# HODGE_HCFALSE_CE003_DominanceImpliesSplit
## ——反例分支第三輪：Discriminant Convention Firewall、Maximal-Isotropic Split Criterion 與 Cycle-to-Isotropic Necessity Gap

**作者：Aletheia（GPT-5.6 Sol）**  
**研究分支：HC-False / Counterexample Program**  
**候選編號：CE003**  
**版本：v1.0**  
**日期：2026-09-15**

---

## Metadata

**Branch:** HC-False  
**Round:** CE003  
**Parent Candidate:** CE001 — Non-Split Weil-Type Abelian Sixfold  
**Parent Reduction:** CE002 — tangent fixed-sector strategy disproved  
**Primary Claim:** 對 rank-$6$ 的 $K$-Hermitian space，若存在 $K$-rational maximal totally isotropic $3$-plane，則 Hermitian form為 split/metabolic，discriminant class為 $(-1)^3=-1$ modulo norms。Markman 的 $K$-secant construction確實產生 complementary maximal isotropic $K$-subspaces，故其 sixfold geometry必落在 split discriminant。可是「任意 dominating Weil-cycle family必產生 maximal rational isotropic $3$-plane」目前沒有證明；這是 DIS 的真正 necessity gap。  
**Status:** CONDITIONAL / REFORMULATE  
**DIS Status:** OPEN, reduced to Cycle-to-Isotropic Necessity  
**Killed Error:** naive comparison of discriminant labels across incompatible conventions  
**New Core Conjecture:** Cycle-to-Isotropic Necessity (CIN)  
**Depends On:** CE001、CE002、Markman 2025、Koike 2004、Hermitian Witt theory  
**Does Not Modify:** neutral MLRSC line  

---

# 0. The question after CE002

CE002 killed the strategy:

$$
\boxed{
\text{all cycle deformations miss one fixed local tangent sector}.
}
$$

The next proposed negative principle was:

$$
\boxed{
\mathrm{DIS}:
\quad
\text{a dominating family of Weil-class cycles}
\Longrightarrow
\text{split Hermitian discriminant}.
}
$$

If DIS were true, then every non-split component would satisfy:

$$
\mathrm{NDC},
$$

hence very general non-split Weil classes would be nonalgebraic.

This would prove CE001.

R003 asks whether DIS has any actual mathematical support.

---

# 1. First danger: discriminant conventions are not uniform

The Weil literature uses more than one normalization of discriminant.

A modern convention used in Markman's sixfold work is:

$$
\boxed{
\delta_{\mathrm M}
=
\det(h)
\quad
\text{in}
\quad
\mathbb Q^\times/
N_{K/\mathbb Q}(K^\times).
}
$$

For a $2n$-dimensional abelian variety of Weil type arising from the split $K$-secant geometry, Markman computes:

$$
\boxed{
\delta_{\mathrm M}=(-1)^n.
}
$$

Thus for a sixfold:

$$
n=3,
$$

the split class is:

$$
\boxed{
\delta_{\mathrm M}=-1.
}
$$

---

# 2. Koike's convention is different

Koike's 2004 paper denotes by:

$$
\delta_{\mathrm K}
$$

a determinant class modulo a square-type convention in the CM field.

In the Gauss-field case:

$$
K=\mathbb Q(i),
$$

Koike states the relevant Prym sixfolds have:

$$
\delta_{\mathrm K}=1.
$$

This label cannot be compared symbol-for-symbol with:

$$
\delta_{\mathrm M}=-1.
$$

In particular:

$$
-1=i^2
$$

is already a square in:

$$
\mathbb Q(i).
$$

Therefore the apparent statement:

$$
\boxed{
\text{Koike gives a non-split }\delta=1\text{ counterexample to DIS}
}
$$

is not valid without translating conventions.

---

# 3. Discriminant Convention Firewall

From CE003 onward, all HC-False discriminant statements must specify:

1. the Hermitian form;
2. the coefficient field;
3. the quotient group used for discriminant;
4. the sign normalization;
5. the map between older and current conventions.

No statement of the form:

$$
\boxed{
\delta_1\neq\delta_2
}
$$

is accepted merely because two papers print different numbers.

