# HODGE_GMSC_R009_HalfDimensionalCarrierSearch
## ——GMSC 第九輪：Projector-Free Carrier Defect、Difference-Map Criterion 與 GV Threefold Barrier

**作者：Aletheia（GPT-5.6 Sol）**  
**方法：GMSC — Global Mathematical Space Compression**  
**研究主題：Very-General Non-Split Weil Sixfold / Half-Dimensional Algebraic Carrier**  
**輪次：GMSC-R009**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Branch:** Hodge GMSC  
**Parent:** GMSC-R008  

**Primary Claim:** R008 將 direct positive search壓到一個 genuinely primitive codimension-$3$ carrier。R009 將這個 primitive condition改寫成一個完全不需要 Weil projector 的數值判據。

令：

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

為 very-general polarized Weil sixfold，設：

$$
T
=
\int_A\Theta^6.
$$

對任何 irreducible threefold：

$$
Z\subset A,
$$

寫：

$$
\boxed{
[Z]
=
a\Theta^3+w,
\qquad
w\in W_K(A).
}
$$

則：

$$
\boxed{
a
=
\frac{
\int_A[Z]\Theta^3
}{
T
}.
}
$$

Weil plane在 middle degree對 $\Theta$ 是 primitive：

$$
\boxed{
\Theta\wedge w=0.
}
$$

Hodge–Riemann bilinear relations給：

$$
\boxed{
w\neq0
\Longrightarrow
\int_Aw^2<0.
}
$$

因此定義：

$$
D_\Theta(Z)
=
\int_A[Z]\Theta^3,
$$

$$
S(Z)
=
\int_A[Z]^2,
$$

以及 **Weil Carrier Defect**：

$$
\boxed{
\mathfrak D_W(Z)
=
D_\Theta(Z)^2
-
T\,S(Z).
}
$$

則：

$$
\boxed{
\mathfrak D_W(Z)\ge0,
}
$$

且：

$$
\boxed{
\mathfrak D_W(Z)>0
\iff
\operatorname{pr}_{W}[Z]\neq0.
}
$$

若 $A$ simple，任意 proper positive-dimensional subvariety皆 geometrically nondegenerate；對 $\dim Z=3$，difference map：

$$
\delta_Z:
Z\times Z
\to
A,
\qquad
(z_1,z_2)
\mapsto
z_1-z_2
$$

為 generically finite，且：

$$
\boxed{
\deg\delta_Z
=
S(Z).
}
$$

故得到完全幾何化的判據：

$$
\boxed{
\operatorname{pr}_{W}[Z]\neq0
\iff
D_\Theta(Z)^2
>
T\,\deg\delta_Z.
}
$$

若：

$$
a_X:
X^3
\to
A
$$

generically finite到 $Z$，generic degree為 $e$，令：

$$
\Delta_X:
X\times X
\to
A,
\qquad
(x,y)
\mapsto
a_X(x)-a_X(y),
$$

則：

$$
\boxed{
\left(
\int_Xa_X^\ast\Theta^3
\right)^2
-
T\,\deg\Delta_X
=
e^2\mathfrak D_W(Z).
}
$$

所以定義 **Albanese Carrier Defect**：

$$
\boxed{
\mathfrak A_W(X,a_X)
=
\left(
\int_Xa_X^\ast\Theta^3
\right)^2
-
T\,\deg\Delta_X,
}
$$

即可得到：

$$
\boxed{
\mathfrak A_W>0
\iff
\operatorname{pr}_{W}[Z]\neq0.
}
$$

這使 Half-Dimensional Albanese Carrier變成一個純數值 geometric search problem。

此外，本輪加入一個 2026-09 最新外部結果。Yuesen Chen 的 codimension-$3$ generic-vanishing classification證明：在 indecomposable principally polarized abelian variety of dimension $g\ge6$ 中，每個 geometrically nondegenerate GV codimension-$3$ subscheme皆為某 Jacobian 中 translate of：

$$
\boxed{
\pm W_{g-3}(C).
}
$$

因此在 dimension $6$：

$$
\boxed{
Z
=
\pm W_3(C),
\qquad
[Z]
=
\frac{\Theta^3}{3!},
}
$$

所以：

$$
\boxed{
\mathfrak D_W(Z)=0.
}
$$

故對 principally polarized simple Weil sixfold：

$$
\boxed{
\text{GV threefold carrier}
\Longrightarrow
\text{zero Weil primitive excess}.
}
$$

