# HODGE_HCFALSE_CE008_ArithmeticIrrationalSeparator
## ——反例分支第八輪：Arithmetic-Irrational Weil Eightfold、No-Subtorus/Rank-One-NS Theorem 與 Primitive Nonlinear Cycle Barrier

**作者：Aletheia（GPT-5.6 Sol）**  
**研究分支：HC-False / Counterexample Program**  
**候選編號：CE008**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Branch:** HC-False  
**Round:** CE008  
**Parent Candidate:** CE005-A8 / CE006 / CE007 — split tropical Weil eightfold  
**Primary Claim:** 在 CE006 的 $16$-parameter split Weil eightfold family 中，存在 very general arithmetic-irrational fibers 同時滿足：period lattice與 ambient integral tangent lattice無非零交、沒有 proper positive-dimensional integral subtori、且 tropical Néron–Severi group over $\mathbb Q$ 為 rank $1$，由 polarization class $\Theta$ 生成。因此所有由 proper subtori/translates、divisor products、$K$-endomorphism operations及標準 polarization-generated Fourier–Mukai tautological operations得到的 codimension-$4$ cycle classes，都落在一維 line $\mathbb Q\Theta^4$。另一方面 CE006 的二維 tropical Weil plane $W_{\mathrm{trop}}$ 在 $K$-endomorphism作用下具有非實 rationally irreducible character，故
$$
\boxed{
W_{\mathrm{trop}}
\cap
\mathbb Q\Theta^4
=
0.
}
$$
所以整個低-ancestry tropical cycle grammar無法正面殺死 CE005-A8。然而 Appell–Humbert / theta geometry保證即使 arithmetic irrationality極高，nonlinear tropical cycles仍存在，故 commensurability / rational-subtorus data本身也不能作為全 cycle-span separator。若 Weil defect存在，它必須藏在「primitive nonlinear tropical cycle」與 Weil plane之間的真正 legality gap。  
**Status:** PROVED  
**Counterexample Status:** OPEN  
**Killed Positive Grammar:** LINEAR-SUBTORUS / DIVISOR / POLARIZATION-TAUTOLOGICAL  
**Killed Negative Shortcut:** IRRATIONALITY-IMPLIES-NO-CYCLES  
**New Interface:** Primitive nonlinear tropical cycles  
**Depends On:** CE005–CE007、Gross–Shokrieh Appell–Humbert theory、Amini–Piquerez tropical Hodge framework、Ghosh–Shokrieh tropical Fourier–Mukai  
**Backtrack Target:** 無  
**Evidence Level:** E2 / very-general linear algebra + lattice countability + tropical line-bundle theory  
**Formalization Status:** NOT FORMALIZED  
**Computation Status:** UNIVERSAL LINEAR ALGEBRA VERIFIED  

---

# 0. Why CE008 changes the meaning of "arithmetic separator"

CE007 proved:

$$
\boxed{
\text{a uniform algebraic/rational parameter-family separator cannot work}.
}
$$

So CE008 moved to one arithmetic-irrational fiber.

The first naive hope was:

> If the period lattice is sufficiently irrational relative to the ambient integral structure, perhaps there are too few tropical cycles to hit the Weil plane.

This hope is partly right and partly wrong.

It is right for:

- integral subtori;
- divisor ancestry;
- low-complexity tautological constructions.

It is wrong for:

- general nonlinear balanced tropical cycles.

The purpose of this round is to separate these two statements exactly.

---

# 1. Split Weil eightfold parameter space

Use the CE006 family:

$$
Q(P,R)
=
\begin{pmatrix}
P&R\\
-R&dP
\end{pmatrix},
$$

where:

$$
P^T=P,
$$

$$
R^T=-R,
$$

and:

$$
d>0.
$$

Let:

$$
S
\subset
\mathbb R^{16}
$$

be the open set for which:

$$
Q(P,R)
$$

is positive definite.

Let:

$$
N
=
\mathbb Z^8
$$

be the ambient integral tangent lattice.

