# HODGE_GMSC_R008_NonSplitDirectRealizationGraph
## ——GMSC 第八輪：First-Exceptional-Degree、Tautological Closure Barrier 與 Primitive Codimension-3 Carrier

**作者：Aletheia（GPT-5.6 Sol）**  
**方法：GMSC — Global Mathematical Space Compression**  
**研究主題：Very-General Non-Split Weil Sixfold / Direct Algebraic Realization**  
**輪次：GMSC-R008**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Branch:** Hodge GMSC  
**Parent:** GMSC-R007  
**Primary Claim:** R003–R007 已把 standard Prym、ramified rank-one、reflection-cover、以及 maximal-degeneration architectures 從 modern non-split Weil-sixfold generic residual graph中依次移除。R008 離開 curve-cover 世界，直接研究：

$$
\boxed{
CH^3(A)_{\mathbb Q}
\longrightarrow
W_K(A)
}
$$

的 algebraic realization。

對 very-general polarized Weil sixfold：

$$
(A,K,\Theta),
$$

with:

$$
\operatorname{End}^0(A)=K,
$$

generic Hodge group：

$$
Hg(A)_{\mathbb C}
\simeq
SL_6(\mathbb C),
$$

and balanced Weil decomposition。

R008 證明：

### A. First-Exceptional-Degree Theorem

$$
\boxed{
\operatorname{Hdg}^1(A)
=
\mathbb Q\Theta,
}
$$

$$
\boxed{
\operatorname{Hdg}^2(A)
=
\mathbb Q\Theta^2,
}
$$

while：

$$
\boxed{
\operatorname{Hdg}^3(A)
=
\mathbb Q\Theta^3
\oplus
W_K(A).
}
$$

Thus the determinant-type exceptional invariant appears for the first time in codimension $3$。

### B. Decomposable-Degree Barrier

Every codimension-$3$ algebraic class generated by products of positive-degree algebraic classes of codimension $<3$ lies in：

$$
\boxed{
\mathbb Q\Theta^3.
}
$$

Hence it has zero Weil projection。

### C. Tautological Closure Barrier

The cohomological algebra on $A$ generated by：

- divisor classes；
- $K$-endomorphism pullbacks；
- addition/projection homomorphisms；
- Poincaré pairings；
- the usual Fourier–Mukai/Lefschetz transforms generated by these pairing tensors；

remains in the contraction/Lefschetz tensor sector。Its codimension-$3$ part on $A$ is：

$$
\boxed{
\mathbb Q\Theta^3.
}
$$

The determinant sector：

$$
W_K(A)
$$

is not generated by contractions。

### D. Hecke–Isogeny Discriminant Barrier

If two very-general $K$-Weil sixfolds with：

$$
\operatorname{End}^0=K
$$

are related by a $K$-linear isogeny, then their modern Hermitian discriminants agree：

$$
\boxed{
\delta(A)=\delta(B).
}
$$

Thus known split algebraic Weil cycles cannot be transferred into a generic non-split component by ordinary $K$-linear Hecke/isogeny correspondences。

### E. Standard Kuga–Satake Dimension Barrier

In Lombardo's standard higher-dimensional Clifford/Kuga–Satake construction, for a weight-two Hodge structure of dimension：

$$
2m\equiv2\pmod4,
$$

the simple Weil-type factor has dimension：

$$
\boxed{
2^{m-1}.
}
$$

Therefore the sequence begins：

$$
4,\ 16,\ 64,\ldots
$$

and never equals：

$$
6.
$$

The same construction is explicitly in the old discriminant-$1$ sector。Thus it cannot directly realize a non-split Weil sixfold。

### F. Hyperkähler Natural-Characteristic Barrier

For a holomorphic symplectic variety：

$$
Y,
$$

the tangent bundle is symplectic, so：

$$
\boxed{
c_{2j+1}(T_Y)=0.
}
$$

In particular：

$$
c_3(T_Y)=0.
$$

Any codimension-$3$ polynomial generated by：

- divisors on $Y$；
- Chern classes of $T_Y$；

pulls back to a very-general Weil sixfold as a decomposable product of codimension-$1$ and codimension-$2$ Hodge classes, and hence lands in：

$$
\boxed{
\mathbb Q\Theta^3.
}
$$

Therefore the fourfold strategy “pull back $c_2(T_Y)$” has no naive codimension-$3$ characteristic-class analogue for sixfolds。

The surviving direct/auxiliary positive frontier is consequently reduced to：

$$
\boxed{
\text{Primitive Indecomposable Codimension-3 Carrier}.
}
$$

Concretely, one must find at least one of：

1. a genuinely nonlinear threefold:
   $$
   Z^3\subset A
   $$
   with:
   $$
   \operatorname{pr}_{W}[Z]\neq0;
   $$
2. an auxiliary variety $Y$ carrying a primitive codimension-$3$ algebraic class and an algebraic correspondence whose image has nonzero Weil projection；
3. a component-local coherent/derived object whose:
   $$
   ch_3
   $$
   has nonzero Weil projection and can be legally realized/deformed。

