# HODGE_HCTRUE_PT007_MixedChannelAlignment
## ——正例證明分支第七輪：Pure-Preimage 修正、Even Lift Rank 與 Generalized Weil-Tangent Lift Criterion

**作者：Aletheia（GPT-5.6 Sol）**  
**研究分支：HC-True / Proof Program**  
**證明目標編號：PT007**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Branch:** HC-True  
**Round:** PT007  
**Parent:** PT005、PT006  
**Primary Claim:** PT006 將 actual Weil tangent alignment暫寫為
$$
D_{\mathrm{Weil}}
=
\Phi_{HT}^{-1}(T_{\mathrm{Weil}})
$$
過於嚴格，因為已成功的 split-sixfold construction 本身使用的是 **commutative-gerby lift**，而不是 pure commutative preimage。正確 first-order object 是一個 source subspace
$$
L\subset HT^2(X\times X)
$$
使
$$
\boxed{
\operatorname{pr}_{\mathrm{comm}}
\circ
\Phi_{HT}|_L
:
L\twoheadrightarrow
T_{\mathrm{Weil}}.
}
$$
若 $G=F_1\boxtimes F_2^\vee$ 的 factor identity-component stabilizers皆有限，PT006 的 mixed-channel injectivity給出
$$
\ker ob_G
\subseteq
C_{20}\oplus C_{02}.
$$
因此任何能讓 $G$ 沿 full Weil tangent deform 的 unobstructed lift都必須從 even obstruction kernel
$$
\boxed{
\mathcal K_{\mathrm{even}}
=
\pi_1^\ast K_1
\oplus
\pi_2^\ast K_2,
\qquad
K_i:=\ker ob_{F_i}\subset HT^2(X)
}
$$
取得，其中第二 factor按 duality involution修正。故 first-order dominance 的 exact necessary-and-sufficient linear criterion是
$$
\boxed{
T_{\mathrm{Weil}}
\subseteq
\operatorname{Im}
\left[
\operatorname{pr}_{\mathrm{comm}}
\circ
\Phi_{HT}
\big|
_{\mathcal K_{\mathrm{even}}}
\right].
}
$$
PT007 定義 **Even Lift Rank**
$$
\mathrm{ELR}(G)
=
\dim_{\mathbb C}
\operatorname{Im}
\left[
\operatorname{pr}_{\mathrm{comm}}
\circ
\Phi_{HT}
\big|
_{\mathcal K_{\mathrm{even}}}
\right]
\cap
T_{\mathrm{Weil}},
$$
以及 stronger full-coverage condition。Split sixfold中 Markman Corollary 8.5.2提供一個 $9$ 維 diagonal even lift，且 commutative projection覆蓋完整 $9$ 維 Weil tangent，因此 split model通過。對 generalized higher-CM construction，現有 paper建立 period domain與 Hodge permanence，但沒有計算 $\mathcal K_{\mathrm{even}}$ 的 commutative image；所以 generalized mixed-channel問題應改寫為 **Even-Lift Coverage**, 而不是要求 pure preimage的 $D_{11}=0$。  
**Status:** PROVED / REFORMULATE PT006 TARGET  
**Supersedes:** PT006 中把 $\Phi_{HT}^{-1}(T_{\mathrm{Weil}})$ 當唯一 alignment object 的過強表述  
**Split Sixfold:** full even lift coverage PROVED  
**Generalized Higher-CM:** Even Lift Coverage OPEN  
**Collision Target:** HC-False 可證 Even Lift Rank 不足以否決 outer-product seed dominance  
**Depends On:** PT005、PT006、Markman 2502.03415 Cor. 8.5.2、Markman 2509.23079 period domain、Toda Fourier–Mukai deformation transport  
**Formalization Status:** NOT FORMALIZED  
**Computation Status:** EXACT FIRST-ORDER LIFT CRITERION  

---

# 0. Why PT006 needs a correction

PT006 introduced the tentative source object:

$$
D_{\mathrm{Weil}}
=
\Phi_{HT}^{-1}
(T_{\mathrm{Weil}}),
$$

where:

$$
T_{\mathrm{Weil}}
\subset
H^1(A,T_A)
$$

is the pure commutative period-domain tangent.