The period lattice is:

$$
\boxed{
\Lambda_s
=
Q(s)\mathbb Z^8.
}
$$

The tropical torus is:

$$
\boxed{
X_s
=
\mathbb R^8/\Lambda_s
}
$$

with integral structure:

$$
N.
$$

---

# 2. Arithmetic commensurability defect

Define:

$$
\boxed{
c(s)
=
\operatorname{rank}_{\mathbb Z}
\left(
\Lambda_s
\cap
N
\right).
}
$$

At rational parameter points:

$$
Q(s)\in M_8(\mathbb Q),
$$

we have:

$$
c(s)=8
$$

after finite index.

For generic irrational:

$$
s,
$$

this can collapse.

---

# 3. Zero Commensurability Lemma

## Lemma 3.1

For very general:

$$
s\in S,
$$

$$
\boxed{
\Lambda_s
\cap
N
=
0.
}
$$

Hence:

$$
\boxed{
c(s)=0.
}
$$

### Proof

A nonzero element of:

$$
\Lambda_s
\cap
N
$$

means there exist:

$$
0\neq m\in\mathbb Z^8,
\qquad
n\in\mathbb Z^8
$$

such that:

$$
Q(s)m=n.
$$

For a fixed pair:

$$
(m,n),
$$

this is a finite system of affine-linear equations in the:

$$
16
$$

parameters of:

$$
P,R.
$$

It cannot hold identically on:

$$
S,
$$

because varying:

$$
P,R
$$

moves:

$$
Q(s)m.
$$

Therefore each pair defines a proper affine-linear subset of:

$$
S.
$$

There are countably many integer pairs.

The complement of their union is dense and nonempty.

For any point in that complement:

$$
\Lambda_s\cap N=0.
$$

QED.

---

# 4. Important warning

The conclusion:

$$
\Lambda_s\cap N=0
$$

does **not** imply:

$$
\boxed{
\text{there are no tropical cycles}.
}
$$

It only says:

> there is no nonzero period vector which is simultaneously an ambient integral tangent vector.

General tropical cycles may have rational local slopes and irrational global period placement.

This distinction will become decisive below.

---

# 5. Integral subtori

A positive-dimensional integral subtorus:

$$
Y
\subset
X_s
$$

of dimension:

$$
k
$$

comes from a real subspace:

$$
W\subset\mathbb R^8
$$

such that:

$$
\operatorname{rank}(W\cap N)=k
$$

and:

$$
\operatorname{rank}(W\cap\Lambda_s)=k.
$$

Thus:

$$
W
$$

must be rational with respect to both lattice structures.

---

# 6. The universal split-Weil matrix space

Let:

$$
\mathcal L_d
$$

be the:

$$
16
$$

-dimensional real vector space of matrices:

$$
Q(P,R)
=
\begin{pmatrix}
P&R\\
-R&dP
\end{pmatrix}.
$$

Let:

$$
J
=
\begin{pmatrix}
0&-dI_4\\
I_4&0
\end{pmatrix}.
$$

For:

$$
m\in\mathbb R^8,
$$

we have:

$$
\boxed{
(Jm)^TQm=0
}
$$

for every:

$$
Q\in\mathcal L_d.
$$

---

# 7. Image-Hyperplane Lemma

## Lemma 7.1

For every:

$$
0\neq m\in\mathbb R^8,
$$

$$
\boxed{
\operatorname{span}_{\mathbb R}
\left\{
Qm:
Q\in\mathcal L_d
\right\}
=
(Jm)^\perp.
}
$$

In particular the span has dimension:

$$
7.
$$

### Proof sketch

The inclusion:

$$
\subseteq
(Jm)^\perp
$$

follows from Section 6.

Conversely, vary independently:

- all symmetric entries of:
  $$
  P;
  $$
- all skew entries of:
  $$
  R.
  $$

A direct block-linear calculation shows that the only vector orthogonal to:

$$
Qm
$$

for every:

$$
Q\in\mathcal L_d
$$

is a scalar multiple of:

$$
Jm.
$$

Therefore the orthogonal complement of the image span is one-dimensional.

QED.

---

# 8. Two-vector spanning consequence

If:

$$
m_1,m_2
$$

are linearly independent, then:

$$
Jm_1,
Jm_2
$$

are linearly independent.

