# HODGE_HCFALSE_CE005_TropicalWeilObstruction
## ——反例分支第五輪：Maximal-Degeneration Accessibility、Rational Tropical Defect 與 Weil Eightfold Pivot

**作者：Aletheia（GPT-5.6 Sol）**  
**研究分支：HC-False / Counterexample Program**  
**候選編號：CE005**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Branch:** HC-False  
**Round:** CE005  
**Parent Candidate:** CE001 — Non-Split Weil-Type Abelian Sixfold  
**Primary Claim:** Kontsevich–Zharkov 的 maximal tropical degeneration 路線不能用來攻 CE001 的 non-split Weil sixfold。Patrick Brosnan 2026 的最新結果指出，Weil-type abelian $2n$-fold 只有在 normalized Hermitian discriminant 為 $(-1)^n$ 時才能有 maximal unipotent degeneration。對 sixfold，$n=3$，故 maximal-degeneration route 只能到達 $\delta=-1$；而此 sixfold regime 的 Weil classes 已被 Markman 證明 algebraic。因此 CE001 的 tropical maximal-degeneration subroute 被 accessibility gate 關閉。另一方面，Amini–Piquerez 已證 tropical Hodge conjecture對 rationally triangulable smooth projective tropical varieties成立，因此任何仍可能產生 rational counterexample 的 Kontsevich tropical target必須同時是 non-rationally-triangulable。再者，classical Hodge conjecture採 $\mathbb Q$ 係數，所以 tropical obstruction 必須造成正的 rational rank defect；proper finite-index sublattice不足以反駁 rational HC。綜合三個 gates，這條方法目前第一個自然未封閉 Weil window 移到 split Weil-type abelian eightfold：$2n=8$、$n=4$、$\delta=+1$。  
**Status:** PROVED  
**CE001 Status:** OPEN, BUT TROPICAL-MAXIMAL SUBROUTE BLOCKED  
**New Tropical Candidate:** CE005-A8 — maximally degenerate split Weil-type abelian eightfold  
**Primary Open Quantity:** Rational Tropical Weil Residual  
**Depends On:** CE001–CE004、Zharkov 2020、Amini–Piquerez 2020、Markman 2025、Brosnan 2026  
**Backtrack Target:** 無  
**Evidence Level:** E2 / route-domain theorem + rationalization theorem  
**Formalization Status:** NOT FORMALIZED  
**Computation Status:** TARGET RECLASSIFICATION / RANK-DEFECT SETUP  

---

# 0. Why CE005 changes target instead of forcing CE001

CE001 chose a very general non-split Weil-type abelian sixfold:

$$
A,
\qquad
\dim A=6,
$$

with:

$$
0\neq
\alpha
\in
W_K(A)
\subset
H^6(A,\mathbb Q)
\cap
H^{3,3}(A).
$$

The tropical route was intended as an independent attack:

$$
\boxed{
\text{classical Weil class}
\to
\text{maximal degeneration}
\to
\text{tropical Weil class}
\to
\text{tropical non-realizability}.
}
$$

CE005 discovers that the first arrow is not available for the non-split sixfold target.

Therefore we do not force the method onto CE001.

We reclassify the method's actual reachable domain.

---

# 1. Zharkov's tropical setup

Let:

$$
X
=
V/\Gamma_1
$$

be a principally polarized tropical abelian variety of dimension:

$$
g.
$$

There are two lattices:

$$
\Gamma_1,
\Gamma_2
\subset
V
\simeq
\mathbb R^g,
$$

and a polarization isomorphism:

$$
Q:
\Gamma_2^\vee
\to
\Gamma_1.
$$

The symmetric positive-definite form:

$$
Q
$$

makes:

$$
X
$$

a tropical principally polarized abelian variety.

Zharkov's maximal-degeneration interpretation says such an:

$$
X
$$

appears as the tropical limit of a family of complex abelian varieties in which half of the first-homology cycles vanish.

This is a maximal degeneration of the abelian variety.

---

# 2. Tropical cohomological carrier

For such a tropical torus, the tropical homology coefficient cosheaf is constant:

$$
\mathcal F_p
\simeq
\bigwedge\nolimits^p\Gamma_2.
$$

Hence:

$$
\boxed{
H_q(X,\mathcal F_p)
\simeq
\bigwedge\nolimits^q\Gamma_1
\otimes
\bigwedge\nolimits^p\Gamma_2.
}
$$

In middle bidegree:

$$
p=q,
$$

the tropical Hodge classes are the kernel of the eigenwave / tropical monodromy operator.

