# RCHM × Hodge Conjecture — Research Round 17
## Gushel–Mukai Fourfolds as a Second Generalization Test: Double EPW Bridges, Multi-Realization Evidence, and Closure-Type Separation

**Status:** Internal research note + executable bridge-selection prototype  
**Version:** Round 17 / v1.7  
**Date:** 2026-09-17  
**Methodological base:** RCHM; Typed Relation & Constraint Algebra; X-Layer Mathematical Completeness; Graph-Native Mathematics  
**Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–16  
**Canonical source:** UTF-8 Markdown

---

# 0. Round 17 Objective

Round 16 showed that for cubic fourfolds the bridge compiler naturally prioritizes the canonical Fano variety of lines over an abstract Kuga–Satake construction.

Round 17 tests whether this principle generalizes to another Fano fourfold of K3 type:

$$
\boxed{
\text{Gushel--Mukai fourfolds}.
}
$$

The key questions are:

1. Does the target Hodge signature again point toward a low-dimensional hyperkähler bridge?
2. Is there a canonical incidence geometry supporting the same bridge?
3. Can the compiler distinguish a Hodge-theoretic bridge from a fully verified algebraic cycle-transfer bridge?
4. Does the bridge candidate receive independent support from classical and categorical constructions?
5. Can the compiler avoid overclaiming even though the generalized Hodge conjecture is already known for all GM varieties?

The answer to all five questions is yes.

---

# 1. GM Fourfold Middle Cohomology

Let:

$$
X
$$

be a smooth complex Gushel--Mukai fourfold.

Fu--Moonen describe the middle cohomology of even-dimensional GM varieties as K3 type.

For a GM fourfold:

$$
\boxed{
h^{3,1}=1,
\qquad
h^{2,2}=22,
\qquad
h^{1,3}=1.
}
$$

Thus:

$$
\boxed{
\operatorname{rank}H^4(X)=24
}
$$

in the middle K3-type block.

There is a natural rank-two algebraic/Tate part.

Removing this algebraic part gives the vanishing/primitive-type Hodge structure:

$$
\boxed{
H^4(X,\mathbb Z)_{00}(1)
}
$$

with Hodge numbers:

$$
\boxed{
(1,20,1)
}
$$

and rank:

$$
\boxed{
22.
}
$$

This is the first bridge signature.

---

# 2. Target Signature

The bridge compiler therefore receives:

```text
target:
  type: smooth_GM_fourfold
  middle_hodge: [1,22,1]
  algebraic_Tate_rank: 2
  vanishing_rank: 22
  shifted_vanishing_hodge: [1,20,1]
  bridge_goal: reduce codimension-2 Hodge problem
```

This signature is strikingly close to the cubic-fourfold primitive signature from Round 16.

The compiler should therefore search first for:

$$
\boxed{
\text{rank-22 hyperkähler primitive }H^2
}
$$

rather than immediately constructing a huge Kuga--Satake variety.

---

# 3. Associated Lagrangian Data

A GM variety determines Lagrangian data:

$$
\boxed{
(V_6,V_5,A),
}
$$

where:

$$
A\subset\wedge^3V_6
$$

is a Lagrangian subspace for the natural symplectic form.

From:

$$
(V_6,A)
$$

one constructs an EPW sextic:

$$
Y_A
$$

and its natural double cover:

$$
\widetilde Y_A.
$$

For general data:

$$
\boxed{
\widetilde Y_A
}
$$

is a smooth projective hyperkähler fourfold of:

$$
K3^{[2]}
$$

type.

A dual double EPW sextic:

$$
\widetilde Y_{A^\perp}
$$

is also naturally present.

Thus the target geometry itself generates the bridge category.

---

# 4. Primitive Hodge Match

Debarre--Kuznetsov establish the analog of the Beauville--Donagi relation:

$$
\boxed{
H^4(X,\mathbb Z)_{00}(1)
\simeq
H^2(\widetilde Y_A,\mathbb Z)_0
}
$$

as polarized integral Hodge structures in the appropriate smooth setting.

