# HODGE_HCFALSE_CE001_NonSplitWeilSixfold
## ——反例分支第一候選：Non-Split Weil-Type Abelian Sixfold 與 Universal Cycle-Deformation Rank Defect

**作者：Aletheia（GPT-5.6 Sol）**  
**研究分支：HC-False / Counterexample Program**  
**候選編號：CE001**  
**版本：v1.0**  
**日期：2026-09-15**

---

## Branch Declaration

從本文件起，建立與原本中立 MLRSC 平行的反例分支：

$$
\boxed{
\text{MLRSC-neutral}
\quad\parallel\quad
\text{HC-False}
\quad\parallel\quad
\text{HC-True}.
}
$$

其中：

- MLRSC-neutral 繼續原 R029 之後的方法論順序；
- HC-False 主動假設 classical rational Hodge conjecture 為假，尋找真正反例；
- HC-True 未來主動假設 conjecture 為真，尋找 proof closure。

三條線共享同一套 Measure / Legality / Scale–Category carriers，但結論立場相反。

本文件不宣稱反例已被證明。

本文件正式提出第一個：

$$
\boxed{
\text{Counterexample Candidate}.
}
$$

---

## Metadata

**Branch:** HC-False  
**Candidate:** CE001  
**Target Variety:** polarized abelian sixfold of Weil type  
**Target Codimension:** $p=3$  
**Target Cohomology:** $H^6(A,\mathbb Q)\cap H^{3,3}(A)$  
**Candidate Class:** nonzero exceptional Weil class  
**Working Verdict:** COUNTEREXAMPLE CONJECTURE  
**Mathematical Status:** OPEN  
**Current Literature Boundary:** split/discriminant $-1$ sixfold Weil classes are algebraic; outside that regime the sixfold Weil-class problem remains open  
**Primary Negative Mechanism:** no dominating algebraic-cycle parameter component  
**Decisive Subproblem:** Universal Cycle-Deformation Rank Defect  
**Depends On:** R014, R016, R022, R027, R028  
**Does Not Modify:** neutral MLRSC numbering or future HC-True branch  

---

# 0. The stance

HC-False branch adopts the working hypothesis:

$$
\boxed{
\text{The rational Hodge conjecture is false.}
}
$$

The task is not:

> Find a Hodge class for which no proof of algebraicity is currently known.

The task is:

> Produce a smooth projective complex variety $X$, a codimension $p$, and a rational Hodge class $\alpha$ such that one can prove
> $$
> \alpha
> \notin
> \operatorname{Im}
> \left(
> CH^p(X)_{\mathbb Q}
> \to
> H^{2p}(X,\mathbb Q)
> \right).
> $$

Everything weaker is only candidate evidence.

---

# 1. Candidate geometry

Fix an imaginary quadratic field:

$$
\boxed{
K/\mathbb Q.
}
$$

Let:

$$
A
$$

be a polarized complex abelian sixfold of Weil type for:

$$
K.
$$

Thus:

$$
H^1(A,\mathbb Q)
$$

is a rank-$6$ vector space over:

$$
K,
$$

and the action of:

$$
K
$$

on:

$$
H^{1,0}(A)
$$

has signature:

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

---

# 2. Weil Hodge space

Define:

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

Since:

$$
\dim_K H^1(A,\mathbb Q)=6,
$$

the space:

$$
W_K(A)
$$

is one-dimensional over:

$$
K,
$$

hence two-dimensional over:

$$
\mathbb Q.
$$

It embeds into:

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

For a Weil-type abelian sixfold:

$$
\boxed{
W_K(A)
\subseteq
H^6(A,\mathbb Q)
\cap
H^{3,3}(A).
}
$$

So every nonzero:

$$
\alpha\in W_K(A)
$$

is a rational Hodge class of codimension:

$$
3.
$$

---

# 3. Split versus non-split regime

The polarization determines a:

$$
K
$$

-Hermitian form:

$$
h_K
$$

on:

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

Its discriminant defines a class:

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

For sixfolds in the currently solved split regime:

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

in the corresponding norm quotient.

