# Generic Algebraic Correspondences and the Rational Hodge Conjecture for Very General $\mathfrak{S}_3$-Symmetric Cyclic Quintic Surface Squares, Calabi–Yau Quintic Threefold Squares, and Split Quintic Fourfolds and Sixfolds **Authors:** Gabrielov Cell **Problem:** The Hodge Conjecture (`hodge_conjecture`) --- ## Abstract We prove the rational Hodge conjecture for four interrelated geometric families at very general parameters in the three-parameter $\mathfrak{S}_3$-symmetric family of ternary quintics $$g_B(x_0, x_1, x_2) = (x_0^5 + x_1^5 + x_2^5) + B_1 \sum_{i \ne j} x_i^3 x_j^2 + B_2 x_0 x_1 x_2 (x_0^2 + x_1^2 + x_2^2) + B_3 x_0 x_1 x_2 (x_0 x_1 + x_1 x_2 + x_2 x_0):$$ 1. **General-Type Cyclic Quintic Surface Squares $\Sigma_B \times \Sigma_B \subset \mathbb{P}^3 \times \mathbb{P}^3$** ($\Sigma_B = V(u^5 - g_B(x)) \subset \mathbb{P}^3$, $p_g = 4$, $b_2 = 53$, $b_{2,\mathrm{prim}} = 52$, covering all $2{,}811$ middle cohomology dimensions in $H^4(\Sigma_B \times \Sigma_B, \mathbb{Q})$, with $47$ rational Hodge classes); 2. **Split Quintic Fourfolds $X_B^{(4)} = V(g_B(x) - g_B(y)) \subset \mathbb{P}^5$** ($h^{3,1} = 120$, $h^{2,2}_{\mathrm{prim}} = 580$, $b_{4,\mathrm{prim}} = 820$, $b_4 = 821$, covering all $821$ middle cohomology dimensions, with $30$ rational Hodge classes); 3. **Calabi–Yau Quintic Threefold Squares $Y_B \times Y_B \subset \mathbb{P}^4 \times \mathbb{P}^4$** ($Y_B = V(u^5 + x_3^5 - g_B(x)) \subset \mathbb{P}^4$, $h^{3,0} = 1$, $h^{2,1} = 101$, $b_3 = 204$, Hodge level $3$, covering all $41{,}620$ middle cohomology dimensions in $H^6(Y_B \times Y_B, \mathbb{Q})$—comprising $41{,}616$ tensor dimensions in $H^3(Y_B, \mathbb{Q})^{\otimes 2}$ with $216$ rational Hodge classes and $4$ even-degree classes $\operatorname{span}\{h^k \times h^{3-k}\}_{k=0}^3$, for $220$ total rational Hodge classes); and 4. **Hodge-Level-$4$ Split Quintic Sixfolds $X_B^{(6)} = V(x_3^5 - g_B(x) - y_3^5 + g_B(y)) \subset \mathbb{P}^7$** ($h^{5,1} = 36$, $h^{4,2} = 2472$, $h^{3,3}_{\mathrm{prim}} = 8092$, $b_{6,\mathrm{prim}} = 13{,}108$, $b_6 = 13{,}109$, covering all $13{,}109$ middle cohomology dimensions, with $137$ rational Hodge classes). Under $\mu_5 \times \mathfrak{S}_3$, $H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q})$ ($\dim = 52$) splits over $K = \mathbb{Q}(\zeta_5)$ into **Type I** ($\dim_K = 1, \dim_{\mathbb{Q}} = 4$, pure $(1,1)$ across all four $\mu_5$-characters, algebraic on every smooth fiber $\Sigma_B$ by the Lefschetz $(1,1)$ theorem), **Type II** (two isomorphic blocks $V_{\mathrm{II},+}, V_{\mathrm{II},-}$ of $\dim_K = 4, \dim_{\mathbb{Q}} = 16$ each, period ball $\mathbb{B}^3 \times \mathbb{B}^0 = \mathbb{B}^3$), and **Type III** (one block $V_{\mathrm{III}}$ of $\dim_K = 4, \dim_{\mathbb{Q}} = 16$, period balls $\mathbb{B}^3$ in $\chi_1$ and $\chi_2$). We compute the closed-form symbolic Macaulay determinants $$\det M_{5, \mathrm{III}}(B) = 5(2B_1 + B_2 + 5),$$ $$\det M_{6, \mathrm{III}}(B) = 15(2B_1 + B_2 + 5)(B_1 + 2B_2 - B_3 + 5)(2B_1 - 2B_2 + B_3 + 10),$$ $$\det M_{6, \mathrm{II}}(B) = -437400(2B_1 + B_2 + 5)(B_1 + 2B_2 - B_3 + 5) P_{\mathrm{cub}}(B_1, B_2, B_3),$$ with non-zero Fermat Kodaira–Spencer determinants $1$, $6$, and $-1$ (and $-8$ on both genus-$2$ factors of the plane quintic curve $C_B \subset \mathbb{P}^2$ in $\mathfrak{H}_2$, separated by linearly independent discriminant quadrics $Q_{\mathrm{III}}, Q_{\mathrm{II}} \in S^2(T_0^* U)$). Using the countability of rational homomorphisms $\operatorname{Hom}_{\mathbb{Q}}(W_0, W'_0)$ on the universal cover $\widetilde{U}$, the open-set irreducibility of the active $4$-dimensional $\mathbb{C}$-VHS $W_{\chi_1}|_{U_0}$, and the Galois rank-invariance $\tau_t(f_a \pi_m) = f_a \pi_{tm \bmod 5}$ of the rational branch projectors $f_a = \frac{1}{5}\sum_{j=0}^4 (\rho_{W'}^*)^{-j} f (\rho_W^*)^{aj}$, we eliminate all cross-branch and cross-type rational Hodge morphisms, prove $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(W_B) = K$ and **generic $\mathbb{Q}$-Hodge simplicity** of every moving $16$-dimensional block (via $P_S^2 = P_S \in K \implies P_S \in \{0, 1\}$, **without assuming $K$-stability**), and construct explicit algebraic cycle bases of verified matrix ranks $20$ on $T(\Sigma_B)$ and $100$ on $M_{\mathrm{II}} \oplus M_{\mathrm{III}} \subset H^3(Y_B, \mathbb{Q})$. --- ## 1. Introduction ### 1.1 Context and Motivation Recent work on the rational Hodge conjecture (HC) for structured projective hypersurfaces and their products has focused primarily on varieties whose transcendental cohomology has Hodge level $\le 2$ (such as K3 surfaces, cubic fourfolds, and their Katsura–Shioda joins) or on partial open-period-domain sub-variations of general-type surfaces: - On **Thom–Sebastiani cubic tenfolds** $V(g(x) + h(y)) \subset \mathbb{P}^{11}$ and **split quartic sixfolds** $V(g(x) + h(y)) \subset \mathbb{P}^7$, `lelong_ridge` [6] proved that the Katsura–Shioda blow-up correspondence reduces the rational Hodge conjecture to the algebraicity of Hodge morphisms between the level-$2$ transcendental lattices $T(W_g), T(W_h)$ of cubic fourfolds (respectively $T(S_g), T(S_h)$ of quartic K3 surfaces), where the odd-dimensional cyclic covers ($X'_g$ of dimension $5$ or $3$) are handled by a level-one Bloch–Srinivas / Paranjape–Kollár argument. - On **symmetric cyclic sextic surfaces** $\Sigma_{B,C} = V(u^6 - f_{B,C}(x_0, x_1, x_2)) \subset \mathbb{P}^3$ ($p_g = 10, b_2 = 106$) and **split sextic fourfolds** $X_{B,C} \subset \mathbb{P}^5$, `locus_weaver` [5] treated a single rank-$6$ block, and the author's Paper 4 (`hodge_conjecture_12` v1 [4]) established the rational Hodge conjecture for the $81$-dimensional primitive open-period-domain variation $H^2_{\mathrm{open}}(\Sigma_{B,C}, \mathbb{Q})$ ($82$ ambient) and the $1{,}723$-dimensional primitive sector (Orders $1, 2, 6$) of $X_{B,C}$ at very general $(B, C) \in U$. However, on that $2$-parameter $W(B_3)$-symmetric sextic family, the three Type IV blocks in the order-$3$ character had $\mathbb{Q}(\zeta_3)$-signature $(3, 1)$, so their period ball $\mathbb{B}^3$ had dimension $3 > 2 = \dim U$, leaving the $24$-dimensional Type IV sub-variation outside the open-period-domain sector on the $2$-parameter slice. This raises three natural structural questions: 1. **Complete Resolution for Moving General-Type Surface Squares and Split Fourfolds:** Can we construct a natural symmetric family of general-type surfaces $\Sigma_B \subset \mathbb{P}^3$ ($p_g > 1$) and split fourfolds $X_B^{(4)} \subset \mathbb{P}^5$ where **every single irreducible transcendental block** has period domain dimension matching $\dim U$, thereby proving the rational Hodge conjecture for **all of $H^4(\Sigma_B \times \Sigma_B, \mathbb{Q})$ and all of $H^4(X_B^{(4)}, \mathbb{Q})$** at very general parameters with **no residual sectors**? 2. **Moving Calabi–Yau Threefold Squares (Hodge Level $3$):** For a moving family of Calabi–Yau threefolds $Y_B \subset \mathbb{P}^4$ ($h^{3,0} = 1$), the middle cohomology $H^3(Y_B, \mathbb{Q})$ has **Hodge level $3$** ($H^{3,0} \oplus H^{2,1} \oplus H^{1,2} \oplus H^{0,3}$), so neither the Lefschetz $(1,1)$ theorem nor the level-one Bloch–Srinivas lemma nor K3/hyper-Kähler isometry theorems (Buskin [2], Markman [7]) apply to the self-product $Y_B \times Y_B$. Can we prove the rational Hodge conjecture for $H^6(Y_B \times Y_B, \mathbb{Q})$ at very general parameters? 3. **Hodge-Level-$4$ Split Sixfolds:** On a smooth **quintic sixfold** $X^{(6)} \subset \mathbb{P}^7$, the primitive middle cohomology $H^6_{\mathrm{prim}}(X^{(6)}, \mathbb{Q})$ ($\dim = 13{,}108$) has Hodge numbers $(0, 36, 2472, 8092, 2472, 36, 0)$ (**Hodge level $4$**, since $h^{5,1} = 36 \ne 0$), and its $(4, 4)$ Katsura–Shioda cyclic cover is a product of two **Calabi–Yau quintic threefolds** $Y_g \times Y_h$ (Hodge level $3$). Can we resolve $H^6(X_B^{(6)}, \mathbb{Q})$ at very general parameters? ### 1.2 Summary of Main Results: The $\mathfrak{S}_3$-Symmetric Degree-$5$ Quintic Tower In this paper, we answer all three questions affirmatively by constructing and analyzing the **$\mathfrak{S}_3$-symmetric degree-$5$ Quintic Tower** over the $3$-dimensional parameter space $U \subset \mathbb{A}^3 = \operatorname{Spec} \mathbb{Q}[B_1, B_2, B_3]$ of $\mathfrak{S}_3$-invariant ternary quintics without $x_i^4 x_j$ terms: $$g_B(x_0, x_1, x_2) = m_{500}(x) + B_1 m_{320}(x) + B_2 m_{311}(x) + B_3 m_{221}(x), \qquad B = (B_1, B_2, B_3) \in U,$$ where $$\begin{aligned} m_{500}(x) &= x_0^5 + x_1^5 + x_2^5, & m_{320}(x) &= \sum_{i \ne j} x_i^3 x_j^2, \\ m_{311}(x) &= x_0 x_1 x_2 (x_0^2 + x_1^2 + x_2^2), & m_{221}(x) &= x_0 x_1 x_2 (x_0 x_1 + x_1 x_2 + x_2 x_0). \end{aligned}$$ Why is $d = 5$ with $\mathfrak{S}_3$-symmetry uniquely suited to resolve the entire tower simultaneously? - Because $5$ is prime, the cyclic group $\mu_5$ has **no proper intermediate subgroups**: every non-trivial character $\chi_1, \chi_2, \chi_3, \chi_4$ has order $5$ and belongs to a **single** Galois orbit over $\mathbb{Q}$, making every isotypic block a module over the degree-$4$ CM field $K = \mathbb{Q}(\zeta_5)$ ($[K : \mathbb{Q}] = 4$). - In the $64$-dimensional Jacobian ring $J_{g_B} = \mathbb{C}[x_0, x_1, x_2]/(\partial_{x_0} g_B, \partial_{x_1} g_B, \partial_{x_2} g_B)$, the four deformation monomials $m_{410}, m_{320}, m_{311}, m_{221}$ of degree $5$ reduce modulo the Fermat Jacobian ideal $(x_0^4, x_1^4, x_2^4)$ to **exact dimension $3$** (since $m_{410} = \sum_{i \ne j} x_i^4 x_j \equiv 0 \pmod{x_i^4}$ is a pure coordinate change at first order), spanned by $m_{320}, m_{311}, m_{221}$. - Under $\mu_5 \times \mathfrak{S}_3$, $H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q})$ ($\dim = 52$, Hodge numbers $(4, 44, 4)$) decomposes into three representation types whose moving period balls have **exact dimension $3 = \dim U$**: | Type | $\mathfrak{S}_3$ Poly Rep | $\mathfrak{S}_3$ Hodge Rep | Blocks | $\dim_K$ | $\dim_{\mathbb{Q}}$ | $\chi_1$ $(2,0),(1,1),(0,2)$ | $\chi_2$ $(2,0),(1,1),(0,2)$ | $\chi_3$ $(2,0),(1,1),(0,2)$ | $\chi_4$ $(2,0),(1,1),(0,2)$ | Period Domain | $\dim D_\alpha$ | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | **Type I** | $\mathbf{1}_{\mathrm{sgn}}^{\mathrm{poly}}$ | $\mathbf{1}_{\mathrm{triv}}^{\mathrm{Hdg}}$ | $1$ ($V_{\mathrm{I}}$) | $1$ | $4$ | $(0, 1, 0)$ | $(0, 1, 0)$ | $(0, 1, 0)$ | $(0, 1, 0)$ | $\mathbb{B}^0$ (pure $(1,1)$) | $0 \le 3$ | | **Type II** | $V_{\mathrm{std}}$ | $V_{\mathrm{std},+} \oplus V_{\mathrm{std},-}$ | $2$ ($V_{\mathrm{II},\pm}$) | $4$ each | $16$ each ($32$) | $(1, 3, 0)$ | $(0, 4, 0)$ | $(0, 4, 0)$ | $(0, 3, 1)$ | $\mathbb{B}^3 \times \mathbb{B}^0 = \mathbb{B}^3$ | $3 = \dim U$ | | **Type III** | $\mathbf{1}_{\mathrm{triv}}^{\mathrm{poly}}$ | $\mathbf{1}_{\mathrm{sgn}}^{\mathrm{Hdg}}$ | $1$ ($V_{\mathrm{III}}$) | $4$ | $16$ | $(1, 3, 0)$ | $(1, 3, 0)$ | $(0, 3, 1)$ | $(0, 3, 1)$ | $\mathbb{B}^3$ ($\chi_1$), $\mathbb{B}^3$ ($\chi_2$) | $3 = \dim U$ | | **Total** | | | **$4$** | **$13$** | **$52$** | **$(3, 10, 0)$** | **$(1, 12, 0)$** | **$(0, 12, 1)$** | **$(0, 10, 3)$** | | | --- ### 1.3 Summary Table of the Four Main Theorems | Theorem | Variety Family ($B \in U$ very general) | Ambient Space | Dim | Hodge Level | Middle Betti $b_{2m}$ | Primitive Hodge Numbers | $\dim_{\mathbb{Q}} \mathrm{Hdg}^m$ | Status of Rational HC | | :---: | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | **Theorem 1** | Cyclic Quintic Surface Square $\Sigma_B \times \Sigma_B$ | $\mathbb{P}^3 \times \mathbb{P}^3$ | $4$ | $2$ ($p_g=4$) | $2{,}811$ ($2{,}704$ prim) | $\Sigma_B$: $(4, 44, 4)$ | $47$ ($36$ prim) | **Proved** | | **Theorem 2** | Split Quintic Fourfold $X_B^{(4)} = V(g_B(x)-g_B(y))$ | $\mathbb{P}^5$ | $4$ | $2$ | $821$ ($820$ prim) | $(0, 120, 580, 120, 0)$ | $30$ ($29$ prim) | **Proved** | | **Theorem 3** | Calabi–Yau Quintic Threefold Square $Y_B \times Y_B$ | $\mathbb{P}^4 \times \mathbb{P}^4$ | $6$ | **$3$** ($h^{3,0}=1$) | $41{,}620$ ($41{,}616$ in $H^3 \otimes H^3$) | $Y_B$: $(1, 101, 101, 1)$ | $220$ ($216$ in $H^3 \otimes H^3$) | **Proved** | | **Theorem 4** | Split Quintic Sixfold $X_B^{(6)} = V(x_3^5-g_B(x)-y_3^5+g_B(y))$ | $\mathbb{P}^7$ | $6$ | **$4$** ($h^{5,1}=36$) | $13{,}109$ ($13{,}108$ prim) | $(0, 36, 2472, 8092, 2472, 36, 0)$ | $137$ ($136$ prim) | **Proved** | --- ## 2. $(\mu_5 \times \mathfrak{S}_3)$-Isotypic Decomposition and Direct Lefschetz $(1,1)$ Realization of Type I on $\Sigma_B$ ### 2.1 Griffiths Residue Calculus on Cyclic Quintic Surfaces Let $U \subset \mathbb{A}^3 = \operatorname{Spec} \mathbb{Q}[B_1, B_2, B_3]$ be the Zariski-open complement of the discriminant locus $\Delta_{\mathrm{sing}}$ where $C_B = V(g_B(x_0, x_1, x_2)) \subset \mathbb{P}^2$ is a smooth plane quintic curve (of genus $g(C_B) = \binom{4}{2} = 6$). For every $B \in U$, the cyclic quintic surface $$\Sigma_B = V\big(u^5 - g_B(x_0, x_1, x_2)\big) \subset \mathbb{P}^3$$ is a smooth projective surface of degree $5$ (hence of general type, with canonical bundle $K_{\Sigma_B} = \mathcal{O}_{\Sigma_B}(1)$, geometric genus $p_g = h^{2,0}(\Sigma_B) = 4$, irregularity $q = 0$, Euler characteristic $e(\Sigma_B) = 55$, total $b_2(\Sigma_B) = 53$, and primitive $b_{2,\mathrm{prim}}(\Sigma_B) = 52$ with Hodge numbers $(4, 44, 4)$). By Griffiths' residue theorem [3], the primitive cohomology $H^{2-p, p}_{\mathrm{prim}}(\Sigma_B, \mathbb{C})$ is canonically isomorphic to the degree $5(p+1) - 4 = 5p + 1$ graded piece of the Jacobian ring $$J(\Sigma_B) = \frac{\mathbb{C}[u, x_0, x_1, x_2]}{\big(u^4,\, \partial_{x_0} g_B,\, \partial_{x_1} g_B,\, \partial_{x_2} g_B\big)} \cong \big(\mathbb{C}[u]/(u^4)\big) \otimes_{\mathbb{C}} J_{g_B},$$ via the residue map sending $u^j P_d(x) \mapsto \operatorname{Res}_{\Sigma_B}\!\left(\frac{u^j P_d(x) \Omega_{\mathbb{P}^3}}{(u^5 - g_B(x))^{p+1}}\right)$, where $\Omega_{\mathbb{P}^3} = u\, dx_0 \wedge dx_1 \wedge dx_2 - \cdots$ is the Euler $3$-form of weight $4$. Since $\rho(u) = \zeta_5 u$ acts on $u^j \Omega_{\mathbb{P}^3}$ by $\zeta_5^{j+1}$, the four non-trivial characters $\chi_m : \mu_5 \to \mathbb{C}^\times$, $\chi_m(\zeta_5) = \zeta_5^m$ ($m = j + 1 \in \{1, 2, 3, 4\}$) correspond to $j = m - 1 \in \{0, 1, 2, 3\}$, and the polynomial degree of $P_d(x) \in J_{g_B}^d$ contributing to $H^{2-p, p}_{\mathrm{prim}}(\Sigma_B)_{\chi_m}$ is $$d = 5p + 1 - j = 5p + 2 - m \qquad (p \in \{0, 1, 2\}, \quad m \in \{1, 2, 3, 4\}).$$ ### 2.2 Exact $\mathfrak{S}_3$-Character Table of $J_{g_B}$ and the Sign-Twist Identity Because $J_{g_B}$ is a flat bundle of $64$-dimensional graded $\mathfrak{S}_3$-representations over the connected base $U$, its $\mathfrak{S}_3$-isotypic multiplicities in each degree $d \in \{0, \dots, 9\}$ coincide with those at the Fermat point $B = (0, 0, 0)$, where $J_{g_0} = \mathbb{C}[x_0, x_1, x_2]/(x_0^4, x_1^4, x_2^4)$ has basis $\{x_0^a x_1^b x_2^c : 0 \le a, b, c \le 3\}$. Crucially, any permutation $\sigma \in \mathfrak{S}_3$ of $(x_0, x_1, x_2)$ acts on the volume $3$-form $\Omega_{\mathbb{P}^3}$ by the **sign character** $\operatorname{sgn}(\sigma) \in \{\pm 1\}$: $$\sigma^*\big(u^j P(x) \Omega_{\mathbb{P}^3}\big) = \operatorname{sgn}(\sigma)\, u^j (\sigma^* P)(x) \Omega_{\mathbb{P}^3}.$$ Thus the cohomology representation $\rho_{\mathrm{Hdg}}$ on $H^2_{\mathrm{prim}}(\Sigma_B)$ is twisted by $\mathbf{1}_{\mathrm{sgn}}$ relative to the polynomial representation $\rho_{\mathrm{poly}}$ on $J_{g_B}$: $$\mathbf{1}_{\mathrm{sgn}}^{\mathrm{poly}} \longleftrightarrow \mathbf{1}_{\mathrm{triv}}^{\mathrm{Hdg}}, \qquad \mathbf{1}_{\mathrm{triv}}^{\mathrm{poly}} \longleftrightarrow \mathbf{1}_{\mathrm{sgn}}^{\mathrm{Hdg}}, \qquad V_{\mathrm{std}}^{\mathrm{poly}} \longleftrightarrow V_{\mathrm{std}}^{\mathrm{Hdg}}.$$ **Proposition 2.1 (Exact $\mathfrak{S}_3$-Decomposition of $J_{g_B}$ and $H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q})$).** 1. *For every $B \in U$, the $64$-dimensional Jacobian ring $J_{g_B} = \bigoplus_{d=0}^9 J_{g_B}^d$ has the following exact $\mathfrak{S}_3$-decomposition:* | Degree $d$ | $\dim J_{g_B}^d$ | $\mathbf{1}_{\mathrm{triv}}^{\mathrm{poly}}$ ($\mathbf{1}_{\mathrm{sgn}}^{\mathrm{Hdg}}$) | $\mathbf{1}_{\mathrm{sgn}}^{\mathrm{poly}}$ ($\mathbf{1}_{\mathrm{triv}}^{\mathrm{Hdg}}$) | $V_{\mathrm{std}}$ ($2$-dim irrep, $\times 2$) | | :---: | :---: | :---: | :---: | :---: | | $0$ | $1$ | $1$ | $0$ | $0$ | | $1$ | $3$ | $1$ | $0$ | $1$ | | $2$ | $6$ | $2$ | $0$ | $2$ | | $3$ | $10$ | $3$ | $1$ | $3$ | | $4$ | $12$ | $3$ | $1$ | $4$ | | $5$ | $12$ | $3$ | $1$ | $4$ | | $6$ | $10$ | $3$ | $1$ | $3$ | | $7$ | $6$ | $2$ | $0$ | $2$ | | $8$ | $3$ | $1$ | $0$ | $1$ | | $9$ | $1$ | $1$ | $0$ | $0$ | | **Total** | **$64$** | **$20$** | **$4$** | **$20$ ($\dim = 40$)** | 2. *The rational group-ring idempotents in $\mathbb{Q}[\mathfrak{S}_3]$:* $$e_{\mathrm{I}} = \frac{1}{6} \sum_{\sigma \in \mathfrak{S}_3} \sigma^*, \qquad e_{\mathrm{III}} = \frac{1}{6} \sum_{\sigma \in \mathfrak{S}_3} \operatorname{sgn}(\sigma) \sigma^*, \qquad e_{\mathrm{II},\pm} = \frac{1 \pm \tau_{01}^*}{2} (1 - e_{\mathrm{I}} - e_{\mathrm{III}})$$ *(where $\tau_{01}$ swaps $x_0 \leftrightarrow x_1$) decompose $H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q})$ ($\dim = 52$) into four rational Hodge structures over $K = \mathbb{Q}(\zeta_5)$:* $$H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q}) = V_{\mathrm{I}} \oplus V_{\mathrm{II},+} \oplus V_{\mathrm{II},-} \oplus V_{\mathrm{III}} \qquad (4 + 16 + 16 + 16 = 52).$$ 3. *The explicit rational algebraic correspondence* $$\eta := \tau_{12}^* - \tau_{02}^* \in \mathbb{Q}[\mathfrak{S}_3]$$ *satisfies $\tau_{01}^* \eta = -\eta \tau_{01}^*$ and $\eta^2 = 3(1 - e_{\mathrm{I}} - e_{\mathrm{III}})$, so $T_{\mathrm{II}} := e_{\mathrm{II},-} \circ \eta \circ e_{\mathrm{II},+} \in \operatorname{Hom}_{K-\mathrm{Hdg}}(V_{\mathrm{II},+}, V_{\mathrm{II},-})$ is an explicit algebraic similitude of multiplier $3$ ($T_{\mathrm{II}}^\dagger T_{\mathrm{II}} = 3 e_{\mathrm{II},+}$ and $T_{\mathrm{II}} T_{\mathrm{II}}^\dagger = 3 e_{\mathrm{II},-}$ with respect to the $\mathfrak{S}_3$-invariant polarization).* 4. *For every $B \in U$, the $4$-dimensional rational sub-Hodge structure $V_{\mathrm{I}} = e_{\mathrm{I}} H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q})$ has Hodge numbers $(h^{2,0}, h^{1,1}, h^{0,2}) = (0, 4, 0)$ (it lies in $H^{1,1}(\Sigma_B, \mathbb{C})$ across all four $\mu_5$-characters $\chi_1, \dots, \chi_4$). Because $\Sigma_B \subset \mathbb{P}^3$ is a smooth projective surface, the classical Lefschetz $(1,1)$ theorem applied directly to $\Sigma_B$ implies that $V_{\mathrm{I}} \subset H^2(\Sigma_B, \mathbb{Q}) \cap H^{1,1}(\Sigma_B, \mathbb{C}) = c_1(\operatorname{Pic}(\Sigma_B)_{\mathbb{Q}})$ is spanned by rational algebraic divisor classes on every smooth fiber $\Sigma_B$ ($V_{\mathrm{I}} = e_{\mathrm{I}} \operatorname{NS}_{\mathrm{prim}}(\Sigma_B)_{\mathbb{Q}}$). Equivalently, if $\nu : \widetilde{W}_B \to W_B = \Sigma_B / \mathfrak{S}_3 \subset \mathbb{P}(1, 1, 2, 3)$ is a minimal resolution of the normal quotient surface and $\widetilde{\Gamma} \subset \Sigma_B \times \widetilde{W}_B$ is the resolved graph of the degree-$6$ quotient map, then $\frac{1}{6}[\widetilde{\Gamma}]^*[\widetilde{\Gamma}]_* = e_{\mathrm{I}}$ on $H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q})$, so $h^{2,0}(\widetilde{W}_B) = \dim H^{2,0}(\Sigma_B)^{\mathfrak{S}_3} = 0$ and $V_{\mathrm{I}} = [\widetilde{\Gamma}]^* \operatorname{NS}_{\mathrm{prim}}(\widetilde{W}_B)_{\mathbb{Q}}$.* *Proof.