# HODGE_HCFALSE_CE007_WeilCycleSpanCertificate
## ——反例分支第七輪：Uniform Flag Certificate No-Go、Dense Rational Specialization 與 Arithmetic-Discontinuity Barrier

**作者：Aletheia（GPT-5.6 Sol）**  
**研究分支：HC-False / Counterexample Program**  
**候選編號：CE007**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Branch:** HC-False  
**Round:** CE007  
**Parent Candidate:** CE005-A8 / CE006 — split tropical Weil eightfold  
**Primary Claim:** Kontsevich–Zharkov 型「在整個 Weil 參數 family 上以 rational/polynomial 方式變化的 uniform local flag separating functional」不可能證明 rational tropical Weil defect。原因是 split Weil eightfold family 內具有 Zariski-dense 的 rational period specializations；這些 tropical tori 與 ambient integral lattice commensurable，因此 rationally triangulable，Amini–Piquerez 定理使其所有 rational tropical Hodge classes，特別是 $W_{\mathrm{trop}}$，都在 tropical cycle-class span 中。故任何 algebraic/rational family $\Psi_s$ 若 uniform annihilate cycle span，必須在 dense rational specializations上 annihilate $W_{\mathrm{trop}}$；若 $\Psi_s(w_s)$ 是 rational function of parameters，則只能 identically zero。  
**Status:** PROVED  
**Killed Certificate Class:** UNIFORM-ALGEBRAIC / POLYNOMIAL FLAG SEPARATOR  
**CE005-A8 Status:** OPEN  
**New Interface:** Arithmetic / discontinuous separator sensitive to irrational parameter relations  
**Depends On:** CE005、CE006、Zharkov 2020、Amini–Piquerez 2020  
**Backtrack Target:** 無  
**Evidence Level:** E2 / dense-specialization theorem + rational tropical Hodge theorem  
**Formalization Status:** NOT FORMALIZED  
**Computation Status:** CERTIFICATE-ARCHITECTURE NO-GO  

---

# 0. The exact question

CE006 reduced the tropical Weil eightfold target to:

$$
\boxed{
\mathcal A_{\mathrm{trop}}^4(X_s;\mathbb Q)
\cap
W_{\mathrm{trop}}(s)
=
0
\quad\text{or}\quad
W_{\mathrm{trop}}(s).
}
$$

Here:

$$
s
$$

denotes a point in the split Weil parameter domain.

A counterexample needs the first alternative:

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

The natural Kontsevich–Zharkov strategy is to construct a functional:

$$
\Psi_s
$$

which:

1. annihilates all tropical cycle classes;
2. is nonzero on the Weil plane.

CE007 asks whether a **uniform algebraic family** of such functionals can exist.

The answer is:

$$
\boxed{
\text{No}.
}
$$

---

# 1. Parameter domain

Recall the split Weil eightfold polarization family:

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

with:

$$
P^T=P,
$$

$$
R^T=-R.
$$

The parameter space has dimension:

$$
10+6=16.
$$

Let:

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

be the nonempty open domain where:

$$
Q(P,R)
$$

is positive definite.

---

# 2. Rational parameter points

Define:

$$
\boxed{
S_{\mathbb Q}
=
S\cap\mathbb Q^{16}.
}
$$

Since positive definiteness is an open condition,

$$
S_{\mathbb Q}
$$

is Euclidean dense in:

$$
S.
$$

Because:

$$
S
$$

contains a nonempty Euclidean open subset of affine:

$$
16
$$

-space defined over:

$$
\mathbb Q,
$$

the rational points:

$$
S_{\mathbb Q}
$$

are also Zariski dense.

---

# 3. Period lattices at rational parameters

Fix:

$$
s=(P,R)
\in
S_{\mathbb Q}.
$$

Then every entry of:

$$
Q(P,R)
$$

is rational.

