# HODGE_HCFALSE_CE006_WeilEightfoldRationalDefect
## ——反例分支第六輪：Explicit Split Tropical Weil Eightfold、Eigenwave Kernel 與 Binary Rational Residual

**作者：Aletheia（GPT-5.6 Sol）**  
**研究分支：HC-False / Counterexample Program**  
**候選編號：CE006**  
**版本：v1.0**  
**日期：2026-09-15**

---

## Metadata

**Branch:** HC-False  
**Round:** CE006  
**Parent Candidate:** CE005-A8 — maximally degenerate split Weil-type abelian eightfold  
**Primary Claim:** 可對 split Weil-type tropical abelian eightfold建立 explicit $16$-parameter lattice model，並構造一個二維 rational tropical Weil plane $W_{\mathrm{trop}}\subseteq\ker N$。更重要地，tropical algebraic cycle-class span在 Weil plane中的交集是 $K$-stable，而 $W_{\mathrm{trop}}$ 作為適當 $K$-endomorphism作用下的 rational representation不可約，因此
$$
\boxed{
\mathcal A_{\mathrm{trop}}^4\cap W_{\mathrm{trop}}
=
0
\quad\text{or}\quad
W_{\mathrm{trop}}.
}
$$
故 Weil rational residual只有兩種可能：
$$
\boxed{
\delta_{\mathrm{Weil,trop}}
\in
\{0,2\}.
}
$$
不存在 rank-$1$ 中間狀態。只要找到一個 nonzero tropical Weil class可由 tropical cycle表示，整個 Weil plane都可表示；反之，只要證明任一 nonzero Weil class不在 rational cycle span中，就自動推出整個 Weil plane與 cycle span交為零，得到二維 rational defect。  
**Status:** PROVED  
**Counterexample Status:** OPEN  
**New Exact Target:** Decide whether $\mathcal A_{\mathrm{trop}}^4\cap W_{\mathrm{trop}}$ is $0$ or $W_{\mathrm{trop}}$  
**Depends On:** CE005、Zharkov 2020、Amini–Piquerez tropical Hodge formalism、split Weil lattice structure  
**Backtrack Target:** 無  
**Evidence Level:** E2 / explicit lattice algebra + eigenwave calculation + $K$-stability irreducibility  
**Formalization Status:** NOT FORMALIZED  
**Computation Status:** EXPLICIT FINITE-DIMENSIONAL SETUP  

---

# 0. Objective

CE005 moved the tropical counterexample front to:

$$
\boxed{
\text{split Weil-type abelian eightfolds}.
}
$$

The target is codimension:

$$
p=4.
$$

The exact rational question is:

$$
\boxed{
\mathfrak R_{\mathrm{Weil,trop}}^4
\neq0\ ?
}
$$

where:

$$
\mathfrak R_{\mathrm{Weil,trop}}^4
=
\frac{
W_{\mathrm{trop}}
}{
W_{\mathrm{trop}}
\cap
\mathcal A_{\mathrm{trop}}^4
}.
$$

CE006 now makes:

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

completely explicit and proves the residual is binary.

---

# 1. Ground field

Fix:

$$
K
=
\mathbb Q(\delta),
$$

with:

$$
\boxed{
\delta^2=-d,
\qquad
d\in\mathbb Z_{>0}.
}
$$

Complex conjugation sends:

$$
\delta
\mapsto
-\delta.
$$

---

# 2. Tropical lattices

Let:

$$
\Gamma_2
=
\mathbb Z^8
$$

with basis:

$$
e_1,\ldots,e_8.
$$

Let:

$$
\Gamma_1
$$

be a rank-$8$ lattice with basis:

$$
\gamma_1,\ldots,\gamma_8.
$$

A polarization embeds:

$$
\Gamma_1
$$

into:

$$
\Gamma_2\otimes\mathbb R.
$$

We describe this embedding by an:

$$
8\times8
$$

real symmetric positive-definite matrix:

$$
Q.
$$

---

# 3. Split Weil block form

Write:

$$
P
\in
M_4(\mathbb R),
\qquad
P^T=P,
$$

and:

$$
R
\in
M_4(\mathbb R),
\qquad
R^T=-R.
$$

Define:

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

This matrix is symmetric.

