# HODGE_HCFALSE_CE010_GlobalSpectralBalancing
## ——反例分支第十輪：Free Tropical Homology、Global Spectral Balancing No-Go 與 Diagonal Framing Barrier

**作者：Aletheia（GPT-5.6 Sol）**  
**研究分支：HC-False / Counterexample Program**  
**候選編號：CE010**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Branch:** HC-False  
**Round:** CE010  
**Parent Candidate:** CE005-A8 / CE006–CE009 — split tropical Weil eightfold  
**Primary Claim:** 在 tropical abelian eightfold 上，global period closure、ordinary chain boundary、balancing 的線性 homological shadow，以及 eigenwave/monodromy condition $N=0$ 本身都不能排除 tropical Weil plane。因為 $\mathcal F_4$ 是 constant coefficient system $\bigwedge^4N_{\mathbb Q}$，free tropical-homology chain complex允許每個 $4$-cell攜帶任意 coefficient $\eta_\sigma\in\bigwedge^4N_{\mathbb Q}$，其 homology為
$$
H_4(X,\mathcal F_4;\mathbb Q)
\cong
\bigwedge^4\Lambda_{\mathbb Q}
\otimes
\bigwedge^4N_{\mathbb Q}.
$$
所以每個 Weil class本來就已有 global closed free-chain representative，且 CE006 已證
$$
W_{\mathrm{trop}}\subseteq\ker N.
$$
真正 tropical cycle則多一個 stronger geometric condition：
$$
\boxed{
\eta_\sigma=w_\sigma\xi_\sigma,
}
$$
其中 $\xi_\sigma$ 是**同一個 cell**的 tangent $4$-plane primitive Plücker volume。這個 support–coefficient diagonal/framing constraint並不是 ordinary $\partial=0$、period closure或 $N=0$ 的線性 consequence。故若 rational tropical Hodge counterexample存在，obstruction 必須來自 **Framed Geometric Realizability**，而非 free global balancing。  
**Status:** PROVED  
**Counterexample Status:** OPEN  
**Killed Negative Shortcut:** GLOBAL-LINEAR-BALANCING-ONLY / FREE-HOMOLOGY SEPARATOR  
**New Exact Interface:** Framed Spectral Realizability  
**Depends On:** CE006–CE009、tropical homology with constant cosheaves on tori、Gross–Shokrieh cycle-class functoriality、Zharkov、Amini–Piquerez  
**Backtrack Target:** 無  
**Evidence Level:** E2 / exact chain-level separation of free coefficients from geometric tangent framing  
**Formalization Status:** NOT FORMALIZED  
**Computation Status:** EXACT CHAIN-MODEL REDUCTION  

---

# 0. The question after CE009

CE009 produced a legality-safe projector:

$$
\boxed{
P_W=p_W([\mu]_\ast)
}
$$

with:

$$
P_W^2=P_W,
$$

$$
\operatorname{Im}P_W=W_{\mathrm{trop}},
$$

and:

$$
\boxed{
P_W\left(\mathcal A_{\mathrm{trop}}^4\right)
=
\mathcal A_{\mathrm{trop}}^4\cap W_{\mathrm{trop}}.
}
$$

The negative target became:

$$
\boxed{
P_Wcl_{\mathrm{trop}}(Z)=0
\quad
\forall Z.
}
$$

A natural next guess was:

> perhaps global balancing and period closure force the extremal $K$-spectral component to cancel.

CE010 shows that this is still too weak.

The free homology problem already contains the Weil plane.

The missing condition is geometric framing.

---

# 1. Tropical torus

Let:

$$
X=V/\Lambda,
$$

where:

$$
V\simeq\mathbb R^8,
$$

$$
\Lambda\simeq\mathbb Z^8
$$

is the period lattice.

Let:

$$
N\simeq\mathbb Z^8
$$

be the integral tangent lattice.

At the CE008 arithmetic-generic target one may have:

$$
\Lambda\cap N=0.
$$

The two lattices play different roles.

---

# 2. Tropical coefficient cosheaf

On a tropical torus the tangent integral structure is globally constant.

