# RCHM × Hodge Conjecture — Research Round 18
## Verra Fourfolds: Parallel K3 / Double-EPW Bridges and Pareto-Optimal Bridge Search

**Status:** Internal research note + executable bridge-frontier prototype  
**Version:** Round 18 / v1.8  
**Date:** 2026-09-17  
**Methodological base:** RCHM; Typed Relation & Constraint Algebra; X-Layer Mathematical Completeness; Graph-Native Mathematics  
**Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–17  
**Canonical source:** UTF-8 Markdown

---

# 0. Round 18 Objective

Rounds 16--17 suggested a strong heuristic:

$$
\text{Fano fourfold of K3 type}
\to
\text{low-degree rational-curve incidence}
\to
\text{low-dimensional hyperkähler bridge}.
$$

Round 18 tests this heuristic on Verra fourfolds.

The result is a correction rather than a simple confirmation.

A general Verra fourfold has:

1. a conic-incidence bridge to a double EPW quartic;
2. a direct correspondence bridge to an associated degree-$2$ K3 surface.

Both bridges transport the same transcendental Hodge structure.

Thus bridge search should not use a fixed total ordering by geometric category.

It should maintain a **Pareto frontier** of valid bridges.

---

# 1. Verra Fourfold

A Verra fourfold is a double cover:

$$
X\to\mathbb P^2\times\mathbb P^2
$$

branched along a smooth divisor of bidegree:

$$
(2,2).
$$

Equivalently it can be realized as a quadric section of the projective cone over:

$$
\mathbb P^2\times\mathbb P^2.
$$

It is Fano.

---

# 2. Middle Hodge Structure

The Hodge diamond has middle line:

$$
\boxed{
h^{3,1}=1,
\qquad
h^{2,2}=21,
\qquad
h^{1,3}=1.
}
$$

Therefore:

$$
\boxed{
\dim H^4(X,\mathbb Q)=23.
}
$$

For a very general Verra fourfold, the algebraic part in degree four has dimension:

$$
\boxed{
2.
}
$$

Hence:

$$
\boxed{
\dim H^4_{\mathrm{tr}}(X,\mathbb Q)=21.
}
$$

After Tate twist:

$$
H^4_{\mathrm{tr}}(X)(1)
$$

has K3-type Hodge numbers:

$$
\boxed{
(1,19,1).
}
$$

---

# 3. Bridge Candidate A — Double EPW Quartic

Let:

$$
F=F_{(1,1)}(X)
$$

be the Hilbert scheme of $(1,1)$-conics on $X$.

For a general Verra fourfold:

$$
\boxed{
\dim F=5.
}
$$

There exists a morphism:

$$
\boxed{
\sigma:F\to Z
}
$$

which is a:

$$
\boxed{
\mathbb P^1\text{-fibration},
}
$$

where:

$$
Z
$$

is the associated double EPW quartic.

The fourfold $Z$ is hyperkähler of:

$$
K3^{[2]}
$$

type.

---

# 4. Conic Abel--Jacobi Correspondence

Let:

$$
C\subset X\times F
$$

be the universal $(1,1)$-conic, with projections:

$$
q:C\to X,
\qquad
p:C\to F.
$$

The conic correspondence gives an isomorphism:

$$
\boxed{
p_*q^*:
H^4_{\mathrm{tr}}(X,\mathbb Q)
\overset{\sim}{\longrightarrow}
\operatorname{Im}
\left(
H^2_{\mathrm{tr}}(Z,\mathbb Q)
\xrightarrow{\sigma^*}
H^2(F,\mathbb Q)
\right).
}
$$

Composing through the $\mathbb P^1$-fibration yields the Abel--Jacobi isomorphism:

$$
\boxed{
\Phi:
H^4_{\mathrm{tr}}(X,\mathbb Q)
\overset{\sim}{\longrightarrow}
H^2_{\mathrm{tr}}(Z,\mathbb Q).
}
$$

The bilinear forms are preserved up to a nonzero rational scalar.

Thus the conic-incidence geometry produces an actual correspondence-level bridge, not merely a dimension match.

