# HODGE_GMSC_R002_WeilCoreAlternativePaths
## ——GMSC 第二輪：Weil Core 的替代合法路徑、Prym Bypass 與 Proof-Cost Pareto Frontier

**作者：Aletheia（GPT-5.6 Sol）**  
**方法：GMSC — Global Mathematical Space Compression**  
**研究主題：Very-General Weil Sixfold / Weil Primitive Algebraicity Core**  
**輪次：GMSC-R002**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Primary Claim:** 對 GMSC-R001 識別出的 very-general Weil sixfold target core

$$
\mathcal C_{\mathrm{Weil6}}^{\mathrm{vg}}
=
[
\text{One Nonzero Weil Cycle},
\text{Full Weil Plane Algebraic},
\text{All-Powers HC}
]_{\mathrm{vg}},
$$

目前存在至少兩種歷史上已真正產生 Weil algebraicity 的不同幾何 engine：

1. Markman 的 secant-sheaf / Fourier–Mukai / semiregularity route；
2. Schoen–Koike 的 Prym / cyclic-cover / dominance route。

此外，Patel–Zhang 2025–2026 的 generalized Prym 工作把 Schoen cycle engine 擴展到一般 finite abelian cover setting，直接構造並證明一類 Weil-type Hodge subspaces 由 algebraic cycles生成。這證明 Prym geometry 不是僅有歷史偶然，而是一個可擴張的獨立 realization architecture。

因此：

$$
\boxed{
\text{Semiregularity}
\notin
R_{\mathrm{global}}
}
$$

在更強的意義下成立：sixfold Weil algebraicity 已有非-semiregularity 的 genuine historical bypass。

對尚未閉合的 general non-split sixfold components，本輪把 positive route space 壓成 provisional Pareto frontier：

$$
\boxed{
\mathcal P_{\mathrm{frontier}}
=
\{
P_{\mathrm{Markman}},
P_{\mathrm{Prym}},
P_{\mathrm{Direct}}
\}.
}
$$

其中：

- $P_{\mathrm{Markman}}$ 的核心 open debt 是 component-local seed / semiregularity；
- $P_{\mathrm{Prym}}$ 的 cycle-realization engine已有 theorem，核心 open debt 是 **Prym Moduli Reach / Dominance**；
- $P_{\mathrm{Direct}}$ 沒有額外 bridge debt，但缺少任何 general non-split explicit cycle construction。

其他看似替代路徑——absolute Hodge、motivated cycles、CM identification、arithmetic/Tate comparison——目前都沒有直接 discharge algebraicity，因此不是 completed bypass，只是 conditional bridge candidates。

**Status:** REFINED / PROVISIONAL PARETO FRONTIER  
**Global Core:** unchanged — Weil primitive legality decision  
**New Genuine Route:** Prym / generalized Prym realization  
**Markman Route:** remains frontier, no longer unique instantiated architecture  
**Prym Route Residual:** target-component reach / dominance  
**Universal Positive Singleton Min-Cut:** NOT FOUND  
**Attack Result:** several apparent bypasses demoted to hidden-realization-debt routes  
**Formalization Status:** NOT FORMALIZED  
**Graph Completeness:** IMPROVED / REBUILD REQUIRED  

---

# 0. Input from GMSC-R001

GMSC-R001 proved that the true target in the very-general Weil sixfold domain is not:

- semiregularity；
- Rank-$20$；
- gluing compensation；
- one specific secant object。

It is the target-equivalent SCC:

$$
\boxed{
\mathcal C_{\mathrm{Weil6}}^{\mathrm{vg}}
}.
$$

R002 therefore asks:

> How many genuinely different legal architectures can enter this SCC?

---

# 1. GMSC path admissibility test

A proposed route:

$$
P
$$

is admitted only if it contains an explicit algebraicity bridge.

A path that ends at:

- absolute Hodge；
- motivated；
- Hodge locus；
- Tate class；
- Mumford–Tate invariant；
- cohomological correspondence；

without a legal lift to:

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

does not reach the core.

---

# 2. True bypass criterion

Call a route a **true semiregularity bypass** if:

1. it can produce a nonzero class in:
   $$
   W_K(A);
   $$
2. that class is represented by an algebraic cycle;
3. its algebraicity proof does not pass through deformation of a Markman-type semiregular seed object.

This is stronger than merely being a different language.

---

# 3. Route P1 — Markman secant / semiregularity

The modern route is:

$$
\boxed{
\text{Secant Sheaf}
\to
\text{Orlov Transform}
\to
\text{Hodge-Permanent Characteristic Class}
\to
\text{Semiregularity}
\to
\text{Relative Algebraic Class}
\to
\text{Weil Core}.
}
$$

For split sixfolds of modern normalized discriminant:

$$
\delta=-1,
$$

Markman closes the route for every imaginary quadratic:

$$
K.
$$

Thus this route is not hypothetical.