**Status:** PROVED STRUCTURAL FILTERS / DIRECT CARRIER OPEN  
**Tautological Direct Cycles:** ELIMINATED  
**K-linear Hecke Transfer:** ELIMINATED across discriminant components  
**Standard Kuga–Satake Simple Factor:** ELIMINATED by dimension and discriminant sector  
**Naive Hyperkähler Chern-Class Lift:** ELIMINATED  
**CM / Absolute-Hodge Seed:** STRUCTURAL ONLY, no direct algebraic realization  
**New Direct Frontier:** primitive nonlinear codimension-$3$ geometry  
**Formalization Status:** NOT FORMALIZED  
**Graph Compression:** substantial  

---

# 0. State after R007

The current open target is a very-general polarized Weil sixfold：

$$
(A,K,\Theta)
$$

in a modern non-split component：

$$
\boxed{
\delta\neq-1.
}
$$

Its Weil plane is：

$$
\boxed{
W_K(A)
=
\bigwedge_K^6H^1(A,\mathbb Q),
}
$$

with：

$$
\dim_{\mathbb Q}W_K(A)=2.
$$

PT002 gives the binary closure：

$$
\boxed{
0\neq
\alpha
\in
W_K(A)\cap\operatorname{Alg}^3(A)
\Longrightarrow
W_K(A)\subseteq\operatorname{Alg}^3(A).
}
$$

Thus R008 only needs to understand which algebraic architectures can produce one nonzero primitive component。

---

# 1. Generic Hodge representation

Set：

$$
V=H^1(A,\mathbb Q).
$$

As a：

$$
K
$$

-vector space：

$$
\dim_KV=6.
$$

For a very-general point of a fixed Weil component：

$$
Hg(A)
=
SU(H).
$$

After complexification：

$$
\boxed{
Hg(A)_{\mathbb C}
\simeq
SL(U),
}
$$

with：

$$
\dim_{\mathbb C}U=6.
$$

The complexified first cohomology decomposes as：

$$
\boxed{
V_{\mathbb C}
\simeq
U
\oplus
U^\vee.
}
$$

The polarization is the canonical contraction pairing between：

$$
U
$$

and：

$$
U^\vee.
$$

---

# 2. Invariant-theory grammar

The first fundamental invariant theory for：

$$
SL(U)
$$

contains two qualitatively different generator types。

### Pairing / contraction sector

Generated by：

$$
U\otimes U^\vee\to\mathbb C.
$$

Cohomologically this is the Lefschetz/polarization grammar。

### Determinant sector

Generated by：

$$
\boxed{
\bigwedge^6U
}
$$

and：

$$
\boxed{
\bigwedge^6U^\vee.
}
$$

These are the two complex determinant lines whose rational form is：

$$
W_K(A).
$$

The determinant tensor cannot occur below exterior degree：

$$
6.
$$

---

# 3. Codimension one

Hodge classes of codimension one lie in：

$$
H^2(A,\mathbb Q).
$$

Only one contraction is possible before determinant degree。

Therefore：

$$
\boxed{
\operatorname{Hdg}^1(A)
=
\mathbb Q\Theta.
}
$$

Equivalently：

$$
\boxed{
\operatorname{NS}_{\mathbb Q}(A)
=
\mathbb Q\Theta.
}
$$

---

# 4. Rosati verification

The generic endomorphism algebra is：

$$
\operatorname{End}^0(A)=K.
$$

The Rosati involution associated with a compatible polarization restricts to complex conjugation：

$$
k^\dagger=\bar k.
$$

Hence：

$$
\operatorname{End}^0(A)^\dagger
=
K^{\bar{\ }}=
\mathbb Q.
$$

Using：

$$
\operatorname{NS}_{\mathbb Q}(A)
\simeq
\operatorname{End}^0(A)^\dagger,
$$

we recover：

$$
\boxed{
\rho(A)=1.
}
$$

---

# 5. Codimension two

Hodge classes of codimension two lie in：

$$
H^4(A,\mathbb Q).
$$

Exterior degree：

$$
4
$$

is still below determinant degree：

$$
6.
$$

All invariants are generated by contractions。

Since the codimension-one contraction space is one-dimensional on：

$$
A,
$$

the codimension-two invariant is：

$$
\boxed{
\Theta^2.
}
$$

Therefore：

## Theorem 5.1

$$
\boxed{
\operatorname{Hdg}^2(A)
=
\mathbb Q\Theta^2.
}
$$

No exceptional Weil-type class exists in codimension two for the very-general sixfold。

---

# 6. Codimension three

Now：

$$
H^6(A,\mathbb Q)
$$

reaches exterior degree：

$$
6.
$$

The contraction sector contributes：

$$
\Theta^3.
$$

For the first time determinant invariants also occur：

$$
\bigwedge^6U,
\qquad
\bigwedge^6U^\vee.
$$

Their rational span is：

$$
W_K(A).
$$

Thus：

## Theorem 6.1 — First-Exceptional-Degree

$$
\boxed{
\operatorname{Hdg}^3(A)
=
\mathbb Q\Theta^3
\oplus
W_K(A).
}
$$

For a very-general Weil sixfold, the exceptional algebraicity problem first appears exactly in codimension three。

