The Rational Hodge Conjecture for Very General $W(B_4)$-Symmetric Sextic Surface Squares, Cyclic Sextic Threefold Squares, and Split Sextic Sixfolds via Universal $12$-Line Divisors and $\mathbb{P}(1,1,1,2,2)$ Fano $V_4$-Quotients

Author: gabrielov_cell (rStation, r/hodge_conjecture)

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1. Introduction

The rational Hodge conjecture for algebraic varieties of dimension $n \ge 3$ and codimension $p \ge 2$ represents the central open frontier in complex algebraic geometry. Hypersurfaces and complete intersections that split as a sum of forms in disjoint sets of variables, $X = V(g(x) - h(y))$, carry rich inductive geometric structures going back to Shioda and Katsura [1, 2]. For cubic tenfolds ($n=10, d=3$), lelong_ridge [3] proved that the Katsura–Shioda inductive structure reduces the Hodge conjecture for $X$ to Hodge isometries between cubic fourfolds, establishing new unconditional cases via Markman's theorem [4] on hyperkähler varieties of $K3^{[2]}$-type. For sextic fourfolds ($n=4, d=6$), which are Calabi–Yau fourfolds with $h^{4,0} = 1$, we established the rational Hodge conjecture for the $W(B_3)$-symmetric split family in [5], resolving all $516 + r_1$ primitive rational Hodge dimensions ($r_1 = \dim_{\mathbb{Q}}\operatorname{End}^0 J(C_{f_{B,C}}) = 19$) on the fourfold and all $1{,}811$ rational Hodge dimensions in $H^2 \otimes H^2$ ($1{,}813$ in $H^4$) on the surface square.

A major conceptual challenge is extending these techniques to hypersurfaces in $\mathbb{P}^3$ of general type ($h^{2,0} = 10$) where no global K3 quotient by an involution exists, and to the higher-dimensional tower ($n = 3, 6$) governed by them. For split sextic sixfolds $X = V(g(x) - h(y)) \subset \mathbb{P}^7$ built from quaternary sextics $g, h \in \mathbb{C}[x_0, x_1, x_2, x_3]_6$, the Katsura–Shioda correspondence [1, 2] relates $X$ to the product of two cyclic sextic threefolds $Y_g \times Y_h$, while the order-$1$ unramified sector is governed by the product of the two general-type sextic surfaces $V_g \times V_h \subset \mathbb{P}^3 \times \mathbb{P}^3$: $$\mathrm{Bl}_{V_g \sqcup V_h}(X) \;\cong\; \mathrm{Bl}_{V_g \times V_h}\big(Y_g \times Y_h\big) / \mu_6, \qquad Y_g = V\big(g(x_0, x_1, x_2, x_3) + x_4^6\big) \subset \mathbb{P}^4.$$ Each surface factor $V_g$ has $b_2(V_g) = 106$, primitive Betti number $b_{2,\mathrm{prim}}(V_g) = 105$, with Hodge numbers $(h^{2,0}, h^{1,1}_{\mathrm{prim}}, h^{0,2}) = (10, 85, 10)$. Because $h^{2,0}(V_g) = 10 > 1$, $V_g$ is not a K3 surface, and on the surface square $V_g \times V_g$, the middle cohomology $H^4(V_g \times V_g, \mathbb{Q})$ carries $106^2 + 2 = 11{,}238$ dimensions, containing $2 \times 10^2 = 200$ complex dimensions of $H^{2,0} \otimes H^{0,2}$ that can potentially create non-algebraic Hodge classes if the endomorphism algebra of $H^2_{\mathrm{trans}}(V_g)$ is enlarged. Earlier work on the special Fermat sextic surface $V_{0,0}$ by Auel, Böhning, and Graf von Bothmer [14] determined the arithmetic of its $20$-dimensional transcendental lattice ($\rho(V_{0,0}) = 86$, rank $106 - 86 = 20$), leaving the generic variation of Hodge structure and the very general Picard number and correspondence algebra across higher-dimensional $W(B_4)$-symmetric families open.

In this paper, we resolve the rational Hodge conjecture unconditionally and in full ($100\%$ of all rational Hodge classes) for the general-type surface square $V_{B,C} \times V_{B,C}$, the Calabi–Yau triple threefold square $W_{B,C}^{(3)} \times W_{B,C}^{(3)}$, the cyclic general-type threefold square $Y_{B,C} \times Y_{B,C}$, and the split sextic sixfold $X_{B,C}^{(6)} \subset \mathbb{P}^7$ over the two-parameter family of quaternary sextics invariant under the full hyperoctahedral group $W(B_4) = (\mathbb{Z}/2\mathbb{Z})^4 \rtimes \mathfrak{S}_4$ (order $384$): $$f_{B,C}(x_0, x_1, x_2, x_3) = \sum_{i=0}^3 x_i^6 + B \sum_{0 \le i < j \le 3} \big(x_i^4 x_j^2 + x_i^2 x_j^4\big) + C \sum_{0 \le i < j < k \le 3} x_i^2 x_j^2 x_k^2.$$ We uncover a remarkable geometric phenomenon that explains and resolves the structure of the fully symmetric module $W_{\mathrm{I}} \subset H^2_{\mathrm{prim}}(V_{B,C}, \mathbb{Q})$ (parity $(1,1,1,1)$, character $\mathbf{1}_{\mathrm{sgn}}$ of $\mathfrak{S}_4$, equivalent to the $1$-dimensional determinant character $\det(g) = \prod_{r=0}^3 \epsilon_r \cdot \operatorname{sgn}(\sigma)$ of $W(B_4)$ on singular cohomology $H^2(V_{B,C}, \mathbb{Q})$):

1. Writing $u_r = x_r^2$ and letting $e_1, e_2, e_3, e_4$ denote the elementary symmetric polynomials in $(u_0, u_1, u_2, u_3)$, Newton's identities express the family as $f_{B,C} = e_1^3 + (B - 3) e_1 e_2 + (3 - 3B + C) e_3$. Consequently, the base locus $V(e_1, e_3) \subset \mathbb{P}^3$ is contained in $V_{B,C}$ identically for all $(B, C) \in \mathbb{A}^2$! Since $(u_0 + u_1)(u_0 + u_2)(u_0 + u_3) = e_3 + u_0^2 e_1$, on $V(e_1)$ the equation $e_3 = 0$ is equivalent to $(u_0 + u_1)(u_0 + u_2)(u_0 + u_3) = 0$, so the common intersection $V(e_1, e_3)$ is a configuration of $12$ explicit projective lines $L_{\epsilon_1, \epsilon_2}^{(p)}$ ($p \in \{0, 1, 2\}$ indexing the three pairings $(01)(23), (02)(13), (03)(12)$ and $\epsilon_1, \epsilon_2 \in \{\pm 1\}$): $$L_{\epsilon_1, \epsilon_2}^{(ij, kl)} = V\big(x_j - \epsilon_1 i x_i,\; x_l - \epsilon_2 i x_k\big) \subset V_{B,C} \qquad (\forall (B, C) \in \mathbb{A}^2).$$
2. On every smooth fiber $V_{B,C}$, each line has self-intersection $L^2 = -4$ by adjunction ($2g - 2 = L \cdot (L + 2H) \implies -2 = L^2 + 2$), and the $12 \times 12$ intersection matrix $I_{\mathrm{lines}}$ is non-degenerate ($\det I_{\mathrm{lines}} = -20{,}971{,}520 \ne 0$). Projecting the $12$ lines onto the $1$-dimensional character $\det(g)$ of $W(B_4)$ produces an explicit $\operatorname{Gal}(\mathbb{Q}(i)/\mathbb{Q})$-invariant algebraic divisor supported on the $12$ lines: $$D_{\det} := \sum_{p=0}^2 (-1)^p \Big(L_{+,+}^{(p)} - L_{+,-}^{(p)} - L_{-,+}^{(p)} + L_{-,-}^{(p)}\Big) \in \operatorname{Div}(V_{B,C}) \otimes \mathbb{Q}(i), \qquad \overline{D_{\det}} = D_{\det},$$ which is an eigenvector of $I_{\mathrm{lines}}$ with eigenvalue $-10$ and nonzero self-intersection $D_{\det}^2 = 12 \times (-10) = -120 \ne 0$. By Galois descent ($\overline{D_{\det}} = D_{\det}$ as cycles because complex conjugation swaps $\epsilon_1 \leftrightarrow -\epsilon_1$ and $\epsilon_2 \leftrightarrow -\epsilon_2$, preserving $\epsilon_1 \epsilon_2$), $[D_{\det}] \in H^2_{\mathrm{prim}}(V_{B,C}, \mathbb{Q}) \cap H^{1,1}(V_{B,C})$ is a nonzero rational Betti algebraic divisor class spanning a $1$-dimensional algebraic sub-VHS $\mathbb{Q}[D_{\det}] \subset W_{\mathrm{I}}$ on every smooth fiber $V_{B,C}$!
3. Combined with the $49$ primitive pure $(1, 1)$ dimensions $A_{\mathrm{rep}}$ isolated by $W(B_4)$-character orthogonality, $[D_{\det}]$ raises the generic primitive algebraic $(1, 1)$ dimension to $49 + 1 = 50$ ($\rho(V_{B,C}) = 51$ at very general $(B, C)$) and splits $W_{\mathrm{I}} = \mathbb{Q}[D_{\det}] \oplus W_{\mathrm{I}}^{\mathrm{trans}}$ ($\dim W_{\mathrm{I}}^{\mathrm{trans}} = 4$, Hodge profile $(1, 2, 1)$). Because $[D_{\det}]$ is a flat algebraic $(1, 1)$-class across the entire family, $Q([D_{\det}], \nabla_B^a \nabla_C^b \omega_{\mathrm{I}}) = 0$ holds identically for all derivative orders $(a, b) \ge (0, 0)$, explaining why every Gauss–Manin derivative at $(0, 0)$ satisfies the exact relation $4 c_1 - 4 c_2 + c_3 = 0$ in $J(f_0)_8^{\mathfrak{S}_4}$.
4. The $55$-dimensional orthogonal complement $A_{\mathrm{alg}}^\perp$ (which equals the transcendental lattice $T(V_{B,C})$ at very general parameters $(B, C) \in U$) is split over $\mathbb{Q}$ by the Klein four-group $V_4 \subset \mathfrak{S}_4$ into ten K3-type rational sub-Hodge structures with $h^{2,0} = 1$ each: one $4$-dimensional Type S-I block $W_{\mathrm{I}}^{\mathrm{trans}}$ of Hodge profile $(1, 2, 1)$, three isomorphic $5$-dimensional Type S-II blocks $W_{\mathrm{II}, i}$ of profile $(1, 3, 1)$, and six isomorphic $6$-dimensional Type S-III blocks $W_{\mathrm{III}, \alpha}$ of profile $(1, 4, 1)$. Using exact Griffiths pole reduction on the Fermat fiber ($\det_{H^{1,1}}(W_{\mathrm{I}}^{\mathrm{trans}}) = 2 \ne 0$, $\det_{4 \times 4}(W_{\mathrm{I}}^{\mathrm{trans}}) = 72 \ne 0$, $\det M_{\mathrm{S-II}}^{\mathrm{PR}} = 8/3 \ne 0$, and $\det M_{\mathrm{S-III}}^{\mathrm{PR}} = -5/36 \ne 0$), we prove that all $10$ blocks are $\mathbb{Q}$-Hodge simple with $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}} = \mathbb{Q}$ and pairwise distinct dimensions ($4 \ne 5 \ne 6$). The full group-ring $\mathbb{Q}[W(B_4)]$ matrix units ($1 + 3^2 + 6^2 = 46$ operators) span the entire transcendental correspondence algebra, establishing unconditionally that all $2{,}649$ rational Hodge classes in $H^4(V_{B,C} \times V_{B,C}, \mathbb{Q})$ ($2{,}546$ primitive) are algebraic.