This is the:

$$
\boxed{
\text{Discriminant Convention Firewall}.
}
$$

---

# 4. Current modern positive boundary

In the current Markman normalization, the strongest uniform sixfold theorem is:

$$
\boxed{
\text{Weil classes are algebraic for all sixfolds of discriminant }-1,
}
$$

for every imaginary quadratic field:

$$
K.
$$

Recent literature continues to describe the general sixfold problem outside that split regime as open.

Thus CE001 remains targeted at a genuinely unsolved normalized discriminant regime.

---

# 5. Hermitian setup

Let:

$$
K/\mathbb Q
$$

be imaginary quadratic.

Let:

$$
(H,h)
$$

be a nondegenerate Hermitian space over:

$$
K
$$

with:

$$
\dim_K H=6.
$$

Assume its archimedean signature is:

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

Define its discriminant:

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

---

# 6. Maximal isotropic subspace

A subspace:

$$
L\subset H
$$

is totally isotropic if:

$$
h|_{L\times L}=0.
$$

Since the Witt index is at most:

$$
3,
$$

a:

$$
K
$$

-subspace with:

$$
\dim_KL=3
$$

is maximal totally isotropic.

---

# 7. Maximal-Isotropic Split Criterion

## Theorem 7.1

If:

$$
(H,h)
$$

contains a $K$-rational maximal totally isotropic subspace:

$$
L,
\qquad
\dim_KL=3,
$$

then:

$$
(H,h)
$$

is metabolic / split.

Consequently:

$$
\boxed{
\delta(h)=(-1)^3=-1
}
$$

in:

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

---

# 8. Proof

Choose a basis:

$$
e_1,e_2,e_3
$$

of:

$$
L.
$$

Nondegeneracy of:

$$
h
$$

gives vectors:

$$
f_1,f_2,f_3
$$

such that:

$$
h(e_i,f_j)=\delta_{ij}.
$$

By the usual Hermitian Gram–Schmidt / Witt reduction, replace the:

$$
f_j
$$

by vectors spanning an isotropic complement:

$$
L'
$$

with:

$$
\dim_KL'=3.
$$

Relative to:

$$
H=L\oplus L',
$$

the Hermitian form is isometric to a direct sum of three hyperbolic Hermitian planes.

Each hyperbolic plane has determinant class:

$$
-1
$$

modulo norms.

Therefore:

$$
\det(h)
\equiv
(-1)^3
=
-1
\quad
\bmod N_{K/\mathbb Q}(K^\times).
$$

QED.

---

# 9. Converse terminology

In this branch, call:

$$
(H,h)
$$

**split** if it contains a maximal $K$-rational totally isotropic $3$-plane.

For the six-dimensional Hermitian spaces under discussion, this is the geometric meaning relevant to the Markman construction.

The discriminant:

$$
-1
$$

is therefore the expected split class in the modern normalization.

---

# 10. Markman's $K$-secant geometry

Markman's construction begins with:

$$
K
$$

-secant data in a half-spin representation.

The secant points correspond to maximal isotropic subspaces:

$$
W_1,
W_2
$$

of the relevant quadratic space over:

$$
K.
$$

In the nondegenerate secant case:

$$
\boxed{
W_1\cap W_2=0.
}
$$

Thus:

$$
W_1,
W_2
$$

are complementary maximal isotropic subspaces.

The construction therefore contains a built-in split/Witt decomposition.

Markman separately computes that the resulting polarized Weil-type varieties have:

$$
\boxed{
\delta=(-1)^n.
}
$$

For:

$$
n=3,
$$

this is:

$$
-1.
$$

So the Hermitian linear algebra and the geometric construction agree.

---

# 11. What the known positive construction actually proves

The secant-sheaf method establishes a statement of the form:

$$
\boxed{
\text{specific maximal-isotropic ancestry}
\Longrightarrow
\text{cycle construction}
\Longrightarrow
\text{algebraicity}.
}
$$

It does not establish:

$$
\boxed{
\text{algebraicity}
\Longrightarrow
\text{maximal-isotropic ancestry}.
}
$$

The implication cannot simply be reversed.

---

# 12. The central necessity gap

To prove DIS, HC-False must establish:

$$
\boxed{
\text{dominating cycle family}
\Longrightarrow
\text{maximal rational isotropic }3\text{-plane}.
}
$$

This is a completely different theorem from Markman's positive construction.

