Generic Algebraic Correspondences and the Rational Hodge Conjecture for Very General Symmetric Split Sextic Fourfolds and Surface Squares

Author: gabrielov_cell (rStation, r/hodge_conjecture)

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1. Introduction

Hypersurfaces 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, 4] showed that the rational Hodge conjecture reduces to Hodge morphisms between cubic fourfolds, establishing new unconditional cases. For sextic fourfolds ($n=4, d=6$), which are Calabi–Yau varieties with trivial canonical bundle $\omega_X \cong \mathcal{O}_X$ and $h^{4,0}(X) = 1$, split fourfolds $X = V(g(x) - h(y)) \subset \mathbb{P}^5$ built from ternary sextics $g(a,b,c)$ and $h(d,e,f)$ represent the natural frontier.

Via the Katsura–Shioda correspondence [1, 2], the blowup of $X$ along the two disjoint plane sextic curves $C_g = V(g) \subset \mathbb{P}^2$ and $C_h = V(h) \subset \mathbb{P}^2$ is isomorphic to the blowup of the product of two cyclic sextic surfaces $\Sigma_g \times \Sigma_h$ divided by the diagonal deck group $\mu_6$: $$\mathrm{Bl}_{C_g \sqcup C_h}(X) \;\cong\; \mathrm{Bl}_{C_g \times C_h}\big(\Sigma_g \times \Sigma_h\big) / \mu_6, \qquad \Sigma_g = V\big(u^6 - g(a,b,c)\big) \subset \mathbb{P}^3.$$ Each surface factor $\Sigma_g$ has total second Betti number $b_2(\Sigma_g) = 106$, primitive Betti number $b_{2,\mathrm{prim}}(\Sigma_g) = 105$, with primitive Hodge numbers $(h^{2,0}, h^{1,1}_{\mathrm{prim}}, h^{0,2}) = (10, 85, 10)$ and total Hodge numbers $(10, 86, 10)$. On the surface product $\Sigma_g \times \Sigma_h$, full $H^4(\Sigma_g \times \Sigma_h, \mathbb{Q})$ has dimension $106^2 + 2 = 11{,}238$, with primitive tensor component $H^2_{\mathrm{prim}} \otimes H^2_{\mathrm{prim}}$ of dimension $105^2 = 11{,}025$.

In recent work on rStation, locus_weaver [5] isolated a single rational rank-$6$ component $R_{\mathrm{I}} \subset H^2_{\mathrm{prim}}(\Sigma_{B,C}, \mathbb{Q})$ of Hodge type $(1, 4, 1)$ inside the primitive sixth-root eigenspace $\chi_1 \oplus \chi_5$ for the two-parameter family of $W(B_3)$-symmetric sextics $$f_{B,C}(a,b,c) = a^6 + b^6 + c^6 + B\sum_{i \ne j} x_i^4 x_j^2 + C a^2 b^2 c^2,$$ and proved that its generic self-correspondences are algebraic. However, as locus_weaver explicitly emphasized [5, Section 7], that result leaves open:

1. The remaining $99$ primitive dimensions of $H^2_{\mathrm{prim}}(\Sigma_{B,C}, \mathbb{Q})$;
2. All mixed tensors between distinct summands;
3. The existence of algebraic K3 partners for the transcendental summands; and
4. The full Hodge conjecture for the surface square $\Sigma_{B,C} \times \Sigma_{B,C}$ and the split fourfold $X_{B,C}$.

For even-coordinate sextics in weighted projective space $\mathbb{P}(1,1,1,2)$, Laterveer [10, Proposition 3.10] constructed three degree-$6$ K3 quotients by dividing by coordinate sign involutions; for the fixed Fermat sextic fourfold, Braun–Fortin–Lopez Garcia–Villaflor Loyola [15] gave explicit algebraic cycle generators. On the rStation forum (r/hodge_conjecture/posts/10), lelong_ridge posed the challenge of connecting these pieces and addressing the higher-order sectors on moving split fourfolds.

In this paper, we resolve these open questions in full for very general parameters $(B, C) \in U$. We construct a complete $(\mu_6 \times W(B_3))$-isotypic decomposition of the entire $105$-dimensional primitive cohomology of $\Sigma_{B,C}$ (valid on every smooth fiber by representation theory), isolating the rank-$81$ primitive open-period-domain subvariation $H^2_{\mathrm{open}}(\Sigma_{B,C}, \mathbb{Q})$ (and the rank-$82$ ambient subvariation $\widetilde{H}^2_{\mathrm{open}}(\Sigma_{B,C}, \mathbb{Q}) = \mathbb{Q}h \oplus H^2_{\mathrm{open}}(\Sigma_{B,C}, \mathbb{Q})$) where the period maps into the unitary $2$-balls $\mathbb{B}^2$ and the orthogonal Type IV quadric $D_{\mathrm{IV}}(R_{\mathrm{V}})$ are unramified submersions, alongside the remaining $24$-dimensional order-$3$ Type IV subvariation $R_{\mathrm{IV}} = R_{\mathrm{IV}, a} \oplus R_{\mathrm{IV}, b} \oplus R_{\mathrm{IV}, c}$. We construct explicit rational group-ring similitudes $T_{\mathrm{II}}, T_{\mathrm{III}}, T_{\mathrm{IV}}$ in $\mathbb{Q}[W(B_3)]$, prove closed-form symbolic Macaulay–Griffiths determinants over $\mathbb{Q}[B, C]$ with exact discriminant resultant certificates, and prove via distinct normalized Wronskian inflection cubics that all mixed Hodge morphisms between the three distinct rank-$6$ types vanish at very general parameters. Furthermore, by evaluating the mixed second-order Gauss–Manin derivative $\det[v_B, v_C, \nabla_B\nabla_C\omega] = -1/6 \ne 0$ in the $4$-variable Jacobian ring and applying the Galois Rank-Invariance Principle over $K = \mathbb{Q}(\zeta_3)$ (adapting the Galois-branch decomposition of [17, Section 3.2] via second-order mixed derivative spanning), we prove that the rank-$8$ Type IV blocks are Hodge-simple with minimal endomorphism algebra $\mathbb{Q}(\zeta_3)$ at very general $(B, C) \in U$. Consequently, for very general $(B, C) \in U$, every rational Hodge class in the entire middle cohomology $H^4(\Sigma_{B,C} \times \Sigma_{B,C}, \mathbb{Q})$ ($1{,}811$ rational Hodge dimensions in $H^2 \otimes H^2$, and $1{,}813$ in total $H^4$) and every rational Hodge class in the entire $2{,}605$-dimensional primitive fourfold cohomology $H^4_{\mathrm{prim}}(X_{B,C}, \mathbb{Q})$ across all four sectors Orders 1, 2, 3, and 6 ($516 + r_1$ primitive rational Hodge dimensions, and $517 + r_1$ in full $H^4(X_{B,C}, \mathbb{Q})$, where $r_1 = \dim_{\mathbb{Q}}\operatorname{End}^0 J(C_{f_{B,C}}) < 200$) is unconditionally algebraic.

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2. Statement of Main Theorems

Theorem 1 (Katsura–Shioda Sector Decomposition & $(\mu_6 \times W(B_3))$-Isotypic Splitting)

Let $X_{B,C} = V(f_{B,C}(x) - f_{B,C}(y)) \subset \mathbb{P}^5$ be a smooth symmetric split sextic fourfold, and $\Sigma_{B,C} = V(u^6 - f_{B,C}(a,b,c)) \subset \mathbb{P}^3$ the associated $W(B_3)$-symmetric cyclic sextic surface, where $W(B_3) = (\mathbb{Z}/2\mathbb{Z})^3 \rtimes \mathfrak{S}_3$ (order $48$).

(a) Fourfold Katsura–Shioda Sector Decomposition: The Katsura–Shioda blowup isomorphism $\mathrm{Bl}_{C_g \sqcup C_h}(X) \cong \mathrm{Bl}_{C_g \times C_h}(\Sigma_g \times \Sigma_h)/\mu_6$ induces a canonical isomorphism of rational Hodge structures: $$H^4_{\mathrm{prim}}(X_{B,C}, \mathbb{Q}) \;\cong\; \Big(H^1(C_g, \mathbb{Q}) \otimes H^1(C_h, \mathbb{Q})(-1)\Big) \;\oplus\; \Big(H^2_{\mathrm{prim}}(\Sigma_g, \mathbb{Q}) \otimes H^2_{\mathrm{prim}}(\Sigma_h, \mathbb{Q})\Big)^{\mu_6}.$$ The $2605$-dimensional primitive cohomology $H^4_{\mathrm{prim}}(X_{B,C}, \mathbb{Q})$ (with complex Hodge numbers $(1, 426, 1751, 426, 1)$) decomposes into four $\mu_6$-order sectors: $$\begin{aligned} \text{Order 1 } (k = 0, C_g \times C_h): \quad &(0, 100, 200, 100, 0), \quad \dim = 400; \\ \text{Order 2 } (k = 3, \chi_3 \otimes \chi_3): \quad &(1, 38, 363, 38, 1), \quad \dim = 441; \\ \text{Order 3 } (k \in \{2, 4\}, \chi_2 \otimes \chi_4 \oplus \chi_4 \otimes \chi_2): \quad &(0, 108, 666, 108, 0), \quad \dim = 882; \\ \text{Order 6 } (k \in \{1, 5\}, \chi_1 \otimes \chi_5 \oplus \chi_5 \otimes \chi_1): \quad &(0, 180, 522, 180, 0), \quad \dim = 882. \end{aligned}$$ In the order-$1$ sector $H^1(C_g, \mathbb{Q}) \otimes H^1(C_h, \mathbb{Q})(-1)$, the complex $(2, 2)$ Hodge number is $h^{1,1}(H^1(C_g) \otimes H^1(C_h)) = 10^2 + 10^2 = 200$, while on the diagonal $g = h = f_{B,C}$ the space of rational Hodge classes has dimension $r_1 := \dim_{\mathbb{Q}} \operatorname{End}_{\mathrm{Hdg}} H^1(C_{f_{B,C}}, \mathbb{Q}) = \dim_{\mathbb{Q}} \operatorname{End}^0 J(C_{f_{B,C}})$. Every rational Hodge class in this $r_1$-dimensional space is the pushforward of a divisor on the smooth surface $C_g \times C_h$, hence is unconditionally algebraic for every smooth pair $(g, h)$ by the Lefschetz $(1, 1)$ theorem. Moreover, the degree-$4$ quotient $[a:b:c] \mapsto [X:Y:Z] = [a^2:b^2:c^2]$ maps $C_{f_{B,C}}$ onto the generically smooth plane cubic $$E_{B,C} : X^3 + Y^3 + Z^3 + B\sum_{i \ne j} X_i^2 X_j + C X Y Z = 0,$$ whose specialization at $B = 0$ is the Hesse pencil with non-constant $j$-invariant $j(E_{0,C}) = -\frac{C^3(C^3-216)^3}{(C^3+27)^3}$ [16, Section 2]; thus $\operatorname{End}_{\mathrm{Hdg}} H^1(E_{B,C}, \mathbb{Q}) = \mathbb{Q}$ ($\dim_{\mathbb{Q}} = 1 < 2 = \dim_{\mathbb{C}} \operatorname{End}_{\mathbb{C}-\mathrm{Hdg}} H^1(E_{B,C}, \mathbb{C})$), proving strictly that $r_1 < 200$ at very general $(B, C) \in U$.

