The Rational Hodge Conjecture for $W(B_4)$-Symmetric General-Type Sextic Surface Squares, Universal $12$-Line Divisors, and Calabi–Yau Sectors of Cyclic Threefold Squares and Split Sixfolds

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 for the general-type surface square $V_{B,C} \times V_{B,C}$ and for the complete Order-$1$ and level-$3$ Calabi–Yau sectors of the threefold and sixfold tower 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 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 $x_4$, all $2 + 2 \cdot 4^2 = 34$ group-ring matrix units ($2$ on $M_{3,\mathrm{CY}}$ plus $8$ diagonal and $24$ off-diagonal on $M_{6,\mathrm{CY}}^{(0..3)}$) commute with the deck transformation and survive the diagonal $\mu_6$-projection: $$\operatorname{Hdg}^3\!\left(\big(M_{\mathrm{CY}} \otimes M_{\mathrm{CY}}\big)^{\mu_6}\right) \;\cong\; K \oplus M_4(K), \qquad \dim_{\mathbb{Q}} = 2 + 2 \cdot 4^2 = 34.$$ All $34$ rational Hodge dimensions in $(M_{\mathrm{CY}} \otimes M_{\mathrm{CY}})^{\mu_6}$ descend through the Katsura–Shioda correspondence as explicit algebraic cycles on $X_{B,C}^{(6)}$, and all $2 \times 66 \times 454 = 59{,}928$ cross-sector dimensions with the $454$-dimensional level-$1$ summand $L \subset H^3(Y_{B,C}, \mathbb{Q})$ vanish identically, reducing the full rational Hodge conjecture for $W_{B,C}^{(3) 2}$, $Y_{B,C}^2$, and $X_{B,C}^{(6)}$ strictly to geometric coniveau-$1$ (curve-correspondence) support of the weight-$3$ level-$1$ summand $L$ [6, 15].

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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) Reduction of the Remaining Level-$1$ Sector to Curve Support: In the Katsura–Shioda blowup decomposition $\mathrm{Bl}_{V_{B,C} \sqcup V_{B,C}}(X_{B,C}^{(6)}) \cong \mathrm{Bl}_{V_{B,C} \times V_{B,C}}(Y_{B,C} \times Y_{B,C}) / \mu_6$, all rational Hodge classes in the Order-$1$ surface-square sector $(H^2_{\mathrm{prim}}(V_{B,C})^{\otimes 2})(-1)$ ($2{,}546$ primitive classes) and in the level-$3$ Calabi–Yau sector $(M_{\mathrm{CY}} \otimes M_{\mathrm{CY}})^{\mu_6}$ ($34$ classes) are unconditionally algebraic at very general $(B, C) \in U$. For any $\mathbb{Q}[W(B_4) \times \mu_6]$-sub-Hodge structure $L' \subset L$ that has geometric coniveau $1$ (i.e., is supported on a family of smooth projective curves via an algebraic correspondence, as in [6, 15]), polarization semisimplicity yields an algebraic retraction onto $L'(1)$ and the Lefschetz $(1, 1)$ theorem on the curve Jacobian square implies that all rational Hodge classes in $L' \otimes L'$ are algebraic.

---

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 $\epsilon_1 \epsilon_2 = \epsilon_1' \epsilon_2'$ ($2 \times 2 = 4$ lines). 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)):
   • Let $\gamma \in W \cap F^1 W$ be 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 and Level-$1$ Curve-Support Condition: 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 $M_{\mathrm{CY}} \otimes M_{\mathrm{CY}} \cong \operatorname{End}_{\mathbb{Q}}(M_{\mathrm{CY}})$ (where $\delta$ acts on $M_{3,\mathrm{CY}}$ as $\zeta_6^2 = \zeta_3$ and on $M_{6,\mathrm{CY}}^{(i)}$ as $\zeta_6$, both preserving the intersection pairing $Q(\delta u, \delta v) = Q(u, v)$), 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 and let $T_{ij} = (ij) \in \mathfrak{S}_4$ be the coordinate transposition exchanging $x_i \leftrightarrow x_j$ ($T_{ii} = 1$). Because both $\epsilon_r$ 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 the deck generator $\delta$ on $Y_{B,C}$. Consequently, for every pair $0 \le i, j \le 3$ and $a \in \{0, 1\}$, the group-ring correspondence $$\mathcal{E}_{ij}^{(a)} := \pm P_i T_{ij} P_j (\delta^*)^a \in \mathbb{Q}\big[W(B_4) \times \mu_6\big]$$ commutes with $\delta^*$, maps $M_{6,\mathrm{CY}}^{(j)}$ isomorphically onto $M_{6,\mathrm{CY}}^{(i)}$, and its graph on $Y_{B,C} \times Y_{B,C}$ is stabilized by the diagonal $\mu_6$-action (after $\mu_6$-averaging $\frac{1}{6}\sum_{k=0}^5 (\delta^k \times \delta^k)_* [\Gamma_g]$), hence descends through the finite quotient $\pi_*$ and blowdown to an explicit algebraic cycle on $X_{B,C}^{(6)}$. In sextic_sixfold_tower.py, we construct the explicit $8 \times 8$ rational matrices of these $32$ operators $\mathcal{E}_{ij}^{(a)}$ on the $4$-copy multiplicity space $\mathbb{Q}^4 \otimes_{\mathbb{Q}} K$ and verify that every operator commutes with $I_4 \otimes J_2$ and that the $32$ operators ($8$ diagonal + $24$ off-diagonal) have exact linear rank $32$ over $\mathbb{Q}$, spanning $M_4(K)$. Together with the $2$ operators $\{I_2, J_2\}$ spanning $K = \operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(M_{3,\mathrm{CY}})$, this establishes: $$\operatorname{Hdg}^3\!\left(\big(M_{\mathrm{CY}} \otimes M_{\mathrm{CY}}\big)^{\mu_6}\right) \;\cong\; K \oplus M_4(K), \qquad \dim_{\mathbb{Q}} = 2 + 2 \cdot 4^2 = 34,$$ all $34$ dimensions of which are unconditionally algebraic on both $Y_{B,C} \times Y_{B,C}$ and $X_{B,C}^{(6)}$. Meanwhile, the cross-term $(M_{\mathrm{CY}} \otimes L) \oplus (L \otimes M_{\mathrm{CY}})$ has zero rational Hodge classes by Schur's lemma. On the remaining level-$1$ summand $L \otimes L$, abstract level-$1$ Hodge type $(2,1) \oplus (1,2)$ implies via the classical Riemann relations that $L(1)$ is polarized-isomorphic to $H^1$ of an abelian variety, and for any sub-Hodge structure $L' \subset L$ of geometric coniveau $1$ (admitting an algebraic correspondence from a curve $C$, as analyzed in [6, 15]), polarization semisimplicity yields an algebraic splitting $L'(1) \hookrightarrow H^1(C, \mathbb{Q}) \twoheadrightarrow L'(1)$ and Lefschetz $(1, 1)$ on $C \times C$ establishes algebraicity of all Hodge classes in $L' \otimes L'$. These family-specific results do not resolve the general Hodge conjecture. $\blacksquare$

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