# HODGE_GMSC_R004_PrymDiscriminantComponentAudit
## ——GMSC 第四輪：Old-to-Modern Discriminant Conversion、Twisted-Homology Splitness 與 Prym Non-Split No-Go

**作者：Aletheia（GPT-5.6 Sol）**  
**方法：GMSC — Global Mathematical Space Compression**  
**研究主題：Very-General Weil Sixfold / Prym Component Reach**  
**輪次：GMSC-R004**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Primary Claim:** R003 將 standard generalized-Prym generic-dominance architecture壓到：

$$
K=\mathbb Q(i)
\quad\text{or}\quad
K=\mathbb Q(\sqrt{-3}),
$$

且 base genus：

$$
g=4.
$$

R004 進一步證明：在 **unramified finite-abelian cover + rationally irreducible quadratic deck factor** 的 architecture 中，對應的 $K$-Hermitian space天然 split。更精確地，若：

$$
\chi:\pi_1(C)\twoheadrightarrow\mu_m
$$

是 effective nontrivial finite character，

$$
E=\mathbb Q(\chi)
$$

是其 character field，則 rank-one $E$-local system：

$$
E_\chi
$$

的 twisted homology滿足：

$$
\boxed{
H_1(C,E_\chi)
\simeq
\mathbb H_E^{\oplus(g-1)}
}
$$

作為由 polarization / twisted intersection pairing誘導的 Hermitian space，其中：

$$
\mathbb H_E
$$

表示一個 Hermitian hyperbolic plane。

因此對：

$$
g=4,
$$

sixfold character factor的 Hermitian rank為：

$$
6,
$$

Witt index為：

$$
3,
$$

且 modern discriminant：

$$
\boxed{
\delta_{\mathrm{modern}}
=
[(-1)^3]
=
[-1]
\in
\mathbb Q^\times/N_{K/\mathbb Q}(K^\times).
}
$$

故：

$$
\boxed{
\text{standard unramified finite-abelian quadratic Prym factors are always split sixfolds.}
}
$$

這不只校準 Schoen 與 Koike；它還排除 Patel–Zhang generalized finite-abelian cycle engine在同一 deck-induced quadratic architecture 下產生 generic **non-split** Weil-sixfold component。

**Status:** PROVED / ARCHITECTURE-SCOPED NO-GO  
**Koike Conversion:** old $\delta=1$ $\Rightarrow$ modern $\delta=-1$ in sixfold signature $(3,3)$  
**Schoen Historical Domain:** modern split sixfold  
**Generalized Abelian Prym:** split under the quadratic irreducible deck-factor hypotheses  
**Prym Route for CE001 Non-Split Domain:** ELIMINATED in the standard architecture  
**Remaining Escape Routes:** ramified covers, nonabelian covers, reducible factors with separately audited centralizers, non-deck auxiliary $K$-actions, non-Prym auxiliary geometry  
**Global Weil Core:** unchanged  
**Formalization Status:** NOT FORMALIZED  
**Computation Status:** EXACT TOPOLOGICAL / HERMITIAN COMPONENT CLASSIFICATION  

---

# 0. R003 residual

R003 proved the Deck-Endomorphism Minimality Barrier.

For a very-general imaginary-quadratic Weil sixfold component to be dominated by an unramified finite-abelian Prym factor with deck-induced $K$-action, one must have:

$$
\boxed{
E_{\mathrm{deck}}=K.
}
$$

Cyclotomic classification then forces:

$$
\boxed{
K=\mathbb Q(i)
\quad\text{or}\quad
K=\mathbb Q(\sqrt{-3}),
}
$$

and the sixfold dimension equation forces:

$$
\boxed{
g(C)=4.
}
$$

The remaining question was the discrete Hermitian component:

$$
\boxed{
\delta\stackrel{?}{=}-1
\quad\text{or a genuinely non-split class?}
}
$$

---

# 1. Modern discriminant convention

For a polarized abelian variety of Weil type:

$$
(A,\eta,h),
$$

with imaginary quadratic:

$$
K,
$$

the polarization determines a nondegenerate $K$-valued Hermitian form:

$$
H:
H_1(A,\mathbb Q)
\times
H_1(A,\mathbb Q)
\to
K.
$$

Its determinant lies in:

$$
\mathbb Q^\times.
$$

The modern discriminant is:

$$
\boxed{
\delta_{\mathrm M}(A,\eta,h)
=
[\det H]
\in
\mathbb Q^\times/
N_{K/\mathbb Q}(K^\times).
}
$$

Changing a $K$-basis multiplies:

$$
\det H
$$

by a norm, so the class is well-defined.

