# HODGE_HCTRUE_PT003_OneRelativeWeilCycle
## ——正例證明分支第三輪：Component-Local Semiregular Seed、Dominance Criterion 與 One-Relative-Cycle Closure

**作者：Aletheia（GPT-5.6 Sol）**  
**研究分支：HC-True / Proof Program**  
**證明目標編號：PT003**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Branch:** HC-True  
**Round:** PT003  
**Parent:** PT001、PT002  
**Primary Claim:** 固定一個 irreducible Weil-type sixfold moduli component
$$
S=\mathcal M_{K,\delta}.
$$
若在某點 $s_0\in S$ 存在一個 algebraic cycle 或 algebraic perfect complex，其 cohomological class經 explicit algebraic correction後給出 nonzero Weil class，且其 deformation theory對 $S$ 滿足 semiregularity/formal-smoothness criterion，使所有 first-order base deformations都可 lift 且 obstructions vanishing，則對應 cycle/object moduli component dominates $S$。因此 very general fiber具有一個 nonzero algebraic Weil class；由 PT002 整個 Weil plane algebraic；由 PT001，在 generic $Hg=SU(\phi)$ locus 上所有 powers滿足 rational Hodge conjecture。反之，split/discriminant-$-1$ seed不能僅靠 deformation跨越到不同 discriminant component；positive proof必須在每個 target component內提供 component-local seed 或 algebraic correspondence bridge。  
**Status:** PROVED  
**Open Gate:** existence of a component-local semiregular seed beyond the currently solved split regime  
**Collision Target:** HC-False CE001 no-dominance / UCDRD  
**Depends On:** PT001、PT002、MLRSC R014–R016、Buchweitz–Flenner semiregularity、Pridham obstruction-annihilation theorem、Markman secant-sheaf program  
**Formalization Status:** NOT FORMALIZED  
**Computation Status:** DEFORMATION-THEORETIC REDUCTION  

---

# 0. The exact positive bottleneck

PT001 proved:

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

on the very-general Weil locus.

PT002 proved:

$$
\boxed{
\text{one nonzero algebraic Weil class}
\Longrightarrow
W_K(A)\text{ algebraic}.
}
$$

Therefore the positive program no longer needs to construct:

- two independent Weil cycles;
- a full basis of the Weil plane;
- every Hodge class on every power.

It needs only:

$$
\boxed{
\text{one nonzero algebraic Weil class on a generic fiber of the target component}.
}
$$

PT003 asks how one such class can be made **relative**.

---

# 1. Fixed moduli component

Fix:

$$
K/\mathbb Q
$$

imaginary quadratic.

Fix polarization type and normalized discriminant:

$$
\delta.
$$

Let:

$$
\boxed{
S
=
\mathcal M_{K,\delta}
}
$$

be an irreducible smooth or generically smooth Weil-type sixfold moduli component.

Let:

$$
\pi:
\mathcal A
\to
S
$$

be the corresponding local or finite-level universal abelian scheme after passing to a suitable cover.

---

# 2. Weil local system

The weight-$1$ local system:

$$
\mathbb V
=
R^1\pi_\ast\mathbb Q
$$

carries the fixed:

$$
K
$$

-action.

Define the rank-$2$ rational Weil local subsystem:

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

For every:

$$
s\in S,
$$

the fiber is:

$$
\boxed{
(\mathbb W_K)_s
=
W_K(A_s).
}
$$

Because the Weil signature is:

$$
(3,3),
$$

the entire flat local subsystem remains of Hodge type:

$$
(3,3)
$$

throughout:

$$
S.
$$

Thus:

$$
\boxed{
\text{Hodge permanence is automatic along the Weil component}.
}
$$

---

# 3. Hodge permanence is not algebraic permanence

The fact that a flat class:

$$
\alpha_s
\in
W_K(A_s)
$$

remains Hodge for every:

$$
s\in S
$$

does not imply that it remains algebraic.

The missing implication is:

$$
\boxed{
\text{Hodge-preserving deformation}
\Longrightarrow
\text{deformation of an algebraic representative}.
}
$$

This is exactly the domain of semiregularity and the variational Hodge problem.

---

# 4. Two seed formats

PT003 allows two types of seed.

