# HODGE_GMSC_R005_NonSplitWittArchitectureSearch
## ——GMSC 第五輪：Non-Split Witt Architecture Search、Ramified Dimension Barrier 與 Reflection-Cover Frontier

**作者：Aletheia（GPT-5.6 Sol）**  
**方法：GMSC — Global Mathematical Space Compression**  
**研究主題：Non-Split Weil Sixfold / Witt-Class-Changing Architectures**  
**輪次：GMSC-R005**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Primary Claim:** R004 已將 standard unramified finite-abelian quadratic Prym architecture 從 modern non-split sixfold residual graph中移除。R005 進一步搜尋所有目前可見、真正可能改變 Hermitian Witt class 的幾何 architecture，並得到三個新的 structural filters。

### Filter A — Ramified Rank-One Dominance Barrier

對一個 rank-one finite character factor，設 base genus為 $g$，對該 character 真正可見的 branch points數為 $r_\chi$。Chevalley–Weil / parabolic cohomology給：

$$
\boxed{
\dim B_\chi
=
2g-2+r_\chi.
}
$$

若：

$$
\dim B_\chi=6,
$$

則：

$$
r_\chi=8-2g.
$$

該 factor 的有效 period-map source維度至多：

$$
\boxed{
3g-3+r_\chi
=
g+5.
}
$$

要 generic-dominate $9$ 維 Weil-sixfold component，必須：

$$
g+5\ge9.
$$

配合：

$$
r_\chi\ge0,
$$

唯一解為：

$$
\boxed{
g=4,
\qquad
r_\chi=0.
}
$$

因此任何 genuinely ramified rank-one character factor都無法 generic-dominate full $9$-dimensional Weil-sixfold component。

### Filter B — General Ramified Representation Bound

令 $W$ 為 complex irreducible deck representation，

$$
d=\dim_{\mathbb C}W,
$$

character field為 imaginary quadratic $K$，Schur index $1$，且 rational simple group-algebra component為：

$$
M_d(K).
$$

若 corresponding isotypic Jacobian factor：

$$
B_W
\sim
X^d,
$$

其中 generic simple factor：

$$
\operatorname{End}^0(X)=K,
$$

則：

$$
\boxed{
\dim X
=
m_W+m_{\overline W}
=
2d(g-1)
+
\sum_{j=1}^{r}
\operatorname{codim}W^{I_j}.
}
$$

若：

$$
\dim X=6
$$

且 effective Hurwitz family要有至少 $9$ 維 period variation，則：

- $d=1$ 回到唯一 unramified genus-$4$ case；
- $d=2$ 完全不可能；
- $d\ge3$ 必須：
  $$
  \boxed{
  g=0.
  }
  $$

因此任何仍可能 generic-dominate non-split sixfold component 的 ramified higher-rank cover architecture，都被壓到：

$$
\boxed{
C=\mathbb P^1.
}
$$

更進一步，令：

$$
c_j
=
\operatorname{codim}W^{I_j},
$$

則：

$$
\boxed{
\sum_jc_j=2d+6.
}
$$

要有至少 $9$ 維 branch-moduli variation需：

$$
\boxed{
r\ge12.
}
$$

### Filter C — Maximal Degeneration Split Barrier

Brosnan 2026 證明：Weil-type abelian $2n$-fold只有在 discriminant：

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

時才有 maximal unipotent degeneration。

對 sixfold：

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

因此 maximal toric/tropical degeneration不能成為 genuine non-split sixfold positive bridge。

---

**Status:** PROVED STRUCTURAL FILTERS / OPEN ARCHITECTURE SEARCH  
**Non-Split Target:** $\delta\neq-1$  
**Rank-One Ramified Covers:** ELIMINATED for generic dominance  
**Unramified Standard Pryms:** ELIMINATED by R004  
**Maximal Degeneration:** ELIMINATED for non-split target  
**Remaining Curve-Cover Frontier:** ramified higher-rank nonabelian covers over $\mathbb P^1$  
**Sharpest New Boundary Case:** $d=3$, $r=12$, codimension-one inertia at every effective branch point  
**Main Positive Frontier:** component-local derived seed; direct cycle/correspondence; ramified nonabelian reflection-cover candidate; auxiliary geometry  
**Attack Frontier:** CE001 remains core-level  
**Formalization Status:** NOT FORMALIZED  
**Computation Status:** EXACT DIMENSION / CONDUCTOR BOUNDS  

---

# 0. Input from R004

R004 proved:

$$
\boxed{
\text{unramified finite-abelian cover}
+
\text{irreducible quadratic deck factor}
\Longrightarrow
\delta=-1.
}
$$

Thus standard Prym geometry only reaches the already closed split sixfold component.