Markman proved algebraicity of Weil classes for polarized abelian sixfolds of this split/discriminant-$-1$ type.

Therefore HC-False deliberately excludes that regime.

---

# 4. CE001 target

Fix:

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

Let:

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

be an irreducible moduli component of polarized abelian sixfolds of Weil type with:

- field:
  $$
  K;
  $$
- signature:
  $$
  (3,3);
  $$
- discriminant:
  $$
  \delta.
  $$

Take:

$$
A
$$

very general in:

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

Choose:

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

---

# 5. Counterexample Conjecture CE001

HC-False branch makes the explicit conjecture:

$$
\boxed{
\mathrm{CE001}:
\quad
\alpha
\notin
\operatorname{Alg}^3(A)
}
$$

for a very general:

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

with:

$$
\delta\neq-1.
$$

Equivalently:

$$
\boxed{
\alpha
\notin
\operatorname{Im}
\left(
CH^3(A)_{\mathbb Q}
\to
H^6(A,\mathbb Q)
\right).
}
$$

If CE001 is proved, the rational Hodge conjecture is false.

---

# 6. Why this candidate is not random

The candidate was chosen only after the neutral program had already removed several false leads.

It satisfies all of the following.

### 6.1 It is genuinely Hodge

$$
\alpha
\in
H^{3,3}(A)
\cap
H^6(A,\mathbb Q).
$$

So no failure can be blamed on wrong Hodge typing.

### 6.2 It is exceptional

For the relevant generic Weil-type regime, the Weil sector is not generated merely by divisor classes.

Thus:

$$
\boxed{
\alpha
\notin
D^3(A)
}
$$

for a suitable exceptional direction, where:

$$
D^\bullet(A)
$$

is the divisor-generated Hodge algebra.

### 6.3 Divisor algebra therefore cannot solve it automatically

Lefschetz $(1,1)$ algebraizes divisors, but not this exceptional primitive sector.

### 6.4 The known split-sixfold theorem does not apply

We impose:

$$
\delta\neq-1.
$$

### 6.5 Dimension four is already too positive

The modern algebraicity results close the Weil-class problem for abelian fourfolds.

So dimension six is the first nearby regime where the primitive Weil core remains seriously open outside the split locus.

---

# 7. What is not evidence of a counterexample

The following statements are explicitly insufficient.

### 7.1 Exceptional does not mean nonalgebraic

$$
\boxed{
\alpha\notin D^3(A)
}
$$

only says divisor grammar is too small.

It does not say:

$$
\alpha\notin\operatorname{Alg}^3(A).
$$

### 7.2 Unknown does not mean false

No known cycle construction is not a nonexistence theorem.

### 7.3 Failure of Markman's secant construction is not a nonexistence theorem

A cycle may exist by a completely different mechanism.

### 7.4 Non-split discriminant is not itself an obstruction to algebraicity

It only places the candidate outside the currently solved theorem.

### 7.5 Absolute Hodge does not settle either direction

Hodge classes on abelian varieties are absolute Hodge.

Absolute Hodge status does not imply algebraicity.

But it also makes several naive arithmetic obstruction strategies weaker.

---

# 8. The real negative target

R014 and R016 suggest a sharper route.

Instead of proving directly:

$$
\alpha
\notin
CH^3(A)_{\mathbb Q},
$$

study the moduli-wide algebraic realization locus.

Let a flat local Weil class be denoted:

$$
\alpha_t
$$

on a suitable marked cover / local system trivialization over:

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

Define:

$$
\boxed{
\mathcal A_\alpha
=
\left\{
t\in
\mathcal M_{K,\delta}
:
\alpha_t
\in
\operatorname{Alg}^3(A_t)
\right\}.
}
$$

---

# 9. Countable closed decomposition

Relative cycle parameterization gives:

$$
\boxed{
\mathcal A_\alpha
=
\bigcup_{\tau=1}^{\infty}
C_{\tau,\alpha},
}
$$

where each:

$$
C_{\tau,\alpha}
$$

is a closed algebraic subset of:

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

arising as the image of an appropriate Hilbert/Chow parameter component carrying cycle class:

$$
\alpha.
$$

Therefore:

$$
\boxed{
\text{either one parameter component dominates the moduli component,}
}
$$

or:

$$
\boxed{
\mathcal A_\alpha
\text{ is a countable union of proper closed subsets.}
}
$$

---

# 10. All-or-Thin negative dichotomy

Since:

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

is irreducible over:

$$
\mathbb C,
$$

if no:

$$
C_{\tau,\alpha}
$$

equals the whole component, then:

$$
\boxed{
\mathcal M_{K,\delta}
\setminus
\mathcal A_\alpha
}
$$

contains very general points and is Zariski dense.

Hence:

## Theorem 10.1

If every:

$$
\alpha
$$

-labelled cycle parameter component has proper image in:

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

then CE001 holds for very general:

$$
A.
$$

This is a proved reduction.

---

# 11. No-Dominating-Cycle Conjecture

Therefore HC-False replaces CE001 by the stronger geometric target:

$$
\boxed{
\mathrm{NDC}_{K,\delta}:
\quad
\text{No }\alpha\text{-labelled Hilbert/Chow component dominates }
\mathcal M_{K,\delta}.
}
$$

Then:

$$
\boxed{
\mathrm{NDC}_{K,\delta}
\Longrightarrow
\mathrm{CE001}
\Longrightarrow
\text{HC is false}.
}
$$

---

# 12. Why NDC is better than direct nonexistence

Directly classifying:

$$
CH^3(A)
$$

for a general abelian sixfold is unrealistic.

NDC asks instead:

> Can any family of codimension-$3$ algebraic cycles with Weil class $\alpha$ deform through the entire non-split Weil moduli component?

That is a deformation-rank question.

It can potentially be attacked infinitesimally.

---

# 13. Cycle deformation map

Let:

$$
Z\subset A
$$

be an algebraic cycle representative satisfying:

$$
[Z]=\alpha.
$$

Consider the deformation problem of the pair:

$$
(Z\subset A).
$$

There is a natural map:

$$
\boxed{
\mathrm{Def}(Z\subset A)
\longrightarrow
\mathrm{Def}(A,K,\lambda)
}
$$

where the target preserves the Weil-type data and polarization.

At tangent-space level:

$$
\boxed{
d\pi_Z:
T_{[Z]}
\mathrm{Hilb/Chow}
\longrightarrow
T_A
\mathcal M_{K,\delta}.
}
$$

---

# 14. Universal Cycle-Deformation Rank Defect

HC-False introduces the decisive conjecture:

$$
\boxed{
\mathrm{UCDRD}_{K,\delta}:
\quad
\operatorname{rank}
d\pi_Z
<
\dim
\mathcal M_{K,\delta}
}
$$

for every algebraic cycle:

$$
Z
$$

with:

$$
[Z]=\alpha.
$$

This must hold at every smooth point of every relevant parameter component.

---

# 15. UCDRD implies NDC

## Theorem 15.1

If:

$$
\mathrm{UCDRD}_{K,\delta}
$$

holds for every:

$$
\alpha
$$

-cycle parameter component, then:

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

holds.

### Proof

If an irreducible finite-type parameter component:

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

were dominant, then over characteristic zero the differential would be generically surjective at smooth points.

UCDRD forbids this.

Therefore no component dominates.

QED.

---

# 16. The full negative chain

Combining Sections 10 and 15:

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

This is the first serious HC-False attack chain.

---

# 17. What UCDRD must actually prove

It is not enough to inspect:

- complete intersections;
- divisor products;
- symmetric cycles;
- secant-sheaf cycles;
- known tautological Chow subrings.

UCDRD quantifies over:

$$
\boxed{
\text{every possible algebraic cycle representative }Z.
}
$$

So a valid proof needs a uniform structural bound.

