# Type-(1,3) double covers, Picard-twelve K3 surfaces and Gaussian Hodge conjectures

## 1. Contribution, dependencies and scope

Let $(B,L)$ be a complex abelian surface with a symmetric polarization of
type $(1,3)$. Normalize the symmetric linearization so that
$h^0(L)^+=2$ and $h^0(L)^-=1$. For a smooth even section
$d\in H^0(B,L^2)^+$, form the double cover $X_d\to B$ given by $z^2=d$.
Write $\delta$ for its sheet involution, and choose the other inversion
lift $\tau$, so that the selected space
$$
U_X=H^2(X_d,\mathbb Q)^{\tau=1,\delta=-1}
$$
has Hodge numbers $(1,8,1)$.

Our principal unconditional calculation determines the integral lattice
underlying a double-Kummer model of this variation. Its full
anti-invariant lattice and the primitive complement of its six persistent
exceptional differences are
$$
P\simeq U_{\mathrm{hyp}}^{\oplus2}\oplus D_{12}(-1),
\qquad
R\simeq U_{\mathrm{hyp}}^{\oplus2}\oplus D_6(-1).
\tag{1.1}
$$
Here $U_{\mathrm{hyp}}$ is the integral hyperbolic plane; $D_n(-1)$ is the
negative root lattice. Both lattices have determinant $4$. The algebraic
differences themselves saturate to a primitive $D_6(-1)\subset P$.
We give an explicit discriminant gluing of $P$ into the K3 lattice and
check every codimension-one event in a general branch pencil.

The comparison conclusion uses precisely stated external inputs.
OpenAI's September 30, 2026 preprint [1], Lemma 5.5 and Theorem 6.1,
assert, respectively, realization of integral K3 period data over the
original curve and an algebraic comparison by two square substitutions.
**Assuming these two statements**, our geometric calculations yield the
following result.

**Conditional comparison theorem.** For a sufficiently general fixed
$(B,L)$ and a sufficiently general pencil in $\mathbb P H^0(B,L^2)^+$,
there is a smooth projective K3 family $S\to O$ over a nonempty open of
the original pencil. At all but countably many points $b$ of a nonempty
open, an algebraic correspondence
$$
\Gamma_b\in\operatorname{CH}^2(S_b\times X_b)_{\mathbb Q}
$$
induces a Hodge similarity
$$
T(S_b)_{\mathbb Q}\longrightarrow U_{X_b},
\qquad
(\Gamma_bx,\Gamma_by)_{X_b}=2(x,y)_{S_b}.
\tag{1.2}
$$
At very general points the integral transcendental and Néron–Severi
lattices of this realization are
$$
T(S_b)\simeq R,\qquad
\operatorname{NS}(S_b)\simeq
U_{\mathrm{hyp}}\oplus D_{10}(-1).
\tag{1.3}
$$
An additional countability argument produces algebraic comparisons with
the same rational normalization over a nonempty Zariski open of the
full ten-dimensional parameter space. This last existence statement
does not specify a simultaneous integral marking.

These generic comparison conclusions do not construct a curve or abelian source for $U_X$.
The rank-eight quadratic-locus theorem of [1] does not apply to this
rank-ten space. Neither the general rational Hodge conjecture nor
algebraicity of a Kuga–Satake tensor on this larger locus follows.

Separately, §9 proves an unconditional source statement on a
four-dimensional locus of the K3 target: its transcendental cohomology
embeds by an algebraic correspondence into the tensor product of a
fixed Gaussian CM elliptic curve and a Prym fivefold. This uses an
explicit classical isotrivial source, Hermitian-form classification
and algebraicity of rational K3 Hodge isometries. It does not depend
on the two preprint inputs. It does not produce a source for
the generic eight-dimensional K3 locus. Section 10 supplies the remaining
Gaussian tenfold by an uniruled quotient; Appendix B gives an explicit
line curve, incidence projector and comparison on a nonempty open.
Section 10 supplies the mixed determinant
classes by a specialization of Schoen's cycles. This proves ordinary HC on all mixed powers of the Gaussian K3,
elliptic curve, Prym fivefold and constructed tenfold, including the
specified even-Clifford Kuga–Satake tensors. Generalized HC follows at full-group points from fundamental sources
and at special points from the complete spin restriction. A direct Prym period calculation and odd-rank Hermitian
similitude transport the abelian statements to any Gaussian fivefold
of signature $(1,4)$, including endomorphism and Picard jumps.
The same K3 isogeny gives a Chow motive of abelian type on this locus.
Within the open set reached by the conditional comparison, the source
also transfers to the selected transcendental part of an original $X$
whenever $U_X$ admits the indicated orthogonal Gaussian action.

The enlargement method itself is not new. Proposition 3.1 of [1]
already retains a rank-sixteen anti-invariant lattice for its rank-eight
seed; its Lemma 5.2 gives a general equivariant completion, and its local
analysis supplies the two roots at an unbranched Kummer node. We extend
that framework to the type-$(1,3)$ family and determine the exact
discriminant, primitive saturation and target. The K3 moduli space in
(1.3), including a concrete double-quadric presentation, is already
described by Ma [2, §9.5 and Appendix A.1].

The rank-ten seed and its rational form already appear in
lelong_ridge [3, Remark 5.2], together with numerical evidence for
period rank eight. That paper proves a different rank-eight seed
result and explains the failure of the indicated Gauss half-sum
construction under its stated monodromy assumption. Our Appendix A
replaces the numerical period evidence with a complete argument;
§§3–4 determine the integral lattices and primitive saturation,
§9 supplies the Gaussian source and motivic conclusion, and §10 gives
the actual missing spin source, determinant transport and mixed support.
Appendix B constructs the explicit Gaussian line cylinder and identifies
it with the specified quotient invariant.
We use none of the conditional rank-eight Hodge-conjecture claims
of [3] as an input.

The finite computations supplied here check the Kummer code,
discriminant gluing, local matrices and one explicit target model.
They do not certify the external comparison theorems. The written
period-rank argument in Appendix A is included because it is needed to
choose pencils without a hidden simplicity assumption.

## 2. The actual double-Kummer cover

The normalized linearization of $L$ has positive sign at ten points
$J_{10}\subset B[2]$ and negative sign at the other six points $J_6$.
This follows from the nondegenerate Arf-zero quadratic refinement on
$B[2]\simeq\mathbb F_2^4$, equivalently from the holomorphic Lefschetz
trace $h^0(L)^+-h^0(L)^-=1$. The lift $\tau$ is the negative of this
linearization on the cover coordinate. It fixes the two points above
each element of $J_6$ and exchanges the sheets above $J_{10}$.

Let $K=\operatorname{Km}(B)$ be the resolved Kummer surface, with
exceptional curves $N_a$, $a\in B[2]$. Resolve the twelve nodes of
$X/\tau$ and blow up its ten smooth points lying over $J_{10}$. The
result $\widehat Y$ has a finite double cover
$$
p:\widehat Y\longrightarrow K
$$
branched on
$$
C+\sum_{a\in J_{10}}N_a=2A,\qquad C^2=12,\quad A^2=-2.
\tag{2.1}
$$
For a general $d$, these eleven branch components are smooth and
disjoint; $g(C)=7$. At $a\in J_6$, the curve $N_a$ lifts to two
disjoint $(-2)$-curves $E_{a,+},E_{a,-}$, exchanged by $\delta$.
At $a\in J_{10}$, the ramification curve $R_a$ has square $-1$ and
$2R_a=p^*N_a$. These assertions follow in local coordinates from the
two possible inversion lifts on $z^2=d(x,y)$; they also explain why
the ten extra blowups add only sheet-invariant cohomology.

The quotient construction gives $q(\widehat Y)=0$: on $X$,
$H^1(X,\mathbb Q)=H^1(B,\mathbb Q)$ by ample vanishing, and inversion
acts as $-1$. The two invariant holomorphic two-forms consist of the
form from $B$ and the selected odd theta form. Thus $p_g(\widehat Y)=2$.
Equation (2.1) gives
$$
e(\widehat Y)=2\cdot24-\bigl(-12+10\cdot2\bigr)=40.
$$
Consequently $b_2(\widehat Y)=38$ and the free integral second
cohomology has signature $(5,33)$.

Write
$$
M=H^2(\widehat Y,\mathbb Z)^\delta/\mathrm{torsion},
\qquad
P=H^2(\widehat Y,\mathbb Z)^{-\delta}/\mathrm{torsion},
$$
where these mean eigensublattices in free cohomology. Pullback identifies
$M_{\mathbb Q}$ with $H^2(K,\mathbb Q)(2)$, so $M$ has rank $22$ and
signature $(3,19)$; $P$ has rank $16$ and signature $(2,14)$.
Put $\beta_a=[E_{a,+}]-[E_{a,-}]$ for $a\in J_6$. They are mutually
orthogonal with square $-4$, and
$$
P_{\mathbb Q}=U_Y\perp\bigoplus_{a\in J_6}\mathbb Q\beta_a.
\tag{2.2}
$$
The resolved quotient graph induces a flat Hodge isomorphism
$\beta:U_Y\to U_X$. The degree-two projection formula gives
$$
(\beta x,\beta y)_X=2(x,y)_{\widehat Y}.
\tag{2.3}
$$
It is this particular similarity, rather than an unspecified rational
isometry of abstract quadratic spaces, that is later made algebraic.

## 3. Exact discriminant and equivariant completion

### 3.1. Saturation of the invariant lattice

Let $L_0=p^*H^2(K,\mathbb Z)\simeq\Lambda_{\mathrm{K3}}(2)$, and let
$M_0$ be generated by $L_0$ and the ten ramification classes $R_a$.
Then
$$
[M_0:L_0]=2^{10}.
\tag{3.1}
$$
Indeed, a relation among these ten classes modulo $L_0$ would make the
half-sum of a nonempty subset of the corresponding $N_a$ integral on
$K$. The primitive Kummer-lattice theorem says that its possible
nonzero supports, modulo the root lattice, are the thirty affine
hyperplanes of $\mathbb F_2^4$ and the full set of sixteen points
[4, Proposition 2.2 and Remark 2.3]. No affine hyperplane lies in the
Arf-zero set $J_{10}$: its intersection with that set has size $4$ or
$6$. Thus no relation exists.

It follows that $|\det M_0|=2^{22}/2^{20}=4$. The only possible strict
integral saturation $M_0\subset M$ would have index $2$ and make $M$
unimodular. But $P$ is even: $K_{\widehat Y}=p^*A$ is invariant, so
$xK_{\widehat Y}=0$ for $x\in P$, and Wu's parity formula gives
$x^2\equiv0\pmod2$. Primitive orthogonal complements in the
unimodular free cohomology have discriminant groups of equal order.
A unimodular $M$ would therefore make $P$ even unimodular of signature
$(2,14)$, contrary to the signature congruence modulo $8$. Hence
$$
M=M_0,\qquad |\det P|=4.
\tag{3.2}
$$

### 3.2. An elementary identification of $P$

The bilinear discriminant group of $L_0$ is
$H^2(K,\mathbb Z)/2H^2(K,\mathbb Z)$. Adjoining the ten $R_a$ takes a
bilinear-isotropic subgroup $W$ of dimension $10$, and
$A_M=W^\perp/W\simeq(\mathbb Z/2)^2$.
Every vector of $M^*$ lies in
$L_0^*=\frac12p^*H^2(K,\mathbb Z)$ and has integral square because
$H^2(K,\mathbb Z)$ is even. Thus the discriminant bilinear form of
$M$ has zero diagonal in $\mathbb Q/\mathbb Z$. Unimodular gluing
gives the same property for $P$.

Since $P$ is even, its discriminant quadratic form $q_P$ takes integral
values modulo $2\mathbb Z$. Its nondegenerate alternating bilinear
form on $(\mathbb Z/2)^2$ ensures that some order-two class $c$ has
$q_P(c)=1$. The subgroup generated by $c$ is isotropic for the
bilinear form, so adjoining it gives an integral index-two
overlattice $P'$. This lattice is odd and unimodular. The classification
of indefinite odd unimodular lattices gives
$P'\simeq I_{2,14}$ [5, Chapter 14, Corollary 1.3(ii)].
Because $P$ is even and of index two, it is exactly the even part of
$P'$. Ma's explicit computation of that even part
[2, Example A.1] gives
$$
P\simeq U_{\mathrm{hyp}}^{\oplus2}\oplus D_{12}(-1).
\tag{3.3}
$$
In particular its discriminant form is the anisotropic form on
$(\mathbb Z/2)^2$: all three nonzero values are $1$ modulo $2\mathbb Z$.
This proof avoids a general classification of indefinite 2-elementary
lattices.

### 3.3. Completion preserving the original pencil

Take $F=U_{\mathrm{hyp}}\oplus D_4(-1)$. It has signature $(1,5)$
and the same anisotropic discriminant form, which also equals its
negative. Gluing $P$ and $F$ along a discriminant anti-isometry produces
an even unimodular lattice of signature $(3,19)$, hence
$\Lambda_{\mathrm{K3}}$. Both factors embed primitively. If $e,f$
is the standard basis of the added hyperbolic plane, then
$h=e+f$ is primitive and has square $2$.

For fixed $B$, the lattice $M$ is a constant local system: it is
generated by pullbacks from the fixed Kummer surface and the ten
labeled ramification curves. Its discriminant action is therefore
trivial. Equivariance of unimodular gluing shows that monodromy of $P$
is trivial on $A_P$ too. It extends to $\Lambda_{\mathrm{K3}}$ by
the identity on $F$, fixing $h$. This construction leaves no finite
cover of the original pencil in the monodromy data.

## 4. The primitive $D_6$ and the rank-ten complement

For every pair $a,b\in J_6$ there is an affine hyperplane whose
intersection with $J_6$ is precisely $\{a,b\}$. Its other six points
lie in $J_{10}$. Pull back the associated integral Kummer half-sum
and subtract the six ramification classes. The result proves that
$p^*(N_a+N_b)/2$ is integral. Hence
$$
\frac{\beta_a+\beta_b}{2}
=E_{a,+}+E_{b,+}-\frac{p^*(N_a+N_b)}2
\tag{4.1}
$$
is integral and anti-invariant. Subtracting $\beta_b$ gives the
half-difference too.