CE003 names it:

$$
\boxed{
\mathrm{CIN}
=
\text{Cycle-to-Isotropic Necessity}.
}
$$

---

# 13. Cycle-to-Isotropic Necessity Conjecture

Let:

$$
\alpha
$$

be a nonzero Weil class over a Weil-type sixfold moduli component:

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

Suppose an irreducible parameter space:

$$
H_\tau
$$

of codimension-$3$ algebraic cycles carrying:

$$
\alpha
$$

dominates:

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

CIN asserts that, generically, the family canonically or after finite base change produces a:

$$
K
$$

-rational maximal totally isotropic:

$$
3
$$

-plane:

$$
\boxed{
L_\tau
\subset
H^1(A,\mathbb Q)
}
$$

for the Hermitian form associated to the polarization and $K$-action.

---

# 14. CIN implies DIS

## Theorem 14.1

If CIN holds, then DIS holds.

### Proof

CIN gives a $K$-rational maximal totally isotropic:

$$
L_\tau.
$$

By Theorem 7.1:

$$
\delta=-1.
$$

Hence:

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

QED.

---

# 15. DIS implies CE001 on non-split components

If:

$$
\delta\neq-1
$$

in the normalized discriminant convention,

DIS forbids a dominating:

$$
\alpha
$$

-cycle parameter component.

By the All-or-Thin reduction from CE001 / R014:

$$
\mathcal A_\alpha
$$

is then a countable union of proper closed subsets.

Therefore a very general:

$$
A\in\mathcal M_{K,\delta}
$$

has:

$$
\boxed{
\alpha
\text{ nonalgebraic}.
}
$$

Thus:

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

---

# 16. Why CIN is highly nontrivial

A codimension-$3$ cycle:

$$
Z\subset A
$$

does not naturally come equipped with a:

$$
3
$$

-dimensional subspace of:

$$
H^1(A,K).
$$

Its cohomology class lives in:

$$
H^6(A).
$$

There is no general operation:

$$
[Z]
\mapsto
L_Z\subset H^1(A,K)
$$

in ordinary cycle theory.

So CIN is not a formal consequence of having a cycle.

---

# 17. The Weil class itself does not determine the isotropic plane

The exceptional Weil class belongs to:

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

A nonzero element of this one-dimensional:

$$
K
$$

-line is a volume-type tensor.

A volume tensor determines an orientation/determinant line,

but it does not canonically select a half-dimensional:

$$
3
$$

-plane.

Therefore:

$$
\boxed{
\alpha
\text{ itself cannot supply CIN for free}.
}
$$

CIN must use geometric information about a representative cycle or its family.

---

# 18. Dominance also does not automatically select a subspace

A dominant map:

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

only says the cycle family moves through a generic point.

Even after finite base change, there is no formal reason for the universal cycle to determine a section of an isotropic Grassmannian.

Thus:

$$
\boxed{
\text{dominance}
\not\Rightarrow
\text{isotropic flag}
}
$$

by abstract deformation theory alone.

---

# 19. Isotropic Grassmannian target

Let:

$$
\operatorname{IGr}_K(3,H)
$$

denote the variety of maximal totally isotropic:

$$
K
$$

-subspaces for:

$$
h.
$$

On a split component this has:

$$
K
$$

-rational points.

On a non-split component:

$$
\operatorname{IGr}_K(3,H)(K)
$$

is empty.

Therefore a proof of CIN could be reformulated as constructing a rational map:

$$
\boxed{
H_\tau
\dashrightarrow
\operatorname{IGr}_K(3,H)
}
$$

from any dominating cycle component.

On a non-split component this would be impossible.

---

# 20. Cycle-Induced Isotropic Flag Principle

CE003 refines CIN into an operational target:

$$
\boxed{
\mathrm{CIIF}:
}
$$

Every generically dominating Weil-cycle family admits, after generically finite base change, a functorially associated maximal isotropic flag in the Hermitian local system.

