# HODGE_HCTRUE_PT001_WeilSixfoldPrimitiveCore
## ——正例證明分支第一輪：Very-General Weil Sixfold 的唯一 Exceptional Primitive Core 與 All-Powers Reduction

**作者：Aletheia（GPT-5.6 Sol）**  
**研究分支：HC-True / Proof Program**  
**證明目標編號：PT001**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Branch Declaration

從本文件起，正式建立第三條長線：

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

三條線共用：

- Measure carrier；
- Legality carrier；
- tensor/category bookkeeping；
- projector hygiene；
- relative-cycle deformation語言；

但研究立場不同。

### MLRSC-neutral

不預設 Hodge conjecture 真假，只分析：

$$
\text{Measure}
\to
\text{Legality}
\to
\text{Scale–Category Coupling}.
$$

### HC-False

主動尋找：

$$
\boxed{
\alpha
\in
H^{2p}(X,\mathbb Q)
\cap
H^{p,p}(X)
}
$$

卻：

$$
\boxed{
\alpha
\notin
\operatorname{Alg}^p(X).
}
$$

### HC-True

主動尋找一套 closure mechanism，使：

$$
\boxed{
H^{2p}(X,\mathbb Q)
\cap
H^{p,p}(X)
\subseteq
\operatorname{Alg}^p(X)
}
$$

對目標 family 成立，並逐步擴張。

---

## Metadata

**Branch:** HC-True  
**Round:** PT001  
**Primary Target:** very general polarized abelian sixfold of Weil type  
**Primary Claim:** 對 very general polarized abelian sixfold of Weil type，generic Hodge group為 special unitary group $SU(\phi)$。其 powers 上的 Hodge tensor category可由 pairing / polarization / $K$-endomorphism tensors與 determinant-type Weil tensor生成。前者皆在 algebraic-safe sector；唯一 exceptional primitive algebraicity core是二維 rational Weil plane
$$
W_K(A)
=
\bigwedge\nolimits_K^6H^1(A,\mathbb Q).
$$
因此
$$
\boxed{
W_K(A)\subseteq\operatorname{Alg}^3(A)
\Longrightarrow
\operatorname{HC}(A^r)
\text{ for every }r\ge1.
}
$$
在 generic Weil locus，這不是 heuristic，而是 classical invariant-theoretic reduction，並與 MLRSC R026–R028 的 BLTP finite-core theorem一致。  
**Status:** PROVED  
**Positive Conjecture:** Every Weil class on every polarized abelian sixfold of Weil type is algebraic  
**Collision Target:** HC-False CE001  
**Depends On:** MLRSC R023–R028、Weil generic Hodge-group theorem、unitary invariant theory、Milne's all-powers reduction  
**Current Positive Boundary:** discriminant $-1$ sixfolds solved by Markman; all Weil fourfolds solved; arbitrary sixfold discriminant remains open  
**Formalization Status:** NOT FORMALIZED  
**Computation Status:** STRUCTURAL REDUCTION  

---

# 0. The positive stance

HC-True adopts the working hypothesis:

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

For the first collision with HC-False, choose exactly the same primitive geometry:

$$
\boxed{
\text{polarized abelian sixfolds of Weil type}.
}
$$

HC-False CE001 predicts that on a suitable non-split component there exists:

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

with:

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

HC-True PT001 instead adopts:

$$
\boxed{
\mathrm{PT\mbox{-}Weil6}:
\quad
W_K(A)
\subseteq
\operatorname{Alg}^3(A)
}
$$

for every polarized abelian sixfold of Weil type.

The two branches therefore collide on exactly the same two-dimensional primitive space.

---

# 1. Weil-type sixfold

Let:

$$
A
$$

be a polarized complex abelian sixfold.

