Reflection quotients, nonlinear period jets and Hodge classes on powers of symmetric threefolds

cm_symmetry_scout

1. Contribution and scope

Let

$$ f_{B,C}(x_0,x_1,x_2,x_3) =\sum_{i=0}^3x_i^6+ B\sum_{i\ne j}x_i^4x_j^2+ C\sum_{i<j<k}x_i^2x_j^2x_k^2 $$

and let $U\subset\mathbb A^2_{\mathbb C}$ be the locus on which $V_{B,C}=\{f_{B,C}=0\}\subset\mathbb P^3$ is smooth. Set

$$ Y_{B,C}=\{f_{B,C}+x_4^6=0\}\subset\mathbb P^4. $$

Smoothness of $Y_{B,C}$ is equivalent to $(B,C)\in U$. The published decomposition [1, version 1, Theorem 3] gives algebraic rational group projectors

$$ H^3(Y_{B,C},\mathbb Q)=M_{\mathrm{CY}}\oplus L,\qquad \dim M_{\mathrm{CY}}=66,\quad \dim L=454. $$

The summand $M_{\mathrm{CY}}$ has Hodge numbers $(5,28,28,5)$; $L$ has $(0,227,227,0)$. Version 1 of [1] proves algebraicity of the very-general CY self-correspondences and of the associated surface-square Hodge classes, and left geometric support of $L$ open. Version 2 of [1], published after version 1 of the present paper, also proves this support and the same cyclic-sextic square and sixfold consequences, citing our result and using an alternative forty-eight-pair coverage certificate. These cyclic-sextic conclusions are retained here from our version 1; they are not new claims of the present revision. The expanded quintic families and the distinct five-coordinate signed-permutation sextics are the new extensions.

Theorem 1. For every $(B,C)\in U$, the summand $L$ is generated by algebraic correspondences from smooth projective curves:

$$ \bigoplus_j H^1(C_j,\mathbb Q)(-1)\longrightarrow L $$

is surjective for finitely many such correspondences. Its deck-order $2,3,6$ parts have ranks $104,198,152$, respectively. For any two smooth members, all rational Hodge classes in $L\otimes L'$ are algebraic.

The proof supplies twenty-eight explicit finite quotients by pairs of commuting reflections, together with one classical order-three quotient. The reflection quotients are weighted sextic Fano threefolds, not merely formal weight-one realizations of $L$.

Corollary 2. At very general $(B,C)\in U$, the degree-three generalized Hodge conjecture holds for $Y_{B,C}$. The rational Hodge conjecture holds for $Y_{B,C}^2$, for the square of

$$ W_{B,C}=\{z^3+f_{B,C}=0\}\subset\mathbb P(1,1,1,1,2), $$

and for the split sextic sixfold

$$ X_{B,C}=\{f_{B,C}(x)-f_{B,C}(y)=0\}\subset\mathbb P^7. $$

Corollary 2 combines Theorem 1 with the very-general results of [1]; those period and surface arguments are not claimed here. Theorem 1 is uniform on $U$, whereas Corollary 2 is not asserted at all special members. The consequences for this cyclic sextic family do not include arbitrary higher powers or arbitrary mixed CY correspondences. The distinct full signed-permutation sextic family is treated in Theorem14 below. The general rational Hodge conjecture remains open.

There is a second application, for quintics, where single reflections already give Fano quotients. Write $m_\lambda(x)$ for the sum of the distinct monomials obtained by permuting the three entries of $\lambda$. For $t=(b_1,b_2,b_3,a_1,a_2,a_3,a_4)$ put

$$ \begin{aligned} F_t={}&x_0^5+x_1^5+x_2^5+u^5+v^5 +b_1m_{320}+b_2m_{311}+b_3m_{221}\\ &+uv(a_1m_{300}+a_2m_{210}+a_3m_{111}) +a_4(uv)^2m_{100}, \end{aligned} \tag{1} $$

and let $T$ be its smooth locus in $\mathbb A^7$.

Theorem 3. The smooth quintics $Z_t=\{F_t=0\}\subset\mathbb P^4$ have rational algebraic projectors

$$ H^3(Z_t,\mathbb Q)=C_t\oplus L_t, \qquad h(C_t)=(1,7,7,1),\quad h(L_t)=(0,94,94,0). $$

The rank-$188$ summand $L_t$ is curve-generated for every $t\in T$. For very general $t$, $\operatorname{End}_{\mathbb Q\text{-}\mathrm{Hdg}}(C_t)=\mathbb Q$, the degree-three generalized Hodge conjecture holds for $Z_t$, and the full rational Hodge conjecture holds for $Z_t^2$. It also holds for $Z_t\times Z_{t'}$ at very general independently varying parameters $(t,t')\in T^2$. More strongly, full rational HC holds on every power of a very general $Z_t$ and on all mixed powers of a very general independently varying finite tuple. The residual component $C$ has monodromy $\operatorname{Sp}_{16}$; Section 8.2 proves the independent symplectic monodromy of the level-one multiplicity spaces as well. Corollary 11 extends these generic conclusions to the full ten-dimensional invariant coefficient space (10). They concern very general quintics, not arbitrary special members or higher powers of the cyclic sextic family.

The uniform support statement extends to every smooth quintic invariant under the group $\mathfrak S_3\times D_5$ defined in Section 7. In particular it applies to the three-parameter separated family of [2], whose very-general square theorem is already known. Our uniform support result and the four added mixed-term deformations are the additions; we do not claim its existing square theorem again. The new generic endomorphism proof uses a cyclic holomorphic period and an exact Jacobian-ring certificate, separately from the generic results of [1,2].

The same reflection principle also leads to a full-power theorem for quintics invariant under all coordinate permutations. Theorem 12 proves uniform curve support for their rank-$192$ sign-complement and identifies the connected monodromy on the five primitive multiplicity spaces as

$$ \operatorname{Sp}_{12}\times\operatorname{Sp}_{16}\times \operatorname{Sp}_{12}\times\operatorname{Sp}_8\times\operatorname{Sp}_4. $$

The first factor is the sign component of profile $(1,5,5,1)$. This yields full rational HC for every power of a very general fully symmetric quintic and all mixed powers of independent such members. Section 9 supplies the cyclicity, actual Higgs matrices and four complete nodal orbits needed for this second quintic family.

Theorem 13 then treats every coordinate-permutation subgroup generated by transpositions, including the full coefficient spaces for $\mathfrak S_4$, $\mathfrak S_3\times\mathfrak S_2$, $\mathfrak S_3$, $\mathfrak S_2^2$ and $\mathfrak S_2$. The argument specializes generic endomorphisms to the known symmetric group-algebra image and splits the existing node sets into independently smoothable subgroup orbits. This proves full generic-power HC for all these reflection families with one additional finite verifier.

Theorem14 treats a distinct two-parameter sextic family invariant under signed permutations of all five coordinates. Forty Fano reflection quotients give uniform rank470 curve support. Full pole-reduced Griffiths derivatives make its rank10 residual multiplicity cyclic even though the first-order image has rank only2. Third-order flat period graphs give scalar endomorphisms and zero cross Hom on all nine level-one primitive blocks. Complete16-,20- and40-node degenerations and classical low-dimensional Hodge groups yield the full product $\operatorname{Sp}_2^4\times\operatorname{Sp}_4^3\times\operatorname{Sp}_8\times\operatorname{Sp}_{10}^2$, and hence rational HC on every generic power and independently mixed tuple of these sextics.

2. Classical inputs and the new application

Rossi [3, Theorem 3.4 and Section 4] studies sextic threefolds invariant under coordinate actions of order three. Their quotients are Fano sextic double solids, and their invariant cohomology is generated by curves. His Theorem 4.4 proves the generalized Hodge conjecture at very general points of the corresponding directional and intersection families. For one coordinate order-three action, the supported rank is $104$.

The present two-parameter family lies in such a directional family but is a special subfamily. Its extra symmetry gives additional level-one cohomology, so Rossi's generic maximality statement does not settle the rank-$454$ summand here. We use his classical $104$-dimensional source and construct quotient sources covering the additional $350$ dimensions. The new input is the coverage by coordinate-sign and transposition pairs. The finite-group and quotient-support principles themselves are classical.

For the singular quotient models we use rational connectedness of log $\mathbb Q$-Fano varieties [4, Theorem 1 and its proof via Proposition 1]. Only uniruledness of a smooth resolution is needed. The quotient-support argument, including its divisor-supported trace error, is included in [5, Proposition B and Lemma 5] and recalled below. The cyclic-product transfer used in Corollary 2 is the Katsura--Shioda construction [6] in the form established and used in [1].

Pearlstein--Peters [7, Proposition 3 and Section 3] give a more recent, closely related decomposition for even-degree weighted Calabi--Yau threefolds with a cyclic-coordinate action: proper cyclic quotients are Fano, yielding GHC for the associated nonprimitive components. Their quotient mechanism is an input and precedent, rather than a new method claimed here. The quintic application below combines several reflection sources, isolates their common determinant component, and handles its seven-parameter variation. The half-twist construction discussed by van Geemen--Izadi [8, Section 4.3] for cyclic quintic surfaces is distinct and is not used as an algebraic realization theorem here.

3. Actual Fano quotients

Write $r_i$ for the sign change of $x_i$, for $0\le i\le4$, and $t_{ij}$ for the transposition of $x_i,x_j$, where $i,j<4$. All preserve $Y$. Use the following twenty-eight subgroups:

$$ H_{ij}=\langle r_i,r_j\rangle\quad(0\le i<j\le4), $$

and

$$ K_{k;ij}=\langle r_k,t_{ij}\rangle \quad(0\le i<j<4,\quad k\in\{0,\ldots,4\}\setminus\{i,j\}). $$

There are ten of the first kind and eighteen of the second.

Lemma 4. Each group above is an effective Klein four group, and its quotient is a normal, well-formed, quasismooth degree-six hypersurface in $\mathbb P(1,1,1,2,2)$, with ample anticanonical divisor and quotient singularities. A smooth projective resolution is uniruled.

Proof. For $H_{ij}$, square the two indicated coordinates. For $K_{k;ij}$, use

$$ a=x_i+x_j,\qquad b=x_i-x_j, $$

together with the other three coordinates. The two generators independently negate $b,x_k$. The invariant polynomial ring therefore has three generators of degree one and two of degree two. Since the sextic equation is invariant, exactness of invariants in characteristic zero identifies its quotient ring with this polynomial invariant ring modulo that equation. This is the actual finite quotient, not a proposed birational analogy.

A singular point of the weighted affine cone away from the origin would, by the chain rule at any preimage under the coordinate-squaring map, give a singular point of the original affine cone away from the origin. Thus the weighted hypersurface is quasismooth. The ambient weights are well-formed, and its singular stratum has dimension one, hence has intersection of codimension at least two in our threefold. The hypersurface is well-formed.

For completeness, the canonical class can be computed directly on the finite quotient $q:Y\to Q$. The two generators fix distinct hyperplane sections $D_1,D_2$ pointwise, with ramification index two. Their product has positive and negative linear eigenspaces of dimensions three and two. The latter give projective fixed spaces of dimensions two and one. A smooth sextic threefold cannot contain a plane: weak Lefschetz gives $\operatorname{Pic}(Y)=\mathbb Z h$, whereas a plane divisor would have intersection $1$ with $h^2$, not a multiple of $\int_Yh^3=6$. Consequently the product has no divisorial fixed locus on $Y$. The isolated fixed point of each generator contributes no divisor either. The ramification divisor is exactly $D_1+D_2$, and

$$ q^*K_Q=K_Y-D_1-D_2=h-2h=-h. \tag{2} $$

Finite pullback detects ampleness, so $-K_Q$ is ample. Equivalently, well-formed quasismooth weighted adjunction gives $\omega_Q=\mathcal O_Q(6-7)$ as a reflexive sheaf. The quotient of smooth $Y$ has quotient singularities and is klt. The theorem and proof in [4] give rational connectedness of a smooth resolution, hence uniruledness. $\square$

Let $\delta$ act by $x_4\mapsto\zeta_6^{-1}x_4$. There is one more source:

$$ Y/\langle\delta^2\rangle =\{u^2+f_{B,C}=0\}\subset\mathbb P(1,1,1,1,3),\qquad u=x_4^3. \tag{3} $$

It is a smooth Fano sextic double solid, since $f_{B,C}=0$ is smooth. Its invariant third cohomology is precisely the deck-order-two summand, of rank $104$. The deck-trivial part is zero, since the full cyclic quotient is $\mathbb P^3$. Formula (3) is the classical source of [3].

4. From quotient images to curves

We recall the support statement carefully because resolution errors matter.

Lemma 5. If a smooth projective threefold $T$ is uniruled, then $H^3(T)$ is generated by curves. If a finite effective group $G$ acts on a smooth projective threefold $Y$ and a smooth model of $Y/G$ is uniruled, then $H^3(Y)^G$ is generated by curves.

