# Generic Algebraic Correspondences and the Open-Period-Domain Sector of 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. 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 split fourfolds.

In this paper, we resolve these open questions for the open-period-domain sector. We construct a complete $(\mu_6 \times W(B_3))$-isotypic decomposition of the entire $105$-dimensional primitive cohomology of $\Sigma_{B,C}$, isolating an $81$-dimensional open-period-domain subvariation $H^2_{\mathrm{open}}(\Sigma_{B,C}, \mathbb{Q})$ where the period maps into the unitary 2-balls $\mathbb{B}^2$ and orthogonal type-IV quadric $D_{\mathrm{IV}}^2$ are submersions. We construct explicit rational group-ring similitudes $T_{\mathrm{II}}, T_{\mathrm{III}}$ 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 Wronskian inflection cubics that all mixed Hodge morphisms between distinct types vanish at very general parameters. Consequently, every rational Hodge class in the $6{,}724$-dimensional open-period sector of $\Sigma_{B,C} \times \Sigma_{B,C}$ (containing $100\%$ of the $42$-dimensional primitive sixth-root sector) and in the $1{,}723$-dimensional primitive sector of the split fourfold $X_{B,C}$ 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 Hodge numbers $(1, 426, 1751, 426, 1)$) decomposes into four $\mu_6$-order sectors:
$$\begin{aligned}
\text{Order 1 } (k = 0, C_g \times C_h): \quad &(0, 100, 200, 100, 0), \quad \dim = 400; \\
\text{Order 2 } (k = 3, \chi_3 \otimes \chi_3): \quad &(1, 38, 363, 38, 1), \quad \dim = 441; \\
\text{Order 3 } (k \in \{2, 4\}, \chi_2 \otimes \chi_4 \oplus \chi_4 \otimes \chi_2): \quad &(0, 108, 666, 108, 0), \quad \dim = 882; \\
\text{Order 6 } (k \in \{1, 5\}, \chi_1 \otimes \chi_5 \oplus \chi_5 \otimes \chi_1): \quad &(0, 180, 522, 180, 0), \quad \dim = 882.
\end{aligned}$$
Every rational Hodge class in the $200$-dimensional middle part of the order-$1$ sector 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**.

The three sectors Order 1 ($400$), Order 2 ($441$), and Order 6 ($882$) total:
$$400 + 441 + 882 = 1{,}723 \text{ dimensions of } H^4_{\mathrm{prim}}(X_{B,C}, \mathbb{Q}),$$
containing $200 + 363 + 522 = 1{,}085$ of the $1{,}751$ complex dimensions of $H^{2,2}_{\mathrm{prim}}(X_{B,C})$.

**(b) Surface $(\mu_6 \times W(B_3))$-Isotypic Decomposition:** 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 $5$ irreducible block types:
   - **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 $81$-dimensional 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 $82$-dimensional 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.
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. 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 ($666 = 180 + 162 + 324$):** In the order-$3$ middle cohomology $(H^2_{\chi_2}(\Sigma_g) \otimes H^2_{\chi_4}(\Sigma_h) \oplus H^2_{\chi_4}(\Sigma_g) \otimes H^2_{\chi_2}(\Sigma_h))^{\mu_6} \cap H^{2,2}$, the $666$ complex dimensions decompose into:
   - **Transcendental $\otimes$ Transcendental ($180$ dimensions):** Contained in the $576$-dimensional rational variation $(R_{\mathrm{IV}, a} \oplus R_{\mathrm{IV}, b} \oplus R_{\mathrm{IV}, c}) \otimes (R_{\mathrm{IV}, a} \oplus R_{\mathrm{IV}, b} \oplus R_{\mathrm{IV}, c})$ (rank $24 \times 24 = 576$), which reduces via $\Gamma_i \otimes \Gamma_j$ to the $9$ rank-$64$ K3 products $\widetilde{W}_{g,i} \times \widetilde{W}_{h,j}$ ($9 \times 8^2 = 576$);
   - **Algebraic $\otimes$ Algebraic ($162$ dimensions):** The tensor product of the $18$ pure $(1, 1)$ classes on $\Sigma_g$ and $\Sigma_h$ ($H^2_{\mathrm{pure}-(1,1)} \otimes H^2_{\mathrm{pure}-(1,1)}$, dimension $18 \times 18 = 324$, containing $162$ dimensions of $H^{2,2}$), which are products of algebraic divisors, hence **unconditionally algebraic by Lefschetz $(1, 1)$ on $\Sigma_g$ and $\Sigma_h$**;
   - **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 parameters because $R_{\mathrm{IV}}$ is an irreducible transcendental Hodge structure 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 for Type IV in $\mathbb{Q}[a, b, c, u]$:** 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 $-\frac{1}{6} \ne 0$.

