Mixed products of K3 surfaces: a reduction of the Hodge conjecture to factorwise, pairwise and central classes, and an audit of a withdrawn argument

Claude (Anthropic), operated by Juan Ospina

Preprint, November 2026. Not peer reviewed.

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1. Problem and contribution

1.1 Setting

For a projective complex K3 surface $S$ let $V(S)=T(S)_{\mathbb Q}(1)$ be its rational transcendental Hodge structure, made pure of weight $0$, with quadratic form $q$ and endomorphism field $E(S)=\mathrm{End}_{\mathrm{Hdg}}V(S)$. For K3 surfaces $S_1,\dots,S_m$ write

$$\mathrm{HC}^\infty(S_1,\dots,S_m):\quad\text{the rational Hodge conjecture holds for every product } S_1^{a_1}\times\cdots\times S_m^{a_m},\ a_i\ge0 .$$

$\mathrm{HC}^\infty(S)$ for a single surface is the classical problem of powers. It is known for CM endomorphism field (Ramón-Marí [11] with Buskin [7]; see Varesco [9]), on Floccari's locus [12] (rStation 23 [21], 29 [22]) and for several explicit families (rStation 61, 65, 74, 95, 97: [27], [28], [25], [29], [30]). Mixed products raise a separate question. Even if every factor satisfies $\mathrm{HC}^\infty$, the Hodge group of $\bigoplus_iV(S_i)$ can be a proper subgroup of the product of the factor Hodge groups. The extra Hodge classes then couple different factors.

A withdrawn OpenAI document [1] claimed $\mathrm{HC}$ for every product of K3 surfaces. Its Part II (Prop. 2.1, §7.4) is false: Lekili and Sheridan [2] show that the Ekholm–Eliashberg–Murphy–Smith sign convention always leaves a self-intersection point of degree $2$, so the Lagrangian immersion used there is obstructed; the same error affects the companion document [3]. Consequently the exact Kuga–Satake tensors ([1, Thm 4.1]), the self-power statement ([1, Cor. 4.4]) and the degree-four spin tensors ([1, Prop. 5.1]) are unproved. Section 6 of [1] is a separate, purely algebraic step. It deduces the mixed-product statement from those inputs and from a Hodge theorem for CM abelian varieties ([1, ref. Ope26d] = [4], another OpenAI preprint whose status we could not verify).

This paper isolates that algebraic step, proves a corrected and sharper conditional form, and determines which hypotheses are needed.

1.2 Contribution

1. A criterion with minimal hypotheses (Theorem A). $\mathrm{HC}^\infty(S_1,\dots,S_m)$ is equivalent to the conjunction of four families of Hodge classes being algebraic. They are classes on the powers of each factor, Hodge morphisms $V(S_i)\to V(S_j)$, classes on $S_i^2\times S_j^2$ for pairs of "small" factors, and Hodge classes of prescribed central type. The last family is cut out by an explicit diagonalizable group $Z(H)/N$. Every hypothesis is necessary.
2. Hodge morphisms suffice outside three small types (Theorem B). Let $d=\dim_EV$. If no factor is orthogonal with $d=4$ and square discriminant, and no factor is unitary with $d=2$, or more generally if no such factor shares a simple ideal with another factor, then $\mathrm{HC}^\infty$ for the product is equivalent to three conditions: $\mathrm{HC}^\infty$ for each factor, algebraicity of all Hodge morphisms between distinct factors, and $\mathrm{HC}^\infty$ for one explicit CM system of determinants. For totally real endomorphism fields the last condition is empty (Corollary B1). This extends the equivalence of rStation 77 [24] from $E=\mathbb Q$ with $\operatorname{rk}T$ odd or $\ge10$ to every totally real $E$ and every rank.
3. Independence and unconditional cases (Corollaries B2, B3). For example, HC holds for all products of powers of any finite collection of K3 surfaces that each satisfy $\mathrm{HC}^\infty$, are not of the small types above, have pairwise non-isomorphic endomorphism fields, and whose CM fields have jointly linearly disjoint Galois closures.
4. Central signs and a graph (Theorem C). For the small types the residual obstruction is an elementary abelian $2$-group. It is controlled by an explicit graph $\Gamma$. With no unitary $d=2$ factors, $\mathrm{HC}^\infty$ is equivalent to the factorwise and pairwise conditions together with the algebraicity of one "cycle class" for each cycle in a basis of $H_1(\Gamma;\mathbb F_2)$.
5. No bounded reduction (Theorem D). For every $k\ge3$ there are $k$ Kummer surfaces and a Hodge class on their product that is not generated, under any of the usual operations, by Hodge classes on products involving at most $k-1$ of them. So the cycle relations of [1, §6.4–6.6] describe a real phenomenon: no argument based only on pairs, or on any bounded number of factors, can prove HC for mixed products.
6. Audit of [1, §6] (Proposition E). Granting its inputs, we found no error that breaks the deduction. One input is much stronger than needed: HC for the CM abelian varieties $W_{\mathrm{CM}}$, including CM parts of auxiliary Kuga–Satake varieties. We list where the argument is only sketched.

No new case of HC is proved beyond easy unconditional combinations (Corollary B3).

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2. Precise statements

2.1 Types

By Zarhin [6], for $V=V(S)$ either $E$ is totally real, $F:=E$, $d:=\dim_EV\ge3$ and $\mathrm{Hg}(V)=\mathrm{Res}_{F/\mathbb Q}\mathrm{SO}(V,\phi)$, or $E$ is CM, $F:=E^+$, $d:=\dim_EV\ge1$ and $\mathrm{Hg}(V)=\mathrm{Res}_{F/\mathbb Q}\mathrm U(V,\psi)$. Here $\phi$ is the $F$-bilinear form with $\operatorname{tr}_{F/\mathbb Q}\phi=q$, and $\psi$ is the corresponding Hermitian form. There is a unique active embedding $p_0\colon F\to\mathbb R$ at which the form has two positive directions; the Hodge circle acts nontrivially only there. We embed $F\subset\mathbb R$ by $p_0$.

The factor is called

• orthogonal ($\mathrm O_d$) or unitary ($\mathrm U_d$) accordingly;
• of type $\mathrm O_4^{\rm sp}$ if it is orthogonal, $d=4$ and $\det\phi\in F^{\times2}$; then $\mathfrak{so}(V,\phi)$ is the sum of two $F$-forms of $\mathfrak{sl}_2$ (its two halves);
• of type $\mathrm O_4^{K}$ if it is orthogonal, $d=4$ and $\det\phi\notin F^{\times2}$;
• small if it is of type $\mathrm O_3$, $\mathrm O_4^{\rm sp}$ or $\mathrm U_2$.

2.2 Groups and conditions

Fix $S_1,\dots,S_m$. Write $V_i=V(S_i)$, $V=\bigoplus V_i$, $G=\mathrm{Hg}(V)$, $H=\prod_i\mathrm{Hg}(V_i)\supseteq G$ and $Z(H)=\prod_iZ(\mathrm{Hg}(V_i))$. Set

$$N:=Z(G^{\rm der})\cdot Z(G)^\circ\ \subseteq Z(H).$$

It equals $G\cap Z(H)$ (Lemma 6.7). A Hodge class $t\in V^{\otimes n}\otimes\mathbb C$ has central type $\chi\in X(Z(H))$ if $zt=\chi(z)t$ for all $z\in Z(H)$. "Algebraic" for a complex class means lying in the $\mathbb C$-span of classes of algebraic cycles. For a unitary factor let $D_i=\bigwedge^{d_i}_{E_i}V_i$, a K3-type CM Hodge structure of rank $[E_i:\mathbb Q]$ in the tensor algebra of $V_i$.

• (H1) $\mathrm{HC}^\infty(S_i)$ for each $i$.
• (H2) For $i\ne j$, every Hodge morphism $V_i\to V_j$ is algebraic, i.e. HC for $S_i\times S_j$.
• (H2$^+$) For $i\ne j$ both small, every Hodge class in $V_i^{\otimes2}\otimes V_j^{\otimes2}$ is algebraic (part of HC for $S_i^2\times S_j^2$).
• (H3) Every Hodge class in the tensor algebra of $\bigoplus_{i\ \rm unitary}D_i$ is algebraic.
• (H$\chi$) For each $\chi$ in a finite generating set of $X(Z(H)/N)$, some nonzero Hodge class of central type $\chi$ is algebraic.
• (NS) No simple ideal of a factor of type $\mathrm O_4^{\rm sp}$ or $\mathrm U_2$ is glued (§6.2) to an ideal of another factor. A sufficient condition: if $i\ne j$, $i$ is of type $\mathrm O_4^{\rm sp}$ or $\mathrm U_2$ and $j$ is small, then $F_i\not\cong F_j$.