---

# 4. Markman route residual outside the split regime

For a target non-split component:

$$
S=\mathcal M_{K,\delta},
\qquad
\delta\neq-1,
$$

the known split seed does not deform across discriminant components.

Therefore the current residual is:

$$
\boxed{
R_{\mathrm{Markman}}
=
\{
\text{component-local seed},
\text{appropriate semiregularity},
\text{nonzero Weil projection}
\}.
}
$$

PT003–PT010 expand these gates.

GMSC compresses them back to this route-level residual.

---

# 5. Route P2 — Schoen / Koike Prym route

There is an older but genuinely independent sixfold route.

Koike studies fourth cyclic étale covers of curves of genus:

$$
4
$$

and the associated Prym varieties.

The Prym map is shown dominant onto a Shimura variety parametrizing a family of Weil-type abelian sixfolds.

Using Schoen's algebraic cycles on the Prym construction, the corresponding Weil classes are algebraic.

Thus:

$$
\boxed{
\text{Prym geometry}
\to
\text{explicit algebraic cycles}
\to
\text{Weil class}
}
$$

is a true bypass.

---

# 6. Historical sixfold Prym closures

Koike records two relevant historical phenomena.

### Cubic cyclotomic field

Schoen constructed algebraic cycles on generalized Pryms and obtained algebraicity for a family of Weil-type abelian sixfolds with:

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

for a specific discriminant regime.

### Gaussian field

Koike's dominance theorem gives algebraicity for a family with:

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

and discriminant:

$$
\delta_{\mathrm{Koike}}=1.
$$

These are actual sixfold positive results independent of Markman's 2025 semiregularity proof.

---

# 7. Discriminant convention firewall

Koike uses an older discriminant convention based on the determinant of the Hermitian form modulo his stated equivalence.

Markman's modern normalized discriminant is written in a different quotient and split rank-$6$ is normalized as:

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

Therefore:

$$
\boxed{
\delta_{\mathrm{Koike}}=1
}
$$

must **not** be identified automatically with:

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

R002 records only the historical domain as stated in each source.

A separate arithmetic normalization audit is required before comparing component labels literally.

---

# 8. Why the Prym route matters to GMSC

Before R002, one could still say:

> Maybe every serious sixfold construction is secretly a semiregularity construction.

Koike/Schoen disproves that.

Prym algebraicity is obtained from explicit cycle geometry plus moduli dominance.

Hence:

$$
\boxed{
\text{Semiregularity is not an articulation point even inside sixfold Weil mathematics}.
}
$$

This is stronger than R001's purely logical alternative-path observation.

---

# 9. The modern generalized Prym engine

Patel–Zhang revisit Schoen's cycle construction using unramified geometric class field theory.

For a finite abelian cover of curves, they construct an algebraic Hodge subspace:

$$
U
$$

from components over a special fiber of a symmetric product cover.

They then identify the corresponding Weil-type Hodge subspace:

$$
U_{\mathrm{Weil}}
$$

in the associated Prym-type abelian variety.

Their theorem gives:

$$
\boxed{
U_{\mathrm{Weil}}
\text{ is generated by algebraic cycles}.
}
$$

---

# 10. Significance of Patel–Zhang

The theorem is important for GMSC because the algebraicity engine is not tied only to:

$$
G=\mathbb Z/3
\quad\text{or}\quad
G=\mathbb Z/4.
$$

The construction is formulated for finite abelian covers and can be applied to rational representation factors.

Thus the route architecture becomes:

$$
\boxed{
\text{Abelian Cover}
\to
\text{Generalized Prym}
\to
\text{Class-Field-Theoretic Cycles}
\to
\text{Weil-Type Hodge Subspace Algebraic}.
}
$$

This is a genuine expandable alternative engine.

---

# 11. Prym route residual

The cycle-realization side of the Prym route is therefore much more closed than it first appears.

For a general target component, the principal missing edge becomes:

$$
\boxed{
\text{Prym Moduli Reach}.
}
$$

That is:

> Can the target Weil component, or a Zariski-dense/open subset of it, be dominated by a family of generalized Prym varieties whose algebraic $U_{\mathrm{Weil}}$ identifies nontrivially with the target $W_K(A)$?

---

# 12. Prym route graph

The route is:

$$
\boxed{
\text{Cover Data}
\to
\text{Generalized Prym Family}
\to
\text{Algebraic Weil-Type Cycles}
\to
\text{Prym Moduli Reach}
\to
\mathcal C_{\mathrm{Weil6}}^{\mathrm{vg}}.
}
$$

Compared with Markman:

$$
\boxed{
\text{Seed Object}
\to
\text{Semiregularity}
\to
\text{Relative Class}
\to
\mathcal C_{\mathrm{Weil6}}^{\mathrm{vg}}.
}
$$

The two routes have different core residual types.

