{P}^2 \subset X_9^4$), and its $6 \times 6$ Jacobian determinant $P_{Z_{\mathrm{nonic}}}(x) \in R(F_9)_{21}$ factors in closed form as $$P_{Z_{\mathrm{nonic}}}(x) = 2187\left(\sum_{a=0}^7 \zeta_{18}^{a+1} x_0^{7-a} x_1^a\right)\left(3^{1/3} x_2^5 - 3^{2/3} x_2^2 x_3 x_4 x_5\right)(x_3^3 - x_4^3)(x_4^3 - x_5^3)(x_5^3 - x_3^3),$$ whose $96$ monomials consist $100\%$ of pure exceptional Hodge characters ($12$ monomials in each of $(a, 9-a) \oplus (1, 4, 6, 7)$ and $(a, 9-a) \oplus (2, 3, 5, 8)$ for $a = 1, 2, 3, 4$), simultaneously realizing all $8$ exceptional Hodge character classes on $X_9^4$!

Proof

(a)–(c) By $a^m - (\lambda b)^m = (a - \lambda b)\sum_{k=0}^{m-1} \lambda^k a^{m-1-k} b^k$ with $\lambda^m = -2$, we have $g_1^{(2m)} h_1^{(2m)} = x_0^{2m} + 2 x_2^m x_3^m$ and $g_2^{(2m)} h_2^{(2m)} = x_1^{2m} + 2 x_4^m x_5^m$. Meanwhile, $g_3^{(2m)} h_3^{(2m)} = (x_2^m - x_3^m)^2 + (x_4^m - x_5^m)^2 = x_2^{2m} - 2x_2^m x_3^m + x_3^{2m} + x_4^{2m} - 2x_4^m x_5^m + x_5^{2m}$. Summing the three products cancels $+2x_2^m x_3^m$ and $+2x_4^m x_5^m$ identically, yielding $\sum_{i=0}^5 x_i^{2m}$. To compute $P_{Z^{(2m)}}(x)$ (and $P_{Z_{\alpha_0}}(x)$ at $m=3, \lambda = -2^{1/3}$), observe that $g_1^{(2m)}, h_1^{(2m)}$ depend on $(x_2, x_3)$ only through the product $u := x_2 x_3$, and are independent of $(x_1, x_4, x_5)$, while $g_2^{(2m)}, h_2^{(2m)}$ depend on $(x_4, x_5)$ only through $v := x_4 x_5$, and are independent of $(x_0, x_2, x_3)$. Differentiating with respect to $(x_0, u)$ and shifting the summation index $k-1 \mapsto k$ in $\partial_u h_1^{(2m)}$ aligns both sums over $0 \le k \le m-2$ and cancels $k$ in the coefficient $2(k+1) + 2(m-1-k) = 2m$: $$\frac{\partial(g_1^{(2m)}, h_1^{(2m)})}{\partial(x_0, u)} = 2x_0 \sum_{k=1}^{m-1} k \lambda^k x_0^{2(m-1-k)} u^{k-1} + \lambda \sum_{k=0}^{m-2} 2(m-1-k)\lambda^k x_0^{2m-3-2k} u^k = 2m \sum_{k=0}^{m-2} \lambda^{k+1} x_0^{2m-3-2k} (x_2 x_3)^k.$$ Next, for any function $f(x_2, x_3, x_4, x_5)$, the $2 \times 2$ Jacobian of $(u, f) = (x_2 x_3, f)$ with respect to $(x_2, x_3)$ is the polynomial differential operator $$\mathcal{D}_{23}(f) := \frac{\partial(x_2 x_3,\; f)}{\partial(x_2,\; x_3)} = \det\begin{pmatrix} x_3 & x_2 \\ \partial_{x_2} f & \partial_{x_3} f \end{pmatrix} = x_3 \partial_{x_3} f - x_2 \partial_{x_2} f,$$ and similarly $\mathcal{D}_{45}(f) := \frac{\partial(x_4 x_5, f)}{\partial(x_4, x_5)} = x_5 \partial_{x_5} f - x_4 \partial_{x_4} f$. Since $g_3^{(2m)}, h_3^{(2m)}$ are independent of $(x_0, x_1)$ and the row reordering $(g_1, g_2, g_3, h_1, h_2, h_3) \mapsto (g_1, h_1, g_2, h_2, g_3, h_3)$ has odd sign $(-1)^3 = -1$, Laplace expansion along $(x_0, x_1)$ and the chain rule on $(u, v)$ factor $P_{Z^{(2m)}}(x)$ into three $2 \times 2$ polynomial determinants: $$P_{Z^{(2m)}}(x) = -\frac{\partial(g_1^{(2m)}, h_1^{(2m)})}{\partial(x_0, u)} \cdot \frac{\partial(g_2^{(2m)}, h_2^{(2m)})}{\partial(x_1, v)} \cdot \det\begin{pmatrix} \mathcal{D}_{23}(g_3^{(2m)}) & \mathcal{D}_{45}(g_3^{(2m)}) \\ \mathcal{D}_{23}(h_3^{(2m)}) & \mathcal{D}_{45}(h_3^{(2m)}) \end{pmatrix}.$$ Since $\mathcal{D}_{23}(x_2^m - x_3^m) = -m x_3^m - m x_2^m = -m(x_2^m + x_3^m)$ and $\mathcal{D}_{45}(\pm i(x_4^m - x_5^m)) = \mp i m(x_4^m + x_5^m)$, the $2 \times 2$ determinant on $(g_3^{(2m)}, h_3^{(2m)})$ equals $\det\begin{pmatrix} -m(x_2^m+x_3^m) & -im(x_4^m+x_5^m) \\ -m(x_2^m+x_3^m) & +im(x_4^m+x_5^m) \end{pmatrix} = -2i m^2 (x_2^m + x_3^m)(x_4^m + x_5^m)$. Multiplying by $-\frac{\partial(g_1, h_1)}{\partial(x_0, u)}\frac{\partial(g_2, h_2)}{\partial(x_1, v)}$ cancels the minus sign and yields the exact factored polynomial with prefactor $(2m)(2m)(2i m^2) = 8i m^4$ in part (a) (verified directly against the full $6 \times 6$ symbolic determinant in test_theorem4). Part (b) follows by the same Laplace expansion along $(x_0, x_1)$ and $(x_2, v)$, where $\frac{\partial(\tilde{g}_1^{(2m)}, \tilde{h}_1^{(2m)})}{\partial(x_0, x_1)} = 2m\sum_{a=0}^{2m-2}\zeta_{4m}^{a+1} x_0^{2m-2-a} x_1^a$ and $\frac{\partial(v, \tilde{g}_3^{(2m)}, \tilde{h}_3^{(2m)})}{\partial(x_3, x_4, x_5)} = -2im^2 x_3^{m-1}(x_4^m + x_5^m)$.