Hence:

$$
\boxed{
\mathcal F_p
\cong
\bigwedge^pN_{\mathbb Q}
}
$$

as a constant rational coefficient system.

For:

$$
p=4,
$$

$$
\boxed{
\mathcal F_4
\cong
\bigwedge^4N_{\mathbb Q}.
}
$$

The fiber dimension is:

$$
\binom84=70.
$$

---

# 3. Free tropical-homology chains

Choose a compatible cellular or singular chain model for:

$$
X.
$$

Define:

$$
\boxed{
C_q^{\mathrm{free}}
=
C_q(X;\mathbb Q)
\otimes
\bigwedge^4N_{\mathbb Q}.
}
$$

A chain has the form:

$$
c
=
\sum_\sigma
[\sigma]\otimes\eta_\sigma,
$$

where:

$$
\eta_\sigma
\in
\bigwedge^4N_{\mathbb Q}
$$

is arbitrary.

The boundary is:

$$
\boxed{
\partial
\left(
[\sigma]\otimes\eta
\right)
=
(\partial[\sigma])\otimes\eta.
}
$$

In this free complex there is no requirement that:

$$
\eta_\sigma
$$

agree with the tangent plane of:

$$
\sigma.
$$

---

# 4. Free homology

Because the coefficient system is constant:

$$
\boxed{
H_q
\left(
C_\bullet^{\mathrm{free}}
\right)
\cong
H_q(X,\mathbb Q)
\otimes
\bigwedge^4N_{\mathbb Q}.
}
$$

For an eight-dimensional torus:

$$
H_q(X,\mathbb Q)
\cong
\bigwedge^q\Lambda_{\mathbb Q}.
$$

Thus:

$$
\boxed{
H_4
\left(
C_\bullet^{\mathrm{free}}
\right)
\cong
\bigwedge^4\Lambda_{\mathbb Q}
\otimes
\bigwedge^4N_{\mathbb Q}.
}
$$

This is exactly the CE006 middle carrier.

---

# 5. Every ambient tensor has a closed free representative

## Theorem 5.1

Every:

$$
h
\in
\bigwedge^4\Lambda_{\mathbb Q}
\otimes
\bigwedge^4N_{\mathbb Q}
$$

has a free chain representative:

$$
c_h
$$

such that:

$$
\partial c_h=0.
$$

### Proof

Write:

$$
h
=
\sum_j
u_j\otimes\eta_j,
$$

with:

$$
u_j
\in
H_4(X,\mathbb Q),
$$

$$
\eta_j
\in
\bigwedge^4N_{\mathbb Q}.
$$

Choose ordinary rational singular cycles:

$$
a_j
$$

representing:

$$
u_j.
$$

Set:

$$
c_h
=
\sum_j
a_j\otimes\eta_j.
$$

Then:

$$
\partial c_h=0,
$$

and its class is:

$$
h.
$$

QED.

---

# 6. Weil consequence

Since:

$$
W_{\mathrm{trop}}
\subset
H_4
\left(
C_\bullet^{\mathrm{free}}
\right),
$$

every:

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

already possesses a globally closed free-chain representative.

Therefore:

$$
\boxed{
\text{ordinary global closure alone cannot exclude Weil classes}.
}
$$

---

# 7. Period-winding × coefficient tensor

There is a useful pairing description.

Let:

$$
\alpha
\in
\left(
\bigwedge^4\Lambda_{\mathbb Q}
\right)^\vee
$$

and:

$$
\beta
\in
\left(
\bigwedge^4N_{\mathbb Q}
\right)^\vee.
$$

For a free chain:

$$
c
=
\sum_\sigma
[\sigma]\otimes\eta_\sigma,
$$

contract coefficients by:

$$
\beta
$$

to obtain an ordinary chain:

$$
\boxed{
c_\beta
=
\sum_\sigma
\beta(\eta_\sigma)
[\sigma].
}
$$

If:

$$
\partial c=0,
$$

then:

$$
\partial c_\beta=0.
$$

Define:

$$
\boxed{
\left\langle
[c],
\alpha\otimes\beta
\right\rangle
=
\left\langle
[c_\beta],
\alpha
\right\rangle.
}
$$

This recovers the tensor:

$$
[c]
\in
\bigwedge^4\Lambda_{\mathbb Q}
\otimes
\bigwedge^4N_{\mathbb Q}.
$$

---

# 8. Meaning of the two factors

The first factor:

$$
\bigwedge^4\Lambda_{\mathbb Q}
$$

records global period winding.