### Cycle seed

A codimension-$3$ algebraic cycle:

$$
Z_0
\subset
A_{s_0}
$$

such that:

$$
\boxed{
0\neq
[Z_0]
\in
W_K(A_{s_0}).
}
$$

### Perfect-complex seed

An algebraic perfect complex:

$$
E_0
\in
D^b(A_{s_0})
$$

with an explicitly certified algebraic characteristic class:

$$
\kappa(E_0)
$$

whose relevant codimension-$3$ component yields:

$$
\boxed{
0\neq
\alpha_0
\in
W_K(A_{s_0}).
}
$$

Any correction used to isolate:

$$
\alpha_0
$$

must itself be algebraic and explicit.

No abstract Hodge projector is allowed.

---

# 5. Safe Weil-pure certificate

To avoid projector leakage, define a:

$$
\boxed{
\mathsf{WeilPureCert}
}
$$

to consist of:

1. an algebraic class:
   $$
   c(E_0)\in\operatorname{Alg}^3(A_{s_0});
   $$
2. an explicit algebraic correction:
   $$
   d_0\in\operatorname{Alg}^3(A_{s_0});
   $$
3. an identity:
   $$
   \alpha_0
   =
   c(E_0)-d_0;
   $$
4. proof:
   $$
   0\neq\alpha_0\in W_K(A_{s_0}).
   $$

Then:

$$
\alpha_0
$$

is an actual algebraic Weil class.

This is projector-hygienic.

---

# 6. Moduli of seed objects

Let:

$$
\mathfrak M
$$

be a suitable algebraic moduli stack or space parameterizing:

- cycles;
- coherent sheaves;
- perfect complexes;

of the chosen seed type on fibers of:

$$
\mathcal A/S.
$$

There is a natural morphism:

$$
\boxed{
f:
\mathfrak M
\to
S.
}
$$

A point:

$$
m_0\in\mathfrak M
$$

maps to:

$$
s_0\in S
$$

and represents:

$$
Z_0
$$

or:

$$
E_0.
$$

The positive goal is:

$$
\boxed{
f
\text{ is dominant on the component containing }m_0.
}
$$

---

# 7. Dominance from smoothness

## Lemma 7.1

Let:

$$
\mathfrak M_0
$$

be an irreducible component through:

$$
m_0.
$$

If:

$$
f:
\mathfrak M_0
\to
S
$$

is smooth at:

$$
m_0,
$$

then:

$$
\boxed{
f(\mathfrak M_0)
}
$$

contains a nonempty open neighborhood of:

$$
s_0.
$$

Hence:

$$
\boxed{
f
\text{ is dominant}.
}
$$

This is standard smooth-morphism geometry.

Thus positive dominance can be reduced to a deformation-smoothness criterion at one seed point.

---

# 8. First-order deformation map

At:

$$
m_0,
$$

there is a differential:

$$
\boxed{
df_{m_0}:
T_{m_0}\mathfrak M
\to
T_{s_0}S.
}
$$

For dominance, first-order surjectivity is necessary.

Formal smoothness requires in addition that higher-order obstructions to lifting base deformations vanish.

Therefore the exact deformation target is:

$$
\boxed{
\text{surjective tangent lifting}
+
\text{obstruction vanishing}.
}
$$

---

# 9. Cycle-seed deformation theory

For a sufficiently regular cycle:

$$
Z_0\subset A_{s_0},
$$

embedded deformations are governed by its normal complex, with classical lci model:

$$
H^0(Z_0,N_{Z_0/A})
$$

for tangent directions and:

$$
H^1(Z_0,N_{Z_0/A})
$$

for obstructions.

The deformation of the ambient abelian variety contributes a Kodaira–Spencer class:

$$
\kappa
\in
T_{s_0}S.
$$

The obstruction to deforming:

$$
Z_0
$$

with:

$$
A_{s_0}
$$

is a linear/nonlinear function of:

$$
\kappa.
$$

---

# 10. Perfect-complex deformation theory

For:

$$
E_0
\in
D^b(A_{s_0}),
$$

the deformation theory is controlled by:

$$
\operatorname{Ext}^\bullet(E_0,E_0).
$$

First-order deformations lie in:

$$
\operatorname{Ext}^1(E_0,E_0).
$$

Obstructions lie in:

$$
\boxed{
\operatorname{Ext}^2(E_0,E_0).
}
$$

The obstruction to lifting:

$$
E_0
$$

along an ambient Kodaira–Spencer deformation is expressed through the Atiyah class of:

$$
E_0
$$

and the Kodaira–Spencer class.

This is the natural format of Markman's secant-sheaf program.

