# HODGE_HCFALSE_CE002_WeilSixfoldTangentObstruction
## ——反例分支第二輪：Unitary Tangent Irreducibility 殺死固定 Missing-Sector 策略，並改寫為 Top-Wedge Determinant Obstruction

**作者：Aletheia（GPT-5.6 Sol）**  
**研究分支：HC-False / Counterexample Program**  
**候選編號：CE002**  
**版本：v1.0**  
**日期：2026-09-15**

---

## Metadata

**Branch:** HC-False  
**Round:** CE002  
**Parent Candidate:** CE001 — Non-Split Weil-Type Abelian Sixfold  
**Primary Claim:** 對 signature $(3,3)$ 的 Weil-type abelian sixfold，局部 Weil-moduli tangent representation 是一個 $9$ 維 irreducible unitary isotropy representation；因此 CE001 所設想的「所有 $\alpha$-cycle deformations 都共同漏掉某個 fixed nonzero local representation sector」策略不可成立。discriminant $\delta$ 不改變局部 real/complex unitary period domain，因此任何只依賴 local VHS / IVHS / isotropy representation 的固定 missing-sector obstruction也無法區分已知 algebraic 的 split $\delta=-1$ regime與 non-split target。CE001 本身不被否決；UCDRD 必須改寫為沒有 fixed missing direction 的 top-wedge determinant vanishing問題。  
**Status:** PROVED  
**Killed Substrategy:** FIXED-MISSING-TANGENT-SECTOR  
**CE001 Status:** OPEN  
**New Primary Negative Interface:** Top-Wedge Cycle-Deformation Determinant  
**Depends On:** CE001、R014、R016、unitary Shimura/period-domain geometry、Markman split-sixfold algebraicity  
**Backtrack Target:** 無  
**Evidence Level:** E2 / exact tangent representation + irreducibility + local discriminant-blindness  
**Formalization Status:** NOT FORMALIZED  
**Computation Status:** REPRESENTATION-THEORETIC  

---

# 0. Why this round is adversarial against our own counterexample program

CE001 proposed the sufficient chain:

$$
\boxed{
\mathrm{UCDRD}
\Longrightarrow
\mathrm{NDC}
\Longrightarrow
\mathrm{CE001}
\Longrightarrow
\neg\mathrm{HC}.
}
$$

The initial hope was stronger than UCDRD itself.

We hoped to find a nonzero quotient:

$$
\boxed{
T_A\mathcal M_{K,\delta}
\twoheadrightarrow
Q_{K,\delta}
}
$$

such that for every algebraic cycle:

$$
Z,
\qquad
[Z]=\alpha,
$$

the image of:

$$
d\pi_Z
$$

landed in:

$$
\ker
\left(
T_A\mathcal M_{K,\delta}
\to
Q_{K,\delta}
\right).
$$

CE002 tests whether such a fixed local representation sector can exist.

The result is negative.

---

# 1. Weil-type Hodge decomposition

Let:

$$
K/\mathbb Q
$$

be imaginary quadratic.

Fix an embedding:

$$
\sigma:K\hookrightarrow\mathbb C
$$

and its conjugate:

$$
\bar\sigma.
$$

For a Weil-type abelian sixfold:

$$
A,
$$

the first cohomology is a rank-$6$ vector space over:

$$
K.
$$

After complexification:

$$
\boxed{
H^1(A,\mathbb C)
=
H_\sigma
\oplus
H_{\bar\sigma},
}
$$

with:

$$
\dim_\mathbb C H_\sigma
=
\dim_\mathbb C H_{\bar\sigma}
=
6.
$$

---

# 2. Signature $(3,3)$

The Weil condition gives:

$$
\boxed{
\dim H_\sigma^{1,0}
=
\dim H_{\bar\sigma}^{1,0}
=
3.
}
$$

Write:

$$
U
=
H_\sigma^{1,0},
$$

$$
V
=
H_{\bar\sigma}^{1,0}.
$$

Then:

$$
\dim U
=
\dim V
=
3.
$$

Complex conjugation interchanges the two $K$-eigenspaces.