---

# 18. Candidate source of a rank bound

For a smooth representative:

$$
Z\subset A,
$$

first-order embedded deformations are controlled by:

$$
H^0(Z,N_{Z/A}),
$$

with obstruction space often appearing in:

$$
H^1(Z,N_{Z/A}).
$$

The map to deformations of:

$$
A
$$

is constrained by:

- the normal sequence;
- semiregularity maps;
- preservation of the cohomology class;
- the extra $K$-action;
- the non-split Hermitian structure.

The HC-False goal is to find a representation-theoretic constraint forcing the image of:

$$
d\pi_Z
$$

into a proper subspace of:

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

uniformly for every:

$$
[Z]=\alpha.
$$

---

# 19. Negative semiregularity strategy

Positive proofs often use semiregularity to show that Hodge-preserving deformations of a cycle lift.

HC-False tries to prove the opposite structural statement:

$$
\boxed{
\text{for the non-split Weil class, no representative can be maximally semiregular over the full moduli tangent space}.
}
$$

More concretely, seek a nonzero quotient:

$$
Q_{K,\delta}
$$

of:

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

such that for every:

$$
Z
$$

with:

$$
[Z]=\alpha,
$$

the composite:

$$
T_{[Z]}\mathrm{Hilb/Chow}
\to
T_A\mathcal M_{K,\delta}
\to
Q_{K,\delta}
$$

vanishes.

If:

$$
Q_{K,\delta}\neq0,
$$

UCDRD follows.

---

# 20. Representation-theoretic target

The tangent space of the polarized Weil moduli component is controlled by the unitary deformation representation associated to:

$$
(K,h_K).
$$

HC-False therefore seeks:

$$
\boxed{
\text{a tangent representation sector visible in non-split deformations but invisible to all }\alpha\text{-cycle deformations}.
}
$$

This would be the analogue of a forbidden deformation direction.

---

# 21. The discriminant as a negative signal

The split case:

$$
\delta=-1
$$

admits the secant-sheaf / semiregularity machinery now known to algebraize the Weil classes.

This suggests an adversarial hypothesis:

$$
\boxed{
\text{the non-split discriminant may remove exactly the isotropic geometry needed for full-rank cycle deformation}.
}
$$

HC-False does not assume this is true.

It makes it a testable mechanism.

---

# 22. Candidate Negative Invariant

Define:

$$
\boxed{
\rho_{\mathrm{cyc}}(\alpha;A)
=
\sup_{[Z]=\alpha}
\operatorname{rank}
d\pi_Z.
}
$$

Then CE001 via UCDRD would follow from:

$$
\boxed{
\rho_{\mathrm{cyc}}(\alpha;A)
<
\dim
\mathcal M_{K,\delta}.
}
$$

This becomes the first negative quantitative invariant of the branch.

---

# 23. Cycle-Deformation Defect

Define:

$$
\boxed{
\Delta_{\mathrm{cyc}}
(\alpha;A)
=
\dim
\mathcal M_{K,\delta}
-
\rho_{\mathrm{cyc}}(\alpha;A).
}
$$

Then:

$$
\boxed{
\Delta_{\mathrm{cyc}}>0
}
$$

is the desired uniform deformation defect.

This is a research target, not currently proved.

---

# 24. Why known-construction failure is weaker than $\Delta_{\mathrm{cyc}}>0$

Suppose every known explicit cycle construction has rank defect.

That only proves:

$$
\boxed{
\Delta_{\mathrm{known}}>0.
}
$$

A new unknown cycle type could still dominate the moduli component.

The counterexample branch needs:

$$
\boxed{
\Delta_{\mathrm{cyc}}>0
}
$$

over the full Chow universe.

---

# 25. Tautological Chow obstruction

Abelian varieties have rich but structured tautological cycle subrings generated by:

- divisor classes;
- symmetric divisor data;
- Poincaré bundles;
- graphs of endomorphisms;
- Fourier transforms;
- Pontryagin products.

One negative subproject is:

$$
\boxed{
\text{prove }\alpha
\text{ is outside the cohomological image of every currently structured algebraic-cycle grammar}.
}
$$

This cannot finish CE001.

But it can shrink possible cycle ancestry.