Both sides have rank:

$$
\boxed{
22.
}
$$

Both have Hodge numbers:

$$
\boxed{
(1,20,1).
}
$$

Thus the bridge has an exact primitive Hodge match.

---

# 5. Rank Check from Hyperkähler Geometry

A hyperkähler fourfold of:

$$
K3^{[2]}
$$

type has:

$$
\boxed{
b_2=23.
}
$$

A primitive polarization removes one rational direction.

Therefore:

$$
\boxed{
\operatorname{rank}H^2_{\mathrm{prim}}=22.
}
$$

This exactly matches:

$$
H^4(X)_{00}.
$$

The match is not a loose dimension analogy.

It is the correct primitive rank on both sides.

---

# 6. Degree-Lowering Property

The bridge shifts:

$$
H^4(X)(1)
\longrightarrow
H^2(\widetilde Y_A).
$$

Therefore:

$$
(2,2)_X
\longrightarrow
(1,1)_{\widetilde Y_A}.
$$

If the Hodge isometry is realized by an algebraic correspondence in the required direction, then the bridge-side witness problem lands in:

$$
H^{1,1},
$$

where Lefschetz $(1,1)$ gives divisors.

Thus the double EPW candidate receives the same major advantage as the Fano bridge in the cubic case:

$$
\boxed{
\text{degree lowering to a solved witness layer}.
}
$$

---

# 7. Critical Closure-Type Distinction

Round 17 deliberately distinguishes:

$$
\boxed{
\text{Hodge isometry}
}
$$

from:

$$
\boxed{
\text{algebraic correspondence inducing the witness transfer}.
}
$$

The existence of:

$$
H^4(X)_{00}(1)
\simeq
H^2(\widetilde Y_A)_0
$$

does not, by itself, authorize the conclusion:

$$
\text{every divisor on }\widetilde Y_A
\text{ transfers algebraically to }X.
$$

Therefore the compiler records:

```text
cohomological_bridge: PASS
degree_lowering: PASS
solved_bridge_layer: PASS
algebraic_cycle_transfer: VERIFY / source-dependent
```

This prevents a Hodge-theoretic bridge from being silently upgraded to a cycle-theoretic proof.

---

# 8. Classical Conic Realization

For a general GM fourfold, the Hilbert scheme of conics:

$$
F_g(X)
$$

is closely related to the associated double EPW sextic.

Classically, it is birational to a:

$$
\boxed{
\mathbb P^1\text{-bundle over a double EPW sextic}.
}
$$

Thus there is a canonical incidence geometry:

$$
\boxed{
\text{conics on }X
\longrightarrow
F_g(X)
\dashrightarrow
\widetilde Y.
}
$$

The universal conic provides an algebraic incidence relation with $X$.

This is a strong geometric reason to prioritize the EPW bridge.

---

# 9. Categorical Realization

The same bridge appears independently through the Kuznetsov component:

$$
\operatorname{Ku}(X).
$$

For very general GM fourfolds, Guo--Liu--Zhang identify the double EPW sextic and double dual EPW sextic with Bridgeland moduli spaces of stable objects in:

$$
\operatorname{Ku}(X).
$$

Schematically:

$$
\boxed{
M_\sigma(\operatorname{Ku}(X),\Lambda_i)
\simeq
\widetilde Y_A
\text{ or }
\widetilde Y_{A^\perp}.
}
$$

The same rank-$22$ primitive Hodge structure is also encoded in the orthogonal complement of the canonical rank-two numerical lattice of the K3 category.

Thus the bridge has a categorical realization in addition to the classical conic realization.

---

# 10. Multi-Realization Bridge Evidence

Define the **realization set** of a bridge candidate:

$$
\boxed{
\mathcal R(\mathcal B)
=
\{
\text{independent geometric/categorical constructions realizing }\mathcal B
\}.
}
$$

For the double EPW bridge one has at least:

1. Lagrangian-data / period construction;
2. classical conic-Hilbert geometry;
3. Bridgeland moduli realization.

Define the realization multiplicity:

$$
\boxed{
r_{\mathrm{real}}(\mathcal B)
=
|\mathcal R(\mathcal B)|.
}
$$

For the present search:

$$
\boxed{
r_{\mathrm{real}}(\mathcal B_{\mathrm{EPW}})
\ge3.
}
$$

This is a search-priority signal.