Identify $\beta_a$ with twice a negative unit coordinate vector.
The vectors in (4.1) and their half-differences generate
$A=D_6(-1)$. Its nonzero discriminant values are
$1,\frac12,\frac12$ modulo $2\mathbb Z$. Thus it has no proper even
overlattice and is primitive in $P$.

Put $R=A^\perp\subset P$ and $k=[P:A\perp R]$. Since
$|\det P|=|\det A|=4$, the determinant relation gives
$|\det R|=k^2$. Primitive gluing makes $k$ divide $4$.
Moreover $\beta_a/2\in P^*$: for $x\in P$,
$$
(\beta_a,x)=2(E_{a,+},x).
$$
Its class in $A^*/A$ is the vector class $v$, with $q_A(v)=1$.
The gluing subgroup projected to $A^*/A$ therefore lies in
$v^\perp=\{0,v\}$. Thus $k\leq2$. The case $k=1$ would make $R$
even unimodular of signature $(2,8)$, which is impossible. Consequently
$$
k=2,\qquad |\det R|=4.
\tag{4.2}
$$

The nonzero glue class has projections $v$ and $c\in A_R$, with
$c$ of order two and $q_R(c)=-q_A(v)=1$ modulo $2\mathbb Z$.
Adjoin $c$ to $R$. As in §3.2, this gives an odd unimodular lattice
$R'\simeq I_{2,8}$ whose even part is $R$. Therefore
$$
R\simeq U_{\mathrm{hyp}}^{\oplus2}\oplus D_6(-1),
\qquad
R_{\mathbb Q}\simeq U_{\mathrm{hyp},\mathbb Q}^{\oplus2}
                 \perp\langle-1\rangle^{\oplus6}.
\tag{4.3}
$$

The full family has period rank eight by Appendix A. At a very
general point $R_{\mathbb Q}=U_Y$ has no rational $(1,1)$-class;
therefore $A$ is the entire algebraic saturation inside $P$.
The primitive embedding $R\subset\Lambda_{\mathrm{K3}}$ has complement
of signature $(1,11)$ with discriminant form $-q_R$. That form
again has an order-two class of value $1$. Its odd unimodular
extension is $I_{1,11}$, so its even part is
$U_{\mathrm{hyp}}\oplus D_{10}(-1)$. This proves the claimed integral
target in the explicit pencil realization.

These are abstract lattice isometries. We do not claim that six
vanishing vectors transported from different singular fibers are
pairwise orthogonal: their intersections depend on the transport
paths, and the local root calculation alone does not determine them.

## 5. Boundary of a general pencil and its Hodge line

The invariant system $H^0(B,2L)^+$ has dimension eight. Its morphism
realizes the singular Kummer surface as a genus-seven model in
$\mathbb P^7$, with precisely the sixteen Kummer nodes contracted
[4, §5.3]. A general hyperplane pencil has only ordinary tangencies
in the smooth locus and transverse passages through individual nodes.
The codimension-two locus of multiple contacts, higher contacts,
nontransverse node passages and simultaneous events is avoided.
This is the usual characteristic-zero Lefschetz-pencil assertion,
applied on the smooth locus and separately to the sixteen node
hyperplanes. A leading member avoiding these loci is chosen at infinity.

There are three finite local models.

**An ordinary contact away from the Kummer nodes.** The local double
surface is $z^2=xy+t$. Its vanishing cycle is an integral $(-2)$-root
in $P$, and monodromy is its Picard–Lefschetz reflection.

**A passage through $a\in J_6$.** Upstairs, equivariant Morse
coordinates give
$$
z^2=x^2+y^2+t,\qquad
\tau(x,y,z)=(-x,-y,z),\qquad \delta(z)=-z.
\tag{5.1}
$$
On a Kummer-resolution chart $a=x^2$, $v=y/x$, the double surface
becomes $z^2=a(1+v^2)+t$. At $t=0$ the branch is $a=0$ together
with a residual curve meeting it at $v=\pm i$. There are two
disjoint ordinary double points, giving two orthogonal integral
anti-invariant roots $r_1,r_2$ in the nearby smooth $\widehat Y$.
Local monodromy is $s_{r_1}s_{r_2}$.

Equivalently the unresolved quotient has the $A_3$ model
$uv=(w^2-t)^2$. In a negative $A_3$ basis
$\alpha_1,\alpha_2,\alpha_3$, with adjacent pairing $1$, one may take
$$
r_1=\alpha_1+\alpha_2,\qquad
r_2=\alpha_2+\alpha_3.
$$
The involution and monodromy on this local lattice are
$$
\alpha_1\longleftrightarrow\alpha_3,\qquad
\alpha_2\longmapsto-\alpha_1-\alpha_2-\alpha_3.
\tag{5.2}
$$
The persistent difference $r_1-r_2$ and transverse vector $r_1+r_2$
each have square $-4$. Keeping only their transverse projection hides
the integral roots needed for the K3 Weyl condition.

To fix the signs geometrically, $\tau$ in (5.1) preserves the
orientation of the real vanishing sphere and $\delta$ reverses it;
$\delta$ also exchanges the two persistent exceptional curves.
The action of $w\mapsto-w$ on the minimal resolution of the central
surface $uv=w^4$ is not a substitute for this nearby calculation:
that action need not lift to the chosen simultaneous resolution
over the parameter, and the Weyl correction changes the marking.

**A passage through $a\in J_{10}$.** Here
$\tau(x,y,z)=(-x,-y,-z)$ in the upstairs quadratic model.
It acts antipodally on the vanishing sphere, so no vanishing class
survives in the $\tau$-invariant cohomology. The quotient smoothing
at the minimal surface is the rational-homology-ball smoothing of
$\frac14(1,1)$. Its added ramification curve is sheet-invariant.
For example, the Kummer chart has equation
$$
\eta^2=a\bigl(a(1+v^2)+t\bigr),\qquad \eta=xz.
$$
Thus the monodromy on $U_Y$ is the identity; the six exceptional
difference lines are unaffected as well. The action on $P$ is the
identity. We do not describe this event as an ADE degeneration of
the double-Kummer model.

These local calculations determine the entire monodromy operator:
outside the disjoint node neighborhoods the original family is
topologically trivial, and (2.2) accounts for all additional
anti-invariant resolution classes. The nontrivial finite operators
are therefore the stated products of integral root reflections.

There are $28$ ordinary contacts, six $J_6$ passages and ten
$J_{10}$ passages. For the first count, the pencil on $K$ has twelve
base points, and the blown-up total surface has Euler number $36$.
The smooth genus-seven fibers contribute $2(-12)=-24$.
At each of the sixteen node passages, the fiber contains a rational
bridge meeting the residual curve in two points; its Euler defect
is $2$. The remaining defect is
$$
36-(-24)-16\cdot2=28.
\tag{5.3}
$$
The count is a consistency check as well as a complete boundary list.

At infinity the monodromy on $P$ is $\delta=-1$; it extends as the
identity on $F$. Its square is the identity on the completed K3
lattice. At finite singular parameters the roots are orthogonal to
$h$ and to the limiting period.

For completeness, interior extension of the period is checked
upstairs. After the quadratic local cover $t=v^2$, an $A_1$
smoothing can be written
$xy=(z-v)(z+v)$. Blowing up $(x,z-v)$ gives charts
$$
z=v+xa,\quad y=a(2v+xa),
\qquad
z=by-v,\quad x=b(by-2v),
$$
which are smooth over $v$. The relative two-form $dx\wedge dz/x$
becomes $dx\wedge da$ in the first chart and $db\wedge dy$ in the
second. Thus the selected theta two-form extends crepantly. It
remains a nonzero global form on the central resolved surface
because it is nonzero away from the node neighborhoods. Its
polarized norm stays positive. The same upstairs invariant
limiting class treats the noncanonical quotient event. The
exceptional difference classes remain algebraic, so (2.2) transfers
this interior period to $P$ and to its K3 completion.

Now make the square substitution $t=s+u^2$, with $s$ away from the
finite boundary. The upstairs compactification over $\mathbb P^1_u$
uses the line bundle $L\boxtimes\mathcal O(1)$ and branch section
$$
(d_0+sd_1)u_0^2+d_1u_1^2.
$$
The selected anti-sheet relative canonical direct image is
$H^0(B,L)^-\otimes\mathcal O(1)$, a line bundle $\mathcal O(1)$.
Explicitly, for $v=1/u$ and $w=z/u$,
$$
w^2=d_1+(d_0+sd_1)v^2,\qquad
\frac{\theta\,dx\wedge dy}{z}
=v\frac{\theta\,dx\wedge dy}{w}.
\tag{5.4}
$$
There are no additional zeros or poles at finite parameters by the
crepant calculation. Finite monodromies fix the nonzero limiting
form, so invariant-frame descent adds no fractional character.
Thus the prescribed extended Hodge line of the completed period
data is exactly $\mathcal O_{\mathbb P^1}(1)$.

## 6. Application of the two external comparison inputs

We specify the imported assertions to separate them from our
geometric work. Lemma 5.5 of [1] takes an integral K3 local system,
a primitive positive flat class, and a positive isotropic
holomorphic period line with finite local monodromy and interior
limits. It realizes these data by polarized ADE K3 surfaces over
the original curve. When the local monodromies are in the Weyl
groups of the limiting perpendicular roots, its extended model
has relative dualizing sheaf pulled back from the Hodge line.
Its final assertion supplies a smooth K3 family on an open of
the original curve, preserving a rational Hodge summand
perpendicular to every persistent root. The period-stack
contraction input is related to the published construction in
Di Lorenzo–Fringuelli–Vistoli [6, Lemma 1.12]; [6] does not itself
assert our comparison.

Appendix A and the ordered choice of a very general abelian
surface followed by a sufficiently general line allow the pencil
to avoid containment in every proper rational Hodge locus.
Consequently $U_Y$ is simple and has no rational $(1,1)$-class
except at countably many parameters. Every locally flat root
that remains algebraic on an open has zero projection to $U_Y$:
its projection would otherwise be a persistent rational
$(1,1)$-class. Analytic continuation and a very general point
force that projection to vanish.

The hypotheses of Lemma 5.5 are now supplied by §§3–5. We obtain
a smooth K3 family $S\to O$ on the original pencil, with an
isometric copy of $U_Y$. Its generic integral transcendental
lattice is $R$, and its generic Picard lattice is (1.3).

For every admissible $s$, square substitution kills the
monodromy at infinity. At the remaining punctures the
monodromies are products of roots perpendicular to the limiting
period and $h$. The same lemma gives an ADE K3 model over
$\mathbb P^1_u$ with
$$
K_Z=f^*\mathcal O_{\mathbb P^1}(-1),
\tag{6.1}
$$
by (5.4). On a smooth projective resolution, intersection of the
canonical class with the square of an ample pullback is negative:
exceptional divisors have zero such intersection because their
images have codimension at least two. Thus the canonical class
is not pseudo-effective, and the uniruledness criterion of
Boucksom–Demailly–Păun–Peternell [7] applies.

It is necessary that this is the square pullback of the fixed
$S$-family, rather than a different family with equal periods.
Use the finite Galois cover chosen in the first application of
Lemma 5.5 and normalize its fiber product with
$t=s+u^2$. On this common cover the extended period maps and
their equivariance data are literal pullbacks. Functoriality of
the period-stack contraction and fully faithful GAGA identify
the algebraic ADE families and their group actions.
Effective Galois descent then identifies their generic fibers.
The generic resolution of the original contraction remains
birational after base change. Hence the uniruledness in (6.1)
belongs to a model of the actual square-pulled-back $S$-family.
This is the common-cover argument of [1, §5.5], applied to our
completed local system.

Theorem 6.1 of [1] compares such a K3 family with a surface family
carrying commuting involutions whose projector annihilates
cohomology outside degree two. It requires a specified rational
Hodge isomorphism, nonconstant simple K3-type variation, and
uniruled threefold models after the square substitutions. It
allows a similarity rather than an isometry.

For $X\to O$ take the literal algebraic projector
$$
e_X=\frac14(1+\tau)(1-\delta).
\tag{6.2}
$$
The involution $\delta$ fixes $H^0,H^4$ and
$H^1(X)=H^1(B)$; it also fixes $H^3$ by duality. Thus (6.2)
annihilates every degree except two and has image $U_X$ there.
After $t=s+u^2$, the total space
$z^2=d_0+(s+u^2)d_1$ is birational to a conic bundle over $B$.
A smooth general conic over $\mathbb C$ is rational, and these
fibers cover the total space; a smooth projective model is
therefore uniruled.

All hypotheses of Theorem 6.1 are satisfied with the specified
map $\beta$ in (2.3). It gives a codimension-two correspondence
inducing exactly $\beta_b$ at all but countably many points of a
nonempty open. Equations (1.2)–(1.3) follow. This is an application
of the two named preprint results, not an independent proof of
their global descent or cylinder arguments.

## 7. A full-base existence statement by countable spread

We explain the quantifier improvement separately, because a
pencil chosen through a prescribed parameter would not by itself
control the exceptional countable set produced afterward.

Take an irreducible fine level parameter space $\mathcal B$ for
the smooth type-$(1,3)$ family. It has dimension $3+7=10$.
The cover and its involutions give the relative projector $e_X$.
Enumerate polarized K3 surfaces by their countably many Hilbert
schemes. Enumerate rational codimension-two cycles on their
products with $X$ using relative Hilbert schemes, finite products
and rational coefficient lists. Smooth connected stratifications
give countably many finite-type strata, each carrying an actual
K3 family and an actual cycle $\Gamma$.

On each stratum impose the equalities of flat cohomological maps
$$
e_X\Gamma=\Gamma,\qquad
\Gamma\Gamma^t=2e_X
\quad\text{on }H^2(X,\mathbb Q).
\tag{7.1}
$$
Flatness means that a connected stratum satisfies (7.1)
everywhere or nowhere. Call the former strata successful.
Their images in $\mathcal B$ are constructible.