Thus the period lattice:

$$
\Gamma_1(s)
=
Q(s)\mathbb Z^8
$$

is commensurable with the ambient lattice:

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

Equivalently, there exists:

$$
N\ge1
$$

such that:

$$
\boxed{
N\Gamma_1(s)
\subseteq
\Gamma_2
}
$$

and another:

$$
M\ge1
$$

such that:

$$
M\Gamma_2
\subseteq
\Gamma_1(s).
$$

---

# 4. Rational Triangulation Lemma

## Lemma 4.1

For:

$$
s\in S_{\mathbb Q},
$$

the tropical torus:

$$
X_s
=
\mathbb R^8/\Gamma_1(s)
$$

admits a rational triangulation with respect to the integral-affine structure induced by:

$$
\Gamma_2.
$$

### Proof

Because:

$$
\Gamma_1(s)
$$

and:

$$
\Gamma_2
$$

are commensurable, choose a common finite-index refinement lattice:

$$
L
\subset
\Gamma_1(s)\cap\Gamma_2.
$$

A fundamental parallelotope for:

$$
\Gamma_1(s)
$$

has vertices rational with respect to:

$$
\Gamma_2.
$$

After passing to a sufficiently fine rational subdivision using:

$$
L,
$$

triangulate that parallelotope into rational simplices.

Extend the triangulation periodically by:

$$
\Gamma_1(s).
$$

It descends to a rational triangulation of:

$$
X_s.
$$

QED.

---

# 5. Projectivity and smoothness

The positive-definite polarization:

$$
Q(s)
$$

makes:

$$
X_s
$$

a projective tropical abelian variety.

As a tropical torus with its standard integral-affine structure, it is smooth in the relevant tropical-manifold sense.

Therefore Amini–Piquerez applies to every:

$$
s\in S_{\mathbb Q}.
$$

---

# 6. Amini–Piquerez theorem at rational points

For every:

$$
s\in S_{\mathbb Q},
$$

and every codimension:

$$
p,
$$

the rational tropical Hodge classes satisfy:

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

For our target:

$$
p=4.
$$

Hence:

$$
\boxed{
W_{\mathrm{trop}}(s)
\subseteq
\mathcal A_{\mathrm{trop}}^4(X_s;\mathbb Q)
}
$$

for every:

$$
s\in S_{\mathbb Q}.
$$

---

# 7. Positive rational specializations are dense

Thus the CE006 binary residual satisfies:

$$
\boxed{
\mathfrak R_{\mathrm{Weil,trop}}^4(X_s)=0
}
$$

for a Zariski-dense set:

$$
s\in S_{\mathbb Q}.
$$

Therefore any possible negative result at a generic irrational parameter must exhibit a jump:

$$
\boxed{
0
\to
2
}
$$

away from a dense positive arithmetic subset.

This already shows the expected defect is not a generic algebraic-family invariant in the ordinary sense.

---

# 8. Uniform separator architecture

Let:

$$
E
$$

be a fixed finite-dimensional:

$$
\mathbb Q
$$

-vector space which trivializes the relevant degree-$4$ tropical homology carrier over a parameter chart.

A **uniform rational separator** is a family:

$$
\boxed{
\Psi_s:
E
\to
Q_0
}
$$

for some finite-dimensional:

$$
\mathbb Q
$$

-vector space:

$$
Q_0,
$$

such that the matrix entries of:

$$
\Psi_s
$$

are rational functions in the:

$$
16
$$

parameters.

We allow finitely many poles.

---

# 9. Uniform cycle annihilation

Assume:

$$
\Psi_s
$$

is defined on a nonempty Zariski-open:

$$
U
\subseteq
S
$$

and satisfies:

$$
\boxed{
\Psi_s
\left(
\mathcal A_{\mathrm{trop}}^4(X_s;\mathbb Q)
\right)
=
0
}
$$

for every:

$$
s\in U.
$$

This is exactly the type of condition a universal local flag identity is designed to enforce.

---

# 10. Weil vectors vary algebraically

CE006 constructs rational Weil generators:

$$
w_1(s),
w_2(s).
$$

Using the universal period matrix:

$$
Q(P,R),
$$

their coordinates in a fixed trivialization are polynomial functions of the entries of:

$$
P,R.
$$

Therefore:

$$
\boxed{
s
\mapsto
\Psi_s(w_i(s))
}
$$

is a rational function on:

$$
U.
$$

---

# 11. Dense-Specialization No-Go Theorem

## Theorem 11.1

Under Sections 8–10:

$$
\boxed{
\Psi_s
\left(
W_{\mathrm{trop}}(s)
\right)
=
0
}
$$

for every:

$$
s\in U.
$$

Therefore no uniform rational separator can distinguish the Weil plane from the tropical cycle span.

---

# 12. Proof

The set:

$$
U\cap S_{\mathbb Q}
$$

is Zariski dense in:

$$
U.
$$

For:

$$
s\in U\cap S_{\mathbb Q},
$$

Section 6 gives:

$$
W_{\mathrm{trop}}(s)
\subseteq
\mathcal A_{\mathrm{trop}}^4(X_s;\mathbb Q).
$$

By uniform cycle annihilation:

$$
\Psi_s(w_1(s))
=
\Psi_s(w_2(s))
=
0.
$$

Thus each rational function:

$$
s\mapsto
\Psi_s(w_i(s))
$$

vanishes on a Zariski-dense subset.