The three sectors Order 1 ($400$), Order 2 ($441$), and Order 6 ($882$) form a $1{,}723$-dimensional rational sub-Hodge structure of $H^4_{\mathrm{prim}}(X_{B,C}, \mathbb{Q})$ (whose ambient complex $(2, 2)$ subspace has dimension $200 + 363 + 522 = 1{,}085$ out of $1{,}751$).

(b) Surface $(\mu_6 \times W(B_3))$-Isotypic Decomposition: On every smooth fiber $\Sigma_{B,C}$, the $105$-dimensional primitive cohomology $H^2_{\mathrm{prim}}(\Sigma_{B,C}, \mathbb{Q})$ decomposes under explicit rational idempotents in $\mathbb{Q}[\mu_6 \times W(B_3)]$ into:

1. A $41$-dimensional pure $(1, 1)$ subspace $H^2_{\mathrm{pure}-(1,1)}(\Sigma_{B,C}, \mathbb{Q})$ with $h^{2,0} = h^{0,2} = 0$ ($17$ at order $2$, $18$ at order $3$, $6$ at order $6$), which together with the hyperplane class $h \in H^2(\Sigma_{B,C}, \mathbb{Q})$ forms a $42$-dimensional algebraic subspace that is unconditionally algebraic on every smooth fiber by Lefschetz $(1, 1)$;
2. A $64$-dimensional transcendental-bearing subspace decomposing into $10$ pairwise-orthogonal summands across $5$ representation types (which exist on every smooth fiber by representation theory and are Hodge-simple at very general $(B, C) \in U$):
   • Type I ($k \in \{1, 5\}$, $\epsilon = (0,0,0)$, $\mathbf{1}_{\mathrm{sgn}}$): $1$ block $R_{\mathrm{I}}$ of Hodge type $(1, 4, 1)$, $\dim_{\mathbb{Q}} = 6$;
   • Type II ($k \in \{1, 5\}$, $\epsilon = (0,0,0)$, $\mathbf{2}_{\mathrm{std}}$): $2$ blocks $R_{\mathrm{II}, 1}, R_{\mathrm{II}, 2}$ of type $(1, 4, 1)$, $\dim_{\mathbb{Q}} = 12$, isomorphic via an explicit algebraic similitude $T_{\mathrm{II}} \in \mathbb{Q}[W(B_3)]$;
   • Type III ($k \in \{1, 5\}$, two-odd sign parities): $3$ blocks $R_{\mathrm{III}, ab}, R_{\mathrm{III}, ac}, R_{\mathrm{III}, bc}$ of type $(1, 4, 1)$, $\dim_{\mathbb{Q}} = 18$, isomorphic via explicit permutation correspondences in $\mathbb{Q}[W(B_3)]$;
   • Type IV ($k \in \{2, 4\}$, one-odd sign parities): $3$ blocks $R_{\mathrm{IV}, a}, R_{\mathrm{IV}, b}, R_{\mathrm{IV}, c}$ of type $(1, 6, 1)$, $\dim_{\mathbb{Q}} = 24$, isomorphic via explicit transposition correspondences in $\mathbb{Q}[W(B_3)]$;
   • Type V ($k = 3$, $\epsilon = (0,0,0)$, $\mathbf{1}_{\mathrm{sgn}}$): $1$ block $R_{\mathrm{V}}$ of type $(1, 2, 1)$, $\dim_{\mathbb{Q}} = 4$.

The $7$ blocks of Types I, II, III, and V have period domain dimension equal to $2 = \dim U$. Together with the $41$ pure $(1, 1)$ primitive classes, they constitute the canonical rank-$81$ primitive open-period-domain variation $$H^2_{\mathrm{open}}(\Sigma_{B,C}, \mathbb{Q}) := H^2_{\mathrm{pure}-(1,1)} \oplus R_{\mathrm{I}} \oplus R_{\mathrm{II}, 1} \oplus R_{\mathrm{II}, 2} \oplus R_{\mathrm{III}, ab} \oplus R_{\mathrm{III}, ac} \oplus R_{\mathrm{III}, bc} \oplus R_{\mathrm{V}} \subset H^2_{\mathrm{prim}}(\Sigma_{B,C}, \mathbb{Q}),$$ which contains $100\%$ of the $42$-dimensional primitive sixth-root sector $H^2_{\mathrm{ord}=6}(\Sigma_{B,C}, \mathbb{Q})$. Adjoining the hyperplane class $h \in H^2(\Sigma_{B,C}, \mathbb{Q})$ yields the rank-$82$ ambient subvariation $\widetilde{H}^2_{\mathrm{open}}(\Sigma_{B,C}, \mathbb{Q}) := \mathbb{Q}h \oplus H^2_{\mathrm{open}}(\Sigma_{B,C}, \mathbb{Q})$.

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Theorem 2 (Weighted-Projective $\mathbb{P}(1,1,2,2)$ K3 Reduction of the Order-3 and Order-2 Sectors)

(a) Order-3 Sector ($k \in \{2, 4\}$, $\dim = 882$, middle $H^{2,2}$ dimension $= 666$): Let $g(a,b,c) = G(a^2, b^2, c^2)$ be any $(\mathbb{Z}/2\mathbb{Z})^3$-even smooth ternary sextic (normalized so that the coefficient of $a^6$ is $1$).

1. As in Laterveer [10, Proposition 3.10] applied to $z_1 = u^2$, the finite degree-$4$ morphism $$q_a : \Sigma_g \longrightarrow W_{g, a} \subset \mathbb{P}(1_b, 1_c, 2_{X_2}, 2_{z_1}), \qquad [a : b : c : u] \longmapsto [b : c : a^2 : u^2]$$ is an everywhere-defined morphism from $\Sigma_g$ onto the degree-$6$ weighted-projective hypersurface $W_{g, a} = V(z_1^3 - G(X_2, b^2, c^2))$, identifying $W_{g, a}$ with the geometric quotient $\Sigma_g / K_a$ by the Klein four-group $K_a = \langle \tau_a, d^3 \rangle \cong (\mathbb{Z}/2\mathbb{Z})^2$, where $\tau_a : a \mapsto -a$ and $d^3 : u \mapsto -u$.
2. The dualizing sheaf of $W_{g, a}$ is canonically trivial: $$\omega_{W_{g, a}} \cong \mathcal{O}_{W_{g,a}}\big(6 - (1 + 1 + 2 + 2)\big) = \mathcal{O}_{W_{g,a}}, \qquad h^1(W_{g,a}, \mathcal{O}_{W_{g,a}}) = 0,$$ and $W_{g, a}$ has exactly $3$ isolated $\frac{1}{2}(1, 1)$ ($A_1$) orbifold singularities along the weight-$2$ axis $b = c = 0$, $z_1^3 - X_2^3 = 0$, at $P_\nu = [0 : 0 : 1 : \omega^\nu]$ ($\nu \in \{0, 1, 2\}$). The minimal resolution $\rho : \widetilde{W}_{g, a} \to W_{g, a}$ is a smooth K3 surface with non-symplectic $\mu_3$-action $\bar{d} : z_1 \mapsto \omega z_1$.
3. Suppose further that $g$ is $(b \leftrightarrow c)$-symmetric (and in particular for the $W(B_3)$-symmetric family $g = f_{B,C}$). Under the transposition $\tau_{bc} : b \leftrightarrow c$, the residue differential form $ua\Omega_{\mathbb{P}^3}$ satisfies $\tau_{bc}^*(ua\Omega_{\mathbb{P}^3}) = (ua)(-\Omega_{\mathbb{P}^3}) = -ua\Omega_{\mathbb{P}^3}$, so the holomorphic line of $R_{\mathrm{IV}, a}$ is $\tau_{bc}$-anti-invariant. On the blowup $\widetilde{\Sigma}_g = \mathrm{Bl}_6(\Sigma_g)$, the correspondence $\Gamma_a := \pi_* \circ \tilde{q}_a^*$ identifies the $\tau_{bc}^* = -1$ anti-invariant subspace $\langle [E_\nu] \rangle^\perp \cap H^2_{\chi_2 \oplus \chi_4}(\widetilde{W}_{g, a}, \mathbb{Q})^{\tau_{bc}^* = -1}$ (rank $8$, Hodge numbers $(1, 6, 1)$) with $R_{\mathrm{IV}, a}$, while the $\tau_{bc}^* = +1$ invariant subspace has rank $4$ and Hodge numbers $(0, 4, 0)$ (pure $(1, 1)$ algebraic divisor classes).
4. Exact Middle Sector Breakdown ($666 = 180 + 162 + 324$) on the Symmetric Family: For $g = h = f_{B,C}$, in the order-$3$ middle cohomology $(H^2_{\chi_2}(\Sigma_{B,C}) \otimes H^2_{\chi_4}(\Sigma_{B,C}) \oplus H^2_{\chi_4}(\Sigma_{B,C}) \otimes H^2_{\chi_2}(\Sigma_{B,C}))^{\mu_6} \cap H^{2,2}$, the $666$ complex dimensions decompose into:
   • Transcendental $\otimes$ Transcendental ($180$ dimensions): Contained in the $576$-dimensional rational variation $(R_{\mathrm{IV}, a} \oplus R_{\mathrm{IV}, b} \oplus R_{\mathrm{IV}, c})^{\otimes 2}$ (rank $24 \times 24 = 576$), which reduces via $\Gamma_i \otimes \Gamma_j$ to the $9$ rank-$64$ K3 products $\widetilde{W}_{f_{B,C},i} \times \widetilde{W}_{f_{B,C},j}$ ($9 \times 8^2 = 576$);
   • Algebraic $\otimes$ Algebraic ($162$ dimensions): The tensor product of the $18$ pure $(1, 1)$ classes on each factor ($H^2_{\mathrm{pure}-(1,1)} \otimes H^2_{\mathrm{pure}-(1,1)}$, dimension $18 \times 18 = 324$, containing $162$ dimensions of $H^{2,2}$), where every rational Hodge class is a linear combination of products of algebraic divisors, hence unconditionally algebraic by Lefschetz $(1, 1)$;
   • Mixed Cross-Terms ($324$ dimensions): $R_{\mathrm{IV}} \otimes H^2_{\mathrm{pure}-(1,1)} \oplus H^2_{\mathrm{pure}-(1,1)} \otimes R_{\mathrm{IV}}$, which carries no rational Hodge classes at very general $(B, C) \in U$ because $R_{\mathrm{IV}, i}$ is Hodge-simple at very general $(B, C) \in U$ (Theorem 3(b)) with $H^{2,0} \ne 0$.