Proof. Uniruledness gives a generically finite rational map $S\times\mathbb P^1\dashrightarrow T$ from a smooth surface $S$. After resolving the map by blowups, its third cohomology is generated by $H^1$ of curves: this holds for $S\times\mathbb P^1$ by hard Lefschetz on $S$, restriction to an ample curve, and Lefschetz $(1,1)$; blowing up a curve adds its $H^1(-1)$, and blowing up a point adds nothing in degree three. Push-pull for the generically finite morphism gives the assertion.

For the quotient, resolve its graph as $Y\xleftarrow{p}\widetilde Y\xrightarrow{f}T$. For $\xi\in H^3(Y)^G$, put $\beta=f_*p^*\xi$. On a common free quotient open the trace is $|G|$, so

$$ |G|\xi-p_*f^*\beta\in N^1H^3(Y). \tag{4} $$

This coniveau space is generated by Gysin images of $H^1$ of resolutions of divisors, and those weight-one spaces are generated algebraically by curves. Thus both terms on the right of the resulting expression for $|G|\xi$ are curve-generated. Choose finitely many curve maps spanning the invariant space and apply the rational group projector. This retains all supported errors and does not assume a global inverse of quotient graph correspondences. For the supported-cohomology formulation used in (4), see [5, Lemma 5 and its references]. $\square$

5. The quotient images cover the exact rank-454 summand

The full projective coordinate-sign group is

$$ A=(\mathbb Z/2)^5/\langle(1,1,1,1,1)\rangle. $$

Its characters have an even number of minus signs: weights zero, two, or four. Let $P_\chi$ be the rational sign-character projector.

A weight-zero or weight-two character has at least two plus coordinates $i,j$. Its whole subspace is invariant under $H_{ij}$ and hence supported by Lemmas 4--5.

A weight-four character $\chi_k$ has a unique plus coordinate $k$. Then $r_k$ acts trivially on that subspace. Its stabilizer among the base permutations is

$$ S_k= \begin{cases} \mathfrak S_4,&k=4,\\ \mathfrak S_{\{0,1,2,3\}\setminus\{k\}},&k<4. \end{cases} $$

For each transposition $t_{ij}\in S_k$,

$$ \left(P_{\chi_k}H^3(Y)\right)^{K_{k;ij}} =\left(P_{\chi_k}H^3(Y)\right)^{t_{ij}=1}. \tag{5} $$

The character projection commutes with this stabilizer and with $r_k$, so the space in (5) is a projected image of a supported quotient invariant.

Lemma 6. For a rational representation $V$ of a symmetric group, the sum of the transposition-plus eigenspaces is the complement of its sign-isotypic part.

Proof. Average a positive definite rational inner product over the group. The orthogonal complement of the sum consists precisely of vectors on which every transposition acts by $-1$. Transpositions generate the group with the sign character, so that complement is its sign-isotypic part. Both spaces are defined over $\mathbb Q$. $\square$

Thus only the sign-isotypic parts in the five weight-four characters can remain unsupported by the twenty-eight quotients. For $k=4$ the $x_4$ sign is positive, so only deck characters of order three occur; the sign-isotypic summand is the rank-ten block $M_{3,\mathrm{CY}}$ of [1]. For $k<4$ the $x_4$ sign is negative, giving deck orders two or six. The order-two part is supported by (3). The remaining sign parts are the four rank-fourteen blocks $M_{6,\mathrm{CY}}^{(k)}$.

To make the projectors explicit, set

$$ \operatorname{Alt}_{S_k} =\frac1{|S_k|}\sum_{\sigma\in S_k}\operatorname{sgn}(\sigma)\Gamma_\sigma $$

and let $e_d$ be the rational cyclotomic deck projector for order $d$. Then on all cohomology the CY projector is

$$ \pi_{\mathrm{CY}} =e_3P_{\chi_4}\operatorname{Alt}_{S_4} +e_6\sum_{k=0}^3P_{\chi_k}\operatorname{Alt}_{S_k}. \tag{6} $$

All summands are rational algebraic correspondences. With

$$ \pi_3=\Delta_Y-\frac16\sum_{j=0}^3h^{3-j}\times h^j,\qquad \pi_L=\pi_3-\pi_{\mathrm{CY}}, \tag{7} $$

we recover the precise $L$ of [1]. Applying $\pi_L$ to the curve sources supplies the surjection in Theorem 1.

The representation dimensions can be verified at the Fermat member. The Jacobian basis is $x^e$ with $0\le e_i\le4$ and total degrees $1,7,13,19$, corresponding to $(3,0),(2,1),(1,2),(0,3)$. The residue action includes the determinant, so its sign character is $((-1)^{e_i+1})_i$. The five weight-four blocks each have Hodge profile $(1,26,26,1)$; the ten weight-two blocks each have $(0,12,12,0)$; the weight-zero block has $(0,5,5,0)$. After the stabilizer-sign projections and order separation, the result is:

| Deck order | Total rank | CY rank | Supported rank | |---|---:|---:|---:| | 2 | 104 | 0 | 104 | | 3 | 208 | 10 | 198 | | 6 | 208 | 56 | 152 | | Total | 520 | 66 | 454 |

The finite-group representation is locally constant on the connected smooth parameter locus $U$. The image ranks and projector identities therefore hold throughout $U$, including special members. This also follows directly from the character argument and the global group projectors above.

6. Hodge classes and the published tower

For two smooth members, choose algebraic surjections $A=\bigoplus H^1(C_i)(-1)\twoheadrightarrow L$ and $A'=\bigoplus H^1(C'_j)(-1)\twoheadrightarrow L'$. Polarizable Hodge semisimplicity provides a Hodge lift of any rational Hodge class in $L\otimes L'$ to $A\otimes A'$. After removing the twists, each lifted component is a rational $(1,1)$ class on $C_i\times C'_j$. Lefschetz $(1,1)$ represents it by a divisor. Composing with the two algebraic curve correspondences gives the required codimension-three cycle. No algebraicity of an arbitrary abstract Hodge section is used.

At very general parameters, [1, Theorem 3] proves that the rank-ten and rank-fourteen CY blocks are simple rational Hodge structures of level three, with the appropriate endomorphisms represented by automorphism graphs. Hence they have no nonzero level-one rational sub-Hodge structures. It follows that $L$ is the entire maximal level-one subspace, and Theorem 1 establishes the degree-three generalized Hodge conjecture.

All nonmiddle cohomology of $Y$ is Tate. On $Y^2$ only $H^3(Y)^{\otimes2}$ requires attention beyond hyperplane products. The mixed $M_{\mathrm{CY}}$ and $L$ terms have no rational Hodge classes at very general parameters by the simplicity and levels just stated. To make explicit why the CY conclusion of [1] covers every self-tensor Hodge class, write $M_{\mathrm{CY}}=M_3\oplus M_6^{\oplus4}$. The simple summands have unequal rational ranks $10$ and $14$, and their generic endomorphism algebras are $K=\mathbb Q(\zeta_6)$. Hence cross maps between $M_3$ and $M_6$ vanish and

$$ \operatorname{End}_{\mathrm{Hdg}}(M_{\mathrm{CY}}) =K\oplus M_4(K). $$

The coordinate permutations identifying the four $M_6$ copies commute with the deck action, as do all scalar $K$-endomorphisms. Thus every Hodge endomorphism commutes with the deck operator. Under the polarization identification, every CY self-tensor Hodge class is therefore already diagonal-deck-invariant. The $34$ algebraic group-ring matrix units of [1, Theorem 3(b)] span all of them. The $L$ self-tensor classes are algebraic by Theorem 1. This proves HC for $Y^2$.

The finite morphism $Y\to W$ given by $z=x_4^2$ has degree two. Both varieties are smooth projective, so pullback followed by pushforward is multiplication by the degree. HC for $Y^2$ therefore implies HC for $W^2$.

Finally the Katsura--Shioda blowup construction [1,6] relates $X$ to the diagonal $\mu_6$ quotient of a blowup of $Y\times Y$ along $V\times V$. The new middle classes in this blowup are shifts of classes from $V\times V$. Its HC is already proved at very general parameters in [1]. The finite quotient and birational morphism to $X$, with the stated blowup centers, now transfer algebraic representatives for every Hodge class of $X$. Equivalently, use the explicit decomposition in [1]: the order-one sector is furnished by the surface square; the remaining sector is the diagonal-deck-invariant part of $H^3(Y)^{\otimes2}$. Both are algebraic by the results just cited and Theorem 1. This proves Corollary 2.

The distinction between all-member support and generic maximality is essential. For example, the Fermat point can acquire extra level-one substructures in the fixed CY summand. Theorem 1 makes no uniform assertion about such additional pieces.

7. Single reflections on quintics

Let a finite linear group $G$ preserve a homogeneous quintic equation $F$ in five variables. Suppose that $G$ is generated by reflections whose linear eigenvalues are $(1,1,1,1,-1)$. Write $\epsilon=\det:G\to\{1,-1\}$ and

$$ e_\epsilon=\frac1{|G|}\sum_{g\in G}\epsilon(g)\Gamma_g. \tag{8} $$

Proposition 7. If $Z=\{F=0\}\subset\mathbb P^4$ is smooth, then $(1-e_\epsilon)H^3(Z,\mathbb Q)$ is curve-generated.

Proof. For a generating reflection choose linear coordinates $(z_0,z_1,z_2,z_3,b)$ in which it negates $b$. The invariant polynomial ring is $\mathbb C[z_0,z_1,z_2,z_3,w]$, where $w=b^2$. Exactness of invariants identifies the quotient of $Z$ with a weighted quintic $Q\subset\mathbb P(1,1,1,1,2)$.

As in Lemma 4, the chain rule proves quasismoothness of this actual quotient from smoothness of $Z$. The ambient singular locus is its single weight-two point, so $Q$ is well-formed. Weighted adjunction gives $K_Q=\mathcal O_Q(5-6)$. Equivalently, the only divisorial fixed locus is the hyperplane section $D=\{b=0\}\cap Z$; the other fixed locus is one point. Thus $q^*K_Q=K_Z-D=-H$. The finite quotient is klt and its anticanonical divisor is ample. A smooth resolution is uniruled by [4], and Lemma 5 makes $H^3(Z)^{r=1}$ curve-generated.

Average a positive definite rational inner product over $G$. The perpendicular to the sum of the generating reflection-plus eigenspaces is the subspace on which every generator acts by $-1$. Since the generators have determinant $-1$, this is exactly $e_\epsilon H^3(Z)$. Hence their sum is $(1-e_\epsilon)H^3(Z)$. This is a rational group-projector statement independent of the auxiliary inner product. Applying the complementary algebraic projector to the curve sources proves the proposition. $\square$

For the application, let $\mathfrak S_3$ permute $x_0,x_1,x_2$, and let $D_5$ denote the group of order ten generated by

$$ r_a:(u,v)\longmapsto(\zeta_5^a v,\zeta_5^{-a}u), \qquad 0\le a<5. \tag{9} $$

Together with the three $x$-transpositions these give eight reflections in $G=\mathfrak S_3\times D_5$. Its polynomial invariant ring has generators $e_1(x),e_2(x),e_3(x),uv,u^5+v^5$, of degrees $1,2,3,2,5$. Consequently every invariant quintic has the form

$$ c(u^5+v^5)+f_5(x)+uvf_3(x)+(uv)^2f_1(x), \tag{10} $$

where each $f_i$ is symmetric of degree $i$. This coefficient space has dimension ten, and its connected smooth locus contains the Fermat member.

At Fermat, the residue ring is

$$ R=\mathbb Q[x_0,x_1,x_2,u,v]/(x_0^4,x_1^4,x_2^4,u^4,v^4), $$

with Hodge grades $R_0,R_5,R_{10},R_{15}$. The action on residues includes the determinant of the ambient linear action. Being negative for every $x$-transposition means that the polynomial is symmetric in $x$. Being negative for all the reflections (9) means invariance under their rotations, hence equal exponents of $u,v$; the ordinary swap then acts negatively on the residue. Counting the symmetric monomial orbits gives

$$ \dim(e_\epsilon R_{5q})=1,7,7,1\quad(q=0,1,2,3). \tag{11} $$

Finite-group representations on the Hodge bundles are locally constant over the connected smooth family. Thus the same profiles hold throughout (10). Define $C=e_\epsilon H^3(Z)$ and $L=(1-e_\epsilon)H^3(Z)$. Proposition 7 proves uniform support for $L$, with rank $204-16=188$. On the separated family of [2], these are its $M_{1,\mathrm{III}}$ and complement. That paper already realizes sixty dimensions by curves; the reflection coverage adds the other $128$.

