# RCHM × Hodge Conjecture — Research Round 14
## Bridge-Object Search, Residual Hodge Defects, and Symmetry-Amplified Closure

**Status:** Internal research note + formalized bridge-closure model  
**Version:** Round 14 / v1.4  
**Date:** 2026-09-17  
**Methodological base:** RCHM; Typed Relation & Constraint Algebra; X-Layer Mathematical Completeness; Graph-Native Mathematics  
**Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–13  
**Canonical source:** UTF-8 Markdown

---

# 0. Round 14 Objective

Rounds 12 and 13 validated two direct carrier-reconstruction branches:

$$
\text{Factor-Ensemble},
$$

and:

$$
\text{Syzygetic-Single}.
$$

Round 14 studies the third branch:

$$
\boxed{
\text{Bridge / Higher-Structure Search}.
}
$$

The motivating 2026 case is the Hodge conjecture for abelian fourfolds of Weil type with discriminant one.

A direct carrier equation on the abelian fourfold is not the starting point.

Instead, one constructs a relation to a six-dimensional hyperkähler manifold and imports a universal algebraic characteristic class.

This gives the model:

$$
\boxed{
\text{hard Hodge class on }A
\leftarrow
\text{known algebraic class on bridge }X.
}
$$

The main methodological result of this round is a general bridge-closure criterion on the residual Hodge defect.

---

# 1. The Published Bridge Geometry

Let:

$$
A'
$$

be a very general abelian fourfold of Weil type with imaginary quadratic field:

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

and discriminant one.

Van Geemen--Rapagnetta prove that $A'$ is isogenous to an abelian fourfold $B$ such that its Kummer fourfold:

$$
K(B)
$$

is a submanifold of a projective hyperkähler sixfold:

$$
X
$$

of:

$$
K3^{[3]}
$$

type.

Thus the geometric bridge has the schematic form:

$$
\boxed{
A'
\dashrightarrow
B
\to
K(B)
\hookrightarrow
X.
}
$$

The first arrow records the isogeny passage; the exact composition used in cohomology is part of the published construction.

The crucial point is that the hard Hodge problem on the abelian fourfold is related to a natural characteristic class on $X$.

---

# 2. Universal Algebraic Seed on the Bridge

Every smooth algebraic variety has algebraic Chern classes.

For the hyperkähler sixfold $X$, consider:

$$
\boxed{
\beta_X
=
c_2(T_X)
\in
H^4(X,\mathbb Q).
}
$$

This is a Hodge class and is algebraic.

The published theorem shows that its pullback to $A'$ under the constructed map is a Hodge-Weil class:

$$
\boxed{
\alpha
=
f^*c_2(T_X)
\in
HW(A',K).
}
$$

Moreover this class is nontrivial in the part not generated merely by intersections of divisor classes.

Therefore the bridge imports a genuinely relevant algebraic class into the difficult Hodge sector.

---

# 3. Residual Hodge Defect

Let:

$$
\operatorname{Hdg}^p(A)
$$

be the rational Hodge classes.

Let:

$$
\mathcal A_{\mathrm{easy}}^p(A)
$$

be the already-understood algebraic sector, for example the subspace generated by divisor intersections and any other classes already certified algebraic.

Define the residual defect space:

$$
\boxed{
D^p(A)
=
\operatorname{Hdg}^p(A)
/
\mathcal A_{\mathrm{easy}}^p(A).
}
$$

The bridge problem should be run on $D^p(A)$ rather than on the entire cohomology group.

This prevents already-solved algebraic directions from obscuring the actual obstruction.

---

# 4. General Algebraic Bridge

Let:

$$
X
$$

be the target variety.

Let:

$$
Y
$$

be a bridge variety.

Let:

$$
\Gamma
$$

be an algebraic morphism or correspondence inducing:

$$
\Gamma_*:
H^\bullet(Y,\mathbb Q)
\to
H^\bullet(X,\mathbb Q).
$$

Let:

$$
B_{\mathrm{alg}}(Y)
$$

be a subspace of cohomology on $Y$ known to be algebraic.