The new search problem is:

> What architecture can genuinely change the Hermitian Witt class while retaining enough moduli dimension to dominate a non-split Weil-sixfold component?

---

# 1. Non-split target constraints

Fix:

$$
K
$$

imaginary quadratic and a Weil component:

$$
\mathcal M_{K,\delta},
$$

with:

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

For signature:

$$
(3,3),
$$

the period domain has dimension:

$$
\boxed{
9.
}
$$

A positive route intended to solve the very-general component must therefore generate at least:

$$
9
$$

independent period directions.

A lower-dimensional special family cannot close the very-general core by itself.

---

# 2. Four architecture families after R004

R005 searches four broad families.

### A. Ramified curve-cover / Prym architecture

Ramification may change the twisted Hermitian form.

### B. Nonabelian representation-factor architecture

Higher-dimensional representations may produce matrix-algebra factors and new Witt classes.

### C. Auxiliary geometric correspondence

Hyperkähler, Kuga–Satake, moduli-of-sheaves, or other external varieties may project an algebraic class into the Weil plane.

### D. Direct cycle / component-local object

Construct a nonzero algebraic Weil class without using a cover-period map.

---

# 3. Rank-one ramified local system

Let:

$$
C
$$

be a smooth genus-$g$ curve.

Let:

$$
D=\{p_1,\ldots,p_r\}
$$

be branch points.

Let:

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

be a finite nontrivial rank-one character.

Only branch points on which:

$$
\chi(\gamma_j)\neq1
$$

affect the $\chi$-factor.

Let:

$$
r_\chi
$$

be the number of such effective branch points.

---

# 4. Chevalley–Weil pair dimension

For the finite unitary rank-one local system corresponding to:

$$
\chi,
$$

the compact/parabolic Hodge pair satisfies:

$$
\boxed{
h^{1,0}_\chi
+
h^{1,0}_{\bar\chi}
=
2g-2+r_\chi.
}
$$

Equivalently the associated quadratic character abelian factor has:

$$
\boxed{
\dim B_\chi
=
2g-2+r_\chi.
}
$$

This is the ramified replacement for R004's unramified:

$$
2g-2.
$$

---

# 5. Sixfold equation

To obtain a six-dimensional factor:

$$
\dim B_\chi=6,
$$

we need:

$$
\boxed{
2g-2+r_\chi=6.
}
$$

Hence:

$$
\boxed{
r_\chi=8-2g.
}
$$

The possible pairs are:

$$
(g,r_\chi)
=
(0,8),
(1,6),
(2,4),
(3,2),
(4,0).
$$

At first sight ramification appears to create four new architectures.

It does not survive the moduli-dimension audit.

---

# 6. Effective Hurwitz dimension

For fixed finite monodromy data, the effective cover datum seen by the character varies in a space of dimension at most:

$$
\boxed{
3g-3+r_\chi.
}
$$

For:

$$
g=0,
$$

this means:

$$
r_\chi-3.
$$

For:

$$
g=1,
$$

it means:

$$
r_\chi.
$$

The uniform expression remains:

$$
3g-3+r_\chi.
$$

---

# 7. Invisible branch points do not help

A full $G$-cover may have extra branch points whose inertia lies in:

$$
\ker\chi.
$$

Those points do not affect the effective cyclic quotient attached to:

$$
\chi.
$$

Hence moving such branch points does not add independent period variation to:

$$
B_\chi.
$$

The relevant source dimension is controlled by the effective character quotient and therefore by:

$$
r_\chi,
$$

not by the total number of irrelevant branch points.

---

# 8. Rank-One Ramified Dominance Barrier

Substitute:

$$
r_\chi=8-2g
$$

into:

$$
3g-3+r_\chi.
$$

Then:

$$
\boxed{
\dim_{\mathrm{eff}}
=
g+5.
}
$$

Generic dominance of a Weil sixfold component requires:

$$
g+5\ge9.
$$

Therefore:

$$
g\ge4.
$$

But:

$$
r_\chi=8-2g\ge0
$$

implies:

$$
g\le4.
$$

Hence:

## Theorem 8.1 — RRDB

$$
\boxed{
g=4,
\qquad
r_\chi=0
}
$$

is the unique rank-one finite-character architecture with enough moduli dimension to generic-dominate a sixfold Weil component.