That definition is natural if one insists that a source generalized deformation map to a **pure** commutative deformation.

But the actual semiregularity strategy does not require this.

It allows gerby/twisted deformation.

The successful split-sixfold proof already uses this additional freedom.

---

# 1. Target generalized deformation carrier

For an abelian variety:

$$
A,
$$

the second Hochschild/HKR deformation carrier is:

$$
\boxed{
HT^2(A)
=
H^2(A,\mathcal O_A)
\oplus
H^1(A,T_A)
\oplus
H^0(A,\wedge^2T_A).
}
$$

Interpret the three summands as:

- gerby;
- commutative complex-structure;
- noncommutative/Poisson.

Let:

$$
\boxed{
\operatorname{pr}_{\mathrm{comm}}
:
HT^2(A)
\to
H^1(A,T_A)
}
$$

be the projection to the ordinary complex-deformation component.

---

# 2. Weil tangent

Let:

$$
S
$$

be a Weil period-domain component.

At:

$$
A,
$$

let:

$$
\boxed{
T_{\mathrm{Weil}}
=
T_{[A]}S
\subset
H^1(A,T_A).
}
$$

This is the ordinary period-domain tangent which must be covered in order to dominate:

$$
S.
$$

The deforming object may nevertheless require a gerby companion direction.

---

# 3. Generalized lift

## Definition 3.1

A subspace:

$$
\boxed{
L
\subset
HT^2(X\times X)
}
$$

is a **generalized Weil lift** if:

$$
\boxed{
\operatorname{pr}_{\mathrm{comm}}
\circ
\Phi_{HT}|_L
:
L
\twoheadrightarrow
T_{\mathrm{Weil}}.
}
$$

We do **not** require:

$$
\Phi_{HT}(L)
\subset
H^1(A,T_A).
$$

Its image may lie in:

$$
H^2(\mathcal O_A)
\oplus
H^1(T_A),
$$

or even in the full:

$$
HT^2(A),
$$

provided its commutative projection covers:

$$
T_{\mathrm{Weil}}.
$$

---

# 4. Why gerby lifts are legal

Markman's semiregularity theorem is formulated for:

- coherent sheaves;
- projective bundles;
- twisted sheaves.

The split-sixfold proof explicitly allows the seed to deform along:

$$
\boxed{
\text{commutative-gerby directions}.
}
$$

Thus a nonzero:

$$
H^2(\mathcal O_A)
$$

component is not a defect.

It is an allowed twist of the derived/algebraic object.

Therefore pure commutative lifting is unnecessarily restrictive.

---

# 5. Split model demonstrates the issue

In the original split-sixfold construction, Markman's Corollary 8.5.2 states that:

$$
\Phi_{HT}
\circ
(id\otimes(\bullet)^\ast)
$$

maps the diagonal embedding of a:

$$
9
$$

-dimensional source kernel to a:

$$
9
$$

-dimensional subspace of:

$$
\boxed{
H^1(T_{X\times\hat X})
\oplus
H^2(\mathcal O_{X\times\hat X})
}
$$

contained in the obstruction kernel of:

$$
E.
$$

Its commutative projection supplies the required Weil tangent directions.

So even the known successful example does not justify replacing the lift by:

$$
\Phi_{HT}^{-1}
(T_{\mathrm{Weil}})
$$

with zero gerby component.

---

# 6. PT006 mixed injectivity survives

The correction does **not** invalidate PT006's structural theorem.

Let:

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

Assume:

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

Then:

$$
ob_G
\big|
_{C_{11}}
$$

is injective.

PT005 also proves channel separation.

Therefore:

$$
\boxed{
\ker ob_G
\cap
C_{11}
=
0.
}
$$

Hence:

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

This statement remains correct.