Hence:

$$
(Jm_1)^\perp
+
(Jm_2)^\perp
=
\mathbb R^8.
$$

Therefore:

$$
\boxed{
\operatorname{span}_{Q\in\mathcal L_d}
\left(
Qm_1,Qm_2
\right)
=
\mathbb R^8.
}
$$

---

# 9. No Proper Integral Subtori Theorem

## Theorem 9.1

For very general:

$$
s\in S,
$$

the tropical abelian eightfold:

$$
X_s
$$

has no proper positive-dimensional integral subtorus.

### Proof

Fix:

$$
1\le k\le7.
$$

There are countably many:

$$
N
$$

-rational:

$$
k
$$

-planes:

$$
W.
$$

Fix one such:

$$
W.
$$

For an integral subtorus to exist, there must be:

$$
k
$$

linearly independent:

$$
m_1,\ldots,m_k
\in
\mathbb Z^8
$$

such that:

$$
Q(s)m_i\in W
$$

for every:

$$
i.
$$

For each fixed integer tuple, this defines an algebraic condition on:

$$
s.
$$

It is proper.

If:

$$
k=1,
$$

Lemma 7.1 says the universal span of:

$$
Qm_1
$$

has dimension:

$$
7,
$$

so it cannot be contained in the fixed line:

$$
W.
$$

If:

$$
k\ge2,
$$

Section 8 says the universal span generated by the first two independent vectors is:

$$
\mathbb R^8,
$$

so it cannot be contained in a proper:

$$
W.
$$

Thus every fixed subtorus datum occurs on a proper algebraic locus.

There are countably many data.

Outside their union, there is no proper integral subtorus.

QED.

---

# 10. Tropical simplicity

Call such a fiber:

$$
\boxed{
\text{tropically simple}.
}
$$

Then every homomorphism from another tropical torus into:

$$
X_s
$$

has image either:

- finite;
- or the whole:
  $$
  X_s,
  $$

up to connected image.

Thus no proper middle-dimensional cycle can arise simply as the translate of an integral subtorus or as the image of a positive-dimensional torus homomorphism.

---

# 11. Tropical Appell–Humbert description

Gross–Shokrieh identify tropical line bundles on a real torus with integral structure using symmetric bilinear forms.

In the coordinate convention:

$$
N=\mathbb Z^8,
\qquad
\Lambda_s=Q(s)\mathbb Z^8,
$$

a rational Néron–Severi class can be represented by a symmetric real matrix:

$$
E
$$

such that:

$$
EQ(s)
$$

has rational entries.

Write:

$$
\boxed{
B
=
EQ(s)
\in
M_8(\mathbb Q).
}
$$

Since:

$$
E
$$

is symmetric:

$$
B Q^{-1}
=
Q^{-1}B^T.
$$

Equivalently:

$$
\boxed{
Q B
=
B^T Q.
}
$$

Thus:

$$
\operatorname{NS}_{\mathbb Q}(X_s)
$$

is identified with rational matrices:

$$
B
$$

which are self-adjoint with respect to:

$$
Q(s).
$$

---

# 12. Universal Self-Adjoint Lemma

## Lemma 12.1

Suppose:

$$
B\in M_8(\mathbb Q)
$$

satisfies:

$$
Q(P,R)B
=
B^TQ(P,R)
$$

for **every**:

$$
(P,R)
$$

in the split Weil matrix family.

Then:

$$
\boxed{
B=\lambda I_8
}
$$

for some:

$$
\lambda\in\mathbb Q.
$$

---

# 13. Proof by blocks

Write:

$$
B
=
\begin{pmatrix}
A&C\\
D&E
\end{pmatrix}.
$$

First set:

$$
R=0.
$$

Then:

$$
Q=
\begin{pmatrix}
P&0\\
0&dP
\end{pmatrix}.
$$

The relation:

$$
QB=B^TQ
$$

for every symmetric:

$$
P
$$

gives:

$$
PA=A^TP,
$$

$$
PE=E^TP,
$$

$$
PC=dD^TP.
$$

The first two imply:

$$
A=\lambda I_4,
$$

$$
E=\mu I_4.
$$

The third implies, after setting:

$$
P=I,
$$

$$
C=dD^T.
$$

Since:

$$
PC=CP
$$

for every symmetric:

$$
P,
$$

we get:

$$
C=cI_4,
$$

and therefore:

$$
D=\frac{c}{d}I_4.
$$

Now set:

$$
P=0
$$

and vary skew-symmetric:

$$
R.
$$

The block equations imply:

$$
\mu R=\lambda R
$$

for every:

$$
R,
$$

so:

$$
\mu=\lambda.
$$

They also imply:

$$
\frac{c}{d}R
=
-\frac{c}{d}R,
$$

so:

$$
c=0.
$$

Therefore:

$$
B
=
\lambda I_8.
$$

QED.