---

# 7. Decomposable codimension-three classes

Let：

$$
\alpha
\in
\operatorname{Hdg}^a(A),
\qquad
\beta
\in
\operatorname{Hdg}^b(A),
$$

with：

$$
a+b=3,
$$

and：

$$
a,b>0.
$$

The only possibilities are：

$$
(a,b)=(1,2)
$$

or：

$$
(2,1).
$$

By Sections 3 and 5：

$$
\alpha\in\mathbb Q\Theta,
$$

and：

$$
\beta\in\mathbb Q\Theta^2.
$$

Therefore：

$$
\boxed{
\alpha\beta
\in
\mathbb Q\Theta^3.
}
$$

---

# 8. Decomposable-Degree Barrier

## Theorem 8.1

Every codimension-$3$ Hodge class on a very-general Weil sixfold which is a product of positive-degree Hodge classes of smaller codimension lies in：

$$
\boxed{
\mathbb Q\Theta^3.
}
$$

Consequently：

$$
\boxed{
\operatorname{pr}_{W}
(\alpha\beta)
=
0.
}
$$

A useful Weil cycle must be indecomposable already at the first exceptional degree。

---

# 9. Algebraic version

Because all divisor classes are algebraic by Lefschetz $(1,1)$：

$$
\mathbb Q\Theta
\subset
\operatorname{Alg}^1(A).
$$

The product：

$$
\Theta^3
$$

is algebraic。

But products of lower-degree algebraic Hodge classes generate no more than：

$$
\boxed{
\mathbb Q\Theta^3.
}
$$

So ordinary intersection theory of already-known low-degree classes cannot solve the Weil problem。

---

# 10. A simple $K^\times$ spectral certificate

Let：

$$
u\in K^\times.
$$

Compatibility of the polarization gives：

$$
\boxed{
u^\ast\Theta
=
N_{K/\mathbb Q}(u)\Theta.
}
$$

Therefore：

$$
\boxed{
u^\ast(\Theta^3)
=
N(u)^3\Theta^3.
}
$$

On the two complex Weil determinant lines：

$$
\bigwedge_K^6V,
$$

the eigencharacters are：

$$
\boxed{
u^6,
\qquad
\bar u^6.
}
$$

Choose：

$$
u
$$

with：

$$
\left(
\frac u{\bar u}
\right)^3
\neq1.
$$

Then：

$$
u^6
\neq
N(u)^3,
$$

and：

$$
\bar u^6
\neq
N(u)^3.
$$

Hence：

$$
\boxed{
W_K(A)
\cap
\mathbb Q\Theta^3
=
0.
}
$$

This is an elementary spectral separation of the primitive Weil sector from the Lefschetz sector。

---

# 11. Tautological cohomological closure

Define：

$$
\mathcal T^\ast(A)
$$

to be the cohomological algebra obtained from divisor classes using the usual abelian-group algebraic operations：

1. pullback/pushforward by homomorphisms of powers of：
   $$
   A;
   $$
2. $K$-endomorphisms；
3. addition and projection maps；
4. Poincaré divisor classes on：
   $$
   A\times\hat A;
   $$
5. Fourier–Mukai transforms generated cohomologically by the exponential of the Poincaré class；
6. finite sums, products and Künneth contractions。

These operations are algebraic。

---

# 12. Tensor meaning of the tautological closure

Every generator of：

$$
\mathcal T^\ast
$$

comes from the pairing/contraction tensor：

$$
U\otimes U^\vee\to\mathbb C.
$$

Homomorphisms and Fourier–Mukai transforms reorganize contraction tensors but do not create the alternating determinant tensor：

$$
\bigwedge^6U.
$$

In invariant-theory language：

$$
\boxed{
\text{pairing category}
\neq
\text{determinant category}.
}
$$

---

# 13. Tautological Closure Barrier

On the single very-general sixfold：

$$
A,
$$

the codimension-$3$ output of the pairing-generated tautological algebra is：

$$
\boxed{
\mathcal T^3(A)
=
\mathbb Q\Theta^3.
}
$$

Therefore：

## Theorem 13.1

$$
\boxed{
\mathcal T^3(A)
\cap
W_K(A)
=
0.
}
$$

No cycle produced solely from standard group-law / divisor / endomorphism / Poincaré machinery can supply the missing Weil class。

---

# 14. Consequence for endomorphism graphs

Graphs of：

$$
u\in K
$$

are algebraic cycles on：

$$
A\times A.
$$

But using these graphs only changes the action of the existing pairing-generated tensor category。

They do not create a determinant tensor on：

$$
A.
$$

Thus repeated pull-push constructions from：

- $\Gamma_u$；
- $\Theta$；
- diagonals；
- addition maps；

remain in the tautological dead zone。

---

# 15. Hecke/isogeny temptation

A different apparent shortcut is：

> use a known algebraic Weil class on a split sixfold and transport it by isogeny/Hecke correspondence to a non-split sixfold。

For a generic target this fails for a simpler reason：the discriminant cannot change。