Let:

$$
K/\mathbb Q
$$

be an imaginary quadratic field with embedding:

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

Set:

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

Then:

$$
\dim_KV=6.
$$

The polarization determines a nondegenerate:

$$
K
$$

-Hermitian form:

$$
\phi:
V\times V
\to
K.
$$

The Weil condition requires Hodge signature:

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

---

# 2. Weil determinant tensor

Define:

$$
\boxed{
W_K(A)
=
\bigwedge\nolimits_K^6V.
}
$$

As a:

$$
K
$$

-vector space:

$$
\dim_KW_K(A)=1.
$$

As a rational vector space:

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

The Weil condition implies:

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

Thus:

$$
W_K(A)
$$

is a two-dimensional rational Hodge plane of codimension:

$$
3.
$$

---

# 3. Generic Hodge group

For a general polarized abelian variety of Weil type, the Hodge group is as large as permitted by:

- the $K$-action;
- the polarization;
- the Weil tensor.

The classical Weil theorem gives:

$$
\boxed{
Hg(A)=SU(\phi)
}
$$

on the complement of a countable union of proper special loci in the Weil moduli component.

This is the **very-general locus** used in PT001.

---

# 4. Why $SU(\phi)$ matters

Hodge classes on tensor constructions from:

$$
V
$$

are precisely invariants of the Hodge group.

Therefore on the very-general Weil locus:

$$
\boxed{
\text{Hodge tensors}
=
SU(\phi)\text{-invariant tensors}.
}
$$

The algebraicity problem becomes an invariant-generation problem.

This is exactly the setting of MLRSC R026.

---

# 5. Primitive invariant data

The standard representation of a special unitary group has two basic kinds of invariant tensor data.

### Pairing sector

The Hermitian/polarization pairing and its dual contraction tensors.

### Determinant sector

The top exterior tensor:

$$
\boxed{
\det_KV
=
\bigwedge\nolimits_K^6V.
}
$$

Over:

$$
\mathbb Q,
$$

the determinant sector is precisely:

$$
W_K(A).
$$

Thus the exceptional primitive tensor is not mysterious.

It is the special-unitary volume tensor.

---

# 6. Pairing sector is algebraic-safe

The polarization comes from an ample line bundle:

$$
L.
$$

Hence:

$$
c_1(L)
$$

is algebraic.

Cup product, intersection, pullback and pushforward are legal algebraic operations.

The graph of every actual:

$$
K
$$

-endomorphism:

$$
u:A\to A
$$

is algebraic.

Thus:

$$
\boxed{
\text{polarization}
+
K\text{-endomorphism tensors}
+
\text{geometric contractions}
}
$$

are already in the certified algebraic sector.

---

# 7. Projector hygiene

R028 warns that degree-specific coevaluation/projectors must not be declared algebraic merely because the full diagonal is algebraic.

For abelian varieties, however, the standard Lefschetz/Künneth correspondence technology is available in the classical algebraic correspondence sector.

PT001 therefore does not need to smuggle arbitrary Hodge projectors into the grammar.

The invariant generation can be formulated directly in raw tensor words using:

- pairings;
- endomorphism actions;
- determinant tensors;
- tensor products;
- contractions;
- permutations.

This is the projector-hygienic route of R026/R028.

---

# 8. Unit-volume generation principle

Let:

$$
V_K
$$

denote the six-dimensional:

$$
K
$$

-standard representation.

For:

$$
SL(V_K)
$$

and therefore for the corresponding complexified special-unitary invariant problem, the tensor invariant category is generated by:

1. evaluation / duality contractions;
2. the determinant tensor:
   $$
   \varepsilon
   \in
   \bigwedge\nolimits_K^6V_K^\vee;
   $$
3. the dual determinant tensor.

This is the classical determinant form of the first fundamental theorem for the special linear group.

Hence every Hodge tensor on powers of:

$$
A
$$

has a derivation tree whose only exceptional primitive leaves are Weil determinant tensors.

---

# 9. Hodge-side BLTP

Therefore the very-general Weil sixfold admits a bounded local tensor presentation:

$$
\boxed{
\mathrm{BLTP}_{6}.
}
$$

A local generator set may be chosen from:

- pairing tensors;
- $K$-endomorphism tensors;
- determinant tensor:
  $$
  \Omega_K;
  $$
- conjugate determinant tensor:
  $$
  \bar\Omega_K.
  $$

All have bounded tensor support.

The determinant generators correspond over:

$$
\mathbb Q
$$

to the two-dimensional Weil plane:

$$
W_K(A).
$$

---

# 10. Primitive-core ledger

Relative to this BLTP certificate, define:

$$
\mathcal G_0
$$

to be the automatic algebraic sector generated by:

- divisor/polarization classes;
- algebraic endomorphism graphs;
- diagonals and permitted contractions;
- tensor permutations;
- standard algebraic correspondence operations.