Project into the residual defect:

$$
\pi_D:
\operatorname{Hdg}^p(X)
\to
D^p(X).
$$

Define the raw bridge image:

$$
\boxed{
I_\Gamma
=
\pi_D
\left(
\Gamma_*
B_{\mathrm{alg}}(Y)
\right).
}
$$

The raw bridge rank is:

$$
\boxed{
b(\Gamma)
=
\dim I_\Gamma.
}
$$

A bridge with:

$$
b(\Gamma)=0
$$

is irrelevant to the unresolved defect.

---

# 5. Direct Bridge Closure

The strongest case is:

$$
\boxed{
I_\Gamma
=
D^p(X).
}
$$

Then the bridge alone closes the residual defect.

Equivalently:

$$
b(\Gamma)
=
\dim D^p(X).
$$

This is **Direct Bridge Closure**.

But the Weil-fourfold example reveals a weaker condition that can still be sufficient.

---

# 6. Symmetry Algebra

Suppose an algebra:

$$
\mathcal E
$$

acts on:

$$
D^p(X)
$$

and the action is algebraic in the sense that algebraic classes are sent to algebraic classes.

For abelian varieties, endomorphisms are algebraic morphisms, so their induced action preserves algebraicity.

Given a bridge image:

$$
I_\Gamma,
$$

define its symmetry closure:

$$
\boxed{
\operatorname{SymCl}_{\mathcal E}(I_\Gamma)
=
\operatorname{span}_{\mathbb Q}
\left\{
e\cdot v:
e\in\mathcal E,\;
v\in I_\Gamma
\right\}.
}
$$

The bridge closes the defect if:

$$
\boxed{
\operatorname{SymCl}_{\mathcal E}(I_\Gamma)
=
D^p(X).
}
$$

Call this:

$$
\boxed{
\textbf{Symmetry-Amplified Bridge Closure}.
}
$$

---

# 7. Amplification Factor

Define:

$$
\boxed{
\operatorname{Amp}(\Gamma,\mathcal E)
=
\frac{
\dim
\operatorname{SymCl}_{\mathcal E}(I_\Gamma)
}{
\dim I_\Gamma
}
}
$$

when:

$$
I_\Gamma\neq0.
$$

This measures how much defect coverage is gained from symmetry after importing the raw bridge image.

A bridge can therefore be weak in raw rank but strong after symmetry closure.

---

# 8. Weil Fourfold Defect Sector

For an abelian fourfold $A$ of Weil type, the Hodge-Weil sector is:

$$
\boxed{
HW(A,K)
\subset
H^{2,2}(A,\mathbb Q),
}
$$

with:

$$
\dim_{\mathbb Q}HW(A,K)=2.
$$

The imaginary quadratic field:

$$
K
$$

acts on this space.

The published result states that if one nontrivial class in:

$$
HW(A,K)
$$

is algebraic, then all Hodge-Weil classes are algebraic.

This is exactly a one-seed symmetry closure.

---

# 9. Linear-Algebra Model of the $K$-Action

Write:

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

with:

$$
\theta^2=-d,
\qquad
d>0.
$$

Model the two-dimensional rational Hodge-Weil space by:

$$
D=\mathbb Q^2.
$$

Let multiplication by $\theta$ act as:

$$
\boxed{
J_d
=
\begin{pmatrix}
0&-d\\
1&0
\end{pmatrix}.
}
$$

Then:

$$
J_d^2=-dI.
$$

Let a nonzero bridge seed be:

$$
v=
\begin{pmatrix}
a\\b
\end{pmatrix}.
$$

Its $K$-orbit contains:

$$
J_dv
=
\begin{pmatrix}
-db\\a
\end{pmatrix}.
$$

The determinant is:

$$
\boxed{
\det
\begin{pmatrix}
a&-db\\
b&a
\end{pmatrix}
=
a^2+db^2.
}
$$

For:

$$
a,b\in\mathbb Q,
\qquad
d>0,
$$

this vanishes only when:

$$
a=b=0.
$$

Therefore every nonzero vector satisfies:

$$
\boxed{
\operatorname{span}_{\mathbb Q}
\{v,J_dv\}
=
D.
}
$$

Hence one nonzero algebraic seed closes the full two-dimensional sector.