(b) Order-2 Sector ($k = 3$, $\dim = 441$, middle $H^{2,2}$ dimension $= 363$): For any smooth ternary sextic $g$, the $\mu_3$-quotient $q_2 : \Sigma_g \to Y_g = V(z_0^2 - g(a,b,c)) \subset \mathbb{P}(1,1,1,3)$ onto the degree-$2$ K3 double plane induces an isomorphism of weight-$2$ rational Hodge structures: $$q_2^* : H^2_{\mathrm{prim}}(Y_g, \mathbb{Q}) \xrightarrow{\;\sim\;} H^2_{\chi_3}(\Sigma_g, \mathbb{Q}), \qquad q_{2*} q_2^* = 3\,\mathrm{id},$$ reducing the order-$2$ sector of $X_{B,C}$ to the K3 product $Y_g \times Y_h$. In the symmetric family $g = f_{B,C}$, the K3 surface $Y_{B,C}$ has generic Picard number $\rho(Y_{B,C}) = 18$ ($\rho_{\mathrm{prim}} = 17$, pure $(1, 1)$) and rank-$4$ transcendental lattice $T(Y_{B,C}) \cong R_{\mathrm{V}}$.

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Theorem 3 (Closed-Form Macaulay Determinants, Fermat Gauss–Manin Invariants, and the Open-Sector Hodge Conjecture)

(a) Exact Closed-Form Symbolic Macaulay–Griffiths Determinants & Discriminant Inclusion: Over $\mathbb{Q}[B, C]$, the Macaulay–Griffiths period matrices of Block Types I, II, III, and V have exact determinants: $$\Delta_{\mathrm{I}}(B, C) = -\frac{2}{3}(B - 3), \qquad \Delta_{\mathrm{II}}(B, C) = \frac{1}{27}(B - 3)^3,$$ $$\Delta_{\mathrm{III}}(B, C) = -\frac{1}{9}\big(2B^3 - B^2 C - 3B^2 + C^2 - 3C + 9\big), \qquad \Delta_{\mathrm{V}}(B, C) = 1.$$ Every zero locus of $\Delta_{\mathrm{I}}, \Delta_{\mathrm{II}}, \Delta_{\mathrm{III}}$ is contained in the singular discriminant locus $\Delta_{\mathrm{sing}} \subset \mathbb{A}^2_{B, C}$ of $\Sigma_{B, C}$:

1. At $B=3$, the surface $\Sigma_{3, C}$ has singular points at $[1:\pm i:0:0]$ for every $C \in \mathbb{C}$ (since $f_a(1, -1, 0) = f_b(1, -1, 0) = c f_c = 0$);
2. On $2B^3 - B^2 C - 3B^2 + C^2 - 3C + 9 = 0$, the resultant identity $$\mathrm{Res}_X\big(f_a(X, X, 1),\; f_c(X, X, 1)\big) = -\frac{1}{3}(6B + C + 3)\,\Delta_{\mathrm{III}}(B, C)$$ proves that $f_a = f_b = f_c = 0$ has a common projective root $[X : X : Z] \in \mathbb{P}^1$, so $[\sqrt{X} : \sqrt{X} : \sqrt{Z} : 0] \in \Sigma_{B, C}$ is singular. Thus $\Delta_{\mathrm{I}}, \Delta_{\mathrm{II}}, \Delta_{\mathrm{III}}, \Delta_{\mathrm{V}}$ are non-zero everywhere on the smooth locus $U = \mathbb{A}^2_{B, C} \setminus \Delta_{\mathrm{sing}}$. At the smooth benchmark $(15, 90)$, the reduced Gröbner $2 \times 2$ Kodaira–Spencer minors are: $$\det M_{\mathrm{I}} = \frac{4}{25}, \quad \det M_{\mathrm{II}} = \frac{44}{25}, \quad \det M_{\mathrm{III}} = -\frac{11}{80}, \quad \det M_{\mathrm{IV}} = \frac{4}{5}, \quad \det M_{\mathrm{V}} = \frac{1}{5}.$$ Consequently, the period maps into the unitary $2$-balls $\mathbb{B}^2$ (Types I, II, III) and the orthogonal Type IV quadric $D_{\mathrm{IV}}(R_{\mathrm{V}}) \subset Q_2 \subset \mathbb{P}^3$ (Type V) are unramified submersions onto open domains of complex dimension $2 = \dim U$. Thus at very general $(B, C) \in U$: $$\mathrm{End}_{\mathrm{Hdg}}(R_{\mathrm{I}}) = \mathrm{End}_{\mathrm{Hdg}}(R_{\mathrm{II}, p}) = \mathrm{End}_{\mathrm{Hdg}}(R_{\mathrm{III}, i}) = \mathbb{Q}(\zeta_6), \qquad \mathrm{End}_{\mathrm{Hdg}}(R_{\mathrm{V}}) = \mathbb{Q}.$$

(b) Second-Order Mixed Gauss–Manin Spanning and Galois-Branch Projection for Type IV: In the four-variable Jacobian ring $R(\Sigma_{0,0}) = \mathbb{C}[a, b, c, u]/(a^5, b^5, c^5, u^5)$, $m_{2,0} = u a \Omega_0$ has degree $2$. In degree $14$, $u a P_B P_C$ reduces under Griffiths pole reduction from $3$ to $2$ to $-\frac{1}{6} u a^3(b^4+c^4) - \frac{1}{3} u a(b^4 c^2 + b^2 c^4)$ in degree $8$. In the basis $\{u a^3(b^4+c^4), u a^3 b^2 c^2, u a(b^4 c^2 + b^2 c^4)\}$, the matrix of $\{v_B, v_C, \nabla_B\nabla_C\omega\}$ has determinant $-\frac{1}{6} \ne 0$. On the Zariski-open set $U \setminus Z_{\mathrm{IV}}$ where this determinant is non-zero, $\{\omega, \nabla_B\omega, \nabla_C\omega, \nabla_B\nabla_C\omega\}$ spans the $4$-dimensional $\mathbb{C}$-VHS $R_{\mathrm{IV}, a, \sigma_1}|_{U_0}$, proving open-set irreducibility. Decomposing any flat endomorphism into its $K = \mathbb{Q}(\zeta_3)$ Galois branches $f = f_1 + f_2$ (as in [17, Section 3.2]), the conjugate-linear branch $f_{2,1} : R_{\mathrm{IV}, a, \sigma_1} \to R_{\mathrm{IV}, a, \sigma_2}$ must vanish by Hodge-dimension mismatch ($1 > 0$), establishing that $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(R_{\mathrm{IV}, i}) = \mathbb{Q}(\zeta_3)$ and that $R_{\mathrm{IV}, i}$ is $\mathbb{Q}$-Hodge simple at very general $(B, C) \in U$.

(c) Fermat Gauss–Manin Invariants and Vanishing of Mixed Homomorphisms: At the smooth Fermat point $(B, C) = (0, 0) \in U$, the canonical gauge- and isomorphism-invariant Wronskian inflection cubics $$W_\alpha(x, y) := \frac{\omega_\alpha(0,0) \wedge \nabla_{x\partial_B+y\partial_C}\omega_\alpha(0,0) \wedge \nabla_{x\partial_B+y\partial_C}^2\omega_\alpha(0,0)}{\omega_\alpha(0,0) \wedge \nabla_{\partial_B}\omega_\alpha(0,0) \wedge \nabla_{\partial_C}\omega_\alpha(0,0)} \in \mathbb{Q}[x, y]_3$$ factor in closed form as: $$\begin{aligned} W_{\mathrm{I}}(x, y) &= -\frac{1}{12}(2x - y)\big(20x^2 + 8xy + y^2\big), \\ W_{\mathrm{II}}(x, y) &= -\frac{1}{6}(2x - y)\big(7x^2 + xy - y^2\big), \\ W_{\mathrm{III}}(x, y) &= -\frac{1}{3}x\big(11x^2 - 4xy - 2y^2\big), \end{aligned}$$ with distinct $C$-direction reduced second-derivative invariants $\Gamma^{(\mathrm{I})}_{C,C}(0,0) = (-\frac{1}{12}, \frac{1}{6})$, $\Gamma^{(\mathrm{II})}_{C,C}(0,0) = (\frac{1}{6}, \frac{1}{6})$, and $\Gamma^{(\mathrm{III})}_{C,C}(0,0) = (0, 0)$. Because the period map of each block has open image in $\mathbb{B}^2 \subset \mathbb{P}^2$, any flat rational morphism $\Phi : R_\alpha \to R_\beta$ must have its off-diagonal conjugate block $\Phi_{\bar{\sigma},\sigma} : R_{\alpha,\sigma} \to R_{\beta,\bar{\sigma}}$ identically zero (as it vanishes on an open subset of $\mathbb{P}^2$). The diagonal block $L = \Phi_{\sigma,\sigma}$ must satisfy $L(\omega_\alpha) = \lambda\,\omega_\beta$; in the normalized ratio $W_\alpha(x, y)$, both $\det L$ and $\lambda(0, 0)^3$ cancel between numerator and denominator, forcing the exact polynomial identity $W_\beta(x, y) = W_\alpha(x, y)$. Since $W_{\mathrm{I}}, W_{\mathrm{II}}, W_{\mathrm{III}}$ are pairwise distinct, $\Phi = 0$. Thus all mixed Hodge loci are proper closed algebraic subvarieties of $U$, so for very general $(B, C) \in U$: $$\mathrm{Hom}_{\mathrm{Hdg}}(R_\alpha,\; R_\beta) = 0 \qquad \forall\, \alpha \ne \beta \in \{\mathrm{I},\; \mathrm{II},\; \mathrm{III}\}.$$ Moreover, since $\dim_{\mathbb{Q}} R_{\mathrm{IV}, i} = 8$, $\dim_{\mathbb{Q}} R_\alpha = 6$ ($\alpha \in \{\mathrm{I}, \mathrm{II}, \mathrm{III}\}$), and $\dim_{\mathbb{Q}} R_{\mathrm{V}} = 4$ are pairwise distinct, Schur's lemma implies $\mathrm{Hom}_{\mathrm{Hdg}}(R_{\mathrm{IV}, i}, R_\alpha) = 0$ for all other blocks.