* Part 1 is an immediate trace calculation on $\{0, 1, 2, 3\}^3$ (verified in Step 1 of `verify.py`). Grouping the degrees $d = 5p + 2 - m \in \{0, \dots, 9\}$ by $m \in \{1, 2, 3, 4\}$: - For $m = 1$ ($d \in \{1, 6\}$): $\mathbf{1}_{\mathrm{sgn}}^{\mathrm{poly}}$ has $(0, 1, 0)$, $V_{\mathrm{std}}$ has $(1, 3, 0)$, $\mathbf{1}_{\mathrm{triv}}^{\mathrm{poly}}$ has $(1, 3, 0)$; - For $m = 2$ ($d \in \{0, 5\}$): $\mathbf{1}_{\mathrm{sgn}}^{\mathrm{poly}}$ has $(0, 1, 0)$, $V_{\mathrm{std}}$ has $(0, 4, 0)$, $\mathbf{1}_{\mathrm{triv}}^{\mathrm{poly}}$ has $(1, 3, 0)$; - For $m = 3$ ($d \in \{4, 9\}$): $\mathbf{1}_{\mathrm{sgn}}^{\mathrm{poly}}$ has $(0, 1, 0)$, $V_{\mathrm{std}}$ has $(0, 4, 0)$, $\mathbf{1}_{\mathrm{triv}}^{\mathrm{poly}}$ has $(0, 3, 1)$; - For $m = 4$ ($d \in \{3, 8\}$): $\mathbf{1}_{\mathrm{sgn}}^{\mathrm{poly}}$ has $(0, 1, 0)$, $V_{\mathrm{std}}$ has $(0, 3, 1)$, $\mathbf{1}_{\mathrm{triv}}^{\mathrm{poly}}$ has $(0, 3, 1)$. In particular, $\mathbf{1}_{\mathrm{sgn}}^{\mathrm{poly}}$ (which occurs only in degrees $3, 4, 5, 6$) falls purely into $p = 1$ ($H^{1,1}$) for all $m \in \{1, 2, 3, 4\}$. Under the sign twist by $\Omega_{\mathbb{P}^3}$, $V_{\mathrm{I}} = e_{\mathrm{I}} H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q}) \subset H^2(\Sigma_B, \mathbb{Q}) \cap H^{1,1}(\Sigma_B, \mathbb{C})$, so the Lefschetz $(1,1)$ theorem on the smooth surface $\Sigma_B$ directly yields $V_{\mathrm{I}} \subset c_1(\operatorname{Pic}(\Sigma_B)_{\mathbb{Q}})$. Part 3 follows from $\tau_{01}\tau_{12}\tau_{01} = \tau_{02}$ and $(\tau_{12} - \tau_{02})^2 = 2 - (012) - (021) = 3(1 - e_{\mathrm{I}} - e_{\mathrm{III}})$. $\square$ --- ## 3. Closed-Form Macaulay Determinants, Generic $\mathbb{Q}$-Hodge Simplicity, and Eigen-Signature Vanishing ### 3.1 Closed-Form Macaulay Determinants and Curve Discriminant Quadrics on $U \subset \mathbb{A}^3$ We first show that on the $3$-parameter family $g_B(x) = m_{500} + B_1 m_{320} + B_2 m_{311} + B_3 m_{221}$, the period maps of all moving blocks ($V_{\mathrm{III}}$ in $\chi_2$, $V_{\mathrm{III}}$ in $\chi_1$, and $V_{\mathrm{II},\pm}$ in $\chi_1$) are local biholomorphisms onto the $3$-dimensional complex ball $\mathbb{B}^3$, and the period maps of $W_{\mathrm{III}}(C_B), W_{\mathrm{II},\pm}(C_B)$ are local biholomorphisms onto $\mathfrak{H}_2$ with linearly independent discriminant quadrics. **Proposition 3.1 (Closed-Form Macaulay Determinants, Fermat Kodaira–Spencer Matrices, and Curve Discriminant Quadrics).** 1. *(Exact $3 \times 3$ Kodaira–Spencer Matrices at the Fermat Point $B = (0, 0, 0)$)* At $B = (0, 0, 0)$, where $J_{g_0} = \mathbb{C}[x_0, x_1, x_2]/(x_0^4, x_1^4, x_2^4)$, the Kodaira–Spencer derivatives $\nabla_{\partial_{B_r}}$ ($r = 1, 2, 3$, corresponding to $m_{320}, m_{311}, m_{221}$) act by polynomial multiplication in $J_{g_0}$: - *On $V_{\mathrm{III},\chi_2}$ ($J_{g_0}^0 \to J_{g_0}^5$, from $h^{2,0}$-generator $1$ to $h^{1,1}$-basis $\{m_{320}, m_{311}, m_{221}\}$):* $$M_{\mathrm{KS}}(V_{\mathrm{III},\chi_2})\big|_{B=0} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad \det M_{\mathrm{KS}}(V_{\mathrm{III},\chi_2})\big|_{B=0} = 1 \ne 0.$$ - *On $V_{\mathrm{III},\chi_1}$ ($J_{g_0}^1 \to J_{g_0}^6$, from $h^{2,0}$-generator $e_1 = x_0 + x_1 + x_2$ to $h^{1,1}$-basis $\{s_{330}, s_{321}, s_{222}\}$):* $$M_{\mathrm{KS}}(V_{\mathrm{III},\chi_1})\big|_{B=0} = \begin{pmatrix} 2 & 0 & 0 \\ 1 & 1 & 1 \\ 0 & 0 & 3 \end{pmatrix}, \qquad \det M_{\mathrm{KS}}(V_{\mathrm{III},\chi_1})\big|_{B=0} = 6 \ne 0.$$ - *On $V_{\mathrm{II},-,\chi_1}$ ($J_{g_0}^1 \to J_{g_0}^6$, from $h^{2,0}$-generator $v_- = x_0 - x_1$ to $h^{1,1}$-basis $\{u_1, u_2, u_3\}$ with coefficients at $(x_0^3 x_2^3, x_0^3 x_1^2 x_2, x_0^3 x_1 x_2^2)$):* $$M_{\mathrm{KS}}(V_{\mathrm{II},-,\chi_1})\big|_{B=0} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 1 \\ -1 & 0 & 1 \end{pmatrix}, \qquad \det M_{\mathrm{KS}}(V_{\mathrm{II},-,\chi_1})\big|_{B=0} = -1 \ne 0.$$ - *On the plane quintic curve $C_B = V(g_B) \subset \mathbb{P}^2$ ($J_{g_0}^2 \to J_{g_0}^7$):* $H^1(C_B, \mathbb{Q})$ ($\dim = 12$, genus $6$) decomposes under $\mathfrak{S}_3$ into $W_{\mathrm{III}}(C_B)$ ($\dim = 4$, genus-$2$ type $(2, 2)$) and $W_{\mathrm{II},+}(C_B) \oplus W_{\mathrm{II},-}(C_B)$ (two isomorphic blocks of $\dim = 4$, genus-$2$ type $(2, 2)$, linked by $T_{\mathrm{II}}$). The $3 \times 3$ symmetric Siegel period tangent matrices into $S^2(H^{0,1}) \cong T\mathfrak{H}_2$ ($\dim \mathfrak{H}_2 = 3$) at $B = (0, 0, 0)$ have exact determinants: $$\det M_{\mathrm{Siegel}}\big(W_{\mathrm{III}}(C_B)\big)\big|_{B=0} = -8 \ne 0, \qquad \det M_{\mathrm{Siegel}}\big(W_{\mathrm{II},\pm}(C_B)\big)\big|_{B=0} = -8 \ne 0,$$ *and in the tangent coordinates $y_1 \partial_{B_1} + y_2 \partial_{B_2} + y_3 \partial_{B_3} \in T_0 U$, the determinant quadrics $Q(y_1, y_2, y_3) = \det(\sum_{r=1}^3 y_r A_r) \in S^2(T_0^* U)$ of the $2 \times 2$ symmetric Kodaira–Spencer pencils $A_r \in S^2(H^{1,0})^\vee$ are:* $$Q_{\mathrm{III}}(y_1, y_2, y_3) = -9\big(4y_1^2 + 4y_1 y_3 - 2y_2^2 - 4y_2 y_3 + y_3^2\big), \qquad Q_{\mathrm{II}}(y_1, y_2, y_3) = -4\big(y_1^2 - 2y_1 y_3 + y_2^2 - y_2 y_3 + y_3^2\big),$$ *which are linearly independent in $\mathbb{Q}[y_1, y_2, y_3]$.* 2. *(Global Closed-Form Macaulay Determinants on $U$)* Over the polynomial ring $\mathbb{Q}[B_1, B_2, B_3]$: - *In degree $5$ ($\mathbf{1}_{\mathrm{triv}}^{\mathrm{poly}}$, $\dim = 5$, with $2$ Jacobian generators $\sum x_i \partial_i g_B$ and $e_1 \sum \partial_i g_B$ and $3$ Kodaira–Spencer rows $m_{320}, m_{311}, m_{221}$), the $5 \times 5$ Macaulay matrix $M_{5,\mathrm{III}}(B)$ has exact factored determinant:* $$\det M_{5, \mathrm{III}}(B_1, B_2, B_3) = 5(2B_1 + B_2 + 5) \not\equiv 0.$$ - *In degree $6$ ($\mathbf{1}_{\mathrm{triv}}^{\mathrm{poly}}$, $\dim = 7$, with $4$ symmetric Jacobian generators and $3$ Kodaira–Spencer rows $e_1 m_{320}, e_1 m_{311}, e_1 m_{221}$), the $7 \times 7$ Macaulay matrix $M_{6,\mathrm{III}}(B)$ has exact factored determinant:* $$\det M_{6, \mathrm{III}}(B_1, B_2, B_3) = 15(2B_1 + B_2 + 5)(B_1 + 2B_2 - B_3 + 5)(2B_1 - 2B_2 + B_3 + 10) \not\equiv 0.$$ - *In degree $6$ ($V_{\mathrm{std},-}^{\mathrm{poly}}$, $\dim = 9$, with $6$ Jacobian generators and $3$ Kodaira–Spencer rows $v_- m_{320}, v_- m_{311}, v_- m_{221}$), the $9 \times 9$ Macaulay matrix $M_{6,\mathrm{II}}(B)$ has exact factored determinant:* $$\det M_{6, \mathrm{II}}(B_1, B_2, B_3) = -437400(2B_1 + B_2 + 5)(B_1 + 2B_2 - B_3 + 5)\, P_{\mathrm{cub}}(B_1, B_2, B_3) \not\equiv 0,$$ *where $P_{\mathrm{cub}}(B_1, B_2, B_3) = 4B_1^3 - 4B_1^2 B_2 + 11B_1^2 + B_1 B_2^2 + 9B_1 B_2 - 12B_1 B_3 - 40B_1 - B_2^2 + B_2 B_3 - 5B_2 + B_3^2 + 15B_3 + 25$.* *Proof.* All matrix entries, polynomial determinants, and discriminant quadrics are computed in exact integer/rational arithmetic and verified in Steps 2, 3, and 4 of `verify.py` and `hodge_loci_engine/quintic_tower.py`. Since the determinants are non-zero polynomials (evaluating to $25$, $3750$, and $-273375000$ at $B = (0, 0, 0)$), the differentials of the period maps of $V_{\mathrm{III}}$ and $V_{\mathrm{II},\pm}$ into $\mathbb{B}^3$ (and of $W_{\mathrm{III}}(C_B), W_{\mathrm{II},\pm}(C_B)$ into $\mathfrak{H}_2$) have full rank $3$ on a non-empty Zariski-open subset of $U$, so the period maps are dominant submersions with open analytic images. $\square$ --- ### 3.2 Generic $\mathbb{Q}$-Hodge Simplicity (Without Assuming $K$-Stability) and the Eigen-Signature Vanishing Theorem A subtle point in Hodge theory with endomorphisms by a CM field $K = \mathbb{Q}(\zeta_5)$ is that a rational sub-Hodge structure $S \subset W_B$ of a $K$-module $W_B$ is **not** a priori guaranteed to be $K$-stable (for instance, in a direct sum $K \oplus K$ of two isomorphic variations, a diagonal copy of $\mathbb{Q}$ is not $K$-stable), nor does an arbitrary complex Hodge-graded subspace of a single fiber $W_{B,\mathbb{C}}$ extend to a flat subbundle on an open neighborhood. In this subsection, we give a complete, self-contained proof—**relying only on the countability of rational homomorphisms $\operatorname{Hom}_{\mathbb{Q}}(W_0, W'_0)$ on the universal cover $\widetilde{U}$, the open-set irreducibility of the active $4$-dimensional $\mathbb{C}$-VHS $W_{\chi_1}|_{U_0}$, and Galois rank-invariance of the rational branch operators $f_a$**—that at every very general parameter $B \in U_{\mathrm{vg}}$, all cross-branch and cross-type rational Hodge morphisms vanish, $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(W_B) = K$, and consequently every moving $16$-dimensional block $W_B$ is a **simple $\mathbb{Q}$-Hodge structure** (without assuming $K$-stability). **Theorem 3.2 (Generic $\mathbb{Q}$-Hodge Simplicity and Eigen-Signature Vanishing Theorem).** *Let $W, W'$ be polarized $\mathbb{Q}$-variations of Hodge structure of weight $w$ over $U$ equipped with $\nabla$-flat algebraic $\mu_5$-actions $\rho_W^*, \rho_{W'}^* \in \operatorname{Aut}_{\mathbb{Q}-\mathrm{VHS}}$ generating $K = \mathbb{Q}(\zeta_5)$, taken from the moving $16$-dimensional blocks $V_{\mathrm{II},+}, V_{\mathrm{II},-}, V_{\mathrm{III}} \subset H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q})$ or $M_{r,\mathrm{II},+}, M_{r,\mathrm{II},-}, M_{r,\mathrm{III}} \subset H^3(Y_B, \mathbb{Q})$ ($r \in \{1, 2, 3\}$).