---

# 10. Weil Bridge Amplification

In the published hyperkähler bridge:

$$
I_f
=
\mathbb Q\cdot
[f^*c_2(T_X)]
$$

has raw rank:

$$
\boxed{
b(f)=1.
}
$$

But:

$$
\dim HW(A,K)=2.
$$

The $K$-action amplifies the one-dimensional seed to all of $HW$:

$$
\boxed{
\operatorname{SymCl}_K(I_f)
=
HW(A,K).
}
$$

Thus:

$$
\boxed{
\operatorname{Amp}(f,K)=2.
}
$$

This is a minimal bridge in raw defect rank.

---

# 11. Bridge Seed Criterion

A general bridge seed $\beta\in B_{\mathrm{alg}}(Y)$ is useful if:

$$
\boxed{
\pi_D(\Gamma_*\beta)\neq0.
}
$$

It does **not** need to span the whole defect directly.

If a symmetry algebra $\mathcal E$ is available, the real success condition is:

$$
\boxed{
\operatorname{SymCl}_{\mathcal E}
\left(
\pi_D(\Gamma_*\beta)
\right)
=
D.
}
$$

This can reduce the required search dimension dramatically.

---

# 12. Characteristic-Class Bridge Strategy

The Weil example suggests a practical bridge-search heuristic.

Instead of searching arbitrary cycles on the target, search bridge spaces $Y$ carrying canonical algebraic classes such as:

$$
c_i(T_Y),
$$

$$
c_i(E),
$$

$$
\operatorname{ch}(E),
$$

$$
[\Delta_Y],
$$

$$
[\text{exceptional locus}],
$$

or other universal geometric cycles.

Then test whether their pushforward or pullback has nonzero residual projection.

Thus:

$$
\boxed{
\text{canonical algebraic seed}
+
\text{bridge correspondence}
+
\text{residual projection}
}
$$

can replace direct carrier guessing.

---

# 13. Why $c_2(T_X)$ Is a Good Seed

The second Chern class has several useful properties.

It is:

1. canonically attached to the bridge variety;
2. algebraic;
3. deformation-compatible as a characteristic class;
4. naturally related to the Beauville--Bogomolov--Fujiki form on a hyperkähler manifold;
5. nontrivial in degree four.

The van Geemen--Rapagnetta construction uses the relation between:

$$
c_2(T_X)
$$

and:

$$
q_X^\vee
$$

to identify the pullback with a Hodge-Weil class.

Therefore the bridge seed is not an arbitrary cycle discovered by brute force.

It is structurally forced by the bridge geometry.

---

# 14. From Bridge Class to Concrete Carrier

The same 2026 paper gives a further geometric route.

Let:

$$
A'_0
$$

be the complement of the $2$-torsion points.

The tangent bundle:

$$
T_{A'_0}
$$

is trivial and occurs as a subbundle of the pullback of:

$$
T_X.
$$

The quotient:

$$
\boxed{
\mathcal N_0
}
$$

is a rank-two bundle.

Its second Chern class is a Hodge-Weil class.

After twisting by a sufficiently ample line bundle, a global section can be chosen.

Its zero locus is an algebraic two-cycle.

Schematically:

$$
\boxed{
\mathcal N_0
\to
\mathcal N_0\otimes L^{\otimes m}
\to
s
\to
Z(s).
}
$$

Up to a complete intersection of divisors, the class of:

$$
Z(s)
$$

realizes the desired Hodge-Weil direction.

Thus the bridge gives a carrier-synthesis program, not only an existence proof.

---

# 15. Bridge-to-Carrier Chain

The full chain is:

$$
\boxed{
A'
\to
K(B)
\hookrightarrow
X
\to
c_2(T_X)
\to
f^*c_2(T_X)
\to
\mathcal N_0
\to
Z(s).
}
$$

Interpretation:

1. find a bridge geometry;
2. import a canonical characteristic class;
3. project into the hard Hodge sector;
4. use bundle geometry to synthesize a concrete cycle.

This is qualitatively different from direct equation reconstruction.

---

# 16. Bridge Search as Defect Matching

Let:

$$
D
$$

be the unresolved Hodge defect.