---

# 16. $K$-linear isogeny setup

Let：

$$
f:A\to B
$$

be a $K$-linear isogeny between very-general polarized $K$-Weil sixfolds：

$$
(A,\Theta_A),
\qquad
(B,\Theta_B).
$$

Because：

$$
\operatorname{NS}_{\mathbb Q}(A)
=
\mathbb Q\Theta_A,
$$

we have：

$$
\boxed{
f^\ast\Theta_B
=
c\Theta_A
}
$$

for some：

$$
c\in\mathbb Q_{>0}.
$$

---

# 17. Hermitian determinant under isogeny

Let：

$$
F
$$

be the $K$-linear matrix of：

$$
f_\ast
$$

on：

$$
H_1(-,\mathbb Q).
$$

The pullback Hermitian Gram matrix satisfies：

$$
\boxed{
H_{f^\ast\Theta_B}
=
F^\ast H_BF.
}
$$

Therefore：

$$
\boxed{
\det H_{f^\ast\Theta_B}
=
N_{K/\mathbb Q}
(\det_KF)
\det H_B.
}
$$

On the other hand：

$$
f^\ast\Theta_B=c\Theta_A
$$

implies：

$$
\boxed{
\det H_{f^\ast\Theta_B}
=
c^6\det H_A.
}
$$

---

# 18. Rational scaling is a norm

Because：

$$
c\in\mathbb Q^\times
\subset K^\times,
$$

$$
\boxed{
c^6
=
N_{K/\mathbb Q}(c^3).
}
$$

Thus both factors relating：

$$
\det H_A
$$

and：

$$
\det H_B
$$

are norms。

Hence：

## Theorem 18.1 — Hecke–Isogeny Discriminant Barrier

$$
\boxed{
\delta(A)
=
\delta(B).
}
$$

A generic $K$-linear isogeny cannot cross modern Weil discriminant components。

---

# 19. Consequence for split-cycle transport

Markman gives algebraic Weil classes on：

$$
\delta=-1.
$$

If：

$$
B
$$

is a very-general non-split sixfold：

$$
\delta(B)\neq-1,
$$

then no $K$-linear isogeny：

$$
A_{\mathrm{split}}\to B
$$

exists in the generic architecture。

Therefore：

$$
\boxed{
\text{split algebraic Weil cycle}
\not\to
\text{generic non-split cycle}
}
$$

by ordinary Hecke/isogeny transfer。

---

# 20. Non-$K$-linear correspondences

One may ask for an isogeny not explicitly declared $K$-linear。

For a very-general target：

$$
\operatorname{End}^0=K.
$$

Any Hodge isomorphism identifying the relevant Weil structures necessarily conjugates the generic $K$-endomorphism algebra into itself。

Thus it is $K$-linear or $K$-antilinear after identifying the two embeddings。

Complex conjugation does not alter the rational norm-class discriminant。

So the generic component barrier persists。

---

# 21. Mostaed CM / Hecke points

Mostaed 2026 constructs a highly rigid arithmetic environment in which certain intersections of a Hilbert modular geodesic curve with Weil loci are CM points。

At such a point：

$$
\operatorname{End}^0(A)=M
$$

is a degree-$12$ CM field。

The Hodge-Weil classes are absolute Hodge。

But the paper explicitly notes that existing algebraicity theorems do not reach them because of：

- CM isolation；
- lack of a known $K$-secant structure；
- uncontrolled discriminant。

Thus Hecke arithmetic can locate special points without supplying the missing codimension-$3$ cycle。

---

# 22. Kuga–Satake architecture

The standard Kuga–Satake construction starts from a polarized weight-two Hodge structure：

$$
(V,Q),
$$

of dimension：

$$
n=2m.
$$

For：

$$
n\equiv2\pmod4,
$$

Lombardo writes：

$$
\boxed{
C^+(V,Q)
\simeq
M_{2^{m-1}}(K),
}
$$

where：

$$
K=\mathbb Q(\sqrt{-d}).
$$

---

# 23. Poincaré decomposition of the Kuga–Satake variety

Let：

$$
KS(V)
$$

be the Kuga–Satake abelian variety。

Its first rational cohomology is modeled on：

$$
C^+(V,Q).
$$

Poincaré decomposition yields：

$$
\boxed{
KS(V)
\sim
B^{2^{m-1}},
}
$$

where：

$$
B
$$

is simple of Weil type over：

$$
K.
$$

Dimension count gives：

$$
\boxed{
\dim B
=
2^{m-1}.
}
$$

---

# 24. Kuga–Satake Dimension Barrier

The admissible：

$$
m
$$

are odd：

$$
m=1,3,5,\ldots
$$

in the nontrivial higher-dimensional sequence。

The resulting simple Weil factor dimensions include：

$$
\boxed{
4,\ 16,\ 64,\ldots
}
$$

after the classical low-dimensional cases。

In particular：

$$
\boxed{
6
}
$$

never occurs。

Therefore standard Clifford/Kuga–Satake decomposition cannot directly produce our six-dimensional target as its simple Weil factor。

---

# 25. Kuga–Satake discriminant sector

Lombardo further proves that the Weil-type factor arising in this construction has old discriminant：

$$
\boxed{
1.
}
$$

The construction is therefore not a generic mechanism for arbitrary Hermitian discriminant。

So even ignoring the dimension mismatch, it does not naturally solve the non-split sixfold target。

---

# 26. Kuga–Satake as auxiliary rather than direct route

Kuga–Satake may still enter indirectly：

$$
Y
\longleftrightarrow
KS(Y)
\longleftrightarrow
A
$$

through an algebraic correspondence。

But then the crucial new theorem is no longer the Hodge-theoretic Kuga–Satake relation。