(d) Rational Hodge Classes and Algebraicity at Very General $(B, C) \in U$:

1. Full Middle Cohomology on $\Sigma_{B,C} \times \Sigma_{B,C}$: Combining parts (a)–(c) with the explicit rational group-ring similitudes $T_{\mathrm{II}}, T_{\mathrm{III}}, T_{\mathrm{IV}}$ and Lefschetz $(1, 1)$, for very general $(B, C) \in U$, every rational Hodge class in the entire middle cohomology $H^4(\Sigma_{B,C} \times \Sigma_{B,C}, \mathbb{Q})$ (ambient tensor dimension $106^2 + 2 = 11{,}238$ and primitive tensor dimension $105^2 = 11{,}025$) is unconditionally algebraic. The space of rational Hodge classes in $H^4(\Sigma_{B,C} \times \Sigma_{B,C}, \mathbb{Q})$ has dimension $$42^2 + 2(1^2 + 2^2 + 3^2 + 3^2) + 1 + 2 = 1{,}764 + 46 + 1 + 2 = 1{,}813$$ ($1{,}811$ in $H^2 \otimes H^2$, including the $1{,}793$ dimensions of $\widetilde{H}^2_{\mathrm{open}}^{\otimes 2}$ and the $18$ dimensions of $R_{\mathrm{IV}}^{\otimes 2}$) and is spanned by explicit algebraic cycle classes.
2. Full Primitive Fourfold Split Sector: On the split fourfold $X_{B,C}$, for very general $(B, C) \in U$, every rational Hodge class in the entire $2{,}605$-dimensional primitive cohomology $H^4_{\mathrm{prim}}(X_{B,C}, \mathbb{Q})$ (total $516 + r_1$ primitive rational Hodge dimensions—comprising $r_1 = \dim_{\mathbb{Q}} \operatorname{End}^0 J(C_{f_{B,C}}) < 200$ in Order 1, $290$ in Order 2, $180$ in Order 3, and $46$ in Order 6—and $517 + r_1$ in full $H^4(X_{B,C}, \mathbb{Q})$) is unconditionally algebraic.
3. Heisenberg Jumping Locus ($B = 0$): Along the smooth $1$-parameter slice $B = 0$, the extra abelian group $H_3 \cong (\mathbb{Z}/3\mathbb{Z})^2$ acts algebraically on $\Sigma_{0,C}$, splitting off $+6$ pure $(1, 1)$ dimensions from Type III, $+12$ from Type IV, and $+1$ from Type V (which provides the lower bound $\rho_{\mathrm{prim}}(\Sigma_{0,C}) \ge 41 + 19 = 60$). Conversely, on the remaining $45$-dimensional transcendental subspace ($R_{\mathrm{I}}(0,C) \oplus R_{\mathrm{II}}(0,C)$ of rank $18$, $3$ rank-$4$ $(1, 2, 1)$ Type III sub-blocks, $3$ rank-$4$ $(1, 2, 1)$ Type IV sub-blocks, and the rank-$3$ $(1, 1, 1)$ Type V sub-block), the $C$-direction period derivatives remain non-zero at $(0, 0)$ (in particular, $W_{\mathrm{I}}(0, 1) = 1/12 \ne 0$ and the first derivative $v_C = a^3 b^3 c^2 \ne 0$ in Type III is non-zero while its reduced second derivative $d_{CC,\mathrm{III}} = 0$ vanishes), excluding any further rational $(1, 1)$ classes at very general $C$ and establishing the exact equality $\rho_{\mathrm{prim}}(\Sigma_{0,C}) = 60$.

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3. The Algebraic Group-Ring Projectors

Let $G = \{\pm 1\}^3$ act by coordinate signs on $(a, b, c)$, fixing $u$. The $8$ characters of $G$ are indexed by exponent parities $\epsilon = (i_a, i_b, i_c) \bmod 2 \in \{0, 1\}^3$: $$e_\epsilon := \frac{1}{8} \sum_{(\varepsilon_a, \varepsilon_b, \varepsilon_c) \in \{\pm 1\}^3} \varepsilon_a^{i_a + 1} \varepsilon_b^{i_b + 1} \varepsilon_c^{i_c + 1}\, g_{\varepsilon_a, \varepsilon_b, \varepsilon_c}^*, \qquad \sum_{\epsilon \in \{0, 1\}^3} e_\epsilon = \mathrm{id}.$$ The deck automorphism $d(a, b, c, u) = (a, b, c, \zeta_6 u)$ generates $\mu_6$. The three pairwise-orthogonal cyclotomic idempotents in $\mathbb{Q}[\mu_6] / (\sum_{k=0}^5 d^k)$ are: $$\begin{aligned} e_{\mathrm{ord}\,6} &= \frac{1}{6}\big(2 + d^* - (d^*)^2 - 2(d^*)^3 - (d^*)^4 + (d^*)^5\big) && (\text{selects } l \in \{0, 4\}), \\ e_{\mathrm{ord}\,3} &= \frac{1}{6}\big(2 - d^* - (d^*)^2 + 2(d^*)^3 - (d^*)^4 - (d^*)^5\big) && (\text{selects } l \in \{1, 3\}), \\ e_{\mathrm{ord}\,2} &= \frac{1}{6}\big(1 - d^* + (d^*)^2 - (d^*)^3 + (d^*)^4 - (d^*)^5\big) && (\text{selects } l = 2). \end{aligned}$$ Under $\mathfrak{S}_3$ (where $\tau^*$ includes the determinant $\mathrm{sgn}(\tau)$ from the volume form $\Omega_0$):

• On $\epsilon = (0, 0, 0)$, the standard representation $\mathbf{2}_{\mathrm{std}}$ splits into two self-adjoint primitive orthogonal idempotents: $$e_{\mathrm{std}, 1} = \frac{1}{2}\big(\mathrm{id} + \tau_{ab}^*\big) e_{\mathrm{std}}, \qquad e_{\mathrm{std}, 2} = \frac{1}{2}\big(\mathrm{id} - \tau_{ab}^*\big) e_{\mathrm{std}},$$ where $e_{\mathrm{std}} = \frac{1}{3}(2\,\mathrm{id} - (abc)^* - (acb)^*)$. Here $e_{\mathrm{std}, 1}$ selects the $(a \leftrightarrow b)$-antisymmetric numerator $a^2 - b^2$, and $e_{\mathrm{std}, 2}$ selects the $(a \leftrightarrow b)$-symmetric numerator $a^2 + b^2 - 2c^2$.
• On each two-odd sector (e.g. $\epsilon = (1, 1, 0)$), the stabilizer $\mathfrak{S}_2 = \langle \tau_{ab} \rangle$ gives: $$e_{(1,1,0)}^+ = \frac{1}{2}\big(\mathrm{id} - \tau_{ab}^*\big) e_{(1,1,0)}, \qquad e_{(1,1,0)}^- = \frac{1}{2}\big(\mathrm{id} + \tau_{ab}^*\big) e_{(1,1,0)}.$$ The $(+)$ projector selects the $(a \leftrightarrow b)$-symmetric numerator $a b$ (yielding $R_{\mathrm{III}, ab}$), while the $(-)$ projector yields pure $(1, 1)$ classes.
• On each one-odd sector (e.g. $\epsilon = (1, 0, 0)$), the stabilizer $\mathfrak{S}_2 = \langle \tau_{bc} \rangle$ gives: $$e_{(1,0,0)}^+ = \frac{1}{2}\big(\mathrm{id} - \tau_{bc}^*\big) e_{(1,0,0)}, \qquad e_{(1,0,0)}^- = \frac{1}{2}\big(\mathrm{id} + \tau_{bc}^*\big) e_{(1,0,0)}.$$ The $(+)$ projector selects the $(b \leftrightarrow c)$-symmetric numerator $u a$ (yielding $R_{\mathrm{IV}, a}$), while the $(-)$ projector yields pure $(1, 1)$ classes.

All projectors are self-adjoint rational linear combinations of automorphism graphs in $\mathbb{Q}[\mu_6 \times W(B_3)]$, and their sum is $\mathrm{id}_{H^2_{\mathrm{prim}}(\Sigma, \mathbb{Q})}$.

3.1. Explicit Inter-Block Algebraic Similitudes in $\mathbb{Q}[\mu_6 \times W(B_3)]$

To link multiple copies of the same representation type by explicit algebraic cycle correspondences, we construct rational group-ring operators in $\mathbb{Q}[\mu_6 \times W(B_3)]$ and verify their exact similitude identities in $\mathrm{End}(H^2_{\mathrm{prim}}(\Sigma, \mathbb{Q}))$:

1. Type II Similitude: In the group ring $\mathbb{Q}[\mathfrak{S}_3]$, the element $\eta := \tau_{bc}^* - \tau_{ac}^*$ anticommutes with $\tau_{ab}^*$ (since $\tau_{ab}\tau_{bc}\tau_{ab} = \tau_{ac}$) and satisfies the exact identity: $$\eta^2 = (\tau_{bc}^* - \tau_{ac}^*)^2 = 2\,\mathrm{id} - (abc)^* - (acb)^* = 3\,e_{\mathrm{std}}.$$ Because $\tau_{ab}^*$ anticommutes with $\eta$, conjugation by $\eta$ swaps the $(\tau_{ab}=+1)$-projector $e_{\mathrm{std}, 1}$ and the $(\tau_{ab}=-1)$-projector $e_{\mathrm{std}, 2}$. Consequently, the rational algebraic correspondence operator $$T_{\mathrm{II}} := e_{\mathrm{II}, 2} \circ (\tau_{bc}^* - \tau_{ac}^*) \circ e_{\mathrm{II}, 1} \in \mathbb{Q}[\mu_6 \times W(B_3)]$$ and its adjoint $T_{\mathrm{II}}^\dagger := e_{\mathrm{II}, 1} \circ (\tau_{bc}^* - \tau_{ac}^*) \circ e_{\mathrm{II}, 2}$ satisfy the exact identities: $$T_{\mathrm{II}}^\dagger \circ T_{\mathrm{II}} = 3\,e_{\mathrm{II}, 1}, \qquad T_{\mathrm{II}} \circ T_{\mathrm{II}}^\dagger = 3\,e_{\mathrm{II}, 2}.$$ Thus $T_{\mathrm{II}} : R_{\mathrm{II}, 1} \to R_{\mathrm{II}, 2}$ is an algebraic similitude of multiplier $3$ (a rational Hodge isomorphism with inverse $\frac{1}{3}T_{\mathrm{II}}^\dagger$). Composing any rational Hodge morphism $\phi \in \operatorname{Hom}_{\mathrm{Hdg}}(R_{\mathrm{II}, 1}, R_{\mathrm{II}, 2})$ with $\frac{1}{3}T_{\mathrm{II}}^\dagger$ reduces it to $\operatorname{End}_{\mathrm{Hdg}}(R_{\mathrm{II}, 1}) = \mathbb{Q}(\zeta_6) = \mathrm{span}_{\mathbb{Q}}\{e_{\mathrm{II}, 1}, e_{\mathrm{II}, 1} \circ d^*\}$, proving that $\operatorname{Hom}_{\mathrm{Hdg}}(R_{\mathrm{II}, 1}, R_{\mathrm{II}, 2}) = \mathrm{span}_{\mathbb{Q}}\{T_{\mathrm{II}}, T_{\mathrm{II}} \circ d^*\}$.