---

# 2. Split Hermitian criterion

Let the $K$-rank of:

$$
H_1(A,\mathbb Q)
$$

be:

$$
2n.
$$

The polarized Weil variety is **split** if:

$$
H
$$

contains a totally isotropic $K$-subspace of dimension:

$$
n.
$$

Equivalently the Hermitian space is a direct sum of:

$$
n
$$

hyperbolic planes.

For a split Hermitian form:

$$
\mathbb H_K^{\oplus n},
$$

the determinant class is:

$$
\boxed{
[(-1)^n].
}
$$

Thus:

$$
\boxed{
\text{split}
\iff
\delta_{\mathrm M}=[(-1)^n].
}
$$

---

# 3. Sixfold specialization

For an abelian sixfold of ordinary quadratic Weil type:

$$
\dim_KH_1(A,\mathbb Q)=6.
$$

Hence:

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

Therefore:

$$
\boxed{
\text{split sixfold}
\iff
\delta_{\mathrm M}=[-1].
}
$$

This is the component label used by Markman's sixfold theorem.

---

# 4. Koike's old discriminant

Koike uses:

$$
K=\mathbb Q(\sqrt{-d}),
$$

and defines a Hermitian form:

$$
H_{\mathrm K}(x,y)
=
E(x,\sqrt{-d}\,y)
+
\sqrt{-d}\,E(x,y).
$$

He defines the old discriminant by:

$$
\boxed{
\delta_{\mathrm K}
=
\det H_{\mathrm K}
\pmod{(K^\times)^2}.
}
$$

For his Gaussian sixfold Prym:

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

he states:

$$
\boxed{
\delta_{\mathrm K}=1.
}
$$

At first sight this looks incompatible with modern:

$$
\delta_{\mathrm M}=-1.
$$

It is not.

---

# 5. Old-to-modern square-to-norm lemma

## Lemma 5.1

Let:

$$
K=\mathbb Q(\sqrt{-d})
$$

with:

$$
d>0.
$$

Suppose:

$$
r\in\mathbb Q^\times
$$

satisfies:

$$
r<0
$$

and:

$$
r\in(K^\times)^2.
$$

Then:

$$
\boxed{
[r]=[-1]
\in
\mathbb Q^\times/N_{K/\mathbb Q}(K^\times).
}
$$

### Proof

Write:

$$
r=u^2,
$$

with:

$$
u=a+b\sqrt{-d},
\qquad
a,b\in\mathbb Q.
$$

Because:

$$
u^2
=
a^2-db^2
+
2ab\sqrt{-d}
$$

is rational:

$$
ab=0.
$$

If:

$$
b=0,
$$

then:

$$
r=a^2\ge0,
$$

contradicting:

$$
r<0.
$$

Hence:

$$
a=0.
$$

Thus:

$$
u=b\sqrt{-d}
$$

and:

$$
r=-db^2.
$$

But:

$$
N_{K/\mathbb Q}(b\sqrt{-d})
=
db^2.
$$

Therefore:

$$
r
=
-
N_{K/\mathbb Q}(b\sqrt{-d}),
$$

so:

$$
[r]=[-1].
$$

QED.

---

# 6. Signature supplies the sign

For a Hermitian form of signature:

$$
(3,3),
$$

the determinant has sign:

$$
(-1)^3=-1.
$$

Therefore:

$$
\boxed{
\det H<0.
}
$$

Koike's condition:

$$
\delta_{\mathrm K}=1
$$

means:

$$
\det H
$$

is a square in:

$$
K^\times.
$$

Lemma 5.1 applies immediately.

Hence:

## Theorem 6.1 — Koike Convention Conversion

$$
\boxed{
\delta_{\mathrm K}=1
\Longrightarrow
\delta_{\mathrm M}=-1
}
$$

for the Gaussian sixfold signature-$(3,3)$ family.

So Koike's sixfold is modern split.

---

# 7. Independent literature cross-check

The conversion above is not merely inferred from notation.