---

# 7. Factor obstruction kernels

Define:

$$
\boxed{
K_1
=
\ker
\left(
ob_{F_1}:
HT^2(X)\to\operatorname{Ext}^2(F_1,F_1)
\right),
}
$$

and:

$$
\boxed{
K_2
=
\ker
\left(
ob_{F_2^\vee}:
HT^2(X)\to\operatorname{Ext}^2(F_2^\vee,F_2^\vee)
\right).
}
$$

Using the duality involution:

$$
(\alpha,\beta,\gamma)^\ast
=
(-\alpha,\beta,-\gamma),
$$

one may equivalently express:

$$
K_2
$$

from the kernel for:

$$
F_2.
$$

---

# 8. Even obstruction kernel

Because the three Ext channels are direct and the mixed map is injective:

$$
\boxed{
\ker ob_G
=
\pi_1^\ast K_1
\oplus
\pi_2^\ast K_2.
}
$$

Define:

$$
\boxed{
\mathcal K_{\mathrm{even}}
=
\pi_1^\ast K_1
\oplus
\pi_2^\ast K_2
\subset
C_{20}\oplus C_{02}.
}
$$

This is the complete first-order unobstructed source carrier under the finite-stabilizer hypothesis.

---

# 9. First-order dominance criterion

The outer-product seed can deform first-order over the full Weil tangent if and only if:

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

This is the correct alignment criterion.

It does not require a pure commutative preimage.

It requires an **unobstructed generalized lift whose commutative shadow is full**.

---

# 10. Even Lift Coverage Theorem

## Theorem 10.1

Assume:

1. $G=F_1\boxtimes F_2^\vee$;
2. the identity-component stabilizers of both factors are trivial;
3. $\Phi$ is the Orlov/Fourier–Mukai equivalence.

Then the following are equivalent at first order.

### A

For every:

$$
v\in T_{\mathrm{Weil}},
$$

there exists:

$$
\tilde v\in HT^2(X\times X)
$$

such that:

$$
ob_G(\tilde v)=0
$$

and:

$$
\operatorname{pr}_{\mathrm{comm}}
\Phi_{HT}(\tilde v)
=
v.
$$

### B

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

### Proof

PT006 gives:

$$
\ker ob_G
=
\mathcal K_{\mathrm{even}}.
$$

Thus every unobstructed source lift belongs to:

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

Condition A is exactly surjectivity of the commutative projection of:

$$
\Phi_{HT}
$$

from that kernel onto:

$$
T_{\mathrm{Weil}}.
$$

QED.

---

# 11. Even Lift Rank

Define the projected image:

$$
\boxed{
\mathcal I_{\mathrm{even}}
=
\operatorname{Im}
\left[
\operatorname{pr}_{\mathrm{comm}}
\circ
\Phi_{HT}
\big|
_{\mathcal K_{\mathrm{even}}}
\right]
\subset
H^1(A,T_A).
}
$$

Define:

$$
\boxed{
\mathrm{ELR}(G;S)
=
\dim_{\mathbb C}
\left(
\mathcal I_{\mathrm{even}}
\cap
T_{\mathrm{Weil}}
\right).
}
$$

This measures how many Weil tangent directions are accessible by unobstructed even lifts.

---

# 12. Rank alone versus full coverage

The condition:

$$
\mathrm{ELR}(G;S)
=
\dim T_{\mathrm{Weil}}
$$

is necessary for full coverage.

It is sufficient if:

$$
\mathcal I_{\mathrm{even}}
\cap
T_{\mathrm{Weil}}
$$

is known to be a subspace of:

$$
T_{\mathrm{Weil}}
$$

of that full dimension.