成功 carrier 必須離開 generic-vanishing/minimal-class architecture。

**Status:** PROVED STRUCTURAL/Numerical REDUCTION  
**Projector-Free Carrier Test:** PROVED  
**Difference-Map Test:** PROVED  
**Albanese Carrier Test:** PROVED  
**Smooth Chern-Number Test:** PROVED  
**Principal GV Carrier:** ELIMINATED  
**Minimal-Degree Nondegenerate Carrier:** ELIMINATED  
**Remaining Direct Frontier:** non-GV, nonminimal, primitive threefold geometry  
**Formalization Status:** NOT FORMALIZED  

---

# 0. Input from R008

R008 proved that on a very-general Weil sixfold：

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

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

and：

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

Thus any useful codimension-$3$ algebraic class must have a nonzero component in：

$$
W_K(A).
$$

R009 asks how to detect that component without constructing a spectral projector。

---

# 1. Middle-degree decomposition

Let：

$$
Z\subset A
$$

be an irreducible algebraic threefold。

Since：

$$
[Z]
\in
H^6(A,\mathbb Q)
\cap
H^{3,3}(A),
$$

the generic Hodge decomposition gives unique：

$$
a\in\mathbb Q,
$$

and：

$$
w\in W_K(A)
$$

such that：

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

The direct-cycle problem is：

$$
\boxed{
w\stackrel{?}{\neq}0.
}
$$

---

# 2. Weil classes are primitive

We first prove：

$$
\boxed{
\Theta\wedge W_K(A)=0.
}
$$

Take the two complex determinant eigenlines：

$$
W_K(A)_{\mathbb C}
=
W_+
\oplus
W_-.
$$

For：

$$
u\in K^\times,
$$

the eigencharacters are：

$$
u^\ast|_{W_+}
=
u^6,
$$

and：

$$
u^\ast|_{W_-}
=
\bar u^6.
$$

The compatible polarization satisfies：

$$
u^\ast\Theta
=
N(u)\Theta.
$$

---

# 3. Spectral proof of primitivity

For：

$$
w_+\in W_+,
$$

the class：

$$
\Theta w_+
$$

is a codimension-$4$ Hodge class。

But for very-general Weil type：

$$
\operatorname{Hdg}^4(A)
=
\mathbb Q\Theta^4.
$$

The $K^\times$ character on：

$$
\Theta^4
$$

is：

$$
N(u)^4.
$$

The character on：

$$
\Theta w_+
$$

would be：

$$
N(u)u^6.
$$

Choose：

$$
u
$$

such that：

$$
N(u)u^6
\neq
N(u)^4.
$$

Then：

$$
\Theta w_+
$$

cannot be a nonzero multiple of：

$$
\Theta^4.
$$

Hence：

$$
\Theta w_+=0.
$$

The same argument gives：

$$
\Theta w_-=0.
$$

Therefore：

## Theorem 3.1

$$
\boxed{
\Theta\wedge w=0
\quad
\forall w\in W_K(A).
}
$$

So every Weil middle class is primitive relative to the compatible polarization。

---

# 4. Orthogonality to the Lefschetz line

Primitivity gives：

$$
\Theta w=0.
$$

Therefore：

$$
\boxed{
\Theta^3w=0.
}
$$

Consequently：

$$
\int_A[Z]\Theta^3
=
a\int_A\Theta^6.
$$

Define：

$$
\boxed{
T
=
\int_A\Theta^6.
}
$$

and：

$$
\boxed{
D_\Theta(Z)
=
\int_A[Z]\Theta^3.
}
$$

Then：

$$
\boxed{
a
=
\frac{D_\Theta(Z)}T.
}
$$

---

# 5. Self-intersection decomposition

Define：

$$
\boxed{
S(Z)
=
\int_A[Z]^2.
}
$$

Using：

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

and：

$$
\Theta^3w=0,
$$

we obtain：

$$
\boxed{
S(Z)
=
a^2T
+
\int_Aw^2.
}
$$

Substituting：

$$
a=D_\Theta(Z)/T,
$$

gives：

$$
\boxed{
S(Z)
=
\frac{D_\Theta(Z)^2}{T}
+
\int_Aw^2.
}
$$

---

# 6. Hodge–Riemann sign

The class：

$$
w
$$

is rational, hence real, primitive, and of Hodge type：

$$
(3,3).
$$

For a primitive middle class of degree：

$$
6
$$

on a complex sixfold, the Hodge–Riemann form is：

$$
(-1)^{6\cdot5/2}
\int_Aw^2
=
-
\int_Aw^2.
$$

Positivity therefore gives：

$$
\boxed{
w\neq0
\Longrightarrow
\int_Aw^2<0.
}
$$

Moreover：

$$
\int_Aw^2=0
$$

if and only if：

$$
w=0.
$$

---

# 7. Weil Carrier Defect

Define：

$$
\boxed{
\mathfrak D_W(Z)
=
D_\Theta(Z)^2
-
T\,S(Z).
}
$$

Using Section 5：

$$
\mathfrak D_W(Z)
=
-T
\int_Aw^2.
$$

Since：

$$
T>0,
$$

Hodge–Riemann gives：

## Theorem 7.1 — Projector-Free Carrier Criterion

$$
\boxed{
\mathfrak D_W(Z)\ge0.
}
$$

Moreover：

$$
\boxed{
\mathfrak D_W(Z)=0
\iff
w=0,
}
$$

and：

$$
\boxed{
\mathfrak D_W(Z)>0
\iff
w\neq0.
}
$$

Thus a nonzero Weil projection is detected by two ordinary intersection numbers。

---

# 8. Hodge-index form

Equivalently：

$$
\boxed{
S(Z)
\le
\frac{D_\Theta(Z)^2}{T}.
}
$$

Equality holds if and only if：

$$
[Z]
\in
\mathbb Q\Theta^3.
$$

Strict inequality is exactly the algebraic carrier condition：

$$
\boxed{
S(Z)
<
\frac{D_\Theta(Z)^2}{T}
\iff
\operatorname{PE}(Z)\neq0.
}
$$

This is the middle-degree Hodge-index shadow of the Weil decomposition。

---

# 9. Simplicity of the target

For very-general Weil sixfold：

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

A field contains no nontrivial idempotents。

Therefore：

$$
\boxed{
A
\text{ is simple}.
}
$$

This has an immediate consequence for every proper positive-dimensional subvariety。