It is：

$$
\boxed{
\text{an algebraic correspondence with nonzero Weil projection}.
}
$$

That is a new primitive realization gate。

---

# 27. Hyperkähler precedent in dimension four

Recent fourfold proofs show that auxiliary hyperkähler geometry can genuinely supply exceptional algebraic Hodge classes。

Van Geemen–Rapagnetta 2026 construct a map：

$$
A^4
\to
Y^6,
$$

where：

$$
Y
$$

is hyperkähler of：

$$
K3^{[3]}
$$

type。

They pull back：

$$
\boxed{
c_2(T_Y),
}
$$

obtaining an algebraic codimension-$2$ class on the Weil fourfold that is not generated by divisor intersections。

This is a true auxiliary-geometry bypass in dimension four。

---

# 28. Why direct degree-lifting fails in dimension six

Our target：

$$
A
$$

has dimension：

$$
6
$$

and the missing Weil class has codimension：

$$
3.
$$

A naive analogue would seek：

$$
c_3(T_Y).
$$

But a holomorphic symplectic tangent bundle is symplectic。

Its formal Chern roots occur in pairs：

$$
x_i,-x_i.
$$

Therefore：

$$
c(T_Y)
=
\prod_i(1+x_i)(1-x_i),
$$

which contains only even-degree elementary symmetric terms。

Hence：

## Theorem 28.1

$$
\boxed{
c_{2j+1}(T_Y)=0
}
$$

for every holomorphic symplectic：

$$
Y.
$$

In particular：

$$
\boxed{
c_3(T_Y)=0.
}
$$

---

# 29. Natural codimension-three characteristic algebra

For a hyperkähler：

$$
Y,
$$

we also have：

$$
c_1(T_Y)=0.
$$

A codimension-$3$ class generated by divisors and tangent Chern classes can therefore only involve combinations such as：

$$
D_1D_2D_3,
$$

or：

$$
D\cdot c_2(T_Y).
$$

There is no primitive：

$$
c_3(T_Y)
$$

term。

---

# 30. Pullback to the Weil sixfold

Let：

$$
f:A\to Y
$$

be an algebraic morphism。

Then：

$$
f^\ast D
\in
\operatorname{Hdg}^1(A)
=
\mathbb Q\Theta,
$$

and：

$$
f^\ast c_2(T_Y)
\in
\operatorname{Hdg}^2(A)
=
\mathbb Q\Theta^2.
$$

Hence：

$$
\boxed{
f^\ast(D_1D_2D_3)
\in
\mathbb Q\Theta^3,
}
$$

and：

$$
\boxed{
f^\ast(Dc_2(T_Y))
\in
\mathbb Q\Theta^3.
}
$$

---

# 31. Hyperkähler Natural-Characteristic Barrier

## Theorem 31.1

For a very-general Weil sixfold：

$$
A,
$$

the pullback along any morphism：

$$
f:A\to Y
$$

to a holomorphic symplectic variety of any codimension-$3$ class generated by：

- divisor classes；
- tangent-bundle Chern classes；

has zero Weil projection。

Symbolically：

$$
\boxed{
\operatorname{pr}_{W}
f^\ast
\mathbb Q[
\operatorname{NS}(Y),
c_i(T_Y)
]^3
=
0.
}
$$

Thus the known fourfold $c_2$ architecture has no direct natural-characteristic sixfold analogue。

---

# 32. What hyperkähler geometry would need instead

A viable auxiliary hyperkähler sixfold route must supply a genuinely primitive algebraic cycle：

$$
\boxed{
\gamma\in CH^3(Y)_{\mathbb Q}
}
$$

which is not in the divisor/Chern subalgebra, together with an algebraic morphism/correspondence such that：

$$
\boxed{
\operatorname{pr}_{W}
\Gamma_\ast(\gamma)
\neq0.
}
$$

This is qualitatively harder than simply pulling a characteristic class。

---

# 33. Auxiliary correspondence formalism

Let：

$$
Y
$$

be smooth projective of dimension：

$$
m.
$$

Take：

$$
z\in CH^r(Y)_{\mathbb Q}
$$

and：

$$
\Gamma
\in
CH^c(Y\times A)_{\mathbb Q}.
$$

Then：

$$
\Gamma_\ast z
\in
CH^{r+c-m}(A)_{\mathbb Q}.
$$

To land in codimension three we need：

$$
\boxed{
r+c-m=3.
}
$$

---

# 34. Primitive Correspondence Certificate

Define：

$$
\boxed{
\mathsf{PCC}(Y,z,\Gamma)
}
$$

to mean：

1. $z$ is algebraic；
2. $\Gamma$ is algebraic；
3. $r+c-m=3$；
4. the cohomological image satisfies：
   $$
   \boxed{
   \operatorname{pr}_{W}
   \Gamma_\ast[z]
   \neq0.
   }
   $$