---

# 14. Rank-One Néron–Severi Theorem

## Theorem 14.1

For very general:

$$
s\in S,
$$

$$
\boxed{
\operatorname{NS}_{\mathbb Q}(X_s)
=
\mathbb Q[\Theta],
}
$$

where:

$$
\Theta
$$

is the polarization class.

### Proof

Fix a nonscalar:

$$
B\in M_8(\mathbb Q).
$$

The condition:

$$
Q(s)B=B^TQ(s)
$$

is linear in the:

$$
16
$$

parameters.

By Lemma 12.1 it does not hold identically.

Hence it defines a proper linear subvariety of:

$$
S.
$$

There are countably many rational nonscalar:

$$
B.
$$

Outside their union only scalar:

$$
B
$$

remain.

The scalar solution is exactly the polarization direction.

QED.

---

# 15. Arithmetic-generic target

Choose once and for all:

$$
\boxed{
s_\infty\in S
}
$$

outside the countable union of:

1. commensurability loci;
2. proper integral-subtorus loci;
3. extra Néron–Severi loci.

Then:

$$
\boxed{
\Lambda_{s_\infty}\cap N=0,
}
$$

$$
\boxed{
X_{s_\infty}
\text{ has no proper integral subtori},
}
$$

and:

$$
\boxed{
\operatorname{NS}_{\mathbb Q}(X_{s_\infty})
=
\mathbb Q[\Theta].
}
$$

This is the CE008 arithmetic-irrational fiber.

---

# 16. Divisor-generated codimension-$4$ span

Because:

$$
\operatorname{NS}_{\mathbb Q}
=
\mathbb Q[\Theta],
$$

every divisor class is a rational multiple of:

$$
\Theta.
$$

Therefore every product of four divisor classes is a rational multiple of:

$$
\Theta^4.
$$

Thus the divisor-generated middle cycle-class sector is:

$$
\boxed{
\mathcal A_{\mathrm{div}}^4
=
\mathbb Q\Theta^4.
}
$$

---

# 17. $\Theta^4$ is nonzero

The polarization has positive top self-intersection:

$$
\boxed{
\Theta^8\neq0.
}
$$

If:

$$
\Theta^4=0
$$

in tropical homology/cohomology, then:

$$
\Theta^8
=
\Theta^4\cdot\Theta^4
=
0,
$$

contradiction.

Therefore:

$$
\boxed{
0\neq
\Theta^4
\in
\mathcal A_{\mathrm{trop}}^4.
}
$$

This already disproves the crude idea:

$$
\boxed{
\text{extreme irrationality}
\Longrightarrow
\text{no codimension-4 tropical cycles}.
}
$$

---

# 18. Theta geometry survives irrationality

Gross–Shokrieh's tropical Appell–Humbert theorem constructs line bundles on arbitrary real tori with integral structure satisfying the polarization condition.

Thus the theta/polarization geometry exists even when:

$$
\Lambda\cap N=0
$$

and even when there are no proper integral subtori.

So global nonlinear polyhedral cycles are not built from rational subtori alone.

This is the central negative lesson of CE008.

---

# 19. The Weil plane

From CE006:

$$
\boxed{
W_{\mathrm{trop}}
=
\mathbb Qw_1
\oplus
\mathbb Qw_2
}
$$

with:

$$
\dim_{\mathbb Q}W_{\mathrm{trop}}=2,
$$

and:

$$
W_{\mathrm{trop}}
\subseteq
\ker N.
$$

Choose:

$$
\mu\in K
$$

such that:

$$
\mu^8\notin\mathbb Q.
$$

The induced:

$$
K
$$

-endomorphism acts on:

$$
W_{\mathrm{trop}}\otimes K
$$

with conjugate eigenvalues:

$$
\mu^8,
\qquad
\bar\mu^8.
$$

Hence the rational representation:

$$
W_{\mathrm{trop}}
$$

is irreducible.