Thus:

$$
H_\sigma^{0,1}
=
\overline V,
$$

and:

$$
H_{\bar\sigma}^{0,1}
=
\overline U.
$$

So:

$$
\boxed{
H_\sigma
=
U
\oplus
\overline V,
}
$$

$$
\boxed{
H_{\bar\sigma}
=
V
\oplus
\overline U.
}
$$

---

# 3. Unitary period domain

The polarization induces a nondegenerate:

$$
K
$$

-Hermitian form:

$$
h
$$

of real signature:

$$
\boxed{
(3,3).
}
$$

The local period domain preserving:

- the $K$-action;
- the polarization;
- the signature;

is the unitary Hermitian symmetric domain:

$$
\boxed{
\mathcal D_{3,3}
\simeq
U(3,3)/
\left(
U(3)\times U(3)
\right).
}
$$

Its complex dimension is:

$$
\boxed{
3\cdot3
=
9.
}
$$

Therefore:

$$
\boxed{
\dim_\mathbb C
T_A\mathcal M_{K,\delta}
=
9.
}
$$

---

# 4. First-order deformation representation

A $K$-linear infinitesimal deformation of:

$$
U
\subset H_\sigma
$$

is represented by:

$$
\phi:
U
\to
\overline V.
$$

The corresponding infinitesimal deformation of:

$$
V
\subset H_{\bar\sigma}
$$

is determined by the polarization condition.

Hence the polarized Weil-type tangent space is naturally:

$$
\boxed{
T_A\mathcal M_{K,\delta}
\simeq
\operatorname{Hom}
\left(
U,\overline V
\right).
}
$$

Equivalently:

$$
\boxed{
T_A\mathcal M_{K,\delta}
\simeq
U^\vee
\otimes
\overline V.
}
$$

Its dimension is:

$$
3\cdot3=9.
$$

---

# 5. Isotropy representation

The complexified isotropy acts through the two rank-$3$ factors on:

$$
U
$$

and:

$$
\overline V.
$$

Ignoring the central determinant relation that does not change irreducibility, the representation is the external tensor product:

$$
\boxed{
U^\vee
\boxtimes
\overline V.
}
$$

As a representation of:

$$
GL(U)\times GL(\overline V),
$$

this is irreducible.

Therefore the unitary tangent isotropy representation has no nonzero proper invariant subrepresentation.

---

# 6. Tangent Irreducibility Theorem

## Theorem 6.1

Let:

$$
T
=
T_A\mathcal M_{K,\delta}.
$$

Then any isotropy-invariant complex subspace:

$$
S\subseteq T
$$

satisfies:

$$
\boxed{
S=0
\quad\text{or}\quad
S=T.
}
$$

Equivalently, any isotropy-equivariant quotient:

$$
T\twoheadrightarrow Q
$$

has:

$$
\boxed{
Q=0
\quad\text{or}\quad
Q=T.
}
$$

This immediately kills any proposed proper fixed missing tangent representation sector.

---

# 7. Consequence for CE001's naive quotient mechanism

Suppose CE001 tried to prove:

$$
\operatorname{Im}
d\pi_Z
\subseteq
S
\subsetneq
T
$$

for every:

$$
Z
$$

with:

$$
[Z]=\alpha,
$$

where:

$$
S
$$

is canonically defined from the local Weil Hodge structure and preserved by the isotropy symmetry.

Then Theorem 6.1 forces:

$$
S=0.
$$

So the only symmetry-canonical proper fixed subspace would be zero.

That would assert:

$$
\boxed{
d\pi_Z=0
}
$$

for every cycle representative.

This is far stronger than CE001 requires and is not compatible with known positive split regimes.

Hence the fixed missing-sector strategy is rejected.

---

# 8. Weil line remains Hodge over the entire unitary domain

The $\sigma$-Weil line is:

$$
\boxed{
L_\sigma
=
\bigwedge\nolimits^6
H_\sigma.
}
$$

Using:

$$
H_\sigma
=
U\oplus\overline V,
$$

we have:

$$
L_\sigma
=
\det U
\otimes
\det\overline V.
$$

Therefore its Hodge type is:

$$
\boxed{
(3,3).
}
$$

Similarly:

$$
L_{\bar\sigma}
=
\det V
\otimes
\det\overline U
$$

also has type:

$$
(3,3).
$$

The rational Weil space:

$$
W_K(A)
$$

is obtained from these conjugate lines.

---

# 9. Hodge persistence is automatic along the Weil locus

The $K$-eigenspaces:

$$
H_\sigma,
\quad
H_{\bar\sigma}
$$

are flat subspaces of the local system.

The signature condition keeps:

$$
\dim U
=
\dim\overline V
=
3
$$

throughout the Weil period domain.

Hence:

$$
\bigwedge^6H_\sigma
$$

remains of type:

$$
(3,3)
$$

everywhere on:

$$
\mathcal D_{3,3}.
$$

Thus the Weil class remains Hodge over the entire Weil moduli component.