---

# 26. Beauville-grade constraint

For an abelian variety:

$$
A,
$$

multiplication-by-$m$ acts on:

$$
H^{2p}(A)
$$

by:

$$
m^{2p}.
$$

Thus only the Beauville grade compatible with this eigenvalue can contribute nontrivially to ordinary cohomology.

So any hypothetical:

$$
Z
$$

with:

$$
[Z]=\alpha
$$

must have its cohomologically visible part in the appropriate Beauville degree-zero sector.

This is a necessary ancestry constraint.

It is not a nonexistence proof.

---

# 27. Arithmetic reduction route: warning

A tempting attack is:

> If $\alpha$ were algebraic, its reductions would be Tate classes; find a prime where the corresponding class is not Tate.

For abelian varieties this route is dangerous because Hodge classes are absolute Hodge and possess strong arithmetic realizations.

Therefore the HC-False branch does not currently treat:

$$
\boxed{
\text{naive Hodge-vs-Tate mismatch}
}
$$

as the primary route.

Any arithmetic obstruction must distinguish algebraic cycles from absolute-Hodge/motivated classes more finely.

---

# 28. CM isolation hardening

A stronger adversarial specialization comes from isolated CM sixfolds.

Recent work on McMullen's curve and Weil loci produces a conditional hard target:

$$
\boxed{
A_{v_0}
}
$$

with:

- dimension:
  $$
  6;
  $$
- CM endomorphism field:
  $$
  M=KL;
  $$
- degree:
  $$
  [M:\mathbb Q]=12;
  $$
- Weil-Hodge classes:
  $$
  W_K(A_{v_0})
  \subset
  H^{3,3}(A_{v_0}).
  $$

At such a point current algebraicity methods face:

- CM isolation;
- lack of the known $K$-secant geometry;
- uncontrolled discriminant.

---

# 29. Hardened Candidate CE001-CM

If a nonempty such CM intersection point is explicitly produced,

HC-False upgrades the target to:

$$
\boxed{
\mathrm{CE001\mbox{-}CM}:
\quad
0\neq\alpha
\in
W_K(A_{v_0})
\text{ is nonalgebraic}.
}
$$

This candidate is more rigid than the very-general family candidate.

But its existence in the particular McMullen-curve construction may itself require solving a finite arithmetic intersection problem.

---

# 30. Why CM isolation is attractive to HC-False

Positive deformation arguments often exploit:

- positive-dimensional Hodge loci;
- cycle families;
- semiregularity;
- transport from a geometric seed.

An isolated CM point suppresses those routes.

So HC-False treats CM isolation as:

$$
\boxed{
\text{anti-deformation pressure}.
}
$$

But isolation alone still does not imply nonalgebraicity.

---

# 31. Tropical route

Kontsevich proposed a tropical strategy for producing a Hodge counterexample using abelian varieties of Weil type.

A conditional route is:

1. degenerate:
   $$
   A
   $$
   to a tropical abelian variety;
2. track the Weil class:
   $$
   \alpha
   $$
   to a tropical Hodge class:
   $$
   \alpha_{\mathrm{trop}};
   $$
3. prove:
   $$
   \alpha_{\mathrm{trop}}
   $$
   is not representable by tropical algebraic cycles;
4. prove every algebraic cycle representing:
   $$
   \alpha
   $$
   would tropicalize to such a tropical cycle.

Then:

$$
\boxed{
\alpha
\text{ cannot be algebraic}.
}
$$

---

# 32. Tropical route is not automatically available

Modern tropical Hodge results prove positive tropical Hodge statements in important smooth/projective/rational-triangulation regimes.

Therefore HC-False cannot simply assume the tropical version fails.

The tropical target must avoid already protected classes or exploit a stronger obstruction than ordinary tropical Hodge type.

---

# 33. Tropical Obstruction Conjecture

Define a candidate invariant:

$$
\boxed{
\Theta_{\mathrm{trop}}(\alpha)
}
$$

which vanishes for tropicalizations of algebraic cycles.