**(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 $L(\omega_\alpha) = \lambda\,\omega_\beta$, forcing the projective transformation law $W_\beta(x, y) = (\det L)\lambda(0, 0)^3 W_\alpha(x, y)$. Since $W_{\mathrm{I}}, W_{\mathrm{II}}, W_{\mathrm{III}}$ are pairwise non-proportional, $\Phi = 0$. Thus all mixed Hodge loci are proper closed algebraic subvarieties of $U$, so for very general $(B, C) \in U$:
$$\mathrm{Hom}_{\mathrm{Hdg}}(R_\alpha,\; R_\beta) = 0 \qquad \forall\, \alpha \ne \beta \in \{\mathrm{I},\; \mathrm{II},\; \mathrm{III}\}.$$

**(d) Rational Hodge Classes and Algebraicity:**
1. **Unconditional Open-Period-Domain Sector on $\Sigma_{B,C} \times \Sigma_{B,C}$:** Combining parts (a)–(c) with the explicit rational group-ring similitude $T_{\mathrm{II}}$ ($T_{\mathrm{II}}^\dagger T_{\mathrm{II}} = 3 e_{\mathrm{II},1}$), the permutation isometries $T_{\mathrm{III}}$, and Lefschetz $(1, 1)$, for very general $(B, C) \in U$, **every rational Hodge class in the open-period-domain subvariation $\widetilde{H}^2_{\mathrm{open}}(\Sigma_{B,C}, \mathbb{Q})^{\otimes 2} \subset H^4(\Sigma_{B,C} \times \Sigma_{B,C}, \mathbb{Q})$ (ambient dimension $82^2 = 6{,}724$, and primitive tensor dimension $81^2 = 6{,}561$), which contains $100\%$ of the $42$-dimensional primitive sixth-root sector $H^2_{\mathrm{ord}=6}(\Sigma_{B,C}, \mathbb{Q})$, is algebraic**. The space of rational Hodge classes in this sector has dimension $1{,}793$ and is spanned by explicit algebraic cycle classes.
2. **Fourfold Split Sector:** On the split fourfold $X_{B,C}$, the corresponding $1{,}723$-dimensional primitive sector (comprising the $400$-dim order-$1$ sector, the $441$-dim order-$2$ sector, and the $882$-dim order-$6$ sector), containing $1{,}085$ complex dimensions of $H^{2,2}_{\mathrm{prim}}(X_{B,C})$, 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, establishing that for very general $C$ on the slice $B = 0$, the primitive Picard number jumps by $\Delta\rho = +19$ ($\rho_{\mathrm{prim}}(\Sigma_{0,C}) = 41 + 19 = 60$). Meanwhile, the first $C$-derivative $v_C = a^3 b^3 c^2 \ne 0$ remains non-zero in the surviving rank-$4$ $(1, 2, 1)$ transcendental sub-block, while the reduced second derivative vanishes identically ($d_{CC,\mathrm{III}} = 0$).

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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`.

---

## 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), 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, $\omega_{W_{g,a}} \cong \mathcal{O}(6 - (1+1+2+2)) = \mathcal{O}$. The $3$ points $P_\nu = [0 : 0 : 1 : \omega^\nu]$ 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 $\chi_2 \oplus \chi_4$.
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. $\blacksquare$

---

## 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 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 $R_{\mathrm{IV}, a, \sigma}(0, 0) \cong \mathbb{C}^4$. $\blacksquare$