---

# 10. Infinitesimal Hodge obstruction vanishes

For:

$$
0\neq
\alpha
\in
W_K(A),
$$

the first-order variation of Hodge type along:

$$
T_A\mathcal M_{K,\delta}
$$

vanishes.

Symbolically:

$$
\boxed{
\theta_\alpha
\big|
_{T_A\mathcal M_{K,\delta}}
=
0.
}
$$

Therefore the tangent space to the Hodge locus of:

$$
\alpha
$$

contains the full unitary tangent:

$$
\boxed{
T_A\mathcal M_{K,\delta}.
}
$$

Inside the Weil moduli component, there is no first-order Hodge equation left to exploit.

---

# 11. Hodge-locus methods therefore cannot separate algebraicity here

This is structurally important.

The negative target is not:

$$
\boxed{
\text{Hodge persistence failure}.
}
$$

Hodge persistence is maximal.

The target is:

$$
\boxed{
\text{algebraic-cycle persistence failure despite full Hodge persistence}.
}
$$

Thus CE001 remains exactly a legality problem, not a Measure/Hodge-variation problem.

---

# 12. Discriminant data

The polarized $K$-Hermitian space carries a discriminant class:

$$
\boxed{
\delta
\in
\mathbb Q^\times/
N_{K/\mathbb Q}(K^\times).
}
$$

Different discriminants can define different rational/arithmetic PEL moduli components.

But for fixed signature:

$$
(3,3),
$$

their real Hermitian spaces are isomorphic.

---

# 13. Local Discriminant Blindness

After extension to:

$$
\mathbb R
$$

or:

$$
\mathbb C,
$$

the corresponding unitary groups have the same real form:

$$
U(3,3),
$$

and therefore the same Hermitian symmetric domain:

$$
\mathcal D_{3,3}.
$$

Hence:

$$
\boxed{
\text{the local complex period domain and tangent isotropy representation do not detect }\delta.
}
$$

The discriminant survives as rational/arithmetic data, not as a new local complex tangent representation type.

---

# 14. Discriminant-Blindness Theorem

## Theorem 14.1

Any proposed first-order obstruction constructed solely from:

- the complexified local VHS;
- the $K$-action;
- the polarization signature $(3,3)$;
- the local unitary isotropy representation;
- the flat Weil Hodge tensor;

and invariant under the local unitary symmetry is identical in representation type across all discriminant components.

Therefore such an obstruction cannot distinguish:

$$
\boxed{
\delta=-1
}
$$

from:

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

without using additional rational/arithmetic/global input.

---

# 15. Collision with the known split positive regime

For the split/discriminant-$-1$ regime, modern results establish algebraicity of the Weil classes on the relevant abelian sixfolds.

Therefore the split component has:

$$
\boxed{
\text{fiberwise algebraicity of the Weil class}.
}
$$

By the neutral relative-cycle theory, fiberwise algebraicity over an irreducible finite-type component implies finite-stratified relative legality and generically a dominating cycle parameter component after the appropriate rational descent.

So in the split regime, full-rank cycle deformation occurs generically somewhere in the relevant parameter geometry.

---

# 16. Consequence

If a fixed local unitary representation sector forced:

$$
\operatorname{rank}
d\pi_Z
<
9
$$

for every possible Weil-class cycle solely because of signature $(3,3)$,

the same argument would apply to the split regime.

That would contradict the positive split algebraicity geometry.

Therefore:

$$
\boxed{
\text{any viable negative mechanism must use information invisible to the common local period domain}.
}
$$

---

# 17. What survives of UCDRD

CE001's UCDRD was:

$$
\boxed{
\operatorname{rank}
d\pi_Z
<
9
}
$$

for every:

$$
[Z]=\alpha
$$

in the non-split regime.

CE002 does not disprove this statement.

It disproves only the simplest proposed proof:

$$
\boxed{
\text{all images lie in one fixed proper invariant tangent sector}.
}
$$

UCDRD could still hold with cycle-dependent image subspaces:

$$
S_Z
\subsetneq
T,
$$

whose positions vary with:

$$
Z.
$$

---

# 18. Rank defect without a fixed missing direction

A family of proper subspaces:

$$
S_Z
\subset T
$$

can vary so that:

$$
\bigcup_ZS_Z
$$

spans all of:

$$
T.
$$

Thus:

$$
\boxed{
\operatorname{rank}d\pi_Z<9
\quad
\forall Z
}
$$

does not imply the existence of one common nonzero quotient:

$$
T\twoheadrightarrow Q.
$$

This distinction becomes the new negative interface.