The very general fixed-pencil comparisons give successful
points. To see this with no extra algebraic component of the
cycle, project the source onto its transcendental cohomology
using the usual algebraic projectors for $H^0,H^4$ and the
Néron–Severi lattice, and project the target by $e_X$. The
resulting map is the similarity $\beta$ on the whole
transcendental part and zero on its orthogonal complement.
Equation (2.3) gives $\beta\beta^t=2e_X$, establishing (7.1).

Suppose all successful images were nondense. Their closures,
together with the proper Hodge loci needed for simplicity,
would be countably many proper algebraic subsets of $\mathcal B$.
First choose very general $(B,L)$ whose branch fiber is not
contained in any of them. Such choices exist: a proper closed
subset cannot contain the generic full branch fiber of the
irreducible total parameter space. Then choose a line in that
fiber contained in none of these subsets and satisfying §5.
Each proper closure meets this line in finitely many points.
Their union covers at most countably many points of the line.
But §6 supplies successful comparisons at all but countably
many points of an open in that line, an uncountable set.
This is a contradiction.

Thus one successful stratum dominates $\mathcal B$. Its
constructible image contains a nonempty Zariski open
$\mathcal B^0$. At every point of $\mathcal B^0$, (7.1) supplies
an algebraic embedding of $U_X$ into K3 cohomology with the
corresponding normalization. Its transpose has multiplier $2$:
$$
(\Gamma^t x,\Gamma^t y)_S=2(x,y)_X
\quad (x,y\in U_X).
$$
The image contains the holomorphic two-form. Its Hodge
orthogonal complement has only type $(1,1)$, so the K3
transcendental part is exactly the nonalgebraic part of this
image. At a very general parameter it has rank ten and the K3
has Picard number twelve.

No simultaneous K3 family on all of $\mathcal B^0$ was assumed.
A finite generically étale cover supporting one can be obtained
after this final spread, by taking a closed point of the generic
fiber of a successful stratum and shrinking the base. This
final cover is different from leaving an unspecified cover in
the original-pencil argument, where uniruledness could be lost.
The spread preserves rational cohomological data; it does not
by itself retain the exact integral lattice marking of §4.

## 8. The known target and a reproducible geometric control

Ma identifies the locus in (1.3) as the two-elementary K3 moduli
space $\mathcal M_{12,2,1}$ [2, §9.5, Appendix A.1].
Let $Q=\mathbb P^1\times\mathbb P^1$. Take a smooth $(3,3)$
curve $C$ with a point $p$ at which one ruling fiber $F_1$ is
triply tangent and the other ruling fiber $F_2$ is transverse.
The resolved double cover branched on $C+F_1+F_2$ is a K3;
the branch has a $D_8$ singularity at $p$ and two further
nodes. Ma proves that this presentation is birational to the
eight-dimensional moduli space. Its genus-four fixed curve
has a totally ramified trigonal point.

For a concrete check, write
$$
\begin{aligned}
C(x,y)&=y^3+xP_1(y)+x^2P_2(y)+x^3P_3(y),\\
P_1(y)&=1+2y+3y^2+5y^3,\\
P_2(y)&=7+11y+13y^2+17y^3,\\
P_3(y)&=19+23y+29y^2+31y^3,
\end{aligned}
$$
and take $w^2=xyC(x,y)$. The attached checker verifies
smoothness of the projective $(3,3)$ curve on all four
quadric charts modulo $1009$, which implies smoothness in
characteristic zero. It checks the total tangency and the
two additional simple intersections. Binary-quartic
invariants for the $x$-ruling give
$$
II^*+2I_2+10I_1,
$$
with smooth infinity and visible lattice
$U_{\mathrm{hyp}}\oplus E_8(-1)\oplus A_1(-1)^{\oplus2}$.
The $j$-invariant is nonconstant.

The second ruling makes the $D_{10}$ lattice visible
directly. The substitution, suggested by gabrielov_cell,
$$
\widehat X=y^4/x,\qquad \widehat W=y^4w/x^2
$$
gives the exact equation
$$
\widehat W^2=\widehat X^3+yP_1(y)\widehat X^2+
             y^5P_2(y)\widehat X+y^9P_3(y).
\tag{8.1}
$$
The checker verifies (8.1), orders $(2,3,12)$ of
$c_4,c_6,\Delta$ at $y=0$, and a squarefree residual
discriminant polynomial of degree twelve. Thus this ruling
has $I_6^*+12I_1$ and trivial lattice
$U_{\mathrm{hyp}}\oplus D_{10}(-1)$.

For this explicit sample the computations prove only
$\rho\geq12$; they do not establish a Picard-rank upper
bound or Mordell–Weil rank zero. For a very general member
those conclusions follow from Ma's moduli theorem, and
the lack of a proper even overlattice of the trivial
lattice excludes torsion. The presentation and the
moduli theorem are known results, not our new K3 family.
No correspondence from this model to an abelian variety
is obtained from these elliptic-fiber calculations alone.

## 9. An algebraic source on the Gaussian locus

The source family used here is classical: its elliptic model is
Artebani–Sarti's Example 8.2(i) [8]. The point of the following
argument is to transport that source to the specified rational
quadratic space and then to an order-four locus in Ma's model.

**Proposition 9.1.** Let $Z$ be a projective K3 surface. Suppose
$V\subset H^2(Z,\mathbb Q)$ is a nondegenerate rational Hodge
subspace containing $H^{2,0}(Z)$, with quadratic form
$$
V\simeq R_{\mathbb Q}
=U_{\mathrm{hyp},\mathbb Q}^{\oplus2}\perp\langle-1\rangle^{\oplus6}.
$$
Suppose $V$ has an orthogonal rational Hodge endomorphism $J$ with
$J^2=-1$. Then an algebraic correspondence embeds $T(Z)_{\mathbb Q}$
into
$$
H^1(E_i,\mathbb Q)\otimes H^1(P,\mathbb Q),
\tag{9.1}
$$
where $E_i$ has complex multiplication by $\mathbb Q(i)$ and $P$
is a five-dimensional Prym of a genus-seven cyclic-four curve over
a genus-two curve. In particular (9.1) embeds all of $V$ when
$V=T(Z)_{\mathbb Q}$. No assertion about embedding every additional
Tate class of $V$ into this particular tensor is needed.

### 9.1. The Hermitian form, with its local invariants

Let $q$ be the cup-product form on $V$, and let $K=\mathbb Q(i)$
act through $J$. Replace $J$ by $-J$ if necessary so that it acts
by $i$ on $H^{2,0}$, using the same convention for the source below. Then
$$
h(x,y)=q(x,y)+i\,q(x,Jy)
$$
is a $K$-Hermitian form, linear in the first argument, of
signature $(1,4)$. Its real part is $q$.
Diagonalize $h=\langle a_1,\ldots,a_5\rangle$ over $K$, with
$a_j\in\mathbb Q^\times$. The underlying rational form is the
orthogonal sum of $\langle a_j,a_j\rangle$. Its Hasse invariant
at every rational place is therefore
$$
\epsilon(q)=\prod_{j=1}^5(a_j,a_j)
            =\left(\prod_{j=1}^5a_j,-1\right).
\tag{9.2}
$$
Here $(\ ,\ )$ denotes the Hilbert symbol; cross terms between
two repeated pairs occur four times and cancel.

But $R_{\mathbb Q}\simeq I_{2,8}$, whose Hasse invariant is
$(-1,-1)^{28}=1$. Thus the determinant of $h$ is a norm from
$\mathbb Q(i)$ at every place and hence globally. Equivalently,
the associated quaternion algebra splits, or the conic
$x^2+y^2=d z^2$ has a rational point for
$d=\prod a_j$. Landherr's classification [9], also stated in
[10, Theorem 70], now gives
$$
h\simeq\langle1,-1,-1,-1,-1\rangle.
\tag{9.3}
$$
Thus all the orthogonal $K$-structures in this proposition have
the same Hermitian form. Matching the rational quadratic
discriminants alone would not have justified this step.

### 9.2. A classical product source with an open period image

Choose seven distinct points $a,b_1,\ldots,b_6$ of $\mathbb P^1$,
and put
$$
f(t)=(t-a)^2\prod_{j=1}^6(t-b_j),\qquad
G:\ v^4=f(t).
$$
The normalization of $G$ has genus seven: the six simple branch
points have ramification defect three and the double point
has defect two, so $2g(G)-2=-8+6\cdot3+2=12$.
The quotient by $v\mapsto-v$ is the genus-two curve
$$
D:\ s^2=\prod_{j=1}^6(t-b_j),\qquad s=v^2/(t-a).
$$
Its Prym $P=\operatorname{Prym}(G/D)$ has dimension five.
The holomorphic eigenspace dimensions for the three
nontrivial fourth-root characters are $1,2,4$, up to
reversing the primitive characters. Thus the primitive
Gaussian part has signature $(1,4)$.

Take $E_i:z^2=u^3+u$. There is an actual rational map of degree four
$$
E_i\times G\dashrightarrow S_f,\qquad
(u,z;t,v)\longmapsto(v^2u,v^3z,t),
\tag{9.4}
$$
where
$$
S_f:\quad Y^2=X^3+f(t)X.
$$
The diagonal order-four action
$(u,z,v)\mapsto(-u,iz,iv)$ fixes the coordinates in (9.4).
The minimal resolution is an elliptic K3 with fibers
$6III+I_0^*$ [8, Example 8.2(i)] and order-four automorphism
$(X,Y,t)\mapsto(-X,iY,t)$.

The visible trivial lattice has rank twelve and rational form
$$
U_{\mathrm{hyp},\mathbb Q}\perp
\langle-2\rangle^{\oplus6}\perp D_4(-1)_{\mathbb Q}
\simeq U_{\mathrm{hyp},\mathbb Q}\perp\langle-1\rangle^{\oplus10}.
\tag{9.5}
$$
For the last equality, $D_4(-1)_{\mathbb Q}\simeq\langle-1\rangle^4$
and each pair $\langle-2,-2\rangle$ is rationally isometric to
$\langle-1,-1\rangle$. Fiber inversion, the square of the
order-four automorphism, fixes this trivial lattice pointwise:
it fixes the components of each $III$ and $I_0^*$ fiber and the
zero section. Its orthogonal complement is anti-invariant.
More precisely, modulo the fiber, section and nonidentity
fiber components, the elliptic Leray decomposition identifies
$H^2$ with $H^1(\mathbb P^1,j_*R^1\pi_*\mathbb Q)$, where $j$
includes the smooth-fiber locus. Inversion is $-1$ on $R^1\pi_*$
and therefore on this quotient. Since it fixes the nondegenerate
trivial lattice pointwise, its orthogonal complement is the
negative eigenspace, including at Picard jumps.
Thus the rank-ten complement carries an orthogonal
$\mathbb Q(i)$-structure.

The moduli of seven points with the double point distinguished
has dimension four. An elliptic isomorphism recovers this
configuration from its singular fibers, up to an automorphism
of $\mathbb P^1$. The map to elliptic K3 moduli is therefore
generically finite. Forgetting the fibration still has
zero-dimensional fibers: a fixed K3 has countably many
Néron–Severi classes and hence countably many choices of fiber
and section classes, and each fiber class determines its
pencil. A finite-type fiber with countably many complex points
has dimension zero. After choosing a local polarization or
level chart these are ordinary algebraic moduli maps.

The unmarked K3 image consequently has dimension four.
K3 local Torelli and the Gaussian period-domain bound show
that the generic transcendental rank is exactly ten and
the period image is open in its four-dimensional complex ball.
Witt cancellation of (9.5) inside
$\Lambda_{\mathrm{K3},\mathbb Q}
=U_{\mathrm{hyp},\mathbb Q}^{\oplus3}\perp\langle-1\rangle^{16}$
gives $R_{\mathbb Q}$ for the rank-ten complement.
Its form stays constant on a local parameter chart, even at
Picard jumps. By §9.1 its Hermitian form is (9.3).

### 9.3. Exact period transport and algebraic correspondences

The rational points of the group $U(h)$ are dense in
$U(1,4)(\mathbb R)$. To see this without a density assumption
on arithmetic subgroups, use the rational Cayley map
$$
A\longmapsto (1-A)(1+A)^{-1}
$$
on the rational vector space of skew-Hermitian endomorphisms.
It parametrizes a neighborhood of the identity. Rational
points in that vector space are real-dense; the closure of
$U(h)(\mathbb Q)$ therefore contains a neighborhood of the
identity. The real group $U(1,4)$ is connected, so the
closure is the whole group.

Choose one simply connected source chart with a fixed
marking whose period image contains a nonempty ball-open
set $\mathcal O$. Given the period $p$ of $V$, transitivity
of the real unitary action and rational density give
$g\in U(h)(\mathbb Q)$ with $g(p)\in\mathcal O$.
There is an actual source $S_f$ with this period.
This produces an exact rational Hodge isometry from $V$
to the rank-ten complement $V_f$ in $H^2(S_f,\mathbb Q)$.
Density is used to enter an open set, not to infer an
algebraic cycle by taking a limit. The resulting correspondence is
existential: this argument does not give an algorithm recovering the
seven branch points from an arbitrary input K3 surface.

Witt's extension theorem extends this isometry to the
full rational K3 cohomology. Both complementary spaces
have only type $(1,1)$, since $V$ and $V_f$ contain the
holomorphic two-forms, so the extension is a Hodge
isometry. Buskin's theorem [11, Theorem 1.1] makes it an
algebraic correspondence.

Resolve (9.4) as $f:\widetilde{E_i\times G}\to S_f$ with
blowdown $p:\widetilde{E_i\times G}\to E_i\times G$.
Pullback is injective because $f_*f^*=4\operatorname{id}$
on $H^2(S_f,\mathbb Q)$. The exceptional summands of the
blowup are Tate, so projection by $p_*$ loses no
transcendental class. The transcendental
part maps into the $H^1(E_i)\otimes H^1(G)$ Künneth
summand: a morphism from the transcendental K3 Hodge
structure to any Tate summand is zero. This follows from
polarizability and the absence of rational $(1,1)$-classes
in $T(S_f)$. Moreover the
involution $v\mapsto-v$ on $G$ lifts fiber inversion
on $S_f$, so the image lies in
$H^1(E_i)\otimes H^1(G)^-$.
The algebraic projector $(1-\tau^2)/2$ on $G$ has
image $H^1(P)$. The usual Poincaré and isogeny
correspondences identify this with the Prym factor
algebraically. This proves (9.1), including at Picard
jumps by restriction to the transcendental summand.