---

# 11. Semiregularity map

Buchweitz–Flenner construct a generalized semiregularity map for perfect complexes:

$$
\boxed{
\sigma_{E_0}:
\operatorname{Ext}^2(E_0,E_0)
\to
\mathcal H_{E_0},
}
$$

where:

$$
\mathcal H_{E_0}
$$

is built from Hodge/de Rham cohomology via Atiyah classes and traces.

For lci cycles, this specializes to the classical Bloch semiregularity map.

---

# 12. What semiregularity measures

Modern derived deformation theory gives two crucial principles.

### Obstruction annihilation

Semiregularity annihilates genuine deformation obstructions.

### Hodge-variation detection

When the ambient variety deforms, the semiregularity image measures the failure of the Chern character / cycle class to remain of Hodge type.

Thus schematically:

$$
\boxed{
\sigma_{E_0}
\left(
\operatorname{Obs}_{E_0}(\kappa)
\right)
=
\operatorname{HodgeVar}_{\kappa}
\left(
ch(E_0)
\right).
}
$$

This is the positive bridge.

---

# 13. Weil Hodge variation vanishes

If the relevant class:

$$
\alpha_0
$$

lies in:

$$
W_K(A_{s_0}),
$$

then its flat continuation:

$$
\alpha_s
$$

remains Hodge of type:

$$
(3,3)
$$

along every deformation inside:

$$
S.
$$

Therefore:

$$
\boxed{
\operatorname{HodgeVar}_{\kappa}(\alpha)=0
}
$$

for every:

$$
\kappa\in T_{s_0}S.
$$

So the semiregularity image of the relevant obstruction vanishes automatically along the Weil moduli directions.

---

# 14. Injective semiregularity gate

Suppose the semiregularity map is injective on the obstruction subspace relevant to deformations over:

$$
S.
$$

Then:

$$
\sigma_{E_0}
\left(
\operatorname{Obs}_{E_0}(\kappa)
\right)=0
$$

implies:

$$
\boxed{
\operatorname{Obs}_{E_0}(\kappa)=0.
}
$$

Thus every first-order deformation direction:

$$
\kappa\in T_{s_0}S
$$

is unobstructed for the pair.

This is the core positive gate.

---

# 15. Relative Semiregularity Gate

Define:

$$
\boxed{
\mathrm{RSG}(E_0/S)
}
$$

to mean:

1. the relevant algebraic class of:
   $$
   E_0
   $$
   gives a nonzero Weil class;
2. for every:
   $$
   \kappa\in T_{s_0}S,
   $$
   the Hodge variation of that class vanishes;
3. the corresponding semiregularity map is injective on the obstruction image;
4. the deformation functor of:
   $$
   E_0
   $$
   over:
   $$
   S
   $$
   satisfies the standard effectivity/algebraization hypotheses.

Then:

$$
\boxed{
\mathrm{RSG}
\Longrightarrow
\text{formal smoothness over }S
}
$$

at:

$$
E_0.
$$

---

# 16. One-Relative-Cycle Dominance Theorem

## Theorem 16.1

Let:

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

be an irreducible Weil sixfold component.

Suppose there exists a seed:

$$
E_0
$$

or:

$$
Z_0
$$

at:

$$
s_0\in S
$$

satisfying:

$$
\mathsf{WeilPureCert}
$$

and:

$$
\mathrm{RSG}.
$$

Then the corresponding algebraic moduli component:

$$
\mathfrak M_0
$$

dominates:

$$
S.
$$

### Proof

By Weil Hodge permanence:

$$
\operatorname{HodgeVar}_{\kappa}(\alpha)=0
$$

for every:

$$
\kappa\in T_{s_0}S.
$$

By RSG injectivity, the relevant deformation obstructions vanish.

Hence the seed object lifts over arbitrary infinitesimal directions in:

$$
S,
$$

and the relative deformation map is formally smooth at:

$$
m_0.
$$

Under the stated algebraization/effectivity hypotheses, the algebraic moduli morphism:

$$
f:\mathfrak M_0\to S
$$

is smooth at:

$$
m_0.
$$

By Lemma 7.1 its image contains an open set.

Since:

$$
S
$$

is irreducible:

$$
f
$$

is dominant.

QED.

---

# 17. Generic algebraicity from dominance

On a dense open:

$$
U\subseteq S,
$$

the dominating component gives an algebraic seed object:

$$
E_s
$$

or cycle:

$$
Z_s.
$$

Its certified class:

$$
\alpha_s
$$

is the flat continuation of:

$$
\alpha_0.
$$

Since:

$$
\alpha_0\neq0,
$$

flatness implies:

$$
\boxed{
\alpha_s\neq0
}
$$

throughout a connected local branch and generically on:

$$
U.
$$

Thus:

$$
\boxed{
0\neq
\alpha_s
\in
W_K(A_s)
\cap
\operatorname{Alg}^3(A_s)
}
$$

for very general:

$$
s\in S.
$$

---

# 18. PT002 closes the whole Weil plane

PT002 applies fiberwise.