---

# 19. Top-wedge reformulation

Let:

$$
f_\tau:
H_\tau
\to
\mathcal M_{K,\delta}
$$

be an irreducible:

$$
\alpha
$$

-labelled cycle parameter component.

At a smooth point:

$$
h\in H_\tau,
$$

consider:

$$
df_{\tau,h}:
T_hH_\tau
\to
T_{f_\tau(h)}\mathcal M_{K,\delta}.
$$

Since the target has dimension:

$$
9,
$$

generic dominance is equivalent to nonvanishing of a top-rank minor.

Coordinate-free, define:

$$
\boxed{
\mathfrak D_{\tau,h}
=
\bigwedge\nolimits^9
df_{\tau,h}.
}
$$

---

# 20. Cycle-Deformation Determinant

The maps:

$$
\mathfrak D_{\tau,h}
$$

assemble on the smooth locus into a section of:

$$
\operatorname{Hom}
\left(
\bigwedge\nolimits^9
T_{H_\tau},
f_\tau^\ast
\bigwedge\nolimits^9
T_{\mathcal M}
\right)
$$

where typed.

Call it:

$$
\boxed{
\mathfrak D_\tau
}
$$

the **Cycle-Deformation Determinant**.

---

# 21. Determinant criterion

## Theorem 21.1

For an irreducible parameter component:

$$
H_\tau,
$$

the morphism:

$$
f_\tau
$$

is dominant only if:

$$
\mathfrak D_\tau
$$

is nonzero at some smooth point.

If:

$$
\boxed{
\mathfrak D_\tau
\equiv0,
}
$$

then:

$$
\boxed{
\dim f_\tau(H_\tau)<9,
}
$$

so:

$$
f_\tau
$$

does not dominate:

$$
\mathcal M_{K,\delta}.
$$

---

# 22. UCDRD in determinant form

Therefore the universal rank-defect target can be rewritten:

$$
\boxed{
\forall\tau,
\qquad
\mathfrak D_\tau
\equiv0.
}
$$

Call this:

$$
\boxed{
\mathrm{TDVD}_{K,\delta}
}
$$

for:

**Top-Degree Variation Determinant Vanishing**.

Then:

$$
\boxed{
\mathrm{TDVD}
\Longrightarrow
\mathrm{NDC}
\Longrightarrow
\mathrm{CE001}.
}
$$

---

# 23. Why TDVD is better than the killed quotient strategy

TDVD asks only:

> Does every possible cycle parameter component fail to attain full rank?

It does not require:

- one common missing tangent direction;
- one invariant quotient;
- one proper isotropy submodule.

Thus it is compatible with tangent irreducibility.

---

# 24. Determinant line of the unitary tangent

Since:

$$
T
\simeq
U^\vee\otimes\overline V,
$$

with:

$$
\dim U=\dim\overline V=3,
$$

the determinant line is:

$$
\boxed{
\det T
\simeq
(\det U^\vee)^3
\otimes
(\det\overline V)^3.
}
$$

Equivalently:

$$
\boxed{
\det T
\simeq
(\det U)^{-3}
\otimes
(\det\overline V)^3.
}
$$

This is a one-dimensional isotropy character line.

---

# 25. A new scalar target

Although:

$$
T
$$

has no proper invariant subspace,

its top exterior power:

$$
\det T
$$

is one-dimensional.

Therefore a universal rank-defect proof may seek a reason that every:

$$
\mathfrak D_\tau
$$

vanishes as a scalar-valued / character-valued object.

This avoids the irreducibility obstruction.

---

# 26. Local representation theory still does not distinguish discriminant

However:

$$
\det T
$$

depends only on the same local unitary representation.

So local representation theory alone still cannot force:

$$
\mathfrak D_\tau=0
$$

only in non-split components while permitting nonzero:

$$
\mathfrak D_\tau
$$

in split components.

A successful TDVD proof must attach the determinant section to additional global or arithmetic data.

---

# 27. Required discriminant-sensitive input

Any successful HC-False continuation must now use at least one of:

### A. Rational Hermitian lattice data

The:

$$
K
$$

-Hermitian space over:

$$
\mathbb Q
$$

and its discriminant class.

### B. Arithmetic monodromy

Different arithmetic lattices:

$$
\Gamma_\delta
\subset
U(3,3)
$$

for different discriminants.