Write:

$$
\boxed{
\mathcal H_{\mathrm{trop}}^p(X;\mathbb Q)
=
\ker
\left(
N:
H_{\mathrm{trop}}^{p,p}(X,\mathbb Q)
\to
H_{\mathrm{trop}}^{p-1,p+1}(X,\mathbb R)
\right).
}
$$

Equivalent homological conventions can be used.

---

# 3. Tropical algebraic-cycle span

Let:

$$
Z_{\mathrm{trop}}^p(X)
$$

denote codimension-$p$ balanced tropical cycles.

Their cycle classes span:

$$
\boxed{
\mathcal A_{\mathrm{trop}}^p(X;\mathbb Q)
=
\operatorname{Span}_{\mathbb Q}
\left\{
cl_{\mathrm{trop}}(Z):
Z\in Z_{\mathrm{trop}}^p(X)
\right\}.
}
$$

By construction:

$$
\mathcal A_{\mathrm{trop}}^p
\subseteq
\mathcal H_{\mathrm{trop}}^p.
$$

---

# 4. Rational Tropical Residual

Define:

$$
\boxed{
\mathfrak R_{\mathrm{trop}}^p(X)
=
\mathcal H_{\mathrm{trop}}^p(X;\mathbb Q)
/
\mathcal A_{\mathrm{trop}}^p(X;\mathbb Q).
}
$$

and:

$$
\boxed{
\delta_{\mathrm{trop}}^p(X)
=
\dim_{\mathbb Q}
\mathfrak R_{\mathrm{trop}}^p(X).
}
$$

Then:

$$
\boxed{
\delta_{\mathrm{trop}}^p(X)>0
}
$$

is the exact kind of tropical defect relevant to the rational Hodge conjecture.

---

# 5. Zharkov specialization principle

In Zharkov's setup, if a tropical Hodge class on a maximal tropical degeneration is not represented by tropical algebraic cycles, then a generic member of the corresponding complex abelian family yields a classical Hodge counterexample.

Thus, under the specialization setup:

$$
\boxed{
\delta_{\mathrm{trop}}^p(X)>0
\Longrightarrow
\text{generic classical fiber violates rational HC}.
}
$$

The converse need not hold.

So tropical failure is a sufficient but not necessary classical counterexample mechanism.

---

# 6. The first firewall: maximal degeneration accessibility

A tropical torus of the Kontsevich–Zharkov type is not an arbitrary boundary object.

It comes from a maximal degeneration.

For a Weil-type family, this imposes arithmetic restrictions on the Hermitian form.

---

# 7. Brosnan 2026 theorem

Patrick Brosnan's September 2026 note establishes:

$$
\boxed{
\text{A family of Weil-type abelian }2n\text{-folds can have maximal unipotent degeneration only if }
\delta=(-1)^n.
}
$$

Here:

$$
\delta
$$

is the normalized Hermitian discriminant class in:

$$
\mathbb Q^\times/
N_{K/\mathbb Q}(K^\times).
$$

This is exactly the arithmetic datum isolated in CE003.

---

# 8. Maximal-Degeneration Accessibility Gate

Define:

$$
\boxed{
\mathrm{MDAG}(A)
=
1
}
$$

if the Weil moduli component containing:

$$
A
$$

admits the maximal unipotent degeneration required by the Kontsevich tropical method.

Then Brosnan gives the necessary condition:

$$
\boxed{
\mathrm{MDAG}(A)=1
\Longrightarrow
\delta(A)=(-1)^n.
}
$$

CE005 calls this the:

**Maximal-Degeneration Accessibility Gate**.

---

# 9. Sixfold consequence

For an abelian sixfold:

$$
2n=6,
$$

so:

$$
n=3.
$$

Therefore:

$$
\boxed{
\mathrm{MDAG}=1
\Longrightarrow
\delta=-1.
}
$$

But CE001 deliberately targets:

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

Hence:

## Theorem 9.1

The Kontsevich–Zharkov maximal-degeneration tropical route cannot directly reach the CE001 non-split sixfold component.