---

# 13. Route P3 — Direct one-cycle construction

PT002 gives an extreme compression:

$$
\boxed{
\text{one nonzero algebraic Weil class}
\Longrightarrow
\text{full Weil plane}.
}
$$

Thus the most conceptually direct route is simply:

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

No family is logically needed.

No semiregularity is logically needed.

No dominance is logically needed.

---

# 14. Direct-cycle route residual

The route has almost no dependency depth:

$$
\boxed{
R_{\mathrm{Direct}}
=
\{
\text{explicit nonzero Weil cycle}
\}.
}
$$

But the constructive difficulty is maximal:

> no general explicit cycle construction is currently known for arbitrary non-split Weil sixfolds.

So low graph depth does not mean low proof cost.

---

# 15. Route P4 — Degeneration / specialization

A second true bridge type would be:

$$
\boxed{
\text{Known Algebraic Seed Fiber}
\to
\text{Degeneration/Specialization}
\to
\text{Target Weil Class}.
}
$$

This is a real mechanism in the fourfold story.

Markman's split sixfold result combines with Schoen's degeneration method to obtain algebraicity for Weil fourfolds beyond the initial split domain.

---

# 16. Why degeneration is not yet a sixfold bypass

For sixfold components, discriminant data is discrete.

There is no currently established degeneration theorem showing:

$$
\boxed{
\text{solved split sixfold component}
\to
\text{arbitrary non-split sixfold component}
}
$$

while preserving a nonzero algebraic Weil class in the required direction.

Therefore the sixfold route residual is:

$$
\boxed{
R_{\mathrm{Deg}}
=
\{
\text{cross-component specialization bridge}
\}.
}
$$

This route is legal as a research architecture but currently uninstantiated at the crucial edge.

---

# 17. Route P5 — Auxiliary hyperkähler geometry

For Weil fourfolds there are now multiple special-geometric bypasses.

Floccari–Fu use singular OG6-type geometry.

Van Geemen–Rapagnetta use a map from the abelian fourfold to a hyperkähler sixfold of:

$$
K3^{[3]}
$$

type, then pull back:

$$
c_2(T_Y).
$$

The resulting algebraic codimension-$2$ class is non-Lefschetz and closes the target.

---

# 18. Hyperkähler route as route-diversity witness

This proves a strong GMSC lesson:

$$
\boxed{
\text{Weil algebraicity can be realized by pulling back an algebraic class from auxiliary hyperkähler geometry}.
}
$$

Thus:

$$
\text{auxiliary geometry}
\to
\text{algebraic correspondence/pullback}
\to
\text{Weil core}
$$

is a genuine route type.

---

# 19. Why auxiliary hyperkähler is not yet a sixfold path

No current theorem supplies for a general non-split abelian sixfold:

- an appropriate auxiliary hyperkähler target；
- an algebraic map/correspondence；
- a characteristic algebraic class whose pullback has nonzero Weil projection。

Hence this is currently:

```text
ROUTE-CANDIDATE
```

not:

```text
LEGAL COMPLETED BYPASS
```

for the sixfold core.

---

# 20. Route P6 — Standard-conjectural route

Milne records a broad implication architecture in which standard-conjectural algebraicity of Lefschetz-type operators implies Hodge conjecture consequences for abelian varieties.

As a graph path this is:

$$
\boxed{
\text{Strong Standard-Conjectural Package}
\to
\operatorname{HC}_{\mathrm{Ab}}
\to
\mathcal C_{\mathrm{Weil6}}^{\mathrm{vg}}.
}
$$

This route is logically powerful.

---

# 21. Why Standard-Conjecture route is not Pareto-efficient

Its target reach is huge.

But its proof debt is at least as broad as the original problem in the relevant formulation.

Thus GMSC marks:

$$
\boxed{
\delta(P_{\mathrm{Std}})
\gg0.
}
$$

It is useful as a non-uniqueness witness.

It is not a low-cost current route.

---

# 22. Route P7 — Absolute Hodge

For abelian varieties, Hodge classes are absolute Hodge.

Therefore every Weil class has strong transport/arithmetic rigidity.

But:

$$
\boxed{
\text{absolute Hodge}
\not\Rightarrow
\text{known algebraic cycle}.
}
$$

So the path:

$$
\text{Weil}
\to
\text{Absolute Hodge}
\to
\text{Algebraic}
$$

contains an absent final edge.

---

# 23. Absolute-Hodge hidden debt

Define:

$$
\boxed{
\delta_{\mathrm{AH}}
=
\text{Absolute-Hodge-to-Algebraic Bridge}.
}
$$

No theorem currently discharges:

$$
\delta_{\mathrm{AH}}
$$

for arbitrary target Weil sixfolds.