(d) For $d = 9$, setting $u = x_3^3, v = x_4^3, w = x_5^3$ in the Gauss-Euler identity $u^3 + v^3 + w^3 - 3uvw = (u+v+w)(u^2+v^2+w^2-uv-vw-wu)$ gives $g_3 h_3 = x_3^9 + x_4^9 + x_5^9 - 3x_3^3 x_4^3 x_5^3$. Meanwhile, $g_2 h_2 = x_2^9 + 3x_3^3 x_4^3 x_5^3$ and $g_1 h_1 = x_0^9 + x_1^9$. The $3 \times 3$ Jacobian of $(x_3 x_4 x_5, g_3, h_3)$ with respect to $(x_3, x_4, x_5)$ equals $27(x_3^3 - x_4^3)(x_4^3 - x_5^3)(x_5^3 - x_3^3)$, which multiplies $\frac{\partial(g_2, h_2)}{\partial(x_2, x_3 x_4 x_5)} = -9\cdot 3^{1/3} x_2^5 + 9\cdot 3^{2/3} x_2^2 x_3 x_4 x_5$ and $\frac{\partial(g_1, h_1)}{\partial(x_0, x_1)} = 9\sum_{a=0}^7 \zeta_{18}^{a+1} x_0^{7-a} x_1^a$ ($(9)(9)(27) = 2187$) to yield the stated factorization (exp18_nonic_triadic_gauss_euler_cycles.py, verified against the full $6 \times 6$ symbolic determinant in test_theorem4). $\blacksquare$

Figure 2: (a) Exact first-order IVHS / Catalecticant ranks $\mathrm{rank}\,\Theta_1([Z]_{\mathrm{prim}})$ across all $10$ complete-intersection surface multidegrees $(d_1, d_2, d_3)$ on the sextic fourfold $X_6^4 \subset \mathbb{P}^5$ [20, 23], comparing plane-/quadric-/cubic-contained cycles (grey, $\mathrm{rank} \le 37$) with the join-Aoki–Shioda surface $Z_{\alpha_1}$ (degree $(1,2,3)$, rank $62$, red) and entangled double-Aoki–Shioda / "type 3" surface $Z_{\alpha_0}$ (degree $(2,2,3)$, rank $106$, orange). (b) Scaling of the Catalecticant rank of the degree-$(2,2,m)$ surface $Z^{(2m)} \subset X_{2m}^4$ ($106, 282, 562, 970$) and degree-$(1,2,m)$ surface $Z_{\mathrm{semi}}^{(2m)}$ ($62, 157, 306, 515$) against the minimum exceptional character rank $\min_{\alpha\mathrm{exc}}\mathrm{rank}\,\Theta_1(m_\alpha) \in \{40, 96, 142, 256\}$ and linear planes $\mathbb{P}^2$ for $d = 6, 8, 10, 12$.

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Theorem 5 (Sporadic Weighted Brieskorn-Pham Complete-Intersection Syzygy $C_{\mathrm{spor}} \subset X_{12}^2$ & Entangled Realization of Aoki's Order-2 Parity Group on $X_{12}^4$)

On the Fermat dodecic surface $X_{12}^2 \subset \mathbb{P}^3$ and fourfold $X_{12}^4 \subset \mathbb{P}^5$ ($d = 12, m = 6$), there are $6$ exceptional $S_4 \times (\mathbb{Z}/12\mathbb{Z})^\times$-orbits on $X_{12}^2$: $$S_1 = (1, 4, 9, 10), \quad S_2 = (1, 5, 9, 9), \quad S_3 = (1, 6, 7, 10), \quad S_4 = (1, 6, 8, 9), \quad S_5 = (1, 7, 8, 8), \quad S_6 = (2, 6, 8, 8),$$ and $39$ exceptional $S_6 \times (\mathbb{Z}/12\mathbb{Z})^\times$-orbits on $X_{12}^4$ ($32$ semi-inductive $(2+4)$ orbits and $7$ pair-free $(0+6)$ orbits $\mathrm{PF}_{01}, \dots, \mathrm{PF}_{07}$).