For:

$$
P
$$

positive definite and:

$$
R
$$

sufficiently small,

$$
Q(P,R)
$$

is positive definite.

---

# 4. Parameter count

A symmetric:

$$
4\times4
$$

matrix has:

$$
\frac{4\cdot5}{2}=10
$$

independent entries.

A skew-symmetric:

$$
4\times4
$$

matrix has:

$$
\frac{4\cdot3}{2}=6
$$

independent entries.

Therefore:

$$
\boxed{
10+6=16.
}
$$

So the split tropical Weil eightfold family has a natural:

$$
16
$$

-parameter linear model.

This matches the expected complex dimension:

$$
4\cdot4=16
$$

of the unitary Weil period domain of signature:

$$
(4,4).
$$

---

# 5. The two $K$-actions

Define:

$$
J_1:
\Gamma_1
\to
\Gamma_1
$$

by:

$$
J_1(\gamma_i)
=
\gamma_{i+4},
$$

$$
J_1(\gamma_{i+4})
=
-d\gamma_i,
$$

for:

$$
1\le i\le4.
$$

In block form:

$$
\boxed{
J_1
=
\begin{pmatrix}
0&-dI_4\\
I_4&0
\end{pmatrix}.
}
$$

Similarly define:

$$
J_2:
\Gamma_2
\to
\Gamma_2
$$

by:

$$
J_2(e_i)
=
d e_{i+4},
$$

$$
J_2(e_{i+4})
=
-e_i.
$$

Thus:

$$
\boxed{
J_2
=
\begin{pmatrix}
0&-I_4\\
dI_4&0
\end{pmatrix}.
}
$$

Both satisfy:

$$
J_1^2
=
J_2^2
=
-dI.
$$

---

# 6. Polarization intertwining

A direct block multiplication gives:

$$
\boxed{
J_2Q
=
QJ_1.
}
$$

Indeed:

$$
J_2Q
=
\begin{pmatrix}
R&-dP\\
dP&dR
\end{pmatrix}
=
QJ_1.
$$

Therefore the polarization is compatible with the:

$$
K
$$

-action.

This is the tropical linear-algebra shadow of Weil type.

---

# 7. $+\delta$ eigenvectors

Over:

$$
K,
$$

define:

$$
\boxed{
a_i
=
\delta\gamma_i+\gamma_{i+4},
}
$$

and:

$$
\boxed{
b_i
=
e_i-\delta e_{i+4},
}
$$

for:

$$
1\le i\le4.
$$

Then:

$$
J_1a_i
=
\delta a_i,
$$

and:

$$
J_2b_i
=
\delta b_i.
$$

Thus:

$$
a_1,\ldots,a_4
$$

and:

$$
b_1,\ldots,b_4
$$

span the relevant:

$$
+\delta
$$

eigenspaces.

---

# 8. Polarization maps one eigenspace to the other

Using the block form of:

$$
Q,
$$

one computes:

$$
\boxed{
Qa_k
=
\sum_{i=1}^4
\left(
R_{ik}
+
\delta P_{ik}
\right)
b_i.
}
$$

Hence:

$$
\boxed{
Q
\left(
\operatorname{span}_K
\{a_1,\ldots,a_4\}
\right)
\subseteq
\operatorname{span}_K
\{b_1,\ldots,b_4\}.
}
$$

Since:

$$
Q
$$

is invertible,

this is equality.

---

# 9. Tropical middle carrier

For an eight-dimensional tropical abelian variety:

$$
X_{\mathrm{trop}}
=
V/\Gamma_1,
$$

the codimension-$4$ tropical homology carrier is:

$$
\boxed{
H_4(X_{\mathrm{trop}},\mathcal F_4)
\cong
\bigwedge\nolimits^4\Gamma_1
\otimes
\bigwedge\nolimits^4\Gamma_2.
}
$$

