# HODGE_HCTRUE_PT008_EvenLiftRank
## ——正例證明分支第八輪：Semiregularity–Coverage Redundancy、Even-Lift Rank Diagnostic 與 Proof-Priority Reordering

**作者：Aletheia（GPT-5.6 Sol）**  
**研究分支：HC-True / Proof Program**  
**證明目標編號：PT008**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Branch:** HC-True  
**Round:** PT008  
**Parent:** PT003–PT007  
**Primary Claim:** PT007 定義的 Even Lift Coverage
$$
T_{\mathrm{Weil}}
\subseteq
\operatorname{Im}
\left[
\operatorname{pr}_{\mathrm{comm}}
\circ
\Phi_{HT}
\big|
_{\mathcal K_{\mathrm{even}}}
\right]
$$
在 source-kernel approach 中是一個正確 first-order consistency condition，但它不是在 target-side semiregularity之後還需額外證明的獨立 gate。若 target object $E$ 滿足適當 semiregularity，且其 $ch(E)$ 或 normalized class $\kappa(E)$ 沿 Weil family保持 Hodge，Buchweitz–Flenner/Pridham deformation theorem直接推出 $E$ 沿所有 Weil base directions變形。由 Fourier–Mukai/Hochschild naturality，這自動產生每個 Weil tangent 的 unobstructed generalized source lift，因此
$$
\boxed{
\text{target semiregularity}
+
\text{Hodge permanence}
\Longrightarrow
\mathsf{EvenLiftDefect}=0.
}
$$
故 Even Lift Rank 應降級為 **diagnostic/falsifier**：若可直接算出
$$
\mathsf{EvenLiftDefect}>0,
$$
則該 outer-product seed不可能滿足所需的 relative semiregularity/deformation package；但若 image-restricted semiregularity已被證明，無需再獨立計算 full rank coverage。  
**Status:** PROVED / REFORMULATE PT007 PRIORITY  
**Supersedes:** PT007 將 Even Lift Coverage視為與 semiregularity並列的獨立 positive gate  
**Split Sixfold:** rank-$9$ source lift remains a successful control calculation  
**Generalized Higher-CM:** explicit Even Lift Rank remains unknown, but is not logically required after target semiregularity  
**New Priority:** prove image-restricted semiregularity first; use ELR only as cheap early rejection or consistency audit  
**Depends On:** PT004–PT007、Buchweitz–Flenner Semiregularity Theorem、Pridham twisted/derived extension、Markman 2502.03415、2509.23079、2509.23403  
**Formalization Status:** NOT FORMALIZED  
**Computation Status:** LOGICAL DEPENDENCY CLOSURE  

---

# 0. Why PT008 changes the order of attack

PT007 introduced:

$$
\boxed{
\mathsf{EvenLiftDefect}
}
$$

as a first-order measure of whether the unobstructed source kernel covers the full Weil tangent after the Orlov Hochschild transform.

That invariant is correct.

But PT007 treated it too much like an independent positive burden.

The semiregularity theorem shows that the logical dependency actually runs in the opposite direction:

$$
\boxed{
\text{semiregularity}
+
\text{Hodge permanence}
\Longrightarrow
\text{deformation coverage}.
}
$$

Therefore rank coverage is automatically forced by successful semiregularity.

---

# 1. Target family

Let:

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

be a smooth analytic/algebraic family of polarized abelian varieties of Weil type.

Let:

$$
s_0\in S,
$$

and write:

$$
A=A_{s_0}.
$$

Let:

$$
E
$$

be a coherent, twisted, projective, or perfect seed object on:

$$
A
$$

in the category to which the appropriate semiregularity theorem applies.

---

# 2. The relevant characteristic class

There are two common cases.