Symbolically:

$$
\boxed{
\mathrm{CE001}
\cap
\mathrm{MDAG}
=
\varnothing.
}
$$

---

# 10. Markman closes the accessible sixfold regime

Markman proves:

$$
\boxed{
\text{Weil classes are algebraic for every abelian sixfold of Weil type with }\delta=-1
}
$$

for every imaginary quadratic field:

$$
K.
$$

Therefore the exact sixfold regime that passes MDAG is already positive.

Combining Brosnan and Markman:

## Theorem 10.1

Within Weil-type abelian sixfolds, the maximal-degeneration tropical route cannot produce a rational Hodge counterexample.

### Proof

Brosnan restricts maximal degeneration to:

$$
\delta=-1.
$$

Markman algebraizes the Weil classes throughout that regime.

Therefore a tropical failure compatible with specialization cannot occur there in a way that would contradict the proven classical algebraicity.

QED.

---

# 11. This does not kill CE001

The conclusion is not:

$$
\boxed{
\text{CE001 is false}.
}
$$

It is:

$$
\boxed{
\text{CE001 cannot be attacked by this maximal tropical degeneration route}.
}
$$

CE001 still has independent possible routes:

- TDVD;
- Chow-level ancestry obstruction;
- coniveau;
- non-maximal degeneration methods;
- arithmetic cycle obstructions.

Only the tropical-maximal subroute is blocked.

---

# 12. Second firewall: rational triangulability

Amini and Piquerez prove the tropical Hodge conjecture for smooth projective tropical varieties that are rationally triangulable.

Thus if:

$$
X
$$

is rationally triangulable:

$$
\boxed{
\mathfrak R_{\mathrm{trop}}^p(X)=0
}
$$

for every:

$$
p.
$$

Therefore no rational tropical Hodge counterexample can live there.

---

# 13. Why this does not kill Kontsevich's original problem

Amini–Piquerez explicitly note that Kontsevich's original problem concerns tropical abelian varieties which are in general not rationally triangulable.

So their theorem is a major firewall but not a complete closure.

The surviving target region is:

$$
\boxed{
\text{smooth projective tropical abelian varieties}
\cap
\text{non-rationally-triangulable}.
}
$$

---

# 14. Rational-Triangulability Gate

Define:

$$
\boxed{
\mathrm{RTG}(X)
=
1
}
$$

if:

$$
X
$$

is rationally triangulable.

Then:

$$
\boxed{
\mathrm{RTG}(X)=1
\Longrightarrow
\delta_{\mathrm{trop}}^p(X)=0.
}
$$

Hence any counterexample target must satisfy:

$$
\boxed{
\mathrm{RTG}(X)=0.
}
$$

---

# 15. Third firewall: rational versus integral obstruction

Classical Hodge conjecture asks for surjectivity:

$$
CH^p(X)_{\mathbb Q}
\to
H^{2p}(X,\mathbb Q)
\cap
H^{p,p}(X).
$$

Therefore a tropical counterexample must survive tensoring with:

$$
\mathbb Q.
$$

---

# 16. Integral lattices

Let:

$$
\mathcal H_{\mathbb Z}
\subset
\mathcal H_{\mathrm{trop}}^p(X;\mathbb Q)
$$

be an integral lattice of tropical Hodge classes.

Let:

$$
\mathcal A_{\mathbb Z}
\subseteq
\mathcal H_{\mathbb Z}
$$

be the subgroup generated by integral tropical cycle classes.

Suppose:

$$
\mathcal A_{\mathbb Z}
\subsetneq
\mathcal H_{\mathbb Z}.
$$

This alone is not enough.

---

# 17. Finite-index defect is rationally invisible

## Theorem 17.1

If:

$$
[\mathcal H_{\mathbb Z}:\mathcal A_{\mathbb Z}]
<
\infty,
$$

then:

$$
\boxed{
\mathcal A_{\mathbb Z}\otimes\mathbb Q
=
\mathcal H_{\mathbb Z}\otimes\mathbb Q.
}
$$

Therefore:

$$
\boxed{
\delta_{\mathrm{trop}}^p(X)=0.
}
$$

### Proof

A finite-index inclusion of lattices becomes an equality after tensoring with:

$$
\mathbb Q.
$$

QED.

---

# 18. Rational Defect Criterion

To disprove rational HC by the tropical route, one needs:

$$
\boxed{
\operatorname{rank}_{\mathbb Z}
\mathcal A_{\mathbb Z}
<
\operatorname{rank}_{\mathbb Z}
\mathcal H_{\mathbb Z},
}
$$

or equivalently:

$$
\boxed{
\dim_{\mathbb Q}
\left(
\mathcal H_{\mathbb Q}/
\mathcal A_{\mathbb Q}
\right)
>0.
}
$$

A proper sublattice of equal rank only produces an integral obstruction.