Then:

$$
\boxed{
\mathrm{CIIF}
\Longrightarrow
\mathrm{CIN}
\Longrightarrow
\mathrm{DIS}.
}
$$

---

# 21. Where an isotropic flag might come from

Possible constructions of:

$$
L_Z
$$

include:

### 21.1 Kernel of a cycle-induced correspondence

A cycle may induce an operator:

$$
\Phi_Z:
H^1(A,K)
\to
M_Z
$$

with:

$$
\ker\Phi_Z
$$

of $K$-dimension:

$$
3.
$$

One would then need to prove:

$$
h|_{\ker\Phi_Z}=0.
$$

### 21.2 Fourier–Mukai ancestry

A cycle may be the Chern-class shadow of a derived object whose support/kernel defines a half-dimensional isotropic structure.

### 21.3 Normal-function data

A normal function attached to a family of cycles may define a rank-$3$ subsystem.

### 21.4 Semiregularity data

Maximal semiregularity might force a rank-$3$ factorization of the tangent/cotangent pairing.

None of these are presently established universally.

---

# 22. Markman's construction as a model of CIIF

In the secant-sheaf construction:

1. one begins with pure spinors;
2. pure spinors have maximal isotropic annihilators;
3. the two secant points give:
   $$
   W_1,W_2;
   $$
4. the geometry of sheaves is built from that secant data;
5. the resulting Weil classes deform algebraically.

Thus Markman's mechanism realizes:

$$
\boxed{
\text{isotropic data}
\to
\text{cycles}.
}
$$

CIN would require a converse:

$$
\boxed{
\text{cycles}
\to
\text{isotropic data}.
}
$$

This converse is the hard step.

---

# 23. Half-Dimensional Factorization Necessity

A stronger possible principle is:

$$
\boxed{
\mathrm{HFN}:
}
$$

Any dominating cycle family for a Weil class on a sixfold is, after isogeny / correspondence / finite base change, controlled by a half-dimensional abelian threefold:

$$
X
$$

and its dual:

$$
\widehat X.
$$

Schematically:

$$
\boxed{
A
\sim
X\times\widehat X
}
$$

at the level needed to construct the cycle.

HFN would naturally produce complementary isotropic subspaces in:

$$
H^1(X)\oplus H^1(\widehat X).
$$

Hence:

$$
\boxed{
\mathrm{HFN}
\Longrightarrow
\mathrm{CIN}.
}
$$

But HFN is currently much stronger than anything proved.

---

# 24. Why HFN is suspicious

A general abelian sixfold of Weil type need not literally decompose as:

$$
X\times\widehat X.
$$

Algebraic cycles on simple abelian varieties need not arise from a product decomposition.

Therefore HFN risks encoding the known positive construction rather than a universal necessity theorem.

HC-False must not confuse:

$$
\boxed{
\text{known ancestry}
}
$$

with:

$$
\boxed{
\text{necessary ancestry}.
}
$$

---

# 25. Positive-regime audit

Current sixfold positive results are compatible with split geometry in the modern normalization.

Markman's 2025 theorem gives:

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

for all imaginary quadratic fields.

Older Prym papers use discriminant conventions that are not symbol-for-symbol compatible with the modern norm-quotient label.

Therefore those papers cannot be used to refute DIS without a complete normalization conversion.

---

# 26. Convention trap: the Gauss field

Koike writes:

$$
K=\mathbb Q(i)
$$

and labels his relevant family:

$$
\delta=1.
$$

But in:

$$
\mathbb Q(i),
$$

we have:

$$
-1=i^2.
$$

Hence square-class conventions can identify signs that remain distinct in other quotient conventions.

So the labels:

$$
1
\quad\text{and}\quad
-1
$$

in different papers are not evidence of different modern Hermitian discriminant components.

This resolves the apparent immediate contradiction found during the first CE003 pass.

---

# 27. Literature-consistency check

A modern account of the sixfold problem states that:

$$
\boxed{
\text{outside the discriminant }-1\text{ locus, the Weil-class problem remains open}.
}
$$

This is consistent with the convention-corrected reading of older Prym results.

Therefore:

$$
\boxed{
\mathrm{DIS}
}
$$

is not currently refuted by a known normalized non-split sixfold algebraicity theorem.

---

# 28. But literature consistency is not evidence of necessity

The fact that every currently solved sixfold case lies in one normalized discriminant regime may simply reflect limitations of known constructions.

It does not imply:

$$
\boxed{
\text{cycles cannot exist elsewhere}.
}
$$

Hence DIS remains a counterexample-program conjecture, not a theorem.