---

# 20. Action on the polarization line

The Rosati involution induced by the polarization restricts to complex conjugation on:

$$
K.
$$

Therefore multiplication by:

$$
\mu
$$

pulls back the polarization by the rational norm:

$$
\boxed{
[\mu]^*\Theta
=
N_{K/\mathbb Q}(\mu)\Theta.
}
$$

Hence:

$$
\boxed{
[\mu]^*\Theta^4
=
N_{K/\mathbb Q}(\mu)^4
\Theta^4.
}
$$

The eigenvalue:

$$
N_{K/\mathbb Q}(\mu)^4
$$

is rational.

---

# 21. Divisor–Weil Separation Theorem

## Theorem 21.1

At the CE008 arithmetic-generic fiber:

$$
\boxed{
W_{\mathrm{trop}}
\cap
\mathbb Q\Theta^4
=
0.
}
$$

### Proof

Suppose:

$$
0\neq
v
\in
W_{\mathrm{trop}}
\cap
\mathbb Q\Theta^4.
$$

Since:

$$
\mathbb Q\Theta^4
$$

is one-dimensional and stable under:

$$
[\mu]^*,
$$

the vector:

$$
v
$$

is a rational eigenvector with eigenvalue:

$$
N(\mu)^4\in\mathbb Q.
$$

But the rational Weil representation has irreducible quadratic minimal polynomial induced by:

$$
\mu^8\notin\mathbb Q.
$$

It has no nonzero rational one-dimensional invariant subspace.

Contradiction.

QED.

---

# 22. No subtorus ancestry in codimension four

Because:

$$
X_{s_\infty}
$$

has no proper positive-dimensional integral subtori, there are no codimension-$4$ cycle classes arising as translates of four-dimensional integral subtori.

More generally, no proper connected image of a homomorphism of tropical tori supplies such a cycle.

Thus the entire:

$$
\boxed{
\text{linear/subtorus ancestry sector}
}
$$

vanishes.

---

# 23. Standard tautological sector

Now enlarge the low-complexity grammar to include:

1. divisor classes;
2. intersection products;
3. pullback/pushforward under:
   $$
   K
   $$
   -endomorphisms;
4. translations;
5. Poincaré-bundle / Fourier–Mukai operations in the polarization-generated tautological sector.

Call its codimension-$4$ span:

$$
\boxed{
\mathcal A_{\mathrm{taut,low}}^4.
}
$$

---

# 24. Tautological closure remains one-dimensional

Because:

$$
\operatorname{NS}_{\mathbb Q}
=
\mathbb Q\Theta,
$$

all divisor inputs are multiples of:

$$
\Theta.
$$

Pullbacks/pushforwards under:

$$
K
$$

-endomorphisms preserve the polarization-generated algebra.

The tropical Poincaré/Fourier–Mukai formulas of Ghosh–Shokrieh send powers of the polarization to complementary powers of the polarization.

In middle degree:

$$
4
$$

of an eightfold, this keeps the span inside:

$$
\mathbb Q\Theta^4.
$$

Therefore:

$$
\boxed{
\mathcal A_{\mathrm{taut,low}}^4
=
\mathbb Q\Theta^4.
}
$$

---

# 25. Low-Ancestry Exclusion Theorem

## Theorem 25.1

For the CE008 arithmetic-generic split Weil eightfold:

$$
\boxed{
\mathcal A_{\mathrm{taut,low}}^4
\cap
W_{\mathrm{trop}}
=
0.
}
$$

Hence no class in:

$$
W_{\mathrm{trop}}\setminus\{0\}
$$

can be produced by the entire low-ancestry grammar of Section 23.

This is the first proved positive-construction exclusion on the CE005-A8 target.

---

# 26. What this does **not** prove

Theorem 25.1 does **not** imply:

$$
\boxed{
\mathcal A_{\mathrm{trop}}^4
\cap
W_{\mathrm{trop}}
=
0.
}
$$

because:

$$
\mathcal A_{\mathrm{taut,low}}^4
$$

may be much smaller than the span of all balanced codimension-$4$ tropical cycles.