---

# 10. Geometric nondegeneracy

A subvariety of a simple abelian variety is geometrically nondegenerate。

Thus every irreducible：

$$
Z^3\subset A^6
$$

satisfies：

$$
\boxed{
Z-Z=A.
}
$$

Because：

$$
\dim(Z\times Z)=6=\dim A,
$$

the difference map：

$$
\delta_Z:
Z\times Z
\to
A
$$

is generically finite。

---

# 11. Difference-map degree

For general：

$$
t\in A,
$$

the fiber：

$$
\delta_Z^{-1}(t)
$$

consists of pairs：

$$
(z_1,z_2)
$$

such that：

$$
z_1-z_2=t.
$$

Equivalently：

$$
z_1
\in
Z\cap(t+Z).
$$

The generic fiber degree therefore equals the intersection number of：

$$
Z
$$

with a general translate of itself。

Since translation is cohomologically trivial：

$$
\boxed{
\deg\delta_Z
=
[Z]^2
=
S(Z).
}
$$

---

# 12. Difference-Map Carrier Criterion

Combining Sections 7 and 11 gives：

## Theorem 12.1

For every irreducible threefold：

$$
Z\subset A
$$

in a very-general Weil sixfold：

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

if and only if：

$$
\boxed{
D_\Theta(Z)^2
>
T\deg\delta_Z.
}
$$

This is a completely geometric criterion。

No Hodge projector and no explicit basis of：

$$
W_K(A)
$$

is needed。

---

# 13. Principal polarization specialization

If：

$$
\Theta
$$

is principal, then：

$$
\boxed{
T
=
\Theta^6
=
6!
=
720.
}
$$

Hence：

$$
\boxed{
\mathfrak D_W(Z)
=
D_\Theta(Z)^2
-
720\deg\delta_Z.
}
$$

A principal carrier succeeds exactly when：

$$
\boxed{
D_\Theta(Z)^2
>
720\deg\delta_Z.
}
$$

---

# 14. Albanese carrier

Now let：

$$
X
$$

be a smooth projective threefold with：

$$
q(X)=6,
$$

and let：

$$
a_X:
X\to A
$$

have three-dimensional image：

$$
Z=a_X(X).
$$

Assume：

$$
a_X
$$

is generically finite onto：

$$
Z
$$

of degree：

$$
e.
$$

Then：

$$
\boxed{
(a_X)_\ast[X]
=
e[Z].
}
$$

---

# 15. Polarization degree on $X$

Define：

$$
\boxed{
D_X
=
\int_Xa_X^\ast\Theta^3.
}
$$

Projection formula gives：

$$
\boxed{
D_X
=
eD_\Theta(Z).
}
$$

So the unknown Albanese degree：

$$
e
$$

appears linearly。

---

# 16. Albanese difference map

Define：

$$
\boxed{
\Delta_X:
X\times X
\to
A,
\qquad
(x,y)
\mapsto
a_X(x)-a_X(y).
}
$$

For a general point：

$$
t\in A,
$$

every pair：

$$
(z_1,z_2)
\in
\delta_Z^{-1}(t)
$$

has：

$$
e
$$

generic preimages in each factor。

Hence：

$$
\boxed{
\deg\Delta_X
=
e^2\deg\delta_Z.
}
$$

---

# 17. Albanese Carrier Defect

Define：

$$
\boxed{
\mathfrak A_W(X,a_X)
=
D_X^2
-
T\deg\Delta_X.
}
$$

Substituting Sections 15 and 16 gives：

$$
\boxed{
\mathfrak A_W(X,a_X)
=
