# HODGE_HCFALSE_CE009_PrimitiveNonlinearCycleSearch
## ——反例分支第九輪：Certified Weil Spectral Projector、Local-Tangent No-Go 與 $K$-Spectral Cycle Reduction

**作者：Aletheia（GPT-5.6 Sol）**  
**研究分支：HC-False / Counterexample Program**  
**候選編號：CE009**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Branch:** HC-False  
**Round:** CE009  
**Parent Candidate:** CE005-A8 / CE006–CE008 — split tropical Weil eightfold  
**Primary Claim:** CE006 的二維 tropical Weil plane $W_{\mathrm{trop}}$ 不只是 abstract $K$-isotypic subspace；可由一個實際 tropical $K$-endomorphism $T=[\mu]_\ast$ 的 rational polynomial 建立 legality-safe spectral projector
$$
P_W=p_W(T),
\qquad
P_W^2=P_W,
\qquad
\operatorname{Im}P_W=W_{\mathrm{trop}}.
$$
因 $T$ 將 tropical cycles push forward 成 tropical cycles，$P_W$ 保持 rational tropical cycle-class span，並滿足
$$
\boxed{
P_W\left(\mathcal A_{\mathrm{trop}}^4\right)
=
\mathcal A_{\mathrm{trop}}^4
\cap
W_{\mathrm{trop}}.
}
$$
因此任何 cycle class只要有非零 Weil component，就可經有限個 $K$-endomorphism iterates 的 rational combination變成 class 完全位於 $W_{\mathrm{trop}}$ 的 $K$-spectral cycle combination。另一方面，所有 rational local $4$-plane primitive volume tensors線性生成 $\bigwedge^4N_{\mathbb Q}$，故不存在只依單一 cell tangent volume且對所有 balanced local germs為零的 nonzero linear separator。真正 obstruction 必須是 global balancing / periodic closure，而非 local tangent restriction。  
**Status:** PROVED  
**Counterexample Status:** OPEN  
**Killed Negative Shortcut:** LOCAL-TANGENT / SINGLE-CELL LINEAR SEPARATOR  
**New Exact Search Domain:** $K$-spectral orbit-symmetrized primitive nonlinear cycles  
**Depends On:** CE006–CE008、tropical cycle pushforward functoriality、imaginary-quadratic endomorphism action  
**Backtrack Target:** 無  
**Evidence Level:** E2 / exact spectral decomposition + CRT projector + local wedge-span theorem  
**Formalization Status:** NOT FORMALIZED  
**Computation Status:** EXACT SPECTRAL REDUCTION  

---

# 0. Where CE008 left the problem

At the arithmetic-generic split Weil tropical eightfold:

$$
X
=
X_{s_\infty},
$$

CE008 proved:

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

and:

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

It also proved:

- no proper integral subtori;
- tropical Néron–Severi rank one;
- low-ancestry constructions miss the Weil plane.

The unresolved question remained:

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

CE009 attacks the primitive nonlinear cycle sector directly.

---

# 1. Tropical middle homology carrier

Let:

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

be the integral tangent lattice.

Let:

$$
\Lambda
=
\Gamma_1
$$

be the period lattice.

The middle tropical homology carrier is:

$$
\boxed{
H
=
H_4(X,\mathcal F_4;\mathbb Q)
\simeq
\bigwedge\nolimits^4\Lambda_{\mathbb Q}
\otimes
\bigwedge\nolimits^4N_{\mathbb Q}.
}
$$

Its dimension is:

$$
\binom84^2
=
4900.
$$

---

# 2. Weil endomorphism

Let:

$$
K
=
\mathbb Q(\delta),
\qquad
\delta^2=-d.
$$

The split Weil lattice model carries a $K$-action.

Choose an algebraic integer:

$$
\mu
\in
\mathcal O_K
$$

such that:

$$
\boxed{
\left(
\frac{\mu}{\bar\mu}
\right)^j
\neq
1
}
$$

for every:

$$
1\le j\le8.
$$

Equivalently, the ratio:

$$
\mu/\bar\mu
$$

is not a root of unity of order at most:

$$
8.
$$

For a generic choice:

$$
\mu=m+\delta
$$

with integer:

$$
m,
$$

this condition holds.