### 9.4. The stronger motivic consequence

**Corollary 9.2.** Under the hypotheses of Proposition 9.1, the rational
Chow motive of $Z$ is of abelian type and is finite-dimensional in
Kimura's sense.

Here ``of abelian type'' means belonging to the pseudoabelian tensor
subcategory of rational Chow motives generated by motives of curves.
The full rational Hodge isometry constructed in §9.3 is an isogeny
of K3 surfaces. Huybrechts's motivic isogeny theorem
[12, Theorem 0.2] lifts it to an isomorphism
$h(Z)\simeq h(S_f)$ of rational Chow motives.
For the resolved product map, the graph identity
$\Gamma_f\circ{}^t\Gamma_f=4\Delta_{S_f}$ holds in the Chow group.
Thus $h(S_f)$ is a direct summand of
$h(\widetilde{E_i\times G})$. The latter is a sum of the product
motive $h(E_i)\otimes h(G)$ and Tate motives from point blowups.
It is of abelian type and Kimura finite-dimensional; these properties
pass to direct summands [13]. This proves the corollary.
The use of [12] is essential here: an arbitrary cohomological injection
alone would not justify the Chow-motive conclusion.

### 9.5. A four-dimensional locus in Ma's family

In the quadric model impose
$$
C=y^3+x(a_0+a_2y^2)+x^2(b_1y+b_3y^3)+x^3(c_0+c_2y^2).
\tag{9.6}
$$
Then
$\sigma(x,y,w)=(-x,-y,iw)$ satisfies $\sigma^2=\delta$
and acts by $-i$ on $dx\wedge dy/w$.
The smooth locus of these curves is nonempty: the
sample $(a_0,a_2,b_1,b_3,c_0,c_2)=(1,2,3,5,7,11)$
is smooth on all four quadric charts modulo $1009$.
It also has a unique totally ramified point for the
$x$-trigonal map. Writing the cubic in $y$ as
$a y^3+b y^2+c y+d$, the common triple-root conditions
$3ac-b^2=27a^2d-b^3=0$ have greatest common divisor
$x$ in this sample; the leading-coefficient and
infinity cases give no additional triple root.
Thus the hypotheses of Ma's open-immersion
presentation [2, Proposition A.5] hold on a
nonempty open.

There are seven coefficients before projectivization,
including the coefficient of $y^3$. The continuous
quadric automorphisms preserving the selected point,
rulings and simultaneous-sign involution are the two
coordinate scalings. Their quotient has dimension four.
Forgetting the selected involution has finite fibers,
because a smooth genus-four curve has finite
automorphism group. Ma's presentation therefore gives
a four-dimensional K3 moduli image.
The anti-invariant lattice is the same rank-ten $R$
as before, and $\sigma$ squares to $-1$ on it.
Its ball has dimension four, so the generic
transcendental rank remains ten and the generic
Picard number is twelve.
Proposition 9.1 supplies the algebraic source for its
transcendental cohomology throughout this smooth open,
also when the Picard rank increases. Corollary 9.2 supplies
the corresponding statement for its Chow motive.

The scope should be distinguished from the following
already known or unproved assertions. Ordinary HC
for powers of a K3 whose transcendental endomorphism
field is CM is already known [14, Corollary 2.3].
The isotrivial family and the classification of
order-four automorphisms are classical [8].
Hermitian classification in geometric source
constructions also has precedents in Paranjape [15]
and Ingalls–Logan–Patashnick [16], in different
orthogonal and unitary ranks.
The half-twist formalism and the product-source method
are classical; van Geemen [17, Theorem 3.10 and Example 3.11]
treats the related Eisenstein isotrivial family. An especially close
Gaussian precedent is Nori's domination of cyclic quartic K3 surfaces
by an elliptic curve times a curve, recorded in van Geemen
[18, Example 11.3]. There a tangent-line construction gives an
isotrivial family of Gaussian elliptic curves, which becomes a product
after finite base change. Our result instead specifies transport for the
rank-ten rational form and Ma sublocus, with the particular genus-seven
cyclic source and its Prym fivefold. It is not a new general
elliptic-product domination or half-twist principle.
We do not identify a prescribed full even-Clifford
Kuga–Satake tensor from (9.1), and we make no
generalized-HC claim from that embedding alone. The additional
geometric constructions of Section10 supply ordinary HC and the specified
KS tensors and generalized HC, including at smooth Gaussian special points.
Finally, the Gaussian sublocus has dimension four;
it does not supply a source at a generic point of
the eight-dimensional orthogonal period domain.

### 9.6. Conditional transfer to the original selected cohomology

Let $\mathcal B^0$ be the nonempty open furnished by §7, under the
two comparison assumptions. Suppose $X=X_b$, $b\in\mathcal B^0$,
and $U_X$ admits an orthogonal rational Hodge endomorphism
$J$ with $J^2=-1$. Choose the actual K3 surface and correspondence
of (7.1), and put $V=\operatorname{im}(\Gamma^t|_{U_X})$.
This is a nondegenerate rank-ten Hodge subspace containing the
K3 holomorphic two-form. The quadratic normalization is
$$
q_V\simeq 2q_{U_X}\simeq4q_R\simeq q_R.
$$
The second equality uses the actual quotient similarity (2.3);
the last is the rational rescaling by $1/2$.
The Gaussian action transports to $V$ by $\Gamma^t$, with inverse
$\Gamma/2$ on its image. Proposition 9.1 therefore applies to this
particular comparison K3, without requiring an integral marking
or identifying it with a member of Ma's displayed chart.

Write $U_X^{\mathrm{tr}}$ for the orthogonal complement in $U_X$ of
its rational $(1,1)$-classes. Its image under $\Gamma^t$ is precisely
the K3 transcendental part. Composition with the correspondence in
Proposition 9.1 yields an algebraic embedding
$$
U_X^{\mathrm{tr}}\hookrightarrow
H^1(E_i,\mathbb Q)\otimes H^1(P,\mathbb Q).
$$
The selected projector and the divisor projectors are algebraic,
so the statement concerns actual correspondences on $X$ as well.
This conclusion is conditional on §7 and applies within
$\mathcal B^0$; it asserts no comparison at an excluded special
fiber and no source for the generic orthogonal variation.

These conditional applications are nonvacuous. On a marked local
chart where the original period map has rank eight, fix one
orthogonal $J_0^2=-1$ on $R_{\mathbb Q}$, using its model $I_{2,8}$.
The positive $J_0$-complex lines form a four-ball inside the
orthogonal period domain. The rational orthogonal group is dense
in the identity component of its real group, by the same Cayley
argument as in §9.3, and that real component acts transitively on
the connected orthogonal period domain. Consequently the union
of rational translates of this four-ball is analytically dense.
Every translate has an orthogonal rational Gaussian action.
Pulling back by the submersive period map gives an analytically
dense countable union of local six-dimensional loci to which the
conditional source statement applies. Density supplies no
correspondence at a generic orthogonal period.

## 10. Actual Gaussian spin sources and mixed Hodge conjectures

This section is independent of the two comparison inputs in §6. We use
$E=E_i$ and the cyclic curve $G$ of §9.2, with primitive Prym $P$.
Write $W=H^1(P,\mathbb Q)$. Choose its complex Gaussian character so
$W_\sigma$ has Hodge numbers $(1,4)$, and let $e=H^{1,0}(E)$.
Thus the selected K3 space is
$$
V_\mathbb C=(e\otimes W_\sigma)\oplus
(\bar e\otimes W_{\bar\sigma}).
\tag{10.1}
$$
The involution, character and alternating projectors used below are
rational algebraic correspondences. A displayed complex eigenspace
always denotes one constituent of its indicated rational conjugate pair.

**Theorem 10.1 (Gaussian mixed support).** There exists an abelian
variety $A$ of dimension ten and an algebraic correspondence inducing
$$
H^1(A,\mathbb Q)(-1)\simeq
\left(e\otimes\bigwedge^2 W_\sigma\right)
\oplus\left(\bar e\otimes\bigwedge^2 W_{\bar\sigma}\right)
\subset H^3(E\times P\times P,\mathbb Q).
\tag{10.2}
$$
The mixed determinant space
$$
\left(\bigwedge^5 W_\sigma\otimes e^{\otimes3}\right)
\oplus\left(\bigwedge^5 W_{\bar\sigma}\otimes\bar e^{\otimes3}\right)
\subset H^8(P\times E^3,\mathbb C)
\tag{10.3}
$$
is the complexification of a rational space of algebraic cycle classes.
Both statements hold for every smooth branch configuration of §9.2.

Put $C=E\times P\times A$. The ordinary rational Hodge conjecture
holds for every $S_f^r\times C^n$, at every smooth branch configuration.
The same holds after adjoining powers of the actual Kuga–Satake variety,
and every specified rational tensor defining its usual Kuga–Satake
correspondence is algebraic. These statements transfer to the K3 surfaces
of Proposition 9.1 through their rational Hodge-isometric source,
including at Picard jumps.

The generalized rational Hodge conjecture holds for all these mixed
products as well, including at Picard and endomorphism jumps.
Here generalized HC means that every rational Hodge substructure of
$H^k$ of level at most $k-2p$ is supported in codimension at least $p$.
We first prove the full-group case in §§10.5–10.7; Section10.8 extends
ordinary HC and the prescribed KS class, and Section10.9 calculates
actual support for every smaller Gaussian Hodge group.

**Corollary 10.2 (arbitrary Gaussian fivefolds).** Every complex abelian
fivefold $B$ with a polarization-compatible Gaussian action of signature
$(1,4)$ is Gaussian-linearly isogenous to a Prym $P$ above. Thus (10.2)
and (10.3) transport to $B$. Ordinary and generalized HC hold for all powers of $E\times B\times A$
and for the associated Gaussian K3 mixed products, without a full-group
hypothesis.

The abstract spin decomposition and half twists are due to van Geemen
[17, Proposition 3.8 and Theorem 3.10]. Our additional geometric steps
are a symmetric-square quotient giving actual support for its tenfold,
and a compact-type specialization of Schoen's existing determinant
cycles [18, Theorem 2.0 and Corollary 3.1]. Related hypersurface half-twist
constructions appear in van Geemen–Izadi [19, §§4–5]. The fully twisted
source strategy belongs to the work of Abdulali [20, Theorem 10]; we
supply the connecting cycles in this case. His later survey [21,
Appendix B.5] corrects the 2004 type-IV theorems by requiring
$|p-q|=1$. Its Appendix A(4) lists fivefold signature $(3,2)$.
Consequently those results do not by themselves imply the present
signature-$(1,4)$ assertion. HC for powers of the K3 alone is already
covered by [14]; the added content concerns the mixed abelian sources,
generalized support and the prescribed Kuga–Satake tensors.

### 10.1. A direct open Prym chart and isogeny transport

The primitive holomorphic forms on $G$ have bases
$$
\omega=\frac{dt}{v},\qquad
\eta_j=\frac{t^j(t-a)\,dt}{v^3},\quad 0\le j\le3,
\tag{10.4}
$$
for the $-i$ and $i$ characters. To check regularity, at a simple branch
point use $t-b=s^4$, $v=s$ times a unit; at a double branch point use
$t-a=s^2$, $v=s$ times a unit. At infinity $v=t^2$ times a unit, which
gives exactly the bound $j\le3$. The remaining two holomorphic forms
have character $-1$ and belong to the genus-two quotient.

For equivariant deformations the cotangent space of the seven-point
configuration modulo $\operatorname{PGL}_2$ is
$H^0(\mathbb P^1,K_{\mathbb P^1}^{\otimes2}(D_7))$, of dimension four.
The dual differential of the curve period map is multiplication of
holomorphic forms; restricting to the primitive Gaussian factors gives
$$
\omega\eta_j=
\frac{t^jdt^2}{(t-a)\prod_{j'=1}^6(t-b_{j'})}.
\tag{10.5}
$$
These four sections are a basis. Descent of invariant quadratic
differentials changes the pairing by a nonzero trace scalar only.
Thus the Prym period map has rank four and supplies an actual open
chart in the unitary ball, without an appeal to the Picard number of
an individual K3 surface.

Let $h_0$ be the rational Hermitian form on this marked chart, and $h$
that of $H^1(B)$ in Corollary 10.2. Both have signature $(1,4)$.
A compatible polarization exists by averaging any polarization under
the order-four action. Their determinants $d_0,d$ in rational
Gaussian bases are positive. Set $c=d_0/d>0$. Then
$$
\frac{\det(ch)}{\det h_0}=c^4=N_{\mathbb Q(i)/\mathbb Q}(c^2).
\tag{10.6}
$$
Scaling a polarization by $c$ preserves positivity. Landherr's
classification [9] and [10, Theorem 70] identifies $ch$ with $h_0$.
The rational unitary density argument of §9.3 sends the target period
into this actual open chart. It gives an exact Gaussian-linear rational
weight-one Hodge isomorphism, hence an algebraic isogeny of abelian
varieties after clearing denominators. Isogeny graphs and rational
inverses transport all the correspondences below. The use of odd
Hermitian rank in (10.6) is essential; this is not a determinant
normalization available in general even rank.

### 10.2. The symmetric-square quotient is uniruled

Let $M=E\times G\times G$, and retain the automorphisms
$\epsilon(u,z)=(-u,iz)$ and $\tau(t,v)=(t,iv)$. The group
$$
H=\langle(\epsilon,\tau,1),(\epsilon,1,\tau),s_{12}\rangle
\simeq(\mathbb Z/4)^2\rtimes S_2
\tag{10.7}
$$
has order $32$. Before the interchange, its quotient has function field
$$
Y^2=X^3+f(t_1)f(t_2)X,qquad
X=(v_1v_2)^2u,quad Y=(v_1v_2)^3z.
\tag{10.8}
$$
Adjoining $v_1,v_2$ recovers the product field and has degree sixteen.
The interchange fixes $X,Y$ and exchanges $t_1,t_2$.
In coordinates $[Z:S:P]$ on
$\operatorname{Sym}^2\mathbb P^1=\mathbb P^2$, its full fixed field is
$$
\mathcal Y:\quad Y^2=X^3+F_8X,qquad
F_8=\ell_a^2\prod_{j=1}^6\ell_{b_j},quad
\ell_c=P-cS+c^2Z.
\tag{10.9}
$$
The lines are tangent to the discriminant conic $S^2-4PZ=0$.