e^2
\mathfrak D_W(Z).
}
$$

Therefore：

## Theorem 17.1 — Albanese Carrier Criterion

$$
\boxed{
\mathfrak A_W(X,a_X)>0
}
$$

if and only if：

$$
\boxed{
\operatorname{pr}_{W}[Z]\neq0.
}
$$

The generic degree：

$$
e
$$

need not be known。

This is the exact numerical realization of the Half-Dimensional Albanese Carrier program introduced in R008。

---

# 18. Smooth embedded carrier

Suppose：

$$
Z\subset A
$$

is smooth。

The self-intersection formula gives：

$$
\boxed{
S(Z)
=
\int_Zc_3(N_{Z/A}).
}
$$

Since：

$$
T_A
$$

is trivial：

$$
c(N_{Z/A})
=
c(T_Z)^{-1}.
$$

---

# 19. Normal-bundle Chern class

For a rank-$3$ normal bundle：

$$
N=N_{Z/A},
$$

the inverse total Chern class gives：

$$
\boxed{
c_3(N)
=
-c_1(T_Z)^3
+
2c_1(T_Z)c_2(T_Z)
-
c_3(T_Z).
}
$$

Write：

$$
K_Z
=
c_1(\omega_Z)
=
-c_1(T_Z).
$$

Then：

$$
\boxed{
c_3(N)
=
K_Z^3
-
2K_Zc_2(Z)
-
c_3(Z).
}
$$

---

# 20. Smooth Chern-Number Carrier Criterion

For smooth embedded：

$$
Z^3\subset A^6,
$$

define：

$$
D_\Theta(Z)
=
\int_Z\Theta^3|_Z.
$$

Then：

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

if and only if：

$$
\boxed{
D_\Theta(Z)^2
>
T
\int_Z
\left(
K_Z^3
-
2K_Zc_2(Z)
-
c_3(Z)
\right).
}
$$

So a candidate can be tested using intrinsic threefold Chern numbers and one polarization degree。

---

# 21. Ran nondegeneracy bound

There is a stronger classical notion of nondegeneracy due to Ran。

For complementary-dimensional nondegenerate subvarieties：

$$
V,W
$$

of an $n$-dimensional abelian variety：

$$
\boxed{
V\cdot W
\ge
\binom nd.
}
$$

Take：

$$
n=6,
\qquad
d=3,
\qquad
V=W=Z.
$$

If：

$$
Z
$$

is nondegenerate in Ran's sense, then：

$$
\boxed{
S(Z)\ge20.
}
$$

---

# 22. Principal minimal-degree bound

Assume principal polarization：

$$
T=720.
$$

The Hodge-index carrier inequality gives：

$$
S(Z)
\le
\frac{D_\Theta(Z)^2}{720}.
$$

If $Z$ is Ran-nondegenerate：

$$
20
\le
\frac{D_\Theta(Z)^2}{720}.
$$

Hence：

$$
\boxed{
D_\Theta(Z)\ge120.
}
$$

This recovers the half-dimensional minimal degree numerically。

---

# 23. Primitive carrier requires super-minimal degree

Suppose now：

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

Then the Hodge inequality is strict：

$$
S(Z)
<
\frac{D_\Theta(Z)^2}{720}.
$$

If $Z$ is also Ran-nondegenerate：

$$
S(Z)\ge20.
$$

Therefore：

$$
20
<
\frac{D_\Theta(Z)^2}{720}.
$$

Hence：

## Theorem 23.1 — Minimal-Degree Carrier Barrier

For a Ran-nondegenerate threefold in a principally polarized very-general Weil sixfold：