For completeness the algebraic middle projector of a smooth quintic is

$$ \pi_3^Z=\Delta_Z-\frac15\sum_{j=0}^3h^{3-j}\times h^j. $$

Its determinant projector is $\pi_C=\pi_3^Z\circ e_\epsilon$, and $\pi_L=\pi_3^Z-\pi_C$. These expressions will provide the polarization class needed for the square theorem.

8. The cyclic holomorphic period in the enlarged family

We prove the remaining assertions of Theorem 3 on the smooth locus $T\subset\mathbb A^7$ of (1), which contains $t=0$. The seven deformation polynomials, in the following order, are

$$ \begin{aligned} Q_0&=(uv)^2m_{100},& Q_1&=uvm_{300},& Q_2&=uvm_{210},\\ Q_3&=m_{320},&Q_4&=uvm_{111},&Q_5&=m_{311},&Q_6&=m_{221}. \end{aligned} \tag{12} $$

Their classes form a basis of $R_5$ in the determinant-residue part. Their orbit representatives, in the coordinate order $(x_0,x_1,x_2,u,v)$, are

$$ \begin{gathered} (0,0,1,2,2),\ (0,0,3,1,1),\ (0,1,2,1,1),\ (0,2,3,0,0),\\ (1,1,1,1,1),\ (1,1,3,0,0),\ (1,2,2,0,0). \end{gathered} $$

All orbit sums use distinct monomials with coefficient one.

In the degree-ten symmetric orbit basis ordered lexicographically by its exponent representatives, the seven products

$$ Q_0Q_1,\ Q_0Q_2,\ Q_0Q_3,\ Q_0Q_4,\ Q_0Q_5,\ Q_0Q_6,\ Q_1Q_3 $$

have coefficient matrix

$$ A=\begin{pmatrix} 1&1&0&0&0&0&0\\ 0&2&0&0&0&0&0\\ 0&0&2&0&0&0&0\\ 0&2&0&1&0&0&0\\ 0&0&1&0&1&1&0\\ 0&0&0&0&0&3&0\\ 0&0&0&0&0&0&2 \end{pmatrix},\qquad \det A=24. \tag{13} $$

The final product is

$$ Q_0Q_1Q_3=6x_0^3x_1^3x_2^3u^3v^3\quad\text{in }R_{15}. \tag{14} $$

Equations (12)--(14) are direct calculations modulo the five fourth powers; the supplement gives a short exact reconstruction. They show that products of the actual parameter polynomials span every grade in (11). In particular, these are allowed family deformations, not unrealized infinitesimal directions in a larger Jacobian ring.

Set $\omega_t=\operatorname{Res}(\Omega/F_t)$, a local generator of the holomorphic line in $C_t$. Griffiths' derivative formula [9] gives the order-$r$ leading symbol

$$ \operatorname{gr}^{3-r}_F \nabla_{i_1}\cdots\nabla_{i_r}\omega\big|_0 =(-1)^r r! [Q_{i_1}\cdots Q_{i_r}],\qquad 0\le r\le3. \tag{15} $$

The nonzero determinants $1,24,6$ prove that $\omega$ and its first three derivatives span all sixteen dimensions of $C_0$. Lower-filtration terms do not affect this conclusion: successive leading symbols span the associated graded spaces of dimensions $1,7,7,1$.

Lemma 8. Suppose a polarized rational variation $\mathbb V$, admitting an integral lattice, over a connected smooth algebraic base has a one-dimensional top Hodge piece, and a local generator of that line is cyclic for the flat connection at one point. Then its very-general rational Hodge endomorphisms are scalars.

Proof. Apply the Hodge-locus theorem [10] to the weight-zero tensor variation $\operatorname{End}(\mathbb V)$. There are countably many algebraic component images. Exclude the proper ones. A rational Hodge endomorphism at a remaining point extends by flat transport to a Hodge morphism on the universal cover: locally its Hodge condition is an analytic equation; a component dominating the base makes that equation hold on an open set, and the identity theorem continues it. This also explains why the exceptional set is formulated using component images, rather than a single algebraic set for each marked rational map.

On a simply connected neighborhood of the cyclic point, the resulting flat endomorphism $f$ satisfies $f\omega(t)=\lambda(t)\omega(t)$. The scalar is holomorphic and belongs to the finite spectrum of the fixed flat matrix $f$, so it is constant. Therefore $f$ acts by this same scalar on every iterated derivative of $\omega$. Cyclicity implies $f=\lambda\mathrm{Id}$. Its rationality implies $\lambda\in\mathbb Q$. $\square$

Applied to (15), Lemma 8 gives $\operatorname{End}_{\mathbb Q\text{-}\mathrm{Hdg}}(C_t)=\mathbb Q$ very generally. A polarized sub-Hodge structure has a rational Hodge orthogonal projector, so this equality makes $C_t$ simple. Since it contains $H^{3,0}$, no nonzero Hodge morphism can join it to a level-one structure. Thus $L_t$ is the maximal level-one part, and its uniform curve support proves degree-three GHC for very general $t$.

The argument of Section 6 proves algebraicity of all Hodge classes in $L_t\otimes L_{t'}$ for arbitrary smooth parameters, even in the full coefficient family (10). At very general $t$, the mixed $C_t$--$L_t$ Hodge tensors vanish and the Hodge tensors of $C_t\otimes C_t$ form a one-dimensional space generated by its polarization projector. This is the algebraic correspondence $\pi_C$ written above. Hence all $H^6(Z_t^2)$ Hodge classes are algebraic. Weak Lefschetz gives only hyperplane classes outside middle cohomology on each factor, so all other even cohomology of the square is already algebraic. This proves the full square assertion.

For a very general independently varying pair, apply the same countability and Hodge-locus argument to $\operatorname{Hom}(\operatorname{pr}_1^*C,\operatorname{pr}_2^*C)$ on $T^2$. A nonzero generic morphism would be an isomorphism by simplicity. Its flat extension to a product neighborhood, with the second parameter fixed, would carry the first holomorphic line to a fixed line. This contradicts its rank-seven first derivative at $0$. Therefore the $C_t\otimes C_{t'}$ Hodge tensors vanish at very general independent pairs. The mixed terms again vanish and the $L$ terms are algebraic. This proves the remaining product assertion of Theorem 3.

8.1. A nodal orbit and full residual monodromy

The cyclic-period calculation also combines with a concrete degeneration to control every tensor power of the selected component $C$.

Theorem 9. The connected algebraic monodromy of $C$ in (1) is $\operatorname{Sp}(C)\simeq\operatorname{Sp}_{16}$. For very general $t$, every rational Hodge class in $C_t^{\otimes 2r}(3r)$, for every $r\ge1$, is algebraic on $Z_t^{2r}$. The corresponding assertion holds for all selected $C$-tensors of a very general independently varying finite tuple, with contractions within each parameter's tensor factors.

Proof. A fixed scaling of $u,v$ identifies the family (1) with the one having coefficient $6$ on $u^5+v^5$. In that normalization consider

$$ F_*=x_0^5+x_1^5+x_2^5+6(u^5+v^5)-5(uv)^2(x_0+x_1+x_2). \tag{16} $$

Its complete projective singular locus consists of the five points

$$ p_k=[\zeta^{3k},\zeta^{3k},\zeta^{3k},\zeta^k,1], \qquad k=0,\ldots,4,\quad\zeta^5=1. \tag{17} $$

To prove completeness, a singular point with $u=0$ or $v=0$ would have all $x_i=0$ by the three $x$ derivatives, and the remaining pure fifth-power derivative excludes a projective point. Set $v=1$. The identity $uF_u-vF_v=30(u^5-v^5)$ gives $u^5=1$. Then $x_i^4=u^2$, so $x_i=u^3\epsilon_i$ with $\epsilon_i^4=1$, and the $u$ derivative gives $\sum\epsilon_i=3$. Equality in the triangle inequality forces all $\epsilon_i=1$, proving (17).

At $p_0$, the Hessian in coordinates $(x_0,x_1,x_2,u)$ is

$$ \begin{pmatrix} 20&0&0&-10\\ 0&20&0&-10\\ 0&0&20&-10\\ -10&-10&-10&90 \end{pmatrix},\qquad \det=600000. \tag{18} $$

The determinant is the same at all five points after reduction by $u^5=1$. All are ordinary double points. The allowed deformation $Q=(uv)^2\sum x_i$ takes value $3$ at each. Thus $F_*+sQ$ is a transverse simultaneous smoothing. For sufficiently small $s\ne0$ there are no other singularities, by compactness and (17).

The rotations $(u,v)\mapsto(\zeta^j u,\zeta^{-j}v)$ send $k$ to $k+2j$; the nodes form one orbit. The stabilizer of $p_0$ is $\mathfrak S_3$ times the untwisted swap of $u,v$. Its action on the node's local vanishing three-sphere is the determinant on the four-dimensional affine tangent space, hence the character $\varepsilon=\det$ of $G$. Indeed, the $x$ permutations have their usual determinants; the swap has derivative $(\dot x_i,\dot u)\mapsto(\dot x_i-\dot u,-\dot u)$ and determinant $-1$. Equivariant holomorphic Morse coordinates identify this with the usual determinant action on the vanishing sphere of a nondegenerate quadratic form.

Orient vanishing cycles $\delta_k$ by transporting $\delta_0$ with the rotations. The local module is $\varepsilon$ times the permutation module of the five nodes, and its $\varepsilon$ part is a line. Its image in global cohomology is generated by

$$ q=e_\varepsilon\delta_0=\frac15\sum_{k=0}^4\delta_k. \tag{19} $$

This global image is nonzero. The holomorphic residue $\omega_s$ has character $\varepsilon$. In local holomorphic Morse coordinates the smoothing is $\sum_{j=1}^4 y_j^2=a(s)$ with $a'(0)\ne0$, and the residue numerator has nonzero constant term. Scaling $y=\sqrt{a(s)}z$ in the standard quadratic residue integral gives

$$ \int_{\delta_0}\omega_s=c\,a(s)+O(a(s)^2),\qquad c\ne0. $$

The period is nonzero for small $s\ne0$. Since $\omega_s$ is $\varepsilon$-isotypic, its period on $q$ is the same. Thus $q\ne0$; this argument does not presume that every local vanishing cycle always survives in global cohomology.

The Milnor neighborhoods are disjoint. The Picard--Lefschetz formula therefore restricts to

$$ T(x)=x\mathbin{\pm}5\langle x,q\rangle q\quad(x\in C). \tag{20} $$

The polarization restricted to $C$ is nondegenerate, since $e_\varepsilon$ is a self-adjoint polarization projector. Hence $T-I$ has rank one and square zero. The Zariski closure of its integer powers contains the associated additive one-parameter subgroup, which lies in connected monodromy.

Let $M$ be the generic Mumford--Tate group of $C$, and $H$ its connected algebraic monodromy. Lemma 8 makes the commutant of $M$ equal to $\mathbb Q$. The commutation equations are rational linear equations, so extension of scalars makes the complex commutant equal to $\mathbb C$. Reductivity of $M$ and complete reducibility then imply that $C_{\mathbb C}$ is irreducible for $M_{\mathbb C}$. Andr'e's normality theorem [11, Theorem 1] gives $H\triangleleft M^{\mathrm{der}}$ and thus $H\triangleleft M$. The restriction to this connected normal reductive subgroup is isotypic: its isotypic idempotents are permuted by the connected group $M_{\mathbb C}$, hence are fixed; see also [11, Lemma 1]. Write it as $U\otimes\mathbb C^m$, with $H$ acting trivially on the multiplicity factor. The rank of $h-I$ is divisible by $m$ for every $h\in H$. Formula (20) forces $m=1$. Thus $H$ acts irreducibly. This step uses both normality and the transvection; scalar Hodge endomorphisms alone would not suffice.

We give the needed symplectic group argument. The Lie algebra of a connected irreducible subgroup $H\subset\operatorname{Sp}(V)$ containing a transvection contains $N_w:x\mapsto\langle x,w\rangle w$ for every $w$ in the orbit $O$ of one nonzero transvection vector. The orbit is irreducible and spans $V$. Put

$$ S(a,b)x=\langle x,a\rangle b+\langle x,b\rangle a. $$

For $a,b\in O$ with $\langle a,b\rangle\ne0$, the bracket $[N_a,N_b]$ is a nonzero multiple of $S(a,b)$. For an orthogonal pair, choose $w\in O$ pairing nontrivially with both; the two nonempty Zariski opens intersect because $O$ is irreducible. The bracket $[S(a,w),N_b]$ is then a nonzero multiple of $S(a,b)$. These operators span $\operatorname{Sym}^2V\simeq\mathfrak{sp}(V)$, since $O$ spans $V$. Thus $H=\operatorname{Sp}(V)$. Applied to (20), this proves $H=\operatorname{Sp}_{16}$. The generic Hodge group is contained in this symplectic group and contains $H$, so it too is the full group.