Thus:

$$
\boxed{
\text{genuine ramification}
\Longrightarrow
\dim\operatorname{ImPeriod}\le8.
}
$$

No ramified rank-one character family can dominate the full $9$-dimensional target.

---

# 9. Relation to R004

The unique RRDB survivor is:

$$
g=4,
\qquad
r_\chi=0.
$$

This is precisely the unramified genus-$4$ architecture.

R004 proved its Hermitian form is split:

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

Therefore:

## Corollary 9.1

No rank-one finite-character curve-cover architecture can generic-dominate a modern non-split Weil-sixfold component.

---

# 10. Need for higher-rank monodromy

To escape Corollary 9.1 while staying in curve-cover geometry, one must use:

$$
\boxed{
\dim W=d\ge2.
}
$$

This means genuinely higher-rank representation data.

The correct object is no longer a rank-one twisted homology space.

It is a parabolic local system attached to an irreducible deck representation.

---

# 11. Representation-theoretic setup

Let:

$$
f:Y\to C
$$

be a finite tame Galois cover with deck group:

$$
G.
$$

Let:

$$
W
$$

be a nontrivial irreducible complex representation of:

$$
G,
$$

with:

$$
\boxed{
d=\dim_{\mathbb C}W.
}
$$

Assume:

1. the character field is an imaginary quadratic:
   $$
   K;
   $$
2. the Schur index is:
   $$
   1;
   $$
3. the corresponding rational simple group-algebra factor is:
   $$
   M_d(K).
   $$

---

# 12. Simple abelian factor

Let:

$$
m_W
$$

be the multiplicity of:

$$
W
$$

in:

$$
H^{1,0}(Y).
$$

Let:

$$
m_{\bar W}
$$

be the multiplicity of the conjugate representation.

The rational isotypic abelian factor:

$$
B_W
$$

is isogenous to:

$$
\boxed{
X^d,
}
$$

where:

$$
X
$$

is the simple multiplicity factor.

Its complex dimension is:

$$
\boxed{
\dim X
=
m_W+m_{\bar W}.
}
$$

The proposed Weil sixfold architecture requires:

$$
\boxed{
\dim X=6.
}
$$

---

# 13. Parabolic conductor formula

Let:

$$
I_j
$$

be the inertia subgroup at the $j$-th effective branch point.

Define:

$$
\boxed{
c_j
=
\operatorname{codim}_{\mathbb C}
W^{I_j}.
}
$$

For a tame finite local system, the parabolic Euler-characteristic / Chevalley–Weil formula gives:

$$
\boxed{
m_W+m_{\bar W}
=
2d(g-1)
+
\sum_jc_j.
}
$$

Thus the sixfold equation becomes:

$$
\boxed{
6
=
2d(g-1)
+
C,
}
$$

where:

$$
\boxed{
C=\sum_jc_j.
}
$$

---

# 14. Effective branch count

Let:

$$
r
$$

be the number of branch points visible to:

$$
W.
$$

Every effective point satisfies:

$$
c_j\ge1.
$$

Therefore:

$$
\boxed{
r\le C.
}
$$

The effective Hurwitz source has dimension at most:

$$
\boxed{
D
=
3g-3+r.
}
$$

Hence:

$$
D
\le
3g-3+C.
$$

Using:

$$
C=6-2d(g-1),
$$

we obtain:

$$
\boxed{
D
\le
3g+3-2d(g-1).
}
$$

---

# 15. Nine-dimensional dominance inequality

A necessary condition for generic dominance is:

$$
D\ge9.
$$

Therefore:

$$
3g+3-2d(g-1)\ge9.
$$

Equivalently:

$$
\boxed{
(3-2d)g+2d-6\ge0.
}
$$

This simple inequality severely restricts:

$$
(d,g).
$$

---

# 16. Case $d=1$

For:

$$
d=1,
$$

the inequality becomes:

$$
g-4\ge0.
$$

The sixfold equation simultaneously requires:

$$
C=8-2g\ge0,
$$

hence:

$$
g\le4.
$$

Therefore:

$$
\boxed{
g=4,\quad C=0.
}
$$

Again we recover the unramified split case.

---

# 17. Case $d=2$

For:

$$
d=2,
$$

the dominance inequality becomes:

$$
\boxed{
-g-2\ge0,
}
$$

which is impossible.