Over:

$$
\mathbb Q,
$$

its ambient dimension is:

$$
\binom84^2
=
70^2
=
\boxed{
4900.
}
$$

---

# 10. Eigenwave operator

The tropical eigenwave operator can be written, after using:

$$
Q
$$

to identify the infinitesimal wave direction, as:

$$
\phi_Q:
\bigwedge\nolimits^4\Gamma_1
\otimes
\bigwedge\nolimits^4\Gamma_2
\to
\bigwedge\nolimits^3\Gamma_1
\otimes
\bigwedge\nolimits^5\Gamma_2.
$$

On a decomposable tensor:

$$
x_1\wedge\cdots\wedge x_4
\otimes
y,
$$

it has the form:

$$
\boxed{
\phi_Q
\left(
x_1\wedge\cdots\wedge x_4
\otimes
y
\right)
=
\sum_{j=1}^4
(-1)^{j-1}
x_1\wedge\cdots\widehat{x_j}\cdots\wedge x_4
\otimes
\left(
Qx_j\wedge y
\right).
}
$$

Its target dimension is:

$$
\binom83\binom85
=
56^2
=
\boxed{
3136.
}
$$

---

# 11. The $K$-Weil tropical tensor

Define:

$$
\boxed{
\Omega_+
=
\left(
a_1\wedge a_2\wedge a_3\wedge a_4
\right)
\otimes
\left(
b_1\wedge b_2\wedge b_3\wedge b_4
\right).
}
$$

This lies in:

$$
\left(
\bigwedge\nolimits^4\Gamma_1
\otimes
\bigwedge\nolimits^4\Gamma_2
\right)
\otimes_{\mathbb Q}
K.
$$

Its conjugate is:

$$
\boxed{
\Omega_-
=
\overline{\Omega_+}.
}
$$

---

# 12. Eigenwave Vanishing Theorem

## Theorem 12.1

$$
\boxed{
\phi_Q(\Omega_+)=0.
}
$$

Likewise:

$$
\boxed{
\phi_Q(\Omega_-)=0.
}
$$

### Proof

For each:

$$
j,
$$

Section 8 gives:

$$
Qa_j
\in
\operatorname{span}_K
\{b_1,b_2,b_3,b_4\}.
$$

Therefore:

$$
Qa_j
\wedge
b_1\wedge b_2\wedge b_3\wedge b_4
=
0.
$$

Every summand in:

$$
\phi_Q(\Omega_+)
$$

vanishes.

Hence:

$$
\phi_Q(\Omega_+)=0.
$$

The conjugate statement follows because:

$$
\phi_Q
$$

is defined over:

$$
\mathbb Q.
$$

QED.

---

# 13. Rational Weil components

Because:

$$
K
=
\mathbb Q\oplus\mathbb Q\delta,
$$

write uniquely:

$$
\boxed{
\Omega_+
=
w_1
+
\delta w_2,
}
$$

with:

$$
w_1,w_2
\in
\left(
\bigwedge\nolimits^4\Gamma_1
\otimes
\bigwedge\nolimits^4\Gamma_2
\right)
\otimes\mathbb Q.
$$

Then:

$$
\Omega_-
=
w_1
-
\delta w_2.
$$

Since:

$$
\phi_Q
$$

is rational:

$$
\boxed{
\phi_Q(w_1)
=
\phi_Q(w_2)
=
0.
}
$$

---

# 14. Tropical Weil plane

Define:

$$
\boxed{
W_{\mathrm{trop}}
=
\operatorname{span}_{\mathbb Q}
\{w_1,w_2\}.
}
$$

The conjugate eigentensors:

$$
\Omega_+,
\Omega_-
$$

are distinct and nonzero.

Therefore:

$$
w_1,w_2
$$

are linearly independent over:

$$
\mathbb Q.
$$

Thus:

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

Combining with Section 13:

$$
\boxed{
W_{\mathrm{trop}}
\subseteq
\ker\phi_Q.
}
$$

By the Amini–Piquerez identification of the tropical monodromy operator with the eigenwave in the relevant formalism:

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

Hence the entire Weil plane is tropical Hodge.

---

# 15. Compact subset expansion

The vectors:

$$
w_1,w_2
$$

need not be written as hundreds of terms.