The standard projective Weierstrass model has fundamental line bundle
$L=\mathcal O_{\mathbb P^2}(2)$; adjunction gives
$$
\omega_{\mathcal Y}=\pi^*(\omega_{\mathbb P^2}\otimes L)
=\pi^*\mathcal O_{\mathbb P^2}(-1).
\tag{10.10}
$$
It is integral and normal. Indeed, on an affine base chart its singular
equations imply $Y=0$, $3X^2+F_8=0$ and $X\,dF_8=0$.
Together with its equation these force $X=F_8=0$, so the singular
locus has codimension at least two. The infinity section is smooth.
As a hypersurface it is Cohen–Macaulay, and Serre's criterion applies.

Let $r:Y\to\mathcal Y$ be a projective resolution and let $D$ be an
ample Cartier divisor on $\mathcal Y$. Every exceptional divisor maps
into codimension at least two, hence intersects $r^*D^2$ trivially.
Since $\pi^*\mathcal O(1)$ is a nonzero effective divisor,
$$
K_Y\cdot r^*D^2=K_{\mathcal Y}\cdot D^2<0.
\tag{10.11}
$$
A pseudoeffective divisor has nonnegative intersection with the square
of a nef divisor. Thus $K_Y$ is not pseudoeffective, and [7, Corollary
0.3] implies that $Y$ is uniruled. No crepant resolution is assumed.

### 10.3. From invariant cohomology to an actual tenfold

The two fourth-root actions in (10.7) impose the character match
$$
H^3(M,\mathbb C)^H=
(e\otimes\bigwedge^2W_\sigma)
\oplus(\bar e\otimes\bigwedge^2W_{\bar\sigma}).
\tag{10.12}
$$
Geometric interchange of two degree-one factors acts by minus tensor
interchange; this is why the exterior square occurs. The other
Künneth summands in degree three have no invariants. Call the rational
space in (10.12) $\mathcal W$. It has dimension twenty, Hodge numbers
$h^{2,1}=h^{1,2}=10$, and Gaussian character dimensions $(4,6)$ after
Tate twist by one. Its projector is the algebraic group average.

For a smooth projective uniruled threefold $Y$,
$H^3(Y)=N^1H^3(Y)$. To see the actual support, take a generically finite
rational domination $Q\times\mathbb P^1\dashrightarrow Y$ by a smooth
projective surface $Q$ and resolve it. The terms $H^3(Q)$ and
$H^1(Q)(-1)$ are supported on divisors: for the first, use hard
Lefschetz and the Gysin map from a hyperplane curve. Resolving a
rational map between smooth threefolds by smooth-center blowups adds
only curve $H^1(-1)$ terms to third cohomology. Pushforward is
surjective because its composite with pullback is the generic degree.
This proves the support assertion by correspondences.

We must transfer that support through a rational quotient without
identifying its invariant cohomology with the cohomology of a
resolution. Let $\Gamma$ be the graph correspondence of
$M\dashrightarrow Y$. Choose nonempty opens $U\subset M$, $V\subset Y$
on which it is the finite étale $H$-quotient. For
$\alpha\in H^3(M)^H$, set $\beta=\Gamma_*\alpha$. On these opens,
$$
q^*(\beta|_V)=\sum_{h\in H}h^*(\alpha|_U)=32\alpha|_U.
\tag{10.13}
$$
Remove a divisor supporting $\beta$, its inverse image, and the
complements of these opens, enlarging the union to a divisor if
necessary. Equation (10.13) shows that $\alpha$ vanishes on the
remaining open. Therefore $\mathcal W\subset N^1H^3(M)$.

In weight three, divisorial support is the sum of Gysin images of
$H^1(\widetilde D)(-1)$ from resolutions of divisors. This follows
from localization and purity of the weight-three piece; strata of
codimension at least two contribute no new degree-three support.
Compose a finite collection of these Gysin correspondences with the
projector onto $\mathcal W$. A finite sum surjects onto $\mathcal W$.
The corresponding quotient of the product of Picard varieties, and
Poincaré complete reducibility, give an abelian tenfold with (10.2).
For each smooth divisor resolution its $H^1$ is algebraically
identified with that of its Picard variety by the Poincaré bundle.
Curve-to-Prym correspondences replace the two $G$ factors by $P$.
This proves the first assertion of Theorem 10.1 for every smooth branch
configuration. Appendix B gives an explicit curve, incidence projector
and cylinder comparison on a nonempty open of that configuration space.

### 10.4. Schoen cycles at a compact-type limit

For fixed distinct $a,b_1,\ldots,b_6$, choose distinct constants in each
pair $r_j,s_j$, and consider the one-parameter family
$$
C_\varepsilon:\quad w^4=(t-a)^2\prod_{j=4}^6(t-b_j)
\prod_{j=1}^3(t-b_j-\varepsilon r_j)^3
(t-b_j-\varepsilon s_j)^2.
\tag{10.14}
$$
For general nonzero $\varepsilon$ its branch tuple is $1^3,3^3,2^4$.
The degree is twenty, so infinity is unramified. The genus is ten,
and the three holomorphic character dimensions are $(4,2,4)$.
Its primitive Prym $B_\varepsilon$ is an eightfold of Gaussian
signature $(4,4)$. The tuple is simple in Schoen's sense: it pairs as
three $(1,3)$ and two $(2,2)$ pairs. Consequently [18, Corollary 3.1]
supplies both rational Weil classes in
$\bigwedge^8_{\mathbb Q(i)}H^1(B_\varepsilon)$ by algebraic cycles.

Blow up the three collisions in the base family and make a finite
base change to obtain the admissible cover. On the core the colliding
exponents sum to five; dividing $w$ by $\prod_{j=1}^3(t-b_j)$ recovers
exactly $G$. A tail has local equation
$w_j^4=(x-r_j)^3(x-s_j)^2$, with exponent three at infinity.
Its tuple $(3,2,3)$ has genus one and holomorphic deck character
$-i$. It is the Gaussian elliptic curve with the inverse of the
original deck action on $E$.

At each attachment the two monodromies are one and three, so there
is exactly one preimage of the node. The covering dual graph is a
tree with a genus-seven core and three elliptic leaves. Thus the
limit is of compact type; its Jacobian extends to an abelian scheme
with special fiber $J(G)\times E^3$. Applying the primitive
idempotent gives, up to equivariant isogeny,
$$
B_0\sim P\times E^3.
\tag{10.15}
$$

Apply Schoen's construction over the geometric generic point. Its
finite collection of cycles descends after a finite extension of the
function field. Their closures on the smooth proper abelian scheme
specialize to algebraic cycles on $B_0$. Smooth proper specialization
identifies the rational cohomology local systems, including both
Gaussian determinant eigenlines. Thus the two independent cycle
classes remain independent. Pulling through the equivariant isogeny
(10.15) gives (10.3). The inverse tail character is essential: the
first line has type $(1,4)+(3,0)=(4,4)$.

This construction works for each smooth core, including cores with
additional Hodge tensors. It asserts algebraicity of the named
space; it does not assert that it generates all new classes at such
special points.

### 10.5. Full-group invariants and algebraic retractions

Assume the full unitary Hodge group. On $W_\sigma$ the Hodge circle
has weights $(1,-1,-1,-1,-1)$, and on $e$ it has weight one.
The determinant character therefore gives
$$
\operatorname{Hg}(E\times P)_\mathbb C
=\{(g,z)\in GL_5\times\mathbb G_m:\det(g)z^3=1\}=:G.
\tag{10.16}
$$
Indeed the derived group is $SL_5$, and the nontrivial projection
to the elliptic Hodge group gives the entire one-dimensional center.
The displayed connected group has precisely this derived group and
center dimension, proving equality. For the K3 in the theorem,
endomorphism field exactly $\mathbb Q(i)$ also gives this full unitary
group by the unitary K3 Hodge-group theorem [23]; the finite central
isogeny between (10.1) and (10.16) preserves the derived group.

The first fundamental theorem for $SL_5$ generates tensor invariants
by contractions and alternating volume tensors. The center of (10.16)
forces each volume tensor to be paired with three elliptic characters;
the conjugate volume uses the opposite characters. Contractions are
polarization classes and endomorphism graphs; the corrected volumes
are (10.3). Hence all tensor invariants are algebraic. Rational
Künneth and alternating projectors yield HC on every power of
$E\times P$. Rational descent causes no extra assertion: the sum of
the two conjugate volume lines is the complexification of the rational
algebraic space already constructed. Base change of its rational
cycle-class span proves the assertion about rational invariants.

Let $u$ denote the correspondence in (10.2). Its polarization adjoint
is algebraic: inverse Lefschetz operators on abelian varieties are
algebraic by Lieberman's theorem [24], and ordinary polarizations and
transposes of correspondences are algebraic. The positive operator
$u^\dagger u$ is invertible; by Cayley–Hamilton its inverse is a
rational polynomial in itself. Thus
$(u^\dagger u)^{-1}u^\dagger$ is an algebraic left inverse to $u$.

All cohomology of $C^n$ is consequently an algebraic direct summand
of a finite sum of Tate twists of cohomology of powers of $E\times P$.
Use the exterior-algebra description of abelian cohomology and its
algebraic projectors. Algebraic embeddings and retractions transfer
HC in both directions needed here. The product-map graph in §9.3
similarly embeds $T(S_f)$ into $H^2(E\times P)$ with an algebraic
retraction; the other K3 cohomology is Tate. This proves ordinary HC
for all $S_f^r\times C^n$. Buskin's algebraic comparison transfers
it to the other K3 surfaces in the theorem.

### 10.6. Fully twisted sources and actual support

For $1\le k\le4$, retain the entire central character in the
weight-one complex modules
$$
F_k=e^{k-1}\otimes\bigwedge^k W_\sigma\,(k-1).
\tag{10.17}
$$
Their Hodge signatures are $(1,4),(4,6),(6,4),(4,1)$.
The first two conjugate pairs are $H^1(P)$ and $H^1(A)$.
The determinant identifies the last two with the conjugates of
the first two, with the Tate weights as in (10.17). Thus every
fundamental module has an actual abelian source in $C$.

Every complex irreducible $G$-module is
$e^b\otimes\mathbb S_\lambda W_\sigma$, with
$\lambda=(\lambda_1,\lambda_2,\lambda_3,\lambda_4,0)$ dominant;
determinant powers are absorbed using $\det W_\sigma=e^{-3}$.
Write $\lambda=\sum_{k=1}^4a_k\omega_k$, and set
$$
m=\sum_k a_k=\lambda_1,qquad
d=b-\sum_k(k-1)a_k.
\tag{10.18}
$$
The exponent of the sole positive circle coordinate in a weight of
$\mathbb S_\lambda W_\sigma$ ranges from zero to $\lambda_1$,
and Weyl-orbit weights attain both endpoints. The circle extrema
are therefore $d-m$ and $d+m$. Including conjugation, its rational
Hodge level is
$$
\ell=m+|d|.
\tag{10.19}
$$
The Cartan component of a tensor of $a_k$ copies of $F_k$ has weight
$m$ and the required derived-group highest weight. Adding $|d|$
elliptic characters of the appropriate sign realizes the given module
at weight $\ell$, with a nonzero extreme Hodge component. Use distinct
factors to place this tensor in ordinary cohomology of some $C^N$.

For clarity, rational descent can be stated without choosing a
complex line as a rational source. Let $U\subset H^j(X,\mathbb Q)$
be an irreducible rational Hodge structure for
$X=S_f^r\times C^n$, and set $p=(j-\ell)/2$.
The rational source $M=H^\ell(C^N,\mathbb Q)$ just constructed and
$U(p)$ share a complex irreducible constituent. Reductivity and
base change for equivariant Hom give
$$
\operatorname{Hom}_G(M,U(p))\otimes\mathbb C
=\operatorname{Hom}_{G_\mathbb C}(M_\mathbb C,U(p)_\mathbb C)\ne0.
\tag{10.20}
$$
A nonzero rational map is surjective since $U$ is irreducible.
Equal weight and equal circle action make it a Hodge morphism.
This argument handles rational multiplicities and Schur indices.

Ordinary HC on $C^N\times X$, proved above, represents this surjection
by a cycle of codimension $\dim C^N+p$. The projection of its support
to $X$ has dimension at most $\dim X-p$. The correspondence therefore
supports $U$ in codimension at least $p$. Summing irreducible summands
proves generalized HC. This is the actual cycle-dimension argument
underlying the domination strategy of [21] and [22].

### 10.7. The prescribed even-Clifford correspondence

For $R_\mathbb Q=I_{2,8}$ the even Clifford algebra has center
$K=\mathbb Q(i)$ and is split over its center: after adjoining $i$
the quadratic form is hyperbolic. Thus its minimal rational spin
module has dimension thirty-two. The central character can be
checked explicitly in the restriction to (10.16). Put
$V_\sigma=eW_\sigma$; then $\det V_\sigma=e^2$. One complex
half-spin representation is
$$
\begin{aligned}
\det(V_\sigma)^{-1/2}
\left(\bigwedge^0 V_\sigma\oplus\bigwedge^2V_\sigma
\oplus\bigwedge^4 V_\sigma\right)
&=e^{-1}\oplus e\bigwedge^2W_\sigma
\oplus e^3\bigwedge^4W_\sigma\\
&=e^{-1}\oplus e\bigwedge^2W_\sigma\oplus W_\sigma^*.
\end{aligned}
\tag{10.21}
$$
With its conjugate and the weight-one normalization, this is exactly
$H^1(E\times P\times A)$, in agreement with [17]. The regular
even-Clifford Hodge structure is a sum of copies of this minimal
module. Hence its Kuga–Satake abelian variety is isogenous to a
power of $C$.