It is **not** a substitute for proof of a particular correspondence.

---

# 11. Convergence of Independent Paths

The bridge graph contains:

$$
\boxed{
X
\to
(V_6,V_5,A)
\to
\widetilde Y_A,
}
$$

and:

$$
\boxed{
X
\to
\operatorname{Hilb}_{2t+1}(X)
\dashrightarrow
\widetilde Y,
}
$$

and:

$$
\boxed{
X
\to
\operatorname{Ku}(X)
\to
M_\sigma(\operatorname{Ku}(X),\Lambda_i)
\simeq
\widetilde Y.
}
$$

All three paths converge on the same hyperkähler bridge type.

This is exactly the kind of graph convergence that an AI-native mathematical graph should preserve.

---

# 12. Candidate A — Double EPW Sextic

The compiler assigns:

```text
candidate: double_EPW_sextic
dimension: 4
hyperkahler_type: K3^[2]
primitive_rank: 22
primitive_hodge: [1,20,1]
cohomological_match: PASS
degree_lowering: PASS
solved_layer: H11
realization_multiplicity: >=3
```

This candidate should rank first.

---

# 13. Candidate B — Associated K3 Surface

GM fourfolds have K3-type middle cohomology and special Noether--Lefschetz loci.

A K3-surface bridge may appear for special GM fourfolds or via equivalences of the Kuznetsov K3 category with an actual K3 surface.

However a generic GM fourfold has vanishing rank:

$$
22
$$

with Hodge numbers:

$$
(1,20,1).
$$

A polarized K3 surface has primitive rank:

$$
21
$$

and Hodge numbers:

$$
(1,19,1).
$$

Therefore the generic direct associated-K3 template has a rank mismatch.

It activates only after extra algebraic structure removes an additional direction.

Thus:

$$
\boxed{
\text{K3 bridge is conditional, not the generic first bridge}.
}
$$

---

# 14. Candidate C — Generic Kuga--Satake

The rank-$22$ K3-type Hodge structure formally admits a Kuga--Satake construction.

As in Round 16:

$$
\boxed{
\dim\mathrm{KS}
=
2^{22-2}
=
1,048,576.
}
$$

This is formally available but extraordinarily large.

Compared with the four-dimensional EPW bridge:

$$
\boxed{
4
\ll
1,048,576.
}
$$

Thus the generic Kuga--Satake bridge should remain a fallback.

---

# 15. Bridge Selection for a Generic GM Fourfold

The bridge policy is conditional rather than a single total ordering.

For a generic GM fourfold, before extra algebraic structure is detected:

$$
\boxed{
\widetilde Y_A
\;\succ\;
\text{generic Kuga--Satake fallback}.
}
$$

The associated-K3 branch remains inactive.

Once an additional Noether--Lefschetz / algebraic direction or a geometricization of the K3 category is detected, activate:

$$
\boxed{
\widetilde Y_A
\;\succ\;
\text{associated K3 refinement}
\;\succ\;
\text{generic Kuga--Satake fallback}.
}
$$

The reasons are:

### Double EPW

- canonical from the target data;
- dimension $4$;
- exact primitive Hodge match;
- degree lowering to $H^2$;
- multiple realizations;
- direct incidence geometry.

### K3

- dimension $2$;
- but generic rank mismatch;
- only activated on special loci / categorical geometricization.

### Kuga--Satake

- formally general;
- dimension $1,048,576$;
- expensive;
- algebraicity of the necessary correspondence is a separate obligation.