The second factor:

$$
\bigwedge^4N_{\mathbb Q}
$$

records the tropical coefficient/tangent-volume data.

Schematically:

$$
\boxed{
\text{homology class}
=
\text{period winding}
\otimes
\text{coefficient volume}.
}
$$

This is the precise carrier behind CE009's heuristic.

---

# 9. Genuine tropical $4$-cycle

Now let:

$$
Z
$$

be a balanced codimension-$4$ tropical cycle.

Its maximal cells have dimension:

$$
4.
$$

After a compatible subdivision, for each oriented maximal cell:

$$
\sigma
$$

let:

$$
L_\sigma
\subset
N_{\mathbb Q}
$$

be its rational tangent:

$$
4
$$

-plane.

Choose its primitive oriented tangent volume:

$$
\boxed{
\xi_\sigma
\in
\bigwedge^4
\left(
L_\sigma\cap N
\right)
\subset
\bigwedge^4N.
}
$$

Let:

$$
w_\sigma
\in
\mathbb Z
$$

be the tropical weight.

---

# 10. Cycle-class chain

The tropical cycle-class map associates the chain:

$$
\boxed{
c_Z
=
\sum_\sigma
w_\sigma
[\sigma]
\otimes
\xi_\sigma.
}
$$

The balancing condition yields the required chain closure after the standard incidence/cosheaf maps are included.

Its homology class is:

$$
\boxed{
cl_{\mathrm{trop}}(Z)
=
[c_Z].
}
$$

The important point is not merely that:

$$
\partial c_Z=0.
$$

It is the form of every coefficient.

---

# 11. Diagonal framing condition

A genuine tropical cycle does **not** allow arbitrary:

$$
\eta_\sigma.
$$

It requires:

$$
\boxed{
\eta_\sigma
=
w_\sigma\xi_\sigma,
}
$$

where:

$$
\xi_\sigma
$$

is the primitive Plücker volume of **the same cell's own tangent plane**.

Thus the support geometry and the coefficient geometry are coupled.

This is the missing condition in the free chain complex.

---

# 12. Grassmannian formulation

Let:

$$
\operatorname{Gr}_{\mathbb Q}(4,N)
$$

be the rational Grassmannian of rational:

$$
4
$$

-planes in:

$$
N_{\mathbb Q}.
$$

The Plücker map is:

$$
\boxed{
\operatorname{Pl}:
\operatorname{Gr}_{\mathbb Q}(4,N)
\to
\mathbb P
\left(
\bigwedge^4N_{\mathbb Q}
\right).
}
$$

Define the diagonal framing graph:

$$
\boxed{
\mathcal D_4
=
\left\{
(L,[\eta]):
[\eta]=\operatorname{Pl}(L)
\right\}.
}
$$

For every genuine maximal cell:

$$
\boxed{
\left(
L_\sigma,
[\eta_\sigma]
\right)
\in
\mathcal D_4.
}
$$

---

# 13. Framing is stronger than decomposability

The Plücker image consists of decomposable:

$$
4
$$

-vectors.

But CE009 already proved that decomposable:

$$
4
$$

-vectors span the entire:

$$
\bigwedge^4N_{\mathbb Q}.
$$

Therefore the condition:

$$
\boxed{
\eta_\sigma
\text{ is decomposable}
}
$$

alone is too weak.

What matters is:

$$
\boxed{
[\eta_\sigma]
=
\operatorname{Pl}(T_\sigma)
}
$$

for the support tangent:

$$
T_\sigma.
$$

This is a support–coefficient diagonal condition.