Hence it vanishes identically.

Therefore:

$$
\Psi_s
\left(
W_{\mathrm{trop}}(s)
\right)
=
0
$$

for all:

$$
s\in U.
$$

QED.

---

# 13. Polynomial version

In particular, if:

$$
\Psi_s
$$

has polynomial dependence on parameters, the same theorem applies.

Thus:

$$
\boxed{
\text{uniform polynomial separator}
=
\text{impossible}.
}
$$

---

# 14. Finite-dimensional linear-ansatz version

Suppose a Kontsevich-style flag map is determined by finitely many coefficients:

$$
c_1,\ldots,c_m
$$

which solve a finite linear system whose matrix entries are polynomial/rational functions of:

$$
P,R.
$$

If the resulting flag identities annihilate all cycle classes uniformly on a Zariski-open family, then the induced separator belongs to the class prohibited by Theorem 11.1.

Therefore:

$$
\boxed{
\text{finite algebraic linear ansatz cannot rationally separate the generic Weil plane}.
}
$$

---

# 15. Relation to Zharkov's dimension-four computation

Zharkov implements a finite ansatz in the four-dimensional Weil family.

He assumes certain differences of local flag maps are linear in parameter-lattice variables and edge directions.

This reduces the compatibility problem to a finite linear system.

The system can be solved modulo the full lattice generated by:

$$
\theta,w_1,w_2,
$$

but not modulo a proper sublattice in the attempted ansatz.

CE007 does not claim to re-prove his exact integral computation.

It gives a structural reason why a **rational** separating solution of a uniform algebraic type should not exist.

---

# 16. Rational versus integral distinction again

Theorem 11.1 is a rational no-go theorem.

It does not forbid a finite-group or torsion-valued invariant:

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

with:

$$
F
$$

finite.

Such an invariant may detect a finite-index cycle lattice defect.

But after tensoring with:

$$
\mathbb Q,
$$

it disappears.

Therefore it cannot refute the classical rational Hodge conjecture.

---

# 17. Why the theorem does not prove tropical HC generically

Theorem 11.1 says:

> no separator varying algebraically/rationally over the full family can detect a rational defect.

It does **not** imply:

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

for irrational:

$$
s.
$$

A cycle-span defect could still occur at irrational parameters.

It would simply be invisible to ordinary algebraic-family separators.

---

# 18. Arithmetic-discontinuity necessity

Suppose CE005-A8 is true for some irrational parameter:

$$
s_\infty.
$$

Then:

$$
\mathfrak R_{\mathrm{Weil,trop}}^4(X_{s_\infty})
\neq0.
$$

But dense rational specializations:

$$
s_j
\to
s_\infty
$$

have:

$$
\mathfrak R_{\mathrm{Weil,trop}}^4(X_{s_j})=0.
$$

Therefore the cycle-span rank is not continuous in the naive parameter topology.

The obstruction must detect arithmetic properties not visible to polynomial functions.

---

# 19. Arithmetic-Discontinuity Barrier

CE007 names this:

$$
\boxed{
\text{Arithmetic-Discontinuity Barrier}.
}
$$

Any viable rational counterexample certificate for the irrational tropical Weil target must depend on data such as:

- rational linear relations among periods;
- the $\mathbb Q$-span of parameter coordinates;
- Diophantine type;
- rational affine subtori;
- commensurability modules;
- failure of rational triangulability;
- another non-algebraic arithmetic invariant.

It cannot depend only on the algebraic point:

$$
(P,R)
$$

through rational functions.

---

# 20. Rational-relation module

For:

$$
s=(s_1,\ldots,s_{16})
\in S,
$$

define the rational relation module:

$$
\boxed{
\operatorname{Rel}_{\mathbb Q}(s)
=
\left\{
q\in\mathbb Q^{16}:
\sum_{i=1}^{16}
q_i s_i
=
0
\right\}.
}
$$

This module is highly discontinuous.

For a sufficiently generic:

$$
s,
$$

one may have:

$$
\operatorname{Rel}_{\mathbb Q}(s)=0.
$$

For rational:

$$
s,
$$

the relation module is very large.