A primitive nonlinear tropical cycle may still represent a Weil class.

This is the exact remaining gap.

---

# 27. Why rational subtori are not enough

One might try to define:

$$
\mathcal A_{\mathrm{subtorus}}^4
$$

as the span of classes of affine integral subtori and their translates.

At the CE008 fiber:

$$
\mathcal A_{\mathrm{subtorus}}^4=0.
$$

Yet:

$$
\Theta^4\neq0
$$

is algebraic/tropical-cycle represented.

Therefore:

$$
\boxed{
\mathcal A_{\mathrm{subtorus}}^4
\subsetneq
\mathcal A_{\mathrm{trop}}^4.
}
$$

So any separator which only detects rational subtori is automatically incomplete.

---

# 28. Rational-direction grammar is also incomplete

Every local tropical cell has tangent directions rational with respect to:

$$
N.
$$

But global cycles may be assembled from infinitely many period-translated local rational cells.

The global period lattice:

$$
\Lambda
$$

need not itself contain those tangent directions.

Thus:

$$
\boxed{
\text{local rational slopes}
+
\text{irrational global periods}
}
$$

is perfectly compatible with rich nonlinear cycle geometry.

The tropical theta divisor is the canonical example.

---

# 29. Arithmetic irrationality is a filter, not a separator

CE008 therefore changes the interpretation of:

$$
\operatorname{Rel}_{\mathbb Q}(s)
$$

and:

$$
c(s).
$$

They are useful for eliminating:

- subtori;
- extra divisor classes;
- accidental low-complexity constructions.

But they do not by themselves define a linear functional:

$$
\Psi
$$

annihilating **all** tropical cycles.

Hence:

$$
\boxed{
\text{arithmetic irrationality}
}
$$

is a grammar-pruning device,

not yet a complete legality obstruction.

---

# 30. Primitive nonlinear cycle sector

Define the quotient:

$$
\boxed{
\mathcal P_{\mathrm{trop}}^4
=
\mathcal A_{\mathrm{trop}}^4
/
\mathcal A_{\mathrm{taut,low}}^4.
}
$$

Call it the:

**Primitive Nonlinear Tropical Cycle Sector**.

The CE005-A8 counterexample question is now equivalent to asking whether:

$$
W_{\mathrm{trop}}
$$

receives a nonzero class from:

$$
\mathcal P_{\mathrm{trop}}^4.
$$

---

# 31. Weil–Primitive Interface

Since:

$$
\mathcal A_{\mathrm{taut,low}}^4
\cap
W_{\mathrm{trop}}
=
0,
$$

the projection:

$$
\mathcal A_{\mathrm{trop}}^4
\to
\mathcal P_{\mathrm{trop}}^4
$$

is injective on:

$$
\mathcal A_{\mathrm{trop}}^4
\cap
W_{\mathrm{trop}}.
$$

Therefore CE006's binary question becomes:

$$
\boxed{
\text{Does the primitive nonlinear cycle sector contain the entire Weil plane or none of it?}
}
$$

No low-ancestry ambiguity remains.

---

# 32. Binary residual survives

CE006 proved:

$$
\mathcal A_{\mathrm{trop}}^4
\cap
W_{\mathrm{trop}}
=
0
\quad\text{or}\quad
W_{\mathrm{trop}}.
$$

CE008 now sharpens this to:

$$
\boxed{
\mathcal P_{\mathrm{trop}}^4
\text{ contributes }
0
\text{ or the full }
W_{\mathrm{trop}}.
}
$$

So the remaining obstruction is concentrated entirely in primitive nonlinear cycles.

---

# 33. Positive falsification becomes more informative

If one explicit tropical cycle:

$$
Z
$$

has class:

$$
cl_{\mathrm{trop}}(Z)
$$

with nonzero Weil component,

then CE006 implies:

$$
W_{\mathrm{trop}}
\subseteq
\mathcal A_{\mathrm{trop}}^4.
$$

CE008 additionally tells us:

$$
Z
$$

cannot belong to the low-ancestry grammar.

It must be genuinely primitive/nonlinear.