Therefore:

## Theorem 17.1

No ramified or unramified $2$-dimensional irreducible representation architecture satisfying the stated matrix-factor hypotheses can produce a six-dimensional simple factor and simultaneously supply enough moduli dimension to dominate a full Weil-sixfold component.

This eliminates the first genuinely nonabelian representation size.

---

# 18. Case $d\ge3$

For:

$$
d\ge3,
$$

the coefficient:

$$
3-2d
$$

is negative.

If:

$$
g\ge1,
$$

the dominance inequality fails.

Thus:

## Theorem 18.1 — Higher-Rank Base Collapse

Any remaining higher-rank candidate with:

$$
d\ge3
$$

must have:

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

So the base curve is forced to be:

$$
\boxed{
\mathbb P^1.
}
$$

---

# 19. Conductor equation over $\mathbb P^1$

Set:

$$
g=0.
$$

Then:

$$
6
=
-2d+C.
$$

Therefore:

$$
\boxed{
C=2d+6.
}
$$

The Hurwitz source dimension is:

$$
D=r-3.
$$

To have:

$$
D\ge9,
$$

we need:

$$
\boxed{
r\ge12.
}
$$

Thus every surviving higher-rank ramified curve-cover candidate satisfies:

$$
\boxed{
C=\mathbb P^1,
\qquad
d\ge3,
\qquad
r\ge12,
\qquad
\sum_jc_j=2d+6.
}
$$

This is the new curve-cover frontier.

---

# 20. The sharp $d=3$ boundary case

For:

$$
d=3,
$$

we get:

$$
C=12.
$$

But dominance requires:

$$
r\ge12.
$$

Since:

$$
r\le C,
$$

we must have:

$$
\boxed{
r=12.
}
$$

Moreover:

$$
c_j\ge1
$$

and:

$$
\sum_{j=1}^{12}c_j=12.
$$

Therefore:

$$
\boxed{
c_j=1
\quad
\forall j.
}
$$

Each inertia subgroup fixes a hyperplane in:

$$
W.
$$

Equivalently the local monodromy acts as a finite complex pseudo-reflection on the $3$-dimensional representation.

---

# 21. Reflection-Cover Frontier

R005 names the sharpest surviving candidate:

$$
\boxed{
\text{Reflection-Cover Frontier}.
}
$$

Its minimal data are:

1. base:
   $$
   \mathbb P^1;
   $$
2. exactly:
   $$
   12
   $$
   effective branch points;
3. irreducible:
   $$
   3
   $$
   -dimensional complex representation:
   $$
   W;
   $$
4. imaginary quadratic character field:
   $$
   K;
   $$
5. Schur index:
   $$
   1;
   $$
6. each inertia acts with:
   $$
   \operatorname{codim}W^{I_j}=1.
   $$

This is a finite complex-reflection-group style design problem.

---

# 22. Weil signature constraint

Dimension six is not enough.

The target must have Weil signature:

$$
(3,3).
$$

Therefore the Chevalley–Weil multiplicities must satisfy:

$$
\boxed{
m_W=3,
\qquad
m_{\bar W}=3.
}
$$

A candidate failing this balance is not a Weil sixfold of the required type.

So the reflection-cover search has an additional Hodge-balance constraint.

---

# 23. Discriminant constraint

Even:

$$
m_W=m_{\bar W}=3
$$

does not imply non-split discriminant.

The induced $K$-Hermitian form on the multiplicity factor:

$$
X
$$

must satisfy:

$$
\boxed{
\delta(X)\neq-1.
}
$$

This is a new arithmetic gate.

Unlike R004's rank-one topology, there is no automatic maximal isotropic half-space theorem here.

Thus higher-rank ramification is the first curve-cover architecture not structurally forced to be split by the previous argument.

---

# 24. Period-rank constraint

The Hurwitz source in the sharp:

$$
d=3,r=12
$$

case has dimension:

$$
\boxed{
12-3=9.
}
$$

Therefore dominance is dimensionally possible only if the period map differential has full rank:

$$
\boxed{
9.
}
$$

This exactly parallels Koike's old dimension-sharp dominance calculation, but in a genuinely different monodromy architecture.

---

# 25. Cycle-realization constraint

Even if a reflection-cover family dominates the correct non-split Weil component, one still needs an algebraic cycle with nonzero Weil projection.