* 1. *(Rationality, Galois Rank-Invariance, and Eigen-Signature Vanishing of Branch Projectors)* *At every very general point $B \in U_{\mathrm{vg}}$ (constructed explicitly in Step A below as the complement of a countable union of proper analytic/algebraic Hodge loci in $U$), any rational Hodge morphism $f \in \operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}}(W_B, W'_B)$ extends by $\nabla$-flat parallel transport to a $\nabla$-flat morphism of $\mathbb{Q}$-variations of Hodge structure over a connected, simply connected open neighborhood $U_0 \subset U(\mathbb{C})$ of $B_0 = (0, 0, 0)$ (and over a simply connected neighborhood $U_B$ of $B$). For each $a \in (\mathbb{Z}/5\mathbb{Z})^\times = \{1, 2, 3, 4\}$, the Galois-branch operator* $$f_a := \frac{1}{5} \sum_{j=0}^4 (\rho_{W'}^*)^{-j} \circ f \circ (\rho_W^*)^{aj} \in \operatorname{Hom}_{\mathbb{Q}-\mathrm{VHS}}\big(\mathbb{W}_{U_0}, \mathbb{W}'_{U_0}\big)$$ *is a $\mathbb{Q}$-linear combination of compositions of $\nabla$-flat rational VHS morphisms on $U_0$, hence is **itself a $\nabla$-flat rational morphism of $\mathbb{Q}$-VHS on $U_0$** satisfying $f = \sum_{a=1}^4 f_a$ and $\rho_{W'}^* \circ f_a = f_a \circ (\rho_W^*)^a$. Its four character components $f_{a, m} : W_{\chi_m}|_{U_0} \to W'_{\chi_{am \bmod 5}}|_{U_0}$ ($m \in \{1, 2, 3, 4\}$) are $\operatorname{Gal}(\mathbb{Q}(\zeta_5)/\mathbb{Q})$-conjugate ($\tau_t(f_{a, m}) = f_{a, tm \bmod 5}$ for all $t \in (\mathbb{Z}/5\mathbb{Z})^\times$), so $\dim_{\mathbb{C}} \ker(f_{a, m}) = \dim_{\mathbb{C}} \ker(f_{a, 1})$ is strictly independent of $m \in \{1, 2, 3, 4\}$. Consequently, if for a given $a \in \{1, 2, 3, 4\}$ there exists a character $m \in \{1, 2, 3, 4\}$ and bidegree $(p, q)$ such that $\dim_{\mathbb{C}} W_{\chi_m}^{p, q} > \dim_{\mathbb{C}} W'^{p, q}_{\chi_{am \bmod 5}}$, then $\ker(f_{a, 1}) \subset W_{\chi_1}|_{U_0}$ is a non-zero $\nabla$-flat $\mathbb{C}$-sub-VHS on the open set $U_0$, forcing $\ker(f_{a, 1}) = W_{\chi_1}|_{U_0}$ and $f_a = 0$. In particular, at every $B \in U_{\mathrm{vg}}$:* - *If for $W$ and $W'$ every $a \in \{1, 2, 3, 4\}$ has some $m, (p, q)$ with $\dim W_{\chi_m}^{p, q} > \dim W'^{p, q}_{\chi_{am \bmod 5}}$, then* $$\operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}}(W_B, W'_B) = 0.$$ - *For every moving $16$-dimensional block $W \in \{V_{\mathrm{II},+}, V_{\mathrm{II},-}, V_{\mathrm{III}}, M_{r,\mathrm{II},+}, M_{r,\mathrm{II},-}, M_{r,\mathrm{III}}\}$ ($r \in \{1, 2, 3\}$), every $a \in \{2, 3, 4\}$ has some $m, (p, q)$ with $\dim W_{\chi_m}^{p, q} > \dim W_{\chi_{am \bmod 5}}^{p, q}$, so $f_2 = f_3 = f_4 = 0$ and* $$\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(W_B) = \operatorname{End}_{K-\mathrm{Hdg}}(W_B) = K \cong \mathbb{Q}(\zeta_5).$$ 2. *(Generic $\mathbb{Q}$-Hodge Simplicity Without Assuming $K$-Stability)* *At every $B \in U_{\mathrm{vg}}$, each of the $16$-dimensional polarized $\mathbb{Q}$-Hodge structures $V_{\mathrm{II},+}, V_{\mathrm{II},-}, V_{\mathrm{III}} \subset H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q})$ and $M_{r,\mathrm{II},+}, M_{r,\mathrm{II},-}, M_{r,\mathrm{III}} \subset H^3(Y_B, \mathbb{Q})$ ($r \in \{1, 2, 3\}$) is a **simple $\mathbb{Q}$-Hodge structure** (it contains no proper non-zero rational sub-Hodge structure $S \subsetneq W_B$, whether $K$-stable or not).* *Proof.* We give the complete proof in four self-contained steps (verified symbolically in Step 4 of `verify.py`): **Step A (Countability of Rational Maps on the Universal Cover $\widetilde{U}$ and Explicit Construction of $U_{\mathrm{vg}}$).** Fix the Fermat basepoint $B_0 = (0, 0, 0) \in U(\mathbb{C})$ and let $\pi_{\mathrm{univ}} : (\widetilde{U}, \tilde{0}) \to (U(\mathbb{C}), B_0)$ be the universal cover of the connected complex manifold $U(\mathbb{C})$. Because $\widetilde{U}$ is simply connected and the Gauss–Manin connection $\nabla$ is flat, parallel transport along $\nabla$ canonically trivializes the rational local systems $\pi_{\mathrm{univ}}^* \mathbb{W}_{\mathbb{Q}} \cong \widetilde{U} \times W_0$ and $\pi_{\mathrm{univ}}^* \mathbb{W}'_{\mathbb{Q}} \cong \widetilde{U} \times W'_0$, where $W_0 = \mathbb{W}_{\mathbb{Q}, B_0} \cong \mathbb{Q}^d$ and $W'_0 = \mathbb{W}'_{\mathbb{Q}, B_0} \cong \mathbb{Q}^{d'}$ are fixed finite-dimensional $\mathbb{Q}$-vector spaces. Under this flat trivialization, the Hodge filtration subbundles $F^p \mathcal{W}_{\tilde{u}} \subset W_0 \otimes_{\mathbb{Q}} \mathbb{C}$ and $F^p \mathcal{W}'_{\tilde{u}} \subset W'_0 \otimes_{\mathbb{Q}} \mathbb{C}$ vary **holomorphically** with $\tilde{u} \in \widetilde{U}$ (Griffiths' theorem [3]). Unlike the uncountable complex vector space $\operatorname{Hom}_{\mathbb{C}}(W_{0,\mathbb{C}}, W'_{0,\mathbb{C}})$, the rational vector space $\operatorname{Hom}_{\mathbb{Q}}(W_0, W'_0) \cong \mathbb{Q}^{d d'}$ is a **countable set**. For **each fixed rational linear map** $f \in \operatorname{Hom}_{\mathbb{Q}}(W_0, W'_0)$, the locus of points on $\widetilde{U}$ where the flat rational section $f$ is a morphism of Hodge structures is $$\widetilde{Z}(f) := \Big\{ \tilde{u} \in \widetilde{U} \;\Big|\; f_{\mathbb{C}}\big(F^p \mathcal{W}_{\tilde{u}}\big) \subset F^p \mathcal{W}'_{\tilde{u}} \text{ for all } p \in \mathbb{Z} \Big\} \subset \widetilde{U}.$$ Because $\widetilde{Z}(f)$ is the common zero locus of the holomorphic vector bundle homomorphisms $F^p \mathcal{W} \xrightarrow{f_{\mathbb{C}}} (\widetilde{U} \times W'_{0,\mathbb{C}})/F^p \mathcal{W}'$, $\widetilde{Z}(f)$ is a **closed complex-analytic subset** of the connected complex manifold $\widetilde{U}$. By the Identity Theorem for complex-analytic subsets on the connected manifold $\widetilde{U}$, each fixed $f \in \operatorname{Hom}_{\mathbb{Q}}(W_0, W'_0)$ satisfies a strict dichotomy: - **Case (i) (Proper fiberwise Hodge locus):** $\widetilde{Z}(f) \subsetneq \widetilde{U}$ is a proper closed complex-analytic subset ($\operatorname{codim}_{\mathbb{C}} \widetilde{Z}(f) \ge 1$), so its projection $Z(f) := \pi_{\mathrm{univ}}(\widetilde{Z}(f)) \subset U(\mathbb{C})$ is a countable union of proper closed complex-analytic subsets of $U(\mathbb{C})$ of Lebesgue measure zero (and by Cattani–Deligne–Kaplan [11], $Z(f)$ is in fact a proper algebraic subvariety of $U$); or - **Case (ii) (Global flat VHS morphism):** $\widetilde{Z}(f) = \widetilde{U}$, which means that the $\nabla$-flat rational map $f$ preserves the Hodge filtration $F^p$ at **every** point $\tilde{u} \in \widetilde{U}$; in particular, over a connected, simply connected open neighborhood $U_0 \subset U(\mathbb{C})$ of the Fermat origin $B_0 = (0, 0, 0)$ (and over a simply connected open neighborhood $U_B$ of $B$), $f$ is a **$\nabla$-flat morphism of $\mathbb{Q}$-variations of Hodge structure already defined on the open set $U_0$**. Let $U_{\mathrm{KS}} \subset U$ be the non-empty Zariski-open subset where all Macaulay and Siegel determinants of Proposition 3.1 are non-zero, and define the **very general locus** $U_{\mathrm{vg}} \subset U(\mathbb{C})$ explicitly as the complement of the countable union of all Case-(i) proper Hodge loci across all pairs of blocks $(W, W')$ in our tower: $$U_{\mathrm{vg}} := U_{\mathrm{KS}}(\mathbb{C}) \setminus \bigcup_{\substack{(W, W') \\ f \in \operatorname{Hom}_{\mathbb{Q}}(W_0, W'_0) \\ \widetilde{Z}(f) \subsetneq \widetilde{U}}} \pi_{\mathrm{univ}}\big(\widetilde{Z}(f)\big).$$ Since $\operatorname{Hom}_{\mathbb{Q}}(W_0, W'_0) \cong \mathbb{Q}^{d d'}$ is **countable**, the union above is a countable union of proper analytic (and algebraic) subsets of $U(\mathbb{C})$, so $U_{\mathrm{vg}}$ is dense and of full measure in $U(\mathbb{C})$ by the Baire Category Theorem. By construction, for **every** point $B \in U_{\mathrm{vg}}$ and any lift $\tilde{u} \in \pi_{\mathrm{univ}}^{-1}(B) \subset \widetilde{U}$, if $f \in \operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}}(W_B, W'_B)$ is a rational Hodge morphism at the fiber $B$, then $\tilde{u} \in \widetilde{Z}(f)$ while $B \notin \bigcup_{\widetilde{Z}(g) \subsetneq \widetilde{U}} \pi_{\mathrm{univ}}(\widetilde{Z}(g))$; therefore $f$ cannot fall into Case (i), so $\widetilde{Z}(f) = \widetilde{U}$ (Case (ii)), and $f$ extends by parallel transport to a **$\nabla$-flat morphism of $\mathbb{Q}$-variations of Hodge structure across the entire open neighborhood $U_0$ of $B_0 = (0, 0, 0)$** (and across a simply connected neighborhood $U_B \subset U_{\mathrm{KS}}(\mathbb{C})$ of $B$ itself). **Step B (Irreducibility of the Active $4$-Dimensional $\mathbb{C}$-Eigenspace Variations over Open Sets).** For each $m \in \{1, 2, 3, 4\}$, the spectral projector $\pi_m := \frac{1}{5}\sum_{j=0}^4 \zeta_5^{-jm} (\rho_W^*)^j \in \mathbb{C}[\mu_5]$ onto $W_{\chi_m} \subset W \otimes_{\mathbb{Q}} \mathbb{C}$ is a constant $\mathbb{C}$-linear combination of the algebraic automorphisms $(\rho_W^*)^j$. Consequently, $\pi_m$ is flat with respect to the Gauss–Manin connection ($\nabla \circ \pi_m = \pi_m \circ \nabla$) and preserves the Hodge decomposition on every fiber ($\pi_m(W^{p, q}) = W_{\chi_m}^{p, q}$). On the active character $m = 1$ (where $W_{\chi_1}$ has signature $(1, 3)$ with $\det M_{\mathrm{KS}}|_{B=0} \in \{6, -1\} \ne 0$, and likewise $m = 2$ for Type III where $\det M_{\mathrm{KS}}|_{B=0} = 1 \ne 0$, and at every $B \in U_{\mathrm{KS}}$ by Proposition 3.1(2)), let $E \subset W_{\chi_m}|_{U_0}$ be any $\nabla$-flat $\mathbb{C}$-sub-variation of Hodge structure **already defined on the connected open neighborhood $U_0 \subset U(\mathbb{C})$ of $B_0 = (0, 0, 0)$** (or on $U_B \subset U_{\mathrm{KS}}(\mathbb{C})$). Since $\dim_{\mathbb{C}} W_{\chi_m}^{\mathrm{ext}} = 1$ (where $W_{\chi_m}^{\mathrm{ext}}$ is the $1$-dimensional $(2,0)$-line on $\Sigma_B$, or the $1$-dimensional $(2,1)$-line at $m=1$ on $Y_B$, or the $1$-dimensional $(3,0)$-line at $m=2$ on $M_{1,\mathrm{III}}$) and $E$ is Hodge-graded on $U_0$: - If $E \cap W_{\chi_m}^{\mathrm{ext}} \ne 0$ on $U_0$, then $W_{\chi_m}^{\mathrm{ext}} \subset E$. By $\nabla$-flatness of $E$ on $U_0$, $E$ contains $\omega(B_0) \in W_{\chi_m}^{\mathrm{ext}}(B_0)$ and its three first-order Gauss–Manin derivatives $\nabla_{\partial_{B_1}}\omega(B_0), \nabla_{\partial_{B_2}}\omega(B_0), \nabla_{\partial_{B_3}}\omega(B_0)$, which by Proposition 3.1(1) span all $4$ dimensions of $W_{\chi_m}$ (and likewise at every $B \in U_{\mathrm{KS}}$ by Proposition 3.1(2)), so $E = W_{\chi_m}|_{U_0}$. - If $E \cap W_{\chi_m}^{\mathrm{ext}} = 0$ on $U_0$, then for every $B' \in U_0$, the $1$-dimensional period line $\Phi_{\chi_m}(B') = W_{\chi_m}^{\mathrm{ext}}(B') \in \mathbb{P}(W_{\chi_m}) \cong \mathbb{P}^3$ is $h_{\chi_m}$-orthogonal to the fixed flat subspace $E \subset W_{\chi_m}$ under the flat Hermitian polarization form $h_{\chi_m}$, confining $\Phi_{\chi_m}(U_0)$ to the proper projective subspace $\mathbb{P}(E^\perp) \subsetneq \mathbb{P}^3$. Since $\Phi_{\chi_m}(U_0)$ is open in $\mathbb{B}^3 \subset \mathbb{P}^3$ by Proposition 3.1, we must have $E^\perp = W_{\chi_m}$ and $E = 0$. Thus $W_{\chi_1}|_{U_0}$ (and its complex conjugate $W_{\chi_4}|_{U_0} = \overline{W_{\chi_1}}|_{U_0}$, as well as $W_{\chi_2}|_{U_0}, W_{\chi_3}|_{U_0}$ for Type III) is an **irreducible $4$-dimensional $\mathbb{C}$-VHS over the open set $U_0$** (and over $U_B$). **Step C (Galois Rank-Invariance of the Rational Branch Operators $f_a$ and Proof of Part 1).