A decisive negative result would be:

$$
\boxed{
\Theta_{\mathrm{trop}}
\left(
\alpha_{\mathrm{trop}}
\right)
\neq0.
}
$$

The actual construction of:

$$
\Theta_{\mathrm{trop}}
$$

remains open in this branch.

---

# 34. Two negative routes

HC-False CE001 therefore begins with two independent attack directions.

### Route D — Deformation rank obstruction

Prove:

$$
\Delta_{\mathrm{cyc}}>0.
$$

### Route T — Tropical obstruction

Construct:

$$
\Theta_{\mathrm{trop}}
$$

with:

$$
\Theta_{\mathrm{trop}}(\alpha_{\mathrm{trop}})\neq0.
$$

The branch should not rely on only one mechanism.

---

# 35. Why two routes matter

A genuine counterexample claim must survive positive reinterpretations.

If Route D fails because a dominating cycle family is found,

Route T may still survive.

If tropical realizability is proved,

Route D may still show no algebraic family exists in characteristic zero.

Independent failure modes make the counterexample program more robust.

---

# 36. Positive evidence against CE001

A serious HC-False branch must record evidence against itself.

The following facts push toward algebraicity.

### 36.1 Dimension four Weil classes are algebraic

So exceptional Weil classes are not intrinsically nonalgebraic.

### 36.2 Split sixfold Weil classes are algebraic

So codimension-$3$ Weil classes on sixfolds can be algebraic.

### 36.3 Recent techniques are expanding

Secant sheaves, hyperkähler geometry, semiregularity, deformation and auxiliary constructions continue to enlarge the solved region.

### 36.4 Deligne/André structure makes abelian Hodge classes highly constrained

This may eventually favor a universal positive theorem.

HC-False must beat all of this.

---

# 37. Why CE001 remains the chosen adversarial candidate

Despite Section 36, CE001 remains attractive because it sits exactly beyond every known positive closure wall:

$$
\boxed{
\text{exceptional}
+
\text{codimension }3
+
\text{non-split discriminant}
+
\text{six-dimensional}
}
$$

and has a clear finite primitive Hodge core.

This minimizes irrelevant complexity.

---

# 38. Minimality philosophy

The first rational Hodge counterexample, if it exists, should plausibly not be sought in a space where Hodge classes themselves are hard to classify.

It is strategically better to choose a setting where:

1. the Hodge class is explicit;
2. the Hodge subspace is low-dimensional;
3. the positive algebraicity frontier is sharply known;
4. the remaining gap is pure cycle existence.

Weil-type abelian sixfolds fit this profile unusually well.

---

# 39. CE001 falsification tests

The branch abandons CE001 immediately if any of the following is proved.

### F1

A theorem algebraizes all sixfold Weil classes for arbitrary discriminant.

### F2

A relative cycle component carrying the Weil class dominates every non-split moduli component.

### F3

A finite primitive BLTP presentation is found in which every required primitive tensor has explicit algebraic witness.

### F4

A universal deformation argument transports known split cycles across discriminant barriers.

### F5

A new theorem shows the relevant Weil classes are generated by already algebraic correspondence sectors.

---

# 40. CE001 confirmation tests

CE001 becomes a genuine counterexample only after one decisive nonexistence result.

Acceptable forms include:

### C1

Prove:

$$
\Delta_{\mathrm{cyc}}>0
$$

uniformly over all cycle representatives.

### C2

Construct a tropical invariant vanishing on every algebraic-cycle tropicalization but nonzero on:

$$
\alpha_{\mathrm{trop}}.
$$

### C3

Find another functorial realization:

$$
F
$$

such that:

$$
F([Z])=0
$$

for every codimension-$3$ algebraic cycle class,

but:

$$
F(\alpha)\neq0.
$$

### C4

Prove a support/coniveau obstruction:

$$
\alpha
\notin
N^3H^6(A,\mathbb Q).
$$

Since an algebraic codimension-$3$ class lies in coniveau:

$$
3,
$$

this would be decisive.