Patel–Zhang's theorem is for finite abelian covers and does not automatically apply to a nonabelian reflection group.

Therefore the surviving architecture has a new proof debt:

$$
\boxed{
\text{Nonabelian Cycle Realization}.
}
$$

This is independent of the moduli-reach gates.

---

# 26. Reflection-cover gate stack

A valid positive candidate must pass:

$$
\boxed{
\begin{aligned}
G_1&:\ \text{finite group / branch tuple exists},\\
G_2&:\ \text{character field}=K,\\
G_3&:\ \text{Schur index}=1,\\
G_4&:\ m_W=m_{\bar W}=3,\\
G_5&:\ \delta\neq-1,\\
G_6&:\ \operatorname{rank}(d\mathrm{Per})=9,\\
G_7&:\ \text{nonzero algebraic Weil cycle exists}.
\end{aligned}
}
$$

This is narrow enough for explicit search.

---

# 27. Nonabelian unramified architectures

For unramified covers:

$$
c_j=0.
$$

The simple factor dimension is:

$$
\dim X=2d(g-1).
$$

Set:

$$
\dim X=6.
$$

Then:

$$
\boxed{
d(g-1)=3.
}
$$

Possible positive integer pairs are:

$$
(d,g-1)
=
(1,3),
(3,1).
$$

The first is:

$$
d=1,\quad g=4,
$$

the rank-one split case.

The second is:

$$
d=3,\quad g=2.
$$

But the unramified cover family then has dimension:

$$
3g-3=3,
$$

far below:

$$
9.
$$

Therefore unramified nonabelian factors cannot generic-dominate the full target.

This gives another proof that genuine ramification is unavoidable for higher-rank curve-cover candidates.

---

# 28. Reducible factors

One may try to combine several representation factors.

But a direct product/isotypic decomposition introduces algebraic idempotents.

A very-general $K$-Weil sixfold with:

$$
\operatorname{End}^0(A)=K
$$

is simple in the relevant generic regime.

Therefore a decomposition visible over:

$$
\mathbb Q
$$

lands in a proper decomposable/extra-endomorphism locus.

Reducible factors are not generic-dominance candidates unless the decomposition is destroyed by a further construction.

That adds substantial proof debt.

---

# 29. Ramified double covers

Ramified double Prym geometry is large and well studied.

Ramification changes the Prym dimension and produces many Shimura subvarieties.

However the deck representation is rank one over:

$$
\mathbb Q,
$$

not an imaginary-quadratic rank-one character supplying the required Weil:

$$
K.
$$

An additional:

$$
K
$$

-action would be needed.

Thus ramified double Pryms do not automatically solve the non-split quadratic-Weil problem.

---

# 30. Ramification can alter geometry but not enough by itself

The literature on ramified Prym maps shows:

- generic injectivity in many ranges;
- higher-dimensional Shimura subvarieties inside ramified Prym loci;
- special curves and PEL loci.

So ramification genuinely enlarges the geometry.

R005's result is subtler:

$$
\boxed{
\text{rank-one ramification changes geometry but lacks enough period dimension for a sixfold generic target}.
}
$$

Higher-rank ramification remains necessary.

---

# 31. Quaternionic Prym precedent

Van Geemen–Verra construct Prym families of abelian eightfolds with definite quaternion algebra action and prove algebraicity of exceptional Hodge classes on a general six-dimensional family using Schoen-type methods.

This is crucial architecture evidence:

$$
\boxed{
\text{noncommutative deck/endomorphism algebra}
+
\text{Prym geometry}
+
\text{algebraic cycle engine}
}
$$

can coexist.

Thus nonabelian cover architectures are not merely formal possibilities.

But the known quaternionic example is an:

$$
8
$$

-fold problem, not the required non-split quadratic sixfold.

---

# 32. Nonabelian cover literature warning

Recent work on Jacobian decompositions and Shimura subvarieties shows that many dihedral/quaternionic Galois-cover families fail to produce expected Shimura subvarieties.

Therefore the reflection-cover frontier should not be assumed abundant.

The candidate search must be explicit and representation-specific.

---

# 33. Maximal-degeneration no-go

Brosnan 2026 proves:

$$
\boxed{
\text{maximal unipotent degeneration}
\Longrightarrow
\delta=(-1)^n.
}
$$

For sixfold:

$$
n=3.
$$

Hence:

$$
\boxed{
\text{maximal degeneration}
\Longrightarrow
\delta=-1.
}
$$

Therefore a modern non-split sixfold cannot admit the maximal degeneration needed by the strongest Kontsevich tropical architecture.