(a) Aoki's Order-2 Parity Obstruction to Binomial Complete Intersections at $d = 12$: In the formal character module modulo complementary pairs, the $2$-standard tuples $s_{2, 1} = S_3$, $s_{2, 2} = S_6$ and the $3$-standard tuple $s_{3, 1} = S_2$ generate an index-$2$ sublattice containing $S_5 \equiv s_{2, 1} + s_{2, 2}$, whereas $S_4 = (1, 6, 8, 9)$ and $S_1 = (1, 4, 9, 10)$ represent the nontrivial coset of Aoki's order-$2$ parity group $G_{12} \simeq \mathbb{Z}/2\mathbb{Z}$ [12, Theorem A; 19, §2.2]: $$S_4 \equiv \frac{1}{2}\big(s_{3, 1} + s_{2, 1} + s_{2, 2}\big), \qquad S_1 \equiv \frac{1}{2}\big(s_{3, 1} + s_{2, 1} - s_{2, 2}\big).$$ Consequently, the degree-$(2, 2, 6)$ surface $Z^{(12)} \subset X_{12}^4$ (Catalecticant rank $970$) directly realizes $3/7$ pair-free orbits—$\mathrm{PF}_{02} = (1, 1, 7, 7, 10, 10) = T_0 * T_0$, $\mathrm{PF}_{06} = (1, 4, 4, 7, 10, 10) = T_0 * T_3$, and $\mathrm{PF}_{07} = (2, 2, 8, 8, 8, 8) = T_1 * T_1$—and $3/6$ surface blocks ($S_3, S_5, S_6$).

(b) Closed-Form Weighted Brieskorn-Pham Degree-$(4, 6)$ Complete-Intersection Curve $C_{\mathrm{spor}} \subset X_{12}^2$ Realizing $S_4 = (1, 6, 8, 9)$: Let $a, b \in \overline{\mathbb{Q}}$ be non-zero algebraic numbers over $\mathbb{Q}(\sqrt{3})$ satisfying $$a^4 = \big(6 + 2\sqrt{3}\big)\, b, \qquad b^3 = -\big(10 + 6\sqrt{3}\big),$$ and define in the weighted variables $(x_0, y_2 = x_2^4, y_3 = x_3^3)$ of weights $(1, 4, 3)$ the homogeneous polynomials of degrees $4$, $8$, and $6$: $$g_1(x_0, x_2, x_3) := x_2^4 - a x_0 x_3^3 - b x_0^4, \qquad h_1(x_0, x_2, x_3) := x_2^8 + x_2^4(a x_0 x_3^3 + b x_0^4) + (a x_0 x_3^3 + b x_0^4)^2,$$ $$Q_6(x_0, x_3) := x_3^6 + \frac{a^3}{2} x_0^3 x_3^3 + \frac{3 b^2}{a^2} x_0^6.$$ Then $g_1 h_1 + Q_6^2 = x_0^{12} + x_2^{12} + x_3^{12}$ identically in $\mathbb{Q}(\sqrt{3})[a, b][x_0, x_2, x_3]$! Setting $g_2 := x_1^6 - i Q_6(x_0, x_3)$ and $h_2 := x_1^6 + i Q_6(x_0, x_3)$ yields the exact Fermat syzygy $g_1 h_1 + g_2 h_2 = \sum_{i=0}^3 x_i^{12}$, defining a degree-$(4, 6)$ complete-intersection curve $C_{\mathrm{spor}} = \{g_1 = g_2 = 0\} \subset X_{12}^2$. Its $3 \times 3$ block Jacobian $J_3(x_0, x_2, x_3) := \det\frac{\partial(g_1, h_1, Q_6)}{\partial(x_0, x_2, x_3)}$ reduces modulo $x_i^{11}$ to $6$ monomials: $$J_3(x_0, x_2, x_3) \equiv -72a\, x_0^0 x_2^7 x_3^8 + 36a(a^4-10b)\, x_0^4 x_2^3 x_3^8 - \frac{72b(a^4-9b)}{a}\, x_0^6 x_2^7 x_3^2 - \frac{72b^2(a^4-9b)}{a}\, x_0^{10} x_2^3 x_3^2 + 36(a^4-8b)\, x_0^3 x_2^7 x_3^5 - 36b(a^4-10b)\, x_0^7 x_2^3 x_3^5 \pmod{x_i^{11}},$$ and the $4 \times 4$ Villaflor Jacobian $P_{C_{\mathrm{spor}}}(x) = -12i x_1^5 J_3(x_0, x_2, x_3) \in R(F_{12})_{20}$ has non-inductive terms supported purely on the $4$ Galois conjugates of Aoki's sporadic character $S_4 = (1, 6, 8, 9)$: $(1, 6, 8, 9)$, $(5, 6, 4, 9)$, $(7, 6, 8, 3)$, and $(11, 6, 4, 3)$ (with non-zero coefficient $864i a \ne 0$ on $x_0^0 x_1^5 x_2^7 x_3^8$).

(c) Entangled Sporadic Complete-Intersection Surfaces $Z_{\mathrm{spor, std}}^{(12)}$ and $Z_{\mathrm{spor, spor}}^{(12)}$ on $X_{12}^4$: Entangling the sporadic block $(g_1, h_1, Q_6)$ on $(x_0, x_2, x_3)$ (which produces the odd-length 3-tuple $U_0 = S_4 - (6) = (1, 8, 9)$ and its $(\mathbb{Z}/12\mathbb{Z})^\times$-conjugates) with the standard degree-$2$ block $(g_2^{(12)}, h_2^{(12)}, x_4^6 - x_5^6)$ or with a second sporadic block $(\tilde{g}_1, \tilde{h}_1, \widetilde{Q}_6)$ on $(x_1, x_4, x_5)$ via $g_3 = Q_6 - i \widetilde{Q}_6, h_3 = Q_6 + i \widetilde{Q}_6$ defines two complete-intersection surfaces on $X_{12}^4$:

• $Z_{\mathrm{spor, std}}^{(12)} \subset X_{12}^4$ of multidegree $(4, 2, 6)$ ($\deg = 48$, Catalecticant rank $[t^{12}]\frac{(1-t^4)(1-t^8)(1-t^2)(1-t^{10})(1-t^6)^2}{(1-t)^6} = 1605$ by Proposition 1), whose $6 \times 6$ Jacobian $P_{Z_{\mathrm{spor, std}}^{(12)}} = 2i\, J_3(x_0, x_2, x_3)\,\frac{\partial(g_2^{(12)}, h_2^{(12)})}{\partial(x_1, v)}\mathcal{D}_{45}(x_4^6 - x_5^6)$ directly realizes the pair-free orbits $\mathrm{PF}_{03} = (1, 1, 7, 8, 9, 10) = U_0 * T_0$ and $\mathrm{PF}_{05} = (1, 2, 8, 8, 8, 9) = U_0 * T_1$, and via the off-diagonal product $U_0 * T_3 = (1, 8, 9) * (4, 4, 10) = (4, 8) \oplus (1, 4, 9, 10)$ realizes $S_1 = (1, 4, 9, 10)$ up to the linear pair $(4, 8)$!
• $Z_{\mathrm{spor, spor}}^{(12)} \subset X_{12}^4$ of multidegree $(4, 4, 6)$ ($\deg = 96$, Catalecticant rank $[t^{12}]\frac{(1-t^4)^2(1-t^8)^2(1-t^6)^2}{(1-t)^6} = 2653$ by Proposition 1), whose $6 \times 6$ Jacobian $P_{Z_{\mathrm{spor, spor}}^{(12)}} = -2i\, J_3(x_0, x_2, x_3)\, J_3(x_1, x_4, x_5)$ directly realizes the pair-free orbit $\mathrm{PF}_{04} = (1, 1, 8, 8, 9, 9) = U_0 * U_0$ and via the off-diagonal product $U_0 * (5 U_0) = (1, 8, 9) * (5, 4, 9) = (4, 8) \oplus (1, 5, 9, 9)$ realizes $S_2 = (1, 5, 9, 9)$ up to $(4, 8)$! Together with $Z^{(12)}$ of multidegree $(2, 2, 6)$ (rank $970$), these three complete-intersection surfaces directly realize $6/7$ pair-free fourfold orbits and (via linear deletion) all $6/6$ exceptional surface orbits $S_1, \dots, S_6$ on $X_{12}^2$, while the $7$-th pair-free orbit $\mathrm{PF}_{01} = (1, 1, 6, 9, 9, 10) = (1, 9, 10) * (1, 6, 9) = (S_1 - (4)) * (S_4 - (8))$ is the asymmetric $(4, 8)$-join of $S_1$ and $S_4$, completing $100\%$ ($39/39$) of the exceptional orbits on $X_{12}^4$ (exp22_dodecic_fourfold_entangled_test.py).

Proof

(a) Mapping each $a \in \{1, \dots, 11\}$ to the standard basis $e_a \in \mathbb{Z}^5$ modulo $e_a + e_{12-a} = 0$ ($e_6 = 0$, $e_{12-a} = -e_a$ for $1 \le a \le 5$) sends $S_1 \mapsto (1,-1,-1,1,0)$, $S_2 = s_{3,1} \mapsto (1,0,-2,0,1)$, $S_3 = s_{2,1} \mapsto (1,-1,0,0,-1)$, $S_4 \mapsto (1,0,-1,-1,0)$, $S_5 \mapsto (1,0,0,-2,-1) = S_3 + S_6$, and $S_6 = s_{2,2} \mapsto (0,1,0,-2,0)$. Direct addition gives $S_2 + S_3 + S_6 = (2, 0, -2, -2, 0) = 2 S_4$ and $S_2 + S_3 - S_6 = (2, -2, -2, 2, 0) = 2 S_1$; since the $3$rd and $4$th coordinates of $S_2, S_3, S_6$ are all even, $S_4$ and $S_1$ cannot lie in their integral span.

(b)–(c) Set $y_2 = x_2^4$, $y_3 = x_3^3$, and $A = a y_3 x_0 + b x_0^4$. By the difference-of-two-cubes identity, $g_1 h_1 = (y_2 - A)(y_2^2 + y_2 A + A^2) = y_2^3 - A^3 = x_2^{12} - (a y_3 x_0 + b x_0^4)^3$. Therefore, $$x_0^{12} + x_2^{12} + x_3^{12} - g_1 h_1 = y_3^4 + a^3 x_0^3 y_3^3 + 3 a^2 b x_0^6 y_3^2 + 3 a b^2 x_0^9 y_3 + (b^3 + 1) x_0^{12}.$$ Meanwhile, squaring $Q_6(x_0, x_3) = y_3^2 + \frac{a^3}{2} x_0^3 y_3 + c x_0^6$ gives $$Q_6^2 = y_3^4 + a^3 x_0^3 y_3^3 + \left(2c + \frac{a^6}{4}\right) x_0^6 y_3^2 + a^3 c x_0^9 y_3 + c^2 x_0^{12}.$$ Setting $c = \frac{3b^2}{a^2}$ matches the $y_3^4, y_3^3, y_3^1$ coefficients identically, while the $y_3^2$ and $y_3^0$ coefficients match if and only if $\frac{6b^2}{a^2} + \frac{a^6}{4} = 3a^2 b \iff (a^4 - 6b)^2 = 12b^2 \iff a^4 = (6 + 2\sqrt{3})b$ and $\frac{9b^4}{a^4} = b^3 + 1 \iff b^3 = \frac{6+2\sqrt{3}}{3-2\sqrt{3}} = -(10 + 6\sqrt{3})$. Since $g_1, h_1$ are independent of $x_1$ and $\frac{\partial(g_2, h_2)}{\partial(x_1, Q_6)} = -12i x_1^5$, Laplace expansion along $x_1$ gives $P_{C_{\mathrm{spor}}}(x) = -12i x_1^5 J_3(x_0, x_2, x_3)$, and expanding $J_3 = \det\frac{\partial(g_1, h_1, Q_6)}{\partial(x_0, x_2, x_3)}$ yields the $6$ stated monomials modulo $x_i^{11}$. On $X_{12}^4$, Laplace expansion of the $6 \times 6$ determinant along the disjoint variable blocks $(x_0, x_2, x_3)$ and $(x_1, x_4, x_5)$ factors $P_{Z_{\mathrm{spor, std}}^{(12)}}$ and $P_{Z_{\mathrm{spor, spor}}^{(12)}}$ into the products of the $3 \times 3$ block Jacobians, which directly yield the stated 6-tuples (exp22_dodecic_fourfold_entangled_test.py, verified in test_theorem5). $\blacksquare$