Hence absolute Hodge is not a true bypass.

---

# 24. Route P8 — Motivated cycles

André's motivated category gives excellent formal behavior:

- tensor operations；
- deformation；
- absolute properties；
- comparison。

But rational Hodge asks for actual algebraic cycle classes.

Thus:

$$
\boxed{
\text{motivated}
\not\Rightarrow
\text{known algebraic}
}
$$

in the needed generality.

The route simply moves the debt to:

$$
\boxed{
\delta_{\mathrm{mot\to alg}}.
}
$$

---

# 25. Route P9 — CM / special-point identification

Mostaed 2026 identifies new CM Weil-sixfold points:

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

of degree:

$$
12.
$$

Their Weil classes are absolute Hodge.

But current algebraicity theorems do not apply due to:

- CM isolation；
- no known $K$-secant structure；
- uncontrolled discriminant。

Thus CM identification gives a highly structured target.

It does not give algebraicity.

---

# 26. CM points as seeds

A CM point could become useful if one adds:

$$
\boxed{
\text{one algebraic Weil class at the CM point}
}
$$

and then a legal relative deformation engine.

At that moment the route returns to:

- Markman-like relative deformation；
- Prym dominance；
- another cycle-family mechanism。

So CM by itself is not a bypass.

---

# 27. Route P10 — Arithmetic / Tate

One may try:

$$
\text{complex target}
\to
\text{good reduction}
\to
\text{Tate class algebraicity}
\to
\text{lift back}.
$$

But the final lift:

$$
\boxed{
\text{cycle in characteristic }p
\to
\text{cycle in characteristic }0
}
$$

is not automatic.

Thus the route contains a major unresolved bridge.

---

# 28. Tate route hidden debt

Define:

$$
\boxed{
\delta_{\mathrm{lift}}
=
\text{Algebraic Cycle Lifting to the Exact Complex Weil Class}.
}
$$

Without discharging:

$$
\delta_{\mathrm{lift}},
$$

the arithmetic route does not enter the core.

---

# 29. Route P11 — Hodge-locus / variational route without semiregularity

Suppose a class remains Hodge along:

$$
S.
$$

That alone gives:

$$
\boxed{
S
\subseteq
\text{Hodge locus}.
}
$$

It does not give a relative algebraic cycle.

Hence a pure Hodge-locus route still needs a realization engine:

- semiregularity；
- Hilbert/Chow dominance；
- normal-function theorem；
- another variational algebraicity theorem。

So this route collapses into the same realization debt.

---

# 30. True bridge versus false bridge table

| Route object | Gives Hodge structure? | Gives actual algebraic cycle? | True bypass status |
|---|---:|---:|---|
| Secant + semiregularity | yes | yes when gate closes | TRUE ENGINE |
| Prym / generalized Prym | yes | yes on covered Prym domain | TRUE ENGINE |
| Direct explicit cycle | yes | yes by definition | TRUE ENGINE |
| Degeneration | yes | yes if specialization bridge is valid | CONDITIONAL TRUE ENGINE |
| Auxiliary hyperkähler map | yes | yes if algebraic map/correspondence exists | CONDITIONAL TRUE ENGINE |
| Absolute Hodge | yes | no general bridge | FALSE BYPASS |
| Motivated cycles | yes | no general bridge | FALSE BYPASS |
| CM classification | yes | no | FALSE BYPASS |
| Tate class mod $p$ | arithmetic analog | not in target characteristic automatically | FALSE BYPASS |
| Hodge locus only | yes | no | FALSE BYPASS |

---

# 31. Current closed sixfold subdomains

The sixfold graph is not uniformly open.

At least the following positive regions are historically closed.

### Markman split region

Modern normalized:

$$
\delta=-1
$$

for every imaginary quadratic:

$$
K.
$$

### Schoen / Koike Prym regions

Specific sixfold families for:

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

and:

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

under their historical discriminant conventions.

The exact relation among old and modern discriminant labels is not assumed.

---

# 32. Consequence for the open-domain graph

The global open domain should therefore be written as:

$$
\boxed{
\mathscr D_{\mathrm{open}}
=
\{
(K,\delta,\text{polarization type})
:
\text{not covered by a certified sixfold algebraicity theorem}
\}.
}
$$

Research should target:

$$
\mathscr D_{\mathrm{open}},
$$

not repeatedly include already-closed components.

This is another GMSC compression.

---

# 33. Path cost vector

Following GMSC, use:

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

R002 uses ordinal scores:

$$
0,1,2,3,4,5,
$$

where smaller means lower current research burden.

These scores are heuristic state variables, not theorems.