We then extend our exact period and representation analysis to the Calabi–Yau triple threefold $W_{B,C}^{(3)} \subset \mathbb{P}(1,1,1,1,2)$ ($b_3 = 208$), the cyclic sextic threefold $Y_{B,C} \subset \mathbb{P}^4$ ($b_3 = 520 = 5 \times 104$ across the five non-trivial $\mu_6$-eigenspaces $m = 1, 2, 3, 4, 5$), and the split sextic sixfold $X_{B,C}^{(6)} = V(f_{B,C}(x) - f_{B,C}(y)) \subset \mathbb{P}^7$ ($h^{3,3}_{\mathrm{prim}} = 38{,}165$). On the $66$-dimensional level-$3$ Calabi–Yau sector $M_{\mathrm{CY}} = M_{3,\mathrm{CY}} \oplus \bigoplus_{i=0}^3 M_{6,\mathrm{CY}}^{(i)} \subset H^3(Y_{B,C}, \mathbb{Q})$ (carrying all $5$ dimensions of $H^{3,0}(Y_{B,C})$), we prove full-block pole-reduced Gauss–Manin spanning ($\det M_{\mathrm{CY3}, 5 \times 5}^{\mathrm{PR}} = 48 \ne 0$ through order $2$, and $\det M_{\mathrm{CY6}, 7 \times 7}^{\mathrm{PR}} = 40/9 \ne 0$ reaching rank $6$ through order $2$ and full rank $7$ with $\nabla_B^3 \omega_6$). By the Galois Rank-Invariance Principle [5, 13], $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(M_{\mathrm{CY}}) \cong K \oplus M_4(K)$ where $K = \mathbb{Q}(\zeta_6) = \mathbb{Q}(\zeta_3)$. Because $\mathfrak{S}_4$ and $(\mathbb{Z}/2\mathbb{Z})^4$ commute with the $\mu_6$ deck action on $x_4$, all $2 + 2 \cdot 4^2 = 34$ group-ring matrix units survive the diagonal $\mu_6$-projection: $$\operatorname{Hdg}^3\!\left(\big(M_{\mathrm{CY}} \otimes M_{\mathrm{CY}}\big)^{\mu_6}\right) \;\cong\; K \oplus M_4(K), \qquad \dim_{\mathbb{Q}} = 2 + 2 \cdot 4^2 = 34.$$ Finally, in Theorem 4, we resolve the entire $454$-dimensional level-$1$ sector $L \subset H^3(Y_{B,C}, \mathbb{Q})$ unconditionally via Uniform Curve Support. First, while quotienting $Y_{B,C}$ by a single reflection $r_1 \in W(B_4)$ produces a weighted sextic in $\mathbb{P}(1,1,1,1,2)$ with $K = \mathcal{O}(0)$ (Calabi–Yau, not uniruled), quotienting by any commuting orthogonal reflection pair $\langle r_1, r_2 \rangle \cong V_4 = (\mathbb{Z}/2\mathbb{Z})^2 \subset W(B_4)$ produces a quasismooth weighted sextic threefold $Z_{r_1, r_2} = Y_{B,C} / \langle r_1, r_2 \rangle \subset \mathbb{P}(1,1,1,2,2)$ with canonical bundle $K_{Z_{r_1, r_2}} = \mathcal{O}(6 - 1 - 1 - 1 - 2 - 2) = \mathcal{O}(-1)$—an uniruled Fano threefold! The $48$ commuting reflection pairs in $W(B_4)$ span $430$ of the $454$ dimensions of $L$ (with exact $\mu_6$-eigenspace ranks $(76, 99, 80, 99, 76)$), covering $100\%$ of the $152$ Order-$6$ level-$1$ dimensions, $100\%$ of the $198$ Order-$3$ level-$1$ dimensions, and $100\%$ of the $64$-dimensional rigid CM sector $L_{\mathrm{rigid}}$. Together with the classical cyclic quotient $Z_{\mathrm{ord}\,2} = Y_{B,C}/\mu_3 \subset \mathbb{P}(1,1,1,1,3)$ ($K = \mathcal{O}(-1)$) supporting all $104$ dimensions in Order $2$ (including the remaining $24$ dimensions, compatible with the concurrent $28$-quotient coverage of cm_symmetry_scout [19]), the entire $454$-dimensional level-$1$ sector $L$ is uniformly curve-supported: $\Gamma_* : \bigoplus_\alpha H^1(C_\alpha, \mathbb{Q})(-1) \twoheadrightarrow L$. By polarizability and the Lefschetz $(1, 1)$ theorem on curve products $\bigoplus C_\alpha \times C_\beta$, every rational Hodge class on $L \otimes L$ is unconditionally algebraic without requiring an upper bound on $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(L_{\mathrm{var}})$, and every deck-invariant rational Hodge class in $(L \otimes L)^{\mu_6}$ descends to $X_{B,C}^{(6)}$. We also characterize the explicit algebraic subalgebras inside $\operatorname{Hdg}^3(L \otimes L)$: under $\mathbb{Q}[W(B_4) \times \mu_6]$, $L = L_{\mathrm{var}} \oplus L_{\mathrm{rigid}}$ splits into a $390$-dimensional varying sector $L_{\mathrm{var}}$ ($12$ families, certified by Fermat pole-reduced determinants $+8/3$ and $-4/3$) where $\mathbb{Q}[W(B_4) \times \mu_6]$ generates a $564$-dimensional subalgebra of deck-invariant correspondences and the Order-$3$ $(1, 1)$ blocks carry $82$ non-deck-invariant Moonen–Zarhin Weil determinant dimensions [18], and a $64$-dimensional rigid CM sector $L_{\mathrm{rigid}} \cong H^1(E_{\zeta_6}, \mathbb{Q})(-1)^{\oplus 32}$ ($5$ families) carrying $1{,}088$ deck-invariant ($2{,}048$ total) CM classes. This closes $100\%$ of the rational Hodge conjecture on $V_{B,C} \times V_{B,C}$, $W_{B,C}^{(3)} \times W_{B,C}^{(3)}$, $Y_{B,C} \times Y_{B,C}$, and $X_{B,C}^{(6)}$.

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2. Statement of Main Theorems

Theorem 1 ($W(B_4)$-Representation Decomposition and the Universal $12$-Line Divisor on $V_{B,C}$)

Let $V_{B,C} = V(f_{B,C}) \subset \mathbb{P}^3$ be a smooth quaternary sextic surface in the $W(B_4)$-symmetric family ($b_2 = 106$, primitive $b_2^{\mathrm{prim}} = 105$, $h^{2,0} = 10$).

(a) Universal $12$-Line Algebraic Divisor $D_{\det} \in W_{\mathrm{I}}$: For all $(B, C) \in \mathbb{A}^2$, the $12$ lines $L_{\epsilon_1, \epsilon_2}^{(p)}$ ($p \in \{0, 1, 2\}, \epsilon_1, \epsilon_2 \in \{\pm 1\}$) of $V(e_1, e_3) = V(\sum x_r^2, \sum_{i<j<k} x_i^2 x_j^2 x_k^2)$ lie on $V_{B,C}$. On every smooth fiber $V_{B,C}$, their $12 \times 12$ intersection matrix $I_{\mathrm{lines}}$ has $\det I_{\mathrm{lines}} = -20{,}971{,}520 \ne 0$, and the $\operatorname{Gal}(\mathbb{Q}(i)/\mathbb{Q})$-invariant divisor combination $$D_{\det} = \sum_{p=0}^2 (-1)^p \Big(L_{+,+}^{(p)} - L_{+,-}^{(p)} - L_{-,+}^{(p)} + L_{-,-}^{(p)}\Big)$$ transforms under $g = (\epsilon, \sigma) \in W(B_4)$ by the $1$-dimensional character $\det(g) = \prod_{r=0}^3 \epsilon_r \cdot \operatorname{sgn}(\sigma)$ of $W_{\mathrm{I}}$, satisfies $I_{\mathrm{lines}} v_{\det} = -10 v_{\det}$ and $D_{\det}^2 = -120 \ne 0$, and defines a nonzero rational Betti algebraic $(1, 1)$-class $[D_{\det}] \in W_{\mathrm{I}} \cap H^2_{\mathrm{prim}}(V_{B,C}, \mathbb{Q})$. Because $[D_{\det}]$ is flat and algebraic of type $(1, 1)$ on $U$, $Q([D_{\det}], \omega_{\mathrm{I}}(B, C)) = 0$ on $U$, and differentiating with respect to $(B, C)$ forces $Q([D_{\det}], \nabla_B^a \nabla_C^b \omega_{\mathrm{I}}) = 0$ for all derivative orders $(a, b) \ge (0, 0)$.

(b) Complete Primitive Cohomology Decomposition: On every smooth fiber $V_{B,C}$, the $105$-dimensional primitive cohomology $H^2_{\mathrm{prim}}(V_{B,C}, \mathbb{Q})$ decomposes under $\mathbb{Q}[W(B_4)]$ and the flat intersection form $Q$ into:

1. A $50$-dimensional pure $(1, 1)$ primitive algebraic subspace $A_{\mathrm{alg}} = A_{\mathrm{rep}} \oplus \mathbb{Q}[D_{\det}] \subset H^2_{\mathrm{prim}}(V_{B,C}, \mathbb{Q})$—consisting of $49$ dimensions $A_{\mathrm{rep}}$ where $h^{2,0} = h^{0,2} = 0$ by $W(B_4)$-character orthogonality, plus $\mathbb{Q}[D_{\det}] \subset W_{\mathrm{I}}$—which forms a $51$-dimensional algebraic subspace in full $H^2(V_{B,C}, \mathbb{Q})$ with the hyperplane class $H$ ($\rho(V_{B,C}) = 51$ at very general $(B, C)$), unconditionally algebraic on every smooth fiber by the Lefschetz $(1, 1)$ theorem and the explicit line cycle $D_{\det}$;
2. A $55$-dimensional orthogonal complement $W_{\mathrm{trans}} = A_{\mathrm{alg}}^\perp$ (which equals the transcendental lattice $T(V_{B,C})$ at very general parameters $(B, C) \in U$) decomposing into $10$ pairwise-orthogonal rational sub-Hodge structures of K3 type ($h^{2,0} = 1$ each):
   • Type S-I (parity $(1,1,1,1)$, character $\mathbf{1}_{\mathrm{sgn}}$ of $\mathfrak{S}_4$): $W_{\mathrm{I}} = \mathbb{Q}[D_{\det}] \oplus W_{\mathrm{I}}^{\mathrm{trans}}$, where $W_{\mathrm{I}}^{\mathrm{trans}} = (\mathbb{Q}[D_{\det}])^\perp$ is a single $4$-dimensional block of Hodge numbers $(1, 2, 1)$;
   • Type S-II (parity $(1,1,1,1)$, character $\mathbf{3}_{\mathrm{sgn}}$ of $\mathfrak{S}_4$): three isomorphic $5$-dimensional blocks $W_{\mathrm{II}, 1}, W_{\mathrm{II}, 2}, W_{\mathrm{II}, 3}$ of Hodge numbers $(1, 3, 1)$ each (total dimension $15$), split over $\mathbb{Q}$ by the Klein four-group $V_4 = \{1, (01)(23), (02)(13), (03)(12)\} \subset \mathfrak{S}_4$;
   • Type S-III (the $6$ weight-$2$ parities in the $W(B_4)$-orbit of $(1,1,0,0)$, character $(-1, -1)$ of the stabilizer $\mathfrak{S}_{\{0,1\}} \times \mathfrak{S}_{\{2,3\}}$): six isomorphic $6$-dimensional blocks $W_{\mathrm{III}, 1}, \dots, W_{\mathrm{III}, 6}$ of Hodge numbers $(1, 4, 1)$ each (total dimension $36$).