---

# 16. Comparison with Cubic Fourfolds

Round 16:

$$
\boxed{
X_{\mathrm{cubic}}
\to
F(X)
}
$$

with:

$$
F(X)
\text{ of }K3^{[2]}\text{ type}.
$$

Round 17:

$$
\boxed{
X_{\mathrm{GM}}
\to
\widetilde Y_A
}
$$

with:

$$
\widetilde Y_A
\text{ of }K3^{[2]}\text{ type}.
$$

Both have:

$$
\boxed{
H^4_{\mathrm{hard}}(1)
\simeq
H^2_{\mathrm{prim}}(\text{hyperkähler fourfold}).
}
$$

Thus a common architecture appears:

$$
\boxed{
\text{Fano fourfold of K3 type}
\to
\text{canonical }K3^{[2]}\text{-type bridge}.
}
$$

---

# 17. Canonical-Incidence Generalization

Round 16 proposed:

$$
\textbf{Canonical-Incidence Heuristic}.
$$

Round 17 supplies a second independent calibration.

For cubic fourfolds:

$$
\text{lines}
\to
F(X).
$$

For GM fourfolds:

$$
\text{conics}
\to
\widetilde Y_A.
$$

Therefore the heuristic can be strengthened:

> For a Fano variety of K3 type, search the natural Hilbert/moduli spaces of minimal or low-degree rational curves before invoking abstract Kuga--Satake geometry.

This is now supported by two distinct fourfold families.

---

# 18. Curve-Degree Is Not Universal

The canonical incidence object is not always the same curve degree.

For cubic fourfolds:

$$
\boxed{
\text{lines}
}
$$

are sufficient.

For GM fourfolds:

$$
\boxed{
\text{conics}
}
$$

lead naturally to the EPW bridge.

Thus the search variable should be:

$$
\boxed{
\text{low-complexity curve class}
}
$$

rather than:

$$
\boxed{
\text{lines only}.
}
$$

The compiler should enumerate low-degree Hilbert components and ask which one produces a symplectic quotient or K3-type Hodge match.

---

# 19. MRC / Quotient Compression

In the GM case, the Hilbert scheme of conics is larger than the hyperkähler bridge.

The EPW fourfold appears as a rational quotient / $\mathbb P^1$-bundle base in the classical geometry.

This suggests another generic operation:

$$
\boxed{
\text{curve parameter space}
\to
\text{MRC / fibration quotient}
\to
\text{hyperkähler bridge}.
}
$$

The bridge need not equal the Hilbert scheme itself.

The compiler should therefore search for quotient geometries of natural parameter spaces.

---

# 20. Categorical Quotient Compression

The categorical side performs a parallel compression.

A family of curve objects on $X$ is projected into:

$$
\operatorname{Ku}(X),
$$

and stable objects with fixed numerical class form a hyperkähler moduli space.

Thus one has two compression mechanisms:

$$
\boxed{
\text{classical curve quotient}
}
$$

and:

$$
\boxed{
\text{categorical moduli quotient}.
}
$$

Their convergence on the same bridge strongly structures the search.

---

# 21. Bridge Realization Redundancy Index

A raw count of realizations may overcount nearly identical constructions.

Define a coarser redundancy index:

$$
\boxed{
R_{\mathrm{ind}}(\mathcal B)
}
$$

as the number of realization paths that use genuinely different mathematical mechanisms.

For the double EPW bridge, reasonable mechanism classes include:

1. period/Lagrangian data;
2. classical Hilbert/incidence geometry;
3. categorical Bridgeland moduli.

Thus:

$$
\boxed{
R_{\mathrm{ind}}\ge3.
}
$$

Again, this is a search metric, not a theorem-strength metric.

---

# 22. Known Generalized Hodge Truth Set

Fu--Moonen prove:

$$
\boxed{
\text{the generalized Hodge conjecture for every complex GM variety}.
}
$$

Thus GM varieties form an excellent truth set for evaluating the bridge compiler.

The compiler is not needed to establish that theorem.

Instead one asks:

> Does the compiler recover geometries and Hodge structures that are genuinely central to the known theory without being told the final proof?

For GM fourfolds, it does.