---

# 34. Current route weights

For the still-open general non-split sixfold domain:

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

Interpretation:

- deep but well-specified theorem chain；
- moderate dependencies；
- semiregularity/seed uncertainty remains；
- nontrivial Ext/deformation computation；
- formalizable local operators；
- modest conceptual novelty because engine exists。

---

# 35. Prym route weight

Assign:

$$
\boxed{
W(P_{\mathrm{Prym}})
=
(3,2,4,2,2,2).
}
$$

The cycle engine is concrete and recently generalized.

But target-component dominance/reach for arbitrary:

$$
(K,\delta)
$$

is substantially uncertain.

Computation burden may be lower because the key gate is a moduli/dominance problem rather than a large Ext calculation.

---

# 36. Direct-cycle route weight

Assign:

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

Dependencies are minimal.

But the cycle itself is unknown, and invention cost is high.

It remains Pareto-relevant because low dependency is a genuine advantage.

---

# 37. Degeneration route weight

Assign:

$$
\boxed{
W(P_{\mathrm{Deg}})
=
(3,2,4,2,2,3).
}
$$

The mechanism is standard and low-compute compared with semiregularity.

But no valid cross-discriminant sixfold bridge is currently known.

---

# 38. Auxiliary hyperkähler route weight

Assign:

$$
\boxed{
W(P_{\mathrm{HK}})
=
(4,2,4,4,3,4).
}
$$

The fourfold precedents are strong.

The sixfold architecture is currently missing.

Thus high novelty and construction cost remain.

---

# 39. High-debt routes

Representative scores:

$$
\boxed{
W(P_{\mathrm{Std}})
=
(5,5,4,4,4,2),
}
$$

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

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

These routes are logically broad but not current low-cost paths into the core.

---

# 40. Provisional Pareto frontier

Using the six GMSC burden coordinates without scalarization, the non-dominated frontier is:

$$
\boxed{
\mathcal P_{\mathrm{frontier}}
=
\{
P_{\mathrm{Markman}},
P_{\mathrm{Prym}},
P_{\mathrm{Direct}}
\}.
}
$$

The degeneration route is near-frontier but currently has a missing cross-component edge.

Hyperkähler, motivic, standard-conjectural, arithmetic, and CM-only routes are dominated or contain larger realization debt in the present graph state.

---

# 41. Meaning of the frontier

The frontier does **not** claim these three paths are equally likely to succeed.

It says none is uniformly cheaper than the others across all GMSC burden dimensions.

### Markman

Best-developed modern general engine.

### Prym

Strongest currently identified non-Markman geometric bypass.

### Direct

Minimal dependency depth, maximal invention burden.

---

# 42. Current articulation points by route

### Markman subgraph

Provisional articulation:

$$
\boxed{
\text{Relative/Object Semiregularity}
}
$$

outside the already solved split domain.

### Prym subgraph

Provisional articulation:

$$
\boxed{
\text{Prym Moduli Reach / Dominance}.
}
$$

### Direct subgraph

The route consists essentially of one construction obligation:

$$
\boxed{
\text{One Nonzero Weil Cycle}.
}
$$

---

# 43. No common positive articulation before the core

Across the three frontier paths:

$$
P_{\mathrm{Markman}},
\quad
P_{\mathrm{Prym}},
\quad
P_{\mathrm{Direct}},
$$

there is no identified nontrivial theorem gate common to all three before entering:

$$
\mathcal C_{\mathrm{Weil6}}^{\mathrm{vg}}.
$$

Therefore:

$$
\boxed{
\text{no pre-core singleton global min-cut is currently established}.
}
$$

---

# 44. Global residual remains target-equivalent

For the very-general target domain:

$$
\boxed{
R_{\mathrm{global}}
=
\{
\mathcal C_{\mathrm{Weil6}}^{\mathrm{vg}}
\}.
}
$$

But positive proof search has at least three independent frontier architectures.

This is precisely the distinction GMSC wants:

$$
\boxed{
\text{core truth state}
\neq
\text{route gate}.
}
$$

---

# 45. Construct result

R002's main Construct discovery is:

$$
\boxed{
\text{Prym Moduli Reach}
}
$$

as a new top-level positive gate.

The cycle engine itself is no longer the main uncertainty thanks to:

- Schoen；
- Koike；
- Patel–Zhang generalized Prym algebraicity。

Thus a new research route is:

$$
\boxed{
\text{Can generalized Prym families dominate new open Weil-sixfold components?}
}
$$

---

# 46. Attack result — false bypasses

R002 attacks several apparent shortcuts.

### Absolute Hodge shortcut

Broken because algebraicity is not obtained.

### Motivic shortcut

Broken because motivated-to-algebraic remains open.

### CM shortcut

Broken because CM structure does not construct a cycle.

### Tate shortcut

Broken because characteristic-$p$ algebraicity does not automatically lift to the exact complex class.