---

# 3. Genuine tropical endomorphism

Multiplication by:

$$
\mu
$$

preserves:

- the period lattice:
  $$
  \Lambda;
  $$
- the integral tangent lattice:
  $$
  N.
  $$

Therefore it defines an integral-affine torus endomorphism:

$$
\boxed{
f_\mu:
X\to X.
}
$$

It acts on tropical cycles by pushforward:

$$
(f_\mu)_\ast.
$$

It acts on tropical homology by:

$$
\boxed{
T
=
(f_\mu)_\ast:
H\to H.
}
$$

The important legality fact is:

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

This is not an abstract Hodge operator.

It is induced by an actual tropical morphism.

---

# 4. $K$-eigenspace decomposition

After extending scalars to:

$$
K,
$$

both:

$$
\Lambda_K
$$

and:

$$
N_K
$$

split into:

$$
+\delta
$$

and:

$$
-\delta
$$

eigenspaces:

$$
\boxed{
\Lambda_K
=
\Lambda_+
\oplus
\Lambda_-,
}
$$

$$
\boxed{
N_K
=
N_+
\oplus
N_-.
}
$$

Each eigenspace has dimension:

$$
4
$$

over:

$$
K.
$$

Multiplication by:

$$
\mu
$$

acts:

- by:
  $$
  \mu
  $$
  on the $+$ eigenspaces;
- by:
  $$
  \bar\mu
  $$
  on the $-$ eigenspaces.

---

# 5. Spectral grading of middle homology

Decompose:

$$
\bigwedge\nolimits^4\Lambda_K
$$

according to the number:

$$
r
$$

of $+$ factors:

$$
0\le r\le4.
$$

Likewise decompose:

$$
\bigwedge\nolimits^4N_K
$$

according to:

$$
s,
\qquad
0\le s\le4.
$$

On the tensor block:

$$
H_{r,s},
$$

the operator:

$$
T
$$

has eigenvalue:

$$
\boxed{
\lambda_{r+s}
=
\mu^{r+s}
\bar\mu^{\,8-r-s}.
}
$$

Thus only the total:

$$
t=r+s
$$

matters:

$$
0\le t\le8.
$$

---

# 6. Extremal eigenvalues

The eigenvalue:

$$
\mu^8
$$

occurs only when:

$$
r=s=4.
$$

Hence its eigenspace is:

$$
\boxed{
\bigwedge\nolimits^4\Lambda_+
\otimes
\bigwedge\nolimits^4N_+,
}
$$

which is one-dimensional over:

$$
K.
$$

This is exactly the line:

$$
K\Omega_+.
$$

Similarly the eigenvalue:

$$
\bar\mu^8
$$

occurs only for:

$$
r=s=0,
$$

with one-dimensional eigenspace:

$$
K\Omega_-.
$$

Therefore the rational span of these conjugate lines is precisely:

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

---

# 7. Distinct spectral factors

For:

$$
0\le t\le8,
$$

define:

$$
\lambda_t
=
\mu^t\bar\mu^{\,8-t}.
$$

If:

$$
\lambda_t=\lambda_u,
$$

then:

$$
\left(
\frac{\mu}{\bar\mu}
\right)^{t-u}
=
1.
$$

By the choice of:

$$
\mu,
$$

this forces:

$$
t=u.
$$

Therefore all:

$$
9
$$

complex eigenvalues:

$$
\lambda_0,\ldots,\lambda_8
$$

are distinct.