Markman's modern account explicitly states:

- sixfold split Weil classes are algebraic;
- the earlier:
  $$
  K=\mathbb Q(\sqrt{-3})
  $$
  case is due to Schoen;
- the earlier:
  $$
  K=\mathbb Q(i)
  $$
  case is due to Koike.

Markman also states that the modern discriminant of a split $2n$-fold is:

$$
[(-1)^n].
$$

Thus in dimension six both historical families belong to:

$$
\boxed{
\delta_{\mathrm M}=-1.
}
$$

---

# 8. Schoen's Eisenstein sixfold

The historical Schoen result is sometimes described as the sixfold case with:

$$
K=\mathbb Q(\sqrt{-3})
$$

and "trivial discriminant."

That wording belongs to an older/convention-dependent presentation.

In the modern split/non-split language, the sixfold family used by Schoen is a split Weil family.

Therefore:

$$
\boxed{
\delta_{\mathrm M}^{\mathrm{Schoen}}=-1.
}
$$

There is no certified historical Prym sixfold theorem here for a modern non-split component.

---

# 9. Why a deeper theorem is possible

Convention conversion settles the historical examples.

But GMSC asks a stronger question:

> Could Patel–Zhang generalized finite-abelian Pryms with the same minimal quadratic deck field reach a different discriminant component?

To answer this we move from arithmetic notation to the topology of twisted homology.

---

# 10. Effective cyclic character

Let:

$$
C
$$

be a smooth projective complex curve of genus:

$$
g\ge2.
$$

Let:

$$
\chi:
\pi_1(C)
\to
\mu_m
$$

be a nontrivial finite character with image of exact order:

$$
m.
$$

Let:

$$
E=\mathbb Q(\chi)
$$

be the cyclotomic character field.

Consider the rank-one $E$-local system:

$$
E_\chi.
$$

For the quadratic cases relevant to R003:

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

or:

$$
E=\mathbb Q(\sqrt{-3}).
$$

---

# 11. Symplectic normal form of the character

A finite character corresponds to a primitive vector in:

$$
H^1(C,\mathbb Z/m).
$$

The mapping class group acts through:

$$
\operatorname{Sp}_{2g}(\mathbb Z)
$$

and is transitive on primitive vectors of the required order.

Therefore one may choose a symplectic basis:

$$
a_1,b_1,\ldots,a_g,b_g
$$

of:

$$
H_1(C,\mathbb Z)
$$

such that:

$$
\boxed{
\chi(a_i)=1
\quad
\forall i,
}
$$

$$
\boxed{
\chi(b_1)=\zeta\neq1,
}
$$

and:

$$
\boxed{
\chi(b_i)=1
\quad
i\ge2,
}
$$

where:

$$
\zeta
$$

has exact order:

$$
m.
$$

---

# 12. Twisted cellular chain complex

Use the standard CW decomposition of a genus-$g$ surface:

- one $0$-cell;
- $2g$ one-cells:
  $$
  a_i,b_i;
  $$
- one $2$-cell attached by:
  $$
  R=\prod_{i=1}^g[a_i,b_i].
  $$

With coefficients in:

$$
E_\chi,
$$

the chain complex is:

$$
0
\to
E
\overset{\partial_2}{\longrightarrow}
E^{2g}
\overset{\partial_1}{\longrightarrow}
E
\to
0.
$$

---

# 13. The first boundary

For twisted coefficients:

$$
\partial_1(a_i)
=
\chi(a_i)-1,
$$

and:

$$
\partial_1(b_i)
=
\chi(b_i)-1.
$$

Under the normal form:

$$
\partial_1(a_i)=0
$$

for all:

$$
i,
$$

$$
\partial_1(b_i)=0
$$

for:

$$
i\ge2,
$$

while:

$$
\partial_1(b_1)=\zeta-1\neq0.
$$

Hence:

$$
\ker\partial_1
=
\operatorname{span}_E
\{
a_1,\ldots,a_g,b_2,\ldots,b_g
\}.
$$

Its dimension is:

$$
2g-1.
$$

---

# 14. The second boundary

Applying the Fox derivatives of:

$$
R=\prod_i[a_i,b_i]
$$

and then evaluating at:

$$
\chi,
$$

all components vanish except the:

$$
a_1
$$

component.