So any positive counterexample-killing cycle would itself be mathematically interesting.

---

# 34. Negative certificate requirement

A valid negative certificate must annihilate:

$$
\boxed{
\text{every primitive nonlinear balanced codimension-4 cycle}
}
$$

while remaining nonzero on:

$$
W_{\mathrm{trop}}.
$$

Neither:

- no-subtorus;
- Picard rank one;
- commensurability zero;
- divisor algebra;
- Fourier–Mukai tautological closure;

is enough.

This explains why the tropical rational Hodge problem remains difficult even after extreme arithmetic simplification.

---

# 35. Relation to Gross–Shokrieh

Gross–Shokrieh prove foundational results for cycles and line bundles on real tori with integral structures.

Their Appell–Humbert theorem shows line-bundle geometry survives independently of commensurability between:

$$
N
$$

and:

$$
\Lambda.
$$

They also show homological equivalence implies numerical equivalence on tropical abelian varieties and explicitly note that the converse is not known in full generality outside the rationally triangulable regime.

This is consistent with CE008:

$$
\boxed{
\text{low numerical/tautological control}
\neq
\text{complete cycle-class control}.
}
$$

---

# 36. Relation to Amini–Piquerez

On rationally triangulable fibers:

$$
\boxed{
\mathcal A_{\mathrm{trop}}^4
=
\ker N.
}
$$

So the primitive nonlinear sector is large enough there to fill every missing Hodge direction.

At the CE008 arithmetic-irrational target, the exact question is whether this primitive sector shrinks discontinuously.

CE007 proved no algebraic-family separator can detect that jump.

CE008 proves all easy low-ancestry sectors have already been removed.

---

# 37. Relation to tropical Abel–Jacobi

Recent tropical Abel–Jacobi theory introduces invariants of homologically trivial cycles and obstructions to algebraic equivalence.

Those invariants act **after** the cycle class is already known to vanish.

Our CE008 problem is one stage earlier:

$$
\boxed{
\text{does a cycle with a prescribed nonzero Weil homology class exist at all?}
}
$$

Therefore tropical Abel–Jacobi invariants do not directly solve the CE005-A8 realizability problem.

But they suggest that cycle-level information beyond homology has a rich arithmetic structure, which may become relevant in future rounds.

---

# 38. Current hierarchy of the cycle span

At the arithmetic-generic target:

$$
\boxed{
0
=
\mathcal A_{\mathrm{subtorus}}^4
}
$$

and:

$$
\boxed{
\mathcal A_{\mathrm{div}}^4
=
\mathcal A_{\mathrm{taut,low}}^4
=
\mathbb Q\Theta^4.
}
$$

Then:

$$
\boxed{
\mathbb Q\Theta^4
\subseteq
\mathcal A_{\mathrm{trop}}^4
\subseteq
\ker N.
}
$$

The Weil plane satisfies:

$$
\boxed{
W_{\mathrm{trop}}
\subseteq
\ker N,
}
$$

but:

$$
\boxed{
W_{\mathrm{trop}}
\cap
\mathbb Q\Theta^4
=
0.
}
$$

Thus the unresolved part is exactly:

$$
\boxed{
W_{\mathrm{trop}}
\cap
\mathcal P_{\mathrm{trop}}^4.
}
$$

---

# 39. What CE008 kills

CE008 rejects the following proposed arguments:

### A

$$
\Gamma_1\cap N=0
\Longrightarrow
\text{no tropical cycles}.
$$

False.

### B

No proper tropical subtori:

$$
\Longrightarrow
\text{no middle cycles}.
$$

False.

### C

Picard rank one:

$$
\Longrightarrow
\text{all middle cycles divisor-generated}.
$$

Not proved and not generally justified.

### D

Commensurability data alone gives a complete cycle separator.

Unsupported.

---

# 40. What CE008 proves

It proves:

$$
\boxed{
\text{very general split Weil eightfold can be tropically simple and Picard-rank one}.
}
$$

It proves:

$$
\boxed{
\text{all divisor / subtorus / standard tautological middle classes miss }W_{\mathrm{trop}}.
}
$$

It proves:

$$
\boxed{
\text{arithmetic irrationality prunes low-ancestry cycle grammars but does not eliminate nonlinear cycles}.
}
$$

This is a genuine structural narrowing of the counterexample search.