More safely, define the coverage defect:

$$
\boxed{
\mathfrak D_{\mathrm{EL}}
=
T_{\mathrm{Weil}}
/
\left(
\mathcal I_{\mathrm{even}}
\cap
T_{\mathrm{Weil}}
\right).
}
$$

The first-order positive target is:

$$
\boxed{
\mathfrak D_{\mathrm{EL}}=0.
}
$$

---

# 13. Split-sixfold control

For the original split-sixfold secant construction:

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

Each secant factor has:

$$
\dim K_i=9.
$$

Markman constructs the diagonal source subspace:

$$
\boxed{
L_{\mathrm{split}}
=
\left\{
\pi_1^\ast\alpha
+
\pi_2^\ast\alpha^\ast
:
\alpha\in K_1
\right\}
\subset
\mathcal K_{\mathrm{even}}.
}
$$

It has dimension:

$$
9.
$$

Corollary 8.5.2 gives:

$$
\boxed{
\operatorname{pr}_{\mathrm{comm}}
\Phi_{HT}(L_{\mathrm{split}})
=
T_{\mathrm{Weil}}.
}
$$

Therefore:

$$
\boxed{
\mathfrak D_{\mathrm{EL}}=0
}
$$

for the split sixfold.

This validates the reformulated criterion against the known successful proof.

---

# 14. Why the old pure-preimage target was too rigid

The image:

$$
\Phi_{HT}(L_{\mathrm{split}})
$$

lies in:

$$
\boxed{
H^1(T_A)
\oplus
H^2(\mathcal O_A).
}
$$

Thus a typical lifted direction may be:

$$
\boxed{
v+\beta,
\qquad
v\in T_{\mathrm{Weil}},
\quad
\beta\in H^2(\mathcal O_A),
}
$$

rather than:

$$
v
$$

alone.

Requiring:

$$
\Phi_{HT}(\tilde v)=v
$$

would discard a lift that the actual twisted-sheaf proof legitimately uses.

So:

$$
\boxed{
\text{pure-preimage alignment}
}
$$

must be replaced by:

$$
\boxed{
\text{projected generalized-lift coverage}.
}
$$

---

# 15. Reformulation of mixed-channel question

PT006 asked:

$$
\operatorname{pr}_{11}
\left(
\Phi_{HT}^{-1}
(T_{\mathrm{Weil}})
\right)
\stackrel{?}{=}0.
$$

PT007 replaces this by the sound question:

> Does the unobstructed **even** source carrier possess a generalized lift whose commutative projection covers the Weil tangent?

That is:

$$
\boxed{
\mathfrak D_{\mathrm{EL}}
\stackrel{?}{=}0.
}
$$

The mixed channel is already excluded from the unobstructed source kernel under finite stabilizers.

No statement about the pure inverse image is required.

---

# 16. Target dimension in general Weil type

For a polarized abelian variety of Weil type with:

$$
e=[K:\mathbb Q],
$$

and:

$$
d=\dim_KH^1(A,\mathbb Q),
$$

the complex dimension of the polarized Weil moduli component is:

$$
\boxed{
\dim_{\mathbb C}T_{\mathrm{Weil}}
=
\frac{ed^2}{8}.
}
$$

This is the correct rank target for:

$$
\mathrm{ELR}.
$$

---

# 17. Quartic-CM genus-$4$ example

In Markman's explicit higher-CM example:

$$
e=4,
$$

$$
d=4.
$$

Hence:

$$
\boxed{
\dim_{\mathbb C}T_{\mathrm{Weil}}
=
\frac{4\cdot16}{8}
=
8.
}
$$

Thus the first-order generalized positive target is only:

$$
\boxed{
8
}
$$

commutative directions.

There is no raw dimension obstruction to covering these directions from the two even factor kernels.

But the required image calculation has not been carried out.

---

# 18. Period domain information is not enough

The generalized paper proves that the period domain is an adjoint orbit:

$$
\Omega_B
$$

of:

$$
{\rm Spin}(V_{\mathbb R})_B
$$

and decomposes the real symmetry group as a product of special unitary factors over the real embeddings of:

$$
F.
$$

This determines:

- the target period geometry;
- the target tangent representation;
- Hodge permanence of:
  $$
  \kappa(E).
  $$

It does **not** determine:

$$
\boxed{
\mathcal I_{\mathrm{even}}
=
\operatorname{Im}
\left(
\operatorname{pr}_{\mathrm{comm}}
\Phi_{HT}
|
_{\mathcal K_{\mathrm{even}}}
\right).
}
$$

The latter depends on the actual factor obstruction kernels:

$$
K_1,
K_2.
$$

So group-theoretic period-domain factorization cannot substitute for deformation calculation of the seed.