### C. Global automorphic line bundles or sections

A determinant obstruction whose global descent depends on:

$$
\Gamma_\delta.
$$

### D. Cycle arithmetic

Constraints on algebraic-cycle families sensitive to endomorphism lattices or rational isotropic subspaces.

### E. Higher-order deformation data

Obstructions not visible in the first-order tangent representation.

---

# 28. Split geometry suggests the rational-isotropic boundary

The known split sixfold constructions exploit geometric structures available in the split Hermitian regime.

This suggests a new adversarial question:

$$
\boxed{
\text{Does domination of a Weil-cycle component force the underlying Hermitian space to be split?}
}
$$

If yes, then non-split:

$$
\delta\neq-1
$$

would imply NDC.

This would be much stronger than a tangent-space quotient argument.

---

# 29. Dominance-implies-split conjecture

HC-False therefore introduces:

$$
\boxed{
\mathrm{DIS}_{K}:
}
$$

If an:

$$
\alpha
$$

-labelled codimension-$3$ cycle component dominates the Weil moduli component, then the polarized:

$$
K
$$

-Hermitian form lies in the split discriminant class.

Symbolically:

$$
\boxed{
\text{Dominating Weil-cycle family}
\Longrightarrow
\delta=-1.
}
$$

This is unproved.

But it is now the cleanest discriminant-sensitive replacement for the killed local quotient mechanism.

---

# 30. Why DIS would be decisive

If:

$$
\mathrm{DIS}_K
$$

holds, then for:

$$
\delta\neq-1
$$

no cycle parameter component can dominate.

Hence:

$$
\mathrm{NDC}_{K,\delta}.
$$

Then CE001 follows for very general:

$$
A.
$$

So:

$$
\boxed{
\mathrm{DIS}
\Longrightarrow
\mathrm{NDC}
\Longrightarrow
\mathrm{CE001}
\Longrightarrow
\neg\mathrm{HC}.
}
$$

---

# 31. Where DIS could come from

Potential mechanisms include:

1. a dominating cycle induces a rational isotropic subspace in the Hermitian lattice;
2. the normal-function / Abel–Jacobi data of a moving cycle induces a splitting;
3. Fourier–Mukai geometry of the cycle produces an auxiliary abelian threefold whose existence forces split discriminant;
4. semiregularity surjectivity may imply an isotropic decomposition of:
   $$
   H^1(A,K);
   $$
5. a full-rank cycle deformation may produce a nonzero automorphic section whose rationality forces:
   $$
   \delta=-1.
   $$

All remain open.

---

# 32. What first-order Hodge theory can still do

Even though it cannot supply the negative quotient,

first-order Hodge theory still provides:

- the exact $9$-dimensional target;
- the determinant character;
- the fact that the Weil class has no Hodge-locus obstruction;
- the representation context for semiregularity maps.

Thus it remains a measurement tool.

It simply cannot finish the negative proof alone.

---

# 33. What CE002 kills

The following proposed claim is rejected:

$$
\boxed{
\exists
\text{ proper nonzero isotropy-invariant }
S
\subset
T_A\mathcal M_{K,\delta}
}
$$

such that every Weil-cycle deformation image lies in:

$$
S.
$$

Status:

$$
\boxed{
\mathrm{DISPROVED}.
}
$$

---

# 34. What CE002 preserves

The following remain open:

$$
\boxed{
\operatorname{rank}
d\pi_Z<9
\text{ for every }Z
}
$$

in the non-split regime.

Also open:

$$
\boxed{
\mathrm{TDVD}_{K,\delta}.
}
$$

Also open:

$$
\boxed{
\mathrm{DIS}_{K}.
}
$$

And therefore:

$$
\boxed{
\mathrm{CE001}.
}
$$

---

# 35. Counterexample-program correction

The HC-False branch must now distinguish:

### Local Hodge obstruction

Fails here.

### Local invariant tangent-sector obstruction

Fails here.

### Cycle-dependent determinantal obstruction

Still viable.

### Global arithmetic/discriminant obstruction

Now prioritized.

### Higher-order deformation obstruction

Still viable.

### Tropical / coniveau obstruction

Still viable as independent branches.

---

# 36. New negative hierarchy

The preferred CE001 attack order becomes:

$$
\boxed{
\mathrm{DIS}
\ \text{or}\
\mathrm{TDVD}
\Longrightarrow
\mathrm{NDC}
\Longrightarrow
\mathrm{CE001}.
}
$$

With backup:

$$
\boxed{
\text{Tropical}
\quad\text{or}\quad
\text{Coniveau}.
}
$$

---

# 37. A useful anti-overfitting lesson

The split and non-split moduli components can have the same local analytic period geometry while differing arithmetically.