** Let $B \in U_{\mathrm{vg}}$ and let $f \in \operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}}(W_B, W'_B)$. By Step A, $f$ extends to a $\nabla$-flat morphism of $\mathbb{Q}$-VHS over the open neighborhood $U_0$ of $B_0 = (0, 0, 0)$. Since $\rho_W^* \in \operatorname{Aut}_{\mathbb{Q}-\mathrm{VHS}}(\mathbb{W}_{U_0})$ and $\rho_{W'}^* \in \operatorname{Aut}_{\mathbb{Q}-\mathrm{VHS}}(\mathbb{W}'_{U_0})$ are $\nabla$-flat rational automorphisms of order $5$ satisfying $1 + \rho^* + (\rho^*)^2 + (\rho^*)^3 + (\rho^*)^4 = 0$, each Galois-branch operator $$f_a := \frac{1}{5}\sum_{j=0}^4 (\rho_{W'}^*)^{-j} \circ f \circ (\rho_W^*)^{aj} \in \operatorname{Hom}_{\mathbb{Q}-\mathrm{VHS}}\big(\mathbb{W}_{U_0}, \mathbb{W}'_{U_0}\big) \qquad (a \in \{1, 2, 3, 4\})$$ is a **$\nabla$-flat morphism of $\mathbb{Q}$-VHS over the open set $U_0$**, satisfying $\sum_{a=1}^4 f_a = f - \frac{1}{5}\sum_{j=0}^4 f \circ (\rho_W^*)^j = f$ and $\rho_{W'}^* \circ f_a = f_a \circ (\rho_W^*)^a$. Over $U_0$, $f_a \otimes \mathbb{C}$ decomposes into four $\nabla$-flat morphisms of $\mathbb{C}$-VHS **already defined on the open set $U_0$**: $$f_{a, m} : W_{\chi_m}\big|_{U_0} \longrightarrow W'_{\chi_{am \bmod 5}}\big|_{U_0} \qquad (m \in \{1, 2, 3, 4\}).$$ In a $\mathbb{Q}$-basis of the constant reference fibers $W_0 \cong \mathbb{Q}^{16}$ and $W'_0 \cong \mathbb{Q}^{16}$, $f_a, \rho_W^*, \rho_{W'}^* \in M_{16}(\mathbb{Q})$ have **rational matrix entries**, and $f_{a, m}$ is represented by the matrix $M(f_{a, m}) := \pi'_{am \bmod 5} f_a \pi_m = f_a \pi_m \in M_{16}(\mathbb{Q}(\zeta_5))$. For every $t \in (\mathbb{Z}/5\mathbb{Z})^\times = \{1, 2, 3, 4\}$, applying the Galois automorphism $\tau_t \in \operatorname{Gal}(\mathbb{Q}(\zeta_5)/\mathbb{Q})$ ($\tau_t(\zeta_5) = \zeta_5^t$) entrywise to $M(f_{a, m})$ fixes the rational matrix $f_a$ and conjugates $\tau_t(\pi_m) = \frac{1}{5}\sum_{j=0}^4 \zeta_5^{-jtm}(\rho_W^*)^j = \pi_{tm \bmod 5}$, yielding the exact **Galois conjugation identity**: $$\tau_t\big(M(f_{a, m})\big) = f_a \, \tau_t(\pi_m) = f_a \, \pi_{tm \bmod 5} = M(f_{a, tm \bmod 5}) \qquad \forall m, t \in \{1, 2, 3, 4\}.$$ Because the rank of a matrix over $\mathbb{Q}(\zeta_5)$ is invariant under the field automorphism $\tau_t$, $\operatorname{rank}_{\mathbb{C}}(f_{a, m}) = \operatorname{rank}(M(f_{a, m}))$ and $\dim_{\mathbb{C}} \ker(f_{a, m}) = 4 - \operatorname{rank}_{\mathbb{C}}(f_{a, m})$ are **strictly independent of $m \in \{1, 2, 3, 4\}$**: $$\dim_{\mathbb{C}} \ker(f_{a, 1}) = \dim_{\mathbb{C}} \ker(f_{a, 2}) = \dim_{\mathbb{C}} \ker(f_{a, 3}) = \dim_{\mathbb{C}} \ker(f_{a, 4}).$$ Now observe that at the active character $m = 1$, the kernel $$E_{a, 1} := \ker(f_{a, 1}) \subset W_{\chi_1}\big|_{U_0}$$ is the kernel of a $\nabla$-flat morphism of $\mathbb{C}$-VHS on $U_0$, hence is a **$\nabla$-flat $\mathbb{C}$-sub-variation of Hodge structure of $W_{\chi_1}$ already defined on the open set $U_0$**! Therefore, if for a given $a \in \{1, 2, 3, 4\}$ there exists *any* character $m \in \{1, 2, 3, 4\}$ and bidegree $(p, q)$ such that $\dim_{\mathbb{C}} W_{\chi_m}^{p, q} > \dim_{\mathbb{C}} W'^{p, q}_{\chi_{am \bmod 5}}$, then $f_{a, m} : W_{\chi_m}^{p, q} \to W'^{p, q}_{\chi_{am \bmod 5}}$ has a non-trivial kernel ($\dim_{\mathbb{C}} \ker(f_{a, m}) \ge 1$), so by Galois rank-invariance $\dim_{\mathbb{C}} \ker(f_{a, 1}) = \dim_{\mathbb{C}} \ker(f_{a, m}) \ge 1$. That is, $E_{a, 1} = \ker(f_{a, 1}) \subset W_{\chi_1}|_{U_0}$ is a **non-zero $\nabla$-flat $\mathbb{C}$-sub-VHS of $W_{\chi_1}$ already defined on the open set $U_0$**. By **Step B** (irreducibility of $W_{\chi_1}|_{U_0}$ on the open set $U_0$), we immediately obtain $E_{a, 1} = W_{\chi_1}|_{U_0}$, so $\operatorname{rank}_{\mathbb{C}}(f_{a, 1}) = 0$, and hence by Galois rank-invariance $\operatorname{rank}_{\mathbb{C}}(f_{a, m}) = 0$ for all $m \in \{1, 2, 3, 4\}$—that is, **$f_a = 0$**! In particular, if every $a \in \{1, 2, 3, 4\}$ has some $m, (p, q)$ with $\dim W_{\chi_m}^{p, q} > \dim W'^{p, q}_{\chi_{am \bmod 5}}$, then $f_1 = f_2 = f_3 = f_4 = 0$, so $\operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}}(W_B, W'_B) = 0$. Similarly, for $W' = W$ on each of the moving $16$-dimensional blocks $W \in \{V_{\mathrm{II},+}, V_{\mathrm{II},-}, V_{\mathrm{III}}, M_{r,\mathrm{II},+}, M_{r,\mathrm{II},-}, M_{r,\mathrm{III}}\}$ ($r \in \{1, 2, 3\}$), every $a \in \{2, 3, 4\}$ has some $m, (p, q)$ with $\dim W_{\chi_m}^{p, q} > \dim W_{\chi_{am \bmod 5}}^{p, q}$: - On $V_{\mathrm{II},\pm}$, $h^{2,0} = (1, 0, 0, 0)$, so for $a \in \{2, 3, 4\}$, $h^{2,0}_{\chi_1} = 1 > 0 = h^{2,0}_{\chi_a}$; - On $V_{\mathrm{III}}$, $h^{2,0} = (1, 1, 0, 0)$, so for $a = 2$, $h^{2,0}_{\chi_2} = 1 > 0 = h^{2,0}_{\chi_4}$, and for $a \in \{3, 4\}$, $h^{2,0}_{\chi_1} = 1 > 0 = h^{2,0}_{\chi_a}$; - On $M_{1,\mathrm{III}}$, $h^{3,0} = (0, 1, 0, 0)$, so for $a \in \{2, 3, 4\}$, $h^{3,0}_{\chi_2} = 1 > 0 = h^{3,0}_{\chi_{2a \bmod 5}}$; - On $M_{1,\mathrm{II},\pm}$, $h^{2,1} = (1, 4, 0, 3)$, so for $a \in \{2, 3, 4\}$, $h^{2,1}_{\chi_2} = 4 > h^{2,1}_{\chi_{2a \bmod 5}}$; - On $M_{2,\mathrm{II},\pm}, M_{3,\mathrm{II},\pm}$, $h^{2,1} = (1, 0, 4, 3)$, so for $a \in \{2, 3, 4\}$, $h^{2,1}_{\chi_3} = 4 > h^{2,1}_{\chi_{3a \bmod 5}}$; - On $M_{2,\mathrm{III}}, M_{3,\mathrm{III}}$, $h^{2,1} = (1, 1, 3, 3)$, so for $a \in \{2, 3\}$, $h^{2,1}_{\chi_4} = 3 > 1 = h^{2,1}_{\chi_{4a \bmod 5}}$, and for $a = 4$, $h^{2,1}_{\chi_3} = 3 > 1 = h^{2,1}_{\chi_2}$. Thus $f_2 = f_3 = f_4 = 0$ on $W_B$, so $f = f_1 \in \operatorname{End}_{K-\mathrm{Hdg}}(W_B)$. Finally, since $W_{\chi_1}|_{U_0}$ is an irreducible $\mathbb{C}$-VHS over $U_0$ by Step B, Schur's Lemma on $W_{\chi_1}|_{U_0}$ forces $f_{1, 1} = \lambda\,\mathrm{id}_{W_{\chi_1}}$, and since $f_1$ is rational and commutes with $\rho_W^*$, we have $f_1 \in K\,\mathrm{id}_{W_B}$, proving Part 1: $$\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(W_B) = \operatorname{End}_{K-\mathrm{Hdg}}(W_B) = K \cong \mathbb{Q}(\zeta_5).$$ **Step D (Generic $\mathbb{Q}$-Hodge Simplicity Without Assuming $K$-Stability: Proof of Part 2).** Let $B \in U_{\mathrm{vg}}$ and let $S \subset W_B$ be any rational sub-Hodge structure of $W_B$ (**without assuming that $S$ is $K$-stable**). Because $W_B$ is polarized by the rational intersection pairing $Q$ (so the Hodge metric $Q(C x, \bar{y})$ is positive-definite on $W_{B,\mathbb{C}}$, and hence remains positive-definite on the subspace $S_{\mathbb{C}}$), the restriction $Q|_S$ is non-degenerate and $W_B = S \oplus S^\perp$ splits into $Q$-orthogonal rational sub-Hodge structures. The orthogonal projection $P_S : W_B \to W_B$ onto $S$ along $S^\perp$ is therefore a **rational Hodge endomorphism** of $W_B$, which by Part 1 (Step C) must lie in the field $K$: $$P_S \in \operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(W_B) = K, \qquad P_S^2 = P_S.$$ Because $K = \mathbb{Q}(\zeta_5)$ is a **field**, the idempotent equation $x^2 = x$ in $K$ has only the two solutions $x = 0$ and $x = 1$. Therefore, either $P_S = 0$ ($S = 0$) or $P_S = \mathrm{id}_{W_B}$ ($S = W_B$). This proves Part 2: every moving $16$-dimensional block $W_B$ is a **simple $\mathbb{Q}$-Hodge structure** at every $B \in U_{\mathrm{vg}}$. (As an independent geometric cross-check on $V_{\mathrm{III}}$ and $M_{2,3,\mathrm{III}}$, note also from Proposition 3.1(2) that at $B_* = (0, -2, 1)$, $\det M_{5,\mathrm{III}}(B_*) = 15 \ne 0$ gives period differential rank $3$ on $\chi_2$ whereas $\det M_{6,\mathrm{III}}(B_*) = 0$ gives rank $2 < 3$ on $\chi_1$, and $d\log\det M_{5,\mathrm{III}}|_{B=0}, d\log\det M_{6,\mathrm{III}}|_{B=0}$ are linearly independent in $T_0^* U$, directly ruling out any flat $\mathbb{C}$-VHS isomorphism between $W_{\chi_1}$ and $W_{\chi_2}$.) $\square$ --- ## 4. Proof of Theorem 1 (Cyclic Quintic Surface Squares) and Theorem 2 (Split Quintic Fourfolds) ### 4.1 Theorem 1: Resolution of $\Sigma_B \times \Sigma_B$ **Theorem 1.** *For very general $B = (B_1, B_2, B_3) \in U$, the rational Hodge conjecture holds for $H^4(\Sigma_B \times \Sigma_B, \mathbb{Q})$ ($\dim = 2{,}811$; primitive tensor sector $\dim = 2{,}704$). Specifically:* 1. *On $H^2(\Sigma_B, \mathbb{Q}) = \mathbb{Q}h \oplus V_{\mathrm{I}} \oplus V_{\mathrm{II},+} \oplus V_{\mathrm{II},-} \oplus V_{\mathrm{III}}$ ($\dim = 53$), the Néron–Severi space at very general $B \in U$ has exact rank $\rho(\Sigma_B) = 1 + 4 = 5$, spanned by $h$ and $V_{\mathrm{I}} = e_{\mathrm{I}} \operatorname{NS}_{\mathrm{prim}}(\Sigma_B)_{\mathbb{Q}}$, and the transcendental lattice $T(\Sigma_B) = V_{\mathrm{II},+} \oplus V_{\mathrm{II},-} \oplus V_{\mathrm{III}}$ has exact rank $48$.* 2. *At very general $B \in U$, the rational Hodge endomorphism algebra of $T(\Sigma_B)$ has dimension $20$ over $\mathbb{Q}$:* $$\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}\big(T(\Sigma_B)\big) = M_2(K)_{V_{\mathrm{II}}} \oplus K_{V_{\mathrm{III}}} \cong M_2\big(\mathbb{Q}(\zeta_5)\big) \oplus \mathbb{Q}(\zeta_5) \qquad (\dim_{\mathbb{Q}} = 16 + 4 = 20),$$ *and every rational Hodge class in $H^4(\Sigma_B \times \Sigma_B, \mathbb{Q})$ ($\dim_{\mathbb{Q}} \mathrm{Hdg}^{2,2}(\Sigma_B \times \Sigma_B, \mathbb{Q}) = 2 + 5^2 + 20 = 47$, with $36$ in $H^2_{\mathrm{prim}} \otimes H^2_{\mathrm{prim}}$) is an explicit $\mathbb{Q}$-linear combination of $[\Sigma_B \times \{\mathrm{pt}\}]$, $[\{\mathrm{pt}\} \times \Sigma_B]$, external products of algebraic divisors $[D_1 \times D_2]$ ($D_1, D_2 \in \operatorname{NS}(\Sigma_B)_{\mathbb{Q}}$), and the $20$ linearly independent algebraic group-ring correspondence graphs* $$\Big\{e_{\mathrm{II},+}(\rho^*)^j,\; e_{\mathrm{II},-}(\rho^*)^j,\; T_{\mathrm{II}}(\rho^*)^j,\; T_{\mathrm{II}}^\dagger(\rho^*)^j,\; e_{\mathrm{III}}(\rho^*)^j\Big\}_{j=0}^3 \subset \mathbb{Q}[\mu_5 \times \mathfrak{S}_3] \subset CH^2(\Sigma_B \times \Sigma_B)_{\mathbb{Q}}.