---

# 5. Bridge Candidate B — Associated Degree-2 K3 Surface

A general Verra fourfold has associated degree-$2$ K3 surfaces arising from its two quadric fibrations.

For a very general Verra fourfold, there is an algebraic correspondence:

$$
\boxed{
\Psi\in A^3(X\times S)
}
$$

inducing:

$$
\boxed{
\Psi_*:
H^4_{\mathrm{tr}}(X)
\overset{\sim}{\longrightarrow}
H^2_{\mathrm{tr}}(S).
}
$$

Again the bilinear forms are respected up to a nonzero rational scalar.

For a very general degree-$2$ K3 surface:

$$
\rho(S)=1.
$$

Since:

$$
b_2(S)=22,
$$

we have:

$$
\boxed{
\dim H^2_{\mathrm{tr}}(S)=21.
}
$$

This exactly matches:

$$
H^4_{\mathrm{tr}}(X).
$$

---

# 6. Universal Family Correspondence

The K3 bridge is not merely fiberwise folklore.

On the open locus where the associated K3 surface is smooth, there exists a relative correspondence:

$$
\boxed{
\Psi\in
A^3
\left(
\mathcal X\times_{B'}\mathcal S
\right)
}
$$

whose restriction to each fiber induces:

$$
\boxed{
H^4_{\mathrm{prim}}(X_b)
\overset{\sim}{\longrightarrow}
H^2_{\mathrm{prim}}(S_b)(1).
}
$$

Thus the K3 bridge has genuine family coverage on the relevant open set.

---

# 7. Relation Between the Two Bridges

The double EPW quartic and the K3 surface are not unrelated guesses.

The double EPW quartic is itself related to a moduli space of twisted sheaves on the associated K3 surface.

At the Hodge level the chain is:

$$
\boxed{
H^4_{\mathrm{tr}}(X)
\overset{\sim}{\longrightarrow}
H^2_{\mathrm{tr}}(Z)
\overset{\sim}{\longrightarrow}
H^2_{\mathrm{tr}}(S).
}
$$

Moreover the published construction shows that the Verra-to-EPW map and the Verra-to-K3 map are induced by algebraic correspondences.

Thus the bridge graph contains a **bridge-of-bridge** relation:

$$
\boxed{
X
\to
Z
\leftrightarrow
S.
}
$$

---

# 8. Hodge-Witness Reduction

Both:

$$
H^2(Z)
$$

and:

$$
H^2(S)
$$

land in the Lefschetz $(1,1)$ regime.

Thus, for rational Hodge classes transported to either bridge:

$$
\boxed{
(1,1)
\Rightarrow
\text{divisor class}.
}
$$

Both candidates therefore satisfy the strongest bridge-search desideratum:

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

---

# 9. The Naive Total-Order Problem

A simplistic compiler might rank candidates only by dimension:

$$
\dim S=2,
\qquad
\dim Z=4.
$$

It would then always choose the K3 surface.

But that loses relevant structure.

The EPW bridge has:

- direct conic incidence from $X$;
- a canonical Hilbert-space construction;
- an intermediate $\mathbb P^1$-fibration;
- rich hyperkähler/Chow geometry.

The K3 bridge has:

- smaller dimension;
- direct correspondence on the smooth-associated-K3 locus;
- simpler divisor theory;
- two natural K3 choices from the two quadric fibrations.

Neither notion of “better” is one-dimensional.

---

# 10. Bridge Pareto Vector

Define a bridge feature vector:

$$
\boxed{
\mathbf P(\mathcal B)
=
(
\text{hard validity},
\text{cycle correspondence},
\text{solved-layer depth},
\text{family coverage},
\text{dimension},
\text{canonical incidence},
\text{denominator},
\text{realization redundancy}
).
}
$$

Some coordinates are ordered upward, some downward.

A bridge dominates another only if it is at least as good in every declared coordinate and strictly better in at least one.

---


# 10.5 Proof-Tier Before Pareto

A bridge with a certified algebraic correspondence and a bridge with only an abstract Hodge construction should not compete in one flat Pareto space.