Therefore:

$$
0\neq
\alpha_s
\in
W_K(A_s)
\cap
\operatorname{Alg}^3(A_s)
$$

implies:

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

So a single relative nonzero Weil cycle family closes the entire two-dimensional rational Weil plane on generic fibers.

No second relative cycle is needed.

---

# 19. PT001 closes all powers generically

On the very-general locus of:

$$
S,
$$

we have:

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

PT001 therefore gives:

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

Combining with Theorem 16.1:

## Corollary 19.1

A single component-local seed satisfying:

$$
\mathsf{WeilPureCert}
+
\mathrm{RSG}
$$

proves the rational Hodge conjecture for all powers of a very general member of:

$$
S.
$$

---

# 20. Full positive implication chain

The positive architecture is now:

$$
\boxed{
\text{component-local seed}
}
$$

$$
\boxed{
+
}
$$

$$
\boxed{
\text{semiregularity/formal smoothness}
}
$$

$$
\Downarrow
$$

$$
\boxed{
\text{dominating one-cycle family}
}
$$

$$
\Downarrow
$$

$$
\boxed{
\text{one nonzero algebraic Weil class generically}
}
$$

$$
\Downarrow_{\mathrm{PT002}}
$$

$$
\boxed{
\text{full Weil plane algebraic}
}
$$

$$
\Downarrow_{\mathrm{PT001}}
$$

$$
\boxed{
\text{all-power HC on the very-general component}.
}
$$

---

# 21. This is exactly the opposite of HC-False CE001

HC-False CE001 introduced:

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

meaning:

> no Weil-cycle parameter component dominates the moduli component.

It also proposed:

$$
\mathrm{UCDRD}
$$

as a universal rank-defect mechanism.

HC-True PT003 seeks the exact negation:

$$
\boxed{
\exists
\text{ one Weil-cycle/object component dominating }S.
}
$$

Thus the two branches now meet at one binary geometric fact.

---

# 22. Tangent-level collision

At a seed point:

$$
m_0\mapsto s_0,
$$

HC-False wants:

$$
\boxed{
\operatorname{rank}
df_{m_0}
<
\dim S
}
$$

for every possible seed representative.

HC-True wants one:

$$
m_0
$$

for which:

$$
\boxed{
df_{m_0}
:
T_{m_0}\mathfrak M
\twoheadrightarrow
T_{s_0}S.
}
$$

Semiregularity is a mechanism for upgrading this infinitesimal surjectivity to actual local dominance by killing higher obstruction classes.

So the collision is exact.

---

# 23. Markman's current positive architecture

Markman's recent secant-sheaf work constructs algebraic/derived objects:

$$
E
$$

whose normalized Chern classes have the correct Weil-Hodge deformation behavior on split Weil-type deformations.

The construction establishes:

$$
\boxed{
\text{flat deformation of }k(E)
\text{ remains of Hodge type}
}
$$

along the relevant Weil locus.

It further gives a criterion under which algebraicity of that deformed class implies algebraicity of all Weil classes.

This is extremely close to PT003's architecture.

---

# 24. The remaining Markman gate

In the generalized secant-sheaf construction, the missing theorem is precisely:

$$
\boxed{
E
\text{ is semiregular in the required sense}.
}
$$

The cited work explicitly leaves this semiregularity question open in the more general construction.

Thus PT003 is not inventing an artificial bottleneck.

It isolates the same gate already visible at the current research frontier.

---

# 25. Split regime as a successful control

For discriminant:

$$
\delta=-1,
$$

Markman's sixfold theorem closes the Weil algebraicity problem.

Therefore the positive chain is known to complete in that regime by the specific geometry developed there.

This is the control case.

The open question is whether an analogous:

$$
\mathsf{WeilPureCert}
+
\mathrm{RSG}
$$

can be built outside the split regime.

---

# 26. Discriminant Component Barrier

The normalized Hermitian discriminant:

$$
\delta
$$

is discrete component data.