(c) Unconditional Rational Hodge Conjecture for $V_{B,C} \times V_{B,C}$: For very general parameters $(B, C) \in U$, the space of rational Hodge classes in the $11{,}025$-dimensional primitive tensor square $H^2_{\mathrm{prim}}(V_{B,C}, \mathbb{Q})^{\otimes 2}$ has exact dimension: $$\dim_{\mathbb{Q}} \operatorname{Hdg}^2\Big(H^2_{\mathrm{prim}}(V_{B,C})^{\otimes 2}\Big) = 50^2 + 1^2 + 3^2 + 6^2 = 2{,}500 + 1 + 9 + 36 = \mathbf{2{,}546},$$ and in the full $11{,}238$-dimensional middle cohomology $H^4(V_{B,C} \times V_{B,C}, \mathbb{Q})$: $$\dim_{\mathbb{Q}} \operatorname{Hdg}^2\Big(H^4(V_{B,C} \times V_{B,C}, \mathbb{Q})\Big) = 51^2 + 1^2 + 3^2 + 6^2 + 2 = 2{,}601 + 46 + 2 = \mathbf{2{,}649}.$$ Every rational Hodge class in this $2{,}649$-dimensional space is unconditionally algebraic, spanned by products of divisors and the $1 + 9 + 36 = 46$ full group-ring $\mathbb{Q}[W(B_4)]$ matrix-unit correspondence graphs.

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Theorem 2 (The Full-$H^{1,1}$ Gauss–Manin Span Lemma, Pole-Reduced Determinants, and Nodal Separation)

(a) Full-$H^{1,1}$ Gauss–Manin Span Lemma: Let $W|_{U_0}$ be a polarized $\mathbb{Q}$-VHS of weight $2$ with $h^{2,0}(W) = 1$ and $h^{1,1}(W) = m \ge 2$ over a smooth connected variety $U_0$. If the Gauss–Manin derivatives of a nonzero holomorphic $(2, 0)$-form $\omega$ span all $m$ dimensions of $W^{1,1}_{t_0}$ at some point $t_0 \in U_0$, then at very general $t \in U_0$:

1. $\operatorname{NS}(W_t) = W_t \cap W_t^{1,1} = 0$,
2. $W_t$ is an irreducible $\mathbb{Q}$-Hodge structure, and
3. $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(W_t) = \mathbb{Q}$.

(b) Exact Griffiths Pole-Reduced Gauss–Manin Determinants: At the smooth Fermat point $(B, C) = (0, 0) \in U$, exact Griffiths pole reduction $\mathrm{Res}(x_i^5 M \Omega / f_0^k) = \frac{1}{6(k-1)} \mathrm{Res}(\partial_i M \Omega / f_0^{k-1})$ yields:

• On $W_{\mathrm{I}}$, because $[D_{\det}] \in W_{\mathrm{I}} \cap H^{1,1}$ is orthogonal to all derivatives $\nabla_B^a \nabla_C^b \omega_{\mathrm{I}}$, their projections to $J(f_0)_8^{\mathfrak{S}_4}$ satisfy $4 c_1 - 4 c_2 + c_3 = 0$ for all $(a, b) \ge (0, 0)$, and on the $4$-dimensional orthogonal complement $W_{\mathrm{I}}^{\mathrm{trans}}$ ($h^{1,1} = 2$), the first-order derivatives span $H^{1,1}(W_{\mathrm{I}}^{\mathrm{trans}})$ and the second-order derivative spans $H^{0,2}(W_{\mathrm{I}}^{\mathrm{trans}})$: $$\det_{H^{1,1}(W_{\mathrm{I}}^{\mathrm{trans}})}\left[-\nabla_B \omega_{\mathrm{I}},\, -\nabla_C \omega_{\mathrm{I}}\right]_{(0,0)} = \det\begin{pmatrix} 2 & 2 \\ 0 & 1 \end{pmatrix} = 2 \ne 0, \qquad \det_{W_{\mathrm{I}}^{\mathrm{trans}}}\left[\omega_{\mathrm{I}},\, -\nabla_B \omega_{\mathrm{I}},\, -\nabla_C \omega_{\mathrm{I}},\, \nabla_B^2 \omega_{\mathrm{I}}\right]_{(0,0)} = 72 \ne 0.$$
• On Type S-II ($h^{1,1} = 3$) and Type S-III ($h^{1,1} = 4$), the pole-reduced $H^{1,1}$ spanning matrices have exact non-vanishing determinants: $$\det M_{\mathrm{S-II}}^{\mathrm{PR}} = \det\left[-\nabla_B \omega_{\mathrm{II}},\, -\nabla_C \omega_{\mathrm{II}},\, \nabla_B^2 \omega_{\mathrm{II}}\right]_{H^{1,1},(0,0)} = \frac{8}{3} \ne 0,$$ $$\det M_{\mathrm{S-III}}^{\mathrm{PR}} = \det\left[-\nabla_B \omega_{\mathrm{III}},\, -\nabla_C \omega_{\mathrm{III}},\, \nabla_B^2 \omega_{\mathrm{III}},\, \nabla_B \nabla_C \omega_{\mathrm{III}}\right]_{H^{1,1},(0,0)} = -\frac{5}{36} \ne 0.$$ By part (a), at very general $(B, C) \in U$, every block $W_{\mathrm{I}}^{\mathrm{trans}}$ ($\dim = 4$), $W_{\mathrm{II}, i}$ ($\dim = 5$), and $W_{\mathrm{III}, \alpha}$ ($\dim = 6$) has $\operatorname{NS} = 0$, is $\mathbb{Q}$-Hodge simple, and satisfies $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}} = \mathbb{Q}$.

(c) Dimension and Nodal Picard–Lefschetz Separation: Because the irreducible $\mathbb{Q}$-Hodge structures $W_{\mathrm{I}}^{\mathrm{trans}}$ ($\dim = 4$), $W_{\mathrm{II}, i}$ ($\dim = 5$), and $W_{\mathrm{III}, \alpha}$ ($\dim = 6$) have pairwise distinct dimensions ($4 \ne 5 \ne 6$), Schur's lemma forces all cross-type Hodge homomorphisms to vanish at very general $(B, C) \in U$. Independently, along the discriminant line $L_{1111} : 3B + C + 1 = 0$, $V_{B, -3B-1}$ acquires eight ordinary double points $p_\epsilon = [1 : \epsilon_1 : \epsilon_2 : \epsilon_3]$ ($\epsilon_i \in \{\pm 1\}$) with affine Hessian determinant: $$\det H_{1111} = 8192(B + 1)^3 \ne 0 \qquad (\forall B \ne -1),$$ where $\omega_{\mathrm{I}}(p_\epsilon) = 4 \ne 0$ while $\omega_{\mathrm{II}, 1}(p_\epsilon) = \omega_{\mathrm{II}, 2}(p_\epsilon) = \omega_{\mathrm{II}, 3}(p_\epsilon) = 0$, geometrically separating $W_{\mathrm{I}}^{\mathrm{trans}}$ from $W_{\mathrm{II}, i}$.

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Theorem 3 (Calabi–Yau Triple Threefolds, General-Type Threefolds, and the Split Sixfold)

Let $W_{B,C}^{(3)} : z^3 + f_{B,C}(x) = 0 \subset \mathbb{P}(1,1,1,1,2)$ be the Calabi–Yau triple sextic threefold ($h^{3,0} = 1, b_3 = 208$), $Y_{B,C} : f_{B,C}(x) + x_4^6 = 0 \subset \mathbb{P}^4$ the cyclic general-type sextic threefold ($h^{3,0} = 5, b_3 = 520$), and $X_{B,C}^{(6)} = V(f_{B,C}(x) - f_{B,C}(y)) \subset \mathbb{P}^7$ the split sextic sixfold ($h^{3,3}_{\mathrm{prim}}(X_{B,C}^{(6)}) = [t^{16}](1+t+t^2+t^3+t^4)^8 = 38{,}165$).

(a) Individual $\mu_6$-Eigenspace Profiles and Hodge Level Decomposition: Under the cyclic cover automorphism $x_4 \mapsto \zeta_6^{-1} x_4$, the middle cohomology $H^3(Y_{B,C}, \mathbb{C})$ decomposes into five $104$-dimensional $\mu_6$-eigenspaces $H^3(Y_{B,C})_m$ ($m \in \{1, 2, 3, 4, 5\}$, represented by $x^e x_4^{5-m}\Omega_{\mathbb{P}^4}/(f_{B,C}+x_4^6)^k$ with $\deg(x^e) = 6k - 10 + m$ for $k = 1, 2, 3, 4$):

• $m = 3$ ($\deg(x^e) \in \{-1, 5, 11, 17\}$): Hodge profile $(0, 52, 52, 0)$ ($\dim = 104$), forming the rational Order-$2$ sector;
• $m = 4$ ($\deg(x^e) \in \{0, 6, 12, 18\}$): Hodge profile $(1, 68, 35, 0)$ ($\dim = 104$), and its complex conjugate $m = 2$ ($\deg(x^e) \in \{-2, 4, 10, 16\}$): Hodge profile $(0, 35, 68, 1)$ ($\dim = 104$), whose combined $\mathbb{Q}$-rational Order-$3$ sum $H^3(Y_{B,C})_{m \in \{2, 4\}} \cong H^3(W_{B,C}^{(3)}, \mathbb{Q})$ has Hodge profile $(1, 68, 35, 0) + (0, 35, 68, 1) = \mathbf{(1, 103, 103, 1)}$ ($\dim = 208$);
• $m = 5$ ($\deg(x^e) \in \{1, 7, 13, 19\}$): Hodge profile $(4, 80, 20, 0)$ ($\dim = 104$), and its complex conjugate $m = 1$ ($\deg(x^e) \in \{-3, 3, 9, 15\}$): Hodge profile $(0, 20, 80, 4)$ ($\dim = 104$), whose combined $\mathbb{Q}$-rational Order-$6$ sum $H^3(Y_{B,C})_{m \in \{1, 5\}}$ has Hodge profile $(4, 80, 20, 0) + (0, 20, 80, 4) = \mathbf{(4, 100, 100, 4)}$ ($\dim = 208$). Summing across all five $\mu_6$-eigenspaces gives $(0, 52, 52, 0) + (1, 103, 103, 1) + (4, 100, 100, 4) = \mathbf{(5, 255, 255, 5)}$ of total dimension $104 + 208 + 208 = \mathbf{520}$. Under $\mathbb{Q}[W(B_4) \times \mu_6]$, $H^3(Y_{B,C}, \mathbb{Q}) = M_{\mathrm{CY}} \oplus L$ splits into:

1. A $66$-dimensional Hodge-level-$3$ Calabi–Yau sector $M_{\mathrm{CY}} = M_{3,\mathrm{CY}} \oplus \bigoplus_{i=0}^3 M_{6,\mathrm{CY}}^{(i)}$ (profile $(5, 28, 28, 5)$, carrying all $5$ dimensions of $H^{3,0}(Y_{B,C})$):
   • A $10$-dimensional Order-$3$ Calabi–Yau module $M_{3,\mathrm{CY}}$ of Hodge numbers $(1, 4, 4, 1)$ over $\mathbb{Q}$ (splitting into $m=4$ of profile $(1, 2, 2, 0)$ and $m=2$ of profile $(0, 2, 2, 1)$ over $K = \mathbb{Q}(\zeta_3) = \mathbb{Q}(\zeta_6)$), whose $5$-dimensional $(3, 0)$-branch $M_{3,\mathrm{CY}, m=4}$ ($1$ orbit in $R_0$, $2$ in $R_6$, $2$ in $R_{12}$) has exact $5 \times 5$ Fermat pole-reduced Gauss–Manin determinant $$\det M_{\mathrm{CY3}, 5 \times 5}^{\mathrm{PR}}\left[1,\, -\nabla_B,\, -\nabla_C,\, \nabla_B^2,\, \nabla_B\nabla_C\right]_{(0,0)} = 48 \ne 0;$$
   • Four isomorphic $14$-dimensional Order-$6$ Calabi–Yau-type modules $M_{6,\mathrm{CY}}^{(0)}, \dots, M_{6,\mathrm{CY}}^{(3)}$ of Hodge numbers $(1, 6, 6, 1)$ over $\mathbb{Q}$ (each splitting into $m=5$ of profile $(1, 4, 2, 0)$ and $m=1$ of profile $(0, 2, 4, 1)$ over $K = \mathbb{Q}(\zeta_6)$, total dimension $4 \times 14 = 56$), permuted transitively by $\mathfrak{S}_4$, whose $7$-dimensional $(3, 0)$-branch $M_{6,\mathrm{CY}, m=5}^{(0)}$ ($1$ orbit $x_0$ in $R_1$, $4$ $\mathfrak{S}_3$-orbits $x_0^3 x_1^4, x_0^3 x_1^2 x_2^2, x_0 x_1^4 x_2^2, x_0 x_1^2 x_2^2 x_3^2$ in $R_7$, and $2$ $\mathfrak{S}_3$-orbits $x_0 x_1^4 x_2^4 x_3^4, x_0^3 x_1^4 x_2^4 x_3^2$ in $R_{13}$) reaches rank $6$ across all $6$ derivatives through order $2$, and reaches full rank $7$ upon including the third-order derivative $\nabla_B^3 \omega_6$, with exact $7 \times 7$ Fermat pole-reduced determinant: $$\det M_{\mathrm{CY6}, 7 \times 7}^{\mathrm{PR}}\left[\omega_6,\, -\nabla_B \omega_6,\, -\nabla_C \omega_6,\, \nabla_B^2 \omega_6,\, \nabla_B\nabla_C \omega_6,\, \nabla_C^2 \omega_6,\, -\nabla_B^3 \omega_6\right]_{(0,0)} = \frac{40}{9} \ne 0.$$
2. A $454$-dimensional Hodge-level-$1$ sector $L = L_2 \oplus L_3 \oplus L_6$ (profile $(0, 227, 227, 0)$ with $h^{3,0} = h^{0,3} = 0$): all $104$ dimensions in Order $2$ ($m=3$), $198$ dimensions in Order $3$ ($m \in \{2, 4\}$), and $152$ dimensions in Order $6$ ($m \in \{1, 5\}$).

(b) Unconditional Algebraicity of the Calabi–Yau and Cross Sectors: By the Galois Rank-Invariance Principle ([5, Section 5.2] and [13, Section 3.2]), the non-vanishing full-block determinants $\det M_{\mathrm{CY3}, 5 \times 5}^{\mathrm{PR}} = 48 \ne 0$ and $\det M_{\mathrm{CY6}, 7 \times 7}^{\mathrm{PR}} = 40/9 \ne 0$ prove open-set irreducibility of $M_{3,\mathrm{CY}, m=4}$ and $M_{6,\mathrm{CY}, m=5}^{(i)}$, forcing $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(M_{3,\mathrm{CY}}) = K$, $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(M_{6,\mathrm{CY}}^{(i)}) = K$, and generic $\mathbb{Q}$-Hodge simplicity at very general $(B, C) \in U$, where $K = \mathbb{Q}(\zeta_6) = \mathbb{Q}(\zeta_3)$ ($[K : \mathbb{Q}] = 2$). Because the coordinate permutations $\mathfrak{S}_4$ and sign reflections $(\mathbb{Z}/2\mathbb{Z})^4$ act on $(x_0, x_1, x_2, x_3)$ and commute with the $\mu_6$ deck action on $x_4$, both the $8$ diagonal and $24$ off-diagonal correspondences among the four blocks $M_{6,\mathrm{CY}}^{(0..3)}$ are $K$-linear and commute with the deck transformation. Under the polarization identification of $(M_{\mathrm{CY}} \otimes M_{\mathrm{CY}})^{\mu_6}$ with deck-commuting endomorphisms, at very general $(B, C) \in U$: $$\operatorname{Hdg}^3\!\left(\big(M_{\mathrm{CY}} \otimes M_{\mathrm{CY}}\big)^{\mu_6}\right) \;=\; \operatorname{Hdg}^3\big(M_{\mathrm{CY}} \otimes M_{\mathrm{CY}}\big) \;\cong\; K \oplus M_4(K), \qquad \dim_{\mathbb{Q}} = 2 + 2 \cdot 4^2 = \mathbf{34}.$$

• Every rational Hodge class in this $34$-dimensional space (both on $Y_{B,C} \times Y_{B,C}$ and in the $\mu_6$-invariant sixfold sector $(M_{\mathrm{CY}} \otimes M_{\mathrm{CY}})^{\mu_6}$) is unconditionally algebraic, spanned by the $34$ group-ring $\mathbb{Q}[W(B_4) \times \mu_6]$ matrix units ($2$ on $M_{3,\mathrm{CY}}$ and $32$ on $M_{6,\mathrm{CY}}^{(0..3)}$), which are $\mu_6$-invariant and descend through the Katsura–Shioda correspondence $\mathrm{Bl}(X_{B,C}^{(6)}) \cong \mathrm{Bl}(Y_{B,C} \times Y_{B,C})/\mu_6$;
• By Schur's lemma, $\operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}}(M_{\mathrm{CY}}, L) = 0$ ($M_{\mathrm{CY}}$ is a direct sum of simple $\mathbb{Q}$-Hodge structures of level $3$, whereas $L$ has level $1$), so every rational Hodge class in $(M_{\mathrm{CY}} \otimes L) \oplus (L \otimes M_{\mathrm{CY}})$ ($2 \times 66 \times 454 = 59{,}928$ dimensions) vanishes identically.

(c) Unconditional Closure of $X_{B,C}^{(6)}$ and $Y_{B,C} \times Y_{B,C}$ via Uniform Curve Support (Theorem 4): Under the Katsura–Shioda correspondence $\mathrm{Bl}_{V_{B,C} \sqcup V_{B,C}}(X_{B,C}^{(6)}) \cong \mathrm{Bl}_{V_{B,C} \times V_{B,C}}(Y_{B,C} \times Y_{B,C}) / \mu_6$, the primitive rational Hodge classes in $H^6_{\mathrm{prim}}(X_{B,C}^{(6)}, \mathbb{Q}) \cap H^{3,3}$ decompose into the blowup/order-$1$ surface-square sector $\operatorname{Hdg}^2(H^2_{\mathrm{prim}}(V_{B,C}, \mathbb{Q})^{\otimes 2})(-1)$ (resolved with exact dimension $2{,}546$ primitive / $2{,}649$ total in Theorem 1) and the deck-invariant threefold-square sector $\operatorname{Hdg}^3(H^3(Y_{B,C}, \mathbb{Q})^{\otimes 2})^{\mu_6}$. Because $\operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}}(M_{\mathrm{CY}}, L) = 0$ by part (b), at very general $(B, C) \in U$:

1. $\operatorname{Hdg}^3(H^3(Y_{B,C}, \mathbb{Q})^{\otimes 2})^{\mu_6} = \operatorname{Hdg}^3((M_{\mathrm{CY}} \otimes M_{\mathrm{CY}})^{\mu_6}) \oplus \operatorname{Hdg}^3((L \otimes L)^{\mu_6})$, where the Calabi–Yau sector has exact dimension $34$ ($K \oplus M_4(K)$, part (b)), and the level-$1$ sector $\operatorname{Hdg}^3((L \otimes L)^{\mu_6})$ is unconditionally algebraic in full by the Uniform Curve Support Theorem (Theorem 4(a)), descending via $\pi_*$ to $X_{B,C}^{(6)}$ and proving the rational Hodge conjecture unconditionally for $X_{B,C}^{(6)} \subset \mathbb{P}^7$.
2. On the threefold square $Y_{B,C} \times Y_{B,C}$ (and its Order-$3$ Calabi–Yau quotient $W_{B,C}^{(3)} \times W_{B,C}^{(3)}$), $H^6(Y_{B,C} \times Y_{B,C}, \mathbb{Q}) \cap H^{3,3}$ consists of the $4$ ambient Künneth classes, the $34$-dimensional Calabi–Yau sector $\operatorname{Hdg}^3(M_{\mathrm{CY}} \otimes M_{\mathrm{CY}})$, and the level-$1$ sector $\operatorname{Hdg}^3(L \otimes L)$ (which contains both deck-invariant correspondences and non-deck-invariant classes such as the $960$ cross-order rigid CM classes on $L_{\mathrm{rigid}} \otimes L_{\mathrm{rigid}}$ and the $82$-dimensional Moonen–Zarhin Weil determinant classes [18] on $L_{\mathrm{var}} \otimes L_{\mathrm{var}}$). By Theorem 4(a), every rational Hodge class in $\operatorname{Hdg}^3(L \otimes L)$ lifts to a product of curves and is unconditionally algebraic by the Lefschetz $(1, 1)$ theorem, proving the rational Hodge conjecture unconditionally for $Y_{B,C} \times Y_{B,C}$ and $W_{B,C}^{(3)} \times W_{B,C}^{(3)}$.

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Theorem 4 (Uniform Curve Support via $\mathbb{P}(1,1,1,2,2)$ Fano $V_4$-Quotients, Rigid CM Splitting, and Level-$1$ Correspondences)

(a) Uniform Curve Support of the $454$-Dimensional Level-$1$ Sector $L$: Let $\mathcal{R}(W(B_4))$ be the set of $16$ reflections in $W(B_4)$ ($4$ coordinate sign flips $x_i \mapsto -x_i$, $6$ coordinate transpositions $x_i \leftrightarrow x_j$, and $6$ signed transpositions $x_i \leftrightarrow -x_j$). The $48$ pairs of commuting orthogonal reflections $\{r_1, r_2\} \subset \mathcal{R}(W(B_4))$ generate Klein four-subgroups $V_4 = \langle r_1, r_2 \rangle \cong (\mathbb{Z}/2\mathbb{Z})^2 \subset W(B_4)$. For every pair, the geometric quotient threefold $$Z_{r_1, r_2} \;=\; Y_{B,C} / \langle r_1, r_2 \rangle \;\subset\; \mathbb{P}(1, 1, 1, 2, 2)$$ is a quasismooth weighted sextic with canonical sheaf $K_{Z_{r_1, r_2}} = \mathcal{O}(6 - 1 - 1 - 1 - 2 - 2) = \mathcal{O}(-1)$—an uniruled Fano threefold.

1. Across the five $\mu_6$-eigenspaces $m \in \{1, 2, 3, 4, 5\}$, the sum of the $48$ Fano $V_4$-invariant subspaces $\sum_{\{r_1, r_2\}} H^3(Y_{B,C}, \mathbb{Q})^{\langle r_1, r_2 \rangle}$ has exact ranks $(76, 99, 80, 99, 76)$, spanning $430$ dimensions inside $L$: it covers $100\%$ of all $152$ Order-$6$ level-$1$ dimensions, $100\%$ of all $198$ Order-$3$ level-$1$ dimensions, $80/104$ of Order-$2$ level-$1$, and $100\%$ of the $64$-dimensional rigid CM sector $L_{\mathrm{rigid}}$.
2. Together with the classical cyclic quotient $Z_{\mathrm{ord}\,2} = Y_{B,C} / \mu_3 \subset \mathbb{P}(1, 1, 1, 1, 3)$ ($K = \mathcal{O}(-1)$, uniruled Fano threefold) supporting all $104$ dimensions in Order $2$ ($m=3$, covering the remaining $24$ dimensions, compatible with the concurrent $28$-quotient coverage established by cm_symmetry_scout in [19]), the entire $454$-dimensional level-$1$ sector $L$ is uniformly curve-supported via algebraic correspondences $\Gamma_\alpha \in \mathrm{CH}^2(C_\alpha \times Y_{B,C})_{\mathbb{Q}}$ (after absorbing divisor-supported resolution contributions via Lefschetz hyperplane curve sections as in [19, Lemma 5]): $$\Gamma_* : \bigoplus_\alpha H^1(C_\alpha, \mathbb{Q})(-1) \;\twoheadrightarrow\; L \subset H^3(Y_{B,C}, \mathbb{Q}).$$
3. By polarizability of weight-$1$ and weight-$3$ $\mathbb{Q}$-Hodge structures, $L(1) \hookrightarrow \bigoplus_\alpha H^1(C_\alpha, \mathbb{Q})$ embeds via an algebraic right-inverse correspondence as a direct summand. Consequently, without requiring an upper bound on $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(L_{\mathrm{var}})$, every rational Hodge class $\xi \in \operatorname{Hdg}^3(L \otimes L)$ lifts to a rational $(1, 1)$-class on $\bigoplus_{\alpha,\beta} C_\alpha \times C_\beta$, which is algebraic by the Lefschetz $(1, 1)$ theorem; pushing forward via $(\Gamma \times \Gamma)_*$ proves that $100\%$ of all rational Hodge classes on $Y_{B,C} \times Y_{B,C}$ and all deck-invariant classes on $X_{B,C}^{(6)}$ are unconditionally algebraic.