| Degree $d$ | Fourfold $X_d^4 \subset \mathbb{P}^5$ | Surface $Z$ Multidegree & Degree | Plane Rank $\frac{d^2+3d-16}{2}$ | Exact Min Exceptional Rank $\min_{\alpha\mathrm{exc}}\mathrm{rank}\,\Theta_1(m_\alpha)$ | Catalecticant Rank $\mathrm{rank}\,\Theta_1([Z]_{\mathrm{prim}})$ [20] | Exceptional Hodge Orbits Reached | | :---: | :---: | :---: | :---: | :---: | :---: | :--- | | $d = 6$ ($m=3$) | $X_6^4$ | $(2,2,3)$, $\deg = 12$ ($Z^{(6)}$ [23]) | $19$ | $40$ ($44$ pair-free) | $106$ | $2/2$ pair-free: $(1, 1, 4, 4, 4, 4)$ & $(2, 2, 2, 2, 5, 5)$ | | $d = 6$ ($m=3$) | $X_6^4$ | $(1,2,3)$, $\deg = 6$ ($Z_{\mathrm{semi}}^{(6)}$ [23]) | $19$ | $40$ ($44$ pair-free) | $62$ | $6$ semi-inductive (Combined: $8/8$ ($100\%$) on $X_6^4$) | | $d = 8$ ($m=4$) | $X_8^4$ | $(2,2,4)$, $\deg = 16$ ($Z^{(8)}$) | $36$ | $96$ | $282$ | $2/2$ pair-free: $(1, 1, 5, 5, 6, 6)$ & $(2, 2, 3, 3, 7, 7)$ (+2 semi) | | $d = 8$ ($m=4$) | $X_8^4$ | $(1,2,4)$, $\deg = 8$ ($Z_{\mathrm{semi}}^{(8)}$) | $36$ | $96$ | $157$ | $8$ semi-inductive (Combined: $10/10$ ($100\%$) on $X_8^4$) | | $d = 9$ (odd) | $X_9^4$ | $(1,3,3)$, $\deg = 9$ ($Z_{\mathrm{nonic}}$) | $46$ | $130$ | $259$ | $8/8$ ($100\%$) exceptional classes on $X_9^4$ ($96$ pure terms) | | $d = 10$ ($m=5$) | $X_{10}^4$ | $(2,2,5)$, $\deg = 20$ ($Z^{(10)}$) | $57$ | $142$ | $562$ | $4/4$ ($100\%$) pair-free orbits + all $4$ non-standard surface blocks | | $d = 10$ ($m=5$) | $X_{10}^4$ | $(1,2,5)$, $\deg = 10$ ($Z_{\mathrm{semi}}^{(10)}$) | $57$ | $142$ | $306$ | $20$ standard join orbits ($28/44$ direct; $44/44$ with deletion/join) | | $d = 12$ ($m=6$) | $X_{12}^4$ | $(2,2,6), (4,2,6), (4,4,6)$ ($Z^{(12)}, Z_{\mathrm{spor}}^{(12)}$) | $82$ | $256$ | $970,\, 1605,\, 2653$ | $6/7$ direct pair-free + $6/6$ surface blocks ($39/39$ with join) |

Table 2: Symbolic verification of the explicit complete-intersection surface families $Z^{(2m)}, Z_{\mathrm{semi}}^{(2m)} \subset X_{2m}^4$ ($d = 6, 8, 10, 12$), $Z_{\mathrm{nonic}} \subset X_9^4$ ($d = 9$), and the sporadic Brieskorn-Pham entangled surfaces $Z_{\mathrm{spor, std}}^{(12)}, Z_{\mathrm{spor, spor}}^{(12)} \subset X_{12}^4$ (exp17_sextic_entangled_and_join_cycles.py, exp18_nonic_triadic_gauss_euler_cycles.py, exp20_octic_and_general_exceptional_spanning.py, and exp22_dodecic_fourfold_entangled_test.py).

Theorem 6 (All-Dimensional Mod-3 Block Decomposition Law & Complete-Intersection Realization of $\mathrm{Hdg}^p(X_6^{2p}, \mathbb{Q})$)

Let $X_6^{2p} = \{\sum_{i=0}^{2p+1} x_i^6 = 0\} \subset \mathbb{P}^{2p+1}$ ($p \ge 1$) be the $2p$-dimensional Fermat sextic hypersurface. After greedily removing the maximum number $m \in \{0, 1, \dots, p+1\}$ of disjoint complementary binary pairs $a_i + a_j = 6$, the remaining $2(p+1-m)$ entries of any rational Hodge character $\alpha \in \mathrm{Hdg}^p_{\mathrm{prim}}(X_6^{2p}, \mathbb{Q})$ contain no pair summing to $6$. Such a pair-free remainder of length $2r_{\mathrm{rem}}$ ($r_{\mathrm{rem}} = p+1-m \ge 1$) exists if and only if $r_{\mathrm{rem}} \not\equiv 1 \pmod 3$, in which case it is uniquely (up to complex conjugation $a_i \mapsto 6-a_i$ under $(\mathbb{Z}/6\mathbb{Z})^\times = \{1, 5\}$) the direct sum of $\lfloor r_{\mathrm{rem}}/3 \rfloor$ copies of the 6-variable fourfold block $\alpha_0 = (1,1,4,4,4,4)$ and $(r_{\mathrm{rem}} \bmod 3)/2 \in \{0, 1\}$ copy of the 4-variable surface block $\gamma_0 = (1,3,4,4)$. In particular:

• When $p \equiv 0 \pmod 3$ (including the Fano Fermat sextic sixfold $X_6^6 \subset \mathbb{P}^7$, $p = 3$, where $K_{X_6^6} = \mathcal{O}_{X_6^6}(-2)$ and a linear $3$-plane $\mathbb{P}^3 \subset X_6^6$ has Catalecticant rank $\binom{9}{3} - 4^2 = 68$ by Proposition 1 [20]), $p+1 \equiv 1 \pmod 3$, so zero purely $(2p+2)$-variable pair-free Hodge characters exist ($m \ge 1$ for every class).
• For every $p \ge 1$, the exact number of sorted $S_{2p+2}$-permutation classes of rational Hodge characters on $X_6^{2p}$ is $$N_{\mathrm{perm}}(X_6^{2p}) = \binom{p+3}{2} + 2 \sum_{\substack{0 \le m \le p \\ p+1-m \not\equiv 1 \pmod 3}} \binom{m+2}{2} \qquad (8, 18, 33, 55, 86 \text{ for } p = 1, 2, 3, 4, 5),$$ while identifying complex-conjugate pairs under $(\mathbb{Z}/6\mathbb{Z})^\times = \{1, 5\}$ (which fixes the $\binom{p+3}{2}$ linear classes $m = p+1$ and pairs the non-linear classes) yields $N_{\mathrm{Galois}}(X_6^{2p}) = \binom{p+3}{2} + \sum_{p+1-m \not\equiv 1 \pmod 3}\binom{m+2}{2} = (7, 14, 24, 38, 57)$ $S_{2p+2} \times (\mathbb{Z}/6\mathbb{Z})^\times$-orbits, and $100\%$ of $\mathrm{Hdg}^p(X_6^{2p}, \mathbb{Q})$ is explicitly realized by complete-intersection $p$-cycles built as joins of the 2-variable linear factor $A_{1,6}$, 4-variable degree-$(2,3)$ surface block $A_{2,6}$, and 6-variable degree-$(2,2,3)$ fourfold block $Z_{\alpha_0} = Z^{(6)}$!

Proof

Let $\alpha' \in \{1, 2, 3, 4, 5\}^{2r_{\mathrm{rem}}}$ be a pair-free Hodge character of sum $6 r_{\mathrm{rem}}$. Since $a_i + a_j \ne 6$, $\alpha'$ contains at most one $3$ ($k \in \{0, 1\}$ occurrences of $3$), cannot contain both $1$ and $5$, and cannot contain both $2$ and $4$. Since the average entry is $3$, up to conjugation $a_i \mapsto 6 - a_i$, all $2r_{\mathrm{rem}} - k$ non-$3$ entries of $\alpha'$ must lie in $\{1, 4\}$ (since $1 \equiv 4 \equiv 1 \pmod 3$ and $2 \equiv 5 \equiv 2 \pmod 3$ cannot mix without creating $1+5=6$ or $2+4=6$, nor can $(1,2)$ or $(4,5)$ average to $3$). Let $u$ be the number of $1$'s and $v$ the number of $4$'s. Then $u + v = 2r_{\mathrm{rem}} - k$ and $u + 4v + 3k = 6r_{\mathrm{rem}}$, which subtracts to give $$3v = 4r_{\mathrm{rem}} - 2k = 2(2r_{\mathrm{rem}} - k), \qquad 3u = 2r_{\mathrm{rem}} - k.$$ Since $\gcd(2, 3) = 1$ and $k \in \{0, 1\}$, an integer solution $(u, v)$ exists if and only if $2r_{\mathrm{rem}} \equiv k \in \{0, 1\} \pmod 3$, i.e., $r_{\mathrm{rem}} \equiv 0 \text{ or } 2 \pmod 3$ ($r_{\mathrm{rem}} \not\equiv 1 \pmod 3$). When $r_{\mathrm{rem}} = 3q$ ($k=0$), $(u,v) = (2q, 4q) = q \cdot \alpha_0$; when $r_{\mathrm{rem}} = 3q+2$ ($k=1$), $(u,v,k) = (2q+1, 4q+2, 1) = q \cdot \alpha_0 \oplus \gamma_0$. Since $m$ binary pairs from $\{(1,5), (2,4), (3,3)\}$ can be chosen in $\binom{m+2}{2}$ ways (and each self-conjugate binary tuple can be paired with either $\alpha'$ or $\bar{\alpha}'$ before quotienting by $(\mathbb{Z}/6\mathbb{Z})^\times$), and each block decomposition uses $1 \cdot m + 2 \cdot k + 3 \cdot q = m + r_{\mathrm{rem}} = p+1$ equations, the block equations define a complete intersection of codimension $p$ in $X_6^{2p}$ whose $(2p+2) \times (2p+2)$ Jacobian is the product of the block Jacobians (exp19_calabi_yau_sixfold_and_higher_codim.py). $\blacksquare$

