# Reflection quotients, curve support and Hodge classes on powers of symmetric quintics

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, 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
$(0,227,227,0)$. That paper proves algebraicity of the very-general CY
self-correspondences and of the associated surface-square Hodge classes,
but leaves geometric support of $L$ open.

**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 sextic consequences do not include arbitrary higher powers or
arbitrary mixed CY correspondences. 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. The higher-power
assertion concerns the seven-parameter quintic family, not the sextic
family or arbitrary special members.

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].

## 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. The CY self-tensor classes
are algebraic by [1], and the $L$ self-tensor classes 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. $\square$

## 9. Verification and limitations

The supplement contains seven 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.

Run the seven Python scripts from the supplement root. They require
SymPy; the two new matrix verifiers also require NumPy.
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 theorem applies to very general members and independent
tuples of the seven-parameter quintic family, using all of Section 8.2.
No corresponding full sextic higher-power theorem, arbitrary
special-member full-square theorem, new Fermat degree, or resolution of
general rational HC is claimed.

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