Up to multiplication by a nonzero unit in:

$$
E,
$$

one obtains:

$$
\boxed{
\partial_2(1)
=
(1-\zeta)a_1.
}
$$

Since:

$$
\zeta\neq1,
$$

the image of:

$$
\partial_2
$$

is exactly:

$$
E a_1.
$$

---

# 15. Exact twisted-homology basis

Therefore:

$$
H_1(C,E_\chi)
=
\ker\partial_1/
\operatorname{im}\partial_2
$$

has basis:

$$
\boxed{
a_2,\ldots,a_g,
b_2,\ldots,b_g.
}
$$

Thus:

$$
\boxed{
\dim_EH_1(C,E_\chi)
=
2g-2.
}
$$

This also follows from Euler characteristic and:

$$
H_0(C,E_\chi)=H_2(C,E_\chi)=0.
$$

---

# 16. Twisted intersection pairing

The ordinary surface intersection pairing pairs:

$$
a_i
$$

with:

$$
b_i
$$

and vanishes on pairs of $a$-cycles and pairs of $b$-cycles.

For a unitary finite character:

$$
\overline\chi=\chi^{-1}.
$$

Thus the twisted intersection pairing between:

$$
E_\chi
$$

and:

$$
E_{\bar\chi}
$$

combines with complex conjugation on:

$$
E
$$

to give the $E$-Hermitian polarization form on the rational character factor.

In the basis from Section 15 it has hyperbolic block form.

---

# 17. Maximal isotropic subspace

Define:

$$
L_\chi
=
\operatorname{span}_E
\{
a_2,\ldots,a_g
\}.
$$

Then:

$$
\dim_E L_\chi
=
g-1.
$$

All pairwise intersections among the:

$$
a_i
$$

vanish.

Therefore:

$$
\boxed{
L_\chi
\text{ is totally isotropic}.
}
$$

Since:

$$
\dim_EH_1(C,E_\chi)=2g-2,
$$

this isotropic subspace has exactly half the total dimension.

Hence it is maximal.

---

# 18. Twisted-Homology Splitness Theorem

## Theorem 18.1 — THST

For every nontrivial finite character:

$$
\chi:
\pi_1(C)\to\mu_m,
$$

the Hermitian space associated with:

$$
H_1(C,E_\chi)
$$

is split:

$$
\boxed{
H_1(C,E_\chi)
\simeq
\mathbb H_E^{\oplus(g-1)}.
}
$$

Equivalently its Witt index is:

$$
\boxed{
g-1.
}
$$

This theorem depends only on the unramified rank-one character topology.

It does not depend on the particular curve in moduli.

---

# 19. Discriminant of the twisted character factor

A Hermitian hyperbolic plane can be represented by:

$$
\begin{pmatrix}
0&1\\
1&0
\end{pmatrix},
$$

whose determinant is:

$$
-1.
$$

Therefore:

$$
\mathbb H_E^{\oplus(g-1)}
$$

has determinant class:

$$
\boxed{
[(-1)^{g-1}].
}
$$

For:

$$
g=4,
$$

this becomes:

$$
\boxed{
[-1].
}
$$

---

# 20. Sixfold consequence

R003 forced:

$$
g=4
$$

for a quadratic irreducible deck factor of dimension six.

THST therefore gives:

$$
\boxed{
\text{Witt index}=3.
}
$$

Hence the associated polarized sixfold is split.

Its modern discriminant is:

$$
\boxed{
\delta_{\mathrm M}=-1.
}
$$

This is independent of whether the quadratic field is Gaussian or Eisenstein.

---

# 21. From cyclic character to abelian cover

Now let:

$$
\widetilde C\to C
$$

be an unramified finite abelian:

$$
G
$$

-cover.

Let:

$$
V
$$

be a rational irreducible representation whose simple field is imaginary quadratic:

$$
E.
$$

Choose a complex character:

$$
\chi
$$

in the Galois orbit defining:

$$
V.
$$

The character factors through its effective cyclic quotient:

$$
\boxed{
G\twoheadrightarrow G/\ker\chi
\simeq C_m.
}
$$

The corresponding isotypic Jacobian/Prym factor is controlled by the same rank-one local system:

$$
E_\chi.
$$

Therefore THST applies.