Define proof tiers:

$$
\boxed{
T_0:
\text{algebraic correspondence / witness transfer certified},
}
$$

$$
\boxed{
T_1:
\text{cohomological bridge certified but cycle transfer unresolved},
}
$$

$$
\boxed{
T_2:
\text{formal/template candidate only}.
}
$$

The active frontier is formed first from the best nonempty proof tier.

Lower proof tiers are retained as:

$$
\boxed{
\text{fallback frontiers},
}
$$

not allowed to block pruning inside a stronger tier merely because they have broader formal scope.

Thus bridge selection is:

$$
\boxed{
\text{proof-tier filtering}
\to
\text{Pareto frontier within tier}.
}
$$

# 11. Pareto Dominance

Let:

$$
\mathcal B_1
$$

and:

$$
\mathcal B_2
$$

be valid bridge candidates.

Write:

$$
\boxed{
\mathcal B_1\succ_P\mathcal B_2
}
$$

if:

1. $\mathcal B_1$ has no weaker hard/proof status;
2. it is no worse on every declared cost/coverage metric;
3. it is strictly better on at least one metric.

The compiler should discard only Pareto-dominated bridges.

It should not force incomparable candidates into an arbitrary scalar ranking.

---

# 12. Verra Pareto Frontier

For a very general Verra fourfold, the two main candidates have schematic profiles.

## K3 bridge

```text
dimension: 2
cohomological_match: PASS
algebraic_correspondence: PASS
solved_layer: H11
family_coverage: open smooth-K3 locus
canonical_incidence_from_X: MEDIUM
bridge_complexity: LOW
```

## Double EPW bridge

```text
dimension: 4
cohomological_match: PASS
algebraic_correspondence: PASS
solved_layer: H11
family_coverage: general Verra locus with smooth bridge
canonical_incidence_from_X: STRONG
bridge_complexity: MEDIUM
```

The K3 bridge wins on dimension.

The EPW bridge wins on direct incidence/canonical curve geometry.

Hence:

$$
\boxed{
\{\mathcal B_{\mathrm{K3}},
\mathcal B_{\mathrm{EPW}}\}
}
$$

should remain on the Pareto frontier.

---

# 13. Candidate C — Generic Kuga--Satake

The transcendental K3-type structure has rank:

$$
\boxed{
21.
}
$$

A generic Kuga--Satake fallback would therefore have:

$$
\boxed{
\dim\mathrm{KS}
=
2^{21-2}
=
2^{19}
=
524,288.
}
$$

Compared with:

$$
\dim S=2,
\qquad
\dim Z=4,
$$

the abstract Kuga--Satake bridge is dramatically more expensive.

Because both low-dimensional bridges already have algebraic correspondences, the generic KS bridge is Pareto-dominated for this target.

Thus it can be pruned from the active frontier.

---

# 14. Exact Bridge Compression Ratio

Define a crude dimension compression:

$$
\boxed{
\operatorname{CompDim}
=
\frac{
\dim\mathrm{KS}_{\mathrm{fallback}}
}{
\dim\mathrm{bridge}
}.
}
$$

For the K3 bridge:

$$
\boxed{
\operatorname{CompDim}_{K3}
=
\frac{524288}{2}
=
262144.
}
$$

For the EPW bridge:

$$
\boxed{
\operatorname{CompDim}_{EPW}
=
\frac{524288}{4}
=
131072.
}
$$

This is only an engineering heuristic.

But it makes clear why direct geometric bridges should be searched before abstract KS constructions.

---

# 15. Third Calibration of Canonical-Incidence Search

We now have:

## Cubic fourfold

$$
\text{lines}
\to
F(X)
\quad
(K3^{[2]}\text{ bridge}).
$$

## GM fourfold

$$
\text{conics}
\to
\text{double EPW sextic}.
$$

## Verra fourfold

$$
(1,1)\text{-conics}
\to
F_{(1,1)}(X)
\to
\text{double EPW quartic}.
$$

Thus the low-curve incidence pattern appears in three distinct Fano-fourfold settings.