$$
\boxed{
\operatorname{PE}(Z)\neq0
\Longrightarrow
D_\Theta(Z)>120.
}
$$

A minimal-degree carrier cannot contain a Weil primitive component。

---

# 24. Jacobian calibration

Let：

$$
(A,\Theta)
=
(J(C),\Theta_C)
$$

with：

$$
g(C)=6.
$$

The Brill–Noether threefold：

$$
W_3(C)
$$

has class：

$$
\boxed{
[W_3(C)]
=
\frac{\Theta^3}{3!}
=
\frac{\Theta^3}{6}.
}
$$

Therefore：

$$
\boxed{
D_\Theta(W_3)
=
\frac{\Theta^6}{6}
=
120.
}
$$

---

# 25. Self-intersection of $W_3$

We have：

$$
\boxed{
S(W_3)
=
\frac{\Theta^6}{36}
=
20.
}
$$

Consequently：

$$
\boxed{
\mathfrak D_W(W_3)
=
120^2
-
720\cdot20
=
0.
}
$$

Thus the minimal Brill–Noether carrier lies exactly on the equality boundary。

---

# 26. Difference-map calibration

The difference map：

$$
W_3(C)\times W_3(C)
\to
J(C)
$$

has generic degree：

$$
\boxed{
20
=
\binom63.
}
$$

Hence the projector-free criterion reads：

$$
\boxed{
20
=
\frac{120^2}{720}.
}
$$

The classical minimal class is therefore exactly the zero-defect calibration point。

---

# 27. Generic-vanishing carrier search

Pareschi–Popa introduced the relation between：

- generic vanishing；
- minimal cohomology class；
- theta duality。

For geometrically nondegenerate GV subschemes, generic vanishing forces minimal cohomological behavior。

Until 2026, the full codimension-$3$ geometric classification remained part of the broader minimal-class program。

That gap has now been closed for the GV hypothesis。

---

# 28. Chen's 2026 codimension-three theorem

Yuesen Chen proves in：

```text
arXiv:2609.15391
Generic vanishing subschemes of codimension three
```

that if：

$$
(A,\Theta)
$$

is an indecomposable principally polarized complex abelian variety of dimension：

$$
g\ge6,
$$

and：

$$
Z\subset A
$$

is a geometrically nondegenerate GV subscheme of codimension：

$$
3,
$$

then there is a smooth curve：

$$
C
$$

of genus：

$$
g
$$

and a polarized isomorphism：

$$
\boxed{
(A,\Theta)
\simeq
(J(C),\Theta_C)
}
$$

such that：

$$
\boxed{
Z
}
$$

is a translate of：

$$
\boxed{
\pm W_{g-3}(C).
}
$$

---

# 29. Dimension-six consequence

Set：

$$
g=6.
$$

Then every geometrically nondegenerate GV codimension-$3$ subscheme is：

$$
\boxed{
\pm W_3(C)
}
$$

up to translation。

Therefore：

$$
\boxed{
[Z]
=
\frac{\Theta^3}{6}.
}
$$

Hence：

$$
\boxed{
\mathfrak D_W(Z)=0.
}
$$

---

# 30. GV Carrier Barrier

## Theorem 30.1

Let：

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

be a simple principally polarized very-general Weil sixfold。

If：

$$
Z\subset A
$$

is a codimension-$3$ GV subscheme, then：

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

### Proof

Because：

$$
A
$$

is simple, every subvariety is geometrically nondegenerate。

Because：

$$
A
$$

is simple, the principal polarization is indecomposable。

Chen's codimension-$3$ theorem applies。

Thus：

$$
Z
$$

is a translated：

$$
\pm W_3(C)
$$

inside a Jacobian。

Its class is：

$$
\Theta^3/6.
$$

Therefore the Weil primitive excess vanishes。

QED.

---

# 31. Stronger geometric interpretation

On a principal target, a successful direct carrier must therefore be：

$$
\boxed{
\text{non-GV}.
}
$$

This is a strong negative design condition。

The most structured low-complexity subvarieties in ppavs often satisfy generic-vanishing conditions。

Those are now eliminated automatically in codimension three。

---

# 32. Jacobian-locus implication

Chen's theorem gives an additional diagnostic。

If an indecomposable ppav：

$$
A^6
$$

contains any geometrically nondegenerate GV codimension-$3$ subscheme, then：

$$
\boxed{
A
\text{ is a genus-}6\text{ Jacobian}.
}
$$

So a GV carrier search on a target component first collapses to a Schottky-locus question。

Even if such a point is found, the carrier itself still has：

$$
\mathfrak D_W=0.
$$

Thus it cannot solve the Weil primitive problem。

---

# 33. Domain firewall

The GV Carrier Barrier in Sections 28–32 requires：

$$
\boxed{
\text{principal polarization}.
}
$$

A general Weil component may have another polarization type。

R009 does **not** claim Chen's theorem for arbitrary polarized abelian varieties。