---

# 19. Why Spin equivariance alone does not prove lift coverage

Orlov's cohomological equivalence is Spin-equivariant.

However the continuous real group:

$$
{\rm Spin}(V_{\mathbb R})_B
$$

governing the period orbit is not the identity component of the derived autoequivalence group of:

$$
X.
$$

Its Lie algebra therefore cannot simply be identified with:

$$
HT^2(X)
$$

factorwise.

Consequently, differentiating Spin-equivariance does not automatically construct an unobstructed source deformation.

This blocks an otherwise tempting but invalid shortcut.

---

# 20. Correct use of Orlov's explicit Hochschild transform

What is needed is the actual linear map:

$$
\boxed{
\operatorname{pr}_{\mathrm{comm}}
\circ
\Phi_{HT}
:
C_{20}\oplus C_{02}
\to
H^1(A,T_A).
}
$$

The original split paper computes this map explicitly and uses a diagonal:

$$
9
$$

-dimensional kernel.

For the higher-CM candidate, the same universal transform exists.

The new unknown is not the formula for:

$$
\Phi_{HT}
$$

itself.

It is the position of:

$$
K_1\oplus K_2
$$

inside its source.

---

# 21. Factor-kernel formulation

Let:

$$
P_1
:
K_1\to H^1(A,T_A)
$$

be the commutative projection of:

$$
\Phi_{HT}
$$

on the first even factor.

Let:

$$
P_2
:
K_2\to H^1(A,T_A)
$$

be the corresponding map on the second even factor.

Then:

$$
\boxed{
\mathcal I_{\mathrm{even}}
=
P_1(K_1)+P_2(K_2).
}
$$

Thus:

$$
\boxed{
\mathfrak D_{\mathrm{EL}}=0
}
$$

is equivalent to:

$$
\boxed{
T_{\mathrm{Weil}}
\subseteq
P_1(K_1)+P_2(K_2).
}
$$

This is the simplest computational form of the positive gate.

---

# 22. Old versus new factor

In the generalized secant program:

$$
F_1
$$

is chosen from an already understood secant construction.

The new search concerns:

$$
F_2.
$$

Therefore one should first compute:

$$
\boxed{
P_1(K_1)\cap T_{\mathrm{Weil}}.
}
$$

The missing tangent directions form:

$$
\boxed{
Q_{\mathrm{miss}}
=
T_{\mathrm{Weil}}
/
\left(
P_1(K_1)\cap T_{\mathrm{Weil}}
\right).
}
$$

The sole responsibility of the new factor is then to cover:

$$
Q_{\mathrm{miss}}
$$

through:

$$
P_2(K_2).
$$

This can be much smaller than all of:

$$
T_{\mathrm{Weil}}.
$$

---

# 23. Minimal new-factor rank

Define:

$$
r_1
=
\dim
\left(
P_1(K_1)\cap T_{\mathrm{Weil}}
\right).
$$

Then any viable:

$$
F_2
$$

must contribute at least:

$$
\boxed{
\dim T_{\mathrm{Weil}}-r_1
}
$$

new independent Weil directions modulo the old image.

For the quartic-CM example:

$$
\boxed{
8-r_1
}
$$

is the minimal required new-factor contribution.

This is a concrete rank target for candidate construction.

---

# 24. Relation to semiregularity

The map:

$$
K_2
=
\ker ob_{F_2}
$$

is exactly where Markman's image-restricted semiregularity condition becomes useful.

Hodge permanence gives a cohomological annihilator of the target characteristic class.