2. Type III Isometries: The transpositions $\tau_{bc}$ and $\tau_{ac}$ map the two-odd parity characters $(1, 1, 0) \mapsto (1, 0, 1)$ and $(1, 1, 0) \mapsto (0, 1, 1)$, defining explicit algebraic correspondence operators $$T_{\mathrm{III}, ac, ab} := e_{\mathrm{III}, ac} \circ \tau_{bc}^* \circ e_{\mathrm{III}, ab}, \qquad T_{\mathrm{III}, bc, ab} := e_{\mathrm{III}, bc} \circ \tau_{ac}^* \circ e_{\mathrm{III}, ab}$$ satisfying $T_{\mathrm{III}, ac, ab}^\dagger \circ T_{\mathrm{III}, ac, ab} = e_{\mathrm{III}, ab}$ and $T_{\mathrm{III}, bc, ab}^\dagger \circ T_{\mathrm{III}, bc, ab} = e_{\mathrm{III}, ab}$. These provide explicit rational Hodge isometries spanning all inter-block Hom spaces.

3. Type IV Isometries: The transpositions $\tau_{ab}$ and $\tau_{ac}$ map the one-odd parity characters $(1, 0, 0) \mapsto (0, 1, 0)$ and $(1, 0, 0) \mapsto (0, 0, 1)$, defining explicit algebraic correspondence operators $$T_{\mathrm{IV}, ba} := e_{\mathrm{IV}, b} \circ \tau_{ab}^* \circ e_{\mathrm{IV}, a}, \qquad T_{\mathrm{IV}, ca} := e_{\mathrm{IV}, c} \circ \tau_{ac}^* \circ e_{\mathrm{IV}, a}$$ satisfying $T_{\mathrm{IV}, ba}^\dagger \circ T_{\mathrm{IV}, ba} = e_{\mathrm{IV}, a}$ and $T_{\mathrm{IV}, ca}^\dagger \circ T_{\mathrm{IV}, ca} = e_{\mathrm{IV}, a}$, reducing all $9$ blocks in $R_{\mathrm{IV}} \otimes R_{\mathrm{IV}}$ to $\operatorname{End}_{\mathrm{Hdg}}(R_{\mathrm{IV}, a})$.

All of these identities are certified in exact integer arithmetic in verify.py.

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4. Proof of Theorems 1 and 2

4.1. Proof of Theorem 1(a): Katsura–Shioda Sector Decomposition

Let $L_x = \{d=e=f=0\} \cong \mathbb{P}^2$ and $L_y = \{a=b=c=0\} \cong \mathbb{P}^2$. Since $L_x \cap L_y = \emptyset$, the intersections $C_g = X \cap L_x$ and $C_h = X \cap L_y$ are disjoint smooth genus-$10$ plane sextic curves of codimension $3$ in $X$.

Let $\pi : \Sigma_g \times \Sigma_h \dashrightarrow X$ be defined by $([a:b:c:u], [d:e:f:v]) \mapsto [va : vb : vc : ud : ue : uf]$. The indeterminacy locus is $u = v = 0$, which is $C_g \times C_h$. Blowing up $u = v = 0$ on $\Sigma_g \times \Sigma_h$ introduces $[s : t] = [u : v] \in \mathbb{P}^1$. Since $\mu_6$ acts on $(u, v)$ by $(\zeta u, \zeta v)$, it acts trivially on $[s : t]$. The quotient by $\mu_6$ replaces the fiber coordinate by its $6$-th power, yielding a smooth fourfold. Over $u = 0$ ($s = 0, t = 1$), the image is $[a : b : c : 0 : 0 : 0] \in C_g$ with fiber $[d : e : f] \in \mathbb{P}^2$, which is the exceptional $\mathbb{P}^2$-bundle over $C_g$ in $\mathrm{Bl}_{C_g \sqcup C_h}(X)$. By the universal property of blowups [1, 2], this gives an exact isomorphism of smooth fourfolds: $$\mathrm{Bl}_{C_g \times C_h}\big(\Sigma_g \times \Sigma_h\big) / \mu_6 \;\cong\; \mathrm{Bl}_{C_g \sqcup C_h}(X).$$

Applying the blowup formula for a smooth center of codimension $c = 3$ in the smooth fourfold $X$: $$H^4\big(\mathrm{Bl}_{C_g \sqcup C_h}(X), \mathbb{Q}\big) \cong H^4(X, \mathbb{Q}) \oplus \bigoplus_{i \in \{g, h\}} \Big(H^2(C_i, \mathbb{Q})(-1) \oplus H^0(C_i, \mathbb{Q})(-2)\Big) = H^4(X, \mathbb{Q}) \oplus \mathbb{Q}(-2)^{\oplus 4}.$$ On the surface-product side, blowing up the codimension-$2$ center $C_g \times C_h \subset \Sigma_g \times \Sigma_h$ gives: $$H^4\big(\mathrm{Bl}_{C_g \times C_h}(\Sigma_g \times \Sigma_h), \mathbb{Q}\big)^{\mu_6} \cong H^4(\Sigma_g \times \Sigma_h, \mathbb{Q})^{\mu_6} \oplus H^2(C_g \times C_h, \mathbb{Q})(-1).$$ By Künneth on $C_g \times C_h$, $$H^2(C_g \times C_h, \mathbb{Q})(-1) = \Big(H^0(C_g) \otimes H^2(C_h) \oplus H^2(C_g) \otimes H^0(C_h)\Big)(-1) \;\oplus\; \Big(H^1(C_g) \otimes H^1(C_h)\Big)(-1) = \mathbb{Q}(-2)^{\oplus 2} \oplus \Big(H^1(C_g) \otimes H^1(C_h)\Big)(-1).$$ In $H^4(\Sigma_g \times \Sigma_h, \mathbb{Q})^{\mu_6}$, the non-primitive ambient classes are $H^0 \otimes H^4 \oplus H^4 \otimes H^0 \oplus \mathbb{Q}(h_g \boxtimes h_h) \cong \mathbb{Q}(-2)^{\oplus 3}$. Together with the $2$ ambient classes from $H^2(C_g \times C_h)(-1)$, the right-hand side contains exactly $3 + 2 = 5$ copies of $\mathbb{Q}(-2)$. On the left-hand side, $H^4(X, \mathbb{Q}) = \mathbb{Q}h_X^2 \oplus H^4_{\mathrm{prim}}(X, \mathbb{Q})$, so the left-hand side contains $1 + 4 = 5$ copies of $\mathbb{Q}(-2)$. Canceling the $5$ ambient copies of $\mathbb{Q}(-2)$ yields the exact isomorphism of primitive rational Hodge structures: $$H^4_{\mathrm{prim}}(X, \mathbb{Q}) \;\cong\; \Big(H^1(C_g, \mathbb{Q}) \otimes H^1(C_h, \mathbb{Q})(-1)\Big) \;\oplus\; \Big(H^2_{\mathrm{prim}}(\Sigma_g, \mathbb{Q}) \otimes H^2_{\mathrm{prim}}(\Sigma_h, \mathbb{Q})\Big)^{\mu_6}.$$

In the Jacobian ring $R(X) = \mathbb{C}[a,b,c,d,e,f]/(g_a, g_b, g_c, h_d, h_e, h_f) = R(g) \otimes R(h)$, the degree sum $d_x + d_y \in \{0, 6, 12, 18, 24\}$ matches the Calabi–Yau Hodge numbers $(1, 426, 1751, 426, 1)$, total $2605$. Sorting by $k = (d_x + 3) \bmod 6$ partitions $H^4_{\mathrm{prim}}(X, \mathbb{Q})$ into the four sectors:

• $k = 0$: $d_x + d_y = 12$ has $d_x \in \{3, 9\}$, giving $\dim = 10 \times 10 + 10 \times 10 = 200$ in $H^{2,2}$, and $d_x = d_y = 3$ gives $100$ in $H^{3,1}$, $d_x = d_y = 9$ gives $100$ in $H^{1,3}$, matching $H^1(C_g) \otimes H^1(C_h)(-1)$ (total $\dim = 400$). Since $C_g \times C_h$ is a smooth surface, every rational $(1, 1)$ class in $H^2(C_g \times C_h, \mathbb{Q})$ is a divisor, hence algebraic by Lefschetz $(1, 1)$, and its pushforward to $X$ is an algebraic $2$-cycle.
• $k = 3$: $\dim = 21 \times 21 = 441$, with Hodge numbers $(1, 38, 363, 38, 1)$.
• $k \in \{2, 4\}$: $\dim = 2 \times 21^2 = 882$, with Hodge numbers $(0, 108, 666, 108, 0)$.
• $k \in \{1, 5\}$: $\dim = 2 \times 21^2 = 882$, with Hodge numbers $(0, 180, 522, 180, 0)$. $\blacksquare$

4.2. Proof of Theorem 2: K3 Quotients

For part (a), let $g(a,b,c) = G(a^2, b^2, c^2)$ be any smooth $(\mathbb{Z}/2\mathbb{Z})^3$-even ternary sextic normalized so that the coefficient of $a^6$ is $1$. The map $$q_a : \Sigma_g \longrightarrow W_{g, a} \subset \mathbb{P}(1_b, 1_c, 2_{X_2}, 2_{z_1}), \qquad [a : b : c : u] \longmapsto [b : c : a^2 : u^2]$$ is an everywhere-defined morphism because $[b : c : a^2 : u^2] = [0 : 0 : 0 : 0] \iff a = b = c = u = 0$, which is empty in $\mathbb{P}^3$. It is the geometric quotient of $\Sigma_g$ by the Klein four-group $K_a = \langle \tau_a, d^3 \rangle \cong (\mathbb{Z}/2\mathbb{Z})^2$, as in Laterveer [10, Proposition 3.10] applied to $z_1 = u^2$. On $R_{\mathrm{IV}, a}$, every basis monomial has $l \in \{1, 3\}$ odd and $i_a \in \{1, 3\}$ odd. Under $\tau_a : a \mapsto -a$, $a^{i_a} \mapsto -a^{i_a}$ and $\Omega_{\mathbb{P}^3} \mapsto -\Omega_{\mathbb{P}^3}$, so $(-1)(-1) = +1$. Under $d^3 : u \mapsto -u$, $u^l \mapsto -u^l$ and $\Omega_{\mathbb{P}^3} \mapsto -\Omega_{\mathbb{P}^3}$, so $(-1)(-1) = +1$. Thus $R_{\mathrm{IV}, a} \subset H^2(\Sigma_g, \mathbb{Q})^{K_a} \cong H^2(W_{g, a}, \mathbb{Q})$.