The projector-free defect：

$$
\mathfrak D_W
$$

and difference-map criterion, however, work for any compatible polarization。

---

# 34. Minimal class versus primitive class

The direct search must avoid confusing two notions。

### Minimal Lefschetz class

$$
\frac{\Theta^3}{3!}.
$$

### Weil primitive class

$$
0\neq w\in W_K(A).
$$

The first is extremal inside the contraction/Lefschetz sector。

The second lies in an orthogonal primitive determinant sector。

The Hodge Carrier Defect measures precisely the separation between these two geometries。

---

# 35. Why low degree is a bad search region

For principal polarization, the natural minimal benchmark is：

$$
D_\Theta=120.
$$

At that benchmark：

- Brill–Noether minimal carriers have:
  $$
  S=20;
  $$
- Ran nondegeneracy forces:
  $$
  S\ge20;
  $$
- Hodge–Riemann primitive excess would require:
  $$
  S<20.
  $$

Therefore these conditions are mutually incompatible。

So a nondegenerate primitive carrier must leave the minimal-degree regime immediately。

---

# 36. Near-minimal arithmetic window

For a principal target：

$$
\mathfrak D_W(Z)
=
D_\Theta(Z)^2
-
720S(Z).
$$

If：

$$
D_\Theta(Z)
=
121
$$

or：

$$
122,
$$

and $Z$ is Ran-nondegenerate, then：

$$
S(Z)\ge20,
$$

while：

$$
S(Z)
<
D_\Theta(Z)^2/720
<
21.
$$

Hence necessarily：

$$
\boxed{
S(Z)=20.
}
$$

So the first two degrees above minimal force equality in Ran's intersection bound even though the Hodge class would have primitive excess。

This places any extremely low-degree candidate into a rigid property-$(P)$ regime。

R009 does not classify that regime further, but records it as a sharp future filter。

---

# 37. Smooth candidate invariant package

For a smooth embedded candidate：

$$
Z^3\subset A,
$$

the full carrier test requires only：

### Polarization degree

$$
D
=
\int_Z\Theta^3.
$$

### Canonical cube

$$
K_Z^3.
$$

### Mixed Chern number

$$
K_Zc_2(Z).
$$

### Euler number

$$
c_3(Z).
$$

Then：

$$
\boxed{
\mathfrak D_W(Z)
=
D^2
-
T
\left[
K_Z^3
-
2K_Zc_2(Z)
-
c_3(Z)
\right].
}
$$

This converts the Hodge problem into a concrete geography-of-threefolds test。

---

# 38. Albanese candidate invariant package

For a smooth threefold：

$$
X
$$

with：

$$
q(X)=6,
$$

and generically finite Albanese map to a six-dimensional abelian variety：

$$
a_X:X\to A,
$$

one needs only：

### Albanese polarization volume

$$
\boxed{
D_X
=
\int_Xa_X^\ast\Theta^3.
}
$$

### Albanese difference degree

$$
\boxed{
M_X
=
\deg
\left[
X\times X
\overset{\Delta_X}{\longrightarrow}
A
\right].
}
$$

Then：

$$
\boxed{
\mathfrak A_W(X)
=
D_X^2
-
T M_X.
}
$$

The target condition is simply：

$$
\boxed{
\mathfrak A_W(X)>0.
}
$$

---

# 39. Why difference degree is useful

The difference-map degree is often accessible from geometric construction data：

- symmetric products；
- incidence correspondences；
- moduli spaces；
- quotient constructions；
- explicit branch/cover data；
- finite group actions。

It can therefore be easier to compute than the cohomology class：

$$
[Z]
$$

itself。

This is precisely the kind of compression GMSC seeks。

---

# 40. Symmetric-power dead zone

The most obvious $q=6$ Albanese carrier is：

$$
X=C^{(3)}
$$

for a genus-$6$ curve。

Its Albanese image is：

$$
W_3(C)\subset J(C).
$$

R009 computes：

$$
\mathfrak A_W=0.
$$

Thus the most canonical half-dimensional Albanese threefold is a zero-defect carrier。

Any successful construction must be less symmetric and less generic-vanishing than the symmetric-power model。

---

# 41. Product-of-curves dead zone

A product：

$$
C_1\times C_2\times C_3
$$

can have irregularity：

$$
q(C_1)+q(C_2)+q(C_3)=6.
$$

But its Albanese variety decomposes as：

$$
J(C_1)\times J(C_2)\times J(C_3).
$$

A very-general Weil sixfold with：

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

is simple。

Therefore such a product cannot provide the target generic Albanese sixfold。

This removes another standard irregular-threefold construction。

---

# 42. Abelian-threefold dead zone

If：

$$
X
$$

itself is an abelian threefold, then its Albanese variety has dimension：

$$
3,
$$

not：

$$
6.
$$

An embedding into：

$$
A^6
$$

would make its image an abelian subvariety。

But：

$$
A
$$

is simple。

Therefore no positive-dimensional abelian-subvariety carrier exists。

---

# 43. Carrier design requirements after R009

A successful half-dimensional carrier on a principal target must therefore satisfy simultaneously：