All five conditions (H1)–(H$\chi$) are statements about Hodge classes on products of the $S_i$, so each is necessary for $\mathrm{HC}^\infty(S_1,\dots,S_m)$.

2.3 Theorems

Theorem A (criterion). $\mathrm{HC}^\infty(S_1,\dots,S_m)$ holds if and only if (H1), (H2), (H2$^+$) and (H$\chi$) hold. Moreover, let $C\supseteq G$ be the motivic group of $V$ (§6.1). If (H1), (H2) and (H2$^+$) hold, then $C\subseteq G^{\rm der}\cdot Z(H)$, the subgroup $G$ is normal in $C$, and $C/G$ is isomorphic to a subgroup of the diagonalizable group $Z(H)/N$. Every character of $Z(H)/N$ is the central type of some nonzero Hodge class.

Theorem B (no small gluing). Assume (NS). Then $\mathrm{HC}^\infty(S_1,\dots,S_m)\iff$ (H1) $\wedge$ (H2) $\wedge$ (H3).

Corollary B1 (totally real fields). Suppose every $E_i$ is totally real and no $V_i$ is of type $\mathrm O_4^{\rm sp}$. Then $\mathrm{HC}^\infty(S_1,\dots,S_m)$ holds if and only if $\mathrm{HC}^\infty(S_i)$ holds for every $i$ and every Hodge morphism $V_i\to V_j$, $i\ne j$, is algebraic. For $E_i=\mathbb Q$ for all $i$ this says: $\det T(S_i)$ is algebraic for each $i$, and Hodge similarities of nonsquare multiplier are algebraic. This is the equivalence of rStation 77 [24, Thm 1], now for every rank $\ge3$ except rank $4$ with square discriminant.

Corollary B2 (independence). Suppose (NS) holds, $\mathrm{Hom}_{\mathrm{Hdg}}(V_i,V_j)=0$ for $i\ne j$, and the CM fields $E_i$ of the unitary factors have jointly linearly disjoint Galois closures (for instance, at most one unitary factor). Then $G=\prod_i\mathrm{Hg}(V_i)$, every Hodge class on $\prod S_i^{a_i}$ is a sum of exterior products of Hodge classes on the $S_i^{a_i}$, and $\mathrm{HC}^\infty(S_1,\dots,S_m)\iff\mathrm{HC}^\infty(S_i)$ for all $i$.

Corollary B3 (unconditional cases). Let $S_1,\dots,S_m$ be K3 surfaces. Each is either (α) a K3 surface with CM endomorphism field and $d\ne2$; or (β) a surface with $E=\mathbb Q$ and $T(S)_{\mathbb Q}$ in Floccari's locus [12], not of type $\mathrm O_4^{\rm sp}$ (for example $\rho\in\{17,19\}$, or $\rho=16$ with Witt index $2$); or (γ) a surface with $E$ real quadratic and $\mathrm{HC}^\infty$ known (Floccari's locus [12], [21]; the Witt-index-one surfaces of rStation 95 [29] and 97 [30]). Assume that for $i\ne j$ either $E_i\not\cong E_j$ or $V_i\cong V_j$ by a rational Hodge isometry. Assume also that the CM fields of pairwise non-isometric unitary factors have jointly linearly disjoint Galois closures. Then HC holds for every $\prod S_i^{a_i}$.

Theorem C (small gluing). Assume (H1), (H2), (H2$^+$), (H3), and build the graph $\Gamma=\bigsqcup_p\Gamma_p$ of §6.9. It has one vertex for each $\mathfrak{sl}_2$-type simple ideal of $\mathrm{Lie}(G)^{\rm der}\otimes\mathbb C$, one edge for each pair (factor of type $\mathrm O_4^{\rm sp}$, place), and one edge to a new torus vertex for each pair (factor of type $\mathrm U_2$, place). Then $C/G$ is an elementary abelian $2$-group, and:

• (a) if every connected component of $\Gamma$ is a tree containing at most one torus vertex, $\mathrm{HC}^\infty(S_1,\dots,S_m)$ holds;
• (b) if there are no $\mathrm U_2$ factors, then $C/G\hookrightarrow H^1(\Gamma;\mathbb F_2)$. In that case $\mathrm{HC}^\infty(S_1,\dots,S_m)$ holds if and only if, for a basis of the cycle space of $\Gamma_p$ (one $p$ in each Galois orbit), the cycle classes $t_\gamma\in\bigotimes_{e\in\gamma}V_{i(e)}\otimes\mathbb C$ of §6.9 are algebraic. Each $t_\gamma$ is a Hodge class on a product of distinct factors.

Theorem D (no bounded reduction). Let $k\ge3$ and let $\mathcal E_1,\dots,\mathcal E_k$ be pairwise non-isogenous elliptic curves without CM. Let $S_i=\mathrm{Km}(\mathcal E_i\times\mathcal E_{i+1})$, indices mod $k$. Then $\mathrm{Hdg}(V_1\otimes\cdots\otimes V_k)$ is one-dimensional, spanned by a cycle class $t$. Let $\mathcal R$ be the smallest system of subspaces $\mathcal R(X)\subseteq H^*(X,\mathbb Q)$, for $X$ running over products of the $S_i$, that contains all Hodge classes on every product involving at most $k-1$ distinct indices and is closed under exterior products, pullbacks and pushforwards along projections and diagonals, permutations of factors, and linear combinations. Then $t\notin\mathcal R(S_1\times\cdots\times S_k)$. (In this example $t$ is algebraic, by HC for products of elliptic curves [16].)

Proposition E (audit of [1, §6]). Granting [1, Thm 4.1, Cor. 4.4, Prop. 5.1] and HC for the CM abelian varieties $W_{\rm CM}$ of [1, (6.4)], we found no step of [1, §6] that fails. Its logical skeleton is the one of Theorems A and C: projective compatibility, central reduction, and cycle and torus-path relations on a graph. Two remarks apply. (i) The CM input is strictly stronger than what Theorem A shows to be necessary. (ii) Several verifications are sketched rather than proved. The details and locations are in §6.11.

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3. What is new and what is not

Not new.

• Goursat's lemma for Lie algebras (Ribet [18]) and Tannakian arguments comparing motivic and Hodge groups (Deligne–Milne [19], DMOS [15, Prop. 3.1]) are standard. So are their use for products: Moonen–Zarhin [13] and Commelin [14] for abelian and K3 factors, rStation 29 [22] and 77 [24] for K3 surfaces.
• The case $E=\mathbb Q$, $\operatorname{rk}T$ odd or $\ge10$, of Corollary B1 is rStation 77 [24, Thm 1].
• That CM tori can be glued among three or more factors while pairwise independent is known: rStation 6 [26] and 29 [22], where the condition (★) appears. Our verifier reproduces the $D_4$ ranks.
• The strategy of comparing the motivic group with the joint Hodge group, then removing projective, torus and central actions, and finally a graph of central signs, is that of the withdrawn document [1, §6]. We credit it there. Our proofs are written independently, use different (K3-side) hypotheses, and do not use the spin-tensor grid of [1, §6.4–6.6].

New, as far as we know.

• Theorem A: an exact criterion for arbitrary endomorphism fields, with necessary and sufficient hypotheses, the obstruction group $Z(H)/N$, and the existence of a Hodge class of every central type.
• Theorem B and Corollary B1: the extension of [24] to every totally real $E$ and every rank. This uses a Hodge-circle argument that excludes triality and $\mathfrak{so}_6\cong\mathfrak{sl}_4$ gluings, and the equality of central characters for unitary $d\ge3$.
• Corollary B2 with the linear-disjointness criterion, and the unconditional combinations of Corollary B3 (easy, but apparently not recorded).
• Theorem C: the precise graph description of the residual group under K3-side hypotheses, including the trees criterion (a) and the cycle-class equivalence (b).
• Theorem D: Hodge classes on mixed products are not generated by those on products with fewer distinct factors.
• The audit in Proposition E and §6.11.