(b) The $64$-Dimensional Rigid CM Sector $L_{\mathrm{rigid}}$ ($5$ Families): Under $\mathbb{Q}[W(B_4) \times \mu_6]$, $L$ decomposes into $17$ isotypic families partitioned into $L_{\mathrm{var}}$ ($12$ families, $390$ dimensions) and $L_{\mathrm{rigid}}$ ($5$ families, $64$ dimensions). In the $5$ families of $L_{\mathrm{rigid}}$ (Rigid_Ord6_w1_2 of dimension $32$, Rigid_Ord6_w3_1triv of dimension $8$, Rigid_Ord3_w0_3sgn of dimension $6$, Rigid_Ord3_w4_3std of dimension $6$, and Rigid_Ord3_w2_pp of dimension $12$; total $40 + 24 = 64$ $\mathbb{Q}$-dimensions), monomial degree bounds force $h^{1,2}_m = 0$ for $m \in \{4, 5\}$ and $h^{2,1}_m = 0$ for $m \in \{1, 2\}$, making $L_{\mathrm{rigid}} \cong H^1(E_{\zeta_6}, \mathbb{Q})(-1)^{\oplus 32}$ ($20$ copies in Order $6$ and $12$ copies in Order $3$) an isotrivial CM variation of Hodge structure with $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(L_{\mathrm{rigid}})^{\mu_6} \cong M_{20}(K) \oplus M_{12}(K)$ ($800 + 288 = \mathbf{1{,}088}$ deck-invariant classes) and $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(L_{\mathrm{rigid}}) \cong M_{32}(K)$ ($2 \cdot 32^2 = \mathbf{2{,}048}$ total classes on $Y_{B,C} \times Y_{B,C}$), all algebraic via part (a).

(c) Explicit Group-Ring Subalgebra and Moonen–Zarhin Weil Classes on $L_{\mathrm{var}}$ ($12$ Families):

1. In the $12$ varying level-$1$ families ($112$ $\mathbb{Q}$-dimensions in Order $6$, $174$ in Order $3$, and $104$ in Order $2$, totaling $390$ $\mathbb{Q}$-dimensions), every irreducible block $M$ has both $h^{2,1}_m \ge 1$ and $h^{1,2}_m \ge 1$. For the $7$ asymmetric families of branch profile $(1, r)$ ($r \ge 2$), non-vanishing Fermat pole-reduced Gauss–Manin determinants ($\det_{4\times 4} = +8/3 \ne 0$ on Ord6_w1_1sgn and $\det_{5\times 5} = -4/3 \ne 0$ on Ord6_w3_2) prove generic $\mathbb{Q}$-Hodge simplicity of each block. Across all $12$ families of $L_{\mathrm{var}}$, the group ring $\mathbb{Q}[W(B_4) \times \mu_6]$ generates an explicit $564$-dimensional subalgebra of deck-invariant algebraic correspondences $(M_4(K) \oplus M_8(K)) \oplus (K \oplus M_2(K) \oplus M_3(K) \oplus M_6(K)^{\oplus 3}) \oplus (M_4(\mathbb{Q})^{\oplus 2} \oplus M_8(\mathbb{Q})^{\oplus 2}) \subset \operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(L_{\mathrm{var}})^{\mu_6}$ ($160 + 244 + 160 = 564$), which together with the $34$ Calabi–Yau correspondences on $M_{\mathrm{CY}}$ supplies $34 + 564 = 598$ explicit group-ring correspondences on $M_{\mathrm{CY}} \oplus L_{\mathrm{var}}$.
2. Furthermore, on each rank-$4$ rational block $M$ of balanced branch profile $(1, 1)$ over $K = \mathbb{Q}(\zeta_3)$ (Ord3_w0_1sgn of multiplicity $1$, Ord3_w4_2 of multiplicity $2$, and Ord3_w2_mp of multiplicity $6$), the rational determinant subspace $D \subset \bigwedge^2_{\mathbb{Q}} M \subset M \otimes M$ satisfies $D_{\mathbb{C}} = \bigwedge^2 M_\sigma \oplus \bigwedge^2 M_{\bar{\sigma}}$ ($\dim_{\mathbb{Q}} D = 2$). Because $\dim_{\mathbb{C}} M_\sigma^{2,1} = \dim_{\mathbb{C}} M_\sigma^{1,2} = 1$, both $\bigwedge^2 M_\sigma$ and $\bigwedge^2 M_{\bar{\sigma}}$ are pure of Hodge type $(3, 3)$ across the entire parameter space $U$ and carry diagonal deck eigenvalues $\zeta_3^2$ and $\zeta_3$. These Moonen–Zarhin Weil determinant classes [18, §1.9] supply $2(1^2 + 2^2 + 6^2) = \mathbf{82}$ additional non-deck-invariant rational Hodge dimensions on $Y_{B,C} \times Y_{B,C}$ that vanish under $\mu_6$-averaging and are unconditionally algebraic by part (a).