1. $X$ has:
   $$
   q(X)=6;
   $$
2. the Albanese image has dimension:
   $$
   3;
   $$
3. the Albanese sixfold is simple and of the target Weil type；
4. the image is not a translated abelian subvariety；
5. it is not a divisor complete intersection；
6. it is not a GV codimension-$3$ subscheme；
7. if Ran-nondegenerate:
   $$
   D_\Theta>120;
   $$
8. its carrier defect satisfies:
   $$
   \boxed{
   \mathfrak A_W>0.
   }
   $$

This is a much sharper target than simply asking for an irregular threefold。

---

# 44. CE001 Attack reformulation

CE001 wants：

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

For irreducible threefold cycles, R009 translates one part of CE001 into：

> prove that every irreducible algebraic threefold $Z\subset A$ satisfies

$$
\boxed{
D_\Theta(Z)^2
=
T\deg\delta_Z.
}
$$

or at least show that no $Z$ with strict defect exists。

This is still extremely strong。

But it gives a geometric target rather than an abstract cycle-space statement。

---

# 45. Construct versus Attack symmetry

### HC-True / Construct

Find：

$$
Z
$$

with：

$$
\boxed{
D_\Theta(Z)^2
>
T\deg\delta_Z.
}
$$

### HC-False / Attack

Show every candidate architecture forces：

$$
\boxed{
D_\Theta(Z)^2
=
T\deg\delta_Z.
}
$$

or show all geometrically available carriers lie in a zero-defect class such as GV/minimal geometry。

Thus both branches now act on the same scalar defect。

---

# 46. GMSC compression result

Before R009 the direct problem was：

$$
\boxed{
\text{find a primitive codimension-3 algebraic cycle}.
}
$$

After R009 it becomes：

$$
\boxed{
\text{find a half-dimensional algebraic carrier with positive numerical defect}.
}
$$

For Albanese carriers：

$$
\boxed{
D_X^2
>
T\deg\Delta_X.
}
$$

This is a substantial reduction in representation complexity。

---

# 47. New route-local bottleneck

Inside the half-dimensional-carrier route, the principal target now has a sharp negative bottleneck：

$$
\boxed{
\text{successful carrier must be non-GV}.
}
$$

The positive search must therefore move into irregular threefolds whose Albanese images fail the usual theta-dual/generic-vanishing package。

This is a qualitative design condition, not merely a numerical one。

---

# 48. Current live carrier classes

After the filters, remaining live possibilities include：

### A. Non-GV Albanese threefolds

$$
q=6,
$$

maximal Albanese dimension, simple Albanese。

### B. Singular half-dimensional images

Their difference degree may be computable even when normal-bundle Chern formulas are unavailable。

### C. Moduli/incidence threefolds

A naturally constructed threefold may map generically finite to a simple sixfold without being minimal class。

### D. Component-local derived carrier

Its：

$$
ch_3
$$

need not be represented initially by a single smooth subvariety, but the same numerical projection principle applies once an algebraic cycle is extracted。

---

# 49. Current dead carrier classes

```text
DEAD / ZERO DEFECT:
  W_3(C) in J(C)
  principal codim-3 GV subschemes
  divisor complete intersections
  translated abelian threefolds
  product-of-curves generic Albanese constructions
  symmetric-cube minimal carrier
```

This eliminates most canonical low-complexity Albanese models。

---

# 50. Bottleneck stability update

Across：

$$
R001\text{--}R009,
$$

the target SCC remains：

$$
\boxed{
\mathrm{BS}_9
\left(
\mathcal C_{\mathrm{Weil6}}^{\mathrm{vg}}
\right)
=
1.
}
$$

A new architecture-level pattern now has three independent manifestations：

1. contraction/tautological algebra collapses to：
   $$
   \Theta^3;
   $$
2. classical Prym/reflection geometries collapse to split components；
3. principal GV half-dimensional carriers collapse to：
   $$
   \Theta^3/3!.
   $$

This strongly suggests that any successful non-split realization must be non-tautological both algebraically and geometrically。

---

# 51. Main Verdict

R009 converts the Half-Dimensional Carrier program into a testable numerical problem。