---

# 22. Generalized Abelian Prym Splitness Theorem

## Theorem 22.1 — GAPS

Assume:

1. $\widetilde C\to C$ is an unramified finite-abelian cover;
2. $V$ is a nontrivial rationally irreducible deck representation;
3. the effective simple field:
   $$
   E_V
   $$
   is imaginary quadratic;
4. the associated isotypic abelian factor has dimension:
   $$
   6.
   $$

Then its polarization Hermitian space is split.

Consequently:

$$
\boxed{
\delta_{\mathrm M}=-1.
}
$$

### Proof

R003 gives:

$$
\dim B_V=(g-1)\dim_{\mathbb Q}V.
$$

Quadratic:

$$
E_V
$$

implies:

$$
\dim_{\mathbb Q}V=2.
$$

Thus six-dimensionality forces:

$$
g=4.
$$

The factor is represented by an effective finite character local system:

$$
E_\chi.
$$

THST gives a:

$$
3
$$

-dimensional isotropic subspace in a rank-$6$ Hermitian space.

Thus it is split.

Modern split discriminant in rank:

$$
6
$$

is:

$$
[-1].
$$

QED.

---

# 23. The $m=6$ Eisenstein possibility

For:

$$
E=\mathbb Q(\sqrt{-3}),
$$

one may obtain the field using primitive characters of order:

$$
3
$$

or:

$$
6.
$$

THST treats both simultaneously.

The discriminant result depends only on:

- unramified finite character topology;
- genus:
  $$
  4;
  $$
- quadratic character field.

Thus:

$$
\boxed{
m=6
}
$$

does not create a new non-split component.

This strengthens the older observation that the order-$6$ case is equivalent, for the relevant Hodge-cycle architecture, to the order-$3$ case.

---

# 24. Polarization scaling does not change the class

A possible concern is that different Prym conventions may scale the induced Hermitian form by a rational scalar:

$$
c\in\mathbb Q^\times.
$$

For Hermitian rank:

$$
6,
$$

scaling:

$$
H
\mapsto
cH
$$

changes the determinant by:

$$
c^6.
$$

But:

$$
c^6
=
N_{K/\mathbb Q}(c^3)
$$

because:

$$
c^3\in\mathbb Q^\times\subset K^\times.
$$

Hence:

$$
[c^6]=1
$$

in:

$$
\mathbb Q^\times/N(K^\times).
$$

Therefore the discriminant class remains:

$$
[-1].
$$

---

# 25. Compatible isogeny does not rescue non-splitness

Passing to an isogenous character factor changes the Hermitian Gram determinant by the determinant of a $K$-linear rational transformation and its conjugate.

That multiplier is a norm.

Thus the norm-class discriminant is unchanged under the relevant compatible isogeny.

So a generalized Prym factor isogenous to the twisted-character realization remains split.

---

# 26. Patel–Zhang factor decomposition

Patel–Zhang use the semisimple decomposition:

$$
\mathbb Q[G]
=
\mathbb Q
\times
\mathbb Q[G]_{\mathrm{nt}},
$$

with:

$$
\mathbb Q[G]_{\mathrm{nt}}
$$

a product of fields corresponding to nontrivial rational irreducible representations.

They explicitly note that one may associate abelian subvarieties:

$$
J(C)_V
$$

to rational representations:

$$
V,
$$

and the analogous algebraicity theorem holds for those factors.

Thus GAPS applies directly to the quadratic irreducible factors relevant to R003.

---

# 27. What Patel–Zhang generalizes

Patel–Zhang proves:

$$
\boxed{
\text{more deck-representation Hodge subspaces are algebraic}.
}
$$

R004 now proves that, for the sixfold quadratic irreducible factors of the standard unramified architecture:

$$
\boxed{
\text{all such factors remain split}.
}
$$

Therefore the generalized cycle engine does not open a new modern non-split sixfold component.

---

# 28. Classical Prym families are now completely located

### Schoen

$$
K=\mathbb Q(\sqrt{-3}),
$$

sixfold Prym family:

$$
\boxed{
\delta_{\mathrm M}=-1.
}
$$

### Koike

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

sixfold Prym family:

$$
\boxed{
\delta_{\mathrm M}=-1.
}
$$

### Standard generalized finite-abelian quadratic factor

$$
\boxed{
\delta_{\mathrm M}=-1.
}
$$

No certified non-split component appears.