---

# 14. Free cycles and framed cycles

Define:

$$
\boxed{
Z_4^{\mathrm{free}}
=
\ker
\left(
\partial:
C_4^{\mathrm{free}}
\to
C_3^{\mathrm{free}}
\right).
}
$$

Let:

$$
\boxed{
Z_4^{\mathrm{fr}}
\subset
Z_4^{\mathrm{free}}
}
$$

denote the subset generated by genuine balanced polyhedral tropical cycles satisfying the diagonal framing condition.

Then:

$$
\boxed{
[Z_4^{\mathrm{free}}]
=
H_4(X,\mathcal F_4;\mathbb Q).
}
$$

Whereas:

$$
\boxed{
\operatorname{Span}_{\mathbb Q}
[Z_4^{\mathrm{fr}}]
=
\mathcal A_{\mathrm{trop}}^4.
}
$$

---

# 15. The tropical Hodge gap

We have:

$$
\boxed{
\mathcal A_{\mathrm{trop}}^4
\subseteq
\ker N
\subseteq
H_4(X,\mathcal F_4;\mathbb Q).
}
$$

Amini–Piquerez proves equality:

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

for rationally triangulable smooth projective tropical varieties.

At our arithmetic-irrational target, equality is unknown.

Hence the unresolved question is precisely:

$$
\boxed{
\text{closed free }N\text{-horizontal class}
\stackrel{?}{\Longrightarrow}
\text{framed geometric cycle span}.
}
$$

---

# 16. Weil free-chain theorem

## Theorem 16.1

Every:

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

has a free chain representative:

$$
c_w
$$

such that:

$$
\boxed{
\partial c_w=0
}
$$

and:

$$
\boxed{
N[c_w]=0.
}
$$

### Proof

The first statement is Theorem 5.1.

The second is CE006:

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

QED.

---

# 17. Global Linear Balancing No-Go Theorem

## Theorem 17.1

No obstruction that is a formal consequence solely of:

1. free chain closure:
   $$
   \partial c=0;
   $$
2. period identification on the torus;
3. linear tropical homology relations;
4. eigenwave/monodromy horizontality:
   $$
   N[c]=0;
   $$

can exclude:

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

### Proof

Every:

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

already has a representative satisfying all four conditions.

Therefore any relation valid for every solution of those linear conditions must also hold on:

$$
w.
$$

It cannot separate the Weil plane.

QED.

---

# 18. What balancing really gives

Balancing is indispensable.

It is what makes the geometric cycle-class chain closed in tropical homology.

But the free equation:

$$
\partial c=0
$$

forgets **why** each coefficient had its value.

A genuine tropical cycle carries an additional memory:

$$
\boxed{
\text{coefficient}
=
\text{primitive tangent volume of support}.
}
$$

That memory is exactly what free homology discards.

---

# 19. Universal-cover formulation

Lift:

$$
Z
$$

to a:

$$
\Lambda
$$

-periodic weighted polyhedral complex:

$$
\widetilde Z
\subset
V.
$$

Each maximal cell carries:

1. a geometric position;
2. a rational tangent:
   $$
   4
   $$
   -plane:
   $$
   L_\sigma;
   $$
3. a weight:
   $$
   w_\sigma;
   $$
4. a tangent Plücker vector:
   $$
   \xi_\sigma.
   $$

Periodicity says:

$$
\widetilde Z+\lambda
=
\widetilde Z
$$

for:

$$
\lambda\in\Lambda.
$$

Balancing couples neighboring tangent planes around codimension-one faces.

The framing condition couples each tangent plane to its coefficient.

---

# 20. Deck-translation data

Choose finitely many cells modulo:

$$
\Lambda.
$$

When a face of one lifted cell is paired with a translated face of another, record:

$$
\lambda\in\Lambda.
$$

These deck labels encode the period-winding component.

Thus a finite periodic cycle has two kinds of data:

$$
\boxed{
\text{deck translations in }\Lambda
}
$$

and:

$$
\boxed{
\text{tangent Plücker data in }\bigwedge^4N.
}
$$

The tropical homology class couples them.

