# RCHM × Hodge Conjecture — Research Round 21
## Küchle $c5$ Ancestry Normalization: Primitive Integral Inheritance and the Re-Emergence of a $K3^{[2]}$ Bridge Profile

**Status:** Internal research note + exact structural verifier  
**Version:** Round 21 / v2.1  
**Date:** 2026-09-17  
**Methodological base:** RCHM; X-Layer Mathematical Completeness; Bridge Search Compiler; Ancestry Reduction  
**Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–20  
**Canonical source:** UTF-8 Markdown

---

# 0. Round 21 Objective

Round 20 isolated an operational rank-$24$ block in a Küchle $c5$ fourfold by removing only the two most visible codimension-two Schubert classes inherited from the ambient Grassmannian.

That was useful for producing an OG10 false-positive regression test, but it was not ancestry-complete.

Round 21 incorporates the full geometry of the Küchle $c5$ fivefold:

$$
X^5_{\lambda,\mu},
$$

of which the fourfold:

$$
X^4_{\lambda,\mu,\nu}
$$

is a smooth hyperplane section.

Kuznetsov proves that the **integral Chow motive** of the fivefold is:

$$
\boxed{
M(X^5)
=
\mathbf 1
\oplus
\mathbf L
\oplus
(\mathbf L^2)^{\oplus4}
\oplus
(\mathbf L^3)^{\oplus4}
\oplus
\mathbf L^4
\oplus
\mathbf L^5.
}
$$

Therefore:

$$
\boxed{
H^4(X^5,\mathbb Z)
\simeq
\mathbb Z^4
}
$$

and all four directions are Tate/Hodge type $(2,2)$.

Since the fourfold is a hyperplane section, integral weak Lefschetz makes their image in $H^4(X^4,\mathbb Z)$ a primitive rank-$4$ sublattice.

The ancestry-normalized new middle block therefore has rank $22$, not $24$.

---

# 1. Fivefold Motive

Kuznetsov's Theorem 5.3 gives:

$$
M(X^5)
=
\mathbf 1
\oplus
\mathbf L
\oplus
(\mathbf L^2)^{\oplus4}
\oplus
(\mathbf L^3)^{\oplus4}
\oplus
\mathbf L^4
\oplus
\mathbf L^5.
$$

Hence the Betti ranks are:

$$
b_0=b_2=b_8=b_{10}=1,
$$

and:

$$
\boxed{
b_4=b_6=4.
}
$$

Corollary 5.4 states equivalently:

$$
\boxed{
h^{2,2}(X^5)=h^{3,3}(X^5)=4,
}
$$

with diagonal Hodge diamond.

The Chow groups are free abelian, in particular:

$$
\boxed{
CH^2(X^5)\simeq\mathbb Z^4.
}
$$

---

# 2. Hyperplane Section

The Küchle fourfold is:

$$
\boxed{
X^4
=
X^5\cap H_\nu.
}
$$

It is a smooth hyperplane section for general $\nu$.

Apply integral weak Lefschetz to:

$$
i:X^4\hookrightarrow X^5.
$$

At middle degree:

$$
\boxed{
i^*:
H^4(X^5,\mathbb Z)
\hookrightarrow
H^4(X^4,\mathbb Z)
}
$$

is injective with torsion-free cokernel.

Therefore its image:

$$
\boxed{
A_{\mathrm{inh}}
=
i^*H^4(X^5,\mathbb Z)
}
$$

is a **primitive integral sublattice** of rank:

$$
\boxed{
4.
}
$$

---

# 3. Torsion-Freeness of the Fourfold Middle Cohomology

We have a short exact sequence:

$$
0
\to
\mathbb Z^4
\to
H^4(X^4,\mathbb Z)
\to
C
\to0,
$$

where:

$$
C
$$

is torsion-free.

Thus:

$$
\boxed{
H^4(X^4,\mathbb Z)
\text{ is torsion-free}.
}
$$

By Poincaré duality, the middle intersection pairing on:

$$
H^4(X^4,\mathbb Z)
$$

is therefore a unimodular integral lattice.

This gives the correct integral ambient setting for future discriminant-form calculations.