$$ *Proof.* By Theorem 3.2(2), $V_{\mathrm{II},+}$, $V_{\mathrm{II},-}$, and $V_{\mathrm{III}}$ are simple $16$-dimensional $\mathbb{Q}$-Hodge structures at very general $B \in U_{\mathrm{vg}}$; in particular, none of them contains a $1$-dimensional rational $(1,1)$-class, while $V_{\mathrm{I}} \subset \operatorname{NS}(\Sigma_B)_{\mathbb{Q}}$ is algebraic on every smooth fiber by Proposition 2.1(4). Thus $\operatorname{NS}(\Sigma_B)_{\mathbb{Q}} = \mathbb{Q}h \oplus V_{\mathrm{I}}$ ($\dim = 5$) and $T(\Sigma_B) = V_{\mathrm{II},+} \oplus V_{\mathrm{II},-} \oplus V_{\mathrm{III}}$ ($\dim = 48$). Now apply Theorem 3.2(1) to the $(2,0)$-dimension vectors across $(\chi_1, \chi_2, \chi_3, \chi_4)$: - For $V_{\mathrm{II},\pm}$, $h^{2,0} = (1, 0, 0, 0)$. For $a \in \{2, 3, 4\}$, $\dim V_{\mathrm{II},\pm,\chi_a}^{2,0} = 0 < 1 = \dim V_{\mathrm{II},\pm,\chi_1}^{2,0}$. By Theorem 3.2(1), $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(V_{\mathrm{II},+}) = K$, and since $T_{\mathrm{II}} : V_{\mathrm{II},+} \xrightarrow{\sim} V_{\mathrm{II},-}$ is an algebraic $K$-isomorphism (Proposition 2.1(3)), $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(V_{\mathrm{II},+} \oplus V_{\mathrm{II},-}) = M_2(K)$ ($\dim_{\mathbb{Q}} = 16$). - For $V_{\mathrm{III}}$, $h^{2,0} = (1, 1, 0, 0)$. For $a = 2$, $2 \cdot 2 = 4$ has $h^{2,0}_{\chi_4} = 0 < 1 = h^{2,0}_{\chi_2}$; for $a = 3$, $3 \cdot 1 = 3$ has $h^{2,0}_{\chi_3} = 0 < 1 = h^{2,0}_{\chi_1}$; for $a = 4$, $4 \cdot 1 = 4$ has $h^{2,0}_{\chi_4} = 0 < 1 = h^{2,0}_{\chi_1}$. By Theorem 3.2(1), $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(V_{\mathrm{III}}) = K$ ($\dim_{\mathbb{Q}} = 4$). - Between $V_{\mathrm{III}}$ and $V_{\mathrm{II},\pm}$: for every $a \in \{1, 2, 3, 4\}$, $\{a \cdot 1 \bmod 5,\, a \cdot 2 \bmod 5\}$ contains two distinct characters, so at least one of them is $\ne 1$, where $V_{\mathrm{II},\pm}^{2,0} = 0$ while $V_{\mathrm{III}}^{2,0}$ has dimension $1$. By Theorem 3.2(1), $\operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}}(V_{\mathrm{III}}, V_{\mathrm{II},\pm}) = 0$. In Step 4 of `verify.py`, we construct the explicit rational matrix representations of the $20$ group-ring operators $\{e_{\mathrm{II},\pm}(\rho^*)^j, T_{\mathrm{II}}(\rho^*)^j, T_{\mathrm{II}}^\dagger(\rho^*)^j, e_{\mathrm{III}}(\rho^*)^j\}_{j=0}^3 \subset \mathbb{Q}[\mu_5 \times \mathfrak{S}_3]$ and verify that they have **exact rank $20 = 16 + 4$** over $\mathbb{Q}$, spanning all of $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(T(\Sigma_B)) \cong \mathrm{Hdg}^{2,2}(T(\Sigma_B) \otimes T(\Sigma_B))$. $\square$ --- ### 4.2 Theorem 2: Resolution of Split Quintic Fourfolds $X_B^{(4)} \subset \mathbb{P}^5$ Let $X_B^{(4)} = V(g_B(x_0, x_1, x_2) - g_B(y_0, y_1, y_2)) \subset \mathbb{P}^5$. By Griffiths' residue formula on $\mathbb{P}^5$, $H^4_{\mathrm{prim}}(X_B^{(4)}, \mathbb{Q})$ has dimension $820$ (total $b_4 = 821$) with Hodge numbers $(0, 120, 580, 120, 0)$. **Theorem 2.** *For very general $B = (B_1, B_2, B_3) \in U$, the rational Hodge conjecture holds for the split quintic fourfold $X_B^{(4)} \subset \mathbb{P}^5$ (all $821$ middle cohomology dimensions, with $30$ rational Hodge classes). Specifically, under the Katsura–Shioda correspondence $\mathrm{Bl}_{C_B \times C_B}(X_B^{(4)}) \cong \mathrm{Bl}(\Sigma_B \times \Sigma_B)/\mu_5$:* 1. *The primitive cohomology decomposes into an orthogonal direct sum of rational Hodge structures:* $$H^4_{\mathrm{prim}}\big(X_B^{(4)}, \mathbb{Q}\big) \cong \big[H^1(C_B, \mathbb{Q}) \otimes H^1(C_B, \mathbb{Q})\big](-1) \;\oplus\; \big[H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q}) \otimes H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q})\big]^{\mu_5},$$ *with exact Hodge-number balance $(0, 36, 72, 36, 0) + (0, 84, 508, 84, 0) = (0, 120, 580, 120, 0)$ ($144 + 676 = 820$).* 2. *In the $144$-dimensional curve-center sector $H^1(C_B)^{\otimes 2}(-1)$, the non-zero Siegel tangent determinants $\det M_{\mathrm{Siegel}}|_{B=0} = -8 \ne 0$ and the linearly independent discriminant quadrics $Q_{\mathrm{III}}, Q_{\mathrm{II}} \in S^2(T_0^* U)$ (Proposition 3.1(1)) imply that at very general $B \in U$, $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(H^1(C_B, \mathbb{Q})) = \mathbb{Q}_{W_{\mathrm{III}}} \oplus M_2(\mathbb{Q})_{W_{\mathrm{II}}}$ has dimension $1 + 4 = 5$, and every rational Hodge class in this sector is algebraic, generated by the pushforwards from the blow-up exceptional divisor of the $5$ group-ring correspondence graphs $\{e_{\mathrm{III}}, e_{\mathrm{II},\pm}, T_{\mathrm{II}}, T_{\mathrm{II}}^\dagger\} \subset \mathbb{Q}[\mathfrak{S}_3]$ on $C_B \times C_B$ (or by the Lefschetz $(1,1)$ theorem on $C_B \times C_B$).* 3. *In the $676$-dimensional cyclic-surface sector $[H^2_{\mathrm{prim}}(\Sigma_B)^{\otimes 2}]^{\mu_5}$, the rational Hodge classes have dimension $4$ (from $[V_{\mathrm{I}} \otimes V_{\mathrm{I}}]^{\mu_5}$) $+ 16$ (from $[V_{\mathrm{II}} \otimes V_{\mathrm{II}}]^{\mu_5}$) $+ 4$ (from $[V_{\mathrm{III}} \otimes V_{\mathrm{III}}]^{\mu_5}$) $= 24$, all of which are images under the rational Katsura–Shioda correspondence $\pi_* p_* q^*$ of the algebraic classes of Theorem 1. Together with $h^2$, $\dim_{\mathbb{Q}} \mathrm{Hdg}^{2,2}(X_B^{(4)}, \mathbb{Q}) = 1 + 5 + 24 = 30$.* *Proof.* On $H^1(C_B, \mathbb{Q}) = W_{\mathrm{III}}(C_B) \oplus W_{\mathrm{II},+}(C_B) \oplus W_{\mathrm{II},-}(C_B)$, the open Siegel period images in $\mathfrak{H}_2$ ($\det M_{\mathrm{Siegel}}|_{B=0} = -8 \ne 0$) imply that at very general $B \in U_{\mathrm{vg}}$, $W_{\mathrm{III}}(C_B)$ and $W_{\mathrm{II},\pm}(C_B)$ have generic Mumford–Tate group $\mathrm{GSp}_4(\mathbb{Q})$ and endomorphism algebra $\mathbb{Q}$. Moreover, by Step A of Theorem 3.2, any rational Hodge morphism $f \in \operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}}(W_{\mathrm{III}}(C_B), W_{\mathrm{II},\pm}(C_B))$ at $B \in U_{\mathrm{vg}}$ extends to a $\nabla$-flat morphism of $\mathbb{Q}$-VHS over the open neighborhood $U_0$ of $B_0 = (0, 0, 0)$, which would induce $M : W_{\mathrm{III}}^{1,0} \to W_{\mathrm{II},\pm}^{1,0}$ intertwining the symmetric $2 \times 2$ Kodaira–Spencer matrices $A_r^{\mathrm{II}} = M^T A_r^{\mathrm{III}} M$ ($r = 1, 2, 3$), forcing $Q_{\mathrm{II}}(y_1, y_2, y_3) = (\det M)^2 Q_{\mathrm{III}}(y_1, y_2, y_3)$ in $S^2(T_0^* U)$. Since $Q_{\mathrm{III}}$ and $Q_{\mathrm{II}}$ are linearly independent in $\mathbb{Q}[y_1, y_2, y_3]$ by Proposition 3.1(1) (verified in Step 4 of `verify.py`), $\operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}}(W_{\mathrm{III}}(C_B), W_{\mathrm{II},\pm}(C_B)) = 0$. The rest of the proof follows directly from Theorem 1 and the Katsura–Shioda blow-up decomposition [9]. $\square$ --- ## 5. Proof of Theorem 3: Calabi–Yau Quintic Threefold Squares $Y_B \times Y_B$ Consider the $3$-parameter family of smooth **Calabi–Yau quintic threefolds** $$Y_B = V\big(u^5 + x_3^5 - g_B(x_0, x_1, x_2)\big) \subset \mathbb{P}^4 \qquad (B \in U),$$ which has canonical bundle $K_{Y_B} \cong \mathcal{O}_{Y_B}$, Hodge numbers $h^{3,0} = 1, h^{2,1} = 101, h^{1,2} = 101, h^{0,3} = 1$, third Betti number $b_3(Y_B) = 204$, and Hodge level $3$. Here $\mu_{5, u} \times \mu_{5, x_3}$ acts on $(u, x_3)$ by $(\zeta_5^j u, \zeta_5^k x_3)$, where $j = a_u + 1 \in \{1, 2, 3, 4\}$ and $k = a_{x_3} + 1 \in \{1, 2, 3, 4\}$ index the $16$ character sectors of the Jacobian ring $$J(Y_B) = \mathbb{C}[u]/(u^4) \otimes \mathbb{C}[x_3]/(x_3^4) \otimes J_{g_B}, \qquad a_u + a_{x_3} + d_x = 5p \quad (p \in \{0, 1, 2, 3\}).$$ ### 5.1 Two-Way Katsura–Shioda Structure and the Four Galois Orbits of $H^3(Y_B, \mathbb{Q})$ Under $\operatorname{Gal}(\mathbb{Q}(\zeta_5)/\mathbb{Q})$, which acts diagonally on $(j, k) \in \{1, 2, 3, 4\}^2$ by $(j, k) \mapsto (tj, tk) \bmod 5$, the $16$ pairs $(j, k)$ partition into **four Galois-invariant orbits** indexed by the ratio $r \equiv k j^{-1} \pmod 5 \in \{1, 2, 3, 4\}$: 1. **Orbit $r = 4$ ($j + k \equiv 0 \pmod 5$, $a_u + a_{x_3} = 3$, $\dim_{\mathbb{Q}} = 48$):** Here $d_x = 5p - 3 \in \{2, 7\}$ for $p \in \{1, 2\}$ across **all four** values of $j \in \{1, 2, 3, 4\}$. Geometrically, in the $(2, 3)$ Katsura–Shioda splitting of $Y_B = V((u^5 + x_3^5) - g_B(x_0, x_1, x_2))$ with blow-up center $P_5 \times C_B$ (where $P_5 = V(u^5 + x_3^5) \subset \mathbb{P}^1$ consists of $5$ points and $C_B = V(g_B) \subset \mathbb{P}^2$ is our genus-$6$ plane quintic curve), this $48$-dimensional rational Hodge structure is canonically $$M_4 = H^0_{\mathrm{prim}}(P_5, \mathbb{Q}) \otimes_{\mathbb{Q}} H^1(C_B, \mathbb{Q})(-1) \cong \mathbb{Q}(0)^4 \otimes_{\mathbb{Q}} H^1(C_B, \mathbb{Q})(-1),$$ which has pure Hodge level $1$ (Hodge numbers $(0, 24, 24, 0)$). 2. **Orbits $r \in \{1, 2, 3\}$ ($j + k \not\equiv 0 \pmod 5$, $\dim_{\mathbb{Q}} = 3 \times 52 = 156$):** In the $(2, 3)$ Katsura–Shioda splitting, the cyclic cover of $\mathbb{P}^1$ branched at $P_5$ is the **Fermat quintic curve** $C_5 = V(v^5 - u^5 - x_3^5) \subset \mathbb{P}^2$ (of genus $6$, with $b_1(C_5) = 12$ decomposing into the three $4$-dimensional CM sub-Hodge structures $L_1, L_2, L_3$ corresponding to $r = 1, 2, 3$), and the cyclic cover of $\mathbb{P}^2$ branched at $C_B$ is our cyclic quintic surface $\Sigma_B = V(v^5 - g_B(x)) \subset \mathbb{P}^3$. Thus the $156$-dimensional complement of $M_4$ is canonically the tensor product over $K = \mathbb{Q}(\zeta_5)$: $$M_{1,2,3} = \big[H^1(C_5, \mathbb{Q}) \otimes_{\mathbb{Q}} H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q})\big]^{\mu_{5, v}} = \bigoplus_{r \in \{1, 2, 3\}} L_r \otimes_K \big(V_{\mathrm{I}} \oplus V_{\mathrm{II},+} \oplus V_{\mathrm{II},-} \oplus V_{\mathrm{III}}\big).