Therefore:

$$
\boxed{
\text{local analytic sameness}
\not\Rightarrow
\text{same algebraic-cycle geometry}.
}
$$

But conversely:

$$
\boxed{
\text{local analytic data alone cannot explain the difference}.
}
$$

Any counterexample mechanism must sit exactly in that gap.

---

# 38. Relation to MLRSC Phase III

This is a direct realization of the neutral framework's prediction:

$$
\boxed{
\text{Measure compatibility}
+
\text{Hodge persistence}
\not\Rightarrow
\text{Legality transport}.
}
$$

Here:

- Measure is completely controlled;
- Hodge persistence is maximal;
- local scale geometry is identical across discriminants;
- legality may still differ globally.

That is a pure Scale–Category coupling problem.

---

# 39. Candidate arithmetic scale

CE002 therefore introduces a new branch-specific scale:

$$
\boxed{
\text{Hermitian Rational Form / Discriminant Scale}.
}
$$

Unlike the local deformation scale,

this scale is discrete and arithmetic.

It distinguishes:

$$
\delta=-1
$$

from:

$$
\delta\neq-1.
$$

The next round must couple this arithmetic scale to cycle-family dominance.

---

# 40. CE002 main result

The strongest proved statement of this round is:

$$
\boxed{
T_A\mathcal M_{K,\delta}
\simeq
U^\vee\otimes\overline V
}
$$

with:

$$
\dim=9
$$

and irreducible isotropy action.

Therefore:

$$
\boxed{
\text{no proper nonzero local isotropy-invariant tangent quotient exists}.
}
$$

Together with local discriminant-blindness:

$$
\boxed{
\text{the naive universal missing-sector strategy cannot prove CE001}.
}
$$

---

# 41. Branch Status

**CE001:** OPEN  
**Fixed Missing Tangent Sector:** DISPROVED  
**UCDRD:** OPEN  
**TDVD:** OPEN  
**DIS:** OPEN  
**Tropical Route:** OPEN  
**Coniveau Route:** OPEN  

Thus the counterexample branch survives,

but after a nontrivial self-correction.

---

# 42. Next Interface

Next counterexample round:

```text
HODGE_HCFALSE_CE003_DominanceImpliesSplit.md
```

Primary target:

$$
\boxed{
\text{Can a dominating family of Weil-class cycles force split Hermitian discriminant?}
}
$$

Subtasks:

1. express splitness in terms of $K$-rational maximal isotropic subspaces;
2. inspect all known split cycle constructions for the exact isotropic datum they require;
3. derive necessary consequences of full-rank cycle deformation;
4. test whether semiregularity surjectivity produces rational isotropic flags;
5. analyze Fourier–Mukai / secant-sheaf ancestry;
6. formulate a precise:
   $$
   \text{dominance}
   \Rightarrow
   \text{rational splitting}
   $$
   theorem candidate;
7. actively search for counterexamples to DIS before trusting it.

Neutral MLRSC remains paused at:

```text
HODGE_MLRSC_R029_C_JointObjectFiniteCore.md
```

---

# 參考文獻

1. G. Shimura, *Automorphic forms and the periods of abelian varieties*. Unitary period domains and PEL-type abelian varieties.

2. B. J. J. Moonen, Y. G. Zarhin, works on Weil classes on abelian varieties. Weil-type Hodge structures and exceptional classes.

3. E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415. Positive algebraicity in the split/discriminant-$-1$ sixfold regime.

4. E. Markman, *Secant sheaves and Weil classes on abelian varieties*, arXiv:2509.23403. Geometry of split Weil-type constructions.

5. P. Griffiths, period mappings and infinitesimal variation of Hodge structure. General tangent/Hodge-locus formalism.

6. Aletheia, *HODGE_HCFALSE_CE001_NonSplitWeilSixfold*, 2026-09-15.

7. Aletheia, *HODGE_MLRSC_R014_L_RelativeCycleLegality*, 2026-09-15.

8. Aletheia, *HODGE_MLRSC_R016_C_FirstCouplingDefect*, 2026-09-15.

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## Canonical Source Declaration

本檔案為 HC-False 分支第二篇正式 UTF-8 Markdown canonical source。

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪否決的是一個反例證明子策略，不是否決 CE001 本身。