---

# 19. Zharkov's $\Phi$ must pass the rational firewall

Zharkov follows Kontsevich by trying to construct a map:

$$
\Phi
$$

whose commutative diagram works modulo a proper sublattice of:

$$
\mathbb Z
\langle
\theta,w_1,w_2
\rangle.
$$

For rational HC, the phrase:

$$
\boxed{
\text{proper sublattice}
}
$$

must be strengthened.

The quotient must have positive:

$$
\mathbb Q
$$

-rank after rationalization.

A merely finite quotient is irrelevant to rational HC.

---

# 20. Zharkov's original computational failure

Zharkov reduces his dimension-$4$ setup to a finite linear system after an ansatz.

He reports that the system can be solved only modulo the full lattice:

$$
\mathbb Z
\langle
\theta,w_1,w_2
\rangle,
$$

not modulo a proper sublattice.

So his 2020 paper does not produce even the proposed integral obstruction, much less a rational counterexample.

---

# 21. Integral Hodge counterexamples do not solve our branch

Engel, de Gaay Fortman, and Schreieder proved in 2025–2026 that the integral Hodge conjecture fails for abelian varieties.

Their proof is tropical/matroid-inspired.

This is an important positive demonstration that tropical methods can detect genuine algebraic-cycle lattice obstructions.

But it does not refute the rational Hodge conjecture.

CE005 therefore imposes:

$$
\boxed{
\text{RATIONAL FIREWALL}.
}
$$

Any torsion or finite-index-only obstruction is rejected.

---

# 22. Dimension ladder under MDAG

Now combine:

$$
\delta=(-1)^n
$$

with currently known positive Weil results.

---

## 22.1 Abelian fourfold

$$
2n=4,
\qquad
n=2.
$$

MDAG requires:

$$
\delta=+1.
$$

But the classical Hodge conjecture is already known for abelian fourfolds, and Weil classes are algebraic for all discriminants.

So this dimension cannot yield a classical rational counterexample.

---

## 22.2 Abelian sixfold

$$
2n=6,
\qquad
n=3.
$$

MDAG requires:

$$
\delta=-1.
$$

Markman proves algebraicity in exactly this regime.

So this dimension also cannot yield a counterexample via maximal degeneration.

---

## 22.3 Abelian eightfold

$$
2n=8,
\qquad
n=4.
$$

MDAG requires:

$$
\boxed{
\delta=+1.
}
$$

The Markman theorem cited above does not close the general Weil-class problem in this dimension.

Thus dimension:

$$
8
$$

is the first natural Weil dimension not already eliminated by the combination:

$$
\boxed{
\text{Brosnan accessibility}
+
\text{Markman low-dimensional algebraicity}.
}
$$

---

# 23. CE005-A8 target

HC-False therefore defines a new tropical candidate:

$$
\boxed{
\mathrm{CE005\mbox{-}A8}.
}
$$

Take a maximally degenerating family of split Weil-type abelian eightfolds:

$$
\mathcal A_t,
$$

with:

$$
\delta=+1.
$$

Let:

$$
X_{\mathrm{trop}}
$$

be its tropical abelian eightfold.

Target codimension:

$$
p=4.
$$

Let:

$$
W_{\mathrm{trop}}
\subseteq
\mathcal H_{\mathrm{trop}}^4(X_{\mathrm{trop}};\mathbb Q)
$$

be the tropical Weil sector.

---

# 24. Tropical Weil Rational Defect Conjecture

CE005-A8 asks whether:

$$
\boxed{
W_{\mathrm{trop}}
\not\subseteq
\mathcal A_{\mathrm{trop}}^4(X_{\mathrm{trop}};\mathbb Q).
}
$$

Equivalently:

$$
\boxed{
\mathfrak R_{\mathrm{Weil,trop}}^4
=
\frac{
W_{\mathrm{trop}}
}{
W_{\mathrm{trop}}
\cap
\mathcal A_{\mathrm{trop}}^4
}
\neq0.
}
$$

This is the exact rational tropical obstruction we need.