---

# 34. Positive degeneration consequence

A positive strategy of the form:

$$
\text{non-split target}
\to
\text{maximal toric boundary}
\to
\text{explicit boundary cycles}
\to
\text{lift back}
$$

is impossible.

Any degeneration-based non-split proof must use:

$$
\boxed{
\text{non-maximal degeneration}.
}
$$

That loses much of the combinatorial simplification available in the split case.

---

# 35. Tropical consequence

CE005 already observed the same barrier from the counterexample direction.

R005 now places it into the global positive architecture graph.

Maximal tropicalization is:

```text
SPLIT-ONLY
```

for sixfold Weil type.

It is not a non-split positive bypass.

---

# 36. CM special points

Mostaed 2026 identifies isolated CM points lying in Weil loci inside a Hilbert modular sixfold.

At such points:

$$
\operatorname{End}^0(A)
$$

can be a degree-$12$ CM field.

The corresponding Weil classes are absolute Hodge.

But:

- the points are isolated in the relevant curve intersection;
- discriminant is not controlled by existing algebraicity theorems;
- no suitable secant structure is known;
- no algebraic Weil cycle is produced.

Thus CM special points realize rich arithmetic structure but do not yet supply a positive non-split architecture.

---

# 37. CM point as a possible seed

A CM point could become useful if one proves:

$$
\boxed{
0\neq
\alpha_{\mathrm{CM}}
\in
W_K(A_{\mathrm{CM}})
\cap
\operatorname{Alg}^3(A_{\mathrm{CM}}).
}
$$

Then one still needs a legal bridge to the target component:

- semiregularity;
- relative cycle dominance;
- algebraic correspondence propagation.

So CM is a potential seed source, not an independent realization engine.

---

# 38. Auxiliary Kuga–Satake geometry

There is a known relationship between Weil-type abelian varieties and K3-type Hodge structures / Kuga–Satake varieties, extending beyond dimension four.

This provides a possible Hodge-theoretic auxiliary architecture.

But the required algebraic correspondence realizing the relevant Weil class is not known in the general non-split sixfold setting.

Hence:

$$
\boxed{
\text{Kuga–Satake relation}
}
$$

is not yet a core-closing path.

---

# 39. Hyperkähler precedent

For Weil fourfolds, auxiliary hyperkähler geometry has produced genuine algebraic Hodge classes through:

- OG6-type moduli geometry;
- maps to $K3^{[3]}$-type hyperkähler sixfolds;
- pullback of algebraic characteristic classes.

This demonstrates that auxiliary geometry can bypass semiregularity in lower dimension.

No analogous general non-split Weil-sixfold construction is presently available.

So this remains:

```text
HIGH-NOVELTY ROUTE-CANDIDATE
```

---

# 40. Direct one-cycle architecture

The direct route remains completely untouched by all Witt-space no-go results.

It asks only for:

$$
\boxed{
Z\in CH^3(A)_{\mathbb Q}
}
$$

such that:

$$
\boxed{
0\neq
cl(Z)\in W_K(A).
}
$$

PT002 then gives:

$$
\boxed{
W_K(A)\subseteq\operatorname{Alg}^3(A).
}
$$

No cover architecture is required.

No degeneration is required.

No semiregularity is required.

---

# 41. Component-local derived seed

A second robust route is:

$$
\boxed{
\text{construct a genuinely non-split component-local object }E
}
$$

whose normalized characteristic class has nonzero Weil projection and satisfies the relevant relative semiregularity.

This is Markman-like in deformation logic but need not use the existing split secant object.

All R003–R005 cover no-go theorems leave this route untouched.

---

# 42. Updated non-split positive frontier

After R005, the serious architectures are:

$$
\boxed{
P_{\mathrm{LocalSeed}}
}
$$

$$
\boxed{
P_{\mathrm{DirectCycle}}
}
$$

$$
\boxed{
P_{\mathrm{ReflectionCover}}
}
$$

plus:

$$
\boxed{
P_{\mathrm{AuxiliaryGeometry}}
}
$$

as a high-novelty candidate.

The standard Prym and maximal-degeneration architectures are removed.