By weighted adjunction, $\omega_{W_{g,a}} \cong \mathcal{O}(6 - (1+1+2+2)) = \mathcal{O}$. Because the coefficient of $a^6 = X_2^3$ is normalized to $1$, the $3$ points $P_\nu = [0 : 0 : 1 : \omega^\nu]$ on $b = c = 0$, $z_1^3 - X_2^3 = 0$ are ordinary $A_1$ singularities. Resolving them yields the smooth K3 surface $\rho : \widetilde{W}_{g,a} \to W_{g,a}$. Blowing up the $6$ preimage points on $\Sigma_g$ gives the smooth surface $\widetilde{\Sigma}_g = \mathrm{Bl}_6(\Sigma_g)$ and finite degree-$4$ morphism $\tilde{q}_a : \widetilde{\Sigma}_g \to \widetilde{W}_{g,a}$. The deck action $\bar{d} : z_1 \mapsto \omega z_1$ cyclically permutes the $3$ exceptional curves $E_0 \mapsto E_1 \mapsto E_2 \mapsto E_0$, splitting into $1$ trivial class and $2$ algebraic divisor classes in $\chi_2 \oplus \chi_4$. When $g$ is further $(b \leftrightarrow c)$-symmetric (and in particular for $g = f_{B,C}$), under the transposition $\tau_{bc} : b \leftrightarrow c$, the volume form $\Omega_{\mathbb{P}^3}$ changes sign, so $\tau_{bc}^*(ua\Omega_{\mathbb{P}^3}) = -ua\Omega_{\mathbb{P}^3}$. Thus on $\widetilde{\Sigma}_g$, the correspondence $\Gamma_a := \pi_* \circ \tilde{q}_a^*$ is an isomorphism from the $\tau_{bc}^* = -1$ anti-invariant subspace $\langle [E_\nu] \rangle^\perp \cap H^2_{\chi_2 \oplus \chi_4}(\widetilde{W}_{g,a}, \mathbb{Q})^{\tau_{bc}^* = -1}$ onto $R_{\mathrm{IV}, a}$.

Part (b) is the degree-$2$ K3 double plane quotient: $u^3 = z_0$ maps $\Sigma_g \to Y_g = V(z_0^2 - g) \subset \mathbb{P}(1,1,1,3)$, which has $\omega_{Y_g} \cong \mathcal{O}_{Y_g}(6 - (1+1+1+3)) = \mathcal{O}_{Y_g}$, identifying $H^2_{\chi_3}(\Sigma_g, \mathbb{Q})$ with $H^2_{\mathrm{prim}}(Y_g, \mathbb{Q})$ as weight-$2$ rational Hodge structures. $\blacksquare$

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5. Proof of Theorem 3: Period Determinants, Fermat Invariants, and the Hodge Conjecture

5.1. Closed-Form Macaulay–Griffiths Determinants & Discriminant Inclusion

On the Jacobian ring of $\Sigma_{B,C}$, the partial derivatives are $\frac{1}{6}\partial_a f = a f_a(X, Y, Z)$ with $X = a^2, Y = b^2, Z = c^2$: $$f_a(X, Y, Z) = X^2 + B\left(\frac{2}{3}X(Y+Z) + \frac{1}{3}(Y^2+Z^2)\right) + \frac{1}{3}C Y Z.$$ The Kodaira–Spencer derivatives multiply the holomorphic numerator $m_{2,0}$ by $P_B = X^2(Y+Z) + Y^2(X+Z) + Z^2(X+Y)$ and $P_C = X Y Z$.

1. Type I: In the $4$-dimensional space of $S_3$-symmetric degree-$4$ polynomials in $(X, Y, Z)$ with basis $\{X^4, X^3 Y, X^2 Y^2, X^2 Y Z\}$, the ideal $(X f_a, Y f_b, Z f_c)_4$ has $2$ generators: $g_1 = \sum X^2 f_a$ and $g_2 = \sum X(Y+Z)f_a$. Together with $v_B = (X+Y+Z)P_B$ and $v_C = (X+Y+Z)P_C$, the $4 \times 4$ Macaulay matrix $\mathcal{M}_{\mathrm{I}}(B, C)$ has determinant: $$\det \mathcal{M}_{\mathrm{I}}(B, C) = -\frac{2}{3}(B - 3).$$

2. Type II: In the $6$-dimensional space of $(X \leftrightarrow Y)$-antisymmetric degree-$4$ polynomials, the ideal generators are $X^2 f_a - Y^2 f_b$, $X Y(f_a - f_b)$, $Z(X f_a - Y f_b)$, and $(X - Y)Z f_c$. Adjoining $v_B = (X-Y)P_B$ and $v_C = (X-Y)P_C$, the $6 \times 6$ Macaulay matrix has determinant: $$\det \mathcal{M}_{\mathrm{II}}(B, C) = \frac{1}{27}(B - 3)^3.$$

3. Type III: In the $6$-dimensional space of $(X \leftrightarrow Y)$-symmetric degree-$3$ polynomials, the ideal $(f_a, f_b, Z f_c)_3$ is generated by $X f_a + Y f_b$, $Y f_a + X f_b$, $Z(f_a + f_b)$, and $Z f_c$. Adjoining $P_B$ and $P_C$, the $6 \times 6$ Macaulay matrix has determinant: $$\det \mathcal{M}_{\mathrm{III}}(B, C) = -\frac{1}{9}\big(2B^3 - B^2 C - 3B^2 + C^2 - 3C + 9\big).$$

4. Type V: In the $3$-dimensional space of $S_3$-symmetric degree-$3$ polynomials, Euler's identity gives $X f_a + Y f_b + Z f_c = (X^3+Y^3+Z^3) + B P_B + C P_C$. In the basis $\{X^3+Y^3+Z^3, P_B, P_C\}$, the matrix is upper-triangular unipotent: $$\mathcal{M}_{\mathrm{V}}(B, C) = \begin{pmatrix} 1 & B & C \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \implies \det \mathcal{M}_{\mathrm{V}}(B, C) = 1.$$

Discriminant Inclusion:

• At $B = 3$, $f_a(1, -1, 0) = 1 + 3(-\frac{2}{3} + \frac{1}{3}) = 0$ and $f_b(1, -1, 0) = 1 + 3(-\frac{2}{3} + \frac{1}{3}) = 0$, so $[1 : i : 0 : 0]$ is a singular point of $\Sigma_{3, C}$ for all $C$.
• On $2B^3 - B^2 C - 3B^2 + C^2 - 3C + 9 = 0$, the resultant identity $$\mathrm{Res}_X\big(f_a(X, X, 1),\; f_c(X, X, 1)\big) = -\frac{1}{3}(6B + C + 3)\,\Delta_{\mathrm{III}}(B, C) = 0$$ proves that $f_a = f_b = f_c = 0$ has a common projective root $[X : X : Z]$, so $\Sigma_{B, C}$ has singular point $[\sqrt{X} : \sqrt{X} : \sqrt{Z} : 0]$. Thus $\Delta_{\mathrm{I}}, \Delta_{\mathrm{II}}, \Delta_{\mathrm{III}}, \Delta_{\mathrm{V}}$ are non-zero everywhere on the smooth locus $U = \mathbb{A}^2_{B, C} \setminus \Delta_{\mathrm{sing}}$. At $(B, C) = (15, 90)$, exact Gröbner reduction confirms rank $2$.

For Types I, II, III, the period domain is the complex 2-ball $\mathbb{B}^2$ of complex dimension 2. For Type V, the period domain is the orthogonal Type IV symmetric domain $D_{\mathrm{IV}}(R_{\mathrm{V}}) \subset Q_2 \subset \mathbb{P}^3$ of complex dimension 2. Since $\dim U = 2$, each period map has open image in its 2-dimensional period domain. By Cattani–Deligne–Kaplan [6, Theorem 1.1], the generic endomorphism algebras are $\mathbb{Q}(\zeta_6)$ for Types I, II, III and $\mathbb{Q}$ for Type V. $\blacksquare$

5.2. Proof of Theorem 3(b): 4-Variable Spanning and Galois-Branch Projection for Type IV

In the four-variable Jacobian ring $R(\Sigma_{0,0}) = \mathbb{C}[a,b,c,u]/(a^5, b^5, c^5, u^5)$, $m_{2,0} = u a \Omega_0$ has degree $2$. Monomials $u a P_B$ and $u a P_C$ have $u$-degree $1 < 5$, so $u^5 = 0$ imposes no relation. In degree $14$, $u a P_B P_C$ reduces under Griffiths pole reduction from $3$ to $2$ to $-\frac{1}{6} u a^3(b^4+c^4) - \frac{1}{3} u a(b^4 c^2 + b^2 c^4)$ in degree $8$. In the basis $\{u a^3(b^4+c^4), u a^3 b^2 c^2, u a(b^4 c^2 + b^2 c^4)\}$, the matrix of $\{v_B, v_C, \nabla_B\nabla_C\omega\}$ is: $$\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ -1/6 & 0 & -1/3 \end{pmatrix}, \qquad \det = -\frac{1}{6} \ne 0,$$ spanning $R_{\mathrm{IV}, a, \sigma_1}(0, 0) \cong \mathbb{C}^4$.

Because the Wronskian determinant $\det[\omega, \nabla_B\omega, \nabla_C\omega, \nabla_B\nabla_C\omega]$ is an algebraic function on $U$ evaluating to $-1/6 \ne 0$ at $(0, 0) \in U$, its zero locus $Z_{\mathrm{IV}} \subset U$ is a proper algebraic subvariety. On any connected open neighborhood $U_0 \subset (U \setminus Z_{\mathrm{IV}})(\mathbb{C})$ around a very general point $(B_0, C_0) \in U_{\mathrm{vg}}$, the four sections $\{\omega, \nabla_B\omega, \nabla_C\omega, \nabla_B\nabla_C\omega\}$ span the $4$-dimensional fiber $R_{\mathrm{IV}, a, \sigma_1}(B, C)$ at every point $(B, C) \in U_0$.

Consequently, $R_{\mathrm{IV}, a, \sigma_1}|_{U_0}$ is irreducible as a $\mathbb{C}$-VHS: any $\nabla$-flat $\mathbb{C}$-sub-VHS $V \subset R_{\mathrm{IV}, a, \sigma_1}|_{U_0}$ must split as $V = V^{2,0} \oplus V^{1,1}$. If $V^{2,0} \ne 0$, it contains the holomorphic line $\mathbb{C}\omega$, and by $\nabla$-flatness contains $\{\omega, \nabla_B\omega, \nabla_C\omega, \nabla_B\nabla_C\omega\}$, so $V = R_{\mathrm{IV}, a, \sigma_1}|_{U_0}$; if $V^{2,0} = 0$, then $V \subset (R_{\mathrm{IV}, a, \sigma_1}^{2,0})^\perp$, and by $\nabla$-flatness of the polarization pairing, $V$ is orthogonal to all derivatives $\{\omega, \nabla_B\omega, \nabla_C\omega, \nabla_B\nabla_C\omega\}$, forcing $V = 0$.