---

# 25. Counterexample implication

If:

1. the eightfold tropical Weil sector is in:
   $$
   \ker N;
   $$
2. it contains:
   $$
   0\neq w
   $$
   outside the rational tropical cycle span;
3. the family satisfies the specialization setup of Kontsevich–Zharkov;

then the generic complex eightfold fiber supplies:

$$
\boxed{
0\neq
\alpha
\in
H^8(A,\mathbb Q)
\cap
H^{4,4}(A)
}
$$

not represented by rational algebraic cycles.

Thus:

$$
\boxed{
\mathrm{CE005\mbox{-}A8}
\Longrightarrow
\neg\mathrm{HC}.
}
$$

---

# 26. Non-rational triangulability is mandatory

Amini–Piquerez implies:

$$
\boxed{
\mathrm{RTG}(X_{\mathrm{trop}})=1
\Longrightarrow
\mathfrak R_{\mathrm{Weil,trop}}^4=0.
}
$$

Therefore CE005-A8 requires:

$$
\boxed{
\mathrm{RTG}(X_{\mathrm{trop}})=0.
}
$$

This is not a nuisance.

It is exactly where Kontsevich's original problem survives modern tropical Hodge theory.

---

# 27. The surviving tropical counterexample chamber

The target must simultaneously satisfy:

$$
\boxed{
\mathrm{MDAG}=1,
}
$$

$$
\boxed{
\mathrm{RTG}=0,
}
$$

and:

$$
\boxed{
\delta_{\mathrm{trop}}^4>0.
}
$$

For Weil type:

$$
\mathrm{MDAG}=1
$$

already forces:

$$
\delta=+1
$$

in dimension eight.

So the candidate chamber is narrow and explicit.

---

# 28. Three independent failure modes

CE005-A8 can fail in three mathematically different ways.

### F1 — Tropical Hodge failure does not occur

$$
\mathfrak R_{\mathrm{trop}}^4=0.
$$

Then tropical HC survives.

### F2 — Only integral index defect occurs

$$
\mathcal A_{\mathbb Z}
\subsetneq
\mathcal H_{\mathbb Z}
$$

but with equal rank.

Then rational HC receives no counterexample.

### F3 — Tropical defect exists but does not specialize correctly

A candidate tropical class may fail to lift as the required persistent Weil Hodge class in the classical family.

Then the tropical failure does not imply classical failure.

All three must be ruled out.

---

# 29. Why Amini–Piquerez does not settle CE005-A8

Their theorem assumes rational triangulability.

They explicitly state that Kontsevich's tropical abelian varieties are generally not rationally triangulable.

Thus:

$$
\boxed{
\text{Amini–Piquerez closes a large positive sector,
not the original irrational torus problem}.
}
$$

CE005-A8 lives precisely outside their theorem.

---

# 30. Why ordinary tropical Hodge type is not enough

It is not enough to prove:

$$
w_{\mathrm{trop}}
\in
\ker N.
$$

That only verifies Hodge type.

The counterexample requires:

$$
\boxed{
w_{\mathrm{trop}}
\notin
\operatorname{Span}_{\mathbb Q}
\{
cl_{\mathrm{trop}}(Z)
\}.
}
$$

So the difficult object is the image of the tropical cycle-class map.

This mirrors the neutral program:

$$
\boxed{
\text{Measure/Hodge membership}
\neq
\text{Legality/cycle realizability}.
}
$$

---

# 31. Current tropical Hodge theorem status

The unrestricted tropical Hodge conjecture remains open.

Known positive theorem:

$$
\boxed{
\text{rationally triangulable}
\Longrightarrow
\text{tropical Hodge}.
}
$$

The irrational tropical abelian case remains the relevant unresolved zone.

So CE005-A8 is not immediately killed by modern tropical Hodge theory.

---

# 32. Gross–Shokrieh control

Modern work on tropical abelian varieties develops:

- tropical cycle classes;
- Appell–Humbert theory;
- homological and numerical equivalence;
- Poincaré formulas;
- real tori with integral structures.

This gives a mature legality language for the target.

But it does not currently identify all rational tropical Hodge classes with tropical cycle classes on arbitrary irrational tropical abelian varieties.

So the exact CE005 residual remains meaningful.

---

# 33. Tropical Fourier–Mukai developments

Recent tropical work constructs Poincaré bundles and cohomological Fourier–Mukai transforms for real tori with integral structures.

This is relevant because classical abelian-cycle proofs often use Fourier transforms.