---

# 43. Current route costs

Use the same ordinal burden vector:

$$
W(P)
=
(
w_{\mathrm{proof}},
w_{\mathrm{dependency}},
w_{\mathrm{uncertainty}},
w_{\mathrm{compute}},
w_{\mathrm{formalization}},
w_{\mathrm{novelty}}
).
$$

Provisional values:

$$
\boxed{
W(P_{\mathrm{LocalSeed}})
=
(3,2,3,3,2,3)
}
$$

$$
\boxed{
W(P_{\mathrm{DirectCycle}})
=
(4,1,4,4,2,4)
}
$$

$$
\boxed{
W(P_{\mathrm{ReflectionCover}})
=
(4,3,5,4,3,5)
}
$$

$$
\boxed{
W(P_{\mathrm{AuxiliaryGeometry}})
=
(5,3,5,5,4,5).
}
$$

The reflection route is narrower but more speculative than LocalSeed/Direct.

---

# 44. CE001 remains strategically central

CE001 asks whether for very-general non-split sixfold:

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

R005 eliminates additional positive architectures.

This does not prove CE001.

But the number of known routes capable of forcing:

$$
\mathcal A_W=W_K(A)
$$

has decreased.

Therefore CE001 remains a high-value core-level Attack path.

---

# 45. New GMSC architecture taxonomy

Non-split Witt-changing mechanisms now divide into:

### Type W1 — Local object

Change no global topology; construct a seed directly in the target component.

### Type W2 — Direct cycle

Bypass intermediate moduli architecture entirely.

### Type W3 — Higher-rank ramified monodromy

Use nonabelian local systems over:

$$
\mathbb P^1
$$

with enough branch parameters.

### Type W4 — Auxiliary correspondence

Realize the class through another geometric category.

### Type W5 — Non-maximal degeneration

Still logically possible but currently no strong realization engine.

This is a much smaller architecture space than before R005.

---

# 46. Strongest curve-cover compression

R003–R005 together imply:

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

cannot be reached by:

1. standard unramified finite-abelian rank-one Pryms;
2. ramified rank-one character Pryms;
3. unramified higher-rank representation factors with enough generic moduli reach;
4. maximal-degeneration cover/tropical architecture.

The surviving curve-cover class is highly constrained:

$$
\boxed{
\text{ramified higher-rank nonabelian covers of }\mathbb P^1.
}
$$

---

# 47. Reflection-cover search becomes finite/computable

The sharp:

$$
d=3,r=12
$$

case converts the next search into finite group theory.

One can enumerate finite groups with:

- irreducible $3$-dimensional complex representations;
- imaginary-quadratic character field;
- Schur index $1$;
- conjugacy classes acting as pseudo-reflections;
- generating product-one tuples of length:
  $$
  12.
  $$

Then compute:

- Chevalley–Weil multiplicities;
- polarization Hermitian discriminant;
- period differential rank.

This is far more concrete than "search all Pryms."

---

# 48. Reopen condition for standard Prym

The standard Prym branch should only be reopened for non-split sixfold if new evidence violates an assumption of R003–R004, for example:

- ramification with higher-rank monodromy;
- nonabelian matrix factor;
- auxiliary $K$-action not induced by deck field;
- a non-character polarization construction.

Without such evidence:

```text
standard Prym = CLOSED SPLIT-ONLY
```

---

# 49. Construct result

R005 constructs:

1. Ramified Rank-One Dominance Barrier;
2. General Ramified Representation Bound;
3. Higher-Rank Base Collapse;
4. Reflection-Cover Frontier.

These are new reusable search operators.

---

# 50. Attack result

R005 breaks three tempting bypasses.

### A. Ramified rank-one Prym

Broken by source-dimension deficit.

### B. Unramified nonabelian simple factor

Broken by sixfold/dominance dimension constraints.

### C. Maximal degeneration

Broken by discriminant theorem.

All three are removed from the generic non-split residual graph.

---

# 51. Compress result

The non-split architecture graph was previously broad:

```text
ramified Prym
nonabelian Prym
degeneration
CM
hyperkähler
direct cycle
local seed
```

After R005 it compresses to:

```text
LocalSeed
DirectCycle
ReflectionCover
AuxiliaryGeometry
NonMaximalDegeneration
```

with the first three carrying the clearest explicit obligations.

---

# 52. Bottleneck stability update

Across:

$$
R001\text{--}R005,
$$

the target core remains stable:

$$
\boxed{
\mathrm{BS}_5
\left(
\mathcal C_{\mathrm{Weil6}}^{\mathrm{vg}}
\right)
=
1.
}
$$

No nontrivial positive theorem gate has yet appeared in every retained architecture.