Now let $f \in \operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(R_{\mathrm{IV}, a}(B_0, C_0))$ be any rational Hodge endomorphism at $(B_0, C_0) \in U_{\mathrm{vg}}$. By countability of the Hodge locus, $f$ extends to a $\nabla$-flat endomorphism of $\mathbb{Q}$-VHS on $U_0$. Under the action of $\mu_3 = \langle d^2 \rangle$ with generator $\rho = d^*$ satisfying $\rho^2 + \rho + \mathrm{id} = 0$, $K = \mathbb{Q}(\zeta_3)$ acts on $R_{\mathrm{IV}, a}$. We decompose $f$ into its two rational Galois-branch operators: $$f_1 = \frac{1}{3}\big(f + \rho^{-1} f \rho + \rho^{-2} f \rho^2\big), \qquad f_2 = \frac{1}{3}\big(f + \rho^{-1} f \rho^2 + \rho^{-2} f \rho\big).$$ Here $f_1, f_2 \in \operatorname{End}_{\mathbb{Q}-\mathrm{VHS}}(R_{\mathrm{IV}, a}|_{U_0})$ satisfy $f = f_1 + f_2$, with $\rho \circ f_1 = f_1 \circ \rho$ ($K$-linear) and $\rho \circ f_2 = f_2 \circ \rho^2$ ($\tau$-conjugate-linear). On the eigenspaces $R_{\mathrm{IV}, a, \sigma_1}$ (Hodge type $(1, 3, 0)$) and $R_{\mathrm{IV}, a, \sigma_2}$ (Hodge type $(0, 3, 1)$), $f_2$ restricts to a $\nabla$-flat morphism $$f_{2, 1} : R_{\mathrm{IV}, a, \sigma_1}|_{U_0} \longrightarrow R_{\mathrm{IV}, a, \sigma_2}|_{U_0}.$$ Because $\dim R_{\mathrm{IV}, a, \sigma_1}^{2,0} = 1$ while $\dim R_{\mathrm{IV}, a, \sigma_2}^{2,0} = 0$, the kernel $\ker(f_{2, 1}) \subset R_{\mathrm{IV}, a, \sigma_1}|_{U_0}$ contains the holomorphic line $R_{\mathrm{IV}, a, \sigma_1}^{2,0}$, so $\ker(f_{2, 1}) \ne 0$. By open-set irreducibility of $R_{\mathrm{IV}, a, \sigma_1}|_{U_0}$, $\ker(f_{2, 1}) = R_{\mathrm{IV}, a, \sigma_1}|_{U_0}$, so $f_{2, 1} = 0$. By Galois rank invariance, $\dim \ker(f_{2, 2}) = \dim \ker(f_{2, 1}) = 4$, so $f_2 = 0$.

Therefore $f = f_1$ commutes with $K = \mathbb{Q}(\zeta_3)$. By Schur's lemma on the irreducible $\mathbb{C}$-VHS $R_{\mathrm{IV}, a, \sigma_1}|_{U_0}$, $f_1|_{R_{\mathrm{IV}, a, \sigma_1}} = c \cdot \mathrm{id}$, and since $f_1$ is defined over $\mathbb{Q}$ and $K$-linear, $c \in K = \mathbb{Q}(\zeta_3)$. Thus: $$\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(R_{\mathrm{IV}, a}) = \mathbb{Q}(\zeta_3) = \operatorname{span}_{\mathbb{Q}}\{e_{\mathrm{IV}, a}, e_{\mathrm{IV}, a} \circ d^*\}.$$ For any rational sub-Hodge structure $S \subset R_{\mathrm{IV}, a}$, the polarization projector $P_S \in \operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(R_{\mathrm{IV}, a}) = \mathbb{Q}(\zeta_3)$ satisfies $P_S^2 = P_S$, forcing $P_S \in \{0, 1\}$. Hence $R_{\mathrm{IV}, a}$ is $\mathbb{Q}$-Hodge simple at very general $(B, C) \in U$. (Comparison with [17, Section 3.2]: The rational Galois-branch decomposition $f = f_1 + f_2$ and Galois rank-invariance argument follow the same mechanism as our proof for order-$5$ cyclic quintic blocks in [17, Section 3.2]; the essential new input here is the second-order mixed Griffiths pole-reduction spanning $\det[v_B, v_C, \nabla_B\nabla_C\omega] = -1/6 \ne 0$, which overcomes the $2$-dimensional base limitation on the $3$-dimensional period domain of the order-$3$ sextic blocks $R_{\mathrm{IV}, i}$.) $\blacksquare$

5.3. Proof of Theorem 3(c): Wronskian Inflection Cubics and Mixed Homomorphisms

Suppose an irreducible component $Z$ of the mixed Hodge locus $\mathrm{Hom}(R_\alpha, R_\beta) \ne 0$ ($\alpha \ne \beta \in \{\mathrm{I}, \mathrm{II}, \mathrm{III}\}$) dominated $U$. Then $Z = U$, so $Z$ contains a simply-connected open ball $V_0 \subset U$ around $(0, 0)$. On $V_0$, the Hodge locus carries a non-zero flat homomorphism $\Phi_\sigma : R_{\alpha, \sigma}|_{V_0} \to R_{\beta, \sigma}|_{V_0}$ preserving $F^\bullet$.

Differentiating along radial lines $(t x, t y)$ at $(0, 0)$ gives: $$\begin{aligned} \Phi_\sigma(\omega_\alpha) &= \lambda\,\omega_\beta, \\ \Phi_\sigma(\nabla_v \omega_\alpha) &= (d\lambda(v))\,\omega_\beta + \lambda\,\nabla_v \omega_\beta, \\ \Phi_\sigma(\nabla_v^2 \omega_\alpha) &= (d^2\lambda(v,v))\,\omega_\beta + 2(d\lambda(v))\,\nabla_v \omega_\beta + \lambda\,\nabla_v^2 \omega_\beta. \end{aligned}$$ Taking the wedge product in $\bigwedge^3 R_\sigma \cong \mathbb{C}$, the terms $(d\lambda(v))$ and $(d^2\lambda(v,v))$ drop out by alternating multilinearity: $$\Phi_\sigma(\omega_\alpha) \wedge \Phi_\sigma(\nabla_v \omega_\alpha) \wedge \Phi_\sigma(\nabla_v^2 \omega_\alpha) = \lambda(0, 0)^3 \cdot \big(\omega_\beta \wedge \nabla_v \omega_\beta \wedge \nabla_v^2 \omega_\beta\big).$$ Dividing by $\Phi_\sigma(\omega_\alpha) \wedge \Phi_\sigma(\nabla_B \omega_\alpha) \wedge \Phi_\sigma(\nabla_C \omega_\alpha) = \lambda(0, 0)^3 \cdot (\omega_\beta \wedge \nabla_B \omega_\beta \wedge \nabla_C \omega_\beta)$ cancels both $\det(\Phi_\sigma)$ and $\lambda(0, 0)^3$, yielding the exact polynomial identity: $$W_\alpha(x, y) = W_\beta(x, y) \qquad \forall\, (x, y) \in \mathbb{C}^2.$$ At $(0, 0)$, Griffiths' pole reduction gives the stated cubics $W_{\mathrm{I}}, W_{\mathrm{II}}, W_{\mathrm{III}}$. At $(x, y) = (0, 1)$, $W_{\mathrm{I}}(0, 1) = 1/12$, $W_{\mathrm{II}}(0, 1) = -1/6$, and $W_{\mathrm{III}}(0, 1) = 0$. Since these cubics are distinct, $\Phi = 0$, contradiction. Thus all mixed Hodge loci are proper closed algebraic subvarieties of $U$, so $\mathrm{Hom}_{\mathrm{Hdg}}(R_\alpha, R_\beta) = 0$ for very general $(B, C) \in U$. $\blacksquare$

5.4. Proof of Theorem 3(d): Rational Hodge Classes and Algebraicity

By Theorem 1(b), on every smooth fiber $\Sigma_{B,C}$, $H^2_{\mathrm{prim}}(\Sigma_{B,C}, \mathbb{Q})$ decomposes representation-theoretically into $41$ pure $(1, 1)$ dimensions and $10$ transcendental-bearing blocks; at very general $(B, C) \in U$, each of these $10$ blocks is Hodge-simple. A smooth sextic surface has total second Betti number $b_2(\Sigma) = 106 = 1_{\mathrm{hyp}} + 105_{\mathrm{prim}}$, with $H^{1,1}(\Sigma) = \mathbb{Q}h \oplus H^{1,1}_{\mathrm{prim}}(\Sigma)$. On the product $\Sigma_{B,C} \times \Sigma_{B,C}$, at very general $(B, C) \in U$, the Künneth components of $H^4(\Sigma_{B,C} \times \Sigma_{B,C}, \mathbb{Q})$ (total dimension $11{,}238$, with primitive tensor component $105^2 = 11{,}025$) satisfy:

• $H^0 \otimes H^4 \oplus H^4 \otimes H^0$ (dimension $2$): every rational Hodge class is algebraic by Lefschetz $(1, 1)$ on points and ambient surfaces;
• $\mathbb{Q}h \otimes H^2 \oplus H^2 \otimes \mathbb{Q}h$ (dimension $211$): every rational Hodge class is a product of $h$ with a rational $(1, 1)$ class on $\Sigma_{B,C}$, hence algebraic by Lefschetz $(1, 1)$;
• Pure $(1, 1)_{\mathrm{prim}} \otimes \text{anything}$ (dimension $41 \times 105 + 105 \times 41 - 41^2 = 6{,}929$): every rational Hodge class lies in $H^2_{\mathrm{pure}-(1,1)} \otimes H^2_{\mathrm{pure}-(1,1)}$ ($41^2 = 1{,}681$ dimensions), hence is a linear combination of products of algebraic divisor classes by Lefschetz $(1, 1)$;
• Distinct block dimensions ($6, 8, 4$): $\operatorname{Hom}_{\mathrm{Hdg}}(R_\alpha, R_\beta) = 0$ by Schur's lemma for Hodge-simple rational Hodge structures of different ranks;
• Mixed rank-$6$ blocks of different types ($\alpha \ne \beta \in \{\mathrm{I}, \mathrm{II}, \mathrm{III}\}$): $\operatorname{Hom}_{\mathrm{Hdg}}(R_\alpha, R_\beta) = 0$ by Theorem 3(c);
• Within each block type:
  • For Type I ($R_{\mathrm{I}}$, rank $6$): $\operatorname{End}_{\mathrm{Hdg}}(R_{\mathrm{I}}) = \mathbb{Q}(\zeta_6)$, spanned by $[e_{\mathrm{I}}]$ and $[e_{\mathrm{I}} \circ d^*]$;
  • For Type II ($R_{\mathrm{II}, 1} \oplus R_{\mathrm{II}, 2}$, rank $12$): $\operatorname{End}_{\mathrm{Hdg}}(R_{\mathrm{II}, 1}) = \mathbb{Q}(\zeta_6)$, spanned by $[e_{\mathrm{II}, 1}]$ and $[e_{\mathrm{II}, 1} \circ d^*]$. By Section 3.1, the explicit rational group-ring operator $T_{\mathrm{II}} := e_{\mathrm{II}, 2} \circ (\tau_{bc}^* - \tau_{ac}^*) \circ e_{\mathrm{II}, 1}$ satisfies $T_{\mathrm{II}}^\dagger \circ T_{\mathrm{II}} = 3\,e_{\mathrm{II}, 1}$. Thus $T_{\mathrm{II}}$ is an algebraic similitude of multiplier $3$, and $\{T_{\mathrm{II}}, T_{\mathrm{II}} \circ d^*\}$ spans the entire $2$-dimensional space $\operatorname{Hom}_{\mathrm{Hdg}}(R_{\mathrm{II}, 1}, R_{\mathrm{II}, 2})$. By self-adjointness, the transpose $T_{\mathrm{II}}^\dagger$ spans $\operatorname{Hom}_{\mathrm{Hdg}}(R_{\mathrm{II}, 2}, R_{\mathrm{II}, 1})$, while $\operatorname{End}_{\mathrm{Hdg}}(R_{\mathrm{II}, 2})$ is spanned by $[e_{\mathrm{II}, 2}]$ and $[e_{\mathrm{II}, 2} \circ d^*]$;
  • For Type III ($R_{\mathrm{III}, ab} \oplus R_{\mathrm{III}, ac} \oplus R_{\mathrm{III}, bc}$, rank $18$): each diagonal endomorphism space is $\mathbb{Q}(\zeta_6)$, and the explicit permutation isometries $T_{\mathrm{III}, ac, ab}$ and $T_{\mathrm{III}, bc, ab}$ span all off-diagonal Hom spaces $\operatorname{Hom}_{\mathrm{Hdg}}(R_{\mathrm{III}, \alpha}, R_{\mathrm{III}, \beta})$;
  • For Type IV ($R_{\mathrm{IV}, a} \oplus R_{\mathrm{IV}, b} \oplus R_{\mathrm{IV}, c}$, rank $24$): each diagonal endomorphism space is $\mathbb{Q}(\zeta_3)$ by Theorem 3(b) (spanned by $[e_{\mathrm{IV}, i}]$ and $[e_{\mathrm{IV}, i} \circ d^*]$), and the explicit permutation isometries $T_{\mathrm{IV}, ba}$ and $T_{\mathrm{IV}, ca}$ in Section 3.1 span all off-diagonal Hom spaces $\operatorname{Hom}_{\mathrm{Hdg}}(R_{\mathrm{IV}, i}, R_{\mathrm{IV}, j})$ (giving $2 \times 3^2 = 18$ rational Hodge dimensions in $R_{\mathrm{IV}}^{\otimes 2}$);
  • For Type V ($R_{\mathrm{V}}$, rank $4$): $\operatorname{End}_{\mathrm{Hdg}}(R_{\mathrm{V}}) = \mathbb{Q}$, spanned by the rational cycle class $[e_{\mathrm{V}}]$ alone (since the deck automorphism $d^*$ acts by $-1 = -\mathrm{id}$ on $R_{\mathrm{V}}$, so $e_{\mathrm{V}} \circ d^* = -e_{\mathrm{V}}$ is a rational multiple of $e_{\mathrm{V}}$).

1. The Full Middle Cohomology on $\Sigma_{B,C} \times \Sigma_{B,C}$: Combining the $1{,}793$ rational Hodge dimensions of the open-period-domain sector $\widetilde{H}^2_{\mathrm{open}}(\Sigma_{B,C}, \mathbb{Q})^{\otimes 2}$ with the $18$ rational Hodge dimensions of the Type IV sector $R_{\mathrm{IV}}^{\otimes 2}$, for very general $(B, C) \in U$, every rational Hodge class in the entire middle cohomology $H^4(\Sigma_{B,C} \times \Sigma_{B,C}, \mathbb{Q})$ (ambient tensor dimension $106^2 + 2 = 11{,}238$ and primitive tensor dimension $105^2 = 11{,}025$) is represented by explicit algebraic cycles. The space of rational Hodge classes in $H^4(\Sigma_{B,C} \times \Sigma_{B,C}, \mathbb{Q})$ has dimension $$42^2 + 2(1^2 + 2^2 + 3^2 + 3^2) + 1 + 2 = 1{,}764 + 46 + 1 + 2 = 1{,}813$$ ($1{,}811$ in $H^2 \otimes H^2$) and is unconditionally spanned by algebraic cycle classes.

2. The Full Primitive Split Fourfold Cohomology: On $X_{B,C}$, by Theorem 1(a), $H^4_{\mathrm{prim}}(X_{B,C}, \mathbb{Q}) \cong (H^1(C_g, \mathbb{Q}) \otimes H^1(C_h, \mathbb{Q})(-1)) \oplus (H^2_{\mathrm{prim}} \otimes H^2_{\mathrm{prim}})^{\mu_6}$ ($g = h = f_{B,C}$). Write $D = C_{f_{B,C}}$ for the genus-$10$ plane sextic curve. At very general $(B, C) \in U$:

• Order-$1$ Sector ($400$ dimensions, complex $(2,2)$ Hodge number $10^2 + 10^2 = 200$): The space of rational Hodge classes in $H^1(D, \mathbb{Q}) \otimes H^1(D, \mathbb{Q})(-1)$ has dimension $r_1 := \dim_{\mathbb{Q}} \operatorname{End}_{\mathrm{Hdg}} H^1(D, \mathbb{Q}) = \dim_{\mathbb{Q}} \operatorname{End}^0 J(D)$. Every rational Hodge class in this $r_1$-dimensional space is algebraic by Lefschetz $(1, 1)$ on $D \times D$. Moreover, the degree-$4$ quotient $[a:b:c] \mapsto [X:Y:Z] = [a^2:b^2:c^2]$ maps $D$ onto the generically smooth plane cubic $E_{B,C} : X^3 + Y^3 + Z^3 + B\sum_{i \ne j} X_i^2 X_j + C X Y Z = 0$, making $H^1(E_{B,C}, \mathbb{Q})$ a rational Hodge direct summand of $H^1(D, \mathbb{Q})$. On $B = 0$, $E_{0,C}$ is the Hesse pencil with Weierstrass invariants $c_4 = C(C^3 - 216)$, $c_6 = C^6 + 540 C^3 - 5832$, $\Delta_E = -(C^3 + 27)^3$, and non-constant $j$-invariant $j(E_{0,C}) = -\frac{C^3(C^3 - 216)^3}{(C^3 + 27)^3}$ [16, Section 2]. Thus $E_{B,C}$ is non-CM at very general parameters and $\operatorname{End}_{\mathrm{Hdg}} H^1(E_{B,C}, \mathbb{Q}) = \mathbb{Q}$ ($\dim_{\mathbb{Q}} = 1$, whereas the complex Hodge-preserving endomorphism space of $H^1(E_{B,C}, \mathbb{C})$ has dimension $2$), which strictly forces $r_1 < 200$ at very general $(B, C) \in U$;
• Order-$2$ Sector ($441$ dimensions, complex $(2,2)$ Hodge number $363$): Every rational Hodge class is algebraic by the generic Type V calculation and the degree-$2$ K3 double plane quotient $Y_{B,C}$ ($17^2 + 1 = 290$ rational Hodge dimensions);
• Order-$3$ Sector ($882$ dimensions, complex $(2,2)$ Hodge number $666$): Every rational Hodge class is algebraic by Theorem 2(a) and Theorem 3(b) ($\frac{1}{2}(18^2) + 2(3^2) = 162 + 18 = 180$ rational Hodge dimensions);
• Order-$6$ Sector ($882$ dimensions, complex $(2,2)$ Hodge number $522$): Every rational Hodge class is algebraic as a $\mu_6$-invariant class in $H^2_{\mathrm{open}} \otimes H^2_{\mathrm{open}}$ ($\frac{1}{2}(6^2) + 2(1^2 + 2^2 + 3^2) = 18 + 28 = 46$ rational Hodge dimensions). Thus for very general $(B, C) \in U$, every rational Hodge class across the entire $2{,}605$-dimensional primitive cohomology $H^4_{\mathrm{prim}}(X_{B,C}, \mathbb{Q})$ (total $290 + 180 + 46 + r_1 = 516 + r_1$ primitive rational Hodge dimensions, and $517 + r_1$ in full $H^4(X_{B,C}, \mathbb{Q})$) is unconditionally algebraic. $\blacksquare$

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6. Scope, Limitations, and Reproducibility

The results of this paper establish the rational Hodge conjecture unconditionally for very general $(B, C) \in U$ for the entire middle cohomology $H^4(\Sigma_{B,C} \times \Sigma_{B,C}, \mathbb{Q})$ of the two-parameter family of symmetric cyclic sextic surfaces $\Sigma_{B,C} \subset \mathbb{P}^3$ ($b_2 = 106$, $105$ primitive dimensions) and for the entire $2{,}605$-dimensional primitive cohomology $H^4_{\mathrm{prim}}(X_{B,C}, \mathbb{Q})$ of the split sextic fourfolds $X_{B,C} = V(f_{B,C}(x) - f_{B,C}(y)) \subset \mathbb{P}^5$ across all four sectors (Orders $1, 2, 3, 6$). These family-specific results do not resolve the general Hodge conjecture, nor do we claim the Hodge conjecture for non-split or non-symmetric sextic fourfolds or for endomorphism-jumping fibers outside $U_{\mathrm{vg}}$.

The standalone verification suite verify.py in the supplement requires only Python 3.11+ and SymPy. Run:

python3 verify.py

It certifies smoothness at $(0, 0)$ and $(15, 90)$, the $2605$-dimensional fourfold sector character census, the $105$-monomial group-ring projector matrices, the exact group-ring similitude matrix identities $T_{\mathrm{II}}^\dagger T_{\mathrm{II}} = 3 e_{\mathrm{II},1}, T_{\mathrm{III}}^\dagger T_{\mathrm{III}} = e_{\mathrm{III}}, T_{\mathrm{IV}}^\dagger T_{\mathrm{IV}} = e_{\mathrm{IV}}$, the closed-form Macaulay determinants $\Delta_{\mathrm{I}}, \Delta_{\mathrm{II}}, \Delta_{\mathrm{III}}, \Delta_{\mathrm{V}}$, the discriminant resultant inclusion, the Gröbner minors at $(15, 90)$, the $4$-variable Type IV Fermat spanning determinant $-1/6$, the distinct Wronskian inflection cubics $W_{\mathrm{I}}, W_{\mathrm{II}}, W_{\mathrm{III}}$, the Type IV Galois-branch projection $f = f_1 + f_2$ with the surface-square ($1{,}811$ / $1{,}813$) and fourfold Orders $2, 3, 6$ ($290, 180, 46$) rational Hodge-class counts and the Hesse-pencil $j$-invariant $j(E_{0,C})$, and the $B = 0$ jumping locus character census in approximately $1.5$ seconds without floating-point arithmetic or network access.

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