Thus:

$$
\operatorname{Rel}_{\mathbb Q}
$$

is the kind of parameter datum capable of distinguishing the dense rationally triangulable locus from a generic irrational target.

---

# 21. Commensurability rank

Another candidate arithmetic invariant is:

$$
\boxed{
c(s)
=
\operatorname{rank}_{\mathbb Z}
\left(
\Gamma_1(s)\cap\Gamma_2
\right).
}
$$

At rational specializations:

$$
c(s)=8.
$$

For sufficiently generic irrational periods:

$$
c(s)
$$

can drop sharply.

Any cycle construction requiring rational affine periodicity may be sensitive to this rank.

CE007 does not yet prove that cycle-span realizability is controlled by:

$$
c(s).
$$

It only identifies it as an admissible discontinuous input.

---

# 22. Why ordinary Zariski geometry cannot see the defect

A rational relation such as:

$$
q_1s_1+\cdots+q_{16}s_{16}=0
$$

for a fixed:

$$
q
$$

defines a hyperplane.

But the statement:

> there exists some nonzero rational relation

is a countable union of rational hyperplanes.

Its complement is not an algebraic-open condition in the finite-type sense.

Therefore arithmetic relation rank can jump on countable dense subsets without contradicting algebraic geometry.

This is precisely the behavior required by CE007.

---

# 23. Countable-union geometry reappears

This is structurally parallel to R014/CE001:

$$
\boxed{
\text{algebraic-cycle realizability}
}
$$

may lie on a countable union of special loci.

In the tropical family, rationally triangulable / commensurable points can be dense yet arithmetically thin.

Thus:

$$
\boxed{
\text{dense}
\neq
\text{generic}
}
$$

in the relevant arithmetic sense.

This leaves room for irrational counterexamples even though positive points are dense.

---

# 24. The $p=4$ local flag architecture

A codimension-$4$ tropical cycle in an eight-dimensional torus has four-dimensional maximal cells.

After rational subdivision, one may work with rational:

$$
4
$$

-simplices.

For a simplex:

$$
\sigma,
$$

consider a complete affine flag:

$$
\boxed{
F_0
\subset
F_1
\subset
F_2
\subset
F_3
\subset
F_4=\sigma.
}
$$

The top cell carries a primitive volume element:

$$
\xi_{F_4}
\in
\bigwedge\nolimits^4\Gamma_2.
$$

A direct generalization of Kontsevich's construction would sum such full flags and seek a local map:

$$
\Phi_F
$$

linear in:

$$
\xi_{F_4}.
$$

---

# 25. Balancing at codimension one

For a balanced four-dimensional tropical cycle, each three-dimensional face:

$$
\tau
$$

satisfies a local normal balancing equation.

Therefore an appropriate boundary/flag map sends a cycle to a combination in which the top-dimensional normal contribution vanishes around each:

$$
\tau.
$$

This is the $p=4$ analogue of the $p=2$ flag cancellation used by Zharkov.

So a local flag architecture is formally available.

---

# 26. Tautological volume map

The tropical cycle class of a four-dimensional cell is built from:

1. its affine parameter-volume contribution;
2. its primitive tangent volume:
   $$
   \xi_{F_4}.
   $$

Thus the universal family gives a degree-$4$ parameter dependence with values in a finite-dimensional tensor space built from:

$$
\bigwedge\nolimits^4\Gamma_2.
$$

The exact Schur-functor decomposition is not needed for CE007.

All that matters is:

$$
\boxed{
\text{the universal cell class depends algebraically on the finite parameter coordinates}.
}
$$

---

# 27. Why the obvious $p=4$ flag program is blocked

Suppose one builds:

$$
\Phi
$$

from finitely many local flag types and rational/polynomial coefficients in:

$$
P,R,
$$

and proves by local balancing identities that:

$$
\Phi
$$

annihilates all tropical cycle classes throughout the family.

Then:

$$
\Phi
$$

induces a uniform algebraic separator of exactly the type forbidden by Theorem 11.1.

Therefore:

$$
\boxed{
\text{the naive higher-dimensional Kontsevich flag generalization cannot prove a rational Weil defect}.
}
$$

---

# 28. What kind of flag certificate could survive

A viable flag certificate must fail at least one hypothesis of Theorem 11.1.

Possible escape routes:

### 28.1 Arithmetic indexing

The local rule depends on:

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

or another arithmetic module.

### 28.2 Non-uniform domain

The certificate exists only for a specific irrational:

$$
s_\infty
$$

and does not extend to a Zariski-open family.