### Hodge-locus shortcut

Broken because Hodge persistence is weaker than algebraic persistence.

These paths remain useful support theories.

They are not current core-closing paths.

---

# 47. Attack result — unverified universal claims

Recent public/preprint claims asserting substantially broader Hodge closure are not admitted into the legal graph merely because they exist.

GMSC requires:

- hypothesis audit；
- domain audit；
- proof-debt audit；
- external verification。

Until then:

```text
status = UNVERIFIED
```

They do not alter the residual graph.

---

# 48. Compress result

R001 listed many alternative route families.

R002 compresses the positive search to:

$$
\boxed{
\text{three frontier architectures}
+
\text{one near-frontier degeneration architecture}
+
\text{high-debt support routes}.
}
$$

This removes another layer of false freedom.

---

# 49. Reopened gate — Prym dominance

Before GMSC, Prym geometry was not in the active sixfold program.

R002 reopens:

$$
\boxed{
\text{Prym Dominance}
}
$$

because:

1. it historically solved genuine sixfold Weil cases；
2. Patel–Zhang supplies a modern generalized algebraic-cycle engine；
3. its residual is qualitatively different from semiregularity。

Status:

```text
REOPENED / HIGH PRIORITY ALTERNATIVE
```

---

# 50. Reopened gate — discriminant reach

The old Prym results use field/discriminant-specific moduli.

Therefore the next exact question is:

$$
\boxed{
\text{Which modern Weil components are in the image/closure of generalized Prym period maps?}
}
$$

This requires a full convention-normalized component map.

Status:

```text
OPEN
```

---

# 51. Reopened gate — direct one-cycle search

PT002 makes direct construction much cheaper conceptually.

We no longer need two independent cycle families.

We need:

$$
\boxed{
\text{one nonzero Weil projection}.
}
$$

Thus any new geometric subvariety, Chern class, correspondence, or auxiliary pullback with a nonzero Weil component is decisive.

Status:

```text
REOPENED
```

---

# 52. Bottleneck stability update

After two GMSC rebuilds:

### Weil primitive legality core

Appears in the target min-cut both rounds.

Thus within its domain:

$$
\boxed{
\mathrm{BS}_2
\left(
\mathcal C_{\mathrm{Weil6}}^{\mathrm{vg}}
\right)
=
1.
}
$$

This is unsurprising because it is target-equivalent.

### Semiregularity

Appears only inside one route.

Thus:

$$
\boxed{
\mathrm{BS}_2(\mathrm{Semiregularity})
<
1
}
$$

for the global positive graph.

### Prym dominance

New in R002.

No stability score yet.

---

# 53. Gate-type entropy

The frontier has three different gate types:

1. deformation/semiregularity；
2. moduli dominance/Prym reach；
3. direct realization/construction。

Therefore gate type entropy remains nonzero.

This means:

$$
\boxed{
\text{do not enter single Core Theorem Mode yet}.
}
$$

---

# 54. Search budget implication

A rational next allocation is not:

$$
100\%
\quad
\text{to PT011}.
$$

A GMSC-consistent allocation is conceptually:

$$
\boxed{
\text{Markman}
\parallel
\text{Prym Reach}
\parallel
\text{Direct Cycle}
\parallel
\text{CE001 Attack}.
}
$$

The exact compute/token distribution may vary by round.

---

# 55. Why Prym deserves immediate attention

Prym route has three unusually favorable properties.

### A. Real sixfold precedent

It has already algebraized Weil classes on actual sixfold families.

### B. Explicit algebraic cycles

It does not rely only on Hodge/motivic avatars.

### C. Modern generalization

Patel–Zhang broadens the cycle engine to generalized Pryms from finite abelian covers.

Therefore the principal unknown has shifted from:

$$
\text{Can Prym produce cycles?}
$$

to:

$$
\boxed{
\text{How far in Weil moduli can Prym reach?}
}
$$

This is an excellent GMSC residual.

---

# 56. Why Markman remains frontier

Despite Prym reopening, Markman remains competitive because:

- it works for every imaginary quadratic $K$ in the split sixfold component；
- its representation-theoretic construction is structurally general；
- semiregularity is a precise local operator problem；
- the route already closed a large nine-dimensional moduli component family。

Thus Prym does not dominate Markman globally.

---

# 57. Why direct cycle remains frontier

Direct cycle has no bridge debt.

If one finds:

$$
Z
$$

with:

$$
0\neq cl(Z)\in W_K(A),
$$

PT002 finishes the Weil plane immediately.

The problem is pure invention.

Therefore it is high-risk but non-dominated.