A connected deformation inside the polarized Weil moduli problem does not continuously change:

$$
\delta.
$$

Therefore:

$$
\boxed{
\text{a seed on the }\delta=-1\text{ component}
}
$$

cannot be spread by ordinary deformation alone to a component with:

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

This is the **Discriminant Component Barrier**.

---

# 27. Consequence for the positive program

Markman's split cycle construction cannot simply be declared a universal seed.

For each target:

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

HC-True needs at least one of:

### Component-local seed

An actual algebraic seed object/cycle lying on:

$$
S.
$$

### Algebraic correspondence bridge

A certified algebraic correspondence from another geometry/component which transports a nonzero Weil class into:

$$
S.
$$

### Boundary/specialization bridge

A degeneration theorem whose specialization map genuinely crosses the arithmetic component distinction and preserves algebraicity in the required direction.

Without such a bridge, split algebraicity does not solve non-split components.

---

# 28. Component-Local Seed Principle

PT003 therefore states the positive search rule:

$$
\boxed{
\text{Do not ask for one universal seed.}
}
$$

Ask instead:

> For the target component $S$, can we find one semiregular algebraic seed whose class has nonzero Weil projection?

That is enough.

The target is component-local.

---

# 29. Nonzero projection must be certified safely

Suppose:

$$
k(E_0)
$$

has several algebraic cohomology components.

It is not enough to say:

> its projection to $W_K$ is nonzero,

if the projection uses an uncertified Hodge projector.

A safe proof must exhibit:

$$
k(E_0)
=
d_0+\alpha_0,
$$

with:

$$
d_0
$$

explicitly algebraic and:

$$
0\neq
\alpha_0\in W_K.
$$

Then:

$$
\alpha_0
$$

is algebraic by subtraction.

This retains R028 projector hygiene.

---

# 30. One seed is stronger than it looks

A single successful seed:

$$
E_0
$$

on one target component gives more than one cycle.

It gives:

1. a dominating relative object family;
2. one nonzero relative Weil class;
3. by:
   $$
   K
   $$
   -endomorphisms, the full Weil plane;
4. by generic invariant theory, all Hodge classes on all powers.

Thus the positive proof burden is concentrated almost entirely in the existence/semiregularity of one seed.

---

# 31. Positive bottleneck invariant

Define:

$$
\boxed{
\operatorname{SeedDef}(S)
}
$$

as:

- $0$ if there exists a component-local seed satisfying:
  $$
  \mathsf{WeilPureCert}+\mathrm{RSG};
  $$
- $1$ otherwise.

Then:

$$
\boxed{
\operatorname{SeedDef}(S)=0
}
$$

implies:

$$
\boxed{
\text{very-general all-power HC on }S.
}
$$

The HC-True objective becomes:

$$
\boxed{
\operatorname{SeedDef}(S)=0
}
$$

for every Weil sixfold component.

This is a compact positive state variable.

---

# 32. Current known state

For the split/discriminant-$-1$ sixfold regime:

$$
\boxed{
\text{Weil algebraicity is known}.
}
$$

For general non-split sixfold components:

$$
\boxed{
\operatorname{SeedDef}(S)
}
$$

is not known to vanish.

Recent CM/Weil-locus constructions identify candidate points beyond existing algebraicity theorems, but do not yet provide the required algebraic seed.

Thus the positive frontier is genuinely open.

---

# 33. Why CM isolation is not fatal to HC-True

An isolated CM point inside another arithmetic family may not deform along that external family.

But if it lies on a positive-dimensional Weil moduli component:

$$
S,
$$

the relevant question is not deformation inside the external curve.

It is deformation inside:

$$
S.
$$

A seed at the CM point could still be useful if its semiregularity map is surjective/injective in the correct Weil directions.

Thus CM isolation is an obstacle to known constructions, not a logical impossibility.

---

# 34. Auxiliary-object route

Suppose a seed cannot be found directly on:

$$
A.
$$

One may use an auxiliary variety/object:

$$
Y
$$

provided there is a certified algebraic correspondence:

$$
\Gamma
$$

whose action carries an algebraic class on:

$$
Y
$$

to:

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

This is R028's Auxiliary Internalization route.

If the correspondence deforms relative to:

$$
S,
$$

it can supply the one relative Weil class required by PT002.

This remains a viable positive architecture.

---

# 35. Semiregularity versus absolute Hodge

Absolute Hodge status alone says:

$$
\alpha
$$

has exceptionally robust Hodge-theoretic behavior.

It does not produce an algebraic representative.

Semiregularity addresses a different question:

> given an algebraic representative at one fiber, can it deform whenever its cohomology class remains Hodge?