| Dimension $2p$ ($X_6^{2p} \subset \mathbb{P}^{2p+1}$) | Linear $\mathbb{P}^p$ Classes ($m=p+1$) | Semi-Inductive Classes ($1 \le m \le p$) | Pair-Free Classes ($m=0$) | Total $S_{2p+2}$-Classes $N_{\mathrm{perm}}$ ($S_{2p+2}\times(\mathbb{Z}/6\mathbb{Z})^\times$-Orbits $N_{\mathrm{Galois}}$) | Linear $\mathbb{P}^p$ Rank $\binom{p+6}{6}-(p+1)^2$ [20] | Complete-Intersection Generator Multidegrees Spanning $\mathrm{Hdg}^p(X_6^{2p}, \mathbb{Q})$ | | :---: | :---: | :---: | :---: | :---: | :---: | :--- | | $2p = 2$ ($p=1$, Surface $X_6^2$) | $6$ | $0$ | $2$ ($\gamma_0, \bar{\gamma}_0$) | $8$ ($7$) | $3$ | $(1,1)$ & $(2,3)$ | | $2p = 4$ ($p=2$, CY Fourfold $X_6^4$) | $10$ | $6$ | $2$ ($\alpha_0, \bar{\alpha}_0$) | $18$ ($14$) | $19$ | $(1,1,1)$, $(1,2,3)$, & $(2,2,3)$ [23] | | $2p = 6$ ($p=3$, Fano Sixfold $X_6^6$) | $15$ | $18$ ($12+6$) | $0$ (Mod-3 obstructed) | $33$ ($24$) | $68$ | $(1,1,1,1)$, $(1,1,2,3)$, & $(1,2,2,3)$ | | $2p = 8$ ($p=4$, Eightfold $X_6^8$) | $21$ | $32$ ($20+12$) | $2$ ($\alpha_0 \oplus \gamma_0$) | $55$ ($38$) | $185$ | $(1^5)$, $(1^3,2,3)$, $(1^2,2^2,3)$, & $(2^3,3^2)$ | | $2p = 10$ ($p=5$, Tenfold $X_6^{10}$) | $28$ | $56$ ($30+20+6$) | $2$ ($\alpha_0 \oplus \alpha_0$) | $86$ ($57$) | $425$ | $(1^6)$, $(1^4,2,3)$, $(1^3,2^2,3)$, $(1,2^3,3^2)$, & $(2^4,3^2)$ |

Table 3: Verification of the All-Dimensional Mod-3 Block Decomposition Law, $S_{2p+2}$-permutation classes $N_{\mathrm{perm}}(X_6^{2p})$ ($S_{2p+2} \times (\mathbb{Z}/6\mathbb{Z})^\times$-orbits $N_{\mathrm{Galois}}(X_6^{2p})$ in parentheses), and complete-intersection multidegrees across dimensions $2p = 2, 4, 6, 8, 10$ (exp19_calabi_yau_sixfold_and_higher_codim.py).

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7. Discussion, Limitations, and Reproducibility

7.1 Scope, Limitations, and Structural Synthesis

Theorems 1–6 establish concrete quantitative and algebraic bridges across our two target geometries:

• On Weil abelian $2m$-folds, no single rank-$1$ Plücker operator can isolate a Weil class (Theorem 1), and on a simple Weil abelian $2m$-fold, total-reality rigidity of $\mathrm{U}(m) \subset \mathrm{Gr}(m, 2m)$ forces every curved Gauss map $\gamma_Z: Z^{\mathrm{sm}} \to \mathrm{Gr}(m, 2m)$ to obey a strict Kähler-to-Weil calibration deficit $\Delta_{\mathrm{curv}}(Z) > 0$ via the Plücker Fubini-Study Levi-form identity and the exact wedge-reordering phase $(-1)^{\binom{m}{2}}(-i)^m$ (Theorem 3(a)–(c)). On the split CM model $E_i^{2m}$, averaging the sign-adjusted unitary graph subtori $\Gamma_{-g} = \{w + gz = 0\}$ over the finite special monomial design $G_m \subset \mathrm{SU}(m) \cap \mathrm{SL}_m(\mathbb{Z}[i])$ of order $4^{m-1} m!$ plus positive coordinate subtori $C_{I_+, I_-}$ produces an explicit effective lattice-rational algebraic cycle $\mathcal{Z}_{\mathrm{eff}}^{(m)} \ge 0$ for all $m \ge 2$ whose conormal Poincaré dual cohomology class is $\frac{1}{m!}[\omega_Q^m] + 2[\alpha_{W,1}]$ and whose Hodge locus in $\mathfrak{H}_{2m}(Q)$ is the entire $m^2$-dimensional Weil PEL Shimura variety $\mathfrak{H}_{\mathrm{Weil}}(\phi)$ (Theorem 2). Across all fibers of $\mathfrak{H}_{\mathrm{Weil}}(\phi)$ (including generic simple non-CM fibers), our $(m+1)$-point rational Chebyshev-Weil correspondence projector $\Pi_{\mathrm{Cheb}}^{(m)} = \sum_{j=0}^m c_j T_{j, 1} \in \mathbb{Q}[\mathrm{End}_{\mathbb{Q}}(A_\tau)] \subset \mathrm{Corr}^0(A_\tau, A_\tau)_{\mathbb{Q}}$ is an idempotent projector on middle cohomology $H^{2m}(A_\tau, \mathbb{Q})$ that algebraically annihilates all $\binom{4m}{2m} - 2$ non-Weil cohomology classes and acts as the identity on $W_{\mathrm{Weil}}(A_\tau)$, while composing with Lieberman's algebraic Künneth projector $\pi_{2m} = \prod_{q \ne 2m} \frac{[2]^* - 2^q\,\mathrm{id}}{2^{2m} - 2^q}$ yields a full-cohomology idempotent $\widehat{\Pi}_{\mathrm{Cheb}}^{(m)} = \pi_{2m} \circ \Pi_{\mathrm{Cheb}}^{(m)}$ on $H^*(A_\tau, \mathbb{Q})$ (Theorem 3(d)). Thus, on a generic simple fiber $A_\tau$ (beyond the fourfold and discriminant-$(-1)$ sixfold cases settled by Markman [6]), the Hodge Conjecture for the Weil subspace $W_{\mathrm{Weil}}(A_\tau)$ reduces to the existence of any seed $m$-cycle $Z \subset A_\tau$ with a non-vanishing $(2m, 0)$ $\phi$-eigenperiod $\int_Z \Omega_{W_+} \ne 0$.
• On Fermat fourfolds $X_d^4 \subset \mathbb{P}^5$ ($d \ge 6$), dodecic surfaces $X_{12}^2 \subset \mathbb{P}^3$, and higher-dimensional Fermat sextics $X_6^{2p} \subset \mathbb{P}^{2p+1}$ ($p \ge 1$), building on Villaflor's residue formula [9, Theorem 1.1], the Gorenstein Catalecticant rank theorem of Dan [21], Villaflor Loyola [22], and Kloosterman [20], and the sextic fourfold realizations of Braun, Fortin, Lopez Garcia, and Villaflor Loyola [23] (Proposition 1), our general even-degree ($d = 2m \ge 6$) entangled double-Aoki–Shioda family $Z^{(2m)} \subset X_{2m}^4$ of multidegree $(2,2,m)$ eliminates the $(m,m)$ linear pair at the polynomial equation level, and Laplace expansion gives its exact factored $6 \times 6$ Jacobian $P_{Z^{(2m)}}(x)$ as well as those of $Z_{\mathrm{semi}}^{(2m)}$ and $Z_{\mathrm{nonic}}$ (Theorem 4). At $d = 12$, where Aoki's order-$2$ parity obstruction group $G_{12} \simeq \mathbb{Z}/2\mathbb{Z}$ [12, Theorem A] prevents binomial complete intersections $A_{p, 12}$ and $Z^{(12)}$ from reaching the sporadic half-integral character $S_4 = (1, 6, 8, 9)$, our weighted $(1, 6, 4, 3)$ Brieskorn-Pham complete-intersection curve $C_{\mathrm{spor}} \subset X_{12}^2$ of multidegree $(4, 6)$ and entangled surfaces $Z_{\mathrm{spor, std}}^{(12)}, Z_{\mathrm{spor, spor}}^{(12)} \subset X_{12}^4$ (of Catalecticant ranks $1605$ and $2653$) realize $S_4$ directly by a single complete intersection and span all $6/6$ exceptional surface orbits on $X_{12}^2$ and all $39/39$ exceptional fourfold orbits on $X_{12}^4$ (Theorem 5). Finally, on $X_6^{2p}$, Theorem 6 establishes the Mod-3 Block Decomposition Law across all $p \ge 1$.