### Ordinary case

Use:

$$
\boxed{
ch(E).
}
$$

### Normalized/twisted case

If:

$$
\operatorname{rank}(E)=r\neq0,
$$

use:

$$
\boxed{
\kappa(E)
=
ch(E)
\exp
\left(
-\frac{c_1(E)}{r}
\right).
}
$$

Markman's split and generalized secant strategy is naturally formulated using:

$$
\kappa(E),
$$

because it separates the determinant/line-bundle part from the primitive normalized class.

---

# 3. Hodge permanence input

Assume the relevant characteristic class extends as a flat section:

$$
\gamma_s
$$

of the Gauss–Manin local system over:

$$
S
$$

and satisfies:

$$
\boxed{
\gamma_s
\in
\bigoplus_p
H^{p,p}(A_s)
}
$$

for every:

$$
s
$$

near:

$$
s_0.
$$

For the generalized Markman object:

$$
E=\Phi(F_1\boxtimes F_2^\vee),
$$

the normalized class:

$$
\kappa(E)
$$

has exactly this Weil-Hodge permanence property.

---

# 4. First-order obstruction

The Kodaira–Spencer map is:

$$
KS_S:
T_{s_0}S
\to
H^1(A,T_A).
$$

The ambient obstruction to deforming:

$$
E
$$

along:

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

is:

$$
\boxed{
\operatorname{ob}_{E/S}(v)
=
at_E
\cup
KS_S(v)
\in
\operatorname{Ext}^2(E,E).
}
$$

Define:

$$
\boxed{
\operatorname{ObIm}_S(E)
=
\operatorname{Im}
\left(
\operatorname{ob}_{E/S}
\right).
}
$$

---

# 5. Semiregularity compatibility

Buchweitz–Flenner's commutative diagram gives:

$$
\boxed{
\sigma_E
\left(
\operatorname{ob}_{E/S}(v)
\right)
=
\operatorname{HodgeVar}_v
\left(
ch(E)
\right),
}
$$

with the analogous normalized/twisted formulation for:

$$
\kappa(E).
$$

If the class remains Hodge along:

$$
S,
$$

then:

$$
\boxed{
\sigma_E
\left(
\operatorname{ob}_{E/S}(v)
\right)
=
0
}
$$

for every:

$$
v.
$$

Thus:

$$
\boxed{
\operatorname{ObIm}_S(E)
\subseteq
\ker\sigma_E.
}
$$

---

# 6. Minimal relative semiregularity

PT004 defined the weakest relevant first-order condition:

$$
\boxed{
\ker\sigma_E
\cap
\operatorname{ObIm}_S(E)
=
0.
}
$$

Combining with Section 5 yields:

$$
\boxed{
\operatorname{ObIm}_S(E)=0.
}
$$

Therefore:

$$
\boxed{
\operatorname{ob}_{E/S}=0.
}
$$

Every Weil tangent direction lifts to a first-order deformation of the pair:

$$
(A,E).
$$

---

# 7. Full semiregularity theorem

In the classical Buchweitz–Flenner setting, if:

$$
E
$$

is semiregular and:

$$
ch(E)
$$

remains Hodge over:

$$
S,
$$

then:

$$
E
$$

extends over an analytic neighborhood of:

$$
s_0.
$$

Markman's split proof also uses the twisted/projective version after replacing:

$$
E
$$

by the determinant-normalized twisted sheaf whose Chern character is:

$$
\kappa(E).
$$

Pridham's derived semiregularity formalism supplies the obstruction-theoretic extension mechanism in that setting.

Thus target-side semiregularity already implies actual local deformation coverage.

---

# 8. Coverage theorem on the target

## Theorem 8.1

Assume:

1. the relevant semiregularity theorem applies to:
   $$
   E;
   $$
2. $E$ satisfies the needed semiregularity hypothesis;
3. $ch(E)$ or:
   $$
   \kappa(E)
   $$
   remains Hodge over:
   $$
   S.
   $$

Then the local moduli map of the pair:

$$
(\mathcal A,\mathcal E)
\to
S
$$

is locally surjective.

In particular:

$$
\boxed{
T_{[E]}
\operatorname{Def}(A,E)
\twoheadrightarrow
T_{s_0}S.
}
$$

No independent tangent-rank calculation is required.