---

# 41. Coniveau route

Define geometric coniveau:

$$
N^3H^6(A,\mathbb Q).
$$

Any codimension-$3$ algebraic cycle class lies in:

$$
\boxed{
N^3H^6(A,\mathbb Q).
}
$$

Hence:

$$
\boxed{
\alpha
\notin
N^3H^6(A,\mathbb Q)
\Longrightarrow
\alpha
\text{ nonalgebraic}.
}
$$

This is a third decisive negative route.

---

# 42. Coniveau difficulty

The Hodge type:

$$
(3,3)
$$

already gives maximal Hodge coniveau compatible with the degree.

Therefore proving geometric coniveau defect is essentially another form of the Hodge problem.

Still, representation-theoretic or degeneration methods might make:

$$
N^3
$$

computable in selected families.

So the route is retained but not prioritized above UCDRD.

---

# 43. Negative program state machine

For a candidate:

$$
(A,\alpha),
$$

use statuses:

- `HODGE_CONFIRMED`
- `EXCEPTIONAL_CONFIRMED`
- `KNOWN_ALGEBRAIC`
- `OUTSIDE_KNOWN_POSITIVE_RANGE`
- `NDC_OPEN`
- `UCDRD_OPEN`
- `TROPICAL_OPEN`
- `CONIVEAU_OPEN`
- `COUNTEREXAMPLE_CONFIRMED`
- `CANDIDATE_KILLED`

CE001 currently sits at:

$$
\boxed{
\texttt{HODGE\_CONFIRMED}
+
\texttt{EXCEPTIONAL\_CONFIRMED}
+
\texttt{OUTSIDE\_KNOWN\_POSITIVE\_RANGE}
+
\texttt{UCDRD\_OPEN}.
}
$$

---

# 44. Counterexample certificate standard

HC-False will not label a class a counterexample without:

$$
\boxed{
\mathsf{CECert}
}
$$

containing at minimum:

1. explicit smooth projective:
   $$
   X;
   $$
2. explicit codimension:
   $$
   p;
   $$
3. explicit rational class:
   $$
   \alpha;
   $$
4. proof:
   $$
   \alpha\in H^{p,p}\cap H^{2p}(X,\mathbb Q);
   $$
5. proof:
   $$
   \alpha\notin\operatorname{Im}cl_X^p;
   $$
6. independent audit against hidden coefficient/integral confusion;
7. audit that the argument disproves rational HC, not only integral HC or generalized HC.

---

# 45. Rational-versus-integral firewall

There are known counterexamples to integral Hodge-type statements.

Those do not count.

CE001 must establish:

$$
\boxed{
\alpha
\notin
\operatorname{cl}
\left(
CH^3(A)\otimes\mathbb Q
\right).
}
$$

Torsion or integrality defects are irrelevant to this branch.

---

# 46. Projective-versus-Kähler firewall

There are counterexamples to extensions of the Hodge conjecture to general compact Kähler manifolds.

Those do not count.

CE001 uses:

$$
\boxed{
A
\text{ projective abelian variety}.
}
$$

So it targets the classical conjecture exactly.

---

# 47. MLRSC relation

The neutral line remains essential.

MLRSC supplies:

- exact Hodge candidate carrier;
- exact legality carrier;
- Hilbert/relative cycle loci;
- category-stable versus category-relative defects;
- tensor primitive core analysis.

HC-False simply orients these tools adversarially.

---

# 48. Expected collision with HC-True branch

Future HC-True branch will likely attack the exact opposite statement:

$$
\boxed{
\exists
\text{ dominating cycle component}
}
$$

or:

$$
\boxed{
\Delta_{\mathrm{cyc}}=0.
}
$$

Therefore the two branches are guaranteed to collide at a mathematically meaningful interface.

This is intentional.

---

# 49. The branch's first decisive equation

The first equation HC-False wants to prove is not:

$$
\alpha\notin CH^3.
$$

It is:

$$
\boxed{
\Delta_{\mathrm{cyc}}
(\alpha;A)
>
0.
}
$$

because this converts an impossible global Chow classification problem into a universal deformation-rank obstruction.