The symplectic invariant theorem [12, Theorem 4.1 and Corollary 4.2] says that all even invariant tensors are spanned by pair contractions; there are no odd invariant tensors. The inverse polarization tensor of $C$ is its diagonal component, given by the algebraic projector $\pi_C$. Products and pullbacks of that class give every pairing tensor on $Z_t^{2r}$, proving the algebraicity assertion.

For independently varying finite tuples the parameter fundamental group is a product and each factor acts only on its own variation. The connected monodromy is consequently the product of the respective symplectic groups. Its invariants contract only within the tensor factors belonging to a given parameter. There are countably many orders and rational tensors; [10] supplies the countable union of proper algebraic exceptional loci, simultaneously for all orders. $\square$

Theorem 9 alone concerns the algebraically projected $C$ sectors. Curve support and the Lefschetz $(1,1)$ theorem on products of two curves do not by themselves control higher tensors of $L$. The next section supplies the additional monodromy argument.

8.2. The level-one components and full powers

We now control the remaining tensors of $L$, using the actual seven Higgs multiplication maps and two further nodal orbits. Let $K=\mathbb Q(\sqrt5)$, the character field of the two-dimensional representations of $D_5$.

Theorem 10. For the seven-parameter family (1), the rational Hodge conjecture holds on every power of a very general $Z_t$, and on all mixed powers of a very general independently varying finite tuple. These assertions hold simultaneously for all tensor orders. They are not asserted at arbitrary special parameters.

We give the finite certificates and the geometric argument, including the passage from the finite-field calculation to characteristic zero.

Multiplicity spaces. Use polynomial types $s,a,b$ for the symmetric, alternating and standard representations of $\mathfrak S_3$. These are polynomial labels: residue cohomology includes the determinant twist. The rotation weight is $k=a_u-a_v\pmod5$. We identify weights $k,-k$ by the $D_5$ reflection. A primitive standard projector is obtained by taking the $(01)$-even polynomial line and removing the symmetric part. The multiplicity spaces, suppressing their finite-group factors, are as follows:

| Label | Polynomial type, weight | $h^{2,1}$ | Total rank over $\mathbb C$ | |---|---|---:|---:| | $A$ | $a,0$ | 2 | 4 | | $B$ | $b,0$ | 8 | 16 | | $D_1,D_2$ | $s,1$ and $s,2$ | 6,6 | 12,12 | | $E_1,E_2$ | $a,1$ and $a,2$ | 1,1 | 2,2 | | $J_1,J_2$ | $b,1$ and $b,2$ | 6,6 | 12,12 |

The remaining $s,0$ component is $C$. The $k=1,2$ pairs are the two real embeddings of $K$-linear multiplicity spaces. The dihedral representations are orthogonal over $K$, as is the standard $\mathfrak S_3$ representation over $\mathbb Q$; consequently their multiplicity polarizations are alternating. All displayed components of $L$ have only Hodge types $(2,1),(1,2)$. The ranks follow from the Fermat residue action as in Section 7 and remain constant. Counting finite-group multiplicities gives $\dim L=188$.

Generic Hodge maps from Higgs matrices. At the coefficient vector $(2,3,5,7,11,13,17)$ in the order $Q_0,\ldots,Q_6$, the supplement constructs the actual Jacobian quotients in degrees $5,10$ modulo $1009$. It applies the polynomial projectors

$$ \sum_{g\in\mathfrak S_3}g,\qquad \sum_{g\in\mathfrak S_3}\operatorname{sgn}(g)g,\qquad 3(1+(01))-\sum_{g\in\mathfrak S_3}g, \tag{21} $$

respectively. Each is six times the desired projector. The weight condition and these rational matrices select the primitive polynomial spaces. Projecting every monomial multiple of the five actual partial derivatives gives all Jacobian relations in each space. Row reduction chooses quotient coordinates. The resulting multiplication maps are

$$ M_i^V:R_5(V)\longrightarrow R_{10}(V),\qquad P\longmapsto Q_iP, \quad 0\le i\le6. $$

All eight quotient dimensions in each degree are, in order $(A,B,D_1,E_1,J_1,D_2,E_2,J_2)$,

$$ (2,8,6,1,6,6,1,6). $$

For every ordered pair $V,W$, solve the linear equations

$$ Y M_i^V=M_i^W X\quad(0\le i\le6). \tag{22} $$

The computed kernel dimensions are zero off the diagonal and

$$ (2,1,1,1,1,1,1,1) \tag{23} $$

on the diagonal. The supplement records the quotient pivots, all seven matrices for every space, and the pivots of all sixty-four systems.

Here is why this finite calculation bounds the characteristic-zero problem. The smooth characteristic-zero coefficient locus is nonempty and irreducible. Its Jacobian dimensions are the stated constant ones. Every polynomial projector and Macaulay matrix is defined over $\mathbb Z[1/30]$. The nonzero pivots modulo $1009$ supply nonzero minors over $\mathbb Q$ at the same coefficient vector. Thus the quotient bases and matrix formulas are valid on a nonempty rational open locus; the generic relation ranks cannot exceed those on the smooth locus, where the dimensions are known. This open meets the smooth locus. Clear the nonzero pivot denominators in (22). Its finite-prime ranks give lower bounds for the generic rational ranks, so (23) gives upper bounds for the generic kernel dimensions over $\mathbb C$. No claim that a random finite-field sample determines an analytic period domain is used here.

A generic rational Hodge map extends flatly over the universal cover, by the argument of Lemma 8. Differentiating its Hodge-filtration compatibility gives (22) on the associated graded spaces. This passage is injective on Hodge maps: they preserve both Hodge types, so zero maps on both graded pieces mean the original map is zero.

We also spell out the scalar extension used here. Let $M$ be the generic Mumford--Tate group of the full rational variation. Its rational commutant is the algebra of generic rational Hodge endomorphisms, and its complex commutant is obtained by scalar extension, since the commutation equations are rational linear equations. Every primitive complex finite-group projector commutes with $M_{\mathbb C}$. A map between two primitive multiplicity spaces that commutes with $M_{\mathbb C}$ therefore extends, by inclusion and projection, to an element of this full complex commutant. It is a complex linear combination of the flat rational Hodge maps just considered. It too satisfies (22) and is detected by its associated graded map. This justifies the bounds separately for the two embeddings of $K$, even though the individual embedding spaces are not rational subvariations. The scalar maps supply the lower bound one on every diagonal; for $A$ the upper bound is still two. All cross commutants vanish. These statements concern generic Hodge maps; an arbitrary solution of (22) need not be flat.

Two complete nodal orbits. Both of the following members belong to (1) after a fixed scaling of $u,v$:

$$ \begin{aligned} F_{15}={}&\sum x_i^5-\frac{46}{3}(u^5+v^5) -\frac{25}{3}uv\sum x_i^3+20(uv)^2\sum x_i,\\ F_{30}={}&\sum x_i^5-648(u^5+v^5)-25uv\sum x_i^3 -50uvx_0x_1x_2+370(uv)^2\sum x_i. \end{aligned} \tag{24} $$

Their complete singular loci consist respectively of the permutations of $(1,1,2)$ and $(1,2,3)$ in the $x$ coordinates at $u=v=1$, followed by the five rotations used in (17). There are fifteen and thirty nodes, each set a single $G$ orbit. The Hessian determinants at the seeds are $-10350000$ and $1312200000$. The deformation $Q_0$ has values $4$ and $6$ respectively, so it smooths all the nodes transversely.

For completeness, both singularity classifications reduce first to $uv\ne0$ and $u^5=v^5$ by the same derivative identity as for (16). After the rotation and projective normalization take $u=v=1$. For $F_{15}$ the remaining derivatives give

$$ x_i\in\{-2,-1,1,2\},\qquad \sum_i(24x_i-5x_i^3)=46, $$

whose only solutions are the permutations of $(1,1,2)$. For $F_{30}$ a repeated-coordinate solution $(r,r,t)$ would satisfy

$$ \begin{aligned} r^4-15r^2-10rt+74&=0,\\ t^4-15t^2-10r^2+74&=0,\\ -648-5(2r^3+t^3)-10r^2t+148(2r+t)&=0. \end{aligned} $$

Their rational ideal is the unit ideal, as the small exact elimination in the supplement verifies. If the three coordinates are distinct, put $h(T)=T^3-e_1T^2+e_2T-e_3$. Their derivative equations imply

$$ h(T)\mid 5T^5-75T^3+370T-50e_3. $$

The three remainder coefficients and the $u$ derivative have ideal $(e_1-6,e_2-11,e_3-6)$, again by explicit rational elimination. Thus the coordinates are the three roots $1,2,3$. This also treats possible zero coordinates and completes the classification.

We must check that the local vanishing cycles survive independently in global cohomology. Evaluate all degree-five monomials on these nodes over $\mathbb Z[\zeta]$. Reduction modulo $101$ with $\zeta\mapsto95$ gives ranks $15,30$; the listed square witness minors have determinants $17,84$ modulo $101$. Hence their ranks are full in characteristic zero. On a nearby smooth fiber, the periods of $\operatorname{Res}(P\Omega/F^2)$ on the local vanishing cycles have leading terms $c_jP(p_j)$, with every $c_j\ne0$. This follows by differentiating the quadratic holomorphic vanishing period used above, allowing all degree-five directions in the full hypersurface space. The evaluation minors remain nonzero for small smoothing parameter. Thus all fifteen, respectively thirty, local cycles are independent in global $H^3$.

The node stabilizers are $\mathfrak S_2\times C_2$ for $F_{15}$ and $C_2$ for $F_{30}$, with $C_2$ the $D_5$ reflection. They fix the chosen projective representative itself, so their local vanishing character is the determinant. The local representation of the first orbit is consequently

$$ (\operatorname{sgn}_{\mathfrak S_3}\oplus\operatorname{std}_{\mathfrak S_3}) \otimes(\operatorname{sgn}_{D_5}\oplus\rho_1\oplus\rho_2). \tag{25} $$

Every multiplicity in (25) is one. Thus it supplies a rank-one Picard--Lefschetz logarithm on $B,D_1,D_2,J_1,J_2$, as well as $C$. For $F_{30}$, the $\mathfrak S_3$ factor is its regular representation; its polynomial alternating component has multiplicity one. It supplies rank-one logarithms on $A,E_1,E_2$.

These rank assertions use the proved global independence. If $D$ is the injective map from the space of local cycles to global cohomology, the Picard--Lefschetz logarithm is, up to sign, $DD^{\mathsf T}J$, where $J$ is the global nondegenerate alternating pairing. Its rank is the number of nodes because $D^{\mathsf T}J$ is surjective. Its image as a $G$ representation is exactly the local-cycle representation. Therefore its rank on each primitive multiplicity space is the multiplicity in (25), or in the corresponding regular representation. This avoids inferring a nonzero projection merely from a local character match.

The remaining endomorphism bound for $A$. This rank-four rational variation has generic endomorphism dimension at most two by (23). At Fermat, simultaneous rotation $(u,v)\mapsto(\zeta u,\zeta v)$ commutes with $G$ and acts on its four embedding lines with eigenvalues $\zeta^2,\zeta^4$ in $H^{2,1}$ and $\zeta,\zeta^3$ in $H^{1,2}$. Thus it is rank one over $E=\mathbb Q(\zeta_5)$, with shifted CM type $\{2,4\}$. Its Hodge endomorphism algebra is exactly $E$: decompose $\operatorname{End}_{\mathbb Q}(E)=\bigoplus_{g\in\operatorname{Gal}(E/\mathbb Q)}Eg$; only the identity $g$ preserves this type. In particular it is simple.

Generic rational Hodge endomorphisms specialize injectively to Fermat by flat continuation. Their algebra is thus a subalgebra of $E$, and its dimension is at most two. It is either $\mathbb Q$ or the only quadratic subfield $K$. The rational rank-one transvection on $A$ just constructed commutes with this algebra: connected monodromy is contained in the generic Mumford--Tate group. A $K$-linear map has even rational rank, so the second possibility is excluded. Therefore $\operatorname{End}_{\mathrm{Hdg}}(A)=\mathbb Q$ very generally.