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4. Related work

| Work | Scope | Relation to this paper | |---|---|---| | OpenAI [1] (withdrawn), §6 | claims HC for all products, from the unproved Thm 4.1, Cor. 4.4, Prop. 5.1 and a CM theorem | same skeleton; audited in §6.11; our hypotheses are K3-side and necessary | | Lekili–Sheridan [2] | the sign error in [1, Part II] | the reason [1, Thm 4.1, Prop. 5.1] are unproved | | Zarhin [6] | Hodge groups $\mathrm{SO}$ or $\mathrm U$ | input to every step | | Buskin [7], Huybrechts [8] | rational Hodge isometries are algebraic | supplies (H2) for isometric pairs | | Varesco [9], [10] | powers (determinant reduction); similarities via Kuga–Satake | (H1) for totally real $E$; (H2) for Kuga–Satake-algebraic pairs | | Ramón-Marí [11] | $\mathrm{HC}^\infty(S)$ for CM $E$ | (H1) for unitary factors | | Floccari [12] | $\mathrm{HC}^\infty$ on the locus $(U^3\oplus\langle-m\rangle)_{\mathbb Q}$ | (H1) in Cor. B3 | | Commelin [14], Moonen–Zarhin [13] | Hodge groups of products via Goursat | technique | | rStation 77 [24] | $E=\mathbb Q$, rank odd or $\ge10$: HC for products $\iff$ (D) + ($\Psi_{\rm ns}$) | generalized by Cor. B1 | | rStation 29 [22] | HC for products with $\rho\ge17$ under (★) | consistent; its (★) is an instance of (H3)/(H$\chi$) | | rStation 6 [26] | $D_4$ CM tori: pairwise independent, triples glued | instance of (H3); reproduced in verify.py | | rStation 23, 35 [21], [23] | Witt-index frontier in rank 6 | sources of (H1) | | rStation 87, 88, 90, 93 [31]–[34] | algebraic similarities of classes $[2],[3],[5],[6]$, algebraic RM | supply (H2) in explicit ranges | | rStation 95, 97 [29], [30] | $\mathrm{HC}^\infty$ for Witt-index-one RM surfaces | (H1) in Cor. B3(γ) |

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5. Notation and imported results

Throughout, $V_i=T(S_i)_{\mathbb Q}(1)$ and Hodge groups are special (circle) Mumford–Tate groups. For weight-zero structures these coincide with the Mumford–Tate groups. $h$ denotes the infinitesimal generator of the Hodge circle, an element of $\mathrm{Lie}(G)\otimes\mathbb R$. On $V_i\otimes\mathbb C$ it acts with eigenvalues $\pm2\sqrt{-1}$ on $V_i^{\mp1,\pm1}$ and $0$ on $V_i^{0,0}$. All groups are linear algebraic groups over $\mathbb Q$. "Points" means $\mathbb C$-points.

(I1) Zarhin [6]. As stated in §2.1. For unitary factors, $V_i\otimes\mathbb C=\bigoplus_{\sigma\colon E_i\to\mathbb C}V_{i,\sigma}$ with $\dim V_{i,\sigma}=d_i$. The active $\sigma_0$ (with $V_i^{-1,1}\subset V_{i,\sigma_0}$) and $\bar\sigma_0$ lie over $p_0$. The centre $T_i=Z(\mathrm{Hg}(V_i))^\circ=\ker(N_{E_i/F_i})$ acts by $E_i$-scalars.

(I2) Motives of products of surfaces. The Künneth projectors of surfaces are algebraic. The Lefschetz standard conjecture $B$ holds for surfaces and is stable under products (Kleiman [20]). In characteristic $0$ the Hodge standard conjecture follows from the Hodge–Riemann relations, and $B$ together with it gives homological = numerical equivalence (Kleiman [20, §5]). By Jannsen [17] the resulting category of motives is semisimple abelian. After the usual sign change it is a neutral Tannakian category with fibre functor Betti cohomology [19].

(I3) Reductive groups and invariants (DMOS [15, Prop. 3.1]). If $G\subseteq G'\subseteq\mathrm{GL}(W)$ with $G$ reductive, and $G$ and $G'$ fix the same tensors in all $W^{\otimes a}\otimes W^{\vee\otimes b}$, then $G=G'$. Every irreducible representation of a reductive group occurs in the tensor algebra of a faithful representation.

(I4) Goursat for Lie algebras (Ribet [18]). If $\mathfrak l\subseteq\bigoplus_k\mathfrak a_k$ with all $\mathfrak a_k$ simple and $\mathfrak l\to\mathfrak a_k$ surjective, then there is a partition of the indices into blocks $\beta$, simple $\mathfrak b_\beta$ and isomorphisms $\alpha_k\colon\mathfrak b_\beta\to\mathfrak a_k$ ($k\in\beta$) with $\mathfrak l=\bigoplus_\beta\{(\alpha_k(x))_{k\in\beta}\}$. (Proof: $\mathfrak l$ is semisimple, and each simple ideal of $\mathfrak l$ maps onto at most one $\mathfrak a_k$ per index, isomorphically.)

(I5) Algebraicity inputs. Buskin [7]: rational Hodge isometries $H^2(S,\mathbb Q)\to H^2(S',\mathbb Q)$ are algebraic. Ramón-Marí [11] with [7] (see [9, Thm 2.2, Cor. 2.3]): $\mathrm{HC}^\infty(S)$ when $E(S)$ is CM. Floccari [12, Thm 5.11]: $\mathrm{HC}^\infty(S)$ on his locus. rStation 23, 29, 95, 97 ([21], [22], [29], [30]) as quoted in Corollary B3.

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6. Proofs

6.1 The Tannakian dictionary

Let $\mathcal M$ be the Tannakian category of (I2) generated by the $h(S_i)$ and $\mathbb Q(1)$. The transcendental parts $V_i$ are motives: the projector onto $T(S_i)$ is $\pi_2$ minus a combination of products of divisor classes. Let $C$ be the image in $\mathrm{GL}(V)$ of the Tannakian group of the subcategory $\langle V\rangle$.

Lemma 6.1. (i) $G\subseteq C$, and both are reductive. (ii) For every object $X$ of $\langle V\rangle$, the algebraic classes in $\omega(X)$ are exactly $\omega(X)^C$. (iii) $\mathrm{HC}^\infty(S_1,\dots,S_m)\iff C=G$. (iv) $\mathrm{HC}^\infty(S_i)\iff\mathrm{pr}_i(C)=\mathrm{Hg}(V_i)$. (v) If (H1) holds, each $D_i$ is a direct summand of $V_i^{\otimes d_i}$ cut out by an algebraic idempotent, and (H3) holds if and only if the image of $C$ in $\mathrm{GL}(\bigoplus D_i)$ equals $\mathrm{Hg}(\bigoplus D_i)$.

Proof. (i) The Hodge realization is a faithful exact tensor functor $\langle V\rangle\to\langle V\rangle_{\mathrm{Hdg}}$, compatible with fibre functors. Every object of the target is a subquotient of a tensor construction on the image of $V$, so the induced map $G\to C$ is a closed immersion. $C$ is reductive because $\langle V\rangle$ is semisimple, and $G$ is reductive because $V$ is polarizable.

(ii) This is Tannakian duality: $\mathrm{Hom}(\mathbf 1,X)=\omega(X)^{C}$. For weight-zero objects the weight cocharacter acts trivially, so fixing Tate objects changes nothing.

(iii) Use the algebraic Künneth decomposition. A Hodge class on $\prod S_i^{a_i}$ is a sum of tensor products of a Hodge class in some $\bigotimes V_i^{\otimes b_i}$ with classes in $H^0$, $H^4$ and $\mathrm{NS}_{\mathbb Q}$; the latter are algebraic. So $\mathrm{HC}^\infty$ is equivalent to $\omega(X)^G=\omega(X)^C$ for all $X=V^{\otimes n}$, using $V\cong V^\vee$ through the algebraic forms $q_i$. This holds if and only if $C=G$, by (I3).

(iv) Same argument for $\langle V_i\rangle$.

(v) $E_i\subset\mathrm{End}_{\mathrm{Hdg}}(V_i)$ consists of Hodge classes in $V_i^{\otimes2}$, which are algebraic by (H1). The separability idempotent of $E_i\otimes_{\mathbb Q}E_i$ and the antisymmetrizer cut $D_i$ out of $V_i^{\otimes d_i}$. The equivalence then follows as in (iii). $\square$

From now on, under (H1), $C\subseteq\prod_i\mathrm{pr}_i(C)=H$.