If the semiregularity map is injective on the relevant Atiyah/Hochschild image, the Hodge-preserving deformation directions belong to:

$$
K_2.
$$

PT007 then asks whether their commutative Orlov image covers:

$$
Q_{\mathrm{miss}}.
$$

So semiregularity and tangent alignment are separate gates:

$$
\boxed{
\text{kernel membership}
}
$$

versus:

$$
\boxed{
\text{period-tangent coverage}.
}
$$

---

# 25. First-order positive certificate

Define:

$$
\boxed{
\mathsf{EvenLiftCert}(F_1,F_2;S)
}
$$

to consist of:

1. finite factor stabilizers;
2. explicit:
   $$
   K_1,K_2;
   $$
3. explicit linear maps:
   $$
   P_1,P_2;
   $$
4. a proof:
   $$
   T_{\mathrm{Weil}}
   \subseteq
   P_1(K_1)+P_2(K_2).
   $$

Then:

$$
\boxed{
\mathsf{EvenLiftCert}
\Longrightarrow
\text{first-order deformability over all Weil tangent directions}.
}
$$

Higher-order formal smoothness remains a separate PT003 gate.

---

# 26. First-order negative certificate

Define:

$$
\boxed{
\mathsf{EvenLiftDefect}
=
\operatorname{codim}_{T_{\mathrm{Weil}}}
\left[
\left(
P_1(K_1)+P_2(K_2)
\right)
\cap
T_{\mathrm{Weil}}
\right].
}
$$

If:

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

then the chosen outer-product seed cannot dominate the full Weil component, regardless of how well its semiregularity behaves on the directions it already deforms.

This gives HC-False an exact seed-killing certificate.

---

# 27. Split control value

For the successful split sixfold:

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

The diagonal:

$$
9
$$

-dimensional lift provides full coverage.

This is a required validation of the invariant.

---

# 28. Generalized current status

For the generalized higher-CM construction, current published/preprint results establish:

- the CM/Weil embedding;
- the period domain:
  $$
  \Omega_B;
  $$
- Hodge permanence of:
  $$
  \kappa(E);
  $$
- a criterion for nonzero Weil projection.

They do not establish:

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

They also do not establish generalized semiregularity.

Therefore both:

$$
\boxed{
\text{Even Lift Coverage}
}
$$

and:

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

remain independent open gates.

---

# 29. Updated 2026 survey signal

Markman's updated ICM survey emphasizes a weaker semiregularity question:

> whether injectivity of the semiregularity map on the image of the Hochschild evaluation map suffices.

It also explicitly notes that full semiregularity can fail in larger constructions even when the semiregularity map remains injective on the invariant obstruction image relevant to deformation.

This strongly supports the PT004–PT007 philosophy:

$$
\boxed{
\text{do not solve all Ext}^{2};
\text{ solve only the actual deformation image}.
}
$$

PT007 applies the same principle to tangent coverage.

---

# 30. Literature firewall

A recent non-peer-reviewed preprint publicly claims a general proof of the Hodge conjecture via "relative secant cycles."

Because that claim is far stronger than the established literature and has not been independently validated, PT007 does **not** use it as a theorem or shortcut.

HC-True continues from verified secant/Fourier–Mukai and semiregularity results.

This prevents a public claim from being mistaken for settled mathematics.

---

# 31. Correction to PT006 terminology

PT006's theorem:

$$
\ker ob_G
\cap
C_{11}
=
0
$$

remains valid under its hypotheses.

The corrected statement is:

> any **unobstructed generalized lift** of Weil tangent directions must be even-channel.

We withdraw the stronger wording that the pure inverse image:

$$
\Phi_{HT}^{-1}(T_{\mathrm{Weil}})
$$

itself must be even.

The latter is unnecessary and may fail because valid lifts can carry gerby components.

---

# 32. Revised alignment hierarchy

The alignment problem now has three levels.

### Pure alignment

$$
\Phi_{HT}^{-1}(T_{\mathrm{Weil}})
\subseteq
C_{20}\oplus C_{02}.
$$

Too strong and unnecessary.

### Generalized even lift

There exists:

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

such that:

$$
\operatorname{pr}_{\mathrm{comm}}\Phi_{HT}(L)
=
T_{\mathrm{Weil}}.
$$

This is the correct first-order target.