HC-False should use these positive structures adversarially:

> compute the full subspace generated from all presently certified tropical cycle operations and test whether the tropical Weil sector escapes it.

Failure to escape is evidence against the counterexample.

---

# 34. Candidate cycle span hierarchy

Define:

$$
\mathcal A_{\mathrm{trop},0}
\subseteq
\mathcal A_{\mathrm{trop},1}
\subseteq
\cdots
\subseteq
\mathcal A_{\mathrm{trop}}
$$

where:

### Level 0

divisors / theta classes.

### Level 1

products, pullbacks, pushforwards, permutations.

### Level 2

tropical Fourier–Mukai / Poincaré-generated classes.

### Level 3

tautological Jacobian/Prym-type cycles.

### Terminal

all balanced tropical cycles.

The target is not merely:

$$
W_{\mathrm{trop}}
\not\subseteq
\mathcal A_{\mathrm{trop},k}
$$

for some small:

$$
k.
$$

It is exclusion from the terminal rational span.

---

# 35. Zharkov's finite-system philosophy survives

Although dimension four is no longer a viable classical target, the structural idea survives:

1. encode balanced polyhedral cycles by finite local cell types;
2. encode their tautological classes;
3. seek a linear functional/certificate vanishing on all balanced-cycle classes;
4. test it on the Weil sector.

The necessary improvement is:

$$
\boxed{
\text{certificate must separate a rational direction, not merely a lattice coset}.
}
$$

---

# 36. Rational separating functional

A decisive certificate would be a:

$$
\mathbb Q
$$

-linear map:

$$
\boxed{
\Psi:
\mathcal H_{\mathrm{trop}}^4
\to
Q
}
$$

to some rational vector space:

$$
Q
$$

such that:

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

but:

$$
\Psi(w)
\neq0
$$

for some:

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

Then:

$$
w
$$

is not in the rational cycle span.

This is stronger and cleaner than a finite-index lattice quotient.

---

# 37. Tropical Legality Functional Conjecture

Define:

$$
\boxed{
\mathrm{TLF}_{8}:
}
$$

There exists a rational separating functional:

$$
\Psi
$$

for a non-rationally-triangulable maximally degenerate split Weil-type tropical abelian eightfold satisfying:

$$
\boxed{
\Psi
\left(
cl_{\mathrm{trop}}(Z)
\right)
=
0
}
$$

for every codimension-$4$ tropical cycle:

$$
Z,
$$

while:

$$
\boxed{
\Psi(w)\neq0
}
$$

for some tropical Weil class:

$$
w.
$$

If TLF$_8$ holds with specialization compatibility, rational HC is false.

---

# 38. Why a finite-index functional is rejected

A map:

$$
\Psi:
\mathcal H_{\mathbb Z}
\to
F
$$

to a finite group:

$$
F
$$

can detect:

$$
\mathcal A_{\mathbb Z}
\subsetneq
\mathcal H_{\mathbb Z}
$$

of finite index.

But:

$$
F\otimes\mathbb Q=0.
$$

Such a certificate cannot prove:

$$
\delta_{\mathrm{trop}}>0.
$$

So TLF$_8$ requires a positive-dimensional rational target.

---

# 39. Relation to the 2025 integral Hodge breakthrough

The recent integral Hodge counterexamples on abelian varieties show that:

$$
\boxed{
\text{integral cycle lattice defects are real and detectable}.
}
$$

But they also demonstrate exactly why CE005 must enforce the rational firewall.

An integral obstruction can coexist with the rational Hodge conjecture.

Hence the HC-False rational branch treats those results as methodological inspiration, not as a solution.

---

# 40. Brosnan validates CE003's arithmetic warning

CE003 found that maximal isotropic $K$-rational ancestry forces the split discriminant.

Brosnan's 2026 theorem independently shows that maximal degeneration itself forces:

$$
\delta=(-1)^n.
$$

Thus the earlier branch insight was pointed at a real global restriction:

$$
\boxed{
\text{arithmetic Hermitian data controls which tropical boundary strata are reachable}.
}
$$

This is a strong external validation of the branch methodology.

---

# 41. But it also kills the sixfold tropical attack

That validation cuts both ways.

For CE001:

$$
\delta\neq-1.
$$

Therefore the desired maximal tropical limit is inaccessible.

So CE005 records a genuine route death:

$$
\boxed{
\text{CE001}
\xleftarrow{\text{maximal tropical}}
\text{BLOCKED}.
}
$$

This is not evidence that CE001 is false.