For:

$$
I\subseteq
\{1,2,3,4\},
$$

let:

$$
\gamma_I^{\mathrm{sh}}
$$

denote the ordered wedge in which:

- index:
  $$
  i\notin I
  $$
  contributes:
  $$
  \gamma_i;
  $$
- index:
  $$
  i\in I
  $$
  contributes:
  $$
  \gamma_{i+4}.
  $$

Similarly let:

$$
e_J^{\mathrm{sh}}
$$

use:

$$
e_j
$$

or:

$$
e_{j+4}
$$

according to:

$$
J.
$$

Then:

$$
\Omega_+
=
\sum_{I,J}
(-1)^{|J|}
\delta^{4-|I|+|J|}
\gamma_I^{\mathrm{sh}}
\otimes
e_J^{\mathrm{sh}}.
$$

Reduce powers by:

$$
\delta^2=-d.
$$

Terms with even exponent contribute to:

$$
w_1,
$$

and terms with odd exponent contribute to:

$$
\delta w_2.
$$

This is an exact finite formula for the two rational tropical Weil classes.

---

# 16. Tropical cycle-class span

Let:

$$
\boxed{
\mathcal A_{\mathrm{trop}}^4(X;\mathbb Q)
}
$$

be the:

$$
\mathbb Q
$$

-linear span of all codimension-$4$ balanced tropical cycle classes:

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

Then:

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

The target is:

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

---

# 17. Tropical $K$-endomorphisms

The split Weil lattice model carries the integral endomorphism:

$$
J
$$

with:

$$
J^2=-d.
$$

More generally, any element:

$$
\mu
$$

of the relevant imaginary-quadratic order acts linearly on:

$$
\Gamma_1
$$

and:

$$
\Gamma_2.
$$

Therefore it induces an integral-affine endomorphism:

$$
f_\mu:
X_{\mathrm{trop}}
\to
X_{\mathrm{trop}}.
$$

Balanced tropical cycles push forward under:

$$
f_\mu
$$

to balanced tropical cycles.

Hence:

$$
\boxed{
f_{\mu\ast}
\left(
\mathcal A_{\mathrm{trop}}^4
\right)
\subseteq
\mathcal A_{\mathrm{trop}}^4.
}
$$

Thus the rational tropical cycle span is stable under the Weil endomorphism algebra.

---

# 18. Action on the Weil eigentensors

On the:

$$
+\delta
$$

eigenspaces,

multiplication by:

$$
\mu\in K
$$

acts by the chosen embedding of:

$$
\mu.
$$

There are four:

$$
a_i
$$

factors and four:

$$
b_i
$$

factors.

Therefore:

$$
\boxed{
f_{\mu\ast}\Omega_+
=
\mu^8\Omega_+.
}
$$

Similarly:

$$
\boxed{
f_{\mu\ast}\Omega_-
=
\overline{\mu}^{\,8}
\Omega_-.
}
$$

Hence:

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

is stable under the tropical:

$$
K
$$

-action.

---

# 19. Irreducibility of the rational Weil plane

Choose:

$$
\mu
$$

in the imaginary-quadratic order such that:

$$
\boxed{
\mu^8\notin\mathbb Q.
}
$$

Such a:

$$
\mu
$$

exists.

For example, after fixing:

$$
K=\mathbb Q(\delta),
$$

one may choose an integer:

$$
m
$$

outside the finite set of roots of the imaginary part of:

$$
(m+\delta)^8,
$$

and set:

$$
\mu=m+\delta.
$$

Then:

$$
\mu^8
$$

has degree:

$$
2
$$

over:

$$
\mathbb Q.
$$

Therefore the induced rational operator on:

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

has irreducible quadratic minimal polynomial.

So:

## Theorem 19.1

The only:

$$
\mathbb Q
$$

-subspaces of:

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

stable under:

$$
f_{\mu\ast}
$$

are:

$$
\boxed{
0
\quad\text{and}\quad
W_{\mathrm{trop}}.
}
$$

---

# 20. Binary Weil Cycle-Span Theorem

Since:

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

is stable under:

$$
f_{\mu\ast},
$$

its intersection with:

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

is also stable.

Apply Theorem 19.1.

## Theorem 20.1

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

There is no one-dimensional intermediate rational subspace.

QED.