For any irreducible threefold：

$$
Z\subset A,
$$

the exact carrier condition is：

$$
\boxed{
D_\Theta(Z)^2
>
T\deg\delta_Z.
}
$$

For any generically finite Albanese threefold：

$$
a_X:X^3\to A^6,
$$

the exact condition is：

$$
\boxed{
\left(
\int_Xa_X^\ast\Theta^3
\right)^2
>
T
\deg\Delta_X.
}
$$

On principal targets, a fresh codimension-$3$ generic-vanishing classification shows that every geometrically nondegenerate GV carrier is a Jacobian $W_3$ and has zero defect。

Therefore the remaining search target is now：

$$
\boxed{
\text{non-GV, nonminimal, half-dimensional irregular geometry with positive carrier defect}.
}
$$

---

# 52. Next Interface

Next GMSC round：

```text
HODGE_GMSC_R010_NonGVAlbaneseCarrierSearch.md
```

Primary target：

$$
\boxed{
\text{Search known and constructible }q=6\text{ threefolds of maximal Albanese dimension whose images are non-GV and may satisfy }\mathfrak A_W>0.
}
$$

Planned tasks：

1. inventory irregular threefolds with:
   $$
   q=6;
   $$
2. remove decomposable Albanese varieties；
3. remove symmetric-product / Brill–Noether models；
4. search Lagrangian and generalized-Lagrangian subvarieties in simple abelian sixfolds；
5. search incidence / Fano / moduli threefolds with six-dimensional Albanese；
6. compute or bound:
   $$
   D_X;
   $$
7. compute or bound:
   $$
   \deg\Delta_X;
   $$
8. use Chern-number formula for smooth embedded candidates；
9. apply Chen's GV barrier immediately when applicable；
10. run CE001 Attack by proving zero-defect or nonexistence for each candidate family；
11. if no known threefold family survives, promote primitive correspondence to the next frontier。

---

# References

1. O. Debarre, *On the geometry of abelian varieties*, lecture notes. Summarizes Ran's nondegeneracy theory, including:
   $$
   V\cdot W
   \ge
   \binom nd
   $$
   for complementary nondegenerate subvarieties, and geometric nondegeneracy of subvarieties in simple abelian varieties.

2. Z. Ran, *On subvarieties of abelian varieties*, Invent. Math. 62 (1981), 459–479. Source of the nondegeneracy intersection bound and property-$(P)$ theory.

3. O. Debarre, *Minimal Cohomology Classes and Jacobians*, J. Algebraic Geom. 4 (1995), 321–335; arXiv:alg-geom/9301002. Classifies minimal classes inside Jacobians and studies the minimal-class locus.

4. G. Pareschi, M. Popa, *Generic vanishing and minimal cohomology classes on abelian varieties*, Math. Ann. 340 (2008), 209–222; arXiv:math/0610166. Introduces the GV/minimal-class framework and theta duality.

5. S. Schreieder, *Decomposable theta divisors and generic vanishing*, arXiv:1602.06226. Develops geometric consequences for nondegenerate GV subschemes.

6. Y. Chen, *Generic vanishing subschemes of codimension three*, arXiv:2609.15391, submitted 2026-09-14 and updated 2026-09-15. Classifies geometrically nondegenerate GV codimension-$3$ subschemes in indecomposable ppavs of dimension at least $6$ as translates of $\pm W_{g-3}(C)$ in Jacobians.

7. O. Debarre, B. Moonen, *Bézout's theorem for abelian varieties*, arXiv:2509.14940. Gives modern semismallness/intersection results for subvarieties in absolutely simple abelian varieties.

8. O. Debarre, *Complex abelian varieties of Weil type*, lecture notes, 2025. Records the very-general Hodge-class dimensions:
   $$
   \operatorname{Hdg}^p(A)
   =
   \mathbb QE^p
   $$
   away from middle degree and:
   $$
   \operatorname{Hdg}^n(A)
   =
   \mathbb QE^n
   \oplus
   WH(A).
   $$

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

10. Aletheia, *HODGE_GMSC_R008_NonSplitDirectRealizationGraph*, 2026-09-16.

---

## Canonical Source Declaration

本檔案為 Hodge GMSC 分支第九篇正式 UTF-8 Markdown canonical source。

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪正式證明：

$$
\boxed{
\operatorname{PE}(Z)\neq0
\iff
D_\Theta(Z)^2
>
\Theta^6\deg\delta_Z
}
$$

for irreducible half-dimensional carriers in a very-general simple Weil sixfold。

對 generically finite Albanese carriers，等價條件為：

$$
\boxed{
\mathfrak A_W(X)>0.
}
$$

本輪亦利用 2026-09 最新 codimension-$3$ GV classification，正式排除 principally polarized target上的 GV threefold carrier。

下一輪只搜尋 non-GV Albanese threefold geometry。