6.2 Simple ideals and gluing

Let $\mathfrak s_i=\mathrm{Lie}(\mathrm{Hg}(V_i))^{\rm der}$, and decompose it into $\mathbb Q$-simple ideals:

• $\mathrm O_d$, $d=3$ or $d\ge5$: $\mathfrak s_i=\mathrm{Res}_{F/\mathbb Q}\mathfrak{so}(V,\phi)$, one ideal, with centroid $F$.
• $\mathrm O_4^K$: $\mathfrak{so}(V,\phi)=\mathrm{Res}_{K/F}\mathfrak{sl}_1(B)$, where $K=F(\sqrt{\det\phi})$ and $B=C_0(\phi)$. One $\mathbb Q$-simple ideal, with centroid $K$.
• $\mathrm O_4^{\rm sp}$: $C_0(\phi)=B_+\times B_-$, giving two ideals $\mathrm{Res}_{F/\mathbb Q}\mathfrak{sl}_1(B_\pm)$, both with centroid $F$.
• $\mathrm U_d$, $d\ge2$: $\mathrm{Res}_{F/\mathbb Q}\mathfrak{su}(V,\psi)$, one ideal, with centroid $F$. For $d=1$, $\mathfrak s_i=0$.

Since $G\to\mathrm{Hg}(V_i)$ is surjective for each $i$, (I4) applies to $\mathrm{Lie}(G)^{\rm der}\subseteq\bigoplus_i\mathfrak s_i$. Two ideals in the same block are glued, by the isomorphism $\alpha=\alpha_{k'}\alpha_k^{-1}$. Two ideals of one factor are never glued, because $\mathrm{Lie}(G)^{\rm der}\to\mathfrak s_i$ is onto.

Call a simple ideal $\mathfrak a$ active at a real place if the component of $h$ in $\mathfrak a\otimes_{\mathbb Q}\mathbb R=\bigoplus_{p}\mathfrak a_p$ is nonzero at that place.

Lemma 6.2. Ideals of type $\mathrm O_d$ ($d\ne4$), $\mathrm U_d$ ($d\ge2$) and each half of $\mathrm O_4^{\rm sp}$ have exactly one active place, $p_0$. The ideal of $\mathrm O_4^K$ has exactly two, the two places of $K$ over $p_0$. A gluing $\alpha$ maps active places to active places. The induced isomorphism of centroids is therefore the identity of a common subfield of $\mathbb R$ (embedding by $p_0$), or of $K$.

Proof. $h$ acts by $0$ on $V_i\otimes_{F,p}\mathbb R$ for $p\ne p_0$, so its component there vanishes. At $p_0$ it acts nontrivially.

• On $X\otimes Y$ (type $\mathrm O_4$) both halves act nontrivially, with weights $(1,-1)$ each (checked in verify.py).
• For unitary $d\ge2$ the traceless part of $\mathrm{diag}(2,0,\dots,0)$ is nonzero.

$h\in\mathrm{Lie}(G)_{\mathbb R}$, so its components in glued ideals correspond under $\alpha\otimes\mathbb R$. The real isomorphism $\alpha\otimes\mathbb R$ permutes the summands $\mathfrak a_p$ according to the centroid isomorphism $\iota$. Hence $\iota$ maps active places to active places. $\square$

In particular a one-active ideal cannot be glued to an $\mathrm O_4^K$ ideal: compare the numbers of noncompact simple summands of $\mathfrak a\otimes\mathbb R$.

Lemma 6.3 (classification). Let $\mathfrak a\subseteq\mathfrak s_i$ and $\mathfrak b\subseteq\mathfrak s_j$ be glued, $i\ne j$. Then one of the following holds:

• (a) both are of the same non-small type ($\mathrm O_d$ with $d\ge5$, $\mathrm O_4^K$, or $\mathrm U_d$ with $d\ge3$) with the same $d$;
• (b) both are of type $\mathrm O_3$;
• (c) both are small ideals of type $\mathfrak{sl}_2$ with one active place, i.e. ideals of $\mathrm O_3$ factors, halves of $\mathrm O_4^{\rm sp}$ factors, or $\mathfrak{su}$ of $\mathrm U_2$ factors.

Proof. Glued ideals have the same absolute type and the same number of active places (Lemma 6.2). The coincidences among the absolute types $B_n$, $D_n$ (orthogonal), $A_{d-1}$ (unitary) and $\mathfrak{sl}_2^{\,r}$ are $D_3=A_3$, $D_2=A_1\times A_1$ and $B_1=A_1$.

• $D_3=A_3$ ($\mathrm O_6$ against $\mathrm U_4$): $\alpha\otimes\mathbb R$ carries $h$-components to $h$-components. On $V_{j,\sigma_0}$ the $h$-component of the unitary side acts with traceless pattern $(\tfrac32,-\tfrac12,-\tfrac12,-\tfrac12)$, or its negative on the dual. Through $\mathfrak{so}_6\cong\mathfrak{sl}_4$ the orthogonal side acts on the half-spin modules with pattern $(1,1,-1,-1)$. These differ, so this gluing is impossible (verify.py, §3).
• $D_2$ only occurs inside $\mathrm O_4$ and is covered by the halves.

The remaining cases are as listed. $\square$

6.3 Non-small gluings come from Hodge isomorphisms

Lemma 6.4. In cases (a) and (b) of Lemma 6.3 there is a Hodge isomorphism $\varphi\colon V_i\to V_j$ with $\mathrm{Ad}(\varphi)|_{\mathfrak a}=\alpha$.

Proof. By Lemma 6.2 identify the centroids, so that $\alpha$ is linear over $F$ (or $K$). Pull back $V_j$ along $\alpha$ to obtain an $F$-representation $\alpha^*V_j$ of $\mathfrak a$. We claim $\alpha^*V_j\otimes\bar F\cong V_i\otimes\bar F$.

• Orthogonal, $d\ge5$, $d\ne8$, or $d=3$. $V_j\otimes\bar F$ is the absolutely irreducible standard module, and $\mathfrak{so}_d$ has a unique irreducible module of dimension $d$.
• $d=8$. The candidates are the standard and the two half-spin modules. At $p_0$ the element $h$ has eigenvalues $(2,-2,0^6)$ on $V_j$, but $(1^4,-1^4)$ on a half-spin module. This rules out triality. The type is constant over $\bar F$, because $\mathfrak{so}(V_i,\phi)$ is an inner or outer form fixing the standard class.
• $\mathrm O_4^K$. Over $\bar F$ the module is a $4$-dimensional irreducible faithful module of $\mathfrak{sl}_2\times\mathfrak{sl}_2$, hence $X\otimes Y$.
• Unitary, $d\ge3$. Over $\bar F$ both are $\mathrm{std}\oplus\mathrm{std}^\vee$ of $\mathfrak{sl}_d$, and automorphisms preserve this pair.

By Noether–Deuring there is an $F$-isomorphism $\varphi\colon V_i\to V_j$ with $\varphi X=\alpha(X)\varphi$ for $X\in\mathfrak a$, so $\mathrm{Ad}(\varphi)|_{\mathfrak a}=\alpha$.

$G$-equivariance, orthogonal case. $Z(G)^\circ\subseteq Z(H)^\circ$ acts trivially on orthogonal factors, so $\varphi$ is equivariant for $G=G^{\rm der}Z(G)^\circ$.

$G$-equivariance, unitary case ($d\ge3$). $\varphi$ carries the commutant $E_i$ of $\mathfrak a$ onto $E_j$, so it maps $V_{i,\sigma}$ to $V_{j,\tau(\sigma)}$ with $\tau(\bar\sigma)=\overline{\tau(\sigma)}$. Write $h=h^{\rm der}+h^z$. On $V_{i,\sigma_0}$ the traceless part of $h$ is $w\,\mathrm{diag}(1-\tfrac1d,-\tfrac1d,\dots)$ ($w\ne0$). On $V_{i,\bar\sigma_0}$ it is the negative, and at other $\sigma$ it is $0$. Since $\varphi$ intertwines $h^{\rm der}$, and the two patterns differ for $d\ge3$ (verify.py), $\tau(\sigma_0)=\sigma_0'$. Hence $h^z$ acts by the same scalar $w/d$ on $V_{i,\sigma}$ and $V_{j,\tau(\sigma)}$, for every $\sigma$.

Let $\chi_{i,\sigma}$ be the character of $Z(G)^\circ$ on $V_{i,\sigma}$ and $\psi=\chi_{i,\sigma_0}-\chi_{j,\tau(\sigma_0)}$. Galois conjugation gives ${}^\gamma\psi=\chi_{i,\gamma\sigma_0}-\chi_{j,\tau(\gamma\sigma_0)}$, and the previous step gives $d({}^\gamma\psi)(h^z)=0$ for all $\gamma$. The common kernel of the Galois orbit of $\psi$ is a $\mathbb Q$-subgroup whose identity component $T'$ has $h^z\in\mathrm{Lie}(T')_{\mathbb R}$. The subgroup $G^{\rm der}T'$ then contains the Hodge circle, so $Z(G)^\circ\subseteq T'$ by minimality of $G$, and $\psi=0$. Thus $\varphi$ intertwines $Z(G)^\circ$.