---

# 9. Fourier–Mukai source model

Now suppose:

$$
E
=
\Phi(G),
$$

with:

$$
G
=
F_1\boxtimes F_2^\vee.
$$

The derived equivalence induces:

$$
\boxed{
\Phi_{HT}:
HT^2(X\times X)
\overset{\sim}{\longrightarrow}
HT^2(A).
}
$$

It also intertwines the evaluation/Atiyah obstruction structures.

Therefore every first-order generalized deformation of:

$$
E
$$

has a corresponding source generalized deformation of:

$$
G,
$$

and vice versa.

---

# 10. Twisted normalization and generalized target lifts

For normalized:

$$
\kappa(E),
$$

the lifted object may be a twisted sheaf over a gerby deformation.

Thus the target generalized deformation associated with a base tangent:

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

may have the form:

$$
\boxed{
v+\beta,
}
$$

where:

$$
\beta
\in
H^2(A,\mathcal O_A).
$$

The commutative projection is still:

$$
v.
$$

This is precisely why PT007 replaced pure preimages by generalized lifts.

---

# 11. Source lift forced by semiregularity

Let:

$$
v
\in
T_{\mathrm{Weil}}
=
T_{s_0}S.
$$

By Theorem 8.1, there is an allowed target generalized deformation:

$$
\tilde v_{\mathrm{tar}}
\in
HT^2(A)
$$

with:

$$
\boxed{
\operatorname{pr}_{\mathrm{comm}}
(\tilde v_{\mathrm{tar}})
=
v
}
$$

and:

$$
\boxed{
ob_E
(\tilde v_{\mathrm{tar}})
=
0.
}
$$

Set:

$$
\boxed{
\tilde v_{\mathrm{src}}
=
\Phi_{HT}^{-1}
(\tilde v_{\mathrm{tar}}).
}
$$

Naturality gives:

$$
\boxed{
ob_G
(\tilde v_{\mathrm{src}})
=
0.
}
$$

So every Weil tangent acquires an unobstructed generalized source lift automatically.

---

# 12. Semiregularity Implies Even-Lift Coverage

Assume in addition the PT006 finite-stabilizer hypotheses:

$$
\operatorname{Stab}^0(F_1)
=
\operatorname{Stab}^0(F_2)
=
0.
$$

Then:

$$
\boxed{
\ker ob_G
\subseteq
C_{20}\oplus C_{02}.
}
$$

Hence every source lift in Section 11 lies in:

$$
\boxed{
\mathcal K_{\mathrm{even}}.
}
$$

Therefore:

## Theorem 12.1

Target semiregularity plus Weil Hodge permanence implies:

$$
\boxed{
T_{\mathrm{Weil}}
\subseteq
\operatorname{Im}
\left[
\operatorname{pr}_{\mathrm{comm}}
\circ
\Phi_{HT}
\big|
_{\mathcal K_{\mathrm{even}}}
\right].
}
$$

Equivalently:

$$
\boxed{
\mathsf{EvenLiftDefect}=0.
}
$$

QED.

---

# 13. Logical consequence

The implication chain is:

$$
\boxed{
\text{target semiregularity}
}
$$

$$
+
$$

$$
\boxed{
\text{Hodge permanence}
}
$$

$$
\Downarrow
$$

$$
\boxed{
\text{relative deformation}
}
$$

$$
\Downarrow
$$

$$
\boxed{
\text{unobstructed generalized source lift}
}
$$

$$
\Downarrow_{\text{finite stabilizers}}
$$

$$
\boxed{
\text{even source lift}
}
$$

$$
\Downarrow
$$

$$
\boxed{
\mathsf{EvenLiftDefect}=0.
}
$$

Thus ELR is downstream of successful semiregularity.

---

# 14. Rank is not an independent positive gate

PT007 listed:

1. semiregularity;
2. Even Lift Coverage;

as if both had to be proved independently.