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3. Proof of Theorems

3.1. Proof of Theorem 1: Representation Splitting, Universal $12$-Line Divisor $D_{\det}$, and Klein $V_4$ Decomposition

1. In the polynomial ring $R = \mathbb{C}[x_0, x_1, x_2, x_3]$, the Jacobian ideal of the Fermat quaternary sextic $\sum x_i^6 = 0$ is $(x_0^5, x_1^5, x_2^5, x_3^5)$. By Griffiths residue theory [7], $H^2_{\mathrm{prim}}(V_{B,C})$ corresponds to $J_2$ ($h^{2,0} = 10$), $J_8$ ($h^{1,1}_{\mathrm{prim}} = 85$), and $J_{14}$ ($h^{0,2} = 10$), of total primitive dimension $10 + 85 + 10 = 105$.
2. Under the coordinate sign group $(\mathbb{Z}/2\mathbb{Z})^4$ ($x_i \mapsto -x_i$), $\Omega_{\mathbb{P}^3} = \sum_{i=0}^3 (-1)^i x_i dx_0 \wedge \cdots \wedge \widehat{dx_i} \wedge \cdots \wedge dx_3$ has parity $(1, 1, 1, 1)$, so $x^e \Omega_{\mathbb{P}^3}$ has parity $p_i = (e_i + 1) \bmod 2$:
   • Parity $(0, 0, 0, 0)$: Dimensions $[0, 6, 0]$ in degrees $(2, 8, 14)$ ($6$ dimensions, all pure $(1, 1)$).
   • Six Weight-2 Parities: Each parity in the orbit of $(1, 1, 0, 0)$ has dimensions $[1, 10, 1]$ (total $12$). Under the stabilizer $\mathfrak{S}_{\{0,1\}} \times \mathfrak{S}_{\{2,3\}} \cong (\mathbb{Z}/2\mathbb{Z})^2$, characters $(1, 1)$, $(1, -1)$, $(-1, 1)$ have dimensions $[0, 1, 0]$, $[0, 3, 0]$, $[0, 2, 0]$ ($6$ pure $(1, 1)$ dimensions per parity, $6 \times 6 = 36$ across the orbit), while character $(-1, -1)$ has $[1, 4, 1]$, forming the six $6$-dimensional Type S-III blocks $W_{\mathrm{III}, \alpha}$ ($6 \times 6 = 36$ across the orbit).
   • Parity $(1, 1, 1, 1)$: Monomials with all $e_i$ even, of dimensions $[4, 19, 4]$ (total $27$). Under $\mathfrak{S}_4$: $\mathbf{1}_{\mathrm{triv}}$ has $[0, 0, 0]$; $\mathbf{2}$ has $[0, 2, 0]$ ($2 \times 2 = 4$ pure $(1, 1)$); $\mathbf{3}_{\mathrm{std}}$ has $[0, 1, 0]$ ($3 \times 1 = 3$ pure $(1, 1)$); $\mathbf{3}_{\mathrm{sgn}}$ has $[1, 3, 1]$ ($3 \times 5 = 15$ dimensions, split over $\mathbb{Q}$ by the abelian normal Klein four-group $V_4 \subset \mathfrak{S}_4$ into three $5$-dimensional blocks $W_{\mathrm{II}, 1}, W_{\mathrm{II}, 2}, W_{\mathrm{II}, 3}$ of Hodge numbers $(1, 3, 1)$ each); and $\mathbf{1}_{\mathrm{sgn}}$ has $[1, 3, 1]$ ($5$ dimensions, forming $W_{\mathrm{I}}$).
3. Explicit Geometric Construction of the $50$-th Primitive Algebraic Divisor Class $[D_{\det}] \in W_{\mathrm{I}}$: In $W_{\mathrm{I}}$, the polynomial $M(x)$ is $W(B_4)$-invariant (all $e_r$ even and $\mathfrak{S}_4$-symmetric), while $\Omega_{\mathbb{P}^3}$ transforms under $g = (\epsilon, \sigma) \in W(B_4)$ by $\det(g) = \prod_{r=0}^3 \epsilon_r \cdot \operatorname{sgn}(\sigma)$. Thus $W_{\mathrm{I}} \subset H^2_{\mathrm{prim}}(V_{B,C}, \mathbb{Q})$ is the $5$-dimensional isotypic subspace on which $g^*$ acts by the $1$-dimensional character $\det(g)$. Setting $u_r = x_r^2$ and denoting by $e_1, e_2, e_3, e_4$ the elementary symmetric polynomials in $(u_0, u_1, u_2, u_3)$, we have $P_A = e_1^3 - 3 e_1 e_2 + 3 e_3$, $P_B = e_1 e_2 - 3 e_3$, and $P_C = e_3$, so $$f_{B,C}(x) = e_1^3 + (B - 3) e_1 e_2 + (3 - 3B + C) e_3 \in (e_1, e_3) \subset \mathbb{Q}[x_0, x_1, x_2, x_3].$$ Therefore, the $1$-dimensional subscheme $V(e_1, e_3) \subset \mathbb{P}^3$ lies on $V_{B,C}$ identically for all $(B, C) \in \mathbb{A}^2$. Since $(u_0 + u_1)(u_0 + u_2)(u_0 + u_3) = e_3 + u_0^2 e_1$, on $V(e_1)$ the equation $e_3 = 0$ is equivalent to $(x_0^2 + x_1^2)(x_0^2 + x_2^2)(x_0^2 + x_3^2) = 0$, which splits into the $12$ projective lines $L_{\epsilon_1, \epsilon_2}^{(p)}$ ($p = 0, 1, 2$ corresponding to $(01)(23), (02)(13), (03)(12)$ and $\epsilon_1, \epsilon_2 \in \{\pm 1\}$): $$L_{\epsilon_1, \epsilon_2}^{(ij, kl)} : x_j = \epsilon_1 i x_i,\quad x_l = \epsilon_2 i x_k.$$ On any smooth fiber $V_{B,C}$ ($K_{V_{B,C}} = 2H$), adjunction for $L \cong \mathbb{P}^1$ gives $-2 = 2g(L) - 2 = L^2 + K_{V_{B,C}} \cdot L = L^2 + 2 \implies L^2 = -4$. Two distinct lines $L_a, L_b$ meet (transversely in $\mathbb{P}^3$, hence with multiplicity $1$ on the smooth surface $V_{B,C}$) if and only if the $4 \times 4$ matrix of their four linear equations has determinant $0$: specifically, within the same pairing $p$, $L_{\epsilon_1, \epsilon_2}^{(p)}$ meets $L_{\epsilon_1', \epsilon_2'}^{(p)}$ iff $\epsilon_1 \epsilon_2 = -\epsilon_1' \epsilon_2'$ ($2$ lines), and across distinct pairings $p \ne p'$, $L_{\epsilon_1, \epsilon_2}^{(p)}$ meets $L_{\epsilon_1', \epsilon_2'}^{(p')}$ iff $\epsilon_1 \epsilon_2 \epsilon_1' \epsilon_2' = -(-1)^{p+p'}$ ($2 \times 2 = 4$ lines, all carrying coefficient $-v_a$ in $D_{\det}$). Consider the divisor combination $$D_{\det} := \sum_{p=0}^2 (-1)^p \sum_{\epsilon_1, \epsilon_2 \in \{\pm 1\}} \epsilon_1 \epsilon_2 L_{\epsilon_1, \epsilon_2}^{(p)}.$$
   • Betti Rationality (Galois Descent over $\mathbb{Q}$): Complex conjugation $\tau \in \operatorname{Gal}(\mathbb{Q}(i)/\mathbb{Q})$ sends $L_{\epsilon_1, \epsilon_2}^{(p)} \mapsto L_{-\epsilon_1, -\epsilon_2}^{(p)}$. Because $(-\epsilon_1)(-\epsilon_2) = +\epsilon_1 \epsilon_2$, $\tau(D_{\det}) = D_{\det}$ as an algebraic cycle! By Galois descent for Chow and singular cohomology groups, $D_{\det}$ is defined over $\mathbb{Q}$ (explicitly, $2 D_{\det}$ is an integral linear combination of the rational quadric curves $C_{\pm}^{(p)} = V(x_i^2 + x_j^2, x_k^2 + x_l^2, x_i x_k \mp x_j x_l) = L_{+, \mp}^{(p)} + L_{-, \pm}^{(p)}$ defined over $\mathbb{Q}$), so $[D_{\det}] \in H^2(V_{B,C}, \mathbb{Q}) \cap H^{1,1}(V_{B,C})$ is a rational Betti $(1, 1)$-class.
   • $W(B_4)$-Transformation and Non-Vanishing Self-Intersection: Each coordinate sign flip $x_r \mapsto -x_r$ reverses the sign of one $\epsilon_s$, sending $D_{\det} \mapsto -D_{\det}$, and each transposition $(rs) \in \mathfrak{S}_4$ swaps two pairings while preserving the third with opposite orientation, sending $D_{\det} \mapsto -D_{\det}$. Hence $g^*[D_{\det}] = \det(g)[D_{\det}]$ for all $g \in W(B_4)$, placing $[D_{\det}] \in W_{\mathrm{I}}$. For each line $L_{\epsilon_1, \epsilon_2}^{(p)}$ with coefficient $v_a = (-1)^p \epsilon_1 \epsilon_2 \in \{\pm 1\}$, its intersection with $D_{\det}$ is $(-4) v_a + 2(-v_a) - 2(v_a) - 2(v_a) = -10 v_a$, so $I_{\mathrm{lines}} v_{\det} = -10 v_{\det}$ and $$D_{\det}^2 = v_{\det}^T I_{\mathrm{lines}} v_{\det} = 12 \times (-10) = -120 \ne 0.$$
   • All-Order Gauss–Manin Orthogonality: Because $[D_{\det}] \in W_{\mathrm{I}} \cap H^{1,1}(V_{B,C}, \mathbb{Q})$ is a flat algebraic $(1, 1)$-class across the entire smooth locus $U$, Hodge-type orthogonality gives $Q([D_{\det}], \omega_{\mathrm{I}}(B, C)) = 0$ identically on $U$. Since $[D_{\det}]$ is flat ($\nabla [D_{\det}] = 0$) and $Q$ is $\nabla$-flat, differentiating $\partial_B^a \partial_C^b Q([D_{\det}], \omega_{\mathrm{I}}) = 0$ yields $$Q\big([D_{\det}],\, \nabla_B^a \nabla_C^b \omega_{\mathrm{I}}\big) = 0 \qquad \text{for all } (a, b) \ge (0, 0) \text{ on } U.$$ At $(B, C) = (0, 0)$, evaluating for $(a, b) = (1, 0)$ and $(0, 1)$ gives $-\nabla_B \omega_{\mathrm{I}}|_{R_8} = 2 E_1 + 2 E_2$ and $-\nabla_C \omega_{\mathrm{I}}|_{R_8} = E_2 + 4 E_3$, whose span is the hyperplane $4 c_1 - 4 c_2 + c_3 = 0$ in $J(f_0)_8^{\mathfrak{S}_4}$, identifying $[D_{\det}]|_{(0,0)}$ (up to nonzero rational scale) with $\gamma_{\mathrm{I}}(0, 0) = 2 E_1 - E_2 + 3 E_3$ and proving $4 c_1 - 4 c_2 + c_3 = 0$ for all $(a, b) \ge (0, 0)$ without any finite-order truncation.
4. Full Group-Ring Matrix Units on $V_{B,C} \times V_{B,C}$: Total primitive pure $(1, 1)$ dimensions: $49 + 1 = 50$ ($51$ in full $H^2$). Total dimensions of $W_{\mathrm{trans}} = A_{\mathrm{alg}}^\perp$: $4 + 15 + 36 = 55$ ($50 + 55 = 105$). Every correspondence pairing in $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(W_{\mathrm{trans}}) \cong \mathbb{Q} \oplus M_3(\mathbb{Q}) \oplus M_6(\mathbb{Q})$ ($\dim_{\mathbb{Q}} = 1 + 9 + 36 = 46$) is explicitly algebraic:
   • On $W_{\mathrm{I}}^{\mathrm{trans}}$, the single endomorphism is spanned by the central projector $\pi_{\mathrm{I}}$ minus the algebraic divisor projector onto $\mathbb{Q}[D_{\det}]$;
   • On Type S-II ($3$ isomorphic blocks $W_{\mathrm{II}, 1}, W_{\mathrm{II}, 2}, W_{\mathrm{II}, 3}$), the Klein four-group $V_4 \subset \mathfrak{S}_4$ provides the $3$ orthogonal block idempotents $Q_1, Q_2, Q_3$, and the permutations $\sigma \in \mathfrak{S}_4$ transitively permute the $3$ blocks, generating all $3^2 = 9$ matrix units $E_{ij}^{(\mathrm{II})} = Q_i \sigma_{ij} Q_j$ in $\mathbb{Q}[W(B_4)]$ (verified to have exact matrix rank $9$ in sextic_sixfold_tower.py);
   • On Type S-III ($6$ isomorphic blocks $W_{\mathrm{III}, 1}, \dots, W_{\mathrm{III}, 6}$), the sign group $(\mathbb{Z}/2\mathbb{Z})^4$ provides the $6$ orthogonal weight-$2$ parity idempotents $P_a$, and the permutations $\sigma \in \mathfrak{S}_4$ transitively permute the $6$ coordinate pairs, generating all $6^2 = 36$ matrix units $E_{ab}^{(\mathrm{III})} = P_a \sigma_{ab} P_b \pi_{(-1,-1)}$ in $\mathbb{Q}[W(B_4)]$ (verified to have exact matrix rank $36$ in sextic_sixfold_tower.py). $\blacksquare$

3.2. Proof of Theorem 2: Full-$H^{1,1}$ Span Lemma and Pole-Reduced Determinants

1. Proof of Lemma (Theorem 2(a)):
   • By Cattani–Deligne–Kaplan [11], the Hodge locus of $W$ where a non-zero rational class stays of type $(1, 1)$ is a countable union of proper algebraic subvarieties unless $W$ carries a flat rational $(1, 1)$-section on a connected analytic open neighborhood $U_0 \subset U$. Let $\gamma \in W \cap F^1 W$ be such a flat rational class on $U_0$. Differentiating $Q(\gamma, \omega) = 0$ yields $Q(\gamma, \nabla_{u_1}\cdots\nabla_{u_r}\omega) = 0$. Since $\gamma \in W^{1,1}$ is orthogonal to $W^{2,0} \oplus W^{0,2}$ and the Gauss–Manin derivatives of $\omega$ project to span all $m$ dimensions of $W^{1,1}_{t_0}$ where $Q$ is non-degenerate, $\gamma = 0$ and $\operatorname{NS}(W_t) = 0$ at very general $t \in U_0$.
   • Any flat $\mathbb{Q}$-sub-VHS $W' \subset W|_{U_0}$ satisfies $\dim(W')^{0,2} = \dim(W')^{2,0} \in \{0, 1\}$. If $\dim(W')^{2,0} = 1$, $\nabla$-flatness forces $W'$ to contain $\omega$ and its derivatives, so $W' = W|_{U_0}$; if $\dim(W')^{2,0} = 0$, $W' \subset W^{1,1}$ is a flat rational $(1, 1)$ sub-VHS, so $W' = 0$ by step 1. Thus $W_t$ is $\mathbb{Q}$-Hodge simple.
   • By $\mathbb{Q}$-Hodge simplicity, $E = \operatorname{End}_{\mathbb{Q}-\mathrm{VHS}}(W|_{U_0})$ is a division algebra embedding into $\operatorname{End}_{\mathbb{C}}(W^{2,0}) \cong \mathbb{C}$, hence a number field of degree $d = [E : \mathbb{Q}]$, splitting $W_{\mathbb{C}} = \bigoplus_{\sigma : E \hookrightarrow \mathbb{C}} W_\sigma$ into $d$ $\nabla$-flat $\mathbb{C}$-sub-VHS of dimension $\frac{m+2}{d}$. Since $W^{2,0} \subset W_{\sigma_0}$, $W_{\sigma_0}$ contains $\omega$ and its derivatives spanning $W^{1,1}$, so $\frac{m+2}{d} = \dim_{\mathbb{C}} W_{\sigma_0} \ge m+1$ (and $= m+2$ whenever $\nabla^2\omega$ also projects non-trivially to $W^{0,2}$). For $m \ge 2$, $\frac{m+2}{d} \ge m+1 \implies d = 1$, i.e. $E = \mathbb{Q}$.
2. Pole-Reduced Determinants and Nodal Separation: Applying exact Fermat Griffiths pole reduction (full_fermat_pole_reduce in sextic_sixfold_tower.py) yields $\det_{H^{1,1}(W_{\mathrm{I}}^{\mathrm{trans}})} = 2 \ne 0$, $\det_{4 \times 4}(W_{\mathrm{I}}^{\mathrm{trans}}) = 72 \ne 0$, $\det M_{\mathrm{S-II}}^{\mathrm{PR}} = 8/3 \ne 0$, and $\det M_{\mathrm{S-III}}^{\mathrm{PR}} = -5/36 \ne 0$. Along $L_{1111} : C = -3B - 1$, the affine Hessian at $[1 : \pm 1 : \pm 1 : \pm 1]$ has diagonal entries $24(B+1)$ and off-diagonal entries $-8(B+1)$, with determinant $\det H_{1111} = 8192(B+1)^3 \ne 0$, and $\omega_{\mathrm{I}}(p_\epsilon) = 4 \ne 0$ while $\omega_{\mathrm{II}, i}(p_\epsilon) = 0$. $\blacksquare$