---

# 16. But the Verra Case Adds a Correction

The Verra case simultaneously provides a cheaper direct K3 bridge.

Therefore the stronger rule is not:

$$
\boxed{
\text{always choose the incidence hyperkähler bridge}.
}
$$

It is:

$$
\boxed{
\text{generate canonical incidence bridges early, then compare all verified low-dimensional bridges on a Pareto frontier}.
}
$$

This is a substantive correction to the Round 16--17 heuristic.

---

# 17. Bridge Redundancy vs. Bridge Diversity

Multiple paths can support the same endpoint.

But multiple distinct endpoints can also represent the same defect.

Define:

$$
\boxed{
\operatorname{BridgeDiversity}(D)
}
$$

as the number of non-equivalent bridge architectures surviving on the Pareto frontier.

For a very general Verra fourfold there are at least two:

$$
\boxed{
\operatorname{BridgeDiversity}\ge2.
}
$$

Namely:

- K3 surface bridge;
- hyperkähler double-EPW bridge.

This is different from realization multiplicity of one bridge.

---

# 18. Why Bridge Diversity Matters

If one bridge fails under degeneration or loses a smooth geometric realization, another may remain available.

Different bridges also expose different structures:

- K3 bridge: lattice/divisor theory;
- EPW bridge: conic incidence, hyperkähler geometry, Chow decomposition;
- twisted-sheaf relation: categorical/moduli structure.

Thus bridge diversity is a robustness resource.

---

# 19. Bridge Failure Domains

Each bridge should carry a failure/scope domain.

For example:

$$
\mathcal B_{\mathrm{K3}}
$$

requires the associated K3 surface to remain in the declared smooth geometric regime.

The EPW construction has its own smoothness/deformation locus.

Therefore AMRAL should store:

$$
\boxed{
\operatorname{Scope}(\mathcal B)
}
$$

and not merely:

$$
\operatorname{Exists}(\mathcal B).
$$

---

# 20. Hodge Conjecture Truth Status

Laterveer notes that Verra fourfolds are Fano and have trivial Chow group of $0$-cycles in the relevant sense, implying the Hodge conjecture for these fourfolds.

This is an external truth status.

The bridge analysis is still useful because it explains the transcendental geometry and gives explicit correspondence-induced reductions.

Again:

$$
\boxed{
\text{target theorem true}
\neq
\text{every bridge proof automatically valid}.
}
$$

---

# 21. Correspondence Strength

Round 17 had to leave the GM cycle-transfer edge explicitly open in the prototype.

The Verra case is stronger.

The conic Abel--Jacobi isomorphism is shown to be induced by a correspondence.

The K3 isomorphism is also induced by a correspondence.

Thus both main bridge candidates receive:

$$
\boxed{
\text{cycle-correspondence status}=PASS
}
$$

in the stated scope.

This makes Verra fourfolds an especially clean bridge-search benchmark.

---

# 22. Correspondence-First Tie Breaker

If two bridge candidates have identical Hodge matches, prefer a candidate with an explicit algebraic correspondence over one supported only by an abstract Hodge isometry.

This is a hard proof-status preference.

Only after proof status ties should dimension or computational cost be used.

Thus:

$$
\boxed{
\text{proof strength}
>
\text{geometric compactness}.
}
$$

---

# 23. Bridge Frontier Update Rule

When a new candidate arrives:

### Step 1

Reject hard-invalid candidates.

### Step 2

Reject candidates with weaker proof status if an otherwise identical candidate has stronger correspondence certification.

### Step 3

Compute Pareto dominance on the remaining features.

### Step 4

Keep all nondominated candidates.

### Step 5

Record scope/failure domains separately.

This prevents premature single-bridge commitment.