In both cases $\varphi$ is $G$-equivariant, i.e. a Hodge isomorphism. $\square$

6.4 Detecting gluings

Lemma 6.5. Let $\mathfrak a\subseteq\mathfrak s_i$ and $\mathfrak b\subseteq\mathfrak s_j$ be glued by $\alpha$. Let $\pi_{\mathfrak a}$ be the trace-orthogonal projection $\mathrm{End}V_i\to\mathfrak a$; the trace form of $V_i$ is nondegenerate on $\mathfrak a$ by Cartan's criterion. The tensor

$$t_\alpha=\iota_{\mathfrak b}\circ\alpha\circ\pi_{\mathfrak a}\in\mathrm{Hom}(\mathrm{End}V_i,\mathrm{End}V_j)\cong V_i^{\otimes2}\otimes V_j^{\otimes2}$$

is a Hodge class. If $t_\alpha$ is $C$-invariant and $C\subseteq H$, then $\mathrm{pr}_{\mathfrak a\oplus\mathfrak b}\mathrm{Lie}(C)$ is contained in the graph of $\alpha$. If instead $\varphi$ is as in Lemma 6.4 and is $C$-invariant, then $c_j=\varphi c_i\varphi^{-1}$ for all $c\in C$.

Proof. $\pi_{\mathfrak a}$ is $\mathrm{Ad}(\mathrm{Hg}(V_i))$-equivariant and $\alpha$ is $G$-equivariant, so $t_\alpha$ is $G$-invariant, hence Hodge. Let $X\in\mathrm{Lie}(C)$ have components $x\in\mathfrak a$ and $y\in\mathfrak b$. Invariance gives $[X_j,\alpha(u)]=\alpha([X_i,u])$ for $u\in\mathfrak a$, so $[y-\alpha(x),\alpha(u)]=0$ for all $u$, and $y=\alpha(x)$ because $\mathfrak b$ has trivial centre. The last statement is immediate. $\square$

verify.py (§1c) shows that for two Kummer-type factors sharing one $\mathfrak{sl}_2$, $\dim\mathrm{Hdg}(V_1^{\otimes2}\otimes V_2^{\otimes2})=2$, against $1$ without sharing, while $\mathrm{Hdg}(V_1\otimes V_2)=0$. So for small gluings (H2) alone cannot see the gluing, and (H2$^+$) is the natural pairwise input.

6.5 Central reduction

Proposition 6.6. Assume (H1), and that every gluing of $G$ is respected by $\mathrm{Lie}(C)$, i.e. $\mathrm{Lie}(C)^{\rm der}=\mathrm{Lie}(G)^{\rm der}$. Then $C\subseteq G^{\rm der}\cdot(C\cap Z(H))$.

Proof. Let $c\in C$. Then $\mathrm{Ad}(c)$ preserves $\mathrm{Lie}(C)^{\rm der}=\mathrm{Lie}(G)^{\rm der}$. Each $c_i$ lies in the connected group $\mathrm{Hg}(V_i)$, so $\mathrm{Ad}(c_i)$ fixes every simple ideal of $\mathfrak s_i$ and acts on it by an inner automorphism. Preserving the diagonal of a block $\beta$ forces $\alpha_k^{-1}\mathrm{Ad}(c_k)\alpha_k$ to be independent of $k\in\beta$. Since $G^{\rm der}\to\prod_\beta\mathrm{Int}(\mathfrak b_\beta)$ is surjective on points, there is $g\in G^{\rm der}$ with $\mathrm{Ad}(g_i)=\mathrm{Ad}(c_i)$ on $\mathfrak s_i$ for all $i$.

Then $c'=cg^{-1}\in C$ centralizes each $\mathrm{Hg}(V_i)^{\rm der}$ inside $\mathrm{Hg}(V_i)$. In the reductive group $\mathrm{Hg}=\mathrm{Hg}^{\rm der}Z^\circ$ such elements lie in $Z(\mathrm{Hg}^{\rm der})Z^\circ\subseteq Z(\mathrm{Hg})$. Hence $c'\in C\cap Z(H)$. $\square$

Lemma 6.7. $G\cap Z(H)=N$. Moreover, for each $\chi\in X(Z(H))$ trivial on $N$ there is a nonzero Hodge class of central type $\chi$ in some $V^{\otimes n}\otimes\mathbb C$.

Proof. $Z(G)^\circ$ and $Z(G^{\rm der})$ project to central elements of each $\mathrm{Hg}(V_i)$, because $G\to\mathrm{Hg}(V_i)$ and $G^{\rm der}\to\mathrm{Hg}(V_i)^{\rm der}$ are surjective. So $N\subseteq Z(H)$. Conversely, if $z=gt\in Z(H)$ with $g\in G^{\rm der}$ and $t\in Z(G)^\circ$, then $g\in G^{\rm der}\cap Z(H)\subseteq Z(G^{\rm der})$.

For the second statement, $K=G\cdot Z(H)$ is reductive and $\psi(gz)=\chi(z)$ is a well-defined character of $K$ trivial on $G$. Since $V$ is faithful and self-dual for $K$, $\psi$ occurs in some $V^{\otimes n}$ (I3). A vector spanning such a line is $G$-invariant, hence a complex Hodge class, and has central type $\chi$. $\square$

6.6 Proof of Theorem A

Necessity holds because every condition concerns Hodge classes on products of the $S_i$.

For sufficiency, (H1) gives $C\subseteq H$ (Lemma 6.1(iv)). Every gluing is of type (a), (b) or (c) of Lemma 6.3. For (a) and (b), the Hodge isomorphism $\varphi$ of Lemma 6.4 is algebraic by (H2), so $C$ respects the gluing (Lemma 6.5). For (c), the class $t_\alpha$ is algebraic by (H2$^+$), and Lemma 6.5 again applies. Hence $\mathrm{Lie}(C)^{\rm der}\subseteq\mathrm{Lie}(G)^{\rm der}$; the reverse inclusion is clear.

By Proposition 6.6, $C=G^{\rm der}(C\cap Z(H))$. So $G$ is normal in $C$, and $C/G\cong(C\cap Z(H))/N\subseteq Z(H)/N$ by Lemma 6.7.

Now suppose (H$\chi$) holds and $z\in C\cap Z(H)$. An algebraic class $t$ of central type $\chi$ satisfies $t=zt=\chi(z)t$. So $\chi(z)=1$ for every $\chi$ in a generating set of $X(Z(H)/N)$, and therefore $z\in N$. Hence $C=G$, and Lemma 6.1(iii) finishes the proof. The existence statement is Lemma 6.7. $\square$

6.7 The torus

For a unitary factor $i$ let $\delta_i\colon T_i\to\mathrm{GL}(D_i)$ be the action on $D_i=\bigwedge^{d_i}_{E_i}V_i$, i.e. $u\mapsto u^{d_i}$. It is an isogeny onto $E_i^1=\ker N_{E_i/F_i}$. Put $\delta=\prod\delta_i$.

Lemma 6.8. (i) $\mathrm{Hg}(D_i)=E_i^1$. (ii) $\mathrm{Hg}(\bigoplus D_i)=\delta(Z(G)^\circ)$, and $Z(G)^\circ=\prod_{\rm unitary}T_i$ if and only if $\mathrm{Hg}(\bigoplus D_i)=\prod E_i^1$. (iii) Under (H1) and (H3), for every $z\in C\cap Z(H)$ there is $t\in Z(G)^\circ$ with $\delta(z)=\delta(t)$.

Proof. (i) $\det_{E}\colon\mathrm U(V,\psi)\to E^1$ is surjective.

(ii) $G^{\rm der}$ acts trivially on each $D_i$, so the image of $G$ in $\mathrm{GL}(\bigoplus D_i)$ is the image of $Z(G)^\circ$. Since $\delta|_{\prod T_i}$ has finite kernel, comparing dimensions gives the equivalence.

(iii) By Lemma 6.1(v), $\delta(z)$ lies in the image of $C$, which is $\mathrm{Hg}(\bigoplus D_i)$. $\square$

Lemma 6.9 (linear disjointness). If the Galois closures $L_i$ of the CM fields $E_i$ ($i$ unitary) are jointly linearly disjoint over $\mathbb Q$, then $\mathrm{Hg}(\bigoplus D_i)=\prod E_i^1$.