### 28.3 Discontinuous choice

The rule requires choosing a:

$$
\mathbb Q
$$

-basis of the period-coordinate span or a Hamel-type arithmetic splitting.

### 28.4 Infinite combinatorial limit

The certificate is not captured by a finite algebraic ansatz and instead depends on an infinite tower of rational approximations.

Any such route is substantially harder than the original finite linear program.

---

# 29. Specific-point separator

The correct negative target should therefore be reformulated.

Fix one explicit irrational split Weil parameter:

$$
s_\infty.
$$

Seek:

$$
\boxed{
\Psi_{s_\infty}
:
\ker N_{s_\infty}
\to
Q_0
}
$$

such that:

$$
\Psi_{s_\infty}
\left(
\mathcal A_{\mathrm{trop}}^4(X_{s_\infty};\mathbb Q)
\right)
=
0
$$

and:

$$
\Psi_{s_\infty}
\left(
W_{\mathrm{trop}}(s_\infty)
\right)
\neq0.
$$

No extension of:

$$
\Psi_{s_\infty}
$$

to nearby rational parameters is required.

---

# 30. Why one explicit irrational point is enough

Zharkov's specialization principle only needs a tropical limit whose tropical Hodge conjecture fails.

Therefore:

$$
\boxed{
\text{one explicit irrational tropical torus}
}
$$

with rational cycle-span defect is sufficient.

There is no need to prove failure on a Zariski-open family.

This is fortunate, because CE007 proves a uniform algebraic-family proof is impossible.

---

# 31. Choice of explicit arithmetic point

A useful target should have parameter coordinates with controlled algebraic independence.

For example choose positive-definite:

$$
P,R
$$

whose:

$$
16
$$

independent entries lie in a field:

$$
F
$$

of high transcendence degree over:

$$
\mathbb Q.
$$

An even stronger symbolic setup uses formal transcendental parameters:

$$
t_1,\ldots,t_{16}
$$

subject only to positivity after a real specialization.

But a purely formal generic point risks re-entering algebraic-family arguments.

To exploit arithmetic discontinuity, the certificate must depend on the actual rational relation structure of a real specialization.

---

# 32. Algebraic-independence target

Choose real numbers:

$$
t_1,\ldots,t_{16}
$$

algebraically independent over:

$$
\mathbb Q.
$$

Then in particular:

$$
\boxed{
\operatorname{Rel}_{\mathbb Q}(t)=0.
}
$$

This maximally separates the target from rational specializations.

A future CE008 can ask whether balancing plus periodicity on such an arithmetic-generic torus forces a smaller tropical cycle-class span.

---

# 33. Important warning

Algebraic independence of period parameters does **not** by itself imply:

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

Tropical cycles can have vertices with rational functions of the parameters, as Kontsevich already emphasized.

Thus the cycle universe remains huge.

Arithmetic genericity is only the candidate input for a new separator.

It is not yet the obstruction.

---

# 34. Relation to Amini–Piquerez

Amini–Piquerez establishes positivity whenever a rational triangulation exists.

CE007 shows that these positive specializations are dense enough to kill algebraically varying separators.

Therefore their theorem has two distinct consequences for HC-False:

1. a direct firewall on rationally triangulable targets;
2. an indirect no-go theorem for uniform algebraic counterexample certificates.

The second consequence is new to this branch.

---

# 35. Relation to CE006 binary residual

CE006 proved:

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

CE007 does not alter that.

It says only:

$$
\boxed{
\text{the decision }0\text{ vs }2
\text{ cannot be made by a uniform rational parameter-family separator}.
}
$$

The Boolean target remains.

---

# 36. Positive falsification remains easy in principle

If at the chosen irrational target one explicitly constructs a single:

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

as a rational combination of tropical cycle classes,

then CE006 forces:

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

So the positive branch still needs only one class.

The negative branch now needs a non-uniform arithmetic certificate.

This asymmetry should guide computation.

---

# 37. Counterexample search should invert priority

Before attempting a complicated arithmetic separator, first search aggressively for a positive tropical cycle with nonzero Weil component.

Why?

Because one explicit positive class kills the whole Weil candidate.

Therefore the computational order should be:

$$
\boxed{
\text{positive falsification search first}
}
$$

then, only if repeated exhaustive structured searches fail:

$$
\boxed{
\text{arithmetic negative certificate}.
}
$$

This is methodologically safer.