---

# 4. Full Fourfold Hodge Numbers

The Küchle $c5$ fourfold has:

$$
\boxed{
h^{3,1}=1,
\qquad
h^{2,2}=24,
\qquad
h^{1,3}=1.
}
$$

Hence:

$$
\boxed{
b_4=26.
}
$$

The inherited rank-$4$ lattice is entirely of type:

$$
(2,2).
$$

---

# 5. Ancestry-Normalized New Block

Define the rational orthogonal new block:

$$
\boxed{
T_{c5}
=
A_{\mathrm{inh},\mathbb Q}^{\perp}
\subset
H^4(X^4,\mathbb Q).
}
$$

Because:

$$
\operatorname{rank}A_{\mathrm{inh}}=4,
$$

we obtain:

$$
\boxed{
\operatorname{rank}T_{c5}
=
26-4
=
22.
}
$$

Removing four $(2,2)$ directions gives Hodge numbers:

$$
\boxed{
T_{c5,\mathbb C}:
(1,20,1).
}
$$

This is the ancestry-normalized Hodge defect.

---

# 6. Why Round 20 Saw Rank 24

Round 20 removed only:

$$
\boxed{
\operatorname{span}
\{\sigma_2,\sigma_{1,1}\},
}
$$

the two codimension-two classes immediately visible from:

$$
\operatorname{Gr}(3,7).
$$

Their exact Gram matrix is:

$$
\begin{pmatrix}
20&16\\
16&14
\end{pmatrix},
$$

so they form a genuine rank-$2$ sublattice.

However:

$$
\boxed{
\operatorname{rank}A_{\mathrm{inh}}=4.
}
$$

Thus two additional inherited Tate directions are present in the fivefold geometry but are not accounted for by the most naive Grassmannian ambient plane.

Therefore the Round-20 rank-$24$ object should be interpreted as:

$$
\boxed{
\text{partial operational defect},
}
$$

not the fully ancestry-normalized new middle block.

---

# 7. False Positive by Incomplete Ancestry

This creates a second kind of bridge false positive.

### Partial view

$$
26-2=24,
$$

so:

$$
(1,22,1),
$$

which superficially matches OG10:

$$
b_2=24.
$$

### Complete inherited view

$$
26-4=22,
$$

so:

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

Therefore OG10 now fails already at the rank gate:

$$
\boxed{
22\neq24.
}
$$

The previous quadratic-signature rejection remains valid for the partial template, but the stronger diagnosis is:

$$
\boxed{
\text{OG10 alert was ancestry-incomplete}.
}
$$

---

# 8. Signature of the Full Middle Form

For a compact Kähler fourfold with Hodge numbers:

$$
(1,24,1)
$$

in middle degree, the middle cup-product form has signature:

$$
\boxed{
(24,2).
}
$$

The four inherited classes lie in:

$$
H^{2,2}(X^4,\mathbb R).
$$

The Hodge--Riemann bilinear relations make the real $(2,2)$ middle sector positive definite for this signature decomposition.

Therefore:

$$
\boxed{
\operatorname{sig}(A_{\mathrm{inh},\mathbb R})
=
(4,0).
}
$$

Hence:

$$
\boxed{
\operatorname{sig}(T_{c5})
=
(20,2).
}
$$

After the conventional global sign reversal used to compare a degree-four K3-type structure with a weight-two Beauville--Bogomolov lattice:

$$
\boxed{
\operatorname{sig}(-T_{c5})
=
(2,20).
}
$$

---

# 9. $K3^{[2]}$ Primitive Profile

For a hyperkähler fourfold of:

$$
K3^{[2]}
$$

type:

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

The Beauville--Bogomolov lattice has signature:

$$
\boxed{
(3,20).
}
$$

For a polarized projective member, removing one positive polarization direction gives the primitive lattice:

$$
H^2_{\mathrm{prim}}
$$

with:

$$
\boxed{
\operatorname{rank}=22,
}
$$

$$
\boxed{
\operatorname{signature}=(2,20),
}
$$

and Hodge numbers:

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

These are exactly the three ancestry-normalized coarse invariants of:

$$
-T_{c5}.
$$

---

# 10. Structural Gate Table

The direct primitive-$K3^{[2]}$ template therefore has:

```text
candidate:
  polarized_K3[2]_primitive_H2

rank_gate:
  PASS
  c5: 22
  K3[2]_prim: 22

Hodge_gate:
  PASS
  c5: [1,20,1]
  K3[2]_prim: [1,20,1]

real_signature_gate:
  PASS
  -c5_cup: [2,20]
  K3[2]_prim_BB: [2,20]

integral_discriminant_gate:
  UNKNOWN

geometric_realization_gate:
  UNKNOWN

algebraic_correspondence_gate:
  UNKNOWN
```

This is a viable bridge **profile**, not yet a bridge theorem.

---

# 11. Comparison with Cubic and GM Fourfolds

The same coarse normalized profile appears in previous families.

### Cubic fourfold

Primitive middle block after removing $h^2$:

$$
\operatorname{rank}=22,
\qquad
h=(1,20,1),
\qquad
\text{sign-reversed signature}=(2,20).
$$

Its canonical bridge is:

$$
F(X),
$$

a $K3^{[2]}$-type hyperkähler fourfold.

### GM fourfold

Vanishing middle block after removing two Tate directions:

$$
\operatorname{rank}=22,
\qquad
h=(1,20,1).
$$

Its bridge is a double EPW sextic, again of:

$$
K3^{[2]}
$$

type.

### Küchle $c5$

After removing the **full inherited rank-$4$ Tate sector**:

$$
\boxed{
\operatorname{rank}=22,
\qquad
h=(1,20,1),
\qquad
\text{signature}=(2,20).
}
$$

Thus the ancestry-normalized compiler places $c5$ in the same coarse bridge-search class.

---

# 12. New Search Hypothesis

The corrected structural hypothesis is:

$$
\boxed{
\text{search for a polarized }K3^{[2]}\text{-type hyperkähler fourfold}
}
$$

or a moduli space deformation equivalent to one.

This is a much sharper target than the previous generic statement:

$$
\text{find some hyperkähler variety}.
$$

It also aligns with historical expectations that suitable moduli spaces of sheaves on the Küchle $c5$ fourfold should be hyperkähler.

That historical expectation is not used as proof.

It is now supported by an independent ancestry-normalized Hodge signature.

---

# 13. Integral Discriminant Gate Remains Open

Matching:

- rank;
- Hodge numbers;
- real signature;

is not enough.

The next exact gate is the integral lattice.

Let:

$$
L=H^4(X^4,\mathbb Z),
$$

a unimodular lattice.

Let:

$$
A=A_{\mathrm{inh}}
$$

be the primitive rank-$4$ inherited lattice.

Let:

$$
T=A^\perp.
$$

Because $A$ is primitive in the unimodular lattice $L$, discriminant-form theory gives:

$$
\boxed{
(A_A,q_A)
\simeq
(A_T,-q_T),
}
$$

where:

$$
A_A=A^\vee/A,
\qquad
A_T=T^\vee/T.
$$

In particular:

$$
\boxed{
|\operatorname{disc}A|
=
|\operatorname{disc}T|.
}
$$

Therefore the remaining integral problem has been reduced to computing the Gram/discriminant form of the four inherited classes.

---

# 14. Why the Rank-$2$ Matrix Is Not Enough

Round 20 computed a rank-$2$ sublattice:

$$
B
\subset
A
$$

with:

$$
\boxed{
\operatorname{disc}B=24.
}
$$

But:

$$
\operatorname{rank}A=4.
$$

Therefore:

$$
\boxed{
\operatorname{disc}B
\neq
\operatorname{disc}A
}
$$

in general.

The determinant $24$ must not be propagated to the new block.

The two missing inherited generators and their pairings are required.

This corrects another possible type-erasure mistake.