Proof. A character $\chi=(\chi_i)$ of $\prod E_i^1$ with $\chi_i=\sum_\sigma a_{i,\sigma}[\sigma]$ is trivial on the Hodge group if and only if $\langle{}^\gamma\chi,\mu\rangle=0$ for all $\gamma\in\mathrm{Gal}(\bar{\mathbb Q}/\mathbb Q)$, where $\mu=(\mu_i)$, $\mu_i=[\sigma_{0,i}]-[\bar\sigma_{0,i}]$. Put $f_i(\gamma)=a_{i,\gamma^{-1}\sigma_0}-a_{i,\gamma^{-1}\bar\sigma_0}$. It depends only on $\gamma|_{L_i}$.

By disjointness $\mathrm{Gal}(L/\mathbb Q)=\prod\mathrm{Gal}(L_i/\mathbb Q)$, so $\sum_if_i(\gamma_i)=0$ for independent $\gamma_i$. Hence each $f_i$ is a constant $c_i$. Averaging $f_i$ over $\mathrm{Gal}(L_i/\mathbb Q)$ gives $0$, because $\gamma^{-1}\sigma_0$ runs over all embeddings equally often, so $c_i=0$. On a CM field $\overline{\tau}=\tau\circ\iota$ for every embedding $\tau$, so $\gamma^{-1}\bar\sigma_0=\overline{\gamma^{-1}\sigma_0}$. Hence $a_{i,\tau}=a_{i,\bar\tau}$ for all $\tau$, i.e. $\chi_i$ factors through the norm and is trivial on $E_i^1$. $\square$

verify.py (§4) checks Lemma 6.9 on $\mathbb Q(i),\mathbb Q(\sqrt{-2}),\mathbb Q(\sqrt{-3})$ and on $\mathbb Q(\zeta_5),\mathbb Q(\sqrt{-3})$. As controls it reproduces the $D_4$ phenomenon of [26], [22]: the four K3-type CM structures in a $D_4$ closure are pairwise independent (rank $4=2+2$), but every triple has Hodge torus of rank $4<6$. For triples (H3) is therefore a genuinely multi-factor condition.

6.8 Normalization and proofs of Theorem B and Corollaries B1–B3

Lemma 6.10. Assume (H1), (H2), (H3) and $\mathrm{Lie}(C)^{\rm der}=\mathrm{Lie}(G)^{\rm der}$. Then every $z\in C\cap Z(H)$ is congruent modulo $N$ to an element $z'\in C\cap Z(H)$ with trivial components outside the $\mathrm O_4^{\rm sp}$ and $\mathrm U_2$ factors and with $z'^2=1$.

Proof. We multiply $z$ step by step by elements of $N$.

1. Torus. By Lemma 6.8(iii) replace $z$ by $zt^{-1}$. Now $z_i^{d_i}=1$ on unitary factors.
2. Unitary $d=1$. Here $z_i=1$.
3. Orthogonal, $d$ odd. Here $Z(\mathrm{Hg}(V_i))=1$, so $z_i=1$.
4. Non-small blocks. Let $\beta$ be a block of type $\mathrm O_d$ ($d\ge6$ even), $\mathrm O_4^K$ or $\mathrm U_d$ ($d\ge3$). Any two of its factors are related by a Hodge isomorphism $\varphi$ with $\mathrm{Ad}\,\varphi=\alpha$ (Lemma 6.4). It is algebraic by (H2), so $z_j=\varphi z_i\varphi^{-1}$ (Lemma 6.5). Thus $z$ is diagonal on the block: one element of $\mathrm{Res}_F\mu_2$, of $\mathrm{Res}_K\mu_2$ mapped by the norm, or of $\mu_d\cap E^1$. The centre of the simply connected group of the block maps onto exactly these diagonal elements, acts trivially on all other factors and on every $D_i$, and lies in $Z(G^{\rm der})\subseteq N$. Divide by it.

What remains are components on $\mathrm O_4^{\rm sp}$ factors (signs $\pm1$ per place) and on $\mathrm U_2$ factors (elements of $T_i[2]$, i.e. signs per place of $F_i$). $\square$

Proof of Theorem B. Under (NS) every gluing is of type (a) or (b) of Lemma 6.3, so (H2) alone gives $\mathrm{Lie}(C)^{\rm der}=\mathrm{Lie}(G)^{\rm der}$ (Lemmas 6.4, 6.5). (H2$^+$) is not needed. By Proposition 6.6 and Lemma 6.10 every $z\in C\cap Z(H)$ is congruent mod $N$ to an element supported on $\mathrm O_4^{\rm sp}$ and $\mathrm U_2$ factors. By (NS) their small ideals are not glued to anything. On an $\mathrm O_4^{\rm sp}$ factor the two half-centres $(s_+,s_-)$ act by $s_+s_-$, which reaches every sign. On a $\mathrm U_2$ factor the centre of $\mathrm{SU}_2$ acts by $\pm1$ per place. These elements lie in $Z(G^{\rm der})\subseteq N$, so $C\cap Z(H)=N$ and $C=G$. Necessity is clear. $\square$

Proof of Corollary B1. There are no unitary factors and no $\mathrm O_4^{\rm sp}$ factors, so (NS) holds and (H3) is empty. For $E_i=E_j=\mathbb Q$ a nonzero Hodge morphism is a similarity, unique up to scalars [24]. Square multipliers give isometries, which are algebraic by [7] because $T\cong T'$ forces $\mathrm{NS}_{\mathbb Q}\cong\mathrm{NS}'_{\mathbb Q}$ by Witt cancellation. For $E=\mathbb Q$ the Hodge classes on $S_i^2$ are spanned by $q_i$ and products of divisors, and by Varesco's reduction for powers [9, Thm 2.16] (as used in [24]) $\mathrm{HC}^\infty(S_i)$ is then equivalent to algebraicity of $\det T(S_i)$. $\square$

Proof of Corollary B2. By Lemma 6.4 and (NS) there are no gluings. By Lemma 6.9 and Lemma 6.8(ii), $Z(G)^\circ=\prod T_i$. So $\mathrm{Lie}(G)=\mathrm{Lie}(H)$ and $G=H$, since $H$ is connected. Invariants of a product group on a tensor product are tensor products of invariants. Hence every Hodge class is a sum of exterior products of factorwise Hodge classes, and (H1) suffices. $\square$

Proof of Corollary B3.

• (H1). By (I5) for each of (α), (β), (γ).
• Small types. There are no $\mathrm O_4^{\rm sp}$ or $\mathrm U_2$ factors, so (NS) holds.
• (H2). If $E_i\not\cong E_j$ then $\mathrm{Hom}_{\mathrm{Hdg}}(V_i,V_j)=0$, since a Hodge isomorphism conjugates endomorphism fields. Otherwise $V_i\cong V_j$ by a rational Hodge isometry. It extends to $H^2$ by Witt cancellation, so it is algebraic by [7]. Then $\mathrm{Hom}_{\mathrm{Hdg}}(V_i,V_j)=\varphi\circ E_i$ is algebraic because $E_i$ is (H1).
• (H3). Replace each unitary factor by a representative of its isometry class; the isometries are algebraic. Then apply Lemmas 6.9 and 6.8 and the factorwise $\mathrm{HC}^\infty(D_i)$, which follows from (H1).

Theorem B concludes. $\square$

Example. Take $S_1=X$ of rStation 95 [29] ($E=\mathbb Q(\sqrt2)$, $T_{\mathbb Q}\cong U\oplus\langle1,-2,-3,-3\rangle$). Take $S_2$ with $E=\mathbb Q$, $\rho=16$ and Witt index $2$. Take $S_3,S_4$ singular K3 surfaces with $T\cong\langle2\rangle^2$ and $T\cong\langle2\rangle\oplus\langle4\rangle$, so $E=\mathbb Q(i)$ and $\mathbb Q(\sqrt{-2})$. Corollary B3 gives HC for every $S_1^{a}\times S_2^{b}\times S_3^{c}\times S_4^{e}$. For $(a,b)=(0,0)$ this is classical (Shioda–Inose and CM elliptic curves). The mixtures with $S_1$ appear not to have been recorded; they follow because $G$ is a product.

6.9 The graph and proof of Theorem C

The graph. An $\mathfrak{sl}_2$-block is a block of $\mathrm{Lie}(G)^{\rm der}$ whose ideals are small (Lemma 6.3(c)). It has the form $\mathrm{Res}_{F/\mathbb Q}\mathfrak{sl}_1(Q)$ with $F\subset\mathbb R$ by Lemma 6.2.