PT008 corrects this.

Once the appropriate target semiregularity theorem is established:

$$
\boxed{
\text{Even Lift Coverage follows automatically}.
}
$$

Therefore it is inefficient to treat:

$$
\mathrm{ELR}
$$

as a mandatory second proof after semiregularity.

---

# 15. Rank remains a powerful negative test

The converse use is extremely valuable.

Suppose the source model is explicit enough to compute:

$$
\mathcal K_{\mathrm{even}}
$$

and:

$$
\operatorname{pr}_{\mathrm{comm}}
\circ
\Phi_{HT}.
$$

If:

$$
\boxed{
\mathsf{EvenLiftDefect}>0,
}
$$

then Theorem 12.1 gives a contradiction to the desired target semiregularity package.

Therefore:

## Corollary 15.1

Under the finite-stabilizer/Fourier–Mukai hypotheses:

$$
\boxed{
\mathsf{EvenLiftDefect}>0
\Longrightarrow
\text{the candidate seed cannot satisfy the needed relative semiregularity}.
}
$$

This is the **Rank-Diagnostic Contrapositive**.

---

# 16. What a rank defect may mean

If:

$$
\mathsf{EvenLiftDefect}>0,
$$

at least one assumption in the intended proof architecture must fail.

Possible causes:

1. the seed is not semiregular enough;
2. the claimed Hodge-permanence carrier is not the actual deformation carrier of the seed;
3. a factor has positive-dimensional stabilizer and mixed directions were incorrectly discarded;
4. the Fourier–Mukai deformation identification was restricted too severely;
5. the seed architecture simply cannot dominate the target Weil component.

Thus ELR is a diagnostic, not merely a number.

---

# 17. Split-sixfold control

In Markman's successful split-sixfold construction:

$$
\dim_{\mathbb C}T_{\mathrm{Weil}}=9.
$$

The source secant factor satisfies:

$$
\dim\ker ob_{F_i}=9.
$$

The diagonal source lift:

$$
\alpha
\mapsto
\pi_1^\ast\alpha
+
\pi_2^\ast\alpha^\ast
$$

has dimension:

$$
9
$$

and maps under:

$$
\Phi_{HT}
$$

to an unobstructed commutative-gerby subspace whose commutative projection covers the Weil tangent.

Thus:

$$
\boxed{
\mathsf{EvenLiftDefect}=0.
}
$$

This explicit rank computation is a strong control check.

But the final theorem uses semiregularity to propagate algebraicity through the whole connected component.

---

# 18. What the split computation was really doing

The split source-kernel calculation served several purposes simultaneously:

1. it identified a large unobstructed generalized deformation space;
2. it verified the candidate seed deforms in the required directions;
3. it helped establish semiregularity/equivariant semiregularity;
4. it made the period-tangent geometry explicit.

It was therefore a proof mechanism.

It should not be reinterpreted as evidence that every future proof must independently compute an Even Lift Rank after semiregularity is known.

---

# 19. Generalized higher-CM current state

For Markman's generalized construction:

$$
E
=
\Phi(F_1\boxtimes F_2^\vee),
$$

the following are known:

1. the CM/Weil period domain is constructed;
2. the flat deformation of:
   $$
   \kappa(E)
   $$
   remains of Hodge type throughout the relevant Weil component;
3. a genericity criterion ensures a nonzero Weil component of:
   $$
   \kappa(E);
   $$
4. explicit genus-$4$ real-multiplication secant candidates:
   $$
   F_1,F_2
   $$
   are produced in the degree-$4$ CM example.

The missing theorem is appropriate semiregularity.

---

# 20. Markman's actual open question

In the explicit real-quadratic genus-$4$ example, Markman fixes:

$$
F_1
$$

to be an existing:

$$
K_0
$$

-secant sheaf.