---

# 23. Independent Truth vs. Bridge-Specific Closure

The database must store separately:

```text
target_generalized_Hodge_status:
  PASS
  source: Fu-Moonen 2024
```

and:

```text
double_EPW_bridge:
  cohomological_match: PASS
  incidence_geometry: PASS
  categorical_realization: PASS
  algebraic_cycle_transfer_for_specific_closure:
      VERIFY_THEOREM_CHAIN
```

This separation prevents a solved target theorem from retroactively certifying every proposed bridge proof.

---

# 24. Bridge Proof Obligation Graph

For the EPW route, the remaining cycle-theoretic obligation can be represented as:

$$
\boxed{
\text{divisor on }\widetilde Y
\to
\text{algebraic correspondence}
\to
\text{codim-2 cycle on }X.
}
$$

The Hodge isometry certifies the cohomological type.

The conic geometry supplies an algebraic incidence source.

The precise composition and normalization required for a complete cycle-level proof must remain explicitly sourced.

Thus the bridge graph contains a legitimate unresolved edge rather than a hidden assumption.

---

# 25. General Bridge Candidate Schema Update

Round 17 adds:

```text
bridge_candidate:
  realization_paths[]
  realization_mechanism_classes[]
  realization_multiplicity
  independent_realization_index
  cohomological_closure_status
  cycle_transfer_status
  target_truth_status
```

This prevents all notions of “bridge exists” from being collapsed into one boolean.

---

# 26. Multi-Realization Priority Principle

Round 17 proposes:

$$
\boxed{
\textbf{Multi-Realization Priority Principle}
}
$$

If two bridge candidates pass the same hard structural gates, prefer early verification of a candidate supported by several independent realization mechanisms.

Reason:

- it is less likely to be an artifact of one representation;
- more tools are available for constructing correspondences;
- alternative paths can repair missing steps;
- graph convergence often signals a canonical object.

This is a search principle, not a proof principle.

---

# 27. External GHC Closure as Calibration

The known generalized Hodge theorem provides a useful diagnostic.

If the compiler produces a residual Hodge sector for a GM variety that it claims is impossible to algebraize, the compiler is wrong.

If it produces only enormous abstract bridges while missing EPW/Kuznetsov/incidence geometry, the ranking is poor.

Thus a solved theorem can serve as:

$$
\boxed{
\text{bridge-search benchmark ground truth}.
}
$$

This is analogous to using solved carriers as blind reconstruction benchmarks in Rounds 12--13.

---

# 28. Prototype Compiler

The accompanying Python prototype evaluates:

1. double EPW bridge;
2. conditional K3 bridge;
3. generic Kuga--Satake bridge.

It verifies:

- total and vanishing Hodge ranks;
- K3-type Hodge-number reduction;
- $K3^{[2]}$ primitive rank;
- K3 primitive rank mismatch;
- Kuga--Satake dimension;
- realization multiplicity metadata;
- separation of target truth from bridge-specific closure.

---

# 29. Exact Structural Checks

The prototype verifies:

$$
1+22+1=24,
$$

and after removing two Tate directions:

$$
\boxed{
1+20+1=22.
}
$$

For the hyperkähler bridge:

$$
b_2(K3^{[2]})=23,
$$

so:

$$
\boxed{
23-1=22.
}
$$

For a polarized K3 surface:

$$
b_2(K3)=22,
$$

so primitive rank:

$$
\boxed{
22-1=21.
}
$$

Finally:

$$
\boxed{
2^{22-2}=1,048,576.
}
$$

These arithmetic checks explain the generic bridge ordering.

---

# 30. New Principles

## Principle A — Multi-Realization Priority

Multiple independent realizations increase bridge-search priority without replacing proof obligations.

---

## Principle B — Truth/Bridge Separation

A target theorem known by another method does not automatically certify a candidate bridge proof.

---

## Principle C — Canonical Low-Curve Search

For Fano varieties of K3 type, search Hilbert/moduli spaces of low-complexity rational curves as bridge generators.

---

## Principle D — Quotient Compression

The useful hyperkähler bridge may be a quotient, MRC base, or Bridgeland moduli compression of the raw curve-parameter space.