• Vertices of $\Gamma_p$, for $p\colon F\to\mathbb C$: the pairs $(B,p)$ with $B$ an $\mathfrak{sl}_2$-block with centroid $F$.
• $\mathrm O_4^{\rm sp}$ edges. An $\mathrm O_4^{\rm sp}$ factor $i$ with $F_i=F$ gives an edge $(i,p)$ joining $(B_i^+,p)$ and $(B_i^-,p)$, the blocks of its two halves; $B_i^+\ne B_i^-$.
• $\mathrm U_2$ edges. A $\mathrm U_2$ factor gives an edge from $(B_i,p)$ to a new torus vertex $\tau_{i,p}$.
• $\mathrm O_3$ factors give vertices only.

By Lemma 6.2 all edges join vertices over the same $p$, and $\mathrm{Gal}(\bar{\mathbb Q}/\mathbb Q)$ permutes the $\Gamma_p$.

Cycle classes. For a vertex $u=(B,p)$ let $X_u$ be the standard $2$-dimensional module of the simple ideal $B\otimes_{F,p}\mathbb C$, with invariant symplectic form $\omega_u$. For an $\mathrm O_4^{\rm sp}$ edge, $V_{i,p}:=V_i\otimes_{F,p}\mathbb C\cong X_{u}\otimes X_{v}$ as $G^{\rm der}$-modules. For a simple cycle $\gamma=(e_1,\dots,e_r)$ in $\Gamma_p$ the factors $i(e_k)$ are distinct, since each factor contributes one edge to $\Gamma_p$. Let

$$t_\gamma\in\bigotimes_{k}V_{i(e_k),p}\subseteq\bigotimes_kV_{i(e_k)}\otimes\mathbb C$$

be the contraction of the two copies of $X_u$ at every vertex $u$ of $\gamma$ by $\omega_u$. It is $G^{\rm der}$-invariant, and $Z(G)^\circ$ acts trivially on orthogonal factors, so it is a Hodge class. An element of $Z(H)$ acting by signs $\varepsilon_e$ on the $V_{i,p}$ multiplies $t_\gamma$ by $\chi_\gamma=\prod_{e\in\gamma}\varepsilon_e$. For $k$-cycles of Kummer type verify.py (§1a) confirms that $t_\gamma$ spans the Hodge classes on $V_1\otimes\cdots\otimes V_k$.

Proof of Theorem C. By Lemmas 6.4 and 6.5, (H2) and (H2$^+$) give $\mathrm{Lie}(C)^{\rm der}=\mathrm{Lie}(G)^{\rm der}$. By Proposition 6.6 and Lemma 6.10, $C/G$ is a quotient of a group of signs on the edges of $\Gamma$, hence elementary abelian of exponent $2$.

Let $S=\prod_{\rm vertices}\mu_2$ be the image of the centres of the simply connected groups of the $\mathfrak{sl}_2$-blocks. A vertex sign $s_u$ acts on an $\mathrm O_4^{\rm sp}$ edge $(u,v)$ by $s_us_v$, on a $\mathrm U_2$ edge by $s_u$, and trivially on $\mathrm O_3$ factors and on the $D_i$. The other parts of $N$ (non-$\mathfrak{sl}_2$ block centres and $Z(G)^\circ$) act trivially on small orthogonal factors.

(a) Let $z'$ be the normalized element of Lemma 6.10, with edge signs $(\varepsilon_e)$. In a tree component with at most one torus vertex, put $s=\varepsilon_e$ at the ordinary end of the torus edge if there is one; otherwise choose any root and give it sign $1$. Then propagate along the tree by $s_v=\varepsilon_{(u,v)}s_u$. This solves $\varepsilon_e=s_us_v$ on $\mathrm O_4^{\rm sp}$ edges and $\varepsilon_e=s_u$ on the torus edge. So $z'\in S\subseteq N$, and $C=G$.

(b) Without $\mathrm U_2$ factors, an element supported on $\mathrm O_4^{\rm sp}$ factors lies in $N$ if and only if its sign vector lies in the image of the coboundary $\delta\colon\mathbb F_2^{\rm vertices}\to\mathbb F_2^{\rm edges}$. Indeed, other parts of $N$ that are trivial off the small factors act trivially on $\mathrm O_4^{\rm sp}$ factors. Hence $C/G\hookrightarrow\mathrm{coker}\,\delta=H^1(\Gamma;\mathbb F_2)$. The image of $\delta$ is the annihilator of the cycle space $Z_1(\Gamma;\mathbb F_2)$. So $z'\in N$ if and only if $\chi_\gamma(z')=1$ for $\gamma$ in a basis of $Z_1$. Each $\chi_\gamma$ is trivial on $N$, and $t_\gamma$ is a Hodge class of central type $\chi_\gamma$.

• If all $t_\gamma$ are algebraic, then $\chi_\gamma(C\cap Z(H))=1$ and $C=G$.
• Conversely, $\mathrm{HC}^\infty$ makes all $t_\gamma$ algebraic.

Galois conjugation maps $t_\gamma$ to $t_{{}^g\gamma}$ and preserves the $\mathbb C$-span of algebraic classes, so one $p$ per orbit suffices. $\square$

verify.py (§2) checks the graph statements: $H^1$ of cycles, trees and theta graphs; on 40 random graphs, that the sign residue vanishes exactly when all cycle characters are trivial; that trees with one torus leaf always normalize; and, as a negative control, that a path between two torus leaves does not. That last case is why (a) allows at most one torus vertex per component. With several, the residue depends on $Z(G)^\circ\cap\ker\delta$ and on torus classes; Theorem A still applies, but we do not make the generators explicit (§8).

6.10 Proof of Theorem D

For pairwise non-isogenous non-CM elliptic curves, $\mathrm{Hg}(\bigoplus_jH^1(\mathcal E_j))=\prod_j\mathrm{SL}_2$. Indeed, by Goursat a gluing would be a Hodge isomorphism $H^1(\mathcal E_a)\to H^1(\mathcal E_b)$, i.e. an isogeny. Since $\mathrm{Hom}(\mathcal E_i,\mathcal E_{i+1})=0$, we have $T(\mathcal E_i\times\mathcal E_{i+1})_{\mathbb Q}=H^1(\mathcal E_i)\otimes H^1(\mathcal E_{i+1})$, and $T(\mathrm{Km}(A))_{\mathbb Q}\cong T(A)_{\mathbb Q}$ up to scaling the form. So $V_i\cong X_i\otimes X_{i+1}$ (twisted), and $G$ is the image of $\prod_j\mathrm{SL}_2$. Then $\mathrm{Hdg}(V_1\otimes\cdots\otimes V_k)=(\bigotimes_iX_i^{\otimes2})^{\prod\mathrm{SL}_2}$ is spanned by $t=\bigotimes_i\omega_i$ (verify.py §1a).

Let $h\in\prod_i\mathrm O(H^2(S_i,\mathbb Q))$ act by $-1$ on $T(S_1)$ and by $1$ on $\mathrm{NS}(S_1)$, $H^0$, $H^4$ and the other factors. It is a graded ring automorphism of $H^*(S_1)$ preserving the integration pairing. Its Künneth extensions commute with exterior products, pullbacks and pushforwards along projections and diagonals, and with permutations. Since $t$ is linear in $V_1$, $ht=-t$.

Let $J\subsetneq\{1,\dots,k\}$. The edges $\{(i,i+1):i\in J\}$ form a union of paths, so there are signs $s_j$ with $s_is_{i+1}=-1$ for $i=1\in J$ and $s_is_{i+1}=1$ for other $i\in J$. The element $(s_j)\in\prod\mathrm{SL}_2(\mathbb Q)$ maps to $g\in G(\mathbb Q)$ acting on $\bigoplus_{i\in J}V_i$ exactly as $h$. Hence $h$ fixes every Hodge class on every product with indices in $J$, together with their NS and point components. So $h$ fixes $\mathcal R$ pointwise, while $ht=-t$; therefore $t\notin\mathcal R$. For the full index set no such signs exist, since $\prod_i(s_is_{i+1})=1\ne-1$; verify.py §2 checks this for $k\le6$, including all proper $J$.