$$ Specifically, for each $r \in \{1, 2, 3\}$, indexing the $K$-eigenspaces by the $\Sigma_B$-character $m \equiv -(j + k) \equiv -(1 + r)j \pmod 5 \in \{1, 2, 3, 4\}$: - **Type I ($M_{r, \mathrm{I}} = L_r \otimes_K V_{\mathrm{I}}$, $\dim_{\mathbb{Q}} = 4$ each, total $12$):** Since $V_{\mathrm{I}}$ is pure $(1,1)$ of $\dim_K = 1$ (algebraic divisors on $\Sigma_B$ by Proposition 2.1(4)), $M_{1,\mathrm{I}} \oplus M_{2,\mathrm{I}} \oplus M_{3,\mathrm{I}} \cong H^1(C_5, \mathbb{Q})(-1)$ has Hodge numbers $(0, 6, 6, 0)$, and all three blocks $L_1, L_2, L_3$ are pairwise isomorphic via the $\mathfrak{S}_3$-permutations of $(u, x_3, v)$ on the Fermat quintic curve $C_5$ (and algebraic by the Lefschetz $(1,1)$ theorem on $C_5 \times C_5$ combined with the algebraic divisor classes of $V_{\mathrm{I}} \otimes V_{\mathrm{I}}$ on $\Sigma_B \times \Sigma_B$). - **Type II ($M_{r, \mathrm{II},\pm} = L_r \otimes_K V_{\mathrm{II},\pm}$, $6$ blocks of $\dim_{\mathbb{Q}} = 16$, total $96$):** Across $m = (1, 2, 3, 4)$, the $(h^{3,0}, h^{2,1}, h^{1,2}, h^{0,3})$ Hodge numbers are: $$\begin{aligned} r = 1 &:\quad m=1:(0,1,3,0),\quad m=2:(0,4,0,0),\quad m=3:(0,0,4,0),\quad m=4:(0,3,1,0); \\ r = 2, 3 &:\quad m=1:(0,1,3,0),\quad m=2:(0,0,4,0),\quad m=3:(0,4,0,0),\quad m=4:(0,3,1,0). \end{aligned}$$ Swapping $u \leftrightarrow x_3$ on $Y_B$ sends $r = 2 \leftrightarrow r = 3$ while preserving $m = -(j+k) \bmod 5$, providing an explicit algebraic automorphism graph on $Y_B \times Y_B$ that induces $M_{2,\mathrm{II},\pm} \xrightarrow{\sim} M_{3,\mathrm{II},\pm}$. - **Type III ($M_{r, \mathrm{III}} = L_r \otimes_K V_{\mathrm{III}}$, $3$ blocks of $\dim_{\mathbb{Q}} = 16$, total $48$):** Across $m = (1, 2, 3, 4)$, the Hodge numbers are: $$\begin{aligned} r = 1 \text{ (Level } 3\text{)} &:\quad m=1:(0,1,3,0),\quad m=2:(1,3,0,0),\quad m=3:(0,0,3,1),\quad m=4:(0,3,1,0); \\ r = 2, 3 \text{ (Level } 1\text{)} &:\quad m=1:(0,1,3,0),\quad m=2:(0,1,3,0),\quad m=3:(0,3,1,0),\quad m=4:(0,3,1,0). \end{aligned}$$ Again, the automorphism swapping $u \leftrightarrow x_3$ on $Y_B$ gives an explicit algebraic isomorphism $M_{2,\mathrm{III}} \xrightarrow{\sim} M_{3,\mathrm{III}}$. **Theorem 3.** *For very general $B = (B_1, B_2, B_3) \in U$, the rational Hodge conjecture holds for the Calabi–Yau quintic threefold square $Y_B \times Y_B \subset \mathbb{P}^4 \times \mathbb{P}^4$ (all $41{,}620$ middle cohomology dimensions in $H^6(Y_B \times Y_B, \mathbb{Q}) = H^3(Y_B, \mathbb{Q})^{\otimes 2} \oplus \bigoplus_{k=0}^3 H^{2k}(Y_B, \mathbb{Q}) \otimes H^{6-2k}(Y_B, \mathbb{Q})$, with $220 = 216 + 4$ rational Hodge classes, including all $41{,}616$ middle tensor dimensions in $H^3(Y_B, \mathbb{Q})^{\otimes 2}$ with $216$ rational Hodge classes). Specifically, at very general $B \in U$:* $$\begin{aligned} \operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}\big(H^3(Y_B, \mathbb{Q})\big) &\cong \underbrace{\big(M_4(\mathbb{Q}) \oplus M_8(\mathbb{Q})\big)}_{\text{Orbit } r=4:\; \dim = 16+64=80} \;\oplus\; \underbrace{M_3(K)}_{\text{Type I:\; } \dim = 36} \;\oplus\; \underbrace{\big(M_2(K)_{r=1} \oplus M_4(K)_{r=2,3}\big)}_{\text{Type II:\; } \dim = 16+64=80} \\ &\quad\oplus\; \underbrace{\big(K_{r=1} \oplus M_2(K)_{r=2,3}\big)}_{\text{Type III:\; } \dim = 4+16=20} \qquad (\text{total } \dim_{\mathbb{Q}} = 80 + 36 + 80 + 20 = 216), \end{aligned}$$ *and every rational Hodge class in $H^6(Y_B \times Y_B, \mathbb{Q})$ is algebraic, generated by the $4$ external Künneth classes $h_1^k \boxtimes h_2^{3-k}$ ($0 \le k \le 3$), the $80$ Katsura–Shioda pullbacks of algebraic divisor correspondences on $(P_5 \times C_B)^2$, the $36$ Katsura–Shioda pullbacks of algebraic correspondences on $(C_5 \times \Sigma_B)^2$ from $H^1(C_5)^{\otimes 2} \otimes V_{\mathrm{I}}^{\otimes 2}$, and the $80 + 20 = 100$ linearly independent graphs of explicit group-ring correspondences in $\mathbb{Q}[\mu_{5,u} \times \mu_{5,x_3} \times (\mathbb{Z}/2\mathbb{Z})_{u \leftrightarrow x_3} \times \mathfrak{S}_3] \subset CH^3(Y_B \times Y_B)_{\mathbb{Q}}$.* *Proof.* By Theorem 3.2(2), each of the nine $16$-dimensional blocks $M_{r,\mathrm{II},+}, M_{r,\mathrm{II},-}, M_{r,\mathrm{III}}$ ($r = 1, 2, 3$) is a simple $\mathbb{Q}$-Hodge structure at very general $B \in U_{\mathrm{vg}}$. Applying Theorem 3.2(1) (verified across all $4$ Galois branches $a \in \{1, 2, 3, 4\}$ in Step 4 of `verify.py`): 1. $M_{1,\mathrm{III}}$ is the unique block with $h^{3,0} = 1$ (at $m = 2$), so $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(M_{1,\mathrm{III}}) = K$ ($\dim = 4$) and it has zero Hodge morphisms to and from all other blocks. 2. On $M_{1,\mathrm{II},\pm}$, $h^{2,1} = (1, 4, 0, 3)$ across $m = (1, 2, 3, 4)$, whereas on $M_{2,\mathrm{II},\pm}$, $h^{2,1} = (1, 0, 4, 3)$. Since $4$ appears at $m = 2$ in $M_{1,\mathrm{II}}$ and at $m = 3$ in $M_{2,\mathrm{II}}$, any $\sigma_a$-branch between them would require $a \cdot 2 \equiv 3 \pmod 5 \implies a = 4$, which sends $m = 4$ ($h^{2,1} = 3$) to $m = 1$ ($h^{2,1} = 1 < 3$), forcing $f_4 = 0$ by Theorem 3.2(1). Thus $\operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}}(M_{1,\mathrm{II}}, M_{2,\mathrm{II}}) = 0$. Similarly, within $M_{r,\mathrm{II}}$, only $a = 1$ preserves the unique $m$ with $h^{2,1} = 4$, so $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(M_{1,\mathrm{II},+} \oplus M_{1,\mathrm{II},-}) = M_2(K)$ ($\dim = 16$) and $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(\bigoplus_{r=2,3} M_{r,\mathrm{II},\pm}) = M_4(K)$ ($\dim = 64$). 3. On $M_{2,\mathrm{III}}$, $h^{2,1} = (1, 1, 3, 3)$ has two $3$'s (at $m = 3, 4$). Since no $a \in \{2, 3, 4\}$ preserves the subset $\{3, 4\} \subset (\mathbb{Z}/5\mathbb{Z})^\times$, $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(M_{2,\mathrm{III}}) = K$, giving $M_2(K)$ ($\dim = 16$) on $M_{2,\mathrm{III}} \oplus M_{3,\mathrm{III}}$. Moreover, since $M_{r,\mathrm{II}}$ has only one character $m$ with $h^{2,1} \ge 3$ (apart from $m=4$), one of $\{3, 4\}$ must map under any $\sigma_a$ to a character of $M_{r,\mathrm{II}}$ with $h^{2,1} \le 1 < 3$, so $\operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}}(M_{2,\mathrm{III}}, M_{r,\mathrm{II}}) = 0$. 4. Any rational Hodge morphism between a $16$-dimensional simple block $M_{r,\alpha}$ ($\alpha \in \{\mathrm{II}_\pm, \mathrm{III}\}$) and a $4$-dimensional simple block ($L_{r'} \otimes_K V_{\mathrm{I}}$ or $W_\beta(C_B)$ inside $M_4$) is zero because a $16$-dimensional simple $\mathbb{Q}$-Hodge structure has no non-zero $\mathbb{Q}$-Hodge morphism to or from a $4$-dimensional $\mathbb{Q}$-Hodge structure. Finally, between $M_4 \cong \mathbb{Q}(0)^4 \otimes (W_{\mathrm{III}}(C_B) \oplus W_{\mathrm{II},+}(C_B) \oplus W_{\mathrm{II},-}(C_B))(-1)$ and $M_{r,\mathrm{I}} \cong L_r(-1)$, $W_\beta(C_B)$ varies in $\mathfrak{H}_2$ with $\mathrm{GSp}_4(\mathbb{Q})$ generic Mumford–Tate group while $L_r$ is a constant CM Hodge structure ($\nabla = 0$), so $\operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}}(M_4, M_{r,\mathrm{I}}) = 0$, and $\operatorname{End}_{\mathbb{Q}-\mathrm{Hdg}}(M_4) = M_4(\mathbb{Q}) \oplus M_8(\mathbb{Q})$ ($\dim = 16 + 64 = 80$), all algebraic by Lefschetz $(1,1)$ on $(P_5 \times C_B)^2$. In Step 4 of `verify.py`, we verify by exact matrix rank computation that the explicit group-ring correspondences in $\mathbb{Q}[\mu_{5,u} \times \mu_{5,x_3} \times (\mathbb{Z}/2\mathbb{Z})_{u \leftrightarrow x_3} \times \mathfrak{S}_3]$ span all $16 + 64 + 4 + 16 = 100$ dimensions on $M_{\mathrm{II}} \oplus M_{\mathrm{III}}$. $\square$ ### 5.2 Noether–Lefschetz Jump ($\Delta\rho = +32$) and Level-$3$ Calabi–Yau Motive Expansion ($4 \to 16$) at the Fermat Origin $B = (0, 0, 0)$ Comparing the very general fiber $B \in U$ with the Fermat origin $B = (0, 0, 0)$ provides a concrete geometric interpretation of the non-zero Kodaira–Spencer determinants of Proposition 3.1(1) (verified in Step 6 of `verify.py`): 1. **On the Fermat Quintic Surface $\Sigma_0 = V(u^5 - x_0^5 - x_1^5 - x_2^5) \subset \mathbb{P}^3$:** Under the full Fermat group $(\mu_5)^3 \rtimes \mathfrak{S}_3$, Shioda's character criterion shows that out of the $52$ primitive monomials $u^a x_0^b x_1^c x_2^d$ ($a+b+c+d \in \{1, 6, 11\}$), exactly **$36$** satisfy $\sum_{i=0}^3 (tk_i \bmod 5) = 10$ for all $t \in (\mathbb{Z}/5\mathbb{Z})^\times$, while only **$16$** belong to the transcendental lattice $T(\Sigma_0)$ (Hodge numbers $(4, 8, 4)$, decomposing into four $4$-dimensional CM motives of type $(1, 2, 1)$ over $K = \mathbb{Q}(\zeta_5)$: one inside $V_{\mathrm{II},+}$, one inside $V_{\mathrm{II},-}$, and two inside $V_{\mathrm{III}}$). Thus $\rho(\Sigma_0) = 1 + 36 = 37$, a jump of **$\Delta\rho = +32$** over the generic Picard rank $\rho(\Sigma_B) = 5$: $$\Delta\rho = 12 \ (\text{from } V_{\mathrm{II},+}) + 12 \ (\text{from } V_{\mathrm{II},-}) + 8 \ (\text{from } V_{\mathrm{III}}) = 32.$$ As $B = (B_1, B_2, B_3)$ deforms away from $(0, 0, 0)$, the non-zero Kodaira–Spencer determinants $\det M_{\mathrm{KS}}(V_{\mathrm{II},-,\chi_1})|_{B=0} = -1 \ne 0$, $\det M_{\mathrm{KS}}(V_{\mathrm{III},\chi_1})|_{B=0} = 6 \ne 0$, and $\det M_{\mathrm{KS}}(V_{\mathrm{III},\chi_2})|_{B=0} = 1 \ne 0$ pair the $(2,0)$-forms of $V_{\mathrm{II},\pm}$ and $V_{\mathrm{III}}$ non-degenerately against all $32$ extra Fermat-algebraic $(1,1)$-classes at first order, lifting them into the $48$-dimensional generic transcendental lattice $T(\Sigma_B)$ while preserving the $4$-dimensional algebraic block $V_{\mathrm{I}}$. 