Every specified rational tensor defining the standard Kuga–Satake
map is now a Hodge class on a mixed product of the K3 and this
abelian variety. Ordinary mixed HC proves its algebraicity, including
the determinant-sensitive middle blocks. Isogeny correspondences
also transfer generalized HC to all the stated mixed products.
This proves the full-group part of Theorem 10.1. The precise object here
is the Kuga–Satake
variety of the rank-ten transcendental form; no choice of a larger
primitive lattice is being silently substituted.

### 10.8. The prescribed class at special Gaussian points

We now remove the full-group hypothesis for ordinary HC and for
algebraicity of the prescribed KS tensor. Section10.9 then treats generalized support by the actual smaller group. The specialization argument is a reuse of the
relative-class method in our earlier paper [24, §§9.5–9.6]; the
additional transfer of ordinary HC uses Floccari [25, Theorem 3.5].

Around any smooth seven-point configuration, the K3 models have the
same six type-III and one type-$I_0^*$ fibers. A finite base change
and simultaneous resolution of their rational double points give a
smooth projective polarized K3 family. Fix a smooth algebraic curve
in the configuration base through the desired parameter and a very
general full-group parameter, removing its finitely many bad points.
A further finite cover supplies the family and the persistent divisor
marking. The rank-ten orthogonal complement $\mathcal V$ of those
divisors is a polarizable variation even when the transcendental rank
jumps. Because the curve contains a full-group point, its very general
point has the full group as well.

The relative KS construction [5, Chapter 6, §4.4 and Proposition 4.10]
gives, after a finite étale cover, an algebraic abelian scheme with
weight-one fibers $C^+(\mathcal V_b)$. Its monodromy is a Spin lift
acting by **left multiplication** on the reference Clifford algebra.
It is not the naive Clifford functor applied to orthogonal monodromy;
that would not in general give the required global polarization.
Fix a nonisotropic reference vector $v_0$ as right Clifford datum.
For $\phi_v(w)=vwv_0$,
$$
L_g\phi_vL_g^{-1}(w)=(gvg^{-1})wv_0.
\tag{10.22}
$$
Thus the specified embedding is monodromy-equivariant even though
$v_0$ need not be a flat vector of $\mathcal V$. Using the global
abelian polarization gives a flat rational codimension-two KS class
$\kappa$ on the relative product of the K3 and two copies of the
abelian scheme. Section 10.7 makes it algebraic at very general points.

For completeness, a fixed flat class algebraic at very general points
of a smooth projective family over a complex curve is algebraic at
every point. Relative Hilbert schemes have countably many components;
finite products and rational coefficient lists parameterize rational
cycle combinations. Their maps to the base are proper. Choose a
point outside the images of all nondominating components and outside
the exceptional set of the very-general assertion. A representing
cycle combination belongs to a dominating component. Its universal
cycle has the specified flat class throughout that connected component.
A finite multisection, followed by normalization and the valuative
criterion of properness, specializes it to every base point. Equally,
the proper dominating component itself surjects onto the base.
This applies to $\kappa$. It does not apply to arbitrary new Hodge
classes appearing only at a special point.

We also record the extraction of the actual smaller transcendental
KS class. At a special point write $V=T\perp N$, with $N$ Tate,
and choose nonisotropic $t_0\in T$. Right multiplication by the even
invertible element $v_0t_0/q(v_0)$ commutes with the left KS complex
structure, hence is an algebraic abelian quasi-isogeny. Composing on
the output changes $\phi^{v_0}$ to $\phi^{t_0}$. In the graded
Clifford decomposition choose inclusion $i:C^+(T)\to C^+(V)$ and
projection $p$ discarding each nonempty $N$-monomial. They commute
with the left KS complex structure in $C^+(T)_\mathbb R$, hence
are actual rational abelian homomorphisms and $pi=1$. For $v\in T$,
$$
p\phi_V^{t_0}(v)i(w)=vwt_0=\phi_T^{t_0}(v)(w).
\tag{10.23}
$$
The source projector onto $T$ is algebraic by the inverse intersection
matrix of a divisor basis. Consequently the specialized cycle extracts
the prescribed smaller KS class, rather than an unspecified nonzero
embedding. These operations are the choice changes of [5, Chapter 4,
Proposition 2.6 and Remark 2.7]. Adjoining a Tate complement gives
isogeny powers of the same KS variety [5, Chapter 4, Example 2.4].

For every smooth source, the order-four action restricts to an
endomorphism of $T(S_f)$ squaring to minus one. Its endomorphism
field is therefore CM. By [14, Corollary 2.3], HC holds on every
power of $S_f$. Floccari's theorem [25, Theorem 3.5] states that,
once the prescribed KS correspondence is algebraic, HC on all K3
powers is equivalent to HC on all powers of its KS variety.
It follows that HC holds for all KS powers at every smooth source.

The spin restriction (10.21) is an identity of representations and
Hodge circles and remains valid when its factors become reducible.
Thus $C_f=E\times P_f\times A_f$ is an isogeny factor of a power of
the KS variety, also at special points. HC for all $C_f^n$ follows,
and the algebraic K3 source/retraction gives HC for all mixed
$S_f^r\times C_f^n$. Isogeny transport in §10.1 proves the ordinary
HC conclusion for every Gaussian fivefold in Corollary 10.2.
For the other K3 surfaces of Proposition 9.1, Buskin's comparison
and the KS functoriality in [25, §3.5 and Remark 3.7] transfer the
prescribed correspondence and ordinary mixed HC, including at
Picard jumps. This strengthening relies on the specified flat class
and the two cited Hodge theorems; it is not a specialization principle
for all Hodge classes.

### 10.9. Generalized support at Hodge-group drops

We prove generalized HC at every smooth Gaussian source by calculating
its actual smaller Hodge group. This is a direct domination argument,
not specialization of generalized HC. The use of all spin constituents
is essential, including both half-spin representations.

Let $T$ be the actual transcendental space, with CM endomorphism field
$F\supset K=\mathbb Q(i)$. Put $d=[F^+:\mathbb Q]$ and
$m=\dim_FT$, so $dm\le5$. Since $F^+$ is totally real,
$$
F=F^+K.
\tag{10.24}
$$
Zarhin's theorem [23], also [5, Chapter 3, Theorem 3.9], gives
$\operatorname{Hg}(T)=\operatorname{Res}_{F^+/\mathbb Q}U(T,h)$.
Over $\mathbb C$ it is $\prod_{\alpha=1}^dGL(U_\alpha)$,
where $\dim U_\alpha=m$. Choose the embeddings of $F$ extending
one fixed embedding of $K$. Their sum is an isotropic half
$U=\bigoplus U_\alpha$ of $T_\mathbb C=U\oplus U^*$.
The Hodge circle has weight two at one coordinate of the selected
factor, and zero at all other coordinates of $U$.

The connected inverse image of this group in the spin group is
$$
\widetilde G_\mathbb C=
\left\{((g_\alpha),z):\prod_\alpha\det(g_\alpha)=z^2\right\}.
\tag{10.25}
$$
Write $e$ for its character $z$. The preimage is connected: on a
maximal torus the defining character has coefficients including one,
so its kernel is a torus, and the derived factors $SL_m$ are connected.
The actual KS Hodge group projects onto $\operatorname{Hg}(T)$,
hence has the same dimension as this connected inverse image and
therefore equals it. In particular its lifted circle sends $-1$ to
the central $-1$, as seen from its weight-one action. We are discussing
Hodge groups; the weight scalar in the Mumford–Tate group is separate.

The full spin module restricts to
$$
e^{-1}\bigwedge^{\bullet}U
=\bigoplus_{0\le k_\alpha\le m}
 e^{-1}\bigotimes_\alpha\bigwedge^{k_\alpha}U_\alpha.
\tag{10.26}
$$
Both complex half-spin modules occur in the left regular
$C^+(T)$ because $\dim_\mathbb Q T$ is even. Thus every summand
in (10.26) occurs in the actual $H^1$ of its KS variety.
At any Galois-selected active factor the summand has only circle
weights $\pm1$: $k_\alpha=0$ gives $-1$, $k_\alpha=m$ gives
$+1$, and an intermediate exterior power gives both. Tate
complements and abelian isogenies place these sources in the mixed
category already covered by ordinary HC in §10.8.

The Galois action on the half-determinant requires care. By (10.24),
Galois automorphisms fixing $K$ permute the $U_\alpha$; conjugating
$K$ dualizes all of them at once. Since
$2e=\sum_\alpha\det(U_\alpha)$ in additive character notation,
and a connected torus has torsion-free character lattice, these
operations respectively fix $e$ or replace it by $-e$. Inner
forms and multiplicities do not change this character calculation.
There is no independent sign change on a subset of the factors.

A complex irreducible representation of (10.25) has the form
$$
R=e^b\bigotimes_\alpha\mathbb S_{\lambda_\alpha}U_\alpha,
\qquad b\in\mathbb Z,\quad
\lambda_{\alpha,1}\ge\cdots\ge\lambda_{\alpha,m}\in\mathbb Z.
\tag{10.27}
$$
Negative entries are allowed. For the circle active at factor $\alpha$,
its extrema are $b+2\lambda_{\alpha,m}$ and
$b+2\lambda_{\alpha,1}$. The real embeddings are Galois-transitive,
and conjugation reverses the extrema. Consequently the rational
irreducible Hodge structure containing $R$ has level
$$
\ell=\max_\alpha\max\left(
 |b+2\lambda_{\alpha,m}|,
 |b+2\lambda_{\alpha,1}|\right).
\tag{10.28}
$$
All extrema have parity $b$. Set $c=(b+\ell)/2\in\mathbb Z$ and
$\lambda'_\alpha=\lambda_\alpha+(c,\ldots,c)$. Then
$$
0\le\lambda'_{\alpha,j}\le\ell,\qquad
R=e^{-\ell}\bigotimes_\alpha
\mathbb S_{\lambda'_\alpha}U_\alpha.
\tag{10.29}
$$
Each partition $\lambda'_\alpha$ fits in an $m$-by-$\ell$ rectangle.
Write it as $\ell$ columns, permitting heights zero and $m$, and
align those columns across $\alpha$. Each aligned column is an
actual weight-one spin summand in (10.26). The Cartan component of
the tensor product of the $\ell$ columns is exactly (10.29).
It lies in $H^\ell(\operatorname{KS}(T)^\ell,\mathbb C)$ and
has an extreme $(\ell,0)$ or $(0,\ell)$ component; its rational
orbit is fully twisted. If $\ell=0$, the representation is trivial
and the source is a point.

For a rational irreducible substructure of degree $j$ in a fixed-source
mixed product, use the rational Hom descent in (10.20) with this
source. It gives a rational Hodge surjection at weight $\ell$.
The ordinary mixed HC theorem of §10.8 represents it by a cycle,
whose projection has codimension at least $(j-\ell)/2$.
Decomposing arbitrary substructures proves generalized HC at every
smooth Gaussian source. The same isogeny and K3 comparisons used for
ordinary HC transfer this statement to Corollary 10.2 and the K3s in
Proposition 9.1. This completes Theorem 10.1 without a full-group
restriction.

The calculation includes CM points ($m=1$), lower-rank points with
endomorphism field $K$, and the non-CM case $d=m=2$. It does not
rely on all real derived factors being noncompact. It treats powers
and mixed products attached to one source at a time, and makes no
assertion about arbitrary tuples of unrelated special sources.

### 10.10. Scope of the advance

The cycle constructions in §§10.2–10.4 and the isogeny transport in
§10.1 require no generic-Hodge-group assumption. Ordinary HC and the prescribed KS correspondence hold also at special
points by Section10.8. Section10.9 proves generalized support there by
calculating the actual smaller group and its full spin representation.
All mixed statements concern a fixed source and its associated varieties;
arbitrary tuples of unrelated special sources are not covered.
Neither a generic $SO(2,8)$ source nor a deformation of these cycles
in the four normal directions is constructed here. In the original
surface family, §9.6 transfers the selected Gaussian cohomology;
we do not claim full HC for arbitrary powers of the entire original
surface from that selected summand alone.

The proof gives actual correspondences by support, specialization and
isogeny. Appendix B additionally describes the tenfold as an incidence
image in the Jacobian of an explicit line curve on a nonempty open.
That geometry does not yet provide the normal deformation of the full
spin source. The general rational Hodge conjecture remains open.

## 11. Reproducibility, credit and unresolved work

The supplement contains eleven Python scripts using SymPy and
python-flint. Run them from its root:

    python code/check_seed13_rank16_Kummer_lattice.py
    python code/check_seed13_Ma12_K3_model.py
    python code/check_seed13_order4_Ma_source.py
    python code/check_Gaussian_tenfold_quotient.py
    python code/check_Gaussian_Schoen_degeneration.py
    python code/check_Gaussian_spin_rectangle_support.py
    python code/check_Gaussian_two_torsion_resolution.py
    python code/check_Gaussian_Eckardt_global_comparison.py
    python code/check_Gaussian_cubic_fibration_boundary.py
    python code/check_Eckardt_line_projector.py
    python code/explore_Gaussian_Eckardt_conic.py

They write JSON certificates to the results directory.
The first checks all affine hyperplanes in
$\mathbb F_2^4$, the relevant index and parity
relations, explicit root-lattice gluing and
the local $A_3$ matrices. The second checks
the explicit quadric model and both elliptic
fibrations. The third checks the Gaussian subfamily,
its smooth sample and the actual cyclic product map.
The fourth checks the quotient field identities, the order-32 group,
its invariant Hodge numbers and the canonical degree. The fifth checks
the balanced smoothing, tail characters, compact-type graph and
central-character level formula. The sixth checks the rectangular spin
source construction at every allowed dimension factorization.
The next four check the ordered-resolution algebra and defect,
Gaussian quotient comparison identities, the cubic-pencil boundary,
and the twenty-seven-line incidence projector. The final script supplies
an exact conic equation certificate. The ten checker scripts have
25,25,21,15,22,5,14,14,6 and8 passing summary checks, totaling155;
these include3328 bounded central-level and30,632 rectangular-partition
controls. The support,
specialization and density arguments are written proofs; finite checks
do not establish them by themselves.
They require no GPU or external
data. Their success is evidence for these
finite consequences; the primitive Kummer
closure theorem, local geometric
identifications and imported comparison
theorems are mathematical inputs proved or
cited in the text.