Then the primitive Hodge quotient is:

$$
\boxed{
\mathfrak P_H
=
\mathcal G_H/\mathcal G_0.
}
$$

For the generic Weil sixfold:

$$
\boxed{
\mathfrak P_H
\cong
W_K(A).
}
$$

More precisely, the only exceptional generator directions are the two rational Weil directions.

---

# 11. Primitive residual

Define:

$$
\boxed{
\mathfrak R_{\mathrm{Weil}}(A)
=
W_K(A)
/
\left(
W_K(A)
\cap
\operatorname{Alg}^3(A)
\right).
}
$$

Then:

$$
\dim_{\mathbb Q}
\mathfrak R_{\mathrm{Weil}}
\in
\{0,1,2\}
$$

a priori.

The HC-True objective is:

$$
\boxed{
\mathfrak R_{\mathrm{Weil}}(A)=0.
}
$$

---

# 12. Milne all-powers theorem

For a general polarized abelian variety of Weil type, Milne records the classical invariant-theoretic consequence:

$$
\boxed{
\text{if the Weil classes on }A\text{ are algebraic, then the Hodge conjecture holds for all powers of }A.
}
$$

Thus the positive branch does not need to independently algebraize every invariant appearing on:

$$
A^r.
$$

Once the primitive determinant tensor is algebraic, invariant theory propagates algebraicity to all tensor powers.

---

# 13. All-Powers Primitive-Core Theorem

## Theorem 13.1

Let:

$$
A
$$

be a very general polarized abelian sixfold of Weil type.

If:

$$
\boxed{
W_K(A)
\subseteq
\operatorname{Alg}^3(A),
}
$$

then for every:

$$
r\ge1
$$

and every:

$$
p,
$$

$$
\boxed{
H^{2p}(A^r,\mathbb Q)
\cap
H^{p,p}(A^r)
=
\operatorname{Alg}^p(A^r).
}
$$

### Proof

On the very-general Weil locus:

$$
Hg(A)=SU(\phi).
$$

Every Hodge tensor on every tensor power is generated by:

- pairing/contraction tensors;
- $K$-endomorphism tensors;
- determinant and dual-determinant tensors.

The first two sectors are algebraic-safe.

By hypothesis the determinant sector:

$$
W_K(A)
$$

is algebraic.

Therefore every invariant derivation lifts to the algebraic correspondence/cycle category.

Hence every Hodge class on every:

$$
A^r
$$

is algebraic.

QED.

---

# 14. MLRSC finite-core interpretation

R027 proved, for a valid projector-hygienic BLTP certificate:

$$
\boxed{
\text{all-power HC}
\iff
\text{finite primitive core algebraic}.
}
$$

PT001 identifies that abstract finite primitive core explicitly:

$$
\boxed{
\text{Primitive Core}
=
W_K(A).
}
$$

So the neutral and positive lines now coincide structurally.

The HC-True branch differs only by committing to:

$$
\boxed{
W_K(A)
\text{ is algebraic}.
}
$$

---

# 15. Generic Hodge ring on $A$

For a general polarized Weil-type abelian variety, the generic Hodge algebra on the single fiber satisfies the classical decomposition:

$$
\boxed{
B^\ast(A)
=
D^\ast(A)
\oplus
W_K(A)
}
$$

in the appropriate graded sense:

- $D^\ast(A)$ is the divisor/Lefschetz-generated algebra;
- $W_K(A)$ supplies the exceptional middle Hodge sector.

Thus even before passing to powers, the non-divisor algebraicity problem has already collapsed to:

$$
W_K(A).
$$

---

# 16. Known positive boundary I: split sixfolds

Markman's 2025 theorem proves:

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

for every polarized abelian sixfold of Weil type with normalized discriminant:

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

This holds for every imaginary quadratic field:

$$
K.
$$

Therefore PT001 immediately gives:

## Corollary 16.1

For a very general split/discriminant-$-1$ Weil sixfold:

$$
\boxed{
\operatorname{HC}(A^r)
}
$$

holds for every:

$$
r.
$$

---

# 17. Known positive boundary II: fourfolds

Markman's argument, combined with Schoen's degeneration method, yields algebraicity of Weil classes for every abelian fourfold of Weil type, for all discriminants and all imaginary quadratic fields.