---

# 15. Integral-Lattice Research Target

The immediate exact task is now:

1. choose an integral basis:

$$
a_1,\ldots,a_4
$$

of:

$$
H^4(X^5,\mathbb Z);
$$

2. compute:

$$
\boxed{
G_{ij}
=
\int_{X^4}
i^*a_i\cup i^*a_j
=
\int_{X^5}
a_i a_j h;
}
$$

3. determine:

$$
\det G;
$$

4. compute the finite discriminant group and quadratic form;

5. compare with primitive polarization complements in known $K3^{[2]}$ lattices.

This is now a finite arithmetic lattice problem.

---

# 16. Polarized $K3^{[2]}$ Discriminant Matching

The full $K3^{[2]}$ lattice is:

$$
\boxed{
\Lambda_{K3^{[2]}}
=
U^3
\oplus
E_8(-1)^2
\oplus
\langle-2\rangle.
}
$$

A primitive polarization:

$$
v\in\Lambda_{K3^{[2]}}
$$

has:

- square $v^2=2d$;
- divisibility $\operatorname{div}(v)$.

Its orthogonal complement:

$$
v^\perp
$$

is a rank-$22$ lattice of signature:

$$
(2,20).
$$

Once:

$$
T_{c5,\mathbb Z}
$$

is known, the bridge compiler can solve:

$$
\boxed{
T_{c5,\mathbb Z}
\stackrel{?}{\simeq}
v^\perp
}
$$

for possible polarization type $(d,\operatorname{div})$.

This will either identify a discrete hyperkähler moduli target or reject the direct $K3^{[2]}$ lattice template.

---

# 17. Updated Degeneration Interpretation

The 2026 conic/quadric degeneration route remains high priority.

But it should now be asked to transport a rank-$22$ block:

$$
\boxed{
T_{c5},
}
$$

not the partial rank-$24$ operational block.

Thus the key degeneration question becomes:

> Does the special-fiber conic/even-nodal geometry account for a rank-$22$ K3-type local system?

This is much closer to the cubic/GM pattern.

---

# 18. Updated Twisted-Cubic / Moduli Interpretation

The historical moduli-space candidate also becomes more constrained.

If a four-dimensional hyperkähler moduli space:

$$
M
$$

is constructed from sheaves or rational curves on $c5$, then one should test:

$$
\boxed{
H^2_{\mathrm{prim}}(M)
\stackrel{?}{\simeq}
T_{c5}(1).
}
$$

If $M$ is of $K3^{[2]}$ type, the ranks and real signatures now match automatically.

Only:

- integral lattice;
- period;
- correspondence;

remain.

---

# 19. Kuga--Satake Cost Correction

Round 20 used the partial rank-$24$ defect and therefore obtained:

$$
\dim KS=2^{22}=4,194,304.
$$

After ancestry normalization:

$$
\operatorname{rank}T_{c5}=22.
$$

Therefore the correct generic fallback dimension is:

$$
\boxed{
\dim KS
=
2^{22-2}
=
2^{20}
=
1,048,576.
}
$$

Still enormous, but four times smaller than the ancestry-incomplete estimate.

---

# 20. Ancestry Completeness Gate

Round 21 motivates a new mandatory gate before bridge matching:

$$
\boxed{
\textbf{Ancestry Completeness Gate}.
}
$$

Before comparing a target defect with an external bridge lattice, ask:

> Have all inherited algebraic/Tate directions from parent geometry been removed?

If not:

$$
\boxed{
\text{bridge rank matching is provisional}.
}
$$

This gate should run before:

- hyperkähler deformation-type matching;
- Kuga--Satake dimension calculation;
- discriminant comparison;
- moduli-dimension heuristics.

---

# 21. Two False-Positive Classes

The c5 experiment now exhibits two distinct failure modes.

## FP-A — Incomplete-Ancestry Rank Alias

Partial subtraction gives:

$$
(1,22,1),
\quad
r=24,
$$

creating an OG10 alert.

Full ancestry subtraction gives:

$$
(1,20,1),
\quad
r=22.
$$

The candidate dies at the rank gate.

---

## FP-B — Quadratic Alias

Even before the ancestry correction, the partial rank-$24$ block had real signature incompatible with OG10.

Thus the candidate also died at the quadratic gate.

The same wrong candidate can therefore have multiple independent rejection certificates.

This is valuable graph information.