The full period-rank proof combines
lelong_ridge's type-$(1,3)$ seed and rank
proposal, gabrielov_cell's cyclic-three
block decomposition and normal map, and our
global logarithmic identification, residue
audit and theta-avoidance argument.
gabrielov_cell also described the actual
double-Kummer cover and supplied (8.1).
The exact determinant and primitive
saturation, odd-unimodular identification,
boundary synthesis and conditional
comparison are developed here. Discussion
replication of finite computations is not
described as an independent geometric proof.
The relevant discussion is preserved at [27]. The new Section10
quotient, compact-type specialization, isogeny transport and mixed support
are developed by cm_symmetry_scout, building on the cited literature.
The coauthors' contributions to the earlier geometric comparison remain
as specified above; no independent endorsement of Section10 or Appendix B is implied.

The remaining generic problem is concrete: construct
an actual algebraic correspondence from a
tensor of curve $H^1$ spaces, or an abelian
variety, carrying the eight-dimensional orthogonal
variation with rank-ten fibers. Sections 9–10 supply such a source, every Gaussian spin factor,
and mixed Hodge conjectures on the four-dimensional Gaussian sublocus
of the K3 target, independently of the conditional comparison.
Section 9.6 makes its conditional transfer precise on the
open of original surfaces reached by the full-base spread.
Rationality of the moduli space,
the presence of the genus-four fixed curve,
and the comparison (1.2) do not provide that
correspondence. The original rational
Hodge conjecture remains unresolved.

## Appendix A. Full period rank of the type-$(1,3)$ family

We include the proof because a numerical Jacobian rank,
without identification with the actual period derivative,
would not suffice. The six-dimensional residue block
adapts [1, §3.4], keeping an order-three line bundle and
its inverse distinct. The cyclic-point formulation and
block decomposition were developed in discussion with
lelong_ridge and gabrielov_cell; the global quotient,
theta-avoidance argument and complete synthesis below
make the geometric rank assertion explicit.

### A.1. Global logarithmic quotient and the actual derivative

For a double cover of an abelian surface with branch
$D=(d=0)$, put $E=T_B(-\log D)$ and fix a nonzero
invariant volume form. Rank-two wedge duality gives
$$
E\otimes L\simeq\Omega_B^1(\log D)\otimes L^{-1}.
\tag{A.1}
$$
The anti-sheet Hodge space $H^{1,1}$ is $H^1$ of the
right-hand side. The logarithmic normal sequence tensored
by $L$ is
$$
0\longrightarrow E L\longrightarrow T_B L
\longrightarrow L^3|_D\longrightarrow0.
\tag{A.2}
$$
Ample vanishing and the restriction sequence identify
$$
H^{1,1}_-(X)\simeq
\frac{H^0(B,L^3)}
{dH^0(B,L)+\text{lifted normal derivatives of }H^0(B,T_BL)}.
\tag{A.3}
$$
There is no assumption here that a collection of
meromorphic exact forms exhausts a kernel.
The map on $H^0(T_BL)$ is injective: a section tangent
to $D$ restricts to a section of
$T_D\otimes L|_D=L^{-1}|_D$, which has negative degree.
It must vanish along $D$ and then lie in
$H^0(T_BL^{-1})=0$.

A branch direction $e\in H^0(L^2)/\langle d\rangle$
has Kodaira–Spencer class given by the connecting map
of the untwisted logarithmic normal sequence.
Multiplication by a theta section $\theta\in H^0(L)$
gives a commutative diagram with (A.2).
Naturality of connecting maps identifies the period
derivative of the form $\theta\,\Omega/z$ with
$\theta e$ in (A.3), up to a universal nonzero
normalizing scalar. Locally a vector field with
$v(d)=a d$ lifts as $v+(a/2)z\partial_z$, which gives
the same contraction formula.

For invariant vector fields $\partial_i$ and
$f\in H^0(L)$, representatives of the normal derivatives are
$$
J_i(f)=d\,\partial_i f-\frac12 f\,\partial_i d.
\tag{A.4}
$$
These are global sections of $L^3$: derivative-of-frame
terms cancel under every line-bundle frame change.
The formulas commute with finite automorphisms and their
rational character projectors.

### A.2. The cyclic-three point

Choose an indecomposable principally polarized abelian
surface $(B_0,M)$, a nontrivial order-three bundle $\chi$,
and the étale cyclic isogeny $\pi:B\to B_0$ trivializing
$\chi$. Put $L=\pi^*M$ and $d=\pi^*d_0$, with
$d_0\in H^0(M^2)$ smooth. Inversion conjugates the
order-three deck transformation to its inverse.
The three theta lines have characters $1,\chi,\chi^{-1}$.
The selected holomorphic form has numerator
$\theta_\chi-j^*\theta_\chi$.

At this point the dihedral isotypic decomposition
splits the selected $U_Y$ into a rank-eight K3-type
piece, of types $(1,6,1)$, and a rank-two algebraic
piece. The latter is the deck-invariant part of the
selected $H^{1,1}$. The six tangent directions
preserving cyclic branch structure consist of three
polarized abelian deformations and three branch
directions. The other four branch directions have
the form $s+j^*s$, for
$s\in H^0(M^2\chi^{-1})$.

### A.3. Rank six in the tangential block

Let $F=(\theta_\chi=0)$ and $K=K_F$.
This is a smooth genus-two theta translate, with
$\operatorname{Jac}(F)\simeq B_0$. In particular
$\eta=\chi^{-2}|_F$ is nontrivial of order three.
Adjunction gives
$$
(M\chi)|_F=K,\qquad M^2|_F=K^2\eta.
$$
The tangent period map is multiplication by
$\theta_\chi$ on
$H^1(T_{B_0}(-\log D_0))$.
It is injective if the cokernel sheaf of
$E\to E M\chi$ has no global section.
Upon restriction to $F$, the relevant normal
derivative map is
$$
A_d:H^0(K)^{\oplus2}\longrightarrow
H^0\bigl(Z,(K^3\eta)|_Z\bigr),
\qquad Z=D_0\cap F,
\tag{A.5}
$$
between spaces of dimension four.

The restriction map
$H^0(M^2)\to H^0(F,K^2\eta)$ is surjective with
kernel $\mathbb C\theta_\chi\theta_{\chi^{-1}}$.
Choose a restriction $s=d|_F$ with four simple
zeros avoiding the zeros of
$t_0=\theta_{\chi^{-1}}|_F\in H^0(K\eta)$.
Such choices exist because $K^2\eta$ is
basepoint-free. Replace
$d$ by $d+\lambda\theta_\chi\theta_{\chi^{-1}}$.
Then $A_d$ changes to $A_d+\lambda A'$.
The derivatives of $\theta_\chi$ in two independent
translation directions form a basis $\theta_1,\theta_2$
of $H^0(K)$: a nonzero kernel vector would give a
translation tangent to an ample theta curve.
Thus
$$
A'(a,b)=t_0(a\theta_1+b\theta_2)|_Z.
$$
The canonical pencil gives
$\operatorname{Sym}^2H^0(K)=H^0(K^2)$.
Restriction to $Z$ is injective since
$K^2(-Z)=\eta^{-1}$ has no section.
Therefore $A'$ has rank three and kernel the
skew-gradient vector $k=(\theta_2,-\theta_1)$.
It is a nowhere-zero section of
$T_F\otimes K=\mathcal O_F$.

For a nonzero constant $c$,
$A_d(k)=c\,d_Fs|_Z$.
In
$$
0\longrightarrow\eta^{-1}
\xrightarrow{\ s\ }K^2\longrightarrow K^2|_Z
\longrightarrow0,
$$
the boundary of $(d_Fs)/t_0|_Z$ pairs with
$t_0\in H^0(K\eta)$ by Serre duality.
Each simple zero contributes
$\operatorname{Res}(ds/s)=1$, so the pairing is $4$.
It is nonzero. Hence $A_d(k)$ is outside
$\operatorname{im}A'$, and the coefficient of
$\lambda^3$ in $\det(A_d+\lambda A')$ is nonzero.
General $\lambda$ makes (A.5) invertible.

Finally $H^1(E)$ is the six-dimensional
polarized pair-deformation tangent space.
In the untwisted logarithmic normal sequence,
$h^0(K_{D_0})=5$, $h^0(T_{B_0})=2$ and
$h^1(T_{B_0})=4$; contraction with $c_1(D_0)$
has one-dimensional image. The quotient of
embedded deformations by translations has
dimension three, and the polarized abelian
deformations have dimension three.
The finite isogeny and torsion choices introduce
no infinitesimal parameters.
This proves rank six in the tangential block.

### A.4. Rank two in the normal block

The deck-invariant odd numerator is
$H^0(M^3)^-$, of dimension four. By (A.3) its
denominator is
$$
J=\langle J_1(\theta_M),J_2(\theta_M)\rangle,
\qquad\dim J=2.
$$
The denominator has no $dH^0(M)$ term in this
parity. The two displayed sections are independent
by the injectivity in A.1.
The projection of the actual normal derivative is
$$
\Pi(s)=\theta_\chi s-j^*(\theta_\chi s),
\qquad s\in H^0(M^2\chi^{-1}).
\tag{A.6}
$$
The two theta translates are distinct irreducible
divisors. If $\Pi(s)=0$, divisibility forces
$\theta_{\chi^{-1}}$ to divide $s$; the remaining
factor is a multiple of the even $\theta_M$.
Conversely this product is in the kernel.
Thus $\ker\Pi$ has dimension one and
$\operatorname{im}\Pi$ has dimension three.

Let
$F_\chi\cap F_{\chi^{-1}}=\{p,-p\}$.
The image of $\Pi$ vanishes at these points.
One can require transverse intersection,
$p\notin B_0[2]$ and $\theta_M(p)\ne0$.
To see nonemptiness, degenerate to
$E_1\times E_2$ with its product theta divisor,
and choose the order-three translation $(a,b)$
with both coordinates nonzero. The translated
theta divisors intersect transversely at
$(a,-b)$ and $(-a,b)$, neither on the product
theta divisor. These open conditions persist
on nearby indecomposable abelian surfaces.

On $\theta_M\ne0$,
$$
\frac{J_i(\theta_M)}{\theta_M^3}
=-\frac12\partial_i(d/\theta_M^2).
$$
The $|2M|$ Kummer morphism is immersive off
$B_0[2]$. For general $d$ at least one of these
derivatives is nonzero at $p$.
Therefore $J$ is not contained in
$\operatorname{im}\Pi$. Since their dimensions
are two and three in a four-dimensional space,
$J+\operatorname{im}\Pi=H^0(M^3)^-$.
The normal block surjects onto the
two-dimensional quotient by $J$.

The rank-six tangential block maps into the
complementary isotypic space, so the full
period derivative has rank $6+2=8$.
All the conditions used are nonempty opens of
the same irreducible section and level family,
so they can be imposed simultaneously.
The rank condition is algebraic on Hodge
bundles; it holds on a nonempty Zariski open
of the type-$(1,3)$ family.

The K3-type period domain for rank ten has
dimension eight, so the image contains an
analytic open set. Outside countably many
proper rational tensor Hodge loci the Hodge
group is the full $\operatorname{SO}(U_Y)$:
the Hodge circles rotating positive
two-planes in an open period ball generate
the connected real orthogonal group. A
proper rational subgroup would impose a
proper rational tensor Hodge locus.
The generic Hodge structure is consequently
simple and has no rational $(1,1)$-class.

## Appendix B. An explicit curve and projector for the Gaussian tenfold

The support construction in §10.3 works at every smooth branch
configuration. On a nonempty open of that configuration space, we can
also specify the supporting curve and identify its cylinder with the
particular invariant space $\mathcal W$ in (10.12). This uses classical
cubic-surface incidence [27, §§1.1–1.13 and 9.1]. The additional point
here is the explicit Gaussian family and its comparison with the
specified exterior-square source; the general Prym–Tyurin construction
is not new.

Send the double branch point $a$ to infinity. On
$\mathbb P^2_{[Z:S:P]}$ put
$$
L_i=P-b_iS+b_i^2Z,\qquad
A_3=L_1L_2L_3,\qquad B_3=L_4L_5L_6.
\tag{B.1}
$$
The seven lines $Z,L_1,\ldots,L_6$ are distinct tangents to a smooth
conic, so no three concur. The quotient threefold of §10.2 is
birational to
$$
\mathcal Z:\quad w^2=(Zt^2)^2-4A_3B_3
\subset\mathbb P(1,1,1,1,3).
\tag{B.2}
$$
For example, in the affine base chart $Z=1$, its Weierstrass coordinates
$x,y$ give $t=y/x$ and $w=2x-t^2$.

**Proposition B.1.** On a nonempty open of the six finite branch-point
configuration, there is a smooth projective threefold $X$ birational to
(B.2), a morphism $f:X\to\mathbb P^1$, and an involution
$\iota:t\mapsto-t$ over the base with
$$
\begin{aligned}
H^3(X,\mathbb Q)^+&\simeq H^1(T,\mathbb Q)(-1),\qquad g(T)=2,\\
H^3(X,\mathbb Q)^-&\simeq\mathcal W.
\end{aligned}
\tag{B.3}
$$
Both isomorphisms are induced by algebraic correspondences. If
$\mathcal C_{27}$ is the normalized curve of lines of its smooth cubic
fibers and $D$ is the incidence correspondence sending a line to the ten
lines meeting it, then
$$
e_{\mathrm{line}}=\frac{(1-D)(1-\iota)}{12},\qquad
A_{\mathrm{line}}=\operatorname{im}
\bigl(e_{\mathrm{line}}:J(\mathcal C_{27})\to J(\mathcal C_{27})\bigr)
\tag{B.4}
$$
define, up to isogeny, an abelian tenfold and the isomorphism (10.2).
The Jacobian of a disconnected normalization means the product of its
component Jacobians. Only the twenty-four non-Eckardt lines contribute
to (B.4).