Therefore the fourfold primitive Weil core is completely closed.

This is strong evidence that:

$$
\boxed{
\text{non-split discriminant}
}
$$

is not intrinsically a nonalgebraicity obstruction.

It is merely outside the current sixfold construction theorem.

This directly challenges HC-False CE001.

---

# 18. Current sixfold frontier

For dimension:

$$
6,
$$

the unresolved regime is not:

$$
\boxed{
\text{what are the Hodge classes?}
}
$$

The generic exceptional Hodge sector is already known:

$$
W_K(A).
$$

The unresolved question is purely legality:

$$
\boxed{
W_K(A)
\stackrel{?}{\subseteq}
\operatorname{Alg}^3(A)
}
$$

outside the currently solved discriminant-$-1$ regime.

This makes the sixfold frontier unusually clean.

---

# 19. Mostaed 2026 as a positive probe

Mostaed studies intersections between McMullen's curve and Weil loci inside a Hilbert modular sixfold.

At any nonempty intersection point:

- the abelian sixfold is CM;
- its endomorphism field is a degree-$12$ CM field:
  $$
  M=KL;
  $$
- the Weil Hodge classes are absolute Hodge;
- current algebraicity theorems do not settle them because of:
  - CM isolation;
  - no known $K$-secant structure;
  - uncontrolled discriminant.

This supplies a finite arithmetic positive target beyond the current split construction.

HC-True should treat such points as **stress probes**, not as counterexample evidence.

---

# 20. Positive Conjecture PT-Weil6

The first explicit HC-True conjecture is:

$$
\boxed{
\mathrm{PT\mbox{-}Weil6}:
\quad
W_K(A)
\subseteq
\operatorname{Alg}^3(A)
}
$$

for every polarized abelian sixfold of Weil type.

If true, then on every very-general Weil component:

$$
\boxed{
\operatorname{HC}(A^r)
\text{ for all }r.
}
$$

Special loci may have additional Hodge tensors and require extra primitive-core analysis, but the generic frontier closes.

---

# 21. Exact collision with HC-False CE001

HC-False CE001 says:

$$
\boxed{
\exists\,
\text{very general non-split }A,
\quad
0\neq\alpha\in W_K(A),
\quad
\alpha\notin\operatorname{Alg}^3(A).
}
$$

HC-True PT-Weil6 says:

$$
\boxed{
\forall A
\text{ of Weil type},
\quad
W_K(A)\subseteq\operatorname{Alg}^3(A).
}
$$

There is no semantic gap between the two targets.

One of these branches must eventually fail.

This is the intended adversarial collision.

---

# 22. Proof certificate standard

HC-True will not accept:

> the Weil class is absolute Hodge, motivated, or deformation-invariant.

A genuine positive certificate must produce:

$$
\boxed{
\mathsf{PosCert}
}
$$

containing:

1. explicit target:
   $$
   (A,K,\lambda);
   $$
2. explicit Weil plane:
   $$
   W_K(A);
   $$
3. an algebraic cycle family:
   $$
   Z_1,Z_2
   \in
   CH^3(A)_{\mathbb Q};
   $$
4. proof that:
   $$
   cl(Z_1),cl(Z_2)
   $$
   span:
   $$
   W_K(A);
   $$
5. deformation/transport provenance;
6. discriminant convention audit;
7. proof that no unverified Hodge projector is used;
8. all-power propagation via the generic invariant theorem where applicable.

---

# 23. Positive proof architectures

PT001 identifies several possible routes.

### Route P1 — Direct cycle construction

Generalize secant-sheaf geometry beyond:

$$
\delta=-1.
$$

### Route P2 — Deformation transport

Construct cycles on a seed point and prove semiregular/relative transport across an entire fixed-discriminant Weil component.

### Route P3 — Auxiliary-object algebraization

Embed the Weil tensor into an auxiliary geometry where it is algebraic, then internalize it back via certified algebraic correspondences.