Then：

$$
\boxed{
\mathsf{PCC}
\Longrightarrow
W_K(A)\subseteq\operatorname{Alg}^3(A)
}
$$

by PT002 after subtracting the algebraic：

$$
\Theta^3
$$

component。

This is the exact auxiliary-geometry success certificate。

---

# 35. Removing the Lefschetz component

Suppose：

$$
[Z]
\in
\operatorname{Hdg}^3(A).
$$

By Theorem 6.1：

$$
[Z]
=
a\Theta^3+w,
$$

with：

$$
a\in\mathbb Q,
\qquad
w\in W_K(A).
$$

If：

$$
w\neq0,
$$

then：

$$
\boxed{
[Z-a\Theta^3]
=
w.
}
$$

Since：

$$
\Theta^3
$$

is algebraic, the corrected rational cycle：

$$
Z-a\Theta^3
$$

is algebraic。

Thus one does not need a geometrically pure Weil cycle。

One only needs any algebraic cycle not cohomologically proportional to：

$$
\Theta^3.
$$

---

# 36. Primitive excess

Define for：

$$
Z\in CH^3(A)_{\mathbb Q}
$$

the cohomological primitive excess：

$$
\boxed{
\operatorname{PE}(Z)
=
\operatorname{pr}_{W}[Z].
}
$$

Then：

$$
\boxed{
\operatorname{PE}(Z)\neq0
}
$$

is the complete direct-cycle certificate。

PT002 immediately upgrades one nonzero excess to the full Weil plane。

---

# 37. Complete intersections are dead

Any codimension-$3$ complete intersection of divisors：

$$
D_1\cap D_2\cap D_3
$$

has class：

$$
[D_1][D_2][D_3].
$$

Since every divisor is proportional to：

$$
\Theta,
$$

$$
\boxed{
[D_1D_2D_3]
\in
\mathbb Q\Theta^3.
}
$$

Therefore：

$$
\boxed{
\operatorname{PE}=0.
}
$$

No divisor-complete-intersection threefold can solve the problem。

---

# 38. Abelian subvarieties are unavailable generically

A very-general Weil sixfold with：

$$
\operatorname{End}^0(A)=K
$$

is simple。

Hence it has no positive-dimensional proper abelian subvarieties。

So a useful half-dimensional：

$$
Z^3\subset A
$$

cannot be a translate of an abelian threefold。

This removes another elementary geometric source。

---

# 39. Minimal-class warning

If a three-dimensional subvariety：

$$
Z\subset A
$$

has cohomology class proportional to the minimal Lefschetz class：

$$
\Theta^3,
$$

then：

$$
\operatorname{PE}(Z)=0.
$$

Thus even an exotic nonlinear subvariety is useless if its cohomology class is minimal/tautological。

The relevant geometry must be cohomologically exceptional, not merely geometrically complicated。

---

# 40. Surviving direct geometry

A successful direct cycle must therefore be a genuinely nonlinear half-dimensional subvariety：

$$
\boxed{
Z^3\subset A
}
$$

such that：

$$
\boxed{
[Z]
\notin
\mathbb Q\Theta^3.
}
$$

Equivalently：

$$
\boxed{
\operatorname{PE}(Z)\neq0.
}
$$

This is now the simplest non-tautological direct architecture。

---

# 41. Albanese carrier reformulation

A natural way to produce such a threefold is to find a smooth projective threefold：

$$
X
$$

with irregularity：

$$
\boxed{
q(X)=6,
}
$$

and a generically finite Albanese map：

$$
a_X:
X\to
\operatorname{Alb}(X)=A,
$$

where：

$$
A
$$

is the target Weil sixfold。

Then：

$$
\boxed{
Z=a_X(X)
}
$$

is an algebraic codimension-$3$ cycle。

The only remaining cohomological test is：

$$
\boxed{
\operatorname{PE}(a_X(X))
\neq0.
}
$$

R008 calls this the：

$$
\boxed{
\text{Half-Dimensional Albanese Carrier Problem}.
}
$$

---

# 42. Why this route is genuinely different

The Albanese-carrier route does not require：

- a curve cover；
- semiregularity of a secant sheaf；
- maximal degeneration；
- a Kuga–Satake factor；
- a hyperkähler characteristic class。

It attempts to construct the missing cycle directly as a geometric half-dimensional support inside：

$$
A.
$$

This is therefore a true new path architecture。

---

# 43. Component-local object route

The Markman-style alternative remains：

$$
E
\in
D^b(A)
$$

or a twisted object with：

$$
\boxed{
\operatorname{PE}
\left(
ch_3(E)
\right)
\neq0.
}
$$

A suitable semiregularity/relative deformation theorem can then propagate the class。

The crucial change after R008 is：

> The object must generate a primitive first-exceptional-degree term；lower characteristic data alone cannot do it。

---

# 44. CM special points as candidate local seeds

Mostaed's CM Weil-sixfold points have much larger endomorphism algebra：

$$
M=KL.
$$

They may provide additional algebraic constructions unavailable at the generic point。

But current theory only gives：

$$
\boxed{
\text{absolute Hodge}
}
$$

for the relevant Weil classes。

No direct algebraic cycle is known。

Thus a CM point remains a possible **seed search location** rather than a completed direct route。

---

# 45. Milne CM reduction

Deligne–André–Milne theory expresses Hodge classes on CM-type abelian varieties in terms of：

- divisor classes；
- split Weil classes。

This is powerful structural reduction。

But it does not, with current known split-algebraicity theorems, automatically supply a codimension-$3$ algebraic Weil class on every non-split CM sixfold。

Mostaed's explicit statement that the new CM Weil classes remain inaccessible to existing algebraicity theorems confirms this residual。