---

# 8. Rational spectral factors

Complex conjugation exchanges:

$$
\lambda_t
\leftrightarrow
\lambda_{8-t}.
$$

Thus over:

$$
\mathbb Q,
$$

the spectral factors are:

$$
\boxed{
m_t(x)
=
(x-\lambda_t)
(x-\lambda_{8-t}),
\qquad
0\le t\le3,
}
$$

together with the middle rational factor:

$$
\boxed{
m_4(x)
=
x-N_{K/\mathbb Q}(\mu)^4.
}
$$

The Weil factor is:

$$
\boxed{
m_W(x)
=
m_0(x)
=
(x-\mu^8)
(x-\bar\mu^8).
}
$$

All these factors are pairwise coprime in:

$$
\mathbb Q[x].
$$

---

# 9. Certified Weil Spectral Projector

Let:

$$
\boxed{
M_{\mathrm{other}}(x)
=
m_1(x)m_2(x)m_3(x)m_4(x).
}
$$

Since:

$$
\gcd
\left(
M_{\mathrm{other}},
m_W
\right)
=
1,
$$

there exists:

$$
u(x)
\in
\mathbb Q[x]
$$

such that:

$$
\boxed{
u(x)
M_{\mathrm{other}}(x)
\equiv
1
\pmod{m_W(x)}.
}
$$

Define:

$$
\boxed{
p_W(x)
=
u(x)M_{\mathrm{other}}(x)
}
$$

modulo the total spectral polynomial:

$$
m_Wm_1m_2m_3m_4.
$$

Then:

$$
p_W(x)
\equiv
1
\pmod{m_W},
$$

and:

$$
p_W(x)
\equiv
0
\pmod{m_t}
$$

for:

$$
1\le t\le4.
$$

---

# 10. Projector Theorem

## Theorem 10.1

Define:

$$
\boxed{
P_W
=
p_W(T).
}
$$

Then:

$$
\boxed{
P_W^2=P_W,
}
$$

and:

$$
\boxed{
\operatorname{Im}P_W
=
W_{\mathrm{trop}}.
}
$$

### Proof

Over:

$$
K,
$$

the operator:

$$
T
$$

is diagonalizable with the distinct eigenvalues:

$$
\lambda_0,\ldots,\lambda_8.
$$

The polynomial:

$$
p_W
$$

is:

- $1$ on:
  $$
  \lambda_0,\lambda_8;
  $$
- $0$ on:
  $$
  \lambda_1,\ldots,\lambda_7.
  $$

Hence:

$$
P_W
$$

is the spectral idempotent onto:

$$
K\Omega_+
\oplus
K\Omega_-.
$$

Its rational form is exactly:

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

QED.

---

# 11. Why this projector is legality-safe

R024 warned that an abstract Hodge projector may not be algebraic.

The present:

$$
P_W
$$

is different.

Write:

$$
p_W(x)
=
\sum_{j=0}^M
c_jx^j,
\qquad
c_j\in\mathbb Q.
$$

Then:

$$
\boxed{
P_W
=
\sum_{j=0}^M
c_j
(f_{\mu^j})_\ast.
}
$$

Every:

$$
(f_{\mu^j})_\ast
$$

comes from an actual tropical endomorphism.

Therefore for any rational tropical cycle class:

$$
z,
$$

$$
\boxed{
P_Wz
}
$$

is a rational linear combination of tropical cycle classes.

Hence:

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

This is a genuine legality-certified projector.

---

# 12. Exact Cycle-Span Retract Theorem

## Theorem 12.1

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

### Proof

If:

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

then legality safety gives:

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

Also:

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

So:

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

Conversely, if:

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

then:

$$
P_Ww=w.
$$

Hence:

$$
w
\in
P_W
\left(
\mathcal A_{\mathrm{trop}}^4
\right).
$$

QED.

---

# 13. Orbit-symmetrized cycle

Let:

$$
Z
$$

be any codimension-$4$ tropical cycle.

Define the rational formal cycle combination:

$$
\boxed{
Z_W
=
p_W(f_{\mu\ast})Z
=
\sum_{j=0}^M
c_j
(f_{\mu^j})_\ast Z.
}
$$

Then:

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

So if:

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

has any nonzero Weil component,

then:

$$
Z_W
$$

is a rational tropical cycle combination whose class lies **entirely** in:

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

---

# 14. Spectral Search Reduction

## Theorem 14.1

The following are equivalent.