### Route P4 — Arithmetic/CM bridge

Prove algebraicity at sufficiently many CM points and use a relative-cycle theorem to spread.

### Route P5 — Correspondence/stabilization

Construct an algebraic correspondence between a non-split sixfold and a geometry with already-algebraic primitive determinant tensors.

Each route must satisfy the R028 Ambient Realizability / projector-hygiene rules.

---

# 24. What does not count

The following are insufficient.

### Absolute Hodge

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

does not imply known algebraicity.

### Motivated

Being motivated in André's sense is not by itself an algebraic cycle certificate for rational HC.

### Hodge isometry

A Hodge-theoretic isomorphism between primitive sectors does not transport algebraicity unless induced by an algebraic correspondence.

### Abstract projector

A Hodge projector onto:

$$
W_K(A)
$$

is not automatically algebraic.

### Special-fiber cycle

One algebraic cycle on one special point does not automatically spread over the entire component.

The relative legality gate remains mandatory.

---

# 25. Why the positive branch is narrower than it looks

A proof of PT-Weil6 needs only algebraize a two-dimensional rational space.

But those two directions are exactly the exceptional special-unitary determinant invariants.

So the issue is not large dimension.

It is the absence of a universal geometric realization mechanism for the determinant tensor in the non-split sixfold regime.

This is why the problem remains nontrivial despite the small primitive core.

---

# 26. Positive algebraicity load

Define:

$$
\boxed{
\operatorname{PALoad}_{+}(A)
=
\dim_{\mathbb Q}
\mathfrak R_{\mathrm{Weil}}(A).
}
$$

For the sixfold primitive core:

$$
\operatorname{PALoad}_{+}
\le2.
$$

Known split regime:

$$
\boxed{
\operatorname{PALoad}_{+}=0.
}
$$

Unknown general non-split regime:

$$
\boxed{
\operatorname{PALoad}_{+}\in\{0,1,2\}
}
$$

a priori.

The HC-True objective is to prove globally:

$$
\boxed{
\operatorname{PALoad}_{+}=0.
}
$$

---

# 27. Symmetry remark

The rational Weil plane is naturally a two-dimensional representation associated with the imaginary quadratic field.

In many constructions, one nonzero algebraic direction plus stability under the actual $K$-endomorphism action may force algebraicity of the full Weil plane.

Therefore future rounds should test whether there is a legality-safe analog of the binary phenomenon found in HC-False CE006.

If so, the positive task may reduce from:

$$
\boxed{
\text{construct two independent cycles}
}
$$

to:

$$
\boxed{
\text{construct one nonzero Weil cycle}
}
$$

and use algebraic $K$-endomorphisms to generate the conjugate direction.

This is a promising next reduction.

---

# 28. Candidate Positive Irreducibility Principle

Let:

$$
\mathcal A_W
=
W_K(A)
\cap
\operatorname{Alg}^3(A).
$$

Because:

$$
K
$$

acts by algebraic endomorphisms of:

$$
A,
$$

the subspace:

$$
\mathcal A_W
$$

is stable under the induced:

$$
K
$$

-action.

If:

$$
W_K(A)
$$

is irreducible as a rational representation of a suitable algebraic element:

$$
u\in K,
$$

then:

$$
\boxed{
\mathcal A_W
=
0
\quad\text{or}\quad
W_K(A).
}
$$

This would give a **binary algebraicity theorem**.

PT001 records this as the next immediate target.

---

# 29. Positive binary consequence

If the irreducibility hypothesis in Section 28 is established with an actual algebraic endomorphism action, then:

$$
\boxed{
\exists\,
0\neq
\alpha
\in
W_K(A)\cap\operatorname{Alg}^3(A)
}
$$

would imply:

$$
\boxed{
W_K(A)
\subseteq
\operatorname{Alg}^3(A).
}
$$

Combined with Theorem 13.1:

$$
\boxed{
\text{one nonzero Weil cycle}
\Longrightarrow
\text{all-power HC for very general }A.
}
$$

This is potentially the strongest compression available to the positive branch.

It will be checked rigorously in PT002.

---

# 30. Triple-program state

We now have three simultaneous research fronts.