---

# 21. Why the Weil tensor is the exact difficult coupling

The complexified Weil tensor has the form:

$$
\boxed{
\Omega_+
=
\omega_+
\otimes\xi_+,
}
$$

with:

$$
\omega_+
\in
\bigwedge^4\Lambda_+,
$$

$$
\xi_+
\in
\bigwedge^4N_+.
$$

So the candidate already has perfect extremal $K$-spectral alignment between:

- the period-winding factor;
- the tangent/coefficient factor.

The question is not whether such a tensor exists.

It does.

The question is whether a **single globally framed periodic polyhedral cycle** can realize that alignment.

---

# 22. Free realization is trivial

At the free-chain level, no obstruction exists.

Choose an ordinary rational:

$$
4
$$

-cycle:

$$
a_+
$$

representing:

$$
\omega_+.
$$

Then:

$$
\boxed{
a_+\otimes\xi_+
}
$$

is a free closed chain representing:

$$
\Omega_+.
$$

The missing condition is:

> can the support of the chain be chosen so that the coefficient on every cell equals the Plücker tangent volume of that same cell?

That is not a free homology question.

---

# 23. Framed Spectral Realizability

Define:

$$
\boxed{
\mathrm{FSR}_W(X)=1
}
$$

if there exists a rational balanced tropical cycle combination:

$$
Z
$$

with:

$$
0\neq
cl_{\mathrm{trop}}(Z)
\in
W_{\mathrm{trop}}.
$$

Set:

$$
\boxed{
\mathrm{FSR}_W(X)=0
}
$$

otherwise.

By CE006:

$$
\boxed{
\mathrm{FSR}_W(X)=1
\Longrightarrow
W_{\mathrm{trop}}
\subseteq
\mathcal A_{\mathrm{trop}}^4.
}
$$

And:

$$
\boxed{
\mathrm{FSR}_W(X)=0
\Longrightarrow
W_{\mathrm{trop}}
\cap
\mathcal A_{\mathrm{trop}}^4=0.
}
$$

So the tropical candidate is now equivalent to one Boolean realizability invariant.

---

# 24. Spectral-projector formulation

CE009 gives:

$$
\boxed{
P_W
\left(
\mathcal A_{\mathrm{trop}}^4
\right)
=
\mathcal A_{\mathrm{trop}}^4
\cap
W_{\mathrm{trop}}.
}
$$

Therefore:

$$
\boxed{
\mathrm{FSR}_W(X)=0
}
$$

if and only if:

$$
\boxed{
P_W
cl_{\mathrm{trop}}(Z)
=
0
}
$$

for every **framed balanced** tropical cycle:

$$
Z.
$$

The word `framed` is now essential.

---

# 25. Free spectral image is maximal

At the free homology level:

$$
\boxed{
P_W
H_4(X,\mathcal F_4;\mathbb Q)
=
W_{\mathrm{trop}}.
}
$$

Thus the only possible loss is:

$$
P_W
\left(
\operatorname{Span}[Z_4^{\mathrm{fr}}]
\right)
$$

being smaller than:

$$
P_W
\left(
[Z_4^{\mathrm{free}}]
\right).
$$

The latter equals:

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

The former is:

$$
0
\quad\text{or}\quad
W_{\mathrm{trop}}.
$$

---

# 26. Framing residual

Define:

$$
\boxed{
\mathfrak R_{\mathrm{frame}}^4(X)
=
\ker N
/
\operatorname{Span}_{\mathbb Q}[Z_4^{\mathrm{fr}}].
}
$$

Its Weil part is:

$$
\boxed{
\mathfrak R_{\mathrm{frame},W}^4
=
W_{\mathrm{trop}}
/
P_W
\operatorname{Span}_{\mathbb Q}[Z_4^{\mathrm{fr}}].
}
$$

By CE006:

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

---

# 27. Rationally triangulable control case

For rationally triangulable smooth projective tropical varieties:

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

Thus every $N$-horizontal class is generated by framed tropical cycle classes.

At the arithmetic-irrational target, that surjectivity theorem is not available.

Therefore the candidate is exactly a possible failure of framed realization across the irrationality boundary.

---

# 28. Relation to CE007

CE007 proved that no rational/algebraic family separator can distinguish our irrational target from dense positive rational specializations.