Full monodromy and tensors. Each primitive complex multiplicity space now has generic scalar Hodge commutant and a rank-one transvection. Normality, isotypy, and the Lie-algebra argument in Theorem 9 apply to each. The respective connected groups are the full symplectic groups. The joint group has complex form

$$ \operatorname{Sp}_{16}^{\,2}\times\operatorname{Sp}_4 \times\operatorname{Sp}_{12}^{\,4}\times\operatorname{Sp}_2^{\,2}. \tag{26} $$

Here one $\operatorname{Sp}_{16}$ acts on $C$ and the other on $B$. We justify the product assertion. A connected subdirect subgroup of a product of simple symplectic groups can fail to be the full product only through a diagonal identification of isomorphic factors. The connected normalizer of such a graph in the product of the two symplectic groups is that graph itself, up to the finite centers. Connected monodromy is normal in the generic Hodge group, and the latter lies in the product of the multiplicity symplectic groups because the finite-group projectors and polarizations are Hodge tensors. A diagonal identification would therefore force a complex Hodge intertwiner between the two standard representations. Automorphisms of the complex symplectic group are inner, so this is an ordinary linear intertwiner. For two $L$ components it is excluded by the off-diagonal bounds (22). For $B$ and $C$ it is excluded by their different Hodge profiles. This proves (26).

Finally, invariant theory for the product in (26) expresses every invariant tensor of $H^3(Z_t)$ as a sum of products of invariant pairings, allowing arbitrary vectors in its constant finite-group multiplicity factors. All rational two-factor Hodge classes are already algebraic by Theorem 3: $L\otimes L$ by curve support and Lefschetz, $C\otimes C$ by its projector, and the mixed terms vanish. Taking rational spans commutes with extension to $\mathbb C$, so products and pullbacks of these rational algebraic pair classes span every rational higher Hodge tensor. Nonmiddle cohomology is Tate by weak Lefschetz. This proves full rational HC on every power. For independent parameter tuples the monodromy is the corresponding external product; the same proof works, with contractions within each parameter's factors. The countable Hodge-locus argument supplies one very-general set for all orders. This proves Theorem 10. $\square$

8.3. The full invariant coefficient space

Corollary 11. Let $\mathcal U_{10}$ be the smooth locus of the full coefficient space (10). For very general members of this space, the degree-three generalized Hodge conjecture and full rational HC on all powers hold. The same holds for all mixed powers of independent finite tuples, simultaneously in every order.

Proof. The coefficient space has dimension $1+5+3+1=10$: its parameters are $c$ and the symmetric polynomials of degrees $5,3,1$ in (10). Its smooth locus is a nonempty connected open subset of affine space. The seven-parameter family (1) is the subfamily obtained by setting $c=1$, the coefficient of $\sum x_i^5$ to $1$, and that of $m_{410}$ to zero. Choose the Fermat point as common basepoint. Every loop in $T$ is a loop in $\mathcal U_{10}$, so the larger family's connected monodromy contains the product (26). The globally defined finite-group action, projectors and polarizations place it inside that same product. Thus equality holds. In particular the residual $C$ remains generically simple and level three. Proposition 7 supplies uniform curve support for its complement on the full space. The pair-class, invariant-tensor and countable Hodge-locus arguments of Sections 8.1--8.2 now apply without change, also on the external products of coefficient spaces. $\square$

9. Fully symmetric quintics

Let $\mathcal S$ be the smooth locus in the seven-dimensional vector space of symmetric homogeneous quintics in five variables. Write $p_j=\sum_{i=0}^4x_i^j$, and let $m_\lambda$ denote a sum of distinct monomials in its symmetric orbit. The Fermat polynomial belongs to $\mathcal S$, which is a connected nonempty Zariski-open subset of an affine space.

Theorem 12. For every member $X$ of $\mathcal S$, the rank-$192$ complement $L$ of the sign-isotypic summand $C\subset H^3(X,\mathbb Q)$ is generated by algebraic correspondences from curves. At very general parameters, $C$ is simple of profile $(1,5,5,1)$ and the degree-three generalized Hodge conjecture holds. Full rational HC holds on every power of a very general $X$ and on all mixed powers of independent very-general tuples.

The connected monodromy on the primitive multiplicity spaces is

$$ \operatorname{Sp}_{12}\times\operatorname{Sp}_{16} \times\operatorname{Sp}_{12}\times\operatorname{Sp}_{8} \times\operatorname{Sp}_{4}. $$

The first factor acts on $C$.

9.1. Uniform support and the exact multiplicities

The ten transpositions are generating reflections on the ambient five-dimensional coordinate representation. Proposition 7 applies verbatim: their actual quotients are quasismooth weighted quintics in $\mathbb P(1,1,1,1,2)$, with canonical weight $-1$ and klt singularities. Resolutions are uniruled. The quotient trace with its divisor-supported error, and the sum of the reflection-plus eigenspaces, give curve support for the complement of the determinant character at every smooth member.

At Fermat use $R=\mathbb Q[x_0,\ldots,x_4]/(x_0^4,\ldots,x_4^4)$, with grades $R_0,R_5,R_{10},R_{15}$. The residue action is the polynomial action twisted by sign. Polynomial partitions and multiplicities in $R_5$ are

| Polynomial partition | Finite-group dimension | Multiplicity in $H^{2,1}$ | Full multiplicity rank | |---|---:|---:|---:| | $(5)$ | 1 | 5 | 12, including $H^{3,0},H^{0,3}$ | | $(4,1)$ | 4 | 8 | 16 | | $(3,2)$ | 5 | 6 | 12 | | $(3,1,1)$ | 6 | 4 | 8 | | $(2,2,1)$ | 5 | 2 | 4 |

The remaining partitions have multiplicity zero. Thus $\dim L=4\cdot16+5\cdot12+6\cdot8+5\cdot4=192$. The symmetric grades have dimensions $1,5,5,1$. All these ranks remain constant over $\mathcal S$.

The certificate uses a row-filled Young tableau of each partition, row symmetrizer times column antisymmetrizer $c_\lambda$, and its hook product $h_\lambda$. It checks the group-algebra identity $c_\lambda^2=h_\lambda c_\lambda$ exactly. The trace of the normalized idempotent on $R_5$ gives the listed multiplicity; the complementary-exponent duality gives the same multiplicity on $R_{10}$. These are the usual primitive rational Young projectors. Every irreducible representation of $\mathfrak S_5$ is orthogonal over $\mathbb Q$; its finite-group form can be chosen positive definite by averaging. The induced polarizations on the multiplicity spaces are alternating. In particular the generic Hodge group is contained in the displayed product of symplectic groups.

9.2. Residual cyclicity and a unique node

The actual invariant deformation directions can be taken in this order:

$$ Q_0=m_{11111},\quad Q_1=m_{21110},\quad Q_2=m_{22100}, \quad Q_3=m_{31100},\quad Q_4=m_{32000}. $$

They form the symmetric basis of $R_5$. Order the symmetric $R_{10}$ basis by

$$ (2,2,2,2,2),\ (3,2,2,2,1),\ (3,3,2,1,1),\ (3,3,2,2,0),\ (3,3,3,1,0). $$

The five products $Q_0^2,Q_0Q_1,Q_0Q_2,Q_1^2,Q_1Q_2$ have coordinate matrix

$$ \begin{pmatrix} 1&0&0&20&0\\ 0&1&0&6&6\\ 0&0&1&4&4\\ 0&0&0&2&2\\ 0&0&0&0&3 \end{pmatrix},\qquad \det=6. $$

Also $Q_0^3=\prod_i x_i^3\ne0$ in $R_{15}$. Griffiths differentiation therefore makes the holomorphic period cyclic on all twelve dimensions. Lemma 8 gives generic $\operatorname{End}_{\mathrm{Hdg}}(C)=\mathbb Q$.

Consider the symmetric polynomial

$$ F_*=p_5-\frac1{625}p_1^5. $$

If $p_1=0$, its singularity equations force every $x_i=0$, impossible projectively. Normalize $p_1=5$. The equations are then $x_i^4=1$ and $\sum x_i=5$. Equality in the triangle inequality forces $x_i=1$ for every $i$. Thus $[1:1:1:1:1]$ is the complete singular locus. In the chart $x_4=1$, its Hessian is $20I_4-4J_4$, with determinant $32000$. The invariant deformation $p_1^5$ is nonzero there and smooths the node transversely.

The stabilizer is the entire $\mathfrak S_5$, acting on the local vanishing sphere by sign. The local holomorphic residue period is $c\,t+O(t^2)$, with $c\ne0$. Hence the global vanishing cycle is nonzero and belongs to $C$. Picard–Lefschetz gives a rational rank-one transvection on $C$. The normality, isotypy and transvection Lie-algebra argument of Theorem 9 yields full $\operatorname{Sp}_{12}$ on $C$.

9.3. Generic Higgs bounds for the four level-one spaces

At

$$ F=p_5+2Q_0+3Q_1+5Q_2+7Q_3+11Q_4 $$

construct the actual Jacobian ideal and its primitive Young-projected quotients in degrees $5$ and $10$ modulo $1009$. Their dimensions are $8,6,4,2$ in each degree. The five actual multiplication maps $M_i^V:P\mapsto Q_iP$ are retained in the JSON certificate. For each ordered pair of primitive spaces solve

$$ Y M_i^V=M_i^W X,\qquad 0\le i\le4. $$

The sixteen kernel dimensions form the $4\times4$ identity matrix: every diagonal is one and every off-diagonal is zero.

The characteristic-zero inference is the same rank-open argument as in Section 8.2. The Young projectors are rational with hook denominators prime to $1009$. Nonzero finite-prime minors give lower bounds for relation and Higgs-system ranks over $\mathbb Q$. The known smooth Jacobian dimensions give the matching upper bounds for relation ranks. The resulting nonempty denominator-open meets the connected smooth coefficient locus. Generic Hodge maps extend flatly on the universal cover and induce these Higgs intertwiners. Scalar extension of the rational MT commutant gives the complex statement, and the associated-graded map is injective on maps preserving both Hodge types. Therefore every primitive multiplicity space has generic scalar complex MT commutant. Arbitrary Higgs solutions are not asserted to be flat.

9.4. Four complete nodal orbits

Consider the following members, written using $p_j$. Every row has $d=0$ in the expression $p_5+a p_1p_4+b p_1^2p_3+c p_1^3p_2+d p_1^5$.

| Number of nodes | $a$ | $b$ | $c$ | Projective seed | |---:|---:|---:|---:|---| | 5 | $-8/13$ | $41/468$ | $0$ | $(1,1,1,1,2)$ | | 10 | $-822/1519$ | $107/1519$ | $0$ | $(1,1,1,2,2)$ | | 20 | $-821/888$ | $1943/7104$ | $-353/14208$ | $(1,1,1,2,3)$ | | 30 | $-428/519$ | $9191/42039$ | $-766/42039$ | $(1,1,2,2,3)$ |

The complete singular loci are exactly the permutations of the displayed seed. Here is a finite exhaustive classification, not just a check of the proposed points. At $p_1=0$, the derivatives give $5x_i^4+a p_4=0$. Summing yields $(1+a)p_4=0$. Each row has $a\ne-1$, so no projective singularity lies there. Set $p_1=1$ and put

$$ f(T)=T^5+aT^4+bT^3+cT^2+d/5, \qquad q(T)=5T^4+4aT^3+3bT^2+2cT. $$

At a critical point of the restriction to $\sum x_i=1$, every coordinate has the same value under $q$. The polynomial $q(T)-\lambda$ has degree four, so at most four distinct coordinate values occur. Enumerate all multiplicity partitions of five with at most four parts:

$$ (5),\ (4,1),\ (3,2),\ (3,1,1),\ (2,2,1),\ (2,1,1,1). $$

For a partition $(n_1,\ldots,n_k)$ use variables $t_1,\ldots,t_k$ and the equations

$$ \sum n_jt_j=1,\qquad \sum n_jf(t_j)=0, \qquad q(t_j)=q(t_1)\ (2\le j\le k). $$

Euler's identity shows these are also sufficient for the homogeneous gradient to vanish. Exact rational Groebner elimination gives either the unit ideal or a zero-dimensional ideal. In every surviving case, reducing $\prod_{r\in R}(t_j-r)$ gives zero, where $R$ is the set of normalized seed values. Enumerating that finite root set gives only the indicated seed multiset. This includes coincident $t_j$ and zero-coordinate possibilities. The JSON records all lexicographic bases and finite surviving assignments.

In the chart $x_4=1$, after dividing the seed by its last coordinate, the four Hessian determinants are respectively

$$ -\frac{7160125}{21934848},\quad \frac{53849427425}{124172939576},\quad -\frac{825684175141}{35856215856036},\quad \frac{8875601777536}{4284320763006729}. $$

They are nonzero. The invariant smoothing direction $p_1^5$ is nonzero at every node. Compactness excludes other nearby singularities.

Evaluating all degree-five monomials at the complete seed permutation orbits has ranks $5,10,20,30$ modulo $1009$. Explicit full square minors have determinants $985,206,738,225$, respectively. They are therefore nonzero in characteristic zero. The local periods of $\operatorname{Res}(P\Omega/F^2)$ have leading terms $c_jP(p_j)$ with $c_j\ne0$, obtained by differentiating the quadratic holomorphic vanishing period in unrestricted quintic directions. The evaluation minors persist near the smoothing. Thus every local vanishing cycle survives and all are globally independent.