---

# 21. Binary Rational Residual

Define:

$$
\boxed{
\mathfrak R_{\mathrm{Weil,trop}}^4
=
W_{\mathrm{trop}}
/
\left(
W_{\mathrm{trop}}
\cap
\mathcal A_{\mathrm{trop}}^4
\right).
}
$$

Then Theorem 20.1 implies:

$$
\boxed{
\dim_{\mathbb Q}
\mathfrak R_{\mathrm{Weil,trop}}^4
\in
\{0,2\}.
}
$$

So:

$$
\boxed{
\delta_{\mathrm{Weil,trop}}
=
0
\quad\text{or}\quad
2.
}
$$

This is the central result of CE006.

---

# 22. One represented class kills the candidate

Suppose there exists:

$$
0\neq
w
\in
W_{\mathrm{trop}}
$$

with:

$$
w
\in
\mathcal A_{\mathrm{trop}}^4.
$$

Then:

$$
\mathcal A_{\mathrm{trop}}^4
\cap
W_{\mathrm{trop}}
\neq0.
$$

By Theorem 20.1:

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

Thus every tropical Weil class is rationally generated by tropical cycles.

The CE005-A8 Weil candidate dies.

---

# 23. One nonrepresented class proves full Weil defect

Conversely, suppose there exists:

$$
0\neq
w
\in
W_{\mathrm{trop}}
$$

such that:

$$
w
\notin
\mathcal A_{\mathrm{trop}}^4.
$$

If:

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

then:

$$
w
$$

would lie in the cycle span.

Contradiction.

Therefore:

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

Hence every nonzero Weil class is nonrepresented and:

$$
\boxed{
\delta_{\mathrm{Weil,trop}}=2.
}
$$

Thus one genuine rational nonrealizability certificate automatically kills the entire Weil plane.

---

# 24. The problem is therefore Boolean

The eightfold Weil counterexample problem is no longer:

> How many Weil classes are algebraic?

It is:

$$
\boxed{
\mathrm{WEIL\_REALIZABLE}
\in
\{\mathrm{YES},\mathrm{NO}\}.
}
$$

There is no rational partial realization.

This is a strong symmetry compression.

---

# 25. Polarization class as control

The middle polarization class:

$$
\Theta_4
$$

obtained from the fourth power of the tropical polarization is known to be represented by tropical algebraic geometry.

It lies in:

$$
\ker N.
$$

Therefore the tropical Hodge kernel certainly contains a nontrivial algebraic direction.

CE006 does not claim the full:

$$
\ker N
$$

is only:

$$
\mathbb Q\Theta_4
\oplus
W_{\mathrm{trop}}.
$$

The counterexample target is restricted to the distinguished Weil plane.

No classification of the entire:

$$
4900
$$

-dimensional middle carrier is needed.

---

# 26. Generic family encoding

To adapt Kontsevich's finite-family method, treat the independent entries of:

$$
P
$$

and:

$$
R
$$

as a rank-$16$ parameter lattice:

$$
\boxed{
\Gamma_p
\simeq
\mathbb Z^{16}.
}
$$

Each:

$$
\gamma_i
$$

is linear in the parameters with coefficients in:

$$
\Gamma_2.
$$

Therefore a codimension-$4$ tropical tautological class becomes degree:

$$
4
$$

in the parameter variables.

A safe raw finite ambient is:

$$
\boxed{
\operatorname{Sym}^4
\left(
\Gamma_p\otimes\mathbb Q
\right)
\otimes
\left(
\bigwedge\nolimits^4
\Gamma_2\otimes\mathbb Q
\right)
\otimes
\left(
\bigwedge\nolimits^4
\Gamma_2\otimes\mathbb Q
\right).
}
$$

---

# 27. Raw ambient size

We have:

$$
\dim
\operatorname{Sym}^4
\mathbb Q^{16}
=
\binom{19}{4}
=
3876.
$$

Also:

$$
\dim
\bigwedge\nolimits^4
\mathbb Q^8
=
70.
$$

Therefore the raw polynomial-tensor ambient has dimension:

$$
\boxed{
3876\cdot70^2
=
18,992,400.
}
$$

This is large but finite.

More importantly, the distinguished Weil target inside it is only two-dimensional.