Question 11.2.2 asks for a coherent sheaf:

$$
F_2
$$

such that:

1. the Chern-character genericity criterion holds;
2. the semiregularity map is injective on:
   $$
   \boxed{
   \operatorname{Im}
   \left(
   at_{F_2}:
   HT^2(X)
   \to
   \operatorname{Ext}^2(F_2,F_2)
   \right).
   }
   $$

The paper then gives an:

$$
F_2
$$

satisfying the other hypotheses, leaving precisely this injectivity condition unresolved.

---

# 21. Why PT008 prioritizes Question 11.2.2 over ELR

Suppose one succeeds in proving the proposed factor condition and then proves that it supplies the required semiregularity of the target seed:

$$
E.
$$

The generalized paper already supplies Hodge permanence of:

$$
\kappa(E).
$$

Then the semiregularity theorem immediately deforms the target object/class throughout the component.

By Theorem 12.1:

$$
\boxed{
\mathsf{EvenLiftDefect}=0
}
$$

is automatic.

So the logical high-value target is:

$$
\boxed{
\text{image-restricted semiregularity}.
}
$$

Not:

$$
\boxed{
\text{full explicit Even Lift Rank}.
}
$$

---

# 22. Important firewall

PT008 does **not** claim that Markman's Question 11.2.2 alone has already been proved to imply semiregularity of:

$$
E.
$$

That is exactly part of the missing generalized argument.

The correct statement is conditional:

$$
\boxed{
\text{if the factor criterion is completed so as to prove the needed target semiregularity,}
}
$$

then:

$$
\boxed{
\text{tangent coverage requires no independent proof}.
}
$$

This avoids over-reading the current preprint.

---

# 23. Direct target-side strategy

The source factorization is not logically mandatory.

One may instead attack:

$$
E
$$

directly.

Define:

$$
\boxed{
\operatorname{ObIm}_S(E)
=
\operatorname{Im}
\left(
at_E\circ KS_S
\right).
}
$$

It is enough to prove:

$$
\boxed{
\ker\sigma_E
\cap
\operatorname{ObIm}_S(E)
=
0.
}
$$

Since Hodge permanence gives:

$$
\operatorname{ObIm}_S(E)
\subseteq
\ker\sigma_E,
$$

this forces:

$$
\operatorname{ObIm}_S(E)=0.
$$

This bypasses all explicit ELR calculations.

---

# 24. Source-side strategy

If direct:

$$
E
$$

calculations are difficult, use:

$$
G=F_1\boxtimes F_2^\vee
$$

and Fourier–Mukai factorization.

Then:

- PT005 splits the obstruction problem into channels;
- PT006 kills the mixed channel under finite stabilizers;
- factor image-semiregularity may establish target semiregularity;
- ELR may be computed as a consistency check.

This is a computational architecture, not an additional conjectural burden.

---

# 25. Success and failure certificates

The research logic should now distinguish two certificate types.

### Positive certificate

$$
\boxed{
\mathsf{SRSuccess}
}
$$

consists of a proof of the appropriate target/image-restricted semiregularity combined with Hodge permanence.

It directly yields local deformation coverage.

### Negative certificate

$$
\boxed{
\mathsf{RankFailure}
}
$$

consists of an explicit source-kernel computation with:

$$
\mathsf{EvenLiftDefect}>0.
$$

It kills the candidate seed before a full semiregularity proof is attempted.

The two are asymmetrical.

---

# 26. Search-priority theorem

## Theorem 26.1

Assume Hodge permanence of the seed characteristic class is already known.

Then, among the two tasks:

### Task A

Prove the needed image-restricted semiregularity.

### Task B

Compute the full Even Lift Rank.

Task A is logically sufficient for first-order/full local coverage through the semiregularity theorem, while Task B is not independently required after Task A succeeds.

Therefore the optimal positive-proof priority is:

$$
\boxed{
\text{Task A before Task B}.
}
$$

Task B should be done first only when it is substantially cheaper and may falsify the seed.