### Formal/algebraic dominance

The even lift extends through all higher obstruction levels and algebraizes.

This is PT003's final deformation gate.

---

# 33. What PT007 proves

PT007 proves the corrected first-order criterion:

$$
\boxed{
T_{\mathrm{Weil}}
\subseteq
P_1(K_1)+P_2(K_2).
}
$$

It proves this criterion is equivalent to existence of an unobstructed generalized lift under the finite-stabilizer hypotheses.

It validates the criterion on the split sixfold.

It defines:

$$
\boxed{
\mathrm{ELR}
}
$$

and:

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

It separates tangent coverage from semiregularity and higher-order effectivity.

---

# 34. What PT007 does not prove

PT007 does not compute:

$$
K_1,K_2
$$

for the generalized higher-CM candidate.

It does not prove:

$$
P_1(K_1)+P_2(K_2)
$$

covers the full generalized Weil tangent.

It does not solve Markman's image-restricted semiregularity question.

It therefore does not yet establish generalized algebraicity.

The next bottleneck is now a finite rank/image computation.

---

# 35. Triple-program interface

### HC-True

Wants:

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

### HC-False

Can kill a proposed seed by proving:

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

### Neutral MLRSC

Sees the defect as:

$$
\boxed{
\text{Hodge/period tangent}
\setminus
\text{legally deformable seed tangent}.
}
$$

This is an exact Phase-III category coupling quantity.

---

# 36. Next Interface

Next HC-True round:

```text
HODGE_HCTRUE_PT008_EvenLiftRank.md
```

Primary target:

$$
\boxed{
T_{\mathrm{Weil}}
\subseteq
P_1(K_1)+P_2(K_2)
\ ?
}
$$

for the explicit generalized secant candidates.

Planned attacks:

1. write the universal dimension-independent matrix of:
   $$
   \operatorname{pr}_{\mathrm{comm}}\circ\Phi_{HT}
   $$
   on:
   $$
   C_{20}\oplus C_{02};
   $$
2. compute:
   $$
   P_1(K_1)
   $$
   for the old secant factor;
3. determine:
   $$
   r_1
   =
   \dim(P_1(K_1)\cap T_{\mathrm{Weil}});
   $$
4. identify the missing quotient:
   $$
   Q_{\mathrm{miss}};
   $$
5. translate Markman's $F_2$ semiregularity condition into a lower bound on:
   $$
   P_2(K_2);
   $$
6. test whether the quartic-CM example can reach all:
   $$
   8
   $$
   Weil tangent directions;
7. if the rank is insufficient, reject that seed;
8. if full rank is achieved, advance to higher-order formal smoothness.

---

# References

1. E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415, especially Sections 8.4–8.5. Corollary 8.5.2 gives a $9$-dimensional source kernel whose Orlov Hochschild transform lies in
   $$
   H^1(T_{X\times\hat X})
   \oplus
   H^2(\mathcal O_{X\times\hat X})
   $$
   and supplies the split Weil deformation directions.

2. E. Markman, *Secant sheaves on abelian $n$-folds with real multiplication and Weil classes on abelian $2n$-folds with complex multiplication*, arXiv:2509.23079. Constructs the generalized CM period domain and proves Hodge permanence, but does not establish generalized semiregularity or the factor-kernel tangent coverage calculation.

3. E. Markman, *Secant sheaves and Weil classes on abelian varieties*, arXiv:2509.23403, updated 2026. The ICM survey emphasizes image-restricted semiregularity rather than full Ext² injectivity in higher constructions.

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

5. Aletheia, *HODGE_HCTRUE_PT006_WeilTangentChannelDecomposition*, 2026-09-16.

---

## Canonical Source Declaration

本檔案為 HC-True 分支第七篇正式 UTF-8 Markdown canonical source。

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪修正 PT006 的 pure-preimage alignment 表述；保留 mixed-channel injectivity theorem，並把真正 first-order dominance gate重寫為 unobstructed even generalized lift 對 Weil commutative tangent 的 full coverage。