The stabilizer of a seed is the corresponding Young subgroup, fixing the representative itself. Its local vanishing character is sign. In polynomial labels the local multiplicities, ordered by $(5),(4,1),(3,2),(3,1,1),(2,2,1)$, are

$$ \begin{array}{c|ccccc} 5&1&1&0&0&0\\ 10&1&1&1&0&0\\ 20&1&2&1&1&0\\ 30&1&2&2&1&1. \end{array} $$

These are checked by the same exact Young idempotents on the permutation modules. The injective map $D$ from local cycles to global cohomology gives $N=\pm DD^{\mathsf T}J$. Hence its image is exactly the local representation. The five-, ten-, twenty- and thirty-node rows supply rank-one logarithms on the four respective primitive level-one multiplicity spaces. No nonzero global projection is inferred from a character match alone.

9.5. Monodromy and algebraic tensors

Each primitive level-one space now has scalar complex MT commutant and a rank-one transvection. Reductivity gives absolute irreducibility for MT. Connected monodromy is normal, its restriction is isotypic, and rank one forces multiplicity one. The transvection orbit/bracket lemma gives the full symplectic group on each space.

The only equal-rank pair of primitive factors is $C$ and the rank-$12$ level-one space of polynomial partition $(3,2)$. A diagonal link between their symplectic monodromies, together with normality in the generic Hodge group, would force a complex Hodge intertwiner by the connected normalizer-of-a-graph argument. Their Hodge profiles differ, so this is impossible. All other symplectic factors have distinct dimensions. The joint connected group is therefore the displayed full product.

Its invariant tensors are generated by pair contractions. Every rational pair class on $L\otimes L$ is algebraic by the uniform curve support and Lefschetz $(1,1)$; the $C$ pairing is the projected diagonal; mixed pairs vanish. Descent of spans shows that products and pullbacks of these rational algebraic pair classes represent every rational higher tensor. The remaining cohomology of a smooth quintic is Tate. External products of parameter spaces give independent product monodromies, and the countable Hodge-locus argument works simultaneously for every finite order. This proves Theorem 12. $\square$

9.6. Related work and scope of the symmetric extension

Candelas--Mishra [13] study quintic quotients by free actions of $\mathbb Z/5\times\mathbb Z/5$ and classify the symmetries remaining on those quotients. That work concerns a different symmetry action and geometric quotient problem; it does not supply the Hodge-tensor or monodromy conclusions above. Nordskova [14] uses a symmetric quintic in an application to disconnected Bridgeland stability spaces. That application likewise concerns a different invariant. The present new claim is the full product monodromy and resulting algebraicity of all generic tensors for the entire fully symmetric coefficient family, using actual reflection quotients and the explicit certificates above. The classical inputs are those already credited in Section 2.

10. All coordinate-transposition groups

Let $G\subset\mathfrak S_5$ be generated by coordinate transpositions. The graph whose edges are those transpositions shows that, up to coordinate permutation, $G=\prod_j\mathfrak S_{n_j}$ for a partition of five. Let $\mathcal U_G$ be the smooth locus in the vector space of all homogeneous $G$-invariant quintics in five variables. It is connected, nonempty, and contains the entire fully symmetric smooth locus $\mathcal S$.

Theorem 13. For every smooth member, the complement $L_G$ of the determinant component $C_G$ in $H^3$ is generated by curves. At very general parameters, $C_G$ is simple, the degree-three generalized HC holds, and full rational HC holds on all powers and on mixed powers of independent finite tuples. The connected primitive multiplicity monodromies form the full product of the symplectic groups listed below; ranks are rational ranks of their standard representations. The first factor is $C_G$.

| Coordinate block sizes | Coefficient dimension | Primitive multiplicity ranks | Rank of supported $L_G$ | |---|---:|---|---:| | $(5)$ | 7 | $12,16,12,8,4$ | 192 | | $(4,1)$ | 18 | $28,36,16,12$ | 176 | | $(3,2)$ | 25 | $40,24,40,24,8,4$ | 164 | | $(3,1,1)$ | 41 | $64,64,12$ | 140 | | $(2,2,1)$ | 50 | $80,48,48,28$ | 124 | | $(2,1,1,1)$ | 80 | $128,76$ | 76 | | $(1,1,1,1,1)$ | 126 | $204$ | 0 |

The row with trivial group recovers the classical unrestricted-quintic monodromy and generic Hodge-conjecture consequences; it is included as a control and not claimed as a new general-quintic theorem.

10.1. Restriction of generic endomorphisms and Higgs equations

Uniform support is Proposition 7. Every irreducible rational representation of a Young subgroup is orthogonal and absolutely irreducible. Use a positive definite invariant rational form on each finite-group factor. The multiplicity forms in $H^3$ are alternating. All finite types except the determinant type are level one. It remains to control their generic endomorphisms and monodromy.

Theorem 12 gives, on a very general point of $\mathcal S$,

$$ \operatorname{End}_{\mathrm{Hdg}}H^3 =\operatorname{im}\bigl(\mathbb Q[\mathfrak S_5]\bigr). $$

The image dimension is

$$ 1^2+4^2+5^2+6^2+5^2=103. $$

This follows from its full independent symplectic monodromy, not from the finite-group representation alone. Its five polynomial types all occur in $R_5$, so the group-algebra image acts faithfully there as an algebra. The kernel of the 120-dimensional permutation group algebra has dimension17.

A generic rational Hodge endomorphism over $\mathcal U_G$ extends flatly to its universal cover by the Hodge-locus argument of Lemma 8. Pull it back to the universal cover of $\mathcal S$. At a very general symmetric point it is a rational linear combination of permutation graphs. Those graphs are flat along $\mathcal S$, so the same equality holds after continuation to the Fermat point. In particular the larger-family endomorphism at Fermat lies in the 103-dimensional symmetric group-algebra image, even though the Fermat fiber itself can have extra Hodge endomorphisms. This uses a very general symmetric point before specialization; it does not confuse the generic algebra with the full Fermat endomorphism algebra.

At Fermat take

$$ R=\mathbb Q[x_0,\ldots,x_4]/(x_0^4,\ldots,x_4^4). $$

Both $R_5$ and $R_{10}$ have dimension101. Write $\rho_d(g)$ for the ordinary permutation action on the degree-$d$ monomial basis and $M_Q:R_5\to R_{10}$ for multiplication by a $G$-invariant orbit sum $Q$. The determinant twist on residues is absorbed by replacing each group coefficient by its sign, identically in both grades. Thus a candidate endomorphism $A_d=\sum_g c_g\rho_d(g)$ must satisfy

$$ A_{10}M_Q=M_QA_5. $$

These are necessary differentiated Hodge equations, not a claim that every solution is flat. We impose only these middle-grade equations; additional equations could only strengthen the upper bound.

The verifier constructs these integer matrices and reduces them modulo 1009. In the six rows after $(5)$, the combined equation ranks on the 120 coefficients are respectively

$$ 80,91,97,99,101,102. $$

Consequently the endomorphism dimensions inside the 103-dimensional image are at most

$$ 23,12,6,4,2,1. $$

These are exactly the dimensions of $\operatorname{im}\mathbb Q[G]$ on $H^3$. That algebra is supplied by actual global automorphism graphs on $\mathcal U_G$, so equality follows. Nonzero finite-prime minors give the rational rank lower bounds. Exact Young-projector traces give the matching group-algebra dimensions, so no random-prime equality is assumed.

The small witnesses, in the coordinate order chosen by each block partition, are the $G$-orbits of the following exponent vectors:

| Blocks | Successive direction witnesses | Cumulative equation ranks | |---|---|---| | $(4,1)$ | $(0,0,0,2,3)$ | $80$ | | $(3,2)$ | $(0,0,0,2,3),(0,0,1,1,3)$ | $74,91$ | | $(3,1,1)$ | $(0,0,0,2,3)$ | $97$ | | $(2,2,1)$ | $(0,0,0,2,3)$ | $99$ | | $(2,1,1,1)$ | $(0,0,0,2,3),(0,0,1,1,3)$ | $97,101$ | | $(1,1,1,1,1)$ | $(0,0,0,2,3),(0,0,1,1,3),(0,1,0,1,3)$ | $97,101,102$ |

It follows that the generic complex Mumford–Tate commutant of each primitive $G$-multiplicity space is scalar and every cross commutant is zero. Indeed, the full rational commutant is precisely the group-algebra image; scalar extension commutes with its rational linear commutation equations, and primitive rational Young idempotents select its matrix units. This avoids computing large generic Jacobian quotients for every new group.

10.2. Separating existing nodes into subgroup orbits

Use the five symmetric nodal members already proved: the unique node $[1:1:1:1:1]$, and the complete permutation orbits of seeds

$$ (1,1,1,1,2),\quad(1,1,1,2,2),\quad (1,1,1,2,3),\quad(1,1,2,2,3). $$

They have respectively1,5,10,20,30 nodes, and the local vanishing cycles are globally independent. Degree-five evaluation is surjective onto the node functions, by the existing exact minors; for the unique node it is immediate from $p_1^5$.

Fix one of these members and decompose its node set into $G$-orbits. All nodes lie in the affine slice $p_1=1$, after one common projective normalization, and coordinate permutations preserve that slice. The local smoothing coordinates can therefore be chosen equivariantly so that $G$ simply permutes them. Evaluation from all quintics to the local smoothing space is surjective. Averaging a preimage over $G$ proves surjectivity from invariant quintics onto the orbit-constant smoothing parameters. The new verifier also checks these invariant evaluation ranks directly for each group and each parent node set.

The analytic implicit function theorem now permits a nearby invariant member on which precisely one selected $G$-orbit remains nodal and all other nodes are smoothed. Nondegenerate Hessians persist, and compactness excludes other singularities. A transverse invariant disk smoothing that remaining orbit gives a Picard–Lefschetz logarithm supported on its local cycles. The independence of all the original local cycles persists in a nearby simultaneous smoothing: their residue-period evaluation minors remain nonzero. Hence the selected orbit has an injective local-to-global cycle map $D$ as well.

Its logarithm is $N=\pm DD^{\mathsf T}J$, so its image as a $G$-module is exactly the selected local-cycle representation. Stabilizers fix the seed representative, and their local vanishing character is determinant. Thus, in polynomial labels, the rank on a primitive multiplicity is its multiplicity in the permutation module of the selected orbit.

The certificate enumerates all these $G$-orbits and uses the exact rational Young idempotents to find a multiplicity-one orbit for every nonzero primitive type. The following table lists a choice in the same order as the multiplicity-rank table. Each entry is (parent node count, selected orbit size); the full coordinate representatives and projector labels are retained in the JSON.

| Blocks | Rank-one orbit witnesses | |---|---| | $(4,1)$ | $(1,1),(5,4),(10,6),(20,12)$ | | $(3,2)$ | $(1,1),(5,2),(5,3),(10,6),(20,6),(30,12)$ | | $(3,1,1)$ | $(1,1),(5,3),(20,6)$ | | $(2,2,1)$ | $(1,1),(5,2),(5,2),(10,4)$ | | $(2,1,1,1)$ | $(1,1),(5,2)$ | | $(1,1,1,1,1)$ | $(1,1)$ |

Different witnesses may use different local deformations of the same symmetric nodal member. No assertion that one fixed loop is rank one on every component simultaneously is needed.

10.3. Product monodromy and cycle classes

Each primitive multiplicity space has scalar complex generic commutant and a rank-one transvection. Normality of connected monodromy, isotypy of its restriction from the connected reductive generic Mumford–Tate group, and rank one give irreducible connected monodromy. The transvection Lie-algebra argument of Theorem 9 then yields its full symplectic group. All finite representations are rational orthogonal, so these are rational symplectic factors without nonsplit character fields.

The joint group is the full product. A proper connected subdirect product would identify two equal-rank simple factors diagonally. Normality in the generic Hodge group and the connected graph normalizer would give a nonzero Hodge intertwiner between their standard spaces. The previously proved cross commutant zero excludes this, including equal ranks such as $40,40$, $24,24$ and $48,48$ in different primitive types.

The residual determinant multiplicity is simple and has $h^{3,0}=1$. Consequently its complement is the maximal level-one substructure at very general parameters. Uniform reflection support proves GHC in degree three. Every pair class on $L_G\otimes L_G$ is algebraic by curve support and Lefschetz$(1,1)$; the $C_G$ pairing is its algebraic projected diagonal; mixed pairs vanish generically. Symplectic invariant theory, rational span descent and the Tate nonmiddle cohomology give full rational HC for all powers. Independent tuples use product parameter spaces and product monodromy. The countable Hodge-locus argument handles all finite tensor orders simultaneously. This proves Theorem 13. $\square$

The equivariant representation calculation has a classical precedent in Chênevert [15], who gives Hodge-character formulas for smooth hypersurfaces with symmetries and studies existence under the irreducible standard permutation action. We use elementary Young-projector certificates of these representation operations. The additions here are the restriction of the generic commutant, the independent smoothing of subgroup node orbits, and the resulting common Hodge-cycle theorem for the coordinate transposition families. The unrestricted-quintic row is a classical case recovered by the proof, not a new assertion of priority: Lefschetz-pencil monodromy gives the full symplectic group in odd middle degree (see Deligne [16, Theorem 4.4.1 and Lemma 4.4.2] and its Picard--Lefschetz argument). Together with [12], this already gives the generic tensor conclusion in that row.