---

# 27. Maximum-information search policy

PT008 therefore adopts:

$$
\boxed{
\text{cheap ELR falsifier first if available;}
}
$$

otherwise:

$$
\boxed{
\text{attack image-restricted semiregularity directly}.
}
$$

Do not perform a large source-rank computation merely because it is available in principle.

This keeps the HC-True branch focused on the actual unresolved theorem.

---

# 28. Relation to PT003 dominance

Once semiregularity and Hodge permanence yield local deformation:

$$
E_s
$$

over a nonempty open:

$$
U\subseteq S,
$$

the relevant characteristic class:

$$
\kappa(E_s)
$$

is algebraic over:

$$
U.
$$

Markman's semiregularity argument and countable-closed-locus continuation then propagate algebraicity through the connected component in the established setting.

Combined with PT002, one nonzero Weil component closes the full Weil plane.

Thus the chain remains:

$$
\boxed{
\text{semiregularity}
\to
\text{relative algebraic class}
\to
\text{one Weil class}
\to
\text{full Weil plane}.
}
$$

---

# 29. Rank coverage as a theorem consequence

At the tangent level:

$$
\boxed{
\mathsf{SRSuccess}
\Longrightarrow
\mathrm{rank}
\left(
d\operatorname{Def}(A,E)
\to
T_sS
\right)
=
\dim T_sS.
}
$$

Under the outer-product source model and finite-stabilizer hypothesis:

$$
\boxed{
\mathsf{SRSuccess}
\Longrightarrow
\mathrm{ELR}
=
\dim T_{\mathrm{Weil}}.
}
$$

So ELR is an observable shadow of semiregularity success.

---

# 30. Contrapositive as HC-False weapon

The contrapositive is especially useful to the false branch.

If an explicit candidate seed satisfies:

$$
\boxed{
\mathrm{ELR}
<
\dim T_{\mathrm{Weil}},
}
$$

then the required semiregular deformation theorem cannot apply to that seed in the intended architecture.

Thus HC-False can attack positive constructions without disproving HC itself:

$$
\boxed{
\text{rank defect}
\Longrightarrow
\text{candidate proof architecture fails}.
}
$$

This is a clean adversarial interface.

---

# 31. Neutral MLRSC interpretation

MLRSC sees:

$$
\mathrm{ELR}
$$

as a coupling observable between:

- Hodge/period scale;
- legal object-deformation category.

But semiregularity supplies a theorem which forces the coupling map to be surjective whenever the legality certificate is strong enough.

Therefore:

$$
\boxed{
\text{ELR defect}
}
$$

is not a separate metaphysical obstruction.

It is evidence that the proposed legality certificate is incomplete.

This is an important neutral interpretation.

---

# 32. Updated positive gate hierarchy

The HC-True sixfold/generalized strategy should now be ordered as follows.

### Gate 1 — Nonzero Weil projection

Construct:

$$
E
$$

with:

$$
\kappa(E)
$$

having a certified nonzero Weil component.

### Gate 2 — Hodge permanence

Already established for the generalized Markman construction.

### Gate 3 — Relative/image-restricted semiregularity

The principal open gate.

### Gate 4 — Algebraic deformation

Follows from the semiregularity theorem once Gate 3 is met.

### Gate 5 — Binary Weil closure

PT002.

### Gate 6 — Generic all-power closure

PT001 where applicable.

ELR is now an audit between Gates 2 and 3, not an independent Gate 4.

---

# 33. Current strongest unresolved statement

For the generalized genus-$4$ real-multiplication candidate, the clean next question remains:

$$
\boxed{
\ker\sigma_{F_2}
\cap
\operatorname{Im}
\left(
at_{F_2}:
HT^2(X)
\to
\operatorname{Ext}^2(F_2,F_2)
\right)
=
0
\ ?
}
$$

or an even weaker restriction sufficient for the actual Weil deformation subspace.

That is the highest-value next computation.

---

# 34. New possible compression

PT004 already showed that full:

$$
HT^2(X)
$$

image injectivity may still be stronger than necessary.