### (A)

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

### (B)

There exists a rational tropical cycle combination:

$$
C
$$

such that:

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

### (C)

There exists an arbitrary tropical cycle:

$$
Z
$$

with:

$$
P_Wcl_{\mathrm{trop}}(Z)\neq0.
$$

Moreover one may take:

$$
C=Z_W.
$$

Thus positive search may be restricted to:

$$
\boxed{
K\text{-spectral orbit-symmetrized cycles}.
}
$$

---

# 15. Binary consequence revisited

CE006 proved:

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

CE009 gives a stronger operational interpretation:

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

or:

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

Thus the unresolved quantity is literally the image rank of one legality-safe projector on the cycle span.

---

# 16. Low-ancestry sector is spectrally killed

CE008 showed:

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

The polarization line transforms under:

$$
f_\mu
$$

with the rational eigenvalue:

$$
N_{K/\mathbb Q}(\mu)^4
=
\lambda_4.
$$

Therefore:

$$
p_W(\lambda_4)=0.
$$

Hence:

$$
\boxed{
P_W
\left(
\mathcal A_{\mathrm{taut,low}}^4
\right)
=
0.
}
$$

So the spectral search automatically deletes every already-excluded low-ancestry class.

Only primitive nonlinear cycles can survive.

---

# 17. Explicit control example

Take:

$$
K=\mathbb Q(i)
$$

and:

$$
\mu=2+i.
$$

Then:

$$
N(\mu)=5.
$$

The nine eigenvalues:

$$
\lambda_t
=
\mu^t\bar\mu^{\,8-t}
$$

are:

$$
-527+336i,
$$

$$
-585-220i,
$$

$$
-175-600i,
$$

$$
375-500i,
$$

$$
625,
$$

$$
375+500i,
$$

$$
-175+600i,
$$

$$
-585+220i,
$$

$$
-527-336i.
$$

The rational spectral factors are:

$$
\boxed{
m_W(x)
=
x^2+1054x+390625,
}
$$

$$
m_1(x)
=
x^2+1170x+390625,
$$

$$
m_2(x)
=
x^2+350x+390625,
$$

$$
m_3(x)
=
x^2-750x+390625,
$$

and:

$$
m_4(x)
=
x-625.
$$

They are pairwise coprime.

Therefore the spectral projector can be computed exactly by the Euclidean algorithm in:

$$
\mathbb Q[x].
$$

This supplies a completely explicit finite projector for computation.

---

# 18. Local tropical cycle germs

Now test the most naive negative strategy.

Let:

$$
L
\subset
N_{\mathbb Q}
$$

be a rational:

$$
4
$$

-plane with primitive tangent lattice basis:

$$
v_1,\ldots,v_4.
$$

Its oriented tangent volume is:

$$
\boxed{
\xi_L
=
v_1\wedge v_2\wedge v_3\wedge v_4
\in
\bigwedge\nolimits^4N_{\mathbb Q}.
}
$$

A rational affine $4$-plane with weight one is locally a balanced tropical cycle germ.

So every such:

$$
\xi_L
$$

is locally admissible.

---

# 19. Local Tangent Span Theorem

## Theorem 19.1

The rational span of primitive tangent volumes of locally balanced rational $4$-plane germs is:

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

### Proof

Choose the standard basis:

$$
e_1,\ldots,e_8
$$

of:

$$
N.
$$

For every:

$$
I
=
\{i_1<i_2<i_3<i_4\},
$$

the coordinate:

$$
4
$$

-plane:

$$
L_I
=
\operatorname{span}_{\mathbb Q}
\{e_{i_1},e_{i_2},e_{i_3},e_{i_4}\}
$$

is a rational balanced local germ.

Its primitive volume is:

$$
e_I
=
e_{i_1}
\wedge
e_{i_2}
\wedge
e_{i_3}
\wedge
e_{i_4}.
$$

The:

$$
\binom84=70
$$

vectors:

$$
e_I
$$

form a basis of:

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

QED.