A bridge candidate should be evaluated by:

$$
\boxed{
\mathcal B
=
(Y,\Gamma,B_{\mathrm{alg}},\mathcal E).
}
$$

Define:

$$
I_{\mathcal B}
=
\pi_D(\Gamma_*B_{\mathrm{alg}}).
$$

Then compute:

$$
C_{\mathcal B}
=
\operatorname{SymCl}_{\mathcal E}
(I_{\mathcal B}).
$$

The bridge residual is:

$$
\boxed{
R_{\mathcal B}
=
D/C_{\mathcal B}.
}
$$

A successful bridge satisfies:

$$
\boxed{
R_{\mathcal B}=0.
}
$$

---

# 17. Bridge Defect

Define the remaining bridge defect dimension:

$$
\boxed{
\delta_{\mathrm{bridge}}(\mathcal B)
=
\dim D
-
\dim C_{\mathcal B}.
}
$$

Then:

$$
\delta_{\mathrm{bridge}}=0
$$

means full closure.

This gives a simple search objective:

$$
\boxed{
\text{maximize symmetry-closed bridge rank}.
}
$$

---

# 18. Bridge Efficiency

Let:

$$
c(\mathcal B)
$$

be an engineering or mathematical complexity cost.

Possible contributions include:

- bridge dimension;
- correspondence complexity;
- number of moduli constraints;
- degree of field extension;
- difficulty of explicit cycle synthesis;
- deformation/globalization cost.

Define:

$$
\boxed{
\eta_{\mathrm{bridge}}
=
\frac{
\dim C_{\mathcal B}
}{
c(\mathcal B)
}.
}
$$

This is a search heuristic, not a mathematical invariant.

It allows AMRAL to compare alternative bridges.

---

# 19. Multiple Bridge Geometries for the Same Defect

The discriminant-one Weil-fourfold Hodge problem now has several known proofs.

Floccari--Fu use a direct relation to singular OG6-type varieties and their resolutions.

Van Geemen--Rapagnetta use a $K3^{[3]}$-type hyperkähler sixfold containing a Kummer fourfold and pull back:

$$
c_2(T_X).
$$

Thus the same residual defect admits multiple bridge paths:

$$
\boxed{
D
\leftarrow
\text{OG6 bridge},
}
$$

and:

$$
\boxed{
D
\leftarrow
K3^{[3]}\text{ bridge}.
}
$$

This motivates a **Bridge Graph** rather than a unique bridge.

---

# 20. Bridge Graph

Define a graph whose nodes include:

- target Hodge sectors;
- bridge varieties;
- canonical algebraic seed classes;
- correspondences;
- symmetry algebras;
- synthesized carriers.

Edges are typed:

$$
\operatorname{EmbedsInto},
$$

$$
\operatorname{IsogenousTo},
$$

$$
\operatorname{PullsBack},
$$

$$
\operatorname{PushesForward},
$$

$$
\operatorname{SymmetryGenerates},
$$

$$
\operatorname{ZeroLocusRealizes}.
$$

A successful bridge proof is then a typed path from a known algebraic seed to the unresolved defect.

---

# 21. Bridge Path Certificate

A bridge path should be accepted only if it certifies:

### B0 — Target Defect

The unresolved sector $D$ is explicitly defined.

### B1 — Algebraic Seed

The bridge class $\beta$ is known algebraic.

### B2 — Algebraic Correspondence

The map/correspondence transporting $\beta$ is algebraic.

### B3 — Residual Nontriviality

$$
\pi_D(\Gamma_*\beta)\neq0.
$$

### B4 — Symmetry Legality

The symmetry algebra preserves algebraicity.

### B5 — Closure

$$
\operatorname{SymCl}_{\mathcal E}(I_\Gamma)=D.
$$

### B6 — Family Coverage

The construction applies to the required moduli locus, or a deformation/specialization argument closes the remaining cases.

### B7 — Carrier Synthesis

If an explicit cycle is requested, a zero-locus, degeneracy-locus, sheaf, correspondence, or other geometric realization is provided.

---

# 22. Bridge Branch Decision

The inverse architecture now has three experimentally/theoretically validated branches.