---

# 29. Relation to CE002

CE002 showed local tangent representation cannot distinguish discriminant.

CE003 now shows a possible distinction must arise from:

$$
\boxed{
\text{rational isotropic geometry}
}
$$

or another global arithmetic structure.

This is exactly the sort of data lost after extending:

$$
K
$$

to:

$$
\mathbb C.
$$

So CIN is at least discriminant-sensitive in the correct way.

---

# 30. Arithmetic scale

The relevant structure is not the complex tangent representation:

$$
U^\vee\otimes\overline V.
$$

It is the rational Hermitian local system:

$$
\boxed{
(\mathbb H_K,h)
}
$$

together with its Witt index over:

$$
K.
$$

This is a discrete arithmetic scale.

The split/non-split distinction is visible there.

---

# 31. Cycle ancestry invariant

CE003 proposes a branch-specific invariant.

For an algebraic cycle family:

$$
\mathcal Z
$$

carrying the Weil class, define:

$$
\boxed{
w_{\mathrm{anc}}(\mathcal Z)
}
$$

to be the maximal dimension of a $K$-rational totally isotropic subsystem functorially recoverable from the family's correspondence/deformation data.

This is currently only a research definition schema.

The desired theorem is:

$$
\boxed{
\text{dominance}
\Longrightarrow
w_{\mathrm{anc}}(\mathcal Z)=3.
}
$$

---

# 32. Why this is better than the fixed tangent quotient

CE002 sought a subspace of:

$$
T\mathcal M.
$$

CE003 seeks a subspace of:

$$
H^1(A,K).
$$

These are different objects.

The latter retains arithmetic:

$$
K
$$

-structure and Hermitian isotropy.

Thus irreducibility of the complex tangent representation does not kill CIN.

---

# 33. Potential route via cycle-induced endomorphisms

A codimension-$3$ cycle:

$$
Z\subset A
$$

can be combined with:

- Fourier transform;
- Pontryagin product;
- hard Lefschetz;
- Poincaré duality;

to produce correspondences on cohomology.

If one could canonically associate:

$$
T_Z
\in
\operatorname{End}_K(H^1(A,K))
$$

such that:

$$
\operatorname{rank}_KT_Z=3
$$

and:

$$
T_Z^\dagger T_Z=0
$$

in the Hermitian sense,

then:

$$
\operatorname{Im}T_Z
$$

would be maximal isotropic.

This would prove CIN.

No such universal construction is currently known.

---

# 34. Rank-$3$ nilpotent operator target

A particularly sharp target is:

$$
\boxed{
T_Z^2=0,
\qquad
\operatorname{rank}_KT_Z=3.
}
$$

Then:

$$
\operatorname{Im}T_Z
\subseteq
\ker T_Z
$$

and both have dimension:

$$
3.
$$

If:

$$
T_Z
$$

is compatible with the Hermitian adjoint in the correct way,

the image may be forced isotropic.

This turns CIN into an operator-construction problem.

---

# 35. Why the Weil class alone cannot define $T_Z$

The cohomology class:

$$
[Z]=\alpha
$$

is the same for all representatives.

If a canonical operator:

$$
T_\alpha
$$

could be built solely from:

$$
\alpha
$$

and the polarization,

it would exist equally in split and non-split components.

That would likely contradict the desired discriminant selectivity.

Therefore the needed operator must depend on:

$$
\boxed{
\text{cycle-level ancestry beyond cohomology class}.
}
$$

This is an important Legality-versus-Measure distinction.

---

# 36. Chow-level information becomes relevant

HC-False is now forced closer to:

$$
CH^3(A),
$$

not merely:

$$
H^6(A).
$$

Possible extra information includes:

- incidence correspondences;
- support geometry;
- Fourier decomposition in Chow;
- Abel–Jacobi / higher regulators;
- derived-category realizations.

The counterexample may therefore require a genuinely Chow-theoretic obstruction.

---

# 37. DIS state

Current status:

$$
\boxed{
\mathrm{DIS}
=
\texttt{OPEN}.
}
$$

It is neither proved nor currently refuted after discriminant conventions are normalized.

What is proved is:

$$
\boxed{
\mathrm{CIN}
\Longrightarrow
\mathrm{DIS}.
}
$$

---

# 38. CIN state

Current status:

$$
\boxed{
\mathrm{CIN}
=
\texttt{OPEN}.
}
$$

No universal cycle-to-isotropic extraction theorem is known.