---

# 29. GMSC consequence for CE001

CE001 targets a modern non-split Weil-sixfold component:

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

The standard Prym architecture now satisfies:

$$
\boxed{
\operatorname{Im}(P_{\mathrm{Prym}})
\subseteq
\{\delta=-1\}.
}
$$

Therefore:

$$
\boxed{
P_{\mathrm{Prym}}
}
$$

cannot be a positive bypass for the exact CE001 domain.

This is stronger than saying no example is known.

It is an architecture-level obstruction.

---

# 30. Prym route demotion

R002 placed:

$$
P_{\mathrm{Prym}}
$$

on the positive Pareto frontier.

R003 restricted it to:

$$
K=\mathbb Q(i),
\mathbb Q(\sqrt{-3}).
$$

R004 further restricts it to:

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

But:

$$
\delta=-1
$$

is already closed for every imaginary quadratic field by Markman's theorem.

Therefore standard Prym no longer belongs to the frontier for **open non-split** sixfold components.

---

# 31. Updated frontier for non-split sixfolds

For:

$$
\delta\neq-1,
$$

the principal current positive architectures become:

$$
\boxed{
P_{\mathrm{Markman\mbox{-}new\ seed}}
\parallel
P_{\mathrm{Direct}}
}
$$

with route candidates:

$$
\boxed{
P_{\mathrm{Degeneration}},
\quad
P_{\mathrm{AuxiliaryGeometry}},
\quad
P_{\mathrm{NonstandardPrym}}.
}
$$

Here:

$$
P_{\mathrm{NonstandardPrym}}
$$

must leave at least one hypothesis of GAPS.

---

# 32. Ways to escape GAPS

A future Prym-like route to non-split sixfolds must use at least one of:

### Escape A — Ramification

Use ramified covers.

Then the twisted chain complex and multiplicity formula change.

### Escape B — Nonabelian deck group

Use higher-dimensional or division-algebra rational representations.

### Escape C — Reducible factor with nontrivial centralizer engineering

Use several character sectors while arranging generic endomorphism centralizer exactly:

$$
K.
$$

This requires a new centralizer theorem.

### Escape D — Auxiliary $K$-action

Let the deck algebra provide one structure and a separate correspondence provide the Weil:

$$
K.
$$

### Escape E — Prym-like but not curve-cover origin

Use other Albanese/intermediate-Jacobian constructions.

---

# 33. Ramification is not automatically an escape

Even with ramification, persistent deck endomorphisms remain.

So R003's minimality barrier still applies qualitatively.

Ramification only changes:

- dimension;
- local system homology;
- possible Witt class.

A new non-split construction must explicitly show that its Hermitian discriminant is not:

$$
[-1].
$$

---

# 34. Nonabelian route burden

A nonabelian deck group may provide a rational simple algebra:

$$
D
$$

with center:

$$
K.
$$

But if:

$$
D\supsetneq K
$$

acts generically, the target again carries too many endomorphisms.

So one needs a representation whose commutant/centralizer in the target Hodge structure is exactly the desired:

$$
K.
$$

This becomes a representation-centralizer design problem.

---

# 35. Direct-cycle route gains relative weight

The standard Prym bypass has now been removed from the open non-split domain.

The direct PT002 route remains unaffected:

$$
\boxed{
\exists
Z\in CH^3(A)_{\mathbb Q},
\quad
0\neq cl(Z)\in W_K(A)
}
$$

immediately closes the full Weil plane.

Hence R004 increases the relative strategic value of:

$$
P_{\mathrm{Direct}}.
$$

---

# 36. Markman route gains relative weight

Markman's architecture is field-uniform and, in principle, not tied to a deck-character hyperbolic form.

Its known split seed still cannot cross discriminant components.

But a genuinely component-local non-split seed would evade GAPS completely.

Thus:

$$
\boxed{
\text{component-local semiregular object}
}
$$

returns as one of the two most serious positive mechanisms for:

$$
\delta\neq-1.
$$

---

# 37. CE001 gains relative weight

The false branch CE001 directly proposes:

$$
\boxed{
\mathcal A_W=0
}
$$

on non-split sixfolds.

R004 removes a major historical positive bypass from that domain.

This does not prove CE001.

But it increases the structural isolation of the non-split component.