The framing interpretation explains the mechanism.

At rationally triangulable fibers:

$$
\boxed{
\text{free }N\text{-horizontal}
=
\text{framed span}.
}
$$

At a hypothetical counterexample fiber:

$$
\boxed{
\text{free }N\text{-horizontal}
\supsetneq
\text{framed span}.
}
$$

The carrier does not jump.

The geometric realizability image does.

---

# 29. Computational no-go

Suppose a solver includes only:

1. a finite cell complex for the torus;
2. arbitrary:
   $$
   \eta_\sigma
   \in
   \bigwedge^4N;
   $$
3. boundary equations;
4. period labels;
5. the matrix of:
   $$
   N;
   $$
6. the spectral projector:
   $$
   P_W.
   $$

Then the solver necessarily sees the Weil plane as feasible.

Such a computation cannot prove CE005-A8.

It is solving free tropical homology, not framed tropical cycle realizability.

---

# 30. Variables required by a sound finite search

A sound finite approximation must include for every maximal cell:

1. affine vertex/position variables;
2. a rational tangent:
   $$
   4
   $$
   -plane;
3. Plücker coordinates:
   $$
   \xi_\sigma;
   $$
4. Plücker quadratic relations;
5. the diagonal matching:
   $$
   \eta_\sigma=w_\sigma\xi_\sigma;
   $$
6. face incidence;
7. balancing;
8. deck-translation labels;
9. periodic closure;
10. final spectral class:
    $$
    P_W[c_Z].
    $$

Without items 1–5, the model is unsound for the counterexample question.

---

# 31. Nonlinearity is unavoidable

The Grassmannian:

$$
\operatorname{Gr}(4,8)
$$

in Plücker coordinates is cut out by quadratic equations.

Cell gluing also constrains which tangent planes may meet along common faces.

Therefore the exact search is not merely:

$$
\boxed{
\text{one large rational linear system}.
}
$$

It is:

$$
\boxed{
\text{combinatorial incidence}
+
\text{quadratic Plücker geometry}
+
\text{linear balancing}
+
\text{global period closure}
+
\text{spectral projection}.
}
$$

This explains why successive purely linear counterexample ansätze keep collapsing.

---

# 32. False proof pattern

The following proof architecture is invalid:

1. write period winding variables;
2. write global boundary equations;
3. write balancing;
4. impose:
   $$
   N=0;
   $$
5. apply:
   $$
   P_W;
   $$
6. solve a linear system;
7. conclude no Weil class.

Unless the support–coefficient diagonal condition is explicitly encoded, the system is a free homology problem and already contains:

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

---

# 33. Valid negative target

A valid negative theorem must prove:

$$
\boxed{
P_W
\operatorname{Span}_{\mathbb Q}
[Z_4^{\mathrm{fr}}]
=
0.
}
$$

Equivalently:

> no globally periodic balanced polyhedral $4$-cycle whose coefficient on every cell is its own tangent Plücker volume can carry the extremal $K$-spectral class.

This is strictly stronger than free closure.

---

# 34. Valid positive falsification target

To kill CE005-A8 it suffices to provide:

1. a finite periodic weighted polyhedral:
   $$
   4
   $$
   -cycle;
2. exact rational tangent planes;
3. exact balancing;
4. exact deck translations;
5. exact cycle class;
6. verification:
   $$
   P_Wcl_{\mathrm{trop}}(Z)\neq0.
   $$

Then CE006 gives:

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

The entire tropical Weil counterexample target dies.

---

# 35. Strongest theorem of CE010

The current situation is:

$$
\boxed{
W_{\mathrm{trop}}
\subseteq
\ker N
\cap
[Z_4^{\mathrm{free}}],
}
$$

while the unknown statement is:

$$
\boxed{
W_{\mathrm{trop}}
\stackrel{?}{\subseteq}
\operatorname{Span}_{\mathbb Q}
[Z_4^{\mathrm{fr}}].
}
$$

Therefore:

$$
\boxed{
\text{global linear balancing alone cannot prove the counterexample}.
}
$$

The hard problem is framed geometric realizability.