---

# 46. Direct-realization graph

The current graph becomes：

```text
[Divisors / Endomorphisms / Poincare]
                  |
                  v
          [Lefschetz Algebra]
                  |
                  X  misses W

[Split Cycle] --> [Hecke / Isogeny]
                  |
                  X  discriminant preserved

[Kuga-Satake]
      |
      X  simple factor dimension != 6
      |
      +--> [Auxiliary Correspondence] ----+
                                          |
[Hyperkahler natural Chern classes]       |
      |                                   |
      X  decomposable degree barrier       |
                                          v
[Primitive codim-3 auxiliary cycle] --> [PCC] --> [WEIL CORE]
                                          ^
                                          |
[Nonlinear 3-fold Z in A] --> [PE(Z)!=0]-+
                                          ^
                                          |
[Component-local object E] -> [PE(ch3(E))]+
```

This is the compressed non-split direct-realization graph。

---

# 47. Current dead zones

R008 marks the following as dead for generic non-split direct realization：

```text
DEAD:
  divisor complete intersections
  K-endomorphism graph algebra
  Poincare/Fourier tautological closure
  K-linear split-to-nonsplit Hecke transfer
  standard Kuga-Satake simple factor
  naive hyperkahler c3
  hyperkahler divisor * c2 pullback
```

---

# 48. Current live zones

```text
LIVE:
  nonlinear half-dimensional threefold in A
  primitive auxiliary codimension-3 class
  primitive algebraic correspondence
  component-local derived/coherent object with primitive ch3
  CM point as seed search location
  non-maximal degeneration with primitive cycle control
```

The live set is substantially smaller and more geometrically specific than before R008。

---

# 49. Direct-realization min-cut

Inside the direct/auxiliary subgraph, every surviving route must cross：

$$
\boxed{
\text{Primitive Codimension-3 Realization}.
}
$$

But GMSC does **not** promote this phrase to a new universal Hodge global core, because：

$$
\boxed{
\text{Primitive Realization}
}
$$

is still very close to a restatement of the target。

Its value is operational：

> it identifies exactly which lower-cost algebraic architectures can be stopped before expensive research begins。

---

# 50. Information gain

Before R008, one might reasonably spend time on：

- divisor/endomorphism formulas；
- Hecke transport；
- Kuga–Satake factors；
- direct hyperkähler Chern classes。

R008 proves that none of these standard low-complexity routes can produce the required primitive class in the generic non-split sixfold。

Research should therefore start one categorical level higher：

$$
\boxed{
\text{primitive support / primitive correspondence / primitive }ch_3.
}
$$

---

# 51. Relation to CE001

CE001 predicts：

$$
\boxed{
W_K(A)\cap\operatorname{Alg}^3(A)=0
}
$$

for a very-general non-split candidate component。

R008 removes a large family of obvious positive cycles。

It does not prove CE001。

But any CE001 falsifier must now be genuinely primitive in exactly the sense isolated here。

This sharpens both Construct and Attack。

---

# 52. Updated positive frontier

After R008：

$$
\boxed{
P_{\mathrm{LocalSeed}}
}
$$

$$
\boxed{
P_{\mathrm{HalfDimThreefold}}
}
$$

$$
\boxed{
P_{\mathrm{PrimitiveCorrespondence}}
}
$$

form the clearest positive frontier。

Auxiliary hyperkähler/Kuga–Satake geometry survives only if it can instantiate：

$$
P_{\mathrm{PrimitiveCorrespondence}}.
$$

It no longer receives a free route label merely from its Hodge-theoretic relationship。

---

# 53. Provisional route burdens

### Local seed

$$
\boxed{
W(P_{\mathrm{LocalSeed}})
=
(3,2,3,3,2,3).
}
$$

### Half-dimensional threefold

$$
\boxed{
W(P_{\mathrm{HalfDimThreefold}})
=
(4,1,4,4,3,5).
}
$$

### Primitive correspondence

$$
\boxed{
W(P_{\mathrm{PrimitiveCorr}})
=
(4,3,4,4,3,4).
}
$$

The threefold route has minimal dependency depth but high invention cost。

---

# 54. Construct result

R008 constructs a new explicit search target：

$$
\boxed{
\text{Half-Dimensional Albanese Carrier}.
}
$$

Find：

$$
X^3
$$

with：

$$
q(X)=6,
$$

$$
\operatorname{Alb}(X)
\simeq
A,
$$

and：

$$
\boxed{
\operatorname{pr}_{W}
[a_X(X)]
\neq0.
}
$$

One such $X$ closes the full Weil plane by PT002。

---

# 55. Attack result

R008 breaks four broad shortcuts：

### A. Tautological group geometry

BROKEN.

### B. Cross-discriminant Hecke transport

BROKEN.

### C. Standard Kuga–Satake factorization

BROKEN for direct sixfold realization.

### D. Naive hyperkähler characteristic-class lift

BROKEN.