---

# 38. No inference from failed finite searches

Even if every cycle in a bounded-complexity enumeration has zero Weil component, that proves only:

$$
\boxed{
\text{bounded grammar failure}.
}
$$

It does not prove:

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

The same warning from R012 applies here.

A genuine negative certificate must quantify over all balanced tropical cycles.

---

# 39. Uniform-Ansatz No-Go is itself useful

Theorem 11.1 prevents wasting enormous compute on the wrong search space.

A solver searching finite polynomial coefficients for a family-wide:

$$
\Phi
$$

may produce increasingly elaborate linear systems.

But no solution of that architecture can yield the desired rational separation.

So:

$$
\boxed{
\text{do not spend the eightfold search budget on uniform polynomial flag ansätze}.
}
$$

This is a concrete algorithmic consequence.

---

# 40. New certificate taxonomy

CE007 introduces:

### UAF — Uniform Algebraic Family certificate

Depends rationally/algebraically on:

$$
P,R.
$$

Status:

$$
\boxed{
\text{NO-GO for rational separation}.
}
$$

### SAF — Specific Arithmetic Fiber certificate

Defined only at one arithmetic irrational target and depends on:

$$
\mathbb Q
$$

-relations / commensurability data.

Status:

$$
\boxed{
\text{VIABLE}.
}
$$

### IFC — Integral Finite-quotient certificate

Detects finite-index lattice defects.

Status:

$$
\boxed{
\text{IRRELEVANT to rational HC}.
}
$$

### PSC — Positive Specific Cycle certificate

Exhibits one nonzero Weil class in the cycle span.

Status:

$$
\boxed{
\text{decisively kills CE005-A8}.
}
$$

---

# 41. Branch status

- **CE001 non-split sixfold:** OPEN
- **CE005-A8 tropical eightfold:** OPEN
- **CE006 binary rational residual:** PROVED
- **Uniform algebraic flag separator:** DISPROVED
- **Uniform polynomial finite ansatz:** DISPROVED as rational-separation architecture
- **Specific arithmetic separator:** OPEN
- **Positive specific cycle:** OPEN
- **Integral finite-index separator:** OUT OF SCOPE for rational HC

---

# 42. Strongest theorem of CE007

The main theorem can be summarized as:

$$
\boxed{
\begin{array}{c}
\text{dense rationally triangulable specializations}\\
+\\
\text{Amini--Piquerez tropical HC}\\
+\\
\text{algebraic parameter dependence}
\end{array}
}
$$

implies:

$$
\boxed{
\text{no family-wide rational separator of the Weil plane}.
}
$$

Therefore any true irrational tropical Hodge failure must be:

$$
\boxed{
\text{arithmetically discontinuous}.
}
$$

---

# 43. Next Interface

Next counterexample round:

```text
HODGE_HCFALSE_CE008_ArithmeticIrrationalSeparator.md
```

Primary target:

$$
\boxed{
\text{Can rational-relation / commensurability data force }
\mathcal A_{\mathrm{trop}}^4
\cap
W_{\mathrm{trop}}
=
0
\text{ at one explicit irrational fiber?}
}
$$

Planned tasks:

1. choose one explicit split Weil eightfold with maximal rational-independence of period parameters;
2. compute:
   $$
   \Gamma_1\cap\Gamma_2;
   $$
3. classify rational affine subtori and rational tangent $4$-planes;
4. determine whether every balanced codimension-$4$ tropical cycle class factors through a countable rational-direction grammar;
5. project that grammar to the two-dimensional Weil plane;
6. search for an arithmetic invariant vanishing on all such rational-direction classes;
7. in parallel search for one explicit positive cycle with nonzero Weil component;
8. abandon CE005-A8 immediately if such a positive cycle is found.

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. Gives the Kontsevich flag/finite-linear-system program and records the failure of the tested finite ansatz to produce the desired proper-lattice separation.

2. O. Amini, M. Piquerez, *Tropical Clemens-Schmid sequence and existence of tropical cycles with a given cohomology class*, arXiv:2012.13142, 2020. Proves that for rationally triangulable smooth projective tropical varieties the tropical cycle-class span equals the kernel of tropical monodromy.

3. Aletheia, *HODGE_HCFALSE_CE005_TropicalWeilObstruction*, 2026-09-16.

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

---

## Canonical Source Declaration

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

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪沒有證明 tropical Weil eightfold 的 rational defect 存在；本輪證明的是一整類 uniform algebraic / polynomial counterexample certificates不可能完成 rational separation。