---

# 38. Historical mystery resolved

Before the audit, the appearance of:

$$
\mathbb Q(\sqrt{-3})
$$

and:

$$
\mathbb Q(i)
$$

in Schoen/Koike could be read as two sporadic arithmetic miracles.

R003 explained the fields.

R004 explains the discriminant.

The whole pattern is:

$$
\boxed{
\text{finite character}
+
\text{unramified genus }4
+
\text{quadratic deck field}
}
$$

$$
\Downarrow
$$

$$
\boxed{
\text{six-dimensional hyperbolic Hermitian factor}
}
$$

$$
\Downarrow
$$

$$
\boxed{
K=\mathbb Q(i)
\text{ or }
\mathbb Q(\sqrt{-3}),
\qquad
\delta=-1.
}
$$

Thus the classical cases are structurally forced.

---

# 39. Construct result

R004 constructs two reusable lemmas/theorems:

### Old-to-Modern Square-to-Norm Conversion

For negative rational determinant in an imaginary quadratic field:

$$
\delta_{\mathrm{old}}=1
\Rightarrow
\delta_{\mathrm M}=-1.
$$

### Twisted-Homology Splitness

For nontrivial finite rank-one character local systems on a genus-$g$ curve:

$$
\boxed{
H_1(C,E_\chi)
\simeq
\mathbb H_E^{\oplus(g-1)}.
}
$$

The second result is the deeper structural operator.

---

# 40. Attack result

R004 attacks the conjectural bypass:

> generalized finite-abelian Pryms might alter the sixfold discriminant while keeping the same minimal quadratic deck field.

Within the unramified irreducible quadratic-factor architecture, this is false.

Status:

$$
\boxed{
\mathrm{BROKEN}.
}
$$

---

# 41. Compress result

The Prym residual for the standard architecture was:

$$
\boxed{
\text{field reach}
+
\text{dimension reach}
+
\text{component reach}.
}
$$

R003 closed the first two.

R004 closes the third:

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

Therefore the standard Prym branch has no residual intersection with the open non-split target.

It can be removed from that residual graph.

---

# 42. Updated non-split residual graph

For:

$$
\delta\neq-1,
$$

the positive graph now compresses to:

```text
[New component-local object] --> [Semiregularity / relative realization] --+
                                                                         |
[Direct algebraic cycle / correspondence] -------------------------------+--> [WEIL CORE]
                                                                         |
[Cross-component degeneration bridge] -----------------------------------+
                                                                         |
[New auxiliary geometry] ------------------------------------------------+
```

The standard unramified finite-abelian Prym branch is absent.

---

# 43. Updated route labels

```text
CLOSED / SPLIT-ONLY:
  Schoen Prym sixfold
  Koike Prym sixfold
  standard unramified finite-abelian quadratic Prym factor

OPEN FRONTIER:
  component-local Markman-like seed
  direct one-cycle construction

ROUTE-CANDIDATE:
  ramified Prym
  nonabelian Prym
  auxiliary K-action Prym
  degeneration
  hyperkähler / moduli correspondence

ATTACK:
  CE001 non-split Weil sixfold
```

---

# 44. Bottleneck stability update

Across:

$$
R001,\ R002,\ R003,\ R004,
$$

the target SCC remains stable:

$$
\boxed{
\mathrm{BS}_4
\left(
\mathcal C_{\mathrm{Weil6}}^{\mathrm{vg}}
\right)
=
1.
}
$$

Semiregularity remains route-only.

Prym dominance was a route-only gate and is now closed only in the split direction.

The non-split graph still has no single positive pre-core articulation shared by all remaining architectures.

---

# 45. Information gain of GMSC

Without GMSC, one might have:

1. continued PT011;
2. separately explored generalized Pryms;
3. treated Schoen/Koike discriminants as notation confusion.

After R003–R004, an entire large branch is compressed to:

$$
\boxed{
\text{standard Prym}
\Rightarrow
\text{split}
\Rightarrow
\text{already closed}.
}
$$

That branch no longer consumes non-split search budget.

---

# 46. Main Verdict

The classical and standard generalized unramified finite-abelian Prym architecture does **not** bypass the modern split/non-split barrier.

It is intrinsically tied to a hyperbolic Hermitian character factor.