---

# 22. Bridge Search Frontier v2

After ancestry normalization:

## Active Candidate A — $K3^{[2]}$ Primitive Bridge

```text
rank: PASS
Hodge: PASS
real_signature: PASS
integral_lattice: UNKNOWN
geometry: UNKNOWN
correspondence: UNKNOWN
priority: HIGH
```

## Active Candidate B — Degeneration / Even-Nodal

```text
target_rank: 22
special_fiber_geometry: ACTIVE
monodromy: UNKNOWN
invariant_cycles: UNKNOWN
hyperkahler_endpoint: UNKNOWN
priority: HIGH
```

## Active Candidate C — Sheaf / Curve Moduli

```text
expected_dimension: 4 candidate desired
K3[2]_type: plausible target
actual_construction: UNKNOWN
Abel-Jacobi/Hodge map: UNKNOWN
priority: HIGH
```

## Scaffold — Lagrangian Grassmannian

```text
birational_geometry: PASS
new_block_transfer: UNKNOWN
priority: MEDIUM
```

## Fallback — Kuga--Satake

```text
dimension: 1048576
correspondence: UNKNOWN
priority: LOW
```

---

# 23. Round 21 Conclusion

Round 21 corrects the first lattice normalization of Küchle $c5$.

The key published input is the integral fivefold motive:

$$
\boxed{
M(X^5)
=
1
\oplus L
\oplus(L^2)^4
\oplus(L^3)^4
\oplus L^4
\oplus L^5.
}
$$

Thus:

$$
\boxed{
H^4(X^5,\mathbb Z)\simeq\mathbb Z^4
}
$$

and consists entirely of Tate classes.

Because the fourfold is a smooth hyperplane section, integral weak Lefschetz gives a primitive embedding:

$$
\boxed{
\mathbb Z^4
\hookrightarrow
H^4(X^4,\mathbb Z).
}
$$

Therefore the ancestry-normalized new block has:

$$
\boxed{
\operatorname{rank}=22,
}
$$

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

and sign-reversed real signature:

$$
\boxed{
(2,20).
}
$$

This eliminates the OG10 direct template at the rank gate.

At the same time it activates a much stronger candidate profile:

$$
\boxed{
T_{c5}(1)
\sim
H^2_{\mathrm{prim}}
(\text{polarized }K3^{[2]}\text{-type fourfold}).
}
$$

The rank, Hodge numbers and real signature all match.

The remaining decisive test is now integral:

$$
\boxed{
\text{compute the full rank-$4$ inherited Gram lattice}
}
$$

and compare its orthogonal discriminant form with primitive polarization complements in:

$$
\Lambda_{K3^{[2]}}.
$$

The corrected global lesson is:

$$
\boxed{
\text{Ancestry completeness must precede bridge-type recognition}.
}
$$

This is the main closure of Round 21.

---

# References

1. Alexander Kuznetsov, **Küchle fivefolds of type c5**, Mathematische Zeitschrift 284 (2016), 1245--1278.  
   Theorem 5.3: integral Chow motive  
   $$M(X^5)=1\oplus L\oplus(L^2)^4\oplus(L^3)^4\oplus L^4\oplus L^5.$$  
   Corollary 5.4: diagonal Hodge diamond with $h^{2,2}=4$.  
   Section 5.1: the Küchle fourfold is a hyperplane section of the fivefold.

2. Integral weak Lefschetz theorem for a smooth hyperplane section: at middle degree the restriction map is injective with torsion-free cokernel.

3. Standard Hodge--Riemann bilinear relations for the signature of middle cohomology of a compact Kähler fourfold.

4. For $K3^{[2]}$ type:  
   $$H^2\simeq U^3\oplus E_8(-1)^2\oplus\langle-2\rangle,$$  
   rank $23$ and Beauville--Bogomolov signature $(3,20)$.

5. Round 20 exact localization result: the visible Grassmannian rank-$2$ sublattice has Gram matrix  
   $$\begin{pmatrix}20&16\\16&14\end{pmatrix}.$$  
   This remains valid but is only a sublattice of the full inherited rank-$4$ sector.