## Direct Factor Branch

$$
\text{signature}
\to
\text{safe envelope}
\to
\text{factorization}
\to
\text{ensemble}.
$$

## Syzygy Branch

$$
\text{signature}
\to
\text{syzygies}
\to
\text{resolution}
\to
\text{single carrier}.
$$

## Bridge Branch

$$
\text{residual defect}
\to
\text{bridge geometry}
\to
\text{canonical algebraic seed}
\to
\text{correspondence}
\to
\text{symmetry closure}
\to
\text{carrier synthesis}.
$$

---

# 23. When to Escalate to a Bridge

A bridge search is justified when:

1. low-degree signature does not isolate a realizable carrier;
2. factorization and syzygy branches fail or remain highly ambiguous;
3. the residual Hodge defect has recognizable symmetry;
4. another geometric category carries a naturally matching Hodge structure;
5. canonical algebraic characteristic classes exist on the candidate bridge.

This is not a proof that a useful bridge always exists.

It is a structured escalation strategy.

---

# 24. Structure Matching Before Geometry Search

Let:

$$
D
$$

carry:

- a Hodge type;
- a polarization;
- an endomorphism algebra;
- monodromy representation;
- field of definition;
- lattice form.

Search first for another cohomological object:

$$
V_Y
$$

with matching structure.

Only after a cohomological match is found should one search for a geometric realization:

$$
Y.
$$

Thus:

$$
\boxed{
\text{defect structure}
\to
\text{cohomological bridge candidate}
\to
\text{geometric bridge}.
}
$$

This can reduce the bridge search space.

---

# 25. Hodge-Structure Bridge Signature

Define:

$$
\boxed{
\operatorname{BridgeSig}(D)
=
(
\text{weight},
\text{Hodge numbers},
\text{polarization},
\text{endomorphism algebra},
\text{monodromy},
\text{field}
).
}
$$

A bridge candidate:

$$
Y
$$

should first pass:

$$
\operatorname{BridgeSig}(V_Y)
\sim
\operatorname{BridgeSig}(D).
$$

This is analogous to type checking before carrier reconstruction.

---

# 26. Symmetry as a Search Multiplier

Suppose:

$$
\dim D=N.
$$

Without symmetry, one might search for:

$$
N
$$

independent algebraic seeds.

If:

$$
D
$$

is cyclic as an $\mathcal E$-module, one seed suffices:

$$
\boxed{
D=\mathcal E\cdot v.
}
$$

Thus the number of bridge seeds required is controlled not by:

$$
\dim_{\mathbb Q}D,
$$

but by the minimal number of generators of $D$ as an $\mathcal E$-module.

Define:

$$
\boxed{
\mu_{\mathcal E}(D)
=
\text{minimal number of }\mathcal E\text{-module generators}.
}
$$

For the Weil Hodge sector:

$$
\boxed{
\mu_K(HW)=1.
}
$$

This is more informative than the raw dimension:

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

---

# 27. Bridge Search Complexity Reduction

The direct search burden scales roughly with:

$$
\dim_{\mathbb Q}D.
$$

The symmetry-aware bridge burden scales with:

$$
\boxed{
\mu_{\mathcal E}(D).
}
$$

Thus the expected reduction factor is:

$$
\boxed{
\frac{
\dim_{\mathbb Q}D
}{
\mu_{\mathcal E}(D)
}.
}
$$

For the Weil example:

$$
\frac21=2.
$$

For higher-dimensional defects with larger symmetry algebras, the gain could be much larger.

---

# 28. Canonical Seed Library

AMRAL should maintain a library of universal algebraic seed classes:

```text
c1(tangent_bundle)
c2(tangent_bundle)
higher_chern_classes
chern_character(universal_sheaf)
diagonal_class
exceptional_divisor
incidence_correspondence
degeneracy_locus_class
tautological_cycles
```

Each seed should be indexed by:

- cohomological degree;
- deformation behavior;
- functoriality;
- known algebraicity;
- natural bridge categories.

This turns bridge search into a typed retrieval problem before it becomes a free-form invention problem.