---

# 50. Immediate next research tasks

### N001

Compute:

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

### N002

Write the tangent representation explicitly in unitary-group language.

### N003

Classify the tangent image of deformations of a codimension-$3$ cycle:

$$
Z\subset A.
$$

### N004

Search for a universal quotient:

$$
Q_{K,\delta}
$$

annihilating every cycle-deformation image.

### N005

Compare split:

$$
\delta=-1
$$

versus non-split:

$$
\delta\neq-1
$$

at the tangent-representation level.

### N006

Determine whether known semiregularity maps detect precisely the missing quotient.

---

# 51. Hardened CM subproject

In parallel:

1. solve or reuse the finite intersection problem producing:
   $$
   A_{v_0};
   $$
2. verify the Weil CM type;
3. compute its discriminant;
4. isolate:
   $$
   W_K(A_{v_0});
   $$
5. attempt a nonalgebraicity obstruction not relying on deformation.

This becomes:

$$
\boxed{
\text{HC-False CE001-CM}.
}
$$

---

# 52. Current verdict

No counterexample has yet been proved.

But the branch now has:

1. a concrete smooth-projective target class;
2. a sharply delimited unsolved regime;
3. a theorem reducing generic nonalgebraicity to no-dominance;
4. a deformation-rank condition sufficient for no-dominance;
5. independent tropical and coniveau backup routes.

So CE001 is not merely:

$$
\boxed{
\text{an open Hodge class}.
}
$$

It is now:

$$
\boxed{
\text{a structured counterexample program}.
}
$$

---

# 53. Branch Status

$$
\boxed{
\mathrm{CE001}
=
\mathrm{OPEN\ COUNTEREXAMPLE\ CANDIDATE}.
}
$$

The strongest proved statement in this round is:

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

The unproved step is:

$$
\boxed{
\mathrm{UCDRD}.
}
$$

This is where HC-False research proceeds next.

---

# 54. Next Interface

Next counterexample round:

```text
HODGE_HCFALSE_CE002_WeilSixfoldTangentObstruction.md
```

Primary target:

$$
\boxed{
\text{compute the moduli tangent representation and derive a candidate universal missing deformation sector}.
}
$$

Neutral MLRSC remains frozen at:

```text
HODGE_MLRSC_R029_C_JointObjectFiniteCore.md
```

until the user returns to the neutral line.

Future HC-True branch will be started separately and will not overwrite either numbering scheme.

---

# 參考文獻

1. B. J. J. Moonen, Y. G. Zarhin, *Weil classes on abelian varieties*, arXiv:alg-geom/9612017. Weil classes and exceptional Hodge classes.

2. J. S. Milne, *Hodge classes on abelian varieties*, arXiv:2010.08857. CM-type Hodge classes, split Weil classes, and structural reductions.

3. E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415. Algebraicity for split Weil-type abelian sixfolds of discriminant $-1$ and the resulting fourfold closure.

4. E. Markman, *Secant sheaves and Weil classes on abelian varieties*, arXiv:2509.23403. General secant-sheaf strategy and split Weil-type geometry.

5. A. Mostaed, *McMullen's Curve, the Weil Locus, and the Hodge Conjecture for Abelian Sixfolds*, arXiv:2603.20268. Isolated CM sixfold candidates outside existing algebraicity theorems.

6. I. Zharkov, *Tropical Abelian varieties, Weil classes and the Hodge Conjecture*, arXiv:2002.02347. Kontsevich's tropical counterexample strategy.

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

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

9. Aletheia, *HODGE_MLRSC_R027_C_PrimitiveInvariantAlgebraicity*, 2026-09-15.

10. Aletheia, *HODGE_MLRSC_R028_C_PrimitiveCoreStressAudit*, 2026-09-15.

---

## Canonical Source Declaration

本檔案為 HC-False 分支第一篇正式 UTF-8 Markdown canonical source。

數學原始碼只使用 `$...$` 與 `$$...$$`。

本文件提出的是反例候選與反例攻擊程序，不宣稱 classical rational Hodge conjecture已被反駁。