---

# 36. Relation to MLRSC

This is an unusually clean Phase-III coupling phenomenon.

The candidate exists in:

- the Measure/Hodge carrier;
- the free homological category;
- the monodromy kernel.

The unresolved legality lives in the smaller category:

$$
\boxed{
\text{framed balanced polyhedral cycles}.
}
$$

The defect is not:

- missing cohomology;
- missing local tangent vectors;
- missing global homology.

It is:

$$
\boxed{
\text{support–coefficient coupling}.
}
$$

This is exactly the kind of scale/category mismatch the neutral MLRSC program was designed to expose.

---

# 37. Status table

- **CE001 non-split sixfold:** OPEN
- **CE005-A8 tropical eightfold:** OPEN
- **CE006 binary Weil residual:** PROVED
- **CE007 uniform algebraic separator:** NO-GO
- **CE008 low-ancestry grammar:** EXCLUDED
- **CE009 legality-safe Weil projector:** PROVED
- **Local tangent-only separator:** NO-GO
- **Global free balancing-only separator:** NO-GO
- **Diagonal Framing Barrier:** IDENTIFIED
- **Framed Spectral Realizability:** OPEN
- **Rational Hodge counterexample:** NOT YET

---

# 38. Strategic conclusion

After CE010, the tropical HC-False branch has reached a sharply defined hard core.

The question is no longer:

> Can the Weil class satisfy global balancing?

At the free homology level, yes.

The exact question is:

> Can the Weil class be represented by a globally periodic balanced polyhedral complex whose coefficient on every maximal cell is the Plücker volume of that same cell's tangent plane?

That is now the whole tropical problem.

---

# 39. Next Interface

Next counterexample round:

```text
HODGE_HCFALSE_CE011_DiagonalFramingObstruction.md
```

Primary target:

$$
\boxed{
P_W
\operatorname{Span}_{\mathbb Q}
[Z_4^{\mathrm{fr}}]
\stackrel{?}{=}
0.
}
$$

Planned attacks:

1. write the diagonal framing condition in explicit Plücker coordinates on:
   $$
   \operatorname{Gr}(4,8);
   $$
2. classify the extremal $K$-spectral component of a framed cell contribution;
3. determine whether a framed cell can have nonzero extremal spectral projection;
4. if yes, derive the codimension-one gluing constraints between such extremal cells;
5. encode a minimal periodic framed complex;
6. search for a local-to-global polynomial identity annihilating all extremal framed cycles;
7. in parallel try to construct one explicit framed cycle with:
   $$
   P_Wcl(Z)\neq0;
   $$
8. whichever side succeeds first determines the fate of CE005-A8.

Neutral MLRSC remains paused at:

```text
HODGE_MLRSC_R029_C_JointObjectFiniteCore.md
```

---

# References

1. A. Gross, F. Shokrieh, *A sheaf-theoretic approach to tropical homology*, arXiv:1906.09245. Develops tropical homology with sheaf/cosheaf coefficients, functoriality and the tropical cycle-class map.

2. I. Zharkov, *Tropical Abelian varieties, Weil classes and the Hodge Conjecture*, arXiv:2002.02347, 2020. Gives the tropical torus carrier, eigenwave condition and Kontsevich counterexample program.

3. O. Amini, M. Piquerez, *Tropical Clemens-Schmid sequence and existence of tropical cycles with a given cohomology class*, arXiv:2012.13142. Proves the tropical Hodge conjecture in the rationally triangulable smooth projective case.

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

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

6. Aletheia, *HODGE_HCFALSE_CE008_ArithmeticIrrationalSeparator*, 2026-09-16.

7. Aletheia, *HODGE_HCFALSE_CE009_PrimitiveNonlinearCycleSearch*, 2026-09-16.

---

## Canonical Source Declaration

本檔案為 HC-False 分支第十篇正式 UTF-8 Markdown canonical source。

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪沒有證明 tropical Weil plane non-realizable；本輪證明的是 global linear balancing / free homology無法排除 Weil plane，並把剩餘問題精確定位到 support–coefficient diagonal framing。