---

# 24. Compiler Example

Input:

```text
target:
  type: Verra_fourfold
  H4_rank: 23
  algebraic_rank_very_general: 2
  transcendental_rank: 21
  shifted_hodge: [1,19,1]
```

Candidates:

```text
K3:
  dimension: 2
  Hodge_match: PASS
  correspondence: PASS
  H11_reduction: PASS

Double_EPW:
  dimension: 4
  Hodge_match: PASS
  correspondence: PASS
  H11_reduction: PASS
  conic_incidence: PASS

Kuga_Satake:
  dimension: 524288
  correspondence: UNKNOWN
```

Output:

```text
Active_proof_tier:
  - K3
  - Double_EPW

Fallback_frontier:
  - Kuga_Satake
```

The Kuga--Satake candidate is not deleted from the global graph. It is removed from the active proof tier because its algebraic correspondence status is weaker.

---

# 25. Structural Verification

The accompanying prototype verifies:

$$
1+21+1=23,
$$

$$
23-2=21,
$$

$$
1+19+1=21.
$$

For a general degree-$2$ K3:

$$
22-1=21.
$$

For the rank-$21$ K3-type fallback:

$$
\boxed{
2^{19}=524288.
}
$$

It then computes the bridge Pareto frontier from typed feature records.

---

# 26. New Principles

## Principle A — Pareto Bridge Search

Do not collapse bridge quality to one scalar score when proof strength, scope, dimension and canonicality trade off.

---

## Principle B — Bridge Diversity

Multiple non-equivalent bridges to the same defect are a robustness asset, not redundant noise.

---

## Principle C — Proof Status Before Cost

An algebraic correspondence is categorically stronger than a bare Hodge isometry for witness transfer.

---

## Principle D — Canonical Incidence as Generator, Not Dictator

Generate incidence bridges early, but allow smaller or stronger bridges to remain on the frontier.

---

## Principle E — Scope-Typed Bridge Validity

Every bridge must record the parameter/deformation locus on which its geometry and correspondence are valid.

---

# 27. Round 18 Conclusion

Round 18 gives the third independent calibration of low-curve incidence bridges and simultaneously corrects their role.

For a very general Verra fourfold:

$$
\boxed{
H^4_{\mathrm{tr}}(X)(1)
:
(1,19,1),
\quad
\operatorname{rank}=21.
}
$$

There are at least two strong low-dimensional bridges.

The conic-incidence route gives:

$$
\boxed{
H^4_{\mathrm{tr}}(X)
\simeq
H^2_{\mathrm{tr}}(Z),
}
$$

where $Z$ is a double EPW quartic.

The associated-K3 route gives:

$$
\boxed{
H^4_{\mathrm{tr}}(X)
\simeq
H^2_{\mathrm{tr}}(S)(1),
}
$$

and this is induced by an algebraic correspondence.

Both land in degree-two Hodge theory.

Both are dramatically smaller than the generic rank-$21$ Kuga--Satake fallback:

$$
\boxed{
\dim\mathrm{KS}=524288.
}
$$

Thus the active bridge frontier should contain:

$$
\boxed{
\{\text{K3},\text{double EPW}\},
}
$$

while Kuga--Satake is pruned.

The stronger search principle is therefore:

$$
\boxed{
\text{generate canonical incidence bridges}
\to
\text{add independently verified bridges}
\to
\text{keep the Pareto frontier}.
}
$$

This is the main closure of Round 18.

---

# References

1. Robert Laterveer, **Algebraic cycles and Verra fourfolds**, Tohoku Mathematical Journal 72 (2020), 451--485.  
   DOI: https://doi.org/10.2748/tmj/1601085625  
   Relevant results: Theorem 2.7, Proposition 2.8, Remark 2.9, Proposition 2.10, Proposition 2.11.

2. Atanas Iliev, Grzegorz Kapustka, Michał Kapustka, Kristian Ranestad, **Hyperkähler fourfolds and Kummer surfaces**, arXiv:1603.00403.  
   Relevant result: Hilbert scheme of $(1,1)$-conics on a Verra fourfold admits a $\mathbb P^1$-fibration over a hyperkähler fourfold.

3. Carl Mazzanti, **On the Chow ring of double EPW quartics**, arXiv:2603.02251 (2026).  
   Relevant context: double EPW quartics, Verra conics and algebraic-cycle geometry.

4. Classical K3 fact: a very general degree-$2$ polarized K3 surface has Picard rank $1$, hence transcendental rank $21$.