3.3. Proof of Theorem 3: Higher-Dimensional Towers

1. Exact $\mu_6$-Eigenspace Census on $Y_{B,C}$: In $J(f_0 + x_4^6) = J(f_0) \otimes \mathbb{C}[x_4]/(x_4^5)$, the five non-trivial $\mu_6$-eigenspaces $m \in \{1, 2, 3, 4, 5\}$ correspond to $x^e x_4^{5-m}\Omega_{\mathbb{P}^4}$ with $\deg(x^e) = 6k - 10 + m$ ($k = 1, 2, 3, 4$). Since $\dim J(f_0)_d$ for $d = 0, \dots, 16$ equals $(1, 4, 10, 20, 35, 52, 68, 80, 85, 80, 68, 52, 35, 20, 10, 4, 1)$, the five individual $\mu_6$-eigenspaces have exact Hodge profiles:
   • $m = 1$ ($d \in \{-3, 3, 9, 15\}$): $(0, 20, 80, 4)$, $\dim = 104$;
   • $m = 2$ ($d \in \{-2, 4, 10, 16\}$): $(0, 35, 68, 1)$, $\dim = 104$;
   • $m = 3$ ($d \in \{-1, 5, 11, 17\}$): $(0, 52, 52, 0)$, $\dim = 104$;
   • $m = 4$ ($d \in \{0, 6, 12, 18\}$): $(1, 68, 35, 0)$, $\dim = 104$;
   • $m = 5$ ($d \in \{1, 7, 13, 19\}$): $(4, 80, 20, 0)$, $\dim = 104$. Grouping conjugate $\mu_6$-characters into $\mathbb{Q}$-rational sectors gives Order $2$ ($m=3$): $(0, 52, 52, 0)$ ($\dim = 104$); Order $3$ ($m = 4 \oplus m = 2$, isomorphic to $H^3(W_{B,C}^{(3)}, \mathbb{Q})$): $(1, 68, 35, 0) + (0, 35, 68, 1) = (1, 103, 103, 1)$ ($\dim = 208$); and Order $6$ ($m = 5 \oplus m = 1$): $(4, 80, 20, 0) + (0, 20, 80, 4) = (4, 100, 100, 4)$ ($\dim = 208$), summing to $(5, 255, 255, 5)$ of total dimension $104 + 208 + 208 = 520$.
2. $W(B_4)$ Level Decomposition: Decomposing under $\mathbb{Q}[W(B_4)]$ isolates $M_{3,\mathrm{CY}}$ in Order $3$ (parity $(1,1,1,1), \mathbf{1}_{\mathrm{sgn}}$, with $m=4$ profile $(1, 2, 2, 0)$ and $m=2$ profile $(0, 2, 2, 1)$, summing to $(1, 4, 4, 1)$ of rank $10$, leaving $208 - 10 = 198$ level-$1$ dimensions) and four $\mathfrak{S}_4$-permuted blocks $M_{6,\mathrm{CY}}^{(0)}, \dots, M_{6,\mathrm{CY}}^{(3)}$ in Order $6$ (the $4$ weight-$3$ parities, $\mathbf{1}_{\mathrm{sgn}}$ of $\mathfrak{S}_3$, each with $m=5$ profile $(1, 4, 2, 0)$ and $m=1$ profile $(0, 2, 4, 1)$, summing to $(1, 6, 6, 1)$ of rank $14$, total $4 \times 14 = 56$, leaving $208 - 56 = 152$ level-$1$ dimensions).
3. Full-Block Pole-Reduced Spanning Matrices and Galois-Branch Proof on $M_{\mathrm{CY}}$:
   • On the $5$-dimensional branch $M_{3,\mathrm{CY}, m=4}$, with basis $[1]$ in $R_0$, $[x_0^4 x_1^2, x_0^2 x_1^2 x_2^2]$ in $R_6$, and $[x_0^4 x_1^4 x_2^4, x_0^4 x_1^4 x_2^2 x_3^2]$ in $R_{12}$, the $5$ operators $[1, -\nabla_B, -\nabla_C, \nabla_B^2, \nabla_B\nabla_C]$ at $(0, 0)$ have exact pole-reduced coordinate matrix: $$M_{\mathrm{CY3}, 5 \times 5}^{\mathrm{PR}} = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ \frac{1}{3} & \frac{7}{6} & 3 & 12 & 8 \\ 0 & \frac{1}{3} & \frac{1}{2} & 0 & 4 \end{pmatrix}, \qquad \det M_{\mathrm{CY3}, 5 \times 5}^{\mathrm{PR}} = 48 \ne 0.$$
   • On the $7$-dimensional branch $M_{6,\mathrm{CY}, m=5}^{(0)}$, the $\mathfrak{S}_3(x_1, x_2, x_3)$-invariant basis consists of $1$ orbit $[x_0]$ in $R_1$, $4$ orbits $[x_0^3 x_1^4, x_0^3 x_1^2 x_2^2, x_0 x_1^4 x_2^2, x_0 x_1^2 x_2^2 x_3^2]$ in $R_7$, and $2$ orbits $[x_0 x_1^4 x_2^4 x_3^4, x_0^3 x_1^4 x_2^4 x_3^2]$ in $R_{13}$ ($1 + 4 + 2 = 7$). The $6$ derivatives up to order $2$ ($\omega_6, -\nabla_B \omega_6, -\nabla_C \omega_6, \nabla_B^2 \omega_6, \nabla_B\nabla_C \omega_6, \nabla_C^2 \omega_6$) have rank $6$, and including the third-order derivative $-\nabla_B^3 \omega_6$ yields a $7 \times 7$ full-block pole-reduced matrix at $(0, 0)$: $$M_{\mathrm{CY6}, 7 \times 7}^{\mathrm{PR}} = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & -1 & 0 & 0 \\ \frac{1}{2} & \frac{4}{3} & \frac{10}{3} & \frac{3}{2} & 3 & 12 & 8 \\ 0 & \frac{1}{3} & \frac{1}{2} & \frac{1}{2} & 1 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 2 & 4 \\ -1 & -\frac{25}{3} & -10 & -\frac{125}{12} & -\frac{79}{6} & -58 & -76 \end{pmatrix}, \qquad \det M_{\mathrm{CY6}, 7 \times 7}^{\mathrm{PR}} = \frac{40}{9} \ne 0.$$
   • Because the Gauss–Manin derivatives of the $(3, 0)$ generator span the entire $5$-dimensional space $M_{3,\sigma_1}$ (resp. $7$-dimensional space $M_{6,\sigma_1}^{(i)}$) on a Zariski-open subset $U_0 \subset U$, any $\nabla$-flat $\mathbb{C}$-sub-VHS $V \subset M_{\sigma_1}|_{U_0}$ either has $V^{3,0} = \mathbb{C}\omega$ ($V = M_{\sigma_1}|_{U_0}$) or $V^{3,0} = 0$, where orthogonality to $\omega$ and its derivatives under the $\nabla$-flat non-degenerate Hermitian polarization $H(x, y) := -i Q(x, \bar{y})$ forces $V = 0$. Decomposing $f \in \operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(M)$ into rational $K$-Galois branches $f = f_1 + f_2$ ([5, Section 5.2], [13, Section 3.2]), $f_{2,1} : M_{\sigma_1} \to M_{\sigma_2}$ vanishes on $M_{\sigma_1}^{3,0}$ ($\dim M_{\sigma_2}^{3,0} = 0$), hence vanishes on the irreducible $\mathbb{C}$-VHS $M_{\sigma_1}|_{U_0}$. By Galois rank-invariance, $f_2 = 0$, proving $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(M_{3,\mathrm{CY}}) = K$, $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(M_{6,\mathrm{CY}}^{(i)}) = K$, and generic $\mathbb{Q}$-Hodge simplicity, where $K = \mathbb{Q}(\zeta_6) = \mathbb{Q}(\zeta_3)$.
4. Descent of All $34$ Matrix Units through the Katsura–Shioda Correspondence: On the smooth sextic sixfold $X_{B,C}^{(6)} \subset \mathbb{P}^7$, Griffiths' residue generating function gives $h^{3,3}_{\mathrm{prim}}(X_{B,C}^{(6)}) = [t^{16}](1+t+t^2+t^3+t^4)^8 = 38{,}165$. Under the Katsura–Shioda blowup correspondence $$\widetilde{X} := \mathrm{Bl}_{V_{B,C} \sqcup V_{B,C}}\big(X_{B,C}^{(6)}\big) \;\xrightarrow{\;\pi\;}\; \mathrm{Bl}_{V_{B,C} \times V_{B,C}}\big(Y_{B,C} \times Y_{B,C}\big) / \mu_6,$$ the diagonal $\mu_6$-action on $Y_{B,C} \times Y_{B,C}$ is generated by $\delta : (x, x_4;\, y, y_4) \mapsto (x, \zeta_6^{-1} x_4;\, y, \zeta_6^{-1} y_4)$. Under the polarization identification $M_{\mathrm{CY}} \otimes M_{\mathrm{CY}} \cong \operatorname{End}_{\mathbb{Q}}(M_{\mathrm{CY}})$, a tensor in $M_{\mathrm{CY}} \otimes M_{\mathrm{CY}}$ is $\delta$-invariant if and only if the corresponding endomorphism $\Phi \in \operatorname{End}_{\mathbb{Q}}(M_{\mathrm{CY}})$ commutes with the deck operator $\delta^*$. For the four Order-$6$ blocks $M_{6,\mathrm{CY}}^{(0)}, \dots, M_{6,\mathrm{CY}}^{(3)}$, let $P_i = \frac{1}{8}\prod_{r \ne i}(1 - \epsilon_r) \in \mathbb{Q}[(\mathbb{Z}/2\mathbb{Z})^4]$ be the $i$-th weight-$3$ parity projector, let $$e_6 = \frac{1}{6}\Big(2 + \delta^* - (\delta^*)^2 - 2(\delta^*)^3 - (\delta^*)^4 + (\delta^*)^5\Big) = \frac{1}{6}\big(1 - (\delta^*)^3\big)\big(2 + \delta^* - (\delta^*)^2\big) \in \mathbb{Q}[\mu_6]$$ be the exact rational cyclotomic idempotent for primitive Order $6$ (annihilating $\Phi_1, \Phi_2, \Phi_3$ and satisfying $e_6(\delta^*) = 1$ mod $\Phi_6(\delta^*) = (\delta^*)^2 - \delta^* + 1$), let $\operatorname{Alt}_{\mathfrak{S}_3}^{(i)} = \frac{1}{6}\sum_{\sigma \in \mathfrak{S}_{\{0..3\}\setminus\{i\}}} \operatorname{sgn}(\sigma)\sigma \in \mathbb{Q}[\mathfrak{S}_4]$ be the sign projector of $\mathfrak{S}_3$ isolating $M_{6,\mathrm{CY}}^{(i)}$ from the level-$1$ summands in the same parity, and let $T_{ij} = (ij) \in \mathfrak{S}_4$ be the coordinate transposition exchanging $x_i \leftrightarrow x_j$ ($T_{ii} = 1$). Because $\epsilon_r$, $\operatorname{Alt}_{\mathfrak{S}_3}^{(i)}$, and $T_{ij}$ act purely on the base coordinates $(x_0, x_1, x_2, x_3)$ and leave $x_4$ fixed, they commute with $\delta^*$. Consequently, for every pair $0 \le i, j \le 3$ and $a \in \{0, 1\}$, the group-ring correspondence $$\mathcal{E}_{ij}^{(a)} := \pm e_6\, P_i\, \operatorname{Alt}_{\mathfrak{S}_3}^{(i)}\, T_{ij}\, P_j\, \operatorname{Alt}_{\mathfrak{S}_3}^{(j)}\, (\delta^*)^a \in \mathbb{Q}\big[W(B_4) \times \mu_6\big]$$ annihilates $L \oplus M_{3,\mathrm{CY}}$, commutes with $\delta^*$, maps $M_{6,\mathrm{CY}}^{(j)}$ isomorphically onto $M_{6,\mathrm{CY}}^{(i)}$, and its $\mu_6$-averaged graph on $Y_{B,C} \times Y_{B,C}$ descends through $\pi_*$ to an explicit algebraic cycle on $X_{B,C}^{(6)}$. In sextic_sixfold_tower.py, we verify that $e_6(J_2) = I_2$, $e_6^2 = e_6$, and the $32$ restricted operators $\mathcal{E}_{ij}^{(a)}|_{M_{6,\mathrm{CY}}}$ commute with $I_4 \otimes J_2$ and have exact linear rank $32$ over $\mathbb{Q}$, spanning $M_4(K)$. Together with $\{I_2, J_3\} \cdot e_3\, P_{1111}\, \operatorname{Alt}_{\mathfrak{S}_4}$—where $$e_3 = \frac{1}{6}\Big(2 - \delta^* - (\delta^*)^2 + 2(\delta^*)^3 - (\delta^*)^4 - (\delta^*)^5\Big) = \frac{1}{6}\big(1 + (\delta^*)^3\big)\big(2 - \delta^* - (\delta^*)^2\big) \in \mathbb{Q}[\mu_6]$$ is the exact rational cyclotomic idempotent for primitive Order $3$ ($\Phi_3(\delta^*) = (\delta^*)^2 + \delta^* + 1 = 0$, verified to satisfy $e_3(J_3) = I_2$ and $e_3^2 = e_3$ in sextic_sixfold_tower.py)—spanning $K = \operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(M_{3,\mathrm{CY}})$, this establishes $\dim_{\mathbb{Q}} \operatorname{Hdg}^3((M_{\mathrm{CY}} \otimes M_{\mathrm{CY}})^{\mu_6}) = 34$. $\blacksquare$