---

# 58. Current positive proof graph

```text
                                      +--> [Semiregularity] --+
                                      |                       |
[Known Geometry] --> [Markman Seed] --+                       |
                                                              v
[Cover Data] --> [Gen. Prym Cycles] --> [Prym Reach] ----> [WEIL CORE]
                                                              ^
[New Subvariety / Correspondence] --> [One Weil Cycle] -------+
                                                              ^
[Degeneration Seed] --> [Cross-Component Bridge] -------------+
```

High-debt routes enter from above but are not frontier paths.

---

# 59. Current Attack graph

HC-False CE001 directly attacks:

$$
\boxed{
\mathcal A_W=W_K(A)
}
$$

by trying to prove:

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

This remains strategically valuable because it attacks the core rather than one positive route.

---

# 60. New constructive research question

The highest-value newly exposed question is:

$$
\boxed{
\text{Generalized Prym Reach Problem}.
}
$$

A precise version:

> Given an open non-split Weil sixfold component $\mathcal M_{K,\delta}$, does there exist a finite abelian group $G$, an étale $G$-cover family of curves, a rational representation factor, and a Prym period map whose image dominates $\mathcal M_{K,\delta}$ and whose Patel–Zhang Weil-type algebraic subspace maps nontrivially to $W_K$?

If yes:

$$
\boxed{
\text{one new component closes without semiregularity}.
}
$$

---

# 61. This is a genuine path-invention problem

The Prym question is not:

> prove semiregularity differently.

It is:

$$
\boxed{
\text{invent a different legal path in theory space}.
}
$$

That is exactly GMSC's alternative-path mode.

---

# 62. Current proof-debt ledger

### Markman

$$
\delta_{\mathrm{M}}
=
\delta_{\mathrm{seed}}
+
\delta_{\mathrm{SR}}.
$$

### Prym

$$
\delta_{\mathrm{P}}
=
\delta_{\mathrm{reach}}.
$$

The algebraic-cycle engine is certified on the generalized Prym domain.

### Direct

$$
\delta_{\mathrm{D}}
=
\delta_{\mathrm{cycle\ invention}}.
$$

### Degeneration

$$
\delta_{\mathrm{Deg}}
=
\delta_{\mathrm{cross-component}}.
$$

---

# 63. Hidden debt ledger

### Absolute Hodge

$$
\delta_{\mathrm{AH}}
=
\delta_{\mathrm{realization}}.
$$

### Motivated

$$
\delta_{\mathrm{Mot}}
=
\delta_{\mathrm{mot\to alg}}.
$$

### Tate

$$
\delta_{\mathrm{Tate}}
=
\delta_{\mathrm{lift}}.
$$

### CM

$$
\delta_{\mathrm{CM}}
=
\delta_{\mathrm{cycle}}.
$$

These are not cheaper than they first appear.

---

# 64. Provisional path ranking by current maturity

Without scalarizing the Pareto vectors, one may still state a maturity ordering.

### Tier A — instantiated engines

$$
\boxed{
P_{\mathrm{Markman}},
\quad
P_{\mathrm{Prym}}.
}
$$

Both have historical positive sixfold results.

### Tier B — pure construction frontier

$$
\boxed{
P_{\mathrm{Direct}},
\quad
P_{\mathrm{Deg}}.
}
$$

### Tier C — analogy-based candidate

$$
\boxed{
P_{\mathrm{HK}}.
}
$$

### Tier D — high hidden-debt theories

$$
\boxed{
P_{\mathrm{Std}},
P_{\mathrm{Mot}},
P_{\mathrm{Tate}},
P_{\mathrm{CM-only}}.
}
$$

---

# 65. Min-cut recomputation

Inside Markman path:

$$
\operatorname{MinCut}_{M}
\sim
\{\mathrm{Semiregularity}\}.
$$

Inside Prym path:

$$
\operatorname{MinCut}_{P}
\sim
\{\mathrm{PrymReach}\}.
$$

Inside Direct path:

$$
\operatorname{MinCut}_{D}
\sim
\{\mathrm{CycleConstruction}\}.
$$

Across all three:

$$
\boxed{
\operatorname{MinCut}_{\mathrm{positive}}
\text{ is not a singleton theorem gate}.
}
$$

Only the target SCC itself is shared.

---

# 66. GMSC verdict on PT011

PT011 Gluing Compensation remains valid research.

But it is no longer the automatic next move.

Its correct label is:

```text
Markman-Route Deep Dive
```

not:

```text
Global Next Core
```

Therefore R002 keeps it paused while alternative-path information gain remains high.

---

# 67. GMSC verdict on CE001

CE001 directly tests the global core truth state.

Therefore it does **not** get demoted simply because alternative positive paths exist.

Its priority remains high.

GMSC sees:

$$
\boxed{
\text{Positive route diversification}
+
\text{Core-level negative attack}
}
$$

as complementary.

---

# 68. Round output checklist

## 1. Graph snapshot

Updated with Prym / generalized Prym engine.

## 2. SCC compression

Unchanged target SCC.