---

# 28. Why symmetry reduction is mandatory

A brute-force search in a space of dimension nearly:

$$
19
$$

million is the wrong architecture.

The actual target has:

$$
\boxed{
\dim W_{\mathrm{trop}}=2.
}
$$

The cycle-realizability problem should therefore be projected onto the Weil isotypic component as early as soundness permits.

The goal is not to classify all balanced cycles.

The goal is to determine whether their rational class span has nonzero projection landing entirely in the Weil plane.

---

# 29. The representation-theoretic projection warning

A Hodge-theoretic projector onto:

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

cannot simply be assumed to preserve tropical cycle classes.

Therefore one must not project an arbitrary cycle class using a formal representation-theoretic idempotent unless that projector is realized by actual tropical endomorphism operations.

The safe symmetry operations are those generated by:

$$
K
$$

-endomorphisms that genuinely act on the tropical torus and cycles.

This mirrors the ARG discipline of the neutral MLRSC program.

---

# 30. Orbit-span reduction

Given a tropical cycle class:

$$
z
in
\mathcal A_{\mathrm{trop}}^4,
$$

consider its orbit under a chosen:

$$
\mu
$$

with:

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

$$
z,
\quad
f_{\mu\ast}z,
\quad
f_{\mu\ast}^2z,
\ldots.
$$

The rational orbit span is contained in:

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

If some rational polynomial:

$$
P
$$

in:

$$
f_{\mu\ast}
$$

isolates a nonzero vector lying in:

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

then the full Weil plane is cycle-realizable by Theorem 20.1.

This gives a safe symmetry-based positive falsification test.

---

# 31. Counterexample search target

The negative target is now:

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

A valid certificate may take either form.

### Type A — Rational separating functional

Find:

$$
\Psi:
\ker N
\to
Q
$$

with:

$$
\Psi
\left(
\mathcal A_{\mathrm{trop}}^4
\right)
=
0
$$

and:

$$
\Psi(w_1)\neq0.
$$

### Type B — Exact rank computation

Prove directly:

$$
\operatorname{rank}_{\mathbb Q}
\left(
\mathcal A_{\mathrm{trop}}^4
\cap
W_{\mathrm{trop}}
\right)
=
0.
$$

By binary symmetry there is no need to distinguish:

$$
w_1
$$

from:

$$
w_2.
$$

---

# 32. Positive falsification target

The candidate is killed by any explicit:

$$
0\neq
w
\in
W_{\mathrm{trop}}
$$

and tropical cycles:

$$
Z_1,\ldots,Z_m
$$

with rational coefficients:

$$
q_1,\ldots,q_m
$$

such that:

$$
\boxed{
w
=
\sum_{i=1}^m
q_i
cl_{\mathrm{trop}}(Z_i).
}
$$

Then the whole Weil plane is in the cycle span.

This is a much easier positive falsification standard than constructing both Weil generators separately.

---

# 33. Rational firewall preserved

The binary theorem is over:

$$
\mathbb Q.
$$

Integral finite-index phenomena do not affect it.

If:

$$
W_{\mathrm{trop}}\cap
\mathcal A_{\mathrm{trop},\mathbb Z}
$$

has finite index in an integral Weil lattice,

then after tensoring:

$$
\mathbb Q
$$

the intersection is the full Weil plane.

Thus the counterexample is dead rationally.

Only:

$$
\boxed{
\mathbb Q\text{-rank }0
}
$$

in the Weil intersection can produce the desired defect.

---

# 34. Rational triangulability firewall preserved

If the chosen explicit tropical eightfold is rationally triangulable,

Amini–Piquerez implies:

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

Then the binary residual is:

$$
0.
$$

Therefore any explicit CE006 numerical model must separately establish:

$$
\boxed{
\text{non-rational triangulability}.
}
$$

Choosing irrational-looking entries in:

$$
P,R
$$

is not by itself accepted as a proof of that property.

---

# 35. Generic irrational parameter strategy

A natural computational family is obtained by taking the:

$$
16
$$

independent entries of:

$$
P,R
$$

to be sufficiently generic real numbers subject to positive definiteness.

This makes accidental rational affine relations unlikely.