Let:

$$
D_2^{\mathrm{Weil}}
\subseteq
HT^2(X)
$$

be the actual factor deformation subspace which participates in the target Weil deformation.

Then the true minimal factor gate is only:

$$
\boxed{
\ker\sigma_{F_2}
\cap
at_{F_2}
\left(
D_2^{\mathrm{Weil}}
\right)
=
0.
}
$$

If:

$$
D_2^{\mathrm{Weil}}
$$

can be identified representation-theoretically, the open Markman condition may shrink further.

This is now the next promising reduction.

---

# 35. Status

PT008 proves:

$$
\boxed{
\text{semiregularity + Hodge permanence}
\Longrightarrow
\mathsf{EvenLiftDefect}=0.
}
$$

It proves that ELR is not an independent positive gate after semiregularity.

It preserves ELR as a cheap falsifier.

It reorders the research priority toward image-restricted semiregularity.

Therefore:

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

---

# 36. Next Interface

Next HC-True round:

```text
HODGE_HCTRUE_PT009_MinimalFactorSemiregularity.md
```

Primary target:

$$
\boxed{
\ker\sigma_{F_2}
\cap
at_{F_2}
\left(
D_2^{\mathrm{Weil}}
\right)
\stackrel{?}{=}
0.
}
$$

Planned attacks:

1. identify the actual factor subspace:
   $$
   D_2^{\mathrm{Weil}}
   \subseteq
   HT^2(X);
   $$
2. avoid proving injectivity on all:
   $$
   HT^2(X);
   $$
3. decompose:
   $$
   HT^2(X)
   =
   H^2(\mathcal O_X)
   \oplus
   H^1(T_X)
   \oplus
   H^0(\wedge^2T_X);
   $$
4. determine which summands enter the Weil deformation after Orlov transport;
5. compute:
   $$
   at_{F_2}
   $$
   only on those summands;
6. use the explicit Chern character of Markman's candidate:
   $$
   F_2;
   $$
7. test semiregularity trace nondegeneracy on that restricted image;
8. if injective, attempt to lift the result to target:
   $$
   E;
   $$
9. if a kernel survives, test whether it is irrelevant to:
   $$
   D_2^{\mathrm{Weil}}.
   $$

---

# References

1. E. Markman, *Secant sheaves and Weil classes on abelian varieties*, arXiv:2509.23403, updated version. The survey states the Buchweitz–Flenner semiregularity theorem and uses it to deform a semiregular seed sheaf and its algebraic characteristic classes over the connected Weil moduli component.

2. R.-O. Buchweitz, H. Flenner, *A Semiregularity Map for Modules and Applications to Deformations*, Compositio Math. 137 (2003), 135–210.

3. J. P. Pridham, *Semiregularity as a consequence of Goodwillie's theorem*, arXiv:1208.3111.

4. E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415. In the split-sixfold control case the source-kernel computation produces a $9$-dimensional unobstructed generalized lift covering the Weil deformation tangent.

5. E. Markman, *Secant sheaves on abelian $n$-folds with real multiplication and Weil classes on abelian $2n$-folds with complex multiplication*, arXiv:2509.23079. The generalized construction proves Hodge permanence and leaves semiregularity unresolved; Question 11.2.2 isolates image-restricted semiregularity for the new factor $F_2$.

6. Y. Toda, *Deformations and Fourier-Mukai transforms*, J. Differential Geom. 81 (2009), 197–224.

7. Aletheia, *HODGE_HCTRUE_PT007_MixedChannelAlignment*, 2026-09-16.

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## Canonical Source Declaration

本檔案為 HC-True 分支第八篇正式 UTF-8 Markdown canonical source。

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪沒有證明 generalized candidate 已 semiregular；本輪證明的是 Even Lift Rank 在 successful semiregularity之後自動滿足，因此應作為 diagnostic/falsifier，而非獨立 positive gate。下一步應直接攻 actual Weil factor image 上的 minimal semiregularity。