---

# 20. No Tangent-Only Linear Separator

## Corollary 20.1

Let:

$$
\ell:
\bigwedge\nolimits^4N_{\mathbb Q}
\to
Q_0
$$

be a rational linear map.

If:

$$
\ell(\xi_L)=0
$$

for every locally admissible rational balanced $4$-plane germ:

$$
L,
$$

then:

$$
\boxed{
\ell=0.
}
$$

Therefore no nonzero linear obstruction depending only on one cell's tangent volume can annihilate all tropical cycles.

---

# 21. Meaning of the local no-go

A local tropical cell is not yet a global cycle.

The counterexample obstruction cannot be:

$$
\boxed{
\text{"this tangent }4\text{-plane is forbidden."}
}
$$

Every rational tangent:

$$
4
$$

-plane is locally legal.

The obstruction, if it exists, must involve how many legal local pieces can or cannot be assembled into a **global periodic balanced cycle**.

---

# 22. Codimension-one balancing is essential but insufficient locally

At a codimension-one face:

$$
\tau,
$$

the adjacent:

$$
4
$$

-cells satisfy a balancing relation in the quotient normal lattice.

This is a relation among several locally legal tangent volumes.

A single rational linear plane already supplies a balanced local fan.

Therefore balancing cannot exclude tangent directions one by one.

Only the global network of balancing relations can constrain the final homology class.

---

# 23. Period closure is the genuinely nonlocal input

A tropical cycle on:

$$
X
=
\mathbb R^8/\Lambda
$$

lifts to a:

$$
\Lambda
$$

-periodic balanced polyhedral complex in:

$$
\mathbb R^8.
$$

The tropical homology class remembers simultaneously:

1. how the lifted cells wind around:
   $$
   \Lambda;
   $$
2. their integral tangent volumes in:
   $$
   N.
   $$

The Weil tensor is precisely an exceptional coupling between these two factors.

Hence a potential obstruction must couple:

$$
\boxed{
\text{period winding}
}
$$

with:

$$
\boxed{
\text{tangent balancing}.
}
$$

Neither factor alone can solve the problem.

---

# 24. Why CE008's no-subtorus theorem was not enough

A flat integral subtorus would use one fixed tangent:

$$
4
$$

-plane globally.

CE008 removed those.

A primitive nonlinear cycle can switch between many rational tangent planes while its lifted support winds through irrational period directions.

The local tangent span theorem shows there is no shortage of legal slope pieces.

So the remaining obstruction is a **global assembly obstruction**.

---

# 25. $K$-spectral cycle group

Let:

$$
Z_{\mathrm{trop}}^4(X)_{\mathbb Q}
$$

be the rational vector space generated by codimension-$4$ tropical cycles.

Define the Weil-spectral cycle group:

$$
\boxed{
Z_W^4(X)
=
p_W(f_{\mu\ast})
Z_{\mathrm{trop}}^4(X)_{\mathbb Q}.
}
$$

Every element is a rational combination of actual tropical cycles.

Its cycle classes satisfy:

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

This is an exact reformulation.

---

# 26. Counterexample criterion

By CE006 and Theorem 12.1:

$$
\boxed{
\mathrm{CE005\mbox{-}A8}
\text{ tropical Weil defect}
}
$$

holds if and only if:

$$
\boxed{
cl_{\mathrm{trop}}
\left(
Z_W^4(X)
\right)
=
0.
}
$$

If there is one:

$$
C\in Z_W^4(X)
$$

with:

$$
cl_{\mathrm{trop}}(C)\neq0,
$$

then:

$$
\boxed{
cl_{\mathrm{trop}}
\left(
Z_W^4(X)
\right)
=
W_{\mathrm{trop}},
}
$$

and the candidate dies.

---

# 27. Positive search becomes finite-orbit symmetrization

Given any primitive nonlinear tropical cycle:

$$
Z,
$$

one does not need to understand its full class.

Compute:

$$
\boxed{
Z_W
=
\sum_j
c_j
(f_{\mu^j})_\ast Z.
}
$$

Then test only:

$$
\boxed{
cl_{\mathrm{trop}}(Z_W)
\stackrel{?}{=}0.
}
$$

This is a major computational reduction.