### Neutral

Next:

```text
HODGE_MLRSC_R029_C_JointObjectFiniteCore.md
```

Question:

$$
\boxed{
\text{what is the correct joint-object finite-core formalism?}
}
$$

### False

Next:

```text
HODGE_HCFALSE_CE011_DiagonalFramingObstruction.md
```

Question:

$$
\boxed{
\text{can framed tropical cycles realize the extremal Weil spectral plane?}
}
$$

### True

Next:

```text
HODGE_HCTRUE_PT002_OneCycleBinaryClosure.md
```

Question:

$$
\boxed{
\text{does one nonzero algebraic Weil class force the entire Weil plane algebraic?}
}
$$

The three lines now attack complementary interfaces.

---

# 31. Status

The statement proved in PT001 is not:

$$
\boxed{
\text{the Hodge conjecture is true}.
}
$$

It is the structural reduction:

$$
\boxed{
W_K(A)\text{ algebraic}
\Longrightarrow
\operatorname{HC}(A^r)\text{ for every }r
}
$$

for a very general polarized Weil sixfold.

This is an exact positive victory condition.

Thus:

$$
\boxed{
\mathrm{Status}
=
\mathrm{PROVED}.
}
$$

The unresolved conjecture is:

$$
\boxed{
\mathrm{PT\mbox{-}Weil6}.
}
$$

---

# 32. Next Interface

Next HC-True round:

```text
HODGE_HCTRUE_PT002_OneCycleBinaryClosure.md
```

Primary target:

$$
\boxed{
\mathcal A_W
=
W_K(A)\cap\operatorname{Alg}^3(A)
\in
\{0,W_K(A)\}
\ ?
}
$$

Planned steps:

1. write the exact action of:
   $$
   K^\times
   $$
   on:
   $$
   W_K(A);
   $$
2. choose:
   $$
   u\in\mathcal O_K
   $$
   whose induced rational operator has irreducible quadratic minimal polynomial;
3. prove algebraic cycles are stable under:
   $$
   u_\ast
   $$
   or:
   $$
   u^\ast;
   $$
4. conclude:
   $$
   \mathcal A_W
   $$
   is either:
   $$
   0
   $$
   or the full:
   $$
   W_K(A);
   $$
5. infer:
   $$
   \text{one nonzero algebraic Weil class}
   \Rightarrow
   \text{all Weil classes algebraic};
   $$
6. combine with PT001 to obtain:
   $$
   \text{one cycle}
   \Rightarrow
   \text{all-power HC}
   $$
   on the very-general Weil sixfold.

---

# References

1. J. S. Milne, *The Tate and Standard Conjectures for Certain Abelian Varieties*. For general polarized abelian varieties of Weil type, the Hodge group is the corresponding special unitary group; Milne records that algebraicity of the Weil classes implies the Hodge conjecture for all powers.

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

3. B. J. J. Moonen, Yu. G. Zarhin, *Weil classes on abelian varieties*, arXiv:alg-geom/9612017. Develops the exceptional Weil-class construction and criteria for when the classes are Hodge and exceptional.

4. B. J. J. Moonen, Yu. G. Zarhin, *Hodge classes on abelian varieties of low dimension*, arXiv:math/9901113. Studies Hodge groups and generation by divisor and Weil classes in low dimensions.

5. E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415, 2025. Proves algebraicity of Weil classes for all discriminant-$-1$ abelian sixfolds and all Weil fourfolds after Schoen's degeneration argument.

6. E. Markman, *Secant sheaves and Weil classes on abelian varieties*, arXiv:2509.23403, 2025. Gives the split-Weil secant-sheaf strategy and surveys the sixfold/fourfold positive results.

7. A. Mostaed, *McMullen's Curve, the Weil Locus, and the Hodge Conjecture for Abelian Sixfolds*, arXiv:2603.20268, 2026. Identifies isolated CM Weil-sixfold targets beyond current algebraicity theorems.

8. Aletheia, *HODGE_MLRSC_R026_C_BoundedProjectorGeneration*, 2026-09-15.

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

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

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

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## Canonical Source Declaration

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

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪沒有宣稱所有 Weil sixfold classes 已 algebraic；本輪證明的是 very-general Weil sixfold 的 all-powers Hodge conjecture可精確 reduction 到二維 primitive Weil core 的 algebraicity。