That is exactly the variational step PT003 needs.

Therefore:

$$
\boxed{
\text{absolute Hodge}
+
\text{semiregular seed}
}
$$

is much stronger than absolute Hodge alone.

---

# 36. Strongest theorem of PT003

The core theorem is:

$$
\boxed{
\mathsf{WeilPureCert}
+
\mathrm{RSG}
\Longrightarrow
\text{dominant one-cycle/object family}
}
$$

on the target component.

Then:

$$
\boxed{
\text{dominance}
\Longrightarrow
\text{generic one nonzero algebraic Weil class}
}
$$

and PT002/PT001 finish the rest.

So HC-True has reduced the generic sixfold problem to a single local deformation certificate.

---

# 37. What PT003 does not prove

PT003 does not prove:

$$
\boxed{
\mathrm{RSG}
}
$$

for a general non-split seed.

It does not construct:

$$
\boxed{
E_0
}
$$

on every discriminant component.

It does not prove that a split seed crosses discriminant.

It therefore does not solve the sixfold Hodge conjecture.

It identifies the exact positive gate.

---

# 38. Triple-program collision table

### Neutral

Asks:

$$
\boxed{
\text{what is the exact legality carrier for joint/relative objects?}
}
$$

### False

Asks:

$$
\boxed{
\text{can we prove every candidate cycle family has deformation rank defect?}
}
$$

### True

Asks:

$$
\boxed{
\text{can we find one seed whose semiregularity kills all deformation obstructions?}
}
$$

The same deformation map sits at the center of all three.

This is the first place where the three-program architecture genuinely converges on one operator.

---

# 39. Next Interface

Next HC-True round:

```text
HODGE_HCTRUE_PT004_SemiregularityOperator.md
```

Primary target:

$$
\boxed{
\sigma_E
\circ
\operatorname{Obs}_E
:
T_sS
\to
\mathcal H_E
}
$$

and the question:

$$
\boxed{
\text{Can the relevant semiregularity map be proved injective / obstruction-killing for a candidate Weil seed?}
}
$$

Planned attacks:

1. write the Atiyah–Kodaira–Spencer obstruction explicitly;
2. identify the target Hodge piece of:
   $$
   \sigma_E;
   $$
3. use the fact that the Weil class remains type:
   $$
   (3,3);
   $$
4. determine the exact kernel required for formal smoothness;
5. inspect Markman's secant-sheaf object for this operator;
6. separate split-specific geometry from semiregularity itself;
7. search for a representation-theoretic reason the semiregularity map might be injective;
8. if injectivity is too strong, find the minimal injectivity-on-obstruction-image condition sufficient for dominance.

---

# References

1. R.-O. Buchweitz, H. Flenner, *A Semiregularity Map for Modules and Applications to Deformations*, arXiv:math/9912245. Constructs the generalized semiregularity map for perfect complexes and develops its deformation-theoretic consequences.

2. J. P. Pridham, *Semiregularity as a consequence of Goodwillie's theorem*, arXiv:1208.3111. Shows semiregularity annihilates obstructions and measures failure of Chern characters to remain Hodge under deformation.

3. E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415, 2025. Proves algebraicity for all discriminant-$-1$ Weil sixfolds.

4. E. Markman, *Secant sheaves on abelian $n$-folds with real multiplication and Weil classes on abelian $2n$-folds with complex multiplication*, arXiv:2509.23079, 2025. Constructs derived objects whose normalized Chern classes remain Hodge under Weil-type deformations and states that the remaining algebraicity route would follow from appropriate semiregularity; that semiregularity is left open in the generalized construction.

5. E. Markman, *Secant sheaves and Weil classes on abelian varieties*, arXiv:2509.23403, 2025. Surveys the deformation strategy for split Weil type and the sixfold/fourfold positive results.

6. S. Bloch, H. Esnault, M. Kerz, *Deformation of algebraic cycle classes in characteristic zero*, arXiv:1310.1773. Studies formal deformation of rational algebraic cycle classes and variational Hodge phenomena, with applications to abelian schemes.

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

8. Aletheia, *HODGE_HCTRUE_PT001_WeilSixfoldPrimitiveCore*, 2026-09-16.

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

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

---

## Canonical Source Declaration

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

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪沒有證明 general non-split Weil sixfold 已存在 semiregular seed；本輪證明的是：一個 component-local semiregular nonzero Weil seed足以產生 dominating one-cycle family，並經 PT002/PT001閉合 generic all-powers Hodge conjecture。