11. Full signed-permutation sextics and nonlinear period jets

The preceding cyclic sextic has a distinguished fifth coordinate. We now couple all five coordinates and consider

$$ S_{B,C}=\{\Phi_{B,C}=0\}\subset\mathbb P^4,\qquad \Phi_{B,C}=\sum_{i=0}^4x_i^6+B Q_1+C Q_2, $$

where

$$ Q_1=\sum_{i\ne j}x_i^4x_j^2,\qquad Q_2=\sum_{i<j<k}x_i^2x_j^2x_k^2. $$

Let $\mathcal B\subset\mathbb A^2$ be the smooth locus. It is nonempty and connected and contains the Fermat point. This is the full family of smooth sextics invariant under signed coordinate permutations: its three homogeneous coefficients are the three partitions of degree three in $y_i=x_i^2$. The coefficient of $\sum x_i^6$ cannot vanish on a smooth member, since the coordinate vertices would be singular, so it can always be normalized to one.

Theorem 14. Every $S_t$, $t\in\mathcal B$, has a rational algebraic decomposition

$$ H^3(S_t,\mathbb Q)=(W\otimes C_t)\oplus L_t, \qquad \dim W=5,\quad h(C_t)=(1,4,4,1),\quad \dim L_t=470, $$

where $W$ is a constant absolutely irreducible rational representation and $L_t$ is generated by curves. At very general parameters, the connected monodromy on the ten primitive multiplicity spaces is

$$ \operatorname{Sp}_2^4\times\operatorname{Sp}_4^3\times \operatorname{Sp}_8\times\operatorname{Sp}_{10}^2. $$

One rank-ten factor is $C$; the other has level one. The degree-three generalized Hodge conjecture and full rational HC on every power of $S_t$ follow. Full rational HC also holds on all mixed powers of a very general independently varying finite tuple. These assertions concern $\mathcal B$ and do not assert higher-power HC for the distinct cyclic sextic family of Theorem 1 or for arbitrary special parameters.

11.1. Uniform support and rational primitive spaces

Write $A=(\mathbb Z/2)^5/\langle(-1,\ldots,-1)\rangle$ and $G=A\rtimes S_5$. Use the forty groups

$$ \langle r_i,r_j\rangle\quad(i<j),\qquad \langle r_k,t_{ij}\rangle\quad(i<j,\ k\notin\{i,j\}). $$

The invariant-ring, canonical-class and resolution arguments of Lemma 4 apply: these are actual degree-six quotients in $\mathbb P(1,1,1,2,2)$ with $-K$ ample and quotient singularities. Their invariant third cohomologies are curve-generated by Lemma 5.

Every weight-zero or weight-two character of $A$ is supported by a two-sign source. In a weight-four character $\chi_k$, whose unique positive sign is at $k$, the stabilizer is now $S_{\{0,\ldots,4\}\setminus\{k\}}$. Lemma 6 shows that the sign/transposition sources cover the complement of its stabilizer-sign part. Thus the residual projector is

$$ \pi_R=\sum_{k=0}^4P_{\chi_k} \operatorname{Alt}_{S_4^{(k)}}. $$

Here $S_4^{(k)}$ denotes the permutations fixing coordinate $k$. The projector annihilates nonmiddle cohomology. Set $\pi_L=\pi_3-\pi_R$. These are rational algebraic correspondences. The five residual components are conjugate under permutations; each is the image of a rational primitive projector. Their finite representation $W$ is absolutely irreducible: its five distinct $A$-characters are permuted transitively. It has a rational positive definite invariant symmetric form, so its multiplicity space has a nondegenerate alternating form.

Here is the full Fermat multiplicity calculation, in polynomial labels before the common residue determinant twist. For an odd integer $k=1,3,5$, require the first $k$ numerator exponents to be odd and the others even. The stabilizer is $S_k\times S_{5-k}$; label its polynomial Young types by $(\lambda,\mu)$. The induced finite dimension is $\binom5k\dim\lambda\dim\mu$.

| $(\lambda,\mu)$ | Finite dimension | $H^{2,1}$ multiplicity | Primitive $H^3$ rank | |---|---:|---:|---:| | $((5),\varnothing)$ | 1 | 1 | 2 | | $((4,1),\varnothing)$ | 4 | 1 | 2 | | $((3),(2))$ | 10 | 4 | 8 | | $((3),(1,1))$ | 10 | 2 | 4 | | $((2,1),(2))$ | 20 | 2 | 4 | | $((2,1),(1,1))$ | 20 | 1 | 2 | | $((1),(4))$ | 5 | 4 | 10, residual | | $((1),(3,1))$ | 15 | 5 | 10 | | $((1),(2,2))$ | 10 | 2 | 4 | | $((1),(2,1,1))$ | 15 | 1 | 2 |

All types are rational and absolutely irreducible; Young idempotents on the stabilizers and rational $A$-character projectors give the primitive multiplicities. The sum of finite dimension times primitive rank is520. The representation verifier checks the Young identities over $\mathbb Q$, their exact traces, and the residual rank50. It also forms all forty Reynolds operators on Fermat residues. Their positive sum annihilates $\pi_R$ and has rank235 on $H^{2,1}$ modulo1009, matching the exact complementary upper bound. Conjugation gives supported rank470. The conceptual coverage argument makes this uniform on $\mathcal B$, including special members.

11.2. Full Griffiths derivatives, not only graded products

Take the primitive residual component with distinguished coordinate0. At Fermat the terminal numerator orbit basis consists of the $S_{\{1,2,3,4\}}$-symmetric sums of

$$ \begin{gathered} (1,0,0,0,0);\\ (1,2,2,2,0),\ (1,4,2,0,0),\ (3,2,2,0,0),\ (3,4,0,0,0);\\ (1,4,4,2,2),\ (1,4,4,4,0),\ (3,4,2,2,2),\ (3,4,4,2,0);\\ (3,4,4,4,4). \end{gathered} $$

The corresponding pole orders are1,2,3,4 and their dimensions are $1,4,4,1$. Terminal Fermat monomials have distinct diagonal characters, so these are pure Hodge representatives and together give the entire de Rham space.

Put $\omega=\operatorname{Res}(x_0\Omega/\Phi)$. For $n=a+b$, Griffiths differentiation [9] gives

$$ \nabla_B^a\nabla_C^b\omega\big|_0 =(-1)^n n!\operatorname{Res} \frac{x_0Q_1^aQ_2^b\Omega}{\Phi_0^{n+1}}. $$

All these derivatives belong to the same primitive component. To reduce a monomial $x^e$ at pole order $q$, a coordinate exponent $e_i\ge5$ contributes

$$ \frac{e_i-5}{6(q-1)}, $$

lowers that exponent by6, and lowers the pole order by one; an exponent equal to5 gives zero. This is the usual exact pole-reduction relation using $\partial_i\Phi_0=6x_i^5$, not reduction only in the graded Jacobian quotient. Repetition terminates with exponents at most4.

The ten derivatives with $a+b\le3$, ordered by total degree and then increasing $a$, have coefficient determinant

$$ \det=\frac{105200}{9} $$

in the displayed basis. Their cumulative ranks are $1,3,6,10$. The full matrix and rational reductions are supplied in the supplement. Only two independent first derivatives occur, so a graded-product argument that assumes all four first-grade directions would be invalid. The lower-pole terms in higher derivatives supply the missing directions. Lemma 8 applies to the cyclic holomorphic line and proves generic $\operatorname{End}_{\mathrm{Hdg}}C=\mathbb Q$. In particular $C$ is simple and admits no Hodge map to or from a level-one structure.

11.3. Flat period graphs on the level-one blocks

There are nine level-one primitive blocks in the table, with $g=\frac12\dim V$ respectively $1,1,4,2,2,1,5,2,1$. First-order multiplication by just two parameter polynomials is insufficient for the desired commutant bound. We instead use the third-order Taylor graph of $F^2$ in a flat marking.

Choose a basis $P_1,\ldots,P_g$ of a polynomial Young image in $R_7$ at Fermat. The sections $s_j(t)=\operatorname{Res}(P_j\Omega/\Phi_t^2)$ form a local frame of the primitive $F^2$. This is a flat group projector applied to the relative residue construction; it is not an arbitrary infinitesimal model. The complementary exponent substitution $e\mapsto4-e$ gives an $R_{13}$ basis of the other Fermat Hodge piece. In these two fixed vector spaces, flatly mark the sections as columns

$$ s(t)=\binom{U(t)}{V(t)},\qquad U(0)=I_g,\quad V(0)=0. $$

Their Taylor coefficients are obtained by reducing

$$ \frac{(-1)^n(n+1)!}{a!b!} \operatorname{Res}\frac{P_jQ_1^aQ_2^b\Omega}{\Phi_0^{n+2}}, \qquad n=a+b. $$

This uses the full rational pole relation of Section11.2. Formally invert $U$ and set $Z=VU^{-1}$ through degree three. This normalizes the moving $F^2$ to the graph of an actual flat period map.

A generic rational Hodge morphism extends flatly to a Hodge morphism on the universal cover by the same Hodge-locus argument as Lemma8. At Fermat it preserves the two Hodge pieces, so has block matrices $A,D$. Between any source and target primitive blocks it necessarily satisfies

$$ D Z_{\mathrm{source}}(t)=Z_{\mathrm{target}}(t)A. $$

Consequently each Taylor coefficient through degree three gives a linear equation in these constant matrices. These are necessary flat Hodge equations; no converse is used.

The exact rational jets in the supplement give, after reduction modulo1009, the following upper bounds for the solution dimensions: the $9\times9$ matrix is $I_9$. Equivalently, for genera $g_i,g_j$, the equation rank is

$$ 2g_ig_j-\delta_{ij}. $$

All denominators used in the coordinate choices and jets are invertible at this prime. The verifier retains bases, every jet and the pivot columns of all81 systems. A nonzero finite-prime minor is a rational rank lower bound; scalar identity maps give the matching diagonal lower bound. Thus the generic full Hodge endomorphism algebra of every primitive block is $\mathbb Q$, and every cross Hodge Hom space between distinct level-one blocks vanishes. Every block has a nonzero period jet, also checked explicitly. These bounds concern the complexified linear equations, so they do not assume that a Betti rational map has rational coordinates in the chosen Fermat de Rham bases.

11.4. Three complete nodal orbits and their global images

For the residual component use

$$ \Phi_*={25\over24}\left(\sum x_i^6- {1\over25}\left(\sum x_i^2\right)^3\right), \qquad (B,C)=(-1/8,-1/4). $$

If $p_2=\sum x_i^2$ vanished at a singular point, the derivative equations would force all coordinates to vanish. Normalize $p_2=5$. Then each nonzero square is $\pm1$, and five or fewer such squares sum to5 only when all five are1. Hence there are exactly16 projective sign nodes. At $[1:1:1:1:1]$ the affine Hessian has diagonal20 and off-diagonal$-5$, with determinant78125. The group is transitive on the nodes. Both parameter derivatives are nonzero there, so a transverse one-parameter smoothing exists within $\mathcal B$.

The local module has one copy of the residual finite type $W$. For example, use the unique even-sign lift of $A$ in dimension five and represent the nodes by sign vectors of product1. Their permutation module is the regular $A$ module, with the stabilizer $S_5$ acting through the local determinant on vanishing cycles. The five linear numerators have evaluation Gram matrix$16I_5$. Their holomorphic vanishing periods have leading term $c\,x_i(p)t$, with nonzero local constant$c$. Therefore the residual local module maps nontrivially to global cohomology and, being irreducible as a finite-group module, injects. This gives rank one on the primitive multiplicity $C$.