### B.1. A global small resolution, including the Gaussian fiber

Write $R=Zt^2$, $u=(R+w)/2$ and $v=(R-w)/2$. Thus
$$
uv=\prod_{i=1}^6L_i,\qquad u+v=R.
\tag{B.5}
$$
Blow up, in the stated order, the proper transforms of the Weil divisors
$(u,L_1),\ldots,(u,L_5)$. We verify smoothness, including the central
Gaussian singularities, before using a deformation argument.

The singularities away from $p=[0:0:0:1]$ are precisely the fifteen
points $t=L_i=L_j=0$. Indeed, a singular point has $w=0$. If
$H=\prod_iL_i\ne0$, then $R\ne0$, so $Z,t\ne0$ and the nonzero
$t$-derivative $4Z^2t^3$ excludes a singularity. If $H=R=0$, the
condition $dH=0$ requires at least two vanishing $L_i$. Their intersection
line meets $R=0$ at $p$ and at the double zero $t=0$.

At the latter point use $x=L_i$, $y=L_j$ and
$H=xyh(x,y)$ with $h(0,0)\ne0$; also $Z=\ell(x,y)$ is a unit. The
local singularity is $uv=xyh$, $u+v=\ell t^2$, of type $cA_3$.
The first ordered blowup meeting either factor has charts
$$
\lambda\ell t^2-x\lambda^2-yh=0,
\qquad
x-\alpha\ell t^2+\alpha^2yh=0.
\tag{B.6}
$$
Along the exceptional curve their respective $y$- and $x$-derivatives
are $-h(0,0)$ and $1$. They are smooth. The remaining factor ideals
are principal there by the transformed equation, so later blowups
are locally trivial.

At $p$, set $t=1$, and put
$A_k=\prod_{j\le k}L_j$, $B_k=\prod_{j>k}L_j$. After $k$ main charts
the equations are
$$
u_kv=B_k,\qquad A_ku_k+v=R.
\tag{B.7}
$$
The next complementary chart is
$$
L_{k+1}-u_k\alpha=0,\qquad
R-A_ku_k-\alpha B_{k+1}=0.
\tag{B.8}
$$
For $k=0$, eliminate $u_0$; the remaining gradient at $p$ is
$L_1-\alpha Z\ne0$. For $1\le k\le4$, above $p$ we have
$u_k\alpha=0$. If $u_k=0$, the second gradient in the three base
coordinates is $Z$, except for $k=4$, when it is $Z-\alpha L_6$.
If $\alpha=0$, it is $Z$, except for $k=1$, when it is $Z-u_1L_1$.
The independence of every triple $Z,L_i,L_j$ makes the two equation
gradients independent, including their common origin. In the final
main chart the equation is
$$
u_5R-A_5u_5^2-L_6=0,
\tag{B.9}
$$
whose gradient at $p$ is $u_5Z-L_6\ne0$. The exceptional locus is
one rational curve over each $cA_3$ point and a chain of five over $p$.
These projective blowups therefore give a smooth projective small
resolution $X$.

The same charts are algebraic in the six plane equations and the cubic
$R$. Their fiber Jacobians have maximal rank at every point of the
resolved Gaussian fiber. Properness and the relative Jacobian criterion
give a smooth projective deformation to general six planes in
$\mathbb P^3$ and a general cubic $R$. The general double sextic has
forty-five nodes $R=L_i=L_j=0$ and no others. The union of its fifteen
pair-intersection lines has ideal generated by $H/L_i$ and resolution
$$
0\longrightarrow S(-6)^5\longrightarrow S(-5)^6
\longrightarrow I_{\mathrm{lines}}\longrightarrow0,
\qquad S=\mathbb C[z_0,z_1,z_2,t].
\tag{B.10}
$$
The coordinate ring is Cohen–Macaulay and $R$ is a nonzerodivisor;
thus the node ideal $I_{\mathrm{lines}}+(R)$ is saturated. Its degree-five
piece has dimension $6+10=16$. The two summands have zero intersection:
if $Rq$ vanishes on each line, $q$ does too, whereas the line ideal has
no degree-two element. The forty-five nodes therefore impose forty
conditions on quintics, so the defect is five by the double-cover
formula [28, Proposition 3.7 and the proof of Theorem 4.6]. A smooth
double sextic has $h^{1,1}=1$, $h^{2,1}=52$, from the degree-five
weighted Jacobian ring. The small-resolution transition gives
$$
h^{1,1}(X)=6,\qquad h^{2,1}(X)=52-45+5=12,
\qquad b_3(X)=24.
\tag{B.11}
$$
These Hodge numbers transfer to the Gaussian fiber by the smooth
projective deformation just verified.

The function $r=u/A_3$ extends to a morphism $f:X\to\mathbb P^1$.
After three main charts, $[u:A_3]=[u_3:1]$. In an earlier complementary
chart, with $k=0,1,2$, it is
$[1:\alpha\prod_{j=k+2}^3L_j]$; neither expression has a base point.
The last two blowups preserve the map. Over $r\ne0,\infty$, $u_3=r$
is a unit, those last blowups are trivial, and eliminating $v$ gives
exactly the cubic-surface family
$$
F_r:\quad A_3r^2+B_3-Zrt^2=0\subset\mathbb P^3.
\tag{B.12}
$$
All ordered blowup ideals are preserved by $t\mapsto-t$.

### B.2. The two line cylinders fill the third cohomology

The smooth fibers of (B.12) have Eckardt point $[0:0:0:1]$. The
involution fixes this point and the plane elliptic section $t=0$.
Lefschetz gives $H^2$ eigendimensions three and four. Removing the
canonical class leaves local systems $L^+$ and $L^-$ of ranks two
and four.

The plane-cubic pencil $B_3+\lambda A_3$ has two triangle fibers,
six other simple nodal fibers, and four simple tangencies to $Z=0$,
with these parameter sets disjoint, on a nonempty open. The supplement
certifies that open at
$(b_1,\ldots,b_6)=(0,1,3,7,11,19)$ modulo $431$: the residual node and
tangency parameter polynomials have respective degrees six and four,
are square-free and coprime, and the node Hessians are units. Both
charts at binary infinity are checked. The pencil has nine transverse
base points, so their blowup is a rational elliptic surface of Euler
number twelve. The two triangles contribute three each; hence the six
nonzero nodal parameters contain exactly one node apiece and exhaust
the remaining singular plane-cubic fibers. These are algebraic open
conditions, so the certificate proves nonemptiness in characteristic zero.

After $\lambda=r^2$, there are twelve single-node cubic-surface fibers
and eight fibers with two nodes exchanged by $\iota$. At a single node,
$\iota$ reverses one coordinate of the local $A_1$ equation and acts
by $-1$ on its vanishing sphere. The reflection has invariant
codimension one on $L^-$ and zero on $L^+$. At an exchanged pair the
two orthogonal vanishing classes give one plus and one minus combination;
the invariant codimension is one on each system.

At $r=0$, the change $t'=\sqrt r\,t$ gives
$B_3+r^2A_3-Zt'^2=0$. The central cubic has three $A_1$ points. The
$r^2$ pullback squares their commuting reflections, killing that
monodromy; the remaining twist around the $r$-loop is $t'\mapsto-t'$,
namely $\iota$. Thus the endpoint monodromy on $L^-$ is $-1$ and on
$L^+$ is $+1$. The same holds at infinity. This is a local-system
calculation; an equivariant simultaneous small resolution at the
endpoint is not assumed.

Let $j:B\hookrightarrow\mathbb P^1$ omit these singular parameters.
The endpoint action excludes global invariants of $L^-$. The parabolic
Euler formula gives
$$
\dim H^1(\mathbb P^1,j_*L^-)=12+8+4+4-2\cdot4=20.
\tag{B.13}
$$
The three Eckardt lines are parametrized by
$$
T:\quad \zeta^2=-A(\beta)B(\beta),\qquad
r=\frac{\zeta}{A(\beta)},\qquad
A(\beta)=\prod_{i=1}^3(\beta-b_i),\quad
B(\beta)=\prod_{i=4}^6(\beta-b_i).
\tag{B.14}
$$
This is the genus-two quotient of $G$; its degree-three map to the
$r$-line has eight simple branch values. Its standard rank-two
permutation summand is $L^+$. The intersection matrix of the three
lines is $-2I+J$, nonsingular on that summand.

The twenty-seven line classes span the second cohomology of a smooth
cubic surface. Their intersection matrix is $D-I$, where $D$ has
eigenvalues $10,-5,1$ with multiplicities $1,6,20$ [28]. Thus (B.4)
projects to $L^-$, on which the intersection pairing is $-6I$.
The closure of the universal line family in
$\mathcal C_{27}\times X$ gives an actual cylinder correspondence.
Over $B$, Leray identifies $H^3(f^{-1}B)$ with
$H^1(B,R^2f_*\mathbb Q)$, since the smooth rational surface fibers
have $H^1=H^3=0$. Restriction of the projective line-cover normalization
to the cover over $B$ is injective on $H^1$, with image the parabolic
cohomology. The fiberwise nondegenerate intersection matrices therefore
show that the cylinders inject
$$
H^1(T)(-1)\hookrightarrow H^3(X)^+,
\qquad
e_{\mathrm{line}}H^1(\mathcal C_{27})(-1)
\hookrightarrow H^3(X)^-.
\tag{B.15}
$$
Their dimensions are four and twenty. Equation (B.11) makes both
isomorphisms. In particular, no assertion about absence of boundary
cohomology is needed to fill these eigenspaces.

### B.3. Comparison with the specified quotient invariant

The dimension count alone would not identify the second space in
(B.15) with the particular $\mathcal W$ from §10.3. We supply that
comparison directly. Write $E:z^2=h^3+h$ and
$G:v^4=\prod_i(q-b_i)$, with normalization at infinity. Delete
$h=0,\infty$ from $E$, and the branch points and points above infinity
from $G$, obtaining $E^\circ,G^\circ$. Set
$$
U=E^\circ\times
\{(c_1,c_2)\in(G^\circ)^2:q(c_1)\ne q(c_2)\}.
\tag{B.16}
$$
The order-thirty-two group $H$ of §10.2 acts freely, and $U/H$ is an
open of $X$. Indeed its coordinates are
$$
\begin{aligned}
S&=q_1+q_2,& P&=q_1q_2,\\
x&=(v_1v_2)^2h,& y&=(v_1v_2)^3z,\\
t&=y/x,& w&=2x-t^2.
\end{aligned}
\tag{B.17}
$$
Here $x\ne0$ and $H=\prod_iL_i\ne0$, so the birational chart is
regular and the sextic and its small resolution agree.

Restriction $\mathcal W\to H^3(U)$ is injective. To prove this, the
complement of $U$ in $M=E\times G\times G$ is a union of product-fiber
divisors and
$E\times\Gamma_{\tau^j}$, $0\le j\le3$, where $\tau(q,v)=(q,iv)$.
The pure weight-three localization kernel is generated by Gysin images
of $H^1$ of the divisor normalizations. Product-fiber terms lie in
non-triple Künneth summands, whose $H$-invariants vanish. On
$E\times\Gamma_{\tau^j}$, the $H^1(G)$ term does likewise. The
$H^1(E)$ term maps to
$H^1(E)\otimes[\Gamma_{\tau^j}]$ and is killed by the group average:
the element $(\epsilon^2,\tau,\tau)$ acts as $-1$ on $H^1(E)$ and
preserves each graph class. Thus the algebraic average $e_H$ annihilates
every generator of this localization kernel.

Let $\Gamma\subset M\times X$ be the closure of the quotient graph.
The finite étale norm identity on (B.16), followed by the injectivity
just proved, gives the exact equality
$$
e_H\Gamma^*\Gamma_*=32\operatorname{id}
\quad\hbox{on }\mathcal W.
\tag{B.18}
$$
Elliptic inversion corresponds under (B.17) to $t\mapsto-t$ and acts
as $-1$ on $\mathcal W$. Hence $\Gamma_*\mathcal W\subset H^3(X)^-$.
Both have dimension twenty, and its inverse is the algebraic
correspondence $e_H\Gamma^*/32$. Composing this inverse with (B.15)
proves (B.3) and the asserted specified identification (10.2).

### B.4. Equations for the twenty-four-line source

Choose an Eckardt line $Z=0$, $P=\beta S$ over (B.14). The planes
through it are $P=\beta S+\mu Z$. Define $a,b,c$ by
$$
A(\beta)B_3-B(\beta)A_3
=Z\bigl(aS^2+bZS+cZ^2\bigr)
\quad\hbox{on }P=\beta S+\mu Z.
\tag{B.19}
$$
The residual conic is
$aS^2+bZS+cZ^2-\zeta t^2=0$. Its four finite degenerate fibers and
their line components give the normalization of
$$
b^2-4ac=0,\qquad
\zeta^2=-A(\beta)B(\beta),\qquad
\eta^2=\zeta a(\beta,\mu),\qquad
r=\zeta/A(\beta).
\tag{B.20}
$$
These equations mean compactification followed by normalization, so
vanishing affine denominators do not discard points. The factor in the
last quadratic equation is $\zeta a$, rather than $ra$, because (B.19)
uses the cleared coefficient. The order-four action is
$(\zeta,\eta)\mapsto(-\zeta,i\eta)$.

Projection from a line on a cubic surface gives five degenerate conics;
see [29, §2.1.2] for the classical construction. The fifth, at
$\mu=\infty$, consists of the other two Eckardt lines. Each is fixed
as a line by $\iota$, so this component disappears only after applying
the anti-involution projector. Every non-Eckardt line meets exactly
one Eckardt line; hence (B.20) describes all twenty-four, while (B.14)
describes the remaining three. Formula (B.4), applied to this normalized
cover, is the explicit tenfold. The entire anti-Prym need not equal its
incidence image.

The external incidence theory is classical, and the separate global
comparison (B.18) is essential to this application. In particular, the
twenty-four-branch uniformization of principally polarized sixfolds in
[28] is not applied to this different fibration. Neither this explicit
Gaussian model nor its small resolution constructs a normal deformation
to the generic orthogonal source: its whole third cohomology has rank
twenty-four, while the minimal generic spin module has rank thirty-two.

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