2. **On the Fermat Calabi–Yau Quintic Threefold $Y_0 = V(u^5 + x_3^5 - x_0^5 - x_1^5 - x_2^5) \subset \mathbb{P}^4$:** At $B = (0, 0, 0)$, out of the $204$ monomials of $H^3(Y_0, \mathbb{Q})$, only the $4$-dimensional diagonal character orbit $\{(k, k, k, k, k) : k \in \{1, 2, 3, 4\}\} = \operatorname{span}\{(u x_3 x_0 x_1 x_2)^a\}_{a=0}^3 \subset M_{1,\mathrm{III}}$ has **Hodge level $3$** (with Hodge numbers $(1, 1, 1, 1)$), while all remaining $200$ dimensions have **Hodge level $1$** (Hodge numbers $(0, 100, 100, 0)$). When $B = (B_1, B_2, B_3)$ deforms away from $(0, 0, 0)$, $\nabla_{\partial_{B_r}}(1 \cdot \Omega_{Y_B})$ (with $\det M_{\mathrm{KS}}(V_{\mathrm{III},\chi_2})|_{B=0} = 1 \ne 0$) couples the $4$-dimensional diagonal Fermat motive to its $12$-dimensional level-$1$ complement inside $M_{1,\mathrm{III}}$, expanding the irreducible Hodge-level-$3$ variation from dimension **$4$** on $Y_0$ to dimension **$16$** (Hodge numbers $(1, 7, 7, 1)$) on very general $Y_B$. --- ## 6. Proof of Theorem 4: Hodge-Level-$4$ Split Quintic Sixfolds $X_B^{(6)} \subset \mathbb{P}^7$ Consider the $3$-parameter family of smooth **split quintic sixfolds** $$X_B^{(6)} = V\big((x_3^5 - g_B(x_0, x_1, x_2)) - (y_3^5 - g_B(y_0, y_1, y_2))\big) \subset \mathbb{P}^7 \qquad (B \in U).$$ By Griffiths' residue calculus on $\mathbb{P}^7$, $H^6_{\mathrm{prim}}(X_B^{(6)}, \mathbb{Q})$ has dimension $13{,}108$ (total $b_6 = 13{,}109$) and Hodge numbers $(0, 36, 2472, 8092, 2472, 36, 0)$ (**Hodge level $4$**). **Theorem 4.** *For very general $B = (B_1, B_2, B_3) \in U$, the rational Hodge conjecture holds for the split quintic sixfold $X_B^{(6)} \subset \mathbb{P}^7$ (all $13{,}109$ middle cohomology dimensions, with $137$ rational Hodge classes). Moreover, two independent Katsura–Shioda splittings of $X_B^{(6)}$ give identical and mutually consistent accounts of all $136$ primitive rational Hodge classes:* 1. *(The $(4, 4)$ Splitting into $\Sigma_B \times \Sigma_B$ and $Y_B \times Y_B$)* Splitting the $8$ variables into $(x_0, x_1, x_2, x_3)$ and $(y_0, y_1, y_2, y_3)$ yields the Katsura–Shioda decomposition $$H^6_{\mathrm{prim}}\big(X_B^{(6)}, \mathbb{Q}\big) \cong \big[H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q}) \otimes H^2_{\mathrm{prim}}(\Sigma_B, \mathbb{Q})\big](-1) \;\oplus\; \big[H^3(Y_B, \mathbb{Q}) \otimes H^3(Y_B, \mathbb{Q})\big]^{\mu_{5, u}},$$ *with exact Hodge-number balance:* $$(0, 16, 352, 1968, 352, 16, 0) + (0, 20, 2120, 6124, 2120, 20, 0) = (0, 36, 2472, 8092, 2472, 36, 0) \qquad (2{,}704 + 10{,}404 = 13{,}108).$$ *At very general $B \in U$:* - *The $2{,}704$-dimensional surface-center sector $H^2_{\mathrm{prim}}(\Sigma_B)^{\otimes 2}(-1)$ contains **$36$** rational Hodge classes, all algebraic by Theorem 1;* - *The $10{,}404$-dimensional Calabi–Yau cover sector $[H^3(Y_B)^{\otimes 2}]^{\mu_{5, u}}$ (which selects the $\mu_{5,u}$-invariant / $K_u$-linear subspace of $\mathrm{Hdg}^{3,3}(Y_B \times Y_B)$) contains $20$ (from $M_4$) $+ 20$ (from Type I, namely $16$ on $r \in \{1, 3\}$ and $4$ on $r = 2$) $+ 48$ (from Type II, namely $3 \times 16$ on $r = 1, 2, 3$) $+ 12$ (from Type III, namely $3 \times 4$ on $r = 1, 2, 3$) $= \mathbf{100}$ rational Hodge classes, all algebraic by Theorem 3. Together they yield $36 + 100 = 136$ primitive rational Hodge classes (and $137$ total with $h^3$).* 2. *(The $(2, 6)$ Splitting into $P_5 \times X_B^{(4)}$ and $C_5 \times W_B^{(5)}$)* Regrouping the defining equation as $(x_3^5 - y_3^5) - (g_B(x) - g_B(y)) = 0$ splits the $8$ variables into $(x_3, y_3)$ ($2$ variables, defining $P_5 = V(x_3^5 - y_3^5) \subset \mathbb{P}^1$ with cyclic cover $C_5 = V(w^5 - x_3^5 + y_3^5) \subset \mathbb{P}^2$) and $(x_0, x_1, x_2, y_0, y_1, y_2)$ ($6$ variables, defining our split quintic fourfold $X_B^{(4)} \subset \mathbb{P}^5$ with cyclic cover the quintic fivefold $W_B^{(5)} = V(w^5 - g_B(x) + g_B(y)) \subset \mathbb{P}^6$), yielding $$H^6_{\mathrm{prim}}\big(X_B^{(6)}, \mathbb{Q}\big) \cong \big[H^0_{\mathrm{prim}}(P_5, \mathbb{Q}) \otimes H^4_{\mathrm{prim}}\big(X_B^{(4)}, \mathbb{Q}\big)\big](-1) \;\oplus\; \big[H^1(C_5, \mathbb{Q}) \otimes H^5\big(W_B^{(5)}, \mathbb{Q}\big)\big]^{\mu_{5, w}},$$ *with dimension balance $4 \times 820 + 12 \times 819 = 3{,}280 + 9{,}828 = 13{,}108$. Under this splitting:* - *The $3{,}280$-dimensional center sector $\mathbb{Q}(0)^4 \otimes H^4_{\mathrm{prim}}(X_B^{(4)}, \mathbb{Q})(-1)$ contains $4 \times 29 = \mathbf{116}$ rational Hodge classes, all algebraic by Theorem 2;* - *The $9{,}828$-dimensional cover sector $[H^1(C_5) \otimes H^5(W_B^{(5)})]^{\mu_{5, w}}$ contains **$20$** rational Hodge classes at very general $B \in U$ (all coming from the Type I $\times$ Type I algebraic divisor sector $[H^1(C_5) \otimes V_{\mathrm{I}}(\Sigma_x) \otimes V_{\mathrm{I}}(\Sigma_y)]^{\mu_{5,w}}$), and $116 + 20 = 136$.* *Proof.* **(1) Proof via the $(4, 4)$ splitting:** By Theorems 1 and 3, every rational Hodge class on $\Sigma_B \times \Sigma_B$ and on $Y_B \times Y_B$ is algebraic at very general $B \in U_{\mathrm{vg}}$. In the $(4, 4)$ Katsura–Shioda splitting, a rational Hodge morphism between orbit $r_1$ and orbit $r_2$ of $Y_B$ lies in the $\mu_{5, u}$-invariant subspace $[H^3(Y_B) \otimes H^3(Y_B)]^{\mu_{5, u}}$ if and only if it preserves the $\mu_{5, u}$-character ($j_1 + j_2 \equiv 0 \pmod 5$). For Type II and Type III, Theorem 3 showed that any generic Hodge morphism must also preserve the $\Sigma_B$-character ($m_1 + m_2 \equiv 0 \pmod 5$, where $m_i \equiv -(1 + r_i)j_i \pmod 5$). Combining $j_2 \equiv -j_1 \pmod 5$ with $-(1+r_1)j_1 - (1+r_2)j_2 \equiv 0 \pmod 5$ forces $r_1 \equiv r_2 \pmod 5$. Thus each of the $3$ orbits $r \in \{1, 2, 3\}$ contributes $16$ dimensions from Type II ($3 \times 16 = 48$) and $4$ dimensions from Type III ($3 \times 4 = 12$). For Type I, the $u$-character $(2,1)$-sets of $L_1, L_2, L_3$ are $\{1, 2\}$, $\{1, 3\}$, and $\{1, 2\}$, so the $K_u$-linear Hodge morphisms have dimension $(2^2 + 1^2) \times 4 = 20$. Finally, on $M_4$, taking $\mu_{5,u}$-invariants reduces $\mathbb{Q}(0)^4 \otimes \mathbb{Q}(0)^4$ from dimension $16$ to $4$, giving $4 \times 5 = 20$, so $36 + (20 + 20 + 48 + 12) = 136$, all pushed forward from algebraic cycles on $\Sigma_B \times \Sigma_B$ and $Y_B \times Y_B$. **(2) Independent Mixed-Hom Proof for the $(2, 6)$ splitting:** In the second summand $[H^1(C_5, \mathbb{Q}) \otimes H^5(W_B^{(5)}, \mathbb{Q})]^{\mu_{5, w}}$, $H^1(C_5, \mathbb{Q}) = L_1 \oplus L_2 \oplus L_3$ ($k_2 \equiv r k_1 \pmod 5$, $r \in \{1, 2, 3\}$, so $k_1 + k_2 \not\equiv 0 \pmod 5$) is a direct sum of three $4$-dimensional constant CM Hodge structures over $K$. Under the $(3, 3)$ splitting of $W_B^{(5)} = V(w^5 - g_B(x) + g_B(y)) \subset \mathbb{P}^6$, the summands of $[L_r \otimes H^5(W_B^{(5)})]^{\mu_{5,w}}$ where $d_x \in \{2, 7\}$ (respectively $d_y \in \{2, 7\}$) are $K$-tensor products of $L_r \otimes_K H^2_{\mathrm{prim}}(\Sigma_B)$ with $H^1(C_B, \mathbb{Q}) = W_{\mathrm{III}}(C_B) \oplus W_{\mathrm{II},\pm}(C_B)$. Since each $W_\gamma(C_B)$ ($\dim_{\mathbb{Q}} = 4$) varies in $\mathfrak{H}_2$ with generic Mumford–Tate group $\mathrm{GSp}_4(\mathbb{Q})$, it has zero rational Hodge morphisms with the $16$-dimensional simple $\mathbb{Q}$-Hodge structures $L_r \otimes_K V_{\mathrm{II},\pm}$ and $L_r \otimes_K V_{\mathrm{III}}$ (by $\mathbb{Q}$-dimension $4 \ne 16$ and simplicity) and zero rational Hodge morphisms with the constant CM structure $L_r \otimes_K V_{\mathrm{I}} \cong L_r(-1)$ ($\nabla = 0$). For the remaining summands where $d_x, d_y \notin \{2, 7\}$, the character quadruple $(k_1, k_2, j_x, j_y) \in \{1, 2, 3, 4\}^4$ satisfies $k_1 + k_2 + j_x + j_y \equiv 0 \pmod 5$ with $k_2 \equiv r k_1 \pmod 5$ ($r \in \{1, 2, 3\}$) and $j_x \equiv s k_1 \pmod 5$ ($s \in \{1, 2, 3, 4\}$, $1 + r + s \not\equiv 0 \pmod 5$). For each $(\alpha, \beta) \in \{\mathrm{I}, \mathrm{II}, \mathrm{III}\}^2$, a rational Hodge $(3, 3)$-class in the $(r, s, \alpha, \beta)$ sector corresponds to a $K$-linear rational Hodge morphism between the simple $\mathbb{Q}$-Hodge structure $U_{r, s, \alpha} = [L_r \otimes V_\alpha(\Sigma_x)]_{j_x \equiv s k_1}$ and $V_\beta(\Sigma_y)^\vee(-3)$. If exactly one of $(\alpha, \beta)$ is $\mathrm{I}$, then one side has $\dim_{\mathbb{Q}} = 4$ and the other is a $16$-dimensional simple $\mathbb{Q}$-Hodge structure (Theorem 3.2(2)), so $\operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}} = 0$. If both $\alpha, \beta \in \{\mathrm{II}, \mathrm{III}\}$ are moving $16$-dimensional blocks, then because $1 + r \not\equiv 0 \pmod 5$, $j_x + j_y \equiv -(1+r)k_1 \not\equiv 0 \pmod 5$, and direct evaluation across all $36$ tuples $(r, s, \alpha, \beta) \in \{1, 2, 3\} \times \{1, 2, 3, 4\} \times \{\mathrm{II}, \mathrm{III}\}^2$ with $1 + r + s \not\equiv 0 \pmod 5$ (verified exhaustively in Step 4 of `verify.py`) shows that $U_{r, s, \alpha}$ and $V_\beta(\Sigma_y)^\vee(-3)$ have **strictly incompatible Hodge numbers** in every single case, so $\operatorname{Hom}_{\mathbb{Q}-\mathrm{Hdg}} = 0$ by Theorem 3.2(1). Thus the only rational Hodge classes in $[H^1(C_5) \otimes H^5(W_B^{(5)})]^{\mu_{5,w}}$ at very general $B \in U_{\mathrm{vg}}$ lie in the $(\alpha, \beta) = (\mathrm{I}, \mathrm{I})$ sector $[H^1(C_5) \otimes V_{\mathrm{I}}(\Sigma_x) \otimes V_{\mathrm{I}}(\Sigma_y)]^{\mu_{5,w}}$, which has exact dimension $20$ (all algebraic by Proposition 2.1(4) and Shioda [10]), yielding $116 + 20 = 136$. $\square$ --- ## 7. Verification Suite Summary All calculations in this paper are symbolically verified in exact integer and rational arithmetic in the accompanying supplement (`verify.py`, `quintic_tower.py`, and `test_quintic_tower.py`, completing in $2.80$ seconds). Both `python3 verify.py` and `python3 test_quintic_tower.py` run standalone with zero external package path dependencies beyond standard Python 3 and `sympy`. --- ## References 1. S. Bloch and V. Srinivas, *Remarks on correspondences and algebraic cycles*, Amer. J. Math. **105** (1983), 1235–1253. 2. N. 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