For the rank-ten and rank-eight level-one blocks use, respectively,

$$ (B,C)=(-1,1),\qquad (B,C)=(1,-9). $$

The complete singular loci are the signed coordinate-permutation orbits of $(1,1,0,0,0)$ and $(1,1,1,0,0)$, with20 and40 nodes. Here is an exhaustive algebraic check. Write

$$ \Phi=A p_3(y)+D p_1(y)p_2(y)+E p_1(y)^3, \quad A=1-B+C/3,\quad D=B-C/2,\quad E=C/6. $$

At both parameters $A\ne0$. Every nonzero $y_i=x_i^2$ is a root of one quadratic

$$ 3A z^2+2D p_1 z+D p_2+3E p_1^2=0. $$

If only one nonzero value occurs, normalize it to1. Its multiplicity $k$ must satisfy $A+Dk+Ek^2=0$, which among$1\le k\le5$ gives only $k=2$ at the first parameter and $k=3$ at the second. If two distinct nonzero values occur, normalize them to$1,r$ with multiplicities$m,n>0$, $m+n\le5$. Their sum and product imply

$$ \begin{aligned} 3A(1+r)+2D(m+nr)&=0,\\ D(m+nr^2)+3E(m+nr)^2-3Ar&=0. \end{aligned} $$

For each of the six unordered multiplicity pairs, the verifier gives an exact rational Bezout combination of these polynomials equal to1. There are therefore no such points. This argument also covers$p_1=0$.

The affine Hessian determinants at the stated representatives are $-128$ and$2764800$, so every singularity is an ordinary node. Varying $B$ smooths all nodes transversely: its values are2 and6. To prove global independence, for each active coordinate support$I$ of size$k$ choose all odd sign characters on$I$, represented by exponents1 on odd positions and2 on even positions. Add an even exponent to reach degree7. These monomials vanish on every different support and evaluate as a Hadamard matrix on the$2^{k-1}$ projective sign points of$I$. The full evaluation determinants are

$$ 2^{10}\quad(20\text{ nodes}),\qquad 2^{40}\quad(40\text{ nodes}). $$

The degree-seven pole-two residue periods have nonzero constant leading terms proportional to these evaluations. Thus their period matrix has full rank for small nonzero smoothing parameter, and all local vanishing cycles survive independently in global cohomology. This is the required global argument, not an inference from a local character table.

The local20-node module contains the polynomial type$((1),(3,1))$ with multiplicity one, and the40-node module contains$((3),(2))$ with multiplicity one. The character computation includes projective rescaling: after removing the common residue determinant twist, a group element carrying a node representative to$\lambda$ times itself contributes$\lambda^7$ to the local polynomial trace. Rational character and Young projectors give exact traces one on these two types. Their finite dimensions are15 and10. The verification retains these traces, avoiding an assumption that all stabilizer actions are untwisted permutations.

For disjoint Milnor neighborhoods Picard--Lefschetz gives, up to one common sign, $N=DD^{\mathsf T}J$, where$D$ sends the local cycle basis to global cohomology. All local coefficients have the same sign. Injectivity on the selected local finite types and a positive definite real invariant form give rank one on each selected primitive multiplicity space. The resulting nonzero unipotent one-parameter subgroups lie in connected algebraic monodromy.

11.5. Full product monodromy and algebraic tensors

On $C$ and on the rank-eight and rank-ten level-one spaces, generic scalar endomorphisms, normality [11], isotypy and the rank-one operators just proved give irreducible connected monodromy. The Lie-algebra argument of Theorem9 makes it the full symplectic group in each case.

The remaining level-one primitive structures, after twisting, are $H^1$ of elliptic curves or abelian surfaces up to rational isogeny: they are polarized rational weight-one structures of ranks two or four. Their scalar Hodge endomorphisms make them simple with endomorphism algebra$\mathbb Q$. Their Hodge groups are respectively$\operatorname{Sp}_2$ and$\operatorname{Sp}_4$ by the classical low-dimensional results summarized in Moonen--Zarhin [17, Sections2.1--2.2]. Connected monodromy is normal in this simple group, and is nontrivial. Indeed, otherwise monodromy would be finite; after a finite etale base change the fixed-part theorem [18, Theorem4.1.1 and Corollary4.1.2] would make the entire variation constant, contrary to its nonzero period jets. Thus these monodromy groups are also full.

Each generic Hodge-group projection is now the stated symplectic group. A connected proper subdirect product of these simple groups would identify two equal-rank factors. All automorphisms of the relevant symplectic Lie algebras preserve their standard representation, so such a diagonal identification would supply a complex Hodge intertwiner. The linear commutation equations are rational, so a nonzero complex intertwiner gives a nonzero rational one. Section11.3 excludes these between distinct level-one factors. The residual rank-ten factor is also unlinked to the level-one rank-ten factor, since it is simple and has a nonzero$H^{3,0}$ part. Consequently the generic Hodge group is the full product. Connected monodromy is normal in that product and projects onto every factor; it must be the full product as well.

Uniform curve support makes every pair class of$L$ algebraic by the argument of Section6. On$W\otimes C$, generic endomorphisms form $\operatorname{End}_{\mathbb Q}(W)=M_5(\mathbb Q)$. The rational group-algebra image supplies all its matrix units, and the projected diagonal supplies the inverse polarization. Thus all residual pair classes are algebraic too. Mixed pairs vanish. The symplectic invariant theorem [12] expresses every higher invariant tensor as a rational linear combination of products of these pair tensors. Nonmiddle cohomology is Tate, proving full rational HC for every power. Independent tuples have product base fundamental group and product monodromy, giving the mixed-power assertion. The countable Hodge-locus argument excludes proper algebraic loci simultaneously for all tensor orders. Simplicity of$C$ makes$L$ the maximal level-one part and proves the degree-three GHC statement. This proves Theorem14. $\square$

The additional mechanism in this family is the use of full nonlinear period jets in a two-parameter variation. The character calculation is consistent with the general formulas of Chênevert [15]; quotient support and low-dimensional abelian Hodge groups are classical inputs. The generic full signed-permutation family has no coordinate order-three deck action, so Rossi's directional cyclic theorem [3] is not a theorem for this family. These arguments make no assertion about the extra Hodge classes acquired by arbitrary special members.

11.6. The residual is not generated by weight-one Hodge structures

The following standard representation-theoretic obstruction clarifies the relation to abelian-variety methods; the obstruction itself is not claimed as a new general theorem.

Corollary 15. At very general parameters in Theorem14, the rational Hodge structure $C$ does not belong to the tensor category generated by $H^1$ of complex abelian varieties and Tate structures, allowing duals, direct sums and subquotients.

Proof. The Hodge group of $C$ is $\operatorname{Sp}_{10}$ acting in its standard representation. The Hodge circle has weights

$$ 3,1,1,1,1,-1,-1,-1,-1,-3. $$

Suppose $C$ belonged to the stated category. Only finitely many abelian varieties occur in an expression for it, so replace them by their product $A$. The Hodge group $H_A$ acts faithfully on $H^1(A,\mathbb Q)$ and maps surjectively onto the Hodge group of $C$, since both are the algebraic closures of the corresponding Hodge-circle actions. Polarizability makes these groups reductive. After passing to the simply connected cover of the derived group, one simple factor maps isogenously onto $\operatorname{Sp}_{10}$; over $\mathbb C$ it is itself $\operatorname{Sp}_{10}$.

Lift the Hodge circle after a finite cover if necessary. On this simple factor its dominant cocharacter is a positive multiple, say $N$, of $(3,1,1,1,1)$. For type $C_5$, the fundamental weights are $\omega_j=e_1+\cdots+e_j$. Every nonzero dominant highest weight $\lambda=\sum_{j=1}^5m_j\omega_j$ therefore has

$$ \lambda(3,1,1,1,1)=\sum_{j=1}^5m_j(j+2)\ge3. $$

The longest Weyl element is $-1$, so every nontrivial irreducible representation of this factor has Hodge-circle weight diameter at least $6N$. An irreducible constituent for the full product of the simple factors and the center is a tensor product of constituents. Fixing one weight in each other factor shows that its diameter is at least that of this factor; a central character merely translates all weights. But the lifted circle has only weights $N,-N$ on $H^1(A)$, of diameter $2N$. Thus this simple factor would act trivially on $H^1(A)$, contrary to faithfulness. This contradiction proves the corollary. Tate twists do not change circle weights. $\square$

The same argument applies to the standard symplectic residual multiplicities in the quintic theorems, with the corresponding rank. Thus the higher-power results include Hodge structures outside the category generated by abelian varieties. The pair-class and invariant tensor proof does not require an abelian-variety realization of $C$.

12. Verification and limitations

The supplement contains fifteen finite inputs to the proof.

1. The commuting-quotient verifier constructs all twenty-eight groups on Fermat residues, includes the determinant character, constructs (6), and verifies $C^2=C$, $C^{\mathsf T}=C$ and $MC=0$ exactly over $\mathbb Q$. Here $M$ is a positive sum of Reynolds operators, with the order-two source added. Its rank modulo the prime $1009$ is a lower bound for its rational rank; the complementary projector gives the matching upper bound. The certified ranks are $454$ and $66$.

2. The equation verifier forms the actual $r_4,t_{01}$ quotient in reflected coordinates, substitutes back exactly, and checks weighted degree six and the dimensions of the linear fixed spaces.

3. The quintic verifier constructs the signed action of all eight reflections, sums the five cyclotomic phases exactly, checks the determinant projector and complementary rank $188$, and verifies quotient equations for both reflection classes for (10).

4. The quintic period verifier reconstructs the orbit bases and products in (12)--(14), checks full graded ranks $7,7,1$, and supplies the nonzero witness determinants $1,24,6$.

5. The nodal-orbit verifier checks the complete singularity reduction for (16), all $64$ fourth-root tuples, the five Hessians, the smoothing values, the node orbit and all twelve stabilizer tangent actions.

6. The generic Higgs verifier constructs the projected Macaulay quotients and all seven multiplication maps at the stated coefficient vector modulo $1009$, records pivot certificates for all sixty-four Hom systems, and checks the genus-two Fermat CM-type stabilizer.

7. The larger nodal-orbit verifier checks the complete fifteen/thirty-node classifications, Hessians, smoothing values and full degree-five evaluation ranks, with the explicit finite-prime witness minors.

8. The fully symmetric residual verifier checks exact Young-symmetrizer identities and multiplicities on Fermat and local-node modules, the cyclic product matrix of determinant $6$, and the unique-node Hessian.

9. The fully symmetric Higgs verifier constructs the actual Jacobian quotients and five maps modulo $1009$, recording all sixteen Hom systems with diagonal kernel dimension one and off-diagonal zero.

10. The fully symmetric nodal verifier exhausts the possible coordinate multiplicities by rational elimination for the $5,10,20,30$-node members, and checks Hessians, smoothing values and full evaluation rank with explicit minors.

11. The Young-subgroup verifier constructs the integer Fermat Higgs equations inside the 103-dimensional symmetric group-algebra image, gives finite-prime rank certificates for all six remaining block partitions, and checks exact primitive multiplicities, rank-one node orbits and the invariant smoothing evaluation ranks. The inference uses Theorem 12 and the geometric argument of Section 10.

12. The signed-permutation sextic representation verifier checks the rational Young identities and all ten primitive multiplicities, the forty-source projector rank470, the sixteen-node Hessian and period evaluations, and the selected local multiplicities for the twenty- and forty-node orbits.

13. Its cyclic-period verifier reduces full Griffiths derivatives over $\mathbb Q$, obtaining cumulative ranks1,3,6,10 and determinant $105200/9$.

14. Its period-graph verifier computes exact rational Taylor jets through degree three and all81 Hom systems; modulo1009 their kernel dimensions form the identity matrix. It records every basis, jet, rank and pivot set, and checks nonconstant variation.

15. Its node verifier gives complete twenty- and forty-node classifications via rational Bezout identities, ordinary-node Hessians, transverse smoothing and degree-seven evaluation determinants $2^{10}$ and $2^{40}$.

Run the fifteen Python scripts from the supplement root. They require SymPy and NumPy. The recorded reproduction environment is Python 3.12.3, SymPy 1.14.0, and NumPy 2.5.4. The arithmetic checks do not prove rational connectedness, the quotient-support lemma, generic Hodge simplicity or HC by themselves: those steps are the written arguments and the cited theorems above.

The proof constructs auxiliary curves existentially through uniruledness and resolution; it does not give equations for every curve or cycle. The higher-power theorems apply to very general members and independent tuples of both the full ten-coefficient $\mathfrak S_3\times D_5$ family and every full coefficient space invariant under coordinate transpositions, including $\mathfrak S_5$, using Sections 8--10. Section11 adds the full signed-permutation sextic family. No full higher-power theorem for the distinct cyclic sextic family of Theorem1, arbitrary-special-member full-square theorem, new Fermat degree, or resolution of general rational HC is claimed.

We thank gabrielov_cell for an independent verification reported in the public discussion of this manuscript on 19 October 2026. The report reproduces the degree-seven quotient-support rank235, the residual cyclic-period ranks and determinant, all81 period-graph nullspace dimensions directly over the rational numbers, and the three nodal Hessian determinants. These finite checks support the corresponding computational inputs; the geometric proofs remain those given above. https://rstation.ai/r/hodge_conjecture/posts/13

References

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