All non-Weil spectral components are automatically removed.

---

# 28. Negative proof target

The negative branch now has one exact theorem to prove:

## Global Spectral Vanishing Conjecture

For the arithmetic-generic split Weil tropical eightfold:

$$
\boxed{
\forall
Z
\in
Z_{\mathrm{trop}}^4(X)_{\mathbb Q},
\qquad
cl_{\mathrm{trop}}
\left(
p_W(f_{\mu\ast})Z
\right)
=
0.
}
$$

Equivalently:

$$
\boxed{
cl_{\mathrm{trop}}
\left(
Z_W^4(X)
\right)
=
0.
}
$$

This would give:

$$
\boxed{
\mathfrak R_{\mathrm{Weil,trop}}^4
=
W_{\mathrm{trop}}
}
$$

of rational dimension:

$$
2.
$$

---

# 29. Why this is stronger than a bounded grammar test

The conjecture quantifies over:

$$
\boxed{
\text{all balanced codimension-4 tropical cycles}.
}
$$

It does not merely exclude:

- subtori;
- divisors;
- theta products;
- Fourier–Mukai tautological cycles;
- bounded-complexity polyhedra.

Therefore proving it would be a genuine cycle-span theorem.

---

# 30. Why a local proof cannot establish it

Corollary 20.1 rules out any argument of the form:

> every allowed local tangent cell has zero Weil contribution.

That statement is false as an obstruction architecture.

A successful proof must use at least one of:

1. global period closure;
2. interactions between multiple slope chambers;
3. monodromy of the lifted support;
4. arithmetic relations among cell translations;
5. a global balancing invariant;
6. a tropical regulator sensitive to winding and tangent data simultaneously.

---

# 31. Chain-level versus homology-level projector

The operator:

$$
p_W(f_{\mu\ast})
$$

is defined on actual cycle combinations before passing to homology.

Therefore:

$$
Z_W
$$

is meaningful at the cycle level.

However:

$$
P_W
$$

being an idempotent on homology does not imply:

$$
p_W(f_{\mu\ast})^2
=
p_W(f_{\mu\ast})
$$

as an identity of raw cycle chains modulo only subdivision/balancing.

Idempotence is a homological statement.

This distinction must be preserved in future formalization.

---

# 32. Spectral relation on the class

For:

$$
C\in Z_W^4(X),
$$

its class satisfies:

$$
\boxed{
m_W(T)
cl_{\mathrm{trop}}(C)
=
0.
}
$$

Explicitly:

$$
\boxed{
\left(
T^2
-
(\mu^8+\bar\mu^8)T
+
N(\mu)^8
\right)
cl_{\mathrm{trop}}(C)
=
0.
}
$$

A positive Weil cycle therefore has a rigid quadratic $K$-spectral signature.

This can be used as a search filter.

---

# 33. Example over $\mathbb Q(i)$

For:

$$
\mu=2+i,
$$

we have:

$$
\mu^8
=
-527-336i,
$$

$$
\bar\mu^8
=
-527+336i.
$$

Thus:

$$
\boxed{
m_W(x)
=
x^2+1054x+390625.
}
$$

Any nonzero Weil cycle class:

$$
c
$$

must satisfy:

$$
\boxed{
T^2c
+
1054Tc
+
390625c
=
0.
}
$$

This is an exact integer-coefficient search condition.

---

# 34. Spectral signature of the theta sector

The middle theta class satisfies:

$$
\boxed{
T\Theta^4
=
N(\mu)^4
\Theta^4.
}
$$

For:

$$
\mu=2+i,
$$

$$
N(\mu)^4
=
5^4
=
625.
$$

Hence:

$$
\Theta^4
$$

lies in the factor:

$$
x-625,
$$

not in:

$$
m_W(x).
$$

The spectral projector therefore kills it exactly.

This recovers CE008's separation with no pairing computation.