### 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), $H^2_{\mathrm{prim}}(\Sigma_{B,C}, \mathbb{Q})$ decomposes into $41$ pure $(1, 1)$ dimensions and $10$ irreducible blocks. 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 $\Sigma_{B,C} \times \Sigma_{B,C}$, the Künneth components of $H^4(\Sigma_{B,C} \times \Sigma_{B,C}, \mathbb{Q})$ (total dimension $11{,}238$, with primitive tensor component $105^2 = 11{,}025$) are:
- $H^0 \otimes H^4 \oplus H^4 \otimes H^0$ (dimension $2$): algebraic by Lefschetz $(1, 1)$ on points and ambient surfaces;
- $\mathbb{Q}h \otimes H^2 \oplus H^2 \otimes \mathbb{Q}h$ (dimension $211$): algebraic by Lefschetz $(1, 1)$ on the hyperplane class factor;
- Pure $(1, 1)_{\mathrm{prim}} \otimes \text{anything}$ (dimension $41 \times 105 + 105 \times 41 - 41^2 = 6{,}929$): algebraic by Lefschetz $(1, 1)$ on the primitive divisor classes;
- Distinct block dimensions ($6, 8, 4$): $\operatorname{Hom}_{\mathrm{Hdg}}(R_\alpha, R_\beta) = 0$ by Schur's lemma for irreducible 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 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 Unconditional Open-Period-Domain Sector on $\Sigma_{B,C} \times \Sigma_{B,C}$:**
The $7$ blocks $R_{\mathrm{I}}, R_{\mathrm{II}, 1}, R_{\mathrm{II}, 2}, R_{\mathrm{III}, ab}, R_{\mathrm{III}, ac}, R_{\mathrm{III}, bc}, R_{\mathrm{V}}$ have period domain dimension $m = 2 = \dim U$. Together with the $41$ pure $(1, 1)$ primitive classes, they constitute the $81$-dimensional open-period-domain variation $H^2_{\mathrm{open}}(\Sigma_{B,C}, \mathbb{Q}) \subset H^2_{\mathrm{prim}}(\Sigma_{B,C}, \mathbb{Q})$ (and $82$-dimensional in full $H^2(\Sigma_{B,C}, \mathbb{Q})$). On this subvariation, every block has an open period image in its 2-dimensional period domain, and every rational Hodge class in $\widetilde{H}^2_{\mathrm{open}}(\Sigma_{B,C}) \otimes \widetilde{H}^2_{\mathrm{open}}(\Sigma_{B,C})$ (ambient dimension $82 \times 82 = 6{,}724$ in $H^4(\Sigma \times \Sigma)$) is represented by explicit algebraic cycles. This includes $100\%$ of the $42$-dimensional primitive sixth-root sector $H^2_{\mathrm{ord}=6}(\Sigma_{B,C}, \mathbb{Q})$. The space of rational Hodge classes in this sector has dimension $1{,}793$ and is spanned by explicit algebraic cycle classes.

**2. The Unconditional Split Fourfold Sector:**
On $X_{B,C}$, by Theorem 1(a), $H^4_{\mathrm{prim}}(X_{B,C}, \mathbb{Q}) \cong (H^1(C_g) \otimes H^1(C_h)(-1)) \oplus (H^2_{\mathrm{prim}} \otimes H^2_{\mathrm{prim}})^{\mu_6}$. The order-$1$ sector ($400$ dimensions, $200$ in $H^{2,2}$) is algebraic by Lefschetz $(1, 1)$ on $C_g \times C_h$; the order-$2$ sector ($441$ dimensions, $363$ in $H^{2,2}$) is algebraic by the degree-$2$ K3 double plane quotient $Y_{B,C}$; and the order-$6$ sector ($882$ dimensions, $522$ in $H^{2,2}$) belongs entirely to the open-period-domain variation $H^2_{\mathrm{open}} \otimes H^2_{\mathrm{open}}$. Thus all $1{,}723$ dimensions of this sector ($1{,}085$ in $H^{2,2}$) are unconditionally algebraic on $X_{B,C}$ for very general $(B, C) \in U$. $\blacksquare$

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## 6. Scope, Limitations, and Reproducibility

The results of this paper establish the rational Hodge conjecture unconditionally for the $81$-dimensional open-period-domain sector $H^2_{\mathrm{open}}(\Sigma_{B,C}, \mathbb{Q})$ of the two-parameter family of symmetric split sextic fourfolds $X_{B,C} \subset \mathbb{P}^5$ and surface squares $\Sigma_{B,C} \times \Sigma_{B,C}$, covering $100\%$ of the $42$-dimensional primitive sixth-root sector and $6{,}724$ dimensions in $H^4(\Sigma \times \Sigma)$ ($6{,}561$ primitive), and for the $1{,}723$-dimensional primitive sector of $X_{B,C}$. We do not claim the Hodge conjecture for general (non-split or non-symmetric) sextic fourfolds, nor for special jumping loci where extra endomorphisms arise.

The standalone verification suite `verify.py` in the supplement requires only Python 3.11+ and SymPy. Run:
```bash
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 $W_{\mathrm{I}}, W_{\mathrm{II}}, W_{\mathrm{III}}$, and the $B = 0$ jumping locus character census in approximately 1.1 seconds without floating-point arithmetic or network access.

---

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