It is evidence that the chosen microscope cannot see it.

---

# 42. Counterexample target ladder

HC-False now maintains separate targets.

### CE001

Very general non-split Weil sixfold.

Main surviving routes:

- TDVD;
- Chow/derived obstruction;
- coniveau;
- non-maximal degeneration.

### CE005-A8

Maximally degenerating split Weil eightfold.

Main route:

- rational tropical cycle-span defect.

The two targets are independent.

---

# 43. Eightfold is a cleaner tropical target

The eightfold target has three advantages.

### 43.1 Accessibility

Brosnan permits maximal degeneration at:

$$
\delta=+1.
$$

### 43.2 Not closed by the cited low-dimensional Weil theorem

The cited Markman theorem closes:

- all fourfolds;
- split sixfolds.

It does not provide a general eightfold closure.

### 43.3 Codimension four gives a nontrivial exceptional middle Weil sector

$$
W_K(A)
\subset
H^8(A,\mathbb Q)
\cap
H^{4,4}(A).
$$

Thus it remains a structurally natural candidate.

---

# 44. First explicit CE005 computation target

Let:

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

be an eight-dimensional tropical abelian variety in the reachable split Weil family.

Compute:

$$
\boxed{
\ker N
\cap
H_{\mathrm{Weil}}
}
$$

and then compute the rational span:

$$
\boxed{
\Xi_{\mathbb Q}
=
\operatorname{Span}_{\mathbb Q}
\left\{
vol(Z):
Z
\text{ balanced codimension-4 tropical cycle}
\right\}
\cap
H_{\mathrm{Weil}}.
}
$$

The decisive quantity is:

$$
\boxed{
\dim_{\mathbb Q}
H_{\mathrm{Weil}}
-
\dim_{\mathbb Q}
\Xi_{\mathbb Q}.
}
$$

---

# 45. Required outcome

A counterexample requires:

$$
\boxed{
\dim_{\mathbb Q}
\Xi_{\mathbb Q}
<
\dim_{\mathbb Q}
H_{\mathrm{Weil}}.
}
$$

If equality holds:

$$
\boxed{
\Xi_{\mathbb Q}
=
H_{\mathrm{Weil}},
}
$$

the tropical Weil route fails rationally even if integral indices remain.

---

# 46. Why this is now computationally sharper

The original question:

> Are there tropical cycles for the Weil classes?

becomes:

$$
\boxed{
\text{What is the rank of a concrete rational cycle-class image inside a finite-dimensional Weil sector?}
}
$$

This is exactly the kind of problem suitable for:

- finite periodic approximations;
- symmetry reduction;
- integer/rational linear algebra;
- SAT/SMT or exact matrix methods;
- formal verification.

The branch should not infer nonexistence from failed random searches.

---

# 47. Counterexample certificate standard for CE005-A8

A successful tropical counterexample certificate must contain:

1. an explicit tropical abelian eightfold:
   $$
   X_{\mathrm{trop}};
   $$
2. proof it comes from a maximal degeneration of split Weil type;
3. explicit:
   $$
   0\neq w\in W_{\mathrm{trop}};
   $$
4. proof:
   $$
   Nw=0;
   $$
5. a rational linear obstruction:
   $$
   \Psi;
   $$
6. proof:
   $$
   \Psi(cl_{\mathrm{trop}}Z)=0
   $$
   for every tropical codimension-$4$ cycle;
7. proof:
   $$
   \Psi(w)\neq0;
   $$
8. specialization theorem mapping classical algebraic cycles to tropical cycle classes in the required family;
9. rational-coefficient audit.

Only then can we declare a classical rational Hodge counterexample.

---

# 48. What CE005 disproves

CE005 disproves the methodological claim:

$$
\boxed{
\text{Kontsevich's maximal tropical method can attack arbitrary Weil discriminant components}.
}
$$

It cannot.

Maximal degeneration itself has a discriminant gate.