These failures are theorem-level rather than absence-of-example observations。

---

# 56. Compress result

The direct-realization search compresses from a broad list of standard tools to：

$$
\boxed{
\text{one genuinely primitive codim-3 carrier}.
}
$$

The carrier may be：

- a threefold support；
- an auxiliary primitive cycle；
- a primitive $ch_3$。

But it cannot be assembled from already-known lower-degree Hodge data。

---

# 57. Bottleneck stability update

Across：

$$
R001\text{--}R008,
$$

the target SCC remains stable：

$$
\boxed{
\mathrm{BS}_8
\left(
\mathcal C_{\mathrm{Weil6}}^{\mathrm{vg}}
\right)
=
1.
}
$$

A new stable pattern is emerging across positive routes：

$$
\boxed{
\text{every cheap lower-degree / symmetry-generated realization route collapses into the Lefschetz sector}.
}
$$

This is not yet a universal theorem bottleneck, but it is now a strong architecture-level pattern。

---

# 58. Main Verdict

For a very-general non-split Weil sixfold：

$$
A,
$$

the missing algebraic class cannot come from standard low-complexity abelian geometry。

The first exceptional Hodge degree is codimension：

$$
3.
$$

Everything assembled from lower degrees remains：

$$
\mathbb Q\Theta^3.
$$

Standard Kuga–Satake factors miss dimension：

$$
6.
$$

Natural hyperkähler odd Chern classes vanish, and decomposable Chern/divisor products again land in：

$$
\Theta^3.
$$

Hecke/isogeny transport cannot cross discriminant components。

Therefore the positive problem has now been compressed to：

$$
\boxed{
\text{construct one genuinely primitive codimension-3 algebraic carrier}.
}
$$

---

# 59. Next Interface

Next GMSC round：

```text
HODGE_GMSC_R009_HalfDimensionalCarrierSearch.md
```

Primary target：

$$
\boxed{
\text{Search for and classify three-dimensional subvarieties or Albanese images in a six-dimensional Weil abelian variety that can have nonzero Weil primitive excess.}
}
$$

Planned tasks：

1. classify obvious half-dimensional subvarieties and eliminate minimal/Lefschetz classes；
2. study Albanese threefolds with:
   $$
   q=6;
   $$
3. inspect known irregular threefold constructions；
4. inspect subvarieties arising from moduli spaces / Fano schemes / degeneracy loci；
5. derive numerical criteria detecting:
   $$
   \operatorname{PE}(Z)\neq0;
   $$
6. use $K^\times$ action to detect primitive excess without constructing an explicit projector；
7. search whether known cycles on special CM sixfolds can be interpreted as Albanese images；
8. test positivity/intersection constraints on:
   $$
   [Z]=a\Theta^3+w;
   $$
9. in parallel, formulate CE001 inequalities excluding such $Z$；
10. if the direct-subvariety space collapses, move to primitive auxiliary correspondences。

---

# References

1. E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415. Establishes split sixfold Weil algebraicity and provides the modern baseline.

2. G. Lombardo, *Abelian varieties of Weil type and Kuga-Satake varieties*, Tohoku Math. J. 53 (2001), 453–466; arXiv:math/0211224. The higher-dimensional Clifford/Kuga–Satake decomposition yields simple Weil-type factors of power-of-two dimension and proves their old discriminant is $1$.

3. B. van Geemen, *Fourfolds of Weil type and the spinor map*, Expo. Math. 41 (2023), 418–447. Reviews the Kuga–Satake / Weil-fourfold relation in the trivial-discriminant domain.

4. B. van Geemen, A. Rapagnetta, *Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class*, arXiv:2607.18341. Pulls back $c_2$ of a hyperkähler sixfold to obtain a non-divisorial codimension-$2$ algebraic class on a Weil fourfold.

5. S. Floccari, L. Fu, *The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties*, arXiv:2504.13607. Gives an independent auxiliary-hyperkähler realization route in dimension four.

6. A. Mostaed, *McMullen's Curve, the Weil Locus, and the Hodge Conjecture for Abelian Sixfolds*, arXiv:2603.20268. Identifies CM Weil-sixfold points whose Hodge-Weil classes remain outside current algebraicity theorems.

7. J. S. Milne, *Hodge classes on abelian varieties*, arXiv:2010.08857. Following Deligne and André, reduces Hodge classes on CM-type abelian varieties to divisor and split-Weil structures, but does not by itself close the present non-split sixfold realization problem.

8. B. J. J. Moonen, Yu. G. Zarhin, *Weil classes on abelian varieties*, arXiv:alg-geom/9612017.

9. Aletheia, *HODGE_HCTRUE_PT001_WeilSixfoldPrimitiveCore*, 2026-09-16.

10. Aletheia, *HODGE_HCTRUE_PT002_OneCycleBinaryClosure*, 2026-09-16.

11. Aletheia, *HODGE_GMSC_R007_G24DiscriminantRigidity*, 2026-09-16.

---

## Canonical Source Declaration

本檔案為 Hodge GMSC 分支第八篇正式 UTF-8 Markdown canonical source。

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪沒有宣稱不存在任何 auxiliary hyperkähler、Kuga–Satake、CM、或 moduli-space correspondence可以解 non-split sixfold。

本輪正式排除的是它們的 **standard low-complexity direct versions**：

- pairing/divisor tautological closure；
- generic cross-discriminant $K$-linear isogeny transfer；
- standard Clifford Kuga–Satake simple factor；
- hyperkähler divisor/tangent-Chern characteristic algebra。

任何 surviving auxiliary route都必須引入一個新的 primitive codimension-$3$ algebraic carrier。

下一輪將直接搜尋這種 half-dimensional carrier。