---

# 35. Generalization of the separation

Any cycle class lying entirely in spectral factors:

$$
m_1,
m_2,
m_3,
m_4
$$

is annihilated by:

$$
P_W.
$$

Therefore future positive construction searches need only examine cycles whose $K$-orbit class has the extremal spectral character:

$$
\mu^8
\quad\text{or}\quad
\bar\mu^8.
$$

This is substantially stronger than merely excluding divisor-generated classes.

---

# 36. What CE009 kills

CE009 rejects:

### A. Tangent-only separator

No nonzero linear tangent-volume functional can vanish on every locally legal cell.

### B. Single-cell Weil prohibition

There is no local rule forbidding all rational $4$-plane germs from contributing to a global Weil class.

### C. Search over all spectral sectors

Unnecessary.

The legality-safe projector removes every irrelevant spectral component.

---

# 37. What CE009 proves

It proves:

$$
\boxed{
\text{Weil plane has an explicit legality-safe spectral retract}.
}
$$

It proves:

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

It proves:

$$
\boxed{
\text{positive realizability can be searched entirely inside }K\text{-spectral orbit symmetrizations}.
}
$$

It proves:

$$
\boxed{
\text{local tangent legality is too large to provide the needed separator}.
}
$$

---

# 38. Counterexample branch state

- **CE001 non-split sixfold:** OPEN
- **CE005-A8 tropical eightfold:** OPEN
- **CE006 explicit Weil plane / binary residual:** PROVED
- **CE007 uniform algebraic separator:** NO-GO
- **CE008 low-ancestry cycle grammar:** EXCLUDED
- **CE009 Certified Weil Spectral Projector:** PROVED
- **Local tangent-only obstruction:** NO-GO
- **Global Spectral Vanishing:** OPEN
- **Rational Hodge counterexample:** NOT YET

---

# 39. The problem is now genuinely global

After nine HC-False rounds, the tropical branch has eliminated nearly every local shortcut.

The remaining statement is no longer:

> Are there special local slopes?

It is:

> Can a globally periodic balanced nonlinear $4$-cycle on an arithmetic-irrational split Weil tropical eightfold carry the extremal $K$-spectral homology character?

That is the exact current hard core.

---

# 40. Next Interface

Next counterexample round:

```text
HODGE_HCFALSE_CE010_GlobalSpectralBalancing.md
```

Primary target:

$$
\boxed{
cl_{\mathrm{trop}}
\left(
p_W(f_{\mu\ast})Z
\right)
=
0
\quad
\forall Z
\ ?
}
$$

Planned attacks:

1. lift a tropical cycle to a $\Lambda$-periodic balanced polyhedral complex in $\mathbb R^8$;
2. encode its slope chambers and deck-translation winding data separately;
3. derive the exact homology tensor as a sum of period-winding $\otimes$ tangent-volume contributions;
4. apply the explicit polynomial spectral projector before quotienting;
5. look for cancellation forced by periodicity and balancing;
6. define a global $K$-spectral winding invariant;
7. search for an explicit primitive nonlinear cycle violating the proposed cancellation;
8. if any such cycle has nonzero spectral class, kill CE005-A8 immediately.

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. Describes tropical homology of abelian tori, eigenwave Hodge classes, cycle classes and Kontsevich's counterexample program.

2. O. Amini, M. Piquerez, *Tropical Clemens-Schmid sequence and existence of tropical cycles with a given cohomology class*, arXiv:2012.13142. Gives the monodromy/eigenwave interpretation and tropical Hodge theorem in the rationally triangulable case.

3. A. Gross, F. Shokrieh, *Tautological cycles on tropical Jacobians*, Algebra & Number Theory 17 (2023), 885–921. Supplies cycle functoriality and real-torus-with-integral-structure framework used by the tropical abelian discussion.

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.

---

## Canonical Source Declaration

本檔案為 HC-False 分支第九篇正式 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；本輪建立 legality-safe Weil spectral projector、將 cycle-span intersection精確化為 spectral image，並證明 local tangent-only separator不可能完成反例。