## 3. Route residual

Markman, Prym, Direct residuals separated.

## 4. Global residual

Weil primitive legality core unchanged.

## 5. Min-cut candidates

No nontrivial singleton pre-core min-cut.

## 6. Articulation points

Semiregularity route-local; PrymReach route-local.

## 7. Bridge edges

PrymReach newly promoted.

## 8. Alternative legal paths

Prym is certified as genuine historical bypass.

## 9. Construct results

Generalized Prym Reach Problem formulated.

## 10. Attack results

False bypasses demoted.

## 11. Reopened gates

Prym dominance, direct one-cycle, discriminant reach.

## 12. Next core targets

Prym reach / component map reconstruction.

---

# 69. Main Verdict

The strongest result of R002 is:

$$
\boxed{
\text{Markman semiregularity is not the unique sixfold realization architecture}.
}
$$

A genuine independent Prym route exists historically, and the cycle engine now has a modern generalized formulation.

Therefore the correct next global-search question is not:

> How do we finish gluing compensation?

It is:

$$
\boxed{
\text{Can generalized Prym geometry reach open non-split Weil-sixfold components?}
}
$$

---

# 70. Next Interface

Next GMSC round:

```text
HODGE_GMSC_R003_GeneralizedPrymReach.md
```

Primary target:

$$
\boxed{
\text{Map the image of Prym / generalized Prym period maps inside modern Weil-sixfold moduli.}
}
$$

Planned tasks:

1. reconstruct Schoen–Koike Prym moduli domains；
2. normalize old discriminant conventions to the modern Hermitian norm-class convention；
3. identify which modern $(K,\delta)$ components are already Prym-closed；
4. express Patel–Zhang generalized Prym data in representation-theoretic terms；
5. compute dimension of generalized Prym parameter spaces versus:
   $$
   \dim\mathcal M_{K,\delta}=9;
   $$
6. derive necessary dimension conditions for dominance；
7. search finite abelian groups / rational representation factors capable of producing sixfold Prym factors；
8. test whether open non-split components can be hit；
9. if dominance is impossible for structural reasons, demote Prym frontier；
10. if one new component is reachable, construct a new semiregularity-free path into the Weil Core.

---

# References

1. GMSC v0.1, *Global Mathematical Space Compression: Theory Graphs, Residual Min-Cuts, and Tri-Directional Closure for Large Mathematical Problems*, Neo.K with Aletheia, 2026-09-16.

2. E. Markman, *Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds*, arXiv:2502.03415.

3. E. Markman, *Secant sheaves and Weil classes on abelian varieties*, arXiv:2509.23403.

4. K. Koike, *Algebraicity of some Weil Hodge Classes*, Canadian Mathematical Bulletin 47 (2004), 566–572; arXiv:math/0211304. Proves dominance of a fourth-cyclic Prym map and obtains algebraicity of Weil classes for a family of abelian sixfolds of Weil type.

5. C. Schoen, *Hodge classes on self-products of a variety with an automorphism*, and subsequent addendum. Supplies the Prym-cycle algebraicity engine used in the classical cyclic-cover route.

6. D. Patel, Y. Zhang, *Algebraicity of Hodge classes on some Generalized Prym Varieties*, arXiv:2506.13729; accepted in Journal of Algebra. Extends Schoen-type algebraic cycle constructions to generalized Prym varieties arising from finite abelian covers.

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

8. B. van Geemen, A. Rapagnetta, *Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class*, arXiv:2607.18341. Gives a further auxiliary-hyperkähler route for general Weil fourfolds in its stated domain.

9. J. S. Milne, *Hodge classes on abelian varieties*, arXiv:2010.08857.

10. F. Charles, C. Schnell, *Notes on absolute Hodge classes*, arXiv:1101.3647.

11. A. Mostaed, *McMullen's Curve, the Weil Locus, and the Hodge Conjecture for Abelian Sixfolds*, arXiv:2603.20268.

12. Aletheia, *HODGE_GMSC_R001_GlobalTheoryGraphCompression*, 2026-09-16.

13. Aletheia, *HODGE_HCTRUE_PT002_OneCycleBinaryClosure*, 2026-09-16.

---

## Canonical Source Declaration

本檔案為 Hodge GMSC 分支第二篇正式 UTF-8 Markdown canonical source。

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪沒有宣稱 generalized Prym 已覆蓋 arbitrary non-split Weil sixfold component。

本輪的正式 closure 是：

$$
\boxed{
\text{Prym geometry is a genuine non-Markman sixfold algebraicity architecture},
}
$$

且 positive search 的 provisional Pareto frontier 已壓為 Markman、Prym、Direct 三條主要路徑。

下一輪將直接研究 Prym route 的真正 route residual：

$$
\boxed{
\text{moduli reach / dominance}.
}
$$