---

## Principle E — Cycle-Transfer Gate

Hodge isometry, incidence geometry, and cycle-transfer correspondence are distinct typed edges.

---

# 31. Round 17 Conclusion

Round 17 provides a second independent generalization test of the Bridge Search Compiler.

A smooth GM fourfold has a rank-$22$ vanishing K3-type Hodge structure:

$$
\boxed{
H^4(X)_{00}(1):
(1,20,1).
}
$$

The target geometry canonically produces a double EPW sextic:

$$
\widetilde Y_A,
$$

a four-dimensional hyperkähler variety of:

$$
K3^{[2]}
$$

type, with primitive:

$$
H^2
$$

of the same rank and Hodge numbers.

Thus:

$$
\boxed{
H^4(X)_{00}(1)
\simeq
H^2(\widetilde Y_A)_0.
}
$$

The same bridge is independently supported by:

$$
\boxed{
\text{Lagrangian/period geometry},
}
$$

$$
\boxed{
\text{Hilbert scheme of conics},
}
$$

and:

$$
\boxed{
\text{Bridgeland moduli in }\operatorname{Ku}(X).
}
$$

The compiler therefore uses a gate-dependent ranking:

For generic GM fourfolds:

$$
\boxed{
\text{double EPW}
\to
\text{generic Kuga--Satake fallback},
}
$$

while the associated-K3 branch remains conditional.

After special algebraic / Noether--Lefschetz structure is detected:

$$
\boxed{
\text{double EPW}
\to
\text{associated K3 refinement}
\to
\text{generic Kuga--Satake fallback}.
}
$$

The Kuga--Satake fallback again has dimension:

$$
\boxed{
1,048,576,
}
$$

while the canonical EPW bridge has dimension:

$$
4.
$$

At the same time, the compiler refuses to identify:

$$
\text{Hodge isometry}
$$

with:

$$
\text{completed algebraic cycle transfer}.
$$

That distinction remains a separately typed proof obligation.

The known theorem of Fu--Moonen that all GM varieties satisfy the generalized Hodge conjecture is used only as an external truth set, not as hidden certification of the EPW route.

Together, Rounds 16 and 17 now support a broader heuristic:

$$
\boxed{
\text{Fano variety of K3 type}
\to
\text{canonical low-curve incidence}
\to
\text{low-dimensional hyperkähler bridge}.
}
$$

This is the main closure of Round 17.

---

# References

1. Olivier Debarre, Alexander Kuznetsov, **Gushel--Mukai varieties: linear spaces and periods**, arXiv:1605.05648.  
   Relevant result: primitive/vanishing Hodge structure of GM fourfolds and associated double EPW sextics.

2. Lie Fu, Ben Moonen, **Algebraic cycles on Gushel--Mukai varieties**, Épijournal de Géométrie Algébrique, Special volume in honour of Claire Voisin, Article 17 (2024).  
   DOI: https://doi.org/10.46298/epiga.2024.9815  
   Relevant results: middle K3-type Hodge structure and generalized Hodge conjecture for all complex GM varieties.

3. Hanfei Guo, Zhiyu Liu, Shizhuo Zhang, **Conics on Gushel--Mukai fourfolds, EPW sextics and Bridgeland moduli spaces**, Mathematical Research Letters 31 (2024), 1061--1106.  
   DOI: https://doi.org/10.4310/MRL.241119003311

4. Soheyla Feyzbakhsh, Hanfei Guo, Zhiyu Liu, Shizhuo Zhang, **Lagrangian families of Bridgeland moduli spaces from Gushel--Mukai fourfolds**, Compositio Mathematica 161 (2025), 2091--2135.  
   DOI: https://doi.org/10.1112/S0010437X25007468

5. Atanas Iliev, Laurent Manivel, **Fano manifolds of degree ten and EPW sextics**, Annales scientifiques de l'École Normale Supérieure 44 (2011), 393--426.

6. Laura Pertusi, Alexander Perry, Xiaolei Zhao and related work on Bridgeland stability/moduli for Kuznetsov components of GM varieties.