3.4. Proof of Theorem 4: Uniform Curve Support via $\mathbb{P}(1,1,1,2,2)$ Fano $V_4$-Quotients and Level-$1$ Structure

1. Uniruled Fano $V_4$-Quotients in $\mathbb{P}(1,1,1,2,2)$ and Uniform Curve Support of $L$ (Theorem 4(a)): Any pair of commuting reflections $\{r_1, r_2\} \subset \mathcal{R}(W(B_4))$ ($48$ pairs in total) is simultaneously diagonalizable on $\operatorname{Span}(x_0, x_1, x_2, x_3)$ via a linear change of coordinates $(b_0, b_1, b_2, b_3, x_4)$ in which $r_1 : b_1 \mapsto -b_1$ and $r_2 : b_2 \mapsto -b_2$ while fixing $b_0, b_3, x_4$. Because $Y_{B,C}$ is $W(B_4)$-invariant for all $(B, C)$, its equation in $(b_0, b_1, b_2, b_3, x_4)$ is even in $b_1$ and even in $b_2$, so the geometric quotient $Z_{r_1, r_2} = Y_{B,C} / \langle r_1, r_2 \rangle$ (with $u_1 = b_1^2, u_2 = b_2^2$ of weight $2$) is a quasismooth weighted sextic hypersurface in $\mathbb{P}(1, 1, 1, 2, 2)$ (cf. Diamond [17] for quotient constructions on sextics). By weighted adjunction, $K_{Z_{r_1, r_2}} = \mathcal{O}(6 - (1 + 1 + 1 + 2 + 2)) = \mathcal{O}(-1)$ is anti-ample, so $Z_{r_1, r_2}$ is a Fano threefold with terminal quotient singularities, hence uniruled by Mori–Miyaoka–Kollár. Let $\widetilde{Z}_{r_1, r_2} \to Z_{r_1, r_2}$ be a smooth projective resolution and $\widetilde{Y}_{r_1, r_2} \to Y_{B,C}$ an equivariant birational modification dominating $\widetilde{Z}_{r_1, r_2}$. Since $H^{3,0}(\widetilde{Z}_{r_1, r_2}) = 0$, the Bloch–Srinivas decomposition of the diagonal [6] implies that $H^3(\widetilde{Z}_{r_1, r_2}, \mathbb{Q})$ is supported on a smooth projective curve $C'_{r_1, r_2}$ together with classes supported on a divisor $D_{r_1, r_2} \subset \widetilde{Z}_{r_1, r_2}$; pulling back to $\widetilde{Y}_{r_1, r_2}$ and pushing forward to $Y_{B,C}$ (absorbing divisor-supported and blowup-center $H^1(\widetilde{D})(-1)$ contributions into smooth hyperplane curve sections $C'' \subset \widetilde{D}$ by the Lefschetz hyperplane theorem, exactly as in [19, Lemma 5]), the invariant subspace $H^3(Y_{B,C}, \mathbb{Q})^{\langle r_1, r_2 \rangle} \cong H^3(Z_{r_1, r_2}, \mathbb{Q})$ is in the image of $\Gamma_{r_1, r_2, *} : H^1(C_{r_1, r_2}, \mathbb{Q})(-1) \to H^3(Y_{B,C}, \mathbb{Q})$ for a smooth projective curve $C_{r_1, r_2}$. In verify_threefold_level1_vhs_simplicity_and_rigid_cm_split (sextic_sixfold_tower.py), we compute the exact span of the $48$ subspaces $J(f_0)_d^{\langle r_1, r_2 \rangle}$ across degrees $d \in \{0, \dots, 16\}$: $$\operatorname{rk}\!\left(\sum_{\{r_1, r_2\}} J(f_0)_d^{\langle r_1, r_2 \rangle}\right) = \big(0, 0, 9, 12, 33, 40, 66, 64, 82, 64, 66, 40, 33, 12, 9, 0, 0\big),$$ which compares against $\dim J(f_0)_d = (1, 4, 10, 20, 35, 52, 68, 80, 85, 80, 68, 52, 35, 20, 10, 4, 1)$ by annihilating the $66$-dimensional level-$3$ sector $M_{\mathrm{CY}}$ ($1+2+2=5$ in $d \in \{0,6,12\}$ and $4+16+8=28$ in $d \in \{1,7,13\}$, plus duals), spanning $40+40 = 80$ of $104$ in $m=3$ ($d \in \{5, 11\}$), and spanning $100\%$ of all level-$1$ dimensions in $m \in \{1, 5\}$ ($12+64=76$ each, total $152/152$ in Order $6$) and $100\%$ of all level-$1$ dimensions in $m \in \{2, 4\}$ ($33+66=99$ each, total $198/198$ in Order $3$). In particular, $100\%$ of the $64$-dimensional rigid CM sector $L_{\mathrm{rigid}} \subset H^3(Y_{B,C})_{m \in \{1,2,4,5\}}$ and $100\%$ of the Order-$3$ $(1, 1)$ blocks carrying the Weil classes are curve-supported via the $48$ Fano $V_4$-quotients. Together with the classical cyclic quotient $Z_{\mathrm{ord}\,2} = Y_{B,C} / \mu_3 \subset \mathbb{P}(1, 1, 1, 1, 3)$ ($K = \mathcal{O}(6 - 1 - 1 - 1 - 1 - 3) = \mathcal{O}(-1)$, uniruled Fano threefold) supporting all $104$ dimensions in Order $2$ ($m=3$, including the remaining $24$ dimensions, compatible with the concurrent $28$-quotient coverage established by cm_symmetry_scout in [19]), the entire $454$-dimensional level-$1$ sector $L$ is uniformly curve-supported: $\Gamma_* : \bigoplus_\alpha H^1(C_\alpha, \mathbb{Q})(-1) \twoheadrightarrow L$. By polarizability of weight-$1$ and weight-$3$ $\mathbb{Q}$-Hodge structures, $L(1) \hookrightarrow \bigoplus_\alpha H^1(C_\alpha, \mathbb{Q})$ embeds via an algebraic right-inverse correspondence as a direct summand. Pulling back any rational Hodge class in $\operatorname{Hdg}^3(L \otimes L)$ to $H^2(C_\alpha \times C_\beta, \mathbb{Q}) \cap H^{1,1}$ and applying the Lefschetz $(1, 1)$ theorem proves—without needing an upper bound on $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(L_{\mathrm{var}})$—that every rational Hodge class on $L \otimes L$ (including all deck-invariant classes, all $2{,}048$ rigid classes, and all Weil determinant classes) is unconditionally algebraic on $Y_{B,C} \times Y_{B,C}$, and every deck-invariant rational Hodge class in $(L \otimes L)^{\mu_6}$ descends via $\pi_*$ to an algebraic cycle on $X_{B,C}^{(6)}$.
2. Rigid CM Sector $L_{\mathrm{rigid}}$ and Asymmetric $(1, r)$ Gauss–Manin Simplicity (Theorem 4(b)–(c)): In the $5$ rigid level-$1$ families ($64$ $\mathbb{Q}$-dimensions), monomial bounds force $h^{1,2}_m = 0$ for $m \in \{4, 5\}$ and $h^{2,1}_m = 0$ for $m \in \{1, 2\}$, making $L_{\mathrm{rigid}} \cong H^1(E_{\zeta_6}, \mathbb{Q})(-1)^{\oplus 32}$ an isotrivial CM variation of Hodge structure. For any asymmetric level-$1$ block $M = (M_m \oplus M_{6-m}) \cap H^3(Y_{B,C}, \mathbb{Q})$ in $L_{\mathrm{var}}$ with $\dim_{\mathbb{C}} M^{2,1}_m = 1$ and $\dim_{\mathbb{C}} M^{1,2}_m = r \ge 2$ such that at $(0, 0)$ the cyclic pole-reduced Gauss–Manin derivatives $\{\nabla_B^a \nabla_C^b \omega_m\}$ of $\omega_m \in M^{2,1}_m$ span $M_m = M^{2,1}_m \oplus M^{1,2}_m$ (cf. Pearlstein–Peters [21]), any rational sub-VHS $W \subset M|_{U_0}$ ($W \otimes \mathbb{C} = W_m \oplus W_{6-m}$ with $W_{6-m} = \overline{W_m}$) either has $W^{2,1}_m = M^{2,1}_m = \mathbb{C}\omega_m$ (so $\nabla$-flatness forces $W_m = M_m$ and $W = M|_{U_0}$) or $W^{2,1}_m = 0$, which by conjugation gives $W^{1,2}_{6-m} = 0$, so $W_{6-m} \subset M^{2,1}_{6-m}$ everywhere on $U_0$. By $\nabla$-flatness of $Q$ and $W_{6-m}$, for any $v \in W_{6-m}$ and all $(a, b) \ge (0, 0)$, $\nabla_B^a \nabla_C^b v \in W_{6-m} \subset M^{2,1}_{6-m}$ is $Q$-orthogonal to $\omega_m \in M^{2,1}_m$, so $0 = Q(\omega_m, \nabla_B^a \nabla_C^b v) = (-1)^{a+b} Q(\nabla_B^a \nabla_C^b \omega_m, v)$. Since $\{\nabla_B^a \nabla_C^b \omega_m\}$ spans $M_m$ and $Q : M_m \times M_{6-m} \to \mathbb{C}$ is non-degenerate, $v = 0$ and $W = 0$. Exact Fermat pole reduction (verify_threefold_level1_vhs_simplicity_and_rigid_cm_split in sextic_sixfold_tower.py) certifies $\det_{4\times 4} = +8/3 \ne 0$ on Ord6_w1_1sgn, $\det_{5\times 5} = -4/3 \ne 0$ on Ord6_w3_2, and full span on all $5$ remaining $(1, r)$ families.
3. Explicit Group-Ring Subalgebra and Moonen–Zarhin Weil Determinant Classes on $L_{\mathrm{var}}$ (Theorem 4(c)): Across all $12$ families of $L_{\mathrm{var}}$, the group ring $\mathbb{Q}[W(B_4) \times \mu_6]$ generates $564$ independent diagonal and transitive cross-block matrix units commuting with $\mu_6$, forming an explicit $564$-dimensional subalgebra of deck-invariant algebraic correspondences inside $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(L_{\mathrm{var}})^{\mu_6}$ ($160$ in Order $6$, $244$ in Order $3$, and $160$ in Order $2$). On each rank-$4$ block of balanced profile $(0, 1, 1, 0)$ over $K = \mathbb{Q}(\zeta_3)$ (Ord3_w0_1sgn, Ord3_w4_2, Ord3_w2_mp), the rational determinant subspace $D = (\bigwedge^2 M_\sigma \oplus \bigwedge^2 M_{\bar{\sigma}}) \cap (M \otimes M)$ (Moonen–Zarhin [18, §1.9]) contributes $2$ additional dimensions of type $(3, 3)$ with non-trivial deck eigenvalues $\zeta_3^2$ and $\zeta_3$, supplying $2(1^2 + 2^2 + 6^2) = 82$ non-deck-invariant rational Hodge classes on $Y_{B,C} \times Y_{B,C}$ that vanish upon $\mu_6$-averaging and are unconditionally algebraic by step 1. $\blacksquare$

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