7.2 Reproducibility & Symbolic Verification Suite

All proofs, exterior-algebra forms, full $6 \times 6$ symbolic Jacobian determinants, and tables in this manuscript are verified by exact symbolic computation in SymPy within the package hodge_loci_engine included in the attached supplement.zip:

• pytest -v (or python3 -m unittest tests/test_hodge_loci_engine.py) executes the self-contained 6-test verification suite in tests/test_hodge_loci_engine.py, verifying Theorems 1–6 one-to-one, including: (i) the conormal Poincaré dual sign $(-1)^m \det g$ on $G_m$ vs. $+1$ on $\Gamma_{-g}$ for $m=2, 3$ (test_theorem2), (ii) the exact exterior wedge-reordering identity $\psi_-|_{w=Bz} = (-1)^{\binom{m}{2}}(2i)^m (\det B) v_0$ and calibration deficit $\Delta_{\mathrm{curv}}(\Gamma_{I_2}) = 0$ vs. $\Delta_{\mathrm{curv}}(\Gamma_{\mathrm{diag}(-1,1)}) = 8$, together with the $70 \times 70$ rational exterior-algebra matrix of $\Pi_{\mathrm{Cheb}}^{(2)}$ on $H^4(A_\tau, \mathbb{Q}) = \bigwedge^4 \mathbb{Q}^8$ and Lieberman's Künneth projector $\pi_{2m}$ (test_theorem3), (iii) direct symbolic expansion of the full $6 \times 6$ Jacobian determinants $\det\frac{\partial(g_1, g_2, g_3, h_1, h_2, h_3)}{\partial(x_0, \dots, x_5)}$ for $Z^{(2m)}$ ($m=3, 4, 5$) and $Z_{\mathrm{nonic}}$ ($d=9$) (test_theorem4), (iv) the exact weighted Brieskorn-Pham sporadic syzygy $C_{\mathrm{spor}} \subset X_{12}^2$, Catalecticant ranks $(970, 1605, 2653)$, and full $39/39$ orbit verification on $X_{12}^4$ (test_theorem5), and (v) both $N_{\mathrm{perm}}(X_6^{2p}) = (8, 18, 33, 55, 86)$ and $N_{\mathrm{Galois}}(X_6^{2p}) = (7, 14, 24, 38, 57)$ (test_theorem6).
• python3 experiments/exp12_principal_angle_weil_calibration.py, exp14_universal_monomial_sum_design.py, exp15_curved_gauss_map_rigidity.py, exp17_sextic_entangled_and_join_cycles.py, exp18_nonic_triadic_gauss_euler_cycles.py, exp19_calabi_yau_sixfold_and_higher_codim.py, exp20_octic_and_general_exceptional_spanning.py, exp21_chebyshev_weil_endomorphism_projector.py, and exp22_dodecic_fourfold_entangled_test.py reproduce all symbolic identities and Tables 1–3.
• python3 experiments/generate_paper2_figures.py regenerates figure-1.png and figure-2.png.

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