---

# 29. AMRAL Bridge Schema

Suggested bridge object:

```text
hodge_bridge:
  target_defect_ref
  bridge_variety_ref
  correspondence_ref
  algebraic_seed_refs[]
  raw_bridge_rank
  symmetry_algebra_ref
  symmetry_closed_rank
  bridge_defect_dimension
  module_generator_count
  family_coverage
  carrier_synthesis_ref
  verification_status
```

Suggested bridge edge:

```text
bridge_edge:
  source
  target
  relation_type
  cohomological_degree
  algebraicity_preserved
  scope
  evidence_ref
```

---

# 30. Exact Linear-Algebra Verification

The accompanying Python script verifies the symmetry-amplification mechanism.

For:

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

with:

$$
d>0,
$$

use:

$$
J_d=
\begin{pmatrix}
0&-d\\
1&0
\end{pmatrix}.
$$

The script checks:

$$
J_d^2=-dI,
$$

and:

$$
\det[v,J_dv]
=
a^2+db^2.
$$

It then checks concrete positive values of $d$ and nonzero rational seeds.

Every nonzero seed spans the entire two-dimensional rational defect under the $K$-action.

Thus:

$$
\boxed{
\text{raw bridge rank }1
\to
\text{symmetry-closed rank }2.
}
$$

---

# 31. Evidence Typing

This round distinguishes:

## Published geometry

- existence of the $K3^{[3]}$ hyperkähler bridge;
- pullback of $c_2(T_X)$ to a Hodge-Weil class;
- one nontrivial algebraic Hodge-Weil class implies algebraicity of the full Hodge-Weil sector;
- rank-two quotient bundle / zero-locus carrier route.

## RCHM abstraction

- residual Hodge defect;
- raw bridge rank;
- symmetry closure;
- amplification factor;
- bridge defect;
- module-generator count;
- bridge graph.

## Exact local verification

- the $2$-dimensional $K$-module linear-algebra amplification model.

No new proof of the published hyperkähler geometry is claimed.

---

# 32. Round 14 Conclusion

Round 14 validates the third reconstruction architecture:

$$
\boxed{
\textbf{Bridge / Higher-Structure Search}.
}
$$

The key example is the 2026 hyperkähler proof for discriminant-one Weil abelian fourfolds.

Instead of constructing the hard algebraic cycle directly on the abelian fourfold, one passes through a hyperkähler sixfold:

$$
X,
$$

imports the canonical algebraic class:

$$
c_2(T_X),
$$

and pulls it into the Hodge-Weil sector.

The raw bridge image is only one-dimensional:

$$
b(f)=1.
$$

But the imaginary quadratic endomorphism field acts on the two-dimensional defect.

A single nonzero seed generates the entire defect:

$$
\boxed{
\operatorname{SymCl}_K(I_f)
=
HW(A,K).
}
$$

Therefore:

$$
\boxed{
\text{one bridge seed}
+
\text{symmetry}
\Rightarrow
\text{two-dimensional Hodge-defect closure}.
}
$$

The correct bridge search target is consequently not:

$$
\text{find }\dim D\text{ independent cycles}.
$$

It is:

$$
\boxed{
\text{find }\mu_{\mathcal E}(D)\text{ algebraic bridge seeds}.
}
$$

This changes the search problem from dimension counting to module generation.

Together, Rounds 12--14 now give three distinct Hodge-witness architectures:

$$
\boxed{
\begin{array}{c}
\text{Factor-Ensemble}\\
\text{Syzygetic-Single}\\
\text{Symmetry-Amplified Bridge}
\end{array}
}
$$

This is the main closure of Round 14.

---

# References

1. Bert van Geemen, Antonio Rapagnetta, **Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class**, arXiv:2607.18341 (submitted 20 July 2026).

2. Salvatore Floccari, Lie Fu, **The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties**, Journal de Mathématiques Pures et Appliquées 210 (2026), Article 103876; arXiv:2504.13607.

3. Salvatore Floccari, Lie Fu, **Algebraic cycles on hyper-Kähler varieties of generalized Kummer type**, Mathematische Annalen (published 2025).

4. Classical fact: Chern classes of algebraic vector bundles are algebraic cycle classes, and algebraic morphisms/correspondences preserve algebraicity under pullback/pushforward in the appropriate setting.