---

# 49. What CE005 leaves open

The following remain open:

$$
\boxed{
\mathrm{CE001}.
}
$$

$$
\boxed{
\mathrm{CE005\mbox{-}A8}.
}
$$

$$
\boxed{
\text{unrestricted tropical Hodge conjecture for irrational tropical abelian varieties}.
}
$$

$$
\boxed{
\mathrm{TLF}_8.
}
$$

---

# 50. Status table

- **CE001 non-split sixfold:** OPEN
- **CE001 maximal-tropical subroute:** BLOCKED
- **MDAG discriminant condition:** PROVED externally by Brosnan
- **Rationally triangulable tropical HC:** PROVED by Amini–Piquerez
- **Unrestricted tropical HC:** OPEN
- **Finite-index lattice obstruction as rational HC counterexample:** DISPROVED
- **CE005-A8:** OPEN
- **TLF$_8$:** OPEN
- **Rational tropical Weil residual:** OPEN

---

# 51. Strategic conclusion

The tropical counterexample program is not dead.

It is much narrower than it looked.

The surviving chamber is:

$$
\boxed{
\text{maximally degenerate}
}
$$

$$
\boxed{
\text{split Weil discriminant }(-1)^n
}
$$

$$
\boxed{
\text{non-rationally-triangulable}
}
$$

$$
\boxed{
\text{positive rational cycle-span defect}.
}
$$

The first dimension not already closed by the cited low-dimensional positive results is naturally:

$$
\boxed{
8.
}
$$

So HC-False moves the tropical front from:

$$
\boxed{
\text{non-split sixfold}
}
$$

to:

$$
\boxed{
\text{split eightfold}.
}
$$

---

# 52. Next Interface

Next counterexample round:

```text
HODGE_HCFALSE_CE006_WeilEightfoldRationalDefect.md
```

Primary target:

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

Planned tasks:

1. construct an explicit maximally degenerate split Weil-type abelian eightfold lattice model;
2. write the tropical Weil sector explicitly;
3. compute the eigenwave/monodromy kernel on that sector;
4. formulate codimension-$4$ balanced-cell generators;
5. quotient by symmetries and periodicity;
6. compute the rational cycle span;
7. distinguish:
   $$
   \text{rank defect}
   $$
   from:
   $$
   \text{finite-index defect};
   $$
8. search for a rational separating functional:
   $$
   \Psi;
   $$
9. reject CE005-A8 if the Weil sector is rationally generated by tropical cycles.

Neutral MLRSC remains paused at:

```text
HODGE_MLRSC_R029_C_JointObjectFiniteCore.md
```

CE001 remains alive on its non-tropical routes.

---

# References

1. I. Zharkov, *Tropical Abelian varieties, Weil classes and the Hodge Conjecture*, arXiv:2002.02347, 2020. Gives Kontsevich's explicit tropical counterexample program, identifies tropical Hodge classes via the eigenwave, and states that tropical Hodge failure in the setup would yield a classical counterexample.

2. O. Amini, M. Piquerez, *Tropical Clemens-Schmid sequence and existence of tropical cycles with a given cohomology class*, arXiv:2012.13142, 2020. Proves the tropical Hodge conjecture for rationally triangulable smooth projective tropical varieties and explicitly notes that Kontsevich's tropical abelian varieties are generally not rationally triangulable.

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

4. P. Brosnan, *Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture*, arXiv:2609.14169, submitted 2026-09-12, updated 2026-09-15. Shows maximal unipotent degeneration of Weil-type abelian $2n$-folds requires discriminant $(-1)^n$.

5. P. Engel, O. de Gaay Fortman, S. Schreieder, *Matroids and the integral Hodge conjecture for abelian varieties*, arXiv:2507.15704. Disproves the integral Hodge conjecture for abelian varieties; relevant here as a warning that integral lattice defects do not imply rational Hodge failure.

6. A. Gross, F. Shokrieh, *Tautological cycles on tropical Jacobians*, Algebra & Number Theory 17 (2023), 885–921. Develops tropical cycle classes, Appell–Humbert theory and cycle equivalences on real tori with integral structures.

7. S. Ghosh, F. Shokrieh, *Tropical Poincaré bundle, Fourier-Mukai transform, and a generalized Poincaré formula*, arXiv:2503.12835, 2025.

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

9. Aletheia, *HODGE_HCFALSE_CE003_DominanceImpliesSplit*, 2026-09-15.

10. Aletheia, *HODGE_HCFALSE_CE004_CycleToIsotropicNecessity*, 2026-09-15.

---

## Canonical Source Declaration

本檔案為 HC-False 分支第五篇正式 UTF-8 Markdown canonical source。

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪沒有宣稱 tropical Weil eightfold 已提供反例；本輪證明的是 counterexample route 的可達域、rational firewall 與新的 eightfold target。