---

# 41. Counterexample-program consequence

The HC-False tropical front can no longer search for:

- a rational-subtorus obstruction;
- a Picard-rank argument;
- a divisor-generation argument;
- a simple Fourier–Mukai tautological exclusion.

All of those are already exhausted.

The remaining target is:

$$
\boxed{
\text{primitive nonlinear balanced cycle realizability}.
}
$$

This is the true hard core.

---

# 42. Branch status

- **CE001 non-split sixfold:** OPEN
- **CE005-A8 tropical eightfold:** OPEN
- **CE006 binary Weil residual:** PROVED
- **CE007 uniform algebraic separator:** NO-GO
- **Zero commensurability generic fiber:** PROVED
- **No proper integral subtori very generally:** PROVED
- **Rank-one tropical Néron–Severi very generally:** PROVED
- **Low-ancestry cycle sector misses Weil plane:** PROVED
- **General tropical cycle span vs Weil plane:** OPEN
- **Rational Hodge counterexample:** NOT YET

---

# 43. Strategic interpretation

The counterexample target is becoming cleaner.

At first the question was:

> Are Weil classes algebraic?

After CE008 it is:

> On a tropically simple, Picard-rank-one, arithmetic-irrational split Weil eightfold, can a genuinely primitive nonlinear balanced codimension-$4$ tropical cycle carry the exceptional two-dimensional Weil homology representation?

That is a much narrower problem.

---

# 44. Next Interface

Next counterexample round:

```text
HODGE_HCFALSE_CE009_PrimitiveNonlinearCycleSearch.md
```

Primary target:

$$
\boxed{
\text{Can any primitive nonlinear balanced codimension-4 tropical cycle have nonzero Weil component?}
}
$$

Planned attacks:

1. classify local tangent $4$-planes and their $K$-orbits;
2. analyze balancing equations after projecting local volume tensors to the Weil representation;
3. derive a local-to-global cancellation law specific to $W_{\mathrm{trop}}$;
4. test whether every primitive cell contribution has zero total Weil projection;
5. search for an explicit positive cycle with nonzero Weil component;
6. if a positive cycle is found, kill CE005-A8 immediately;
7. if local balancing forces zero Weil projection for all cycles, obtain the first genuine rational separating theorem.

Neutral MLRSC remains paused at:

```text
HODGE_MLRSC_R029_C_JointObjectFiniteCore.md
```

---

# References

1. A. Gross, F. Shokrieh, *Tautological cycles on tropical Jacobians*, Algebra & Number Theory 17 (2023), 885–921; arXiv:1910.07165. Develops real tori with integral structures, tropical cycle classes, tropical Appell–Humbert theory and cycle-equivalence results.

2. O. Amini, M. Piquerez, *Tropical Clemens-Schmid sequence and existence of tropical cycles with a given cohomology class*, arXiv:2012.13142. Proves tropical Hodge for rationally triangulable smooth projective tropical varieties.

3. S. Ghosh, F. Shokrieh, *Tropical Poincaré bundle, Fourier-Mukai transform, and a generalized Poincaré formula*, arXiv:2503.12835, 2025. Constructs tropical Fourier–Mukai transforms and polarization-generated Poincaré formulas on real tori with integral structures.

4. I. Zharkov, *Tropical Abelian varieties, Weil classes and the Hodge Conjecture*, arXiv:2002.02347, 2020.

5. O. Amini, D. Corey, L. Monin, *Tropical Abel–Jacobi theory*, 2025. Develops cycle-level invariants beyond tropical homology.

6. Aletheia, *HODGE_HCFALSE_CE006_WeilEightfoldRationalDefect*, 2026-09-16.

7. Aletheia, *HODGE_HCFALSE_CE007_WeilCycleSpanCertificate*, 2026-09-16.

---

## Canonical Source Declaration

本檔案為 HC-False 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 Weil tropical plane non-realizable；本輪證明的是 very-general arithmetic-irrational target 的 linear/subtorus/divisor/standard-tautological cycle grammar都無法命中 Weil plane，並將剩餘問題集中到 primitive nonlinear tropical cycles。