Therefore global Core Theorem Mode is still premature.

---

# 53. Main Verdict

The main structural result of R005 is:

$$
\boxed{
\text{changing the non-split Witt class is much harder than merely adding ramification or noncommutativity.}
}
$$

For curve-cover architectures, generic non-split sixfold reach is forced into a very narrow higher-rank ramified regime over:

$$
\mathbb P^1.
$$

The sharpest candidate is:

$$
\boxed{
d=3,
\quad
r=12,
\quad
c_j=1,
\quad
m_W=m_{\bar W}=3.
}
$$

Everything simpler has now been structurally eliminated.

---

# 54. Next Interface

Next GMSC round:

```text
HODGE_GMSC_R006_ReflectionCoverCandidateSearch.md
```

Primary target:

$$
\boxed{
\text{Determine whether the }d=3,\ r=12\text{ reflection-cover frontier is nonempty and can realize }\delta\neq-1.
}
$$

Planned tasks:

1. enumerate finite complex reflection groups / finite groups with suitable $3$-dimensional irreducible representations;
2. require imaginary-quadratic character field;
3. require Schur index $1$;
4. enumerate pseudo-reflection conjugacy classes;
5. search product-one generating Nielsen tuples of length $12$;
6. compute Chevalley–Weil:
   $$
   m_W,\ m_{\bar W};
   $$
7. retain only:
   $$
   (3,3);
   $$
8. compute the induced Hermitian discriminant;
9. test period-map dimension / special-subvariety constraints;
10. if no candidates survive, eliminate the last explicit curve-cover frontier;
11. if a candidate survives, search a nonabelian algebraic-cycle realization engine.

---

# References

1. L. Candelori, *The Chevalley-Weil Formula for Orbifold Curves*, arXiv:1712.02437. Gives Chevalley–Weil decompositions for ramified Galois/orbifold covers.

2. H. Lange, A. Ortega, *Prym varieties of cyclic coverings*, arXiv:0805.1020. Studies ramified cyclic Prym maps and image dimensions.

3. J. C. Naranjo, A. Ortega, A. Verra, *Generic injectivity of the Prym map for double ramified coverings*, arXiv:1708.06512.

4. P. Frediani, G. P. Grosselli, *Shimura curves in the Prym loci of ramified double covers*, arXiv:2007.09646.

5. P. Frediani, G. P. Grosselli, A. Mohajer, *Higher dimensional Shimura varieties in the Prym loci of ramified double covers*, arXiv:2101.09016.

6. B. van Geemen, A. Verra, *Quaternionic Pryms and Hodge classes*, arXiv:math/0103111. Gives a nonabelian/quaternionic Prym precedent where exceptional Hodge classes are algebraic.

7. A. Mohajer, *Decomposition of Jacobian varieties and Shimura subvarieties in $A_g$*, arXiv:2311.15789. Gives nonexistence results for some dihedral/quaternionic cover constructions.

8. P. Brosnan, *Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture*, arXiv:2609.14169. Proves maximal unipotent degeneration requires discriminant $(-1)^n$.

9. A. Mostaed, *McMullen's Curve, the Weil Locus, and the Hodge Conjecture for Abelian Sixfolds*, arXiv:2603.20268. Produces isolated CM Weil-sixfold points beyond current algebraicity theorems.

10. G. Lombardo, *Abelian varieties of Weil type and Kuga-Satake varieties*, arXiv:math/0211224. Relates Weil-type abelian varieties to K3-type/Kuga–Satake Hodge structures in arbitrary dimension.

11. S. Floccari, L. Fu, *The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties*, arXiv:2504.13607.

12. B. van Geemen, A. Rapagnetta, *Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class*, arXiv:2607.18341.

13. Aletheia, *HODGE_GMSC_R004_PrymDiscriminantComponentAudit*, 2026-09-16.

---

## Canonical Source Declaration

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

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪沒有宣稱所有 ramified nonabelian covers 都能產生 non-split Weil sixfold。

本輪正式證明的是 architecture-level dimension/conductor filters，將所有 curve-cover候選壓縮到高階 ramified nonabelian regime，並識別：

$$
\boxed{
d=3,\ r=12,\ c_j=1
}
$$

為最尖銳的 first candidate frontier。

下一輪將直接把此 frontier轉成 finite-group / branch-tuple search。