For sixfolds:

$$
\boxed{
\delta_{\mathrm M}=-1.
}
$$

Therefore:

$$
\boxed{
\text{Prym geometry explains and realizes the split core;}
}
$$

it does not currently provide a route into:

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

---

# 47. Strategic next move

GMSC should now stop spending rounds on standard Prym moduli reach for CE001.

The highest-information next move is to search for an architecture that can **change the Hermitian Witt class**.

This is more precise than saying "find another cycle."

The new question is:

$$
\boxed{
\text{What legal geometric mechanisms can realize a non-hyperbolic signature-}(3,3)\text{ Hermitian space?}
}
$$

---

# 48. Candidate mechanisms for changing Witt class

Potential directions:

1. ramified rank-one local systems;
2. nonabelian local systems;
3. auxiliary correspondences modifying the polarization form;
4. non-Prym intermediate Jacobians;
5. moduli spaces of sheaves / hyperkähler correspondences;
6. direct subvarieties whose cycle classes do not arise from a curve-cover character factor;
7. degeneration with controlled discriminant jump.

These are qualitatively different from standard Prym.

---

# 49. Next Interface

Next GMSC round:

```text
HODGE_GMSC_R005_NonSplitWittArchitectureSearch.md
```

Primary target:

$$
\boxed{
\text{Enumerate and eliminate/retain geometric architectures capable of producing }
\delta\neq-1.
}
$$

Planned tasks:

1. classify how ramification changes twisted Hermitian Witt class;
2. inspect whether rank-one ramified local systems can be non-split in rank six;
3. examine nonabelian representation factors with center $K$;
4. test centralizer constraints for generic dominance;
5. search auxiliary moduli / hyperkähler sixfold constructions;
6. search algebraic correspondences that can alter the effective Hermitian discriminant;
7. compare with Mostaed CM non-split points;
8. identify a new Pareto frontier specifically for:
   $$
   \delta\neq-1;
   $$
9. keep CE001 Attack running against every retained positive architecture;
10. enter a new Core Theorem Mode only if one Witt-changing gate becomes stable across rebuilds.

---

# References

1. E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415. Uses the modern discriminant
   $$
   \mathbb Q^\times/N_{K/\mathbb Q}(K^\times)
   $$
   and identifies Koike's Gaussian sixfold theorem with discriminant $-1$.

2. E. Markman, *Secant sheaves and Weil classes on abelian varieties*, arXiv:2509.23403v2. Defines split Weil type by a half-dimensional isotropic subspace, states that split components have discriminant $[(-1)^n]$, and identifies Schoen and Koike as the earlier quadratic-imaginary split sixfold cases.

3. K. Koike, *Algebraicity of some Weil Hodge Classes*, arXiv:math/0211304. Defines the older square-class discriminant and proves algebraicity for the Gaussian sixfold family with old discriminant $1$.

4. C. Schoen, *Hodge classes on self-products of a variety with an automorphism*, Compositio Math. 65 (1988), 3–32, and Addendum, Compositio Math. 114 (1998), 329–336.

5. D. Patel, Y. Zhang, *Algebraicity of Hodge classes on some Generalized Prym Varieties*, arXiv:2506.13729v2. Decomposes the generalized Prym by rational representations of the finite abelian deck group and proves algebraicity of the corresponding Weil-type Hodge classes.

6. Z. Zhang, *On geometric and motivic realizations of variations of Hodge structure over Hermitian symmetric domains*, dissertation, 2014. Records the classical genus-$4$ Prym realizations of the Eisenstein and Gaussian sixfold split/discriminant-$-1$ families.

7. Aletheia, *HODGE_GMSC_R003_GeneralizedPrymReach*, 2026-09-16.

---

## Canonical Source Declaration

本檔案為 Hodge GMSC 分支第四篇正式 UTF-8 Markdown canonical source。

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪沒有宣稱所有 Prym-like、ramified、nonabelian、或 auxiliary-correspondence constructions 都必然 split。

本輪正式證明的 architecture scope 是：

$$
\boxed{
\text{unramified finite-abelian cover}
+
\text{rationally irreducible quadratic deck factor}.
}
$$

在此 scope 中，sixfold factor必為 split：

$$
\boxed{
\delta_{\mathrm M}=-1.
}
$$

因此 standard generalized Prym route 可從 modern non-split sixfold residual graph中移除。