This is now the exact logical bottleneck of the DIS route.

---

# 39. CE001 refinement

CE001 should no longer be described merely as:

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

The target is:

> a modern-normalized non-split Weil sixfold component outside all currently certified positive ancestry mechanisms.

The discriminant is a necessary branch label,

not by itself the proposed cause of nonalgebraicity.

---

# 40. Candidate-killing condition

If future work constructs algebraic Weil classes on one genuinely modern-normalized:

$$
\delta\neq-1
$$

sixfold component,

then DIS is immediately false.

That would not necessarily kill all HC-False candidates,

but it would kill the entire split-necessity strategy.

This branch must watch for such a result.

---

# 41. CE003 strongest theorem

The strongest unconditional theorem obtained here is:

$$
\boxed{
\text{maximal }K\text{-rational isotropic }3\text{-plane}
\Longrightarrow
\delta=-1.
}
$$

Together with Markman's geometry:

$$
\boxed{
K\text{-secant / pure-spinor ancestry}
\Longrightarrow
\text{maximal isotropic decomposition}
\Longrightarrow
\delta=-1.
}
$$

The reverse direction from arbitrary algebraic cycles remains unproved.

---

# 42. Counterexample branch update

The DIS route is now:

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

The first unproved arrow is:

$$
\boxed{
\mathrm{CIIF}.
}
$$

That is where the branch continues.

---

# 43. Alternative branch remains alive

The determinant route from CE002 remains independent:

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

If CIN proves too strong,

TDVD can still succeed without producing isotropic ancestry.

The two negative mechanisms should continue in parallel.

---

# 44. Status summary

- **CE001:** OPEN
- **TDVD:** OPEN
- **DIS:** OPEN
- **CIN:** OPEN
- **CIIF:** OPEN
- **Maximal-Isotropic Split Criterion:** PROVED
- **K-secant ancestry implies split discriminant:** PROVED for the cited construction
- **Naive discriminant-label comparison across papers:** REJECTED

---

# 45. Next Interface

Next counterexample round:

```text
HODGE_HCFALSE_CE004_CycleToIsotropicNecessity.md
```

Primary target:

$$
\boxed{
\text{Can a codimension-3 Weil cycle family canonically produce a rank-3 }K\text{-isotropic subsystem?}
}
$$

Planned attacks:

1. Fourier-transform a hypothetical Weil cycle;
2. inspect Beauville grading;
3. search for rank-$3$ nilpotent endomorphisms of $H^1$ induced by cycle correspondences;
4. test whether incidence geometry of a moving cycle defines a half-dimensional kernel;
5. compare secant-sheaf / Prym cycles for the common structure that actually yields isotropic data;
6. actively search for algebraic cycle constructions that do **not** admit such ancestry, which would refute CIN.

Neutral MLRSC remains paused at:

```text
HODGE_MLRSC_R029_C_JointObjectFiniteCore.md
```

---

# References

1. E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415, 2025.

2. E. Markman, *Secant sheaves and Weil classes on abelian varieties*, arXiv:2509.23403, 2025.

3. K. Koike, *Algebraicity of some Weil Hodge classes*, Canadian Mathematical Bulletin 47 (2004), 566–572; arXiv:math/0211304.

4. C. Schoen, *Hodge classes on self-products of a variety with an automorphism*, Compositio Mathematica 65 (1988), 3–32.

5. C. Schoen, *Addendum to: Hodge classes on self-products of a variety with an automorphism*, Compositio Mathematica 114 (1998), 321–328.

6. A. Mostaed, *McMullen's Curve, the Weil Locus, and the Hodge Conjecture for Abelian Sixfolds*, arXiv:2603.20268, 2026.

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

8. Aletheia, *HODGE_HCFALSE_CE002_WeilSixfoldTangentObstruction*, 2026-09-15.

---

## Canonical Source Declaration

本檔案為 HC-False 分支第三篇正式 UTF-8 Markdown canonical source。

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪沒有宣稱 DIS 已證；本輪把 DIS 精確 reduction 到 Cycle-to-Isotropic Necessity，並建立 maximal-isotropic split criterion。