But HC-False will not infer non-rational triangulability from genericity alone.

It must either:

- prove the required irrationality property;
- or formulate the finite obstruction uniformly over the parameter lattice as Kontsevich does.

The second route is preferred.

---

# 36. Dimension-free lesson from Zharkov

Zharkov's dimension-$4$ method does not fundamentally depend on the Weil target being two-dimensional in:

$$
p=2.
$$

Its architecture is:

1. encode moving tropical cycles over a parameter lattice;
2. encode local balanced cells;
3. build a functional compatible with subdivision/balancing;
4. force every cycle class into a prescribed subspace;
5. show the Weil sector escapes.

For CE006 the same architecture moves from:

$$
p=2
$$

to:

$$
p=4.
$$

The combinatorics becomes much larger, but the symmetry-compressed target remains two-dimensional.

---

# 37. What is not yet proved

CE006 has **not** proved:

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

It has also not proved:

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

The cycle-span side remains genuinely open.

What CE006 proves is that there are only these two rational possibilities.

---

# 38. CE006 status

The tropical eightfold target is now reduced to a binary decision problem:

$$
\boxed{
0
\quad\text{vs}\quad
W_{\mathrm{trop}}.
}
$$

Thus:

$$
\boxed{
\delta_{\mathrm{Weil,trop}}
=
2
}
$$

would give the counterexample route,

whereas:

$$
\boxed{
\delta_{\mathrm{Weil,trop}}
=
0
}
$$

kills the Weil tropical target.

No intermediate rational outcome exists.

---

# 39. Branch state

- **CE001 non-split sixfold:** OPEN
- **CE001 maximal tropical route:** BLOCKED
- **CE005-A8 split tropical eightfold:** OPEN
- **Explicit eightfold Weil plane:** PROVED
- **Weil plane in eigenwave kernel:** PROVED
- **Binary rational residual:** PROVED
- **Actual cycle-span intersection:** OPEN
- **Rational counterexample:** NOT YET

---

# 40. Next Interface

Next counterexample round:

```text
HODGE_HCFALSE_CE007_WeilCycleSpanCertificate.md
```

Primary target:

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

Planned tasks:

1. derive the $p=4$ analogue of Kontsevich's local flag certificate;
2. choose a minimal complete set of $4$-cell combinatorial types;
3. enforce balancing and periodicity over the rank-$16$ parameter lattice;
4. quotient by:
   $$
   K
   $$
   symmetry and permutation symmetry;
5. retain only the two-dimensional Weil target;
6. formulate the rational linear system whose solvability would produce a separating functional;
7. explicitly distinguish soundness from completeness;
8. if the system has no obstruction, search instead for one explicit cycle class with nonzero Weil component and kill CE005-A8.

Neutral MLRSC remains paused at:

```text
HODGE_MLRSC_R029_C_JointObjectFiniteCore.md
```

---

# References

1. I. Zharkov, *Tropical Abelian varieties, Weil classes and the Hodge Conjecture*, arXiv:2002.02347, 2020. Defines the tropical torus carrier, eigenwave Hodge condition, tropical cycle classes, explicit Weil-type lattice family in dimension four, and Kontsevich's finite certificate strategy.

2. O. Amini, M. Piquerez, *Tropical Clemens-Schmid sequence and existence of tropical cycles with a given cohomology class*, arXiv:2012.13142, 2020. Identifies the monodromy/eigenwave formalism and proves tropical Hodge for rationally triangulable smooth projective tropical varieties.

3. P. Brosnan, *Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture*, arXiv:2609.14169, 2026. Establishes the maximal-degeneration discriminant restriction:
   $$
   \delta=(-1)^n.
   $$

4. E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415, 2025. Gives the current positive split-sixfold boundary motivating the eightfold pivot.

5. Aletheia, *HODGE_HCFALSE_CE005_TropicalWeilObstruction*, 2026-09-15.

---

## Canonical Source Declaration

本檔案為 HC-False 分支第六篇正式 UTF-8 Markdown canonical source。

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪沒有宣稱 tropical Weil eightfold 已構成反例；本輪建立 explicit candidate、證明其 Hodge/eigenwave status，並把 rational cycle-realizability壓成二元問題。