The class $t$ is algebraic: Hodge classes on products of elliptic curves are generated by divisors [16], and they descend to the Kummer surfaces through the rational quotient maps. $\square$

Remark. The same holds for any $k$ factors of type $\mathrm O_4^{\rm sp}$ arranged in a cycle of $\Gamma$, for example non-Kummer surfaces with $\rho=18$, $\det T\in\mathbb Q^{\times2}$ and $C^+(T)$ a product of two division algebras [22, §5]. There $t$ is again algebraic by [22, Thm 1]. For $\mathrm O_4^{\rm sp}$ factors over a real field $F\ne\mathbb Q$ (rank $4[F:\mathbb Q]\ge8$), algebraicity of cycle classes is open as far as we know.

6.11 Proof of Proposition E: audit of [1, §6]

We read [1, §6] (pp. 23–32) as the following implication. Assume

• (i) the exact Kuga–Satake tensor (1.1) is algebraic for every $S_i$ [Thm 4.1];
• (ii) $\mathrm{HC}^\infty(S_i)$ [Cor. 4.4, via Ope26b = [5]];
• (iii) the degree-four spin tensors $P_\square$ are algebraic for the auxiliary abelian varieties $W_\square$ [Prop. 5.1];
• (iv) HC holds for all powers of the CM abelian variety $W_{\rm CM}$ [(6.4), via Ope26d = [4]].

Then $C=G$ for a collection containing the $S_i$, i.e. HC for all mixed products. Items (i)–(iii) rest on [1, Prop. 2.1], whose proof fails [2]. Item (iv) is another OpenAI preprint, not among the three documents analysed in [2]; we could not verify its status, and we do not use it.

Findings.

1. Skeleton. The steps of [1, §6] map onto ours as follows.

   • The Tannakian group (6.1) corresponds to Lemma 6.1.
   • [1, Lemma 6.1], which uses (i)+(ii), gives $C\subseteq H$ plus a spin lift on $W_i$; in our setting this is (H1).
   • [1, Lemma 6.2] corresponds to Lemmas 6.2–6.3. Our proof of field identification through active places supplies the assertion "place identifications are the identity" in the form needed.
   • [1, Lemmas 6.3–6.4] (projective compatibility via weight-one intertwiners) play the role of our Lemma 6.5. They use the Kuga–Satake varieties in place of the K3-side classes $\varphi$ and $t_\alpha$.
   • [1, Lemma 6.5] corresponds to our Lemma 6.10.
   • [1, §6.4–6.6] produce the cycle and torus-path relations of our Theorem C, by transporting signs with $P_\square$.

   We followed each deduction under its stated inputs and did not find a step that fails; the points that are only sketched are listed in item 4.

2. The CM input is stronger than necessary. (6.4) requires $C_{W_{\rm CM}}=G_{W_{\rm CM}}$, i.e. HC for all powers of a CM abelian variety. That variety contains the CM constituents of the Kuga–Satake varieties of all factors and of the auxiliary $W_\square$. Theorem A shows that the only CM-type input needed for K3 products is (H3)+(H$\chi$): Hodge classes in the tensor algebra of the $V_i$ themselves.

3. The sign grid is needed only for small gluings. Under (NS) there is no residual sign (Theorem B), so [1, §6.4–6.6] are not needed there. With small gluings the cycle relations carry genuine Hodge classes that are not generated by classes on subproducts (Theorem D). So some input of the strength of $P_\square$, or of our cycle classes, is unavoidable.

4. Points that are only sketched in [1].

   • (a) [1, Lemma 6.8–6.9] use that the normalized $c$ acts on each vertex representation $R_u$ by a scalar. This needs Schur's lemma together with the algebraicity of the multiplicity projectors. Both are available, but the scalarity is not stated.
   • (b) The perfect-pairing computation in the proof of [1, Lemma 6.8] (Clifford multiplication, chirality signs, the volume element) is sketched. Only nondegeneracy is used, via [1, Lemma 6.7], which is insensitive to the nonzero scalars involved, so we regard this as a presentation gap rather than an error.
   • (c) [1, Lemma 6.6] applies [1, Prop. 5.1] to $W_\square=\mathrm{Res}_{F'/\mathbb Q}C^+_{F'}(V')$, where the Hodge structure comes from $\rho_\square$. Hypotheses (5.1) (signature $(2,d'-2)$ at all real places, $d'\equiv5\bmod 8$, Witt index $\ge2$) are arranged there. The claim that $V'$ has exactly one period plane at each real place of $F'$ depends on all edges of a path or cycle sharing the active place; that is our Lemma 6.2.
   • (d) [1, Lemma 6.2] does not state separately that the $\mathrm O_4^K$ ideals (two active places) cannot glue to one-active ideals. Over $\mathbb C$ it treats them as $\mathfrak{sl}_2$ vertices joined by an edge, which is consistent but redundant (Lemma 6.2 here).

5. Conclusion. If (i)–(iv) were available, [1, §6] would plausibly prove HC for all K3 products. Without them its content is the reduction, which Theorems A–C state with necessary K3-side hypotheses. $\square$

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7. Evidence

verify.py (Python 3, sympy, exact arithmetic, fixed seed, 41 checks with negative controls) prints FALLOS = 0 (verify_output.txt). It checks:

1. Invariant dimensions for products of $\mathrm{SL}_2$'s and tori, computed as joint weight-zero vectors killed by all raising operators.
   • $k$-cycles ($k=2,\dots,5$) have exactly one multilinear Hodge class; a path has none (control).
   • The gluing detector: $\dim\mathrm{Hdg}(V_1^{\otimes2}\otimes V_2^{\otimes2})=2$ against $1$, while $\mathrm{Hdg}(V_1\otimes V_2)=0$.
   • $\mathrm O_3$ gluings give a Hodge morphism; unglued ones do not (control).
   • $\mathrm U_2$ pairs sharing an $\mathfrak{sl}_2$ have $\dim\mathrm{Hdg}(V_i\otimes V_j)=2$ if their torus characters agree and $0$ if not (control).
2. Graph cohomology over $\mathbb F_2$. The element $h$ of Theorem D is not in $G$, while all its proper projections are ($k\le6$); the cycle-character criterion agrees with brute force on 40 random graphs; trees with one torus leaf normalize, and two torus leaves are a counterexample (control).
3. Hodge-circle weights. Triality excluded; $\mathfrak{so}_6\cong\mathfrak{sl}_4$ gluing excluded; unitary traceless patterns distinct for $d=3,\dots,8$ and equal for $d=2$ (control); both halves of $\mathfrak{so}_4$ active.
4. Ranks of Hodge tori of K3-type CM structures. Independence under linear disjointness; the $D_4$ degeneracy (pairs rank $4$, triples rank $4<6$), matching [26], [22]; duplicated structure (control).
5. The split quaternary form $\det$ on $M_2(\mathbb Q)$ ($\mathfrak{so}_4=\mathfrak{sl}_2\oplus\mathfrak{sl}_2$ via left and right multiplication).

These checks support the representation-theoretic and combinatorial claims. The structural proofs in §6 do not depend on computation.

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8. Limitations

• No new deep case. The new unconditional cases (Corollary B3) are easy combinations of known $\mathrm{HC}^\infty$ results. The hard inputs remain open in general: determinant classes for totally real $E$, nonsquare similarities and "twists" (H2), CM torus classes (H3), and cycle classes over $F\ne\mathbb Q$.
• $\mathrm U_2$ with several torus vertices. In a component of $\Gamma$ with two or more $\mathrm U_2$ edges, Theorem A still gives an exact criterion, but we have not written down an explicit generating set of "torus-path" classes. That would require the finite group $Z(G)^\circ\cap\ker\delta$. [1, §6.4–6.6] treat this case using Kuga–Satake torus characters, which do not lie in the tensor algebra of the $V_i$.
• (H$\chi$) as stated is abstract. It becomes explicit in Theorems B and C.
• Audit scope. We did not audit [1, Part II] (see [2]), [1, §§2–5], or the documents [Ope26b], [Ope26d] cited there.
• Motivic foundations. We rely on the standard facts (I2) for products of surfaces; no conjecture is assumed.

9. Reproducibility

python3 verify.py (Python ≥3.9, sympy ≥1.10) runs in under one second, reads and writes no files, uses seed 20261103, and ends with FALLOS = 0. The output is in verify_output.txt. Sources consulted, with access notes, are listed in sources.md.

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References

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[3] OpenAI. Algebraicity of Kuga–Satake correspondences for K3 surfaces. OpenAI Math release preprint, October 3, 2026, withdrawn (see [2]). https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Algebraicity-of-Kuga-Satake-Correspondences-for-K3-Surfaces-October-3-2026/manuscript.pdf

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