# RCHM × Hodge Conjecture — Research Round 19
## Ancestry Reduction for K3-Type Fano Families: Blow-Up Source Classification, Bridge Reuse, and Novelty Frontier Detection

**Status:** Internal research note + executable ancestry scanner  
**Version:** Round 19 / v1.9  
**Date:** 2026-09-17  
**Methodological base:** RCHM; Typed Relation & Constraint Algebra; X-Layer Mathematical Completeness; Graph-Native Mathematics  
**Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–18  
**Canonical source:** UTF-8 Markdown

---

# 0. Round 19 Objective

Rounds 16--18 developed bridge discovery for K3-type Fano fourfolds.

Round 19 adds a more primitive compiler stage:

$$
\boxed{
\textbf{Ancestry Reduction}.
}
$$

Before searching for a new bridge, ask:

> Is the K3-type cohomology genuinely new, or is it inherited from a parent variety or injected by a blow-up center?

This question is forced by the modern classification program.

A 2026 study produces $64$ families of Fano fourfolds of K3 type and finds that the K3 structure of most of them can be related to cubic fourfolds, Gushel--Mukai fourfolds, or actual K3 surfaces.

Thus a bridge compiler that treats all $64$ families as unrelated targets would massively duplicate work.

---

# 1. Blow-Up Cohomology

Let:

$$
\pi:
\widetilde Y
=
\operatorname{Bl}_Z Y
\to
Y
$$

be the blow-up of a smooth variety $Y$ along a smooth center $Z$ of codimension:

$$
c.
$$

The rational Hodge structure satisfies:

$$
\boxed{
H^k(\widetilde Y)
\simeq
H^k(Y)
\oplus
\bigoplus_{i=1}^{c-1}
H^{k-2i}(Z)(-i).
}
$$

For a codimension-two center:

$$
c=2,
$$

this simplifies to:

$$
\boxed{
H^k(\widetilde Y)
\simeq
H^k(Y)
\oplus
H^{k-2}(Z)(-1).
}
$$

In degree four:

$$
\boxed{
H^4(\widetilde Y)
\simeq
H^4(Y)
\oplus
H^2(Z)(-1).
}
$$

This formula is the basic ancestry compiler rule.

---

# 2. Hodge Shift from the Center

If:

$$
H^2(Z)
=
H^{2,0}(Z)
\oplus
H^{1,1}(Z)
\oplus
H^{0,2}(Z),
$$

then after the Tate shift:

$$
(-1),
$$

the contribution to $H^4(\widetilde Y)$ is:

$$
\boxed{
H^{3,1}
\oplus
H^{2,2}
\oplus
H^{1,3}.
}
$$

Therefore the center contributes a K3-type off-diagonal block exactly when:

$$
\boxed{
h^{2,0}(Z)\neq0.
}
$$

A Tate center such as:

$$
\mathbb P^2
$$

adds only:

$$
(2,2)
$$

cohomology.

A K3 center adds:

$$
(3,1),
\quad
(2,2),
\quad
(1,3).
$$

---

# 3. K3-Source Classifier

Define two binary source indicators for a fourfold blow-up:

$$
\kappa_{\mathrm{parent}}
=
h^{3,1}(Y),
$$

and:

$$
\kappa_{\mathrm{center}}
=
h^{2,0}(Z).
$$

Then classify:

### Parent-Inherited

$$
\boxed{
\kappa_{\mathrm{parent}}>0,
\qquad
\kappa_{\mathrm{center}}=0.
}
$$

The K3 block comes from the parent.

---

### Center-Injected

$$
\boxed{
\kappa_{\mathrm{parent}}=0,
\qquad
\kappa_{\mathrm{center}}>0.
}
$$

The K3 block comes from the center.

---

### Mixed

$$
\boxed{
\kappa_{\mathrm{parent}}>0,
\qquad
\kappa_{\mathrm{center}}>0.
}
$$

Both sources contribute.

---

### No K3 Source

$$
\kappa_{\mathrm{parent}}
=
\kappa_{\mathrm{center}}
=
0.
$$

A K3-type target cannot arise from this codimension-two blow-up formula alone.

---

# 4. Küchle $c7$: Parent-Inherited Case

A Küchle fourfold of type $c7$ is isomorphic to the blow-up:

$$
\boxed{
X_{c7}
\simeq
\operatorname{Bl}_{V}
Y,
}
$$

where:

$$
Y\subset\mathbb P^5
$$

is a cubic fourfold containing a Veronese surface:

$$
V\simeq\mathbb P^2.
$$

The corresponding cubic lies in Hassett's divisor:

$$
\mathcal C_{20}.
$$

Since:

$$
h^{2,0}(\mathbb P^2)=0,
$$

the center contributes no $(3,1)$ or $(1,3)$ classes.

Therefore:

$$
\boxed{
H^{3,1}(X_{c7})
\simeq
H^{3,1}(Y),
}
$$

and similarly for:

$$
H^{1,3}.
$$

The K3-type transcendental block is inherited from the cubic parent.

---

# 5. Exact Degree-Four Decomposition for $c7$

For the Veronese center:

$$
H^2(\mathbb P^2,\mathbb Q)
=
\mathbb Q(-1).
$$

After the blow-up Tate shift:

$$
H^2(\mathbb P^2)(-1)
=
\mathbb Q(-2),
$$

which is pure of type:

$$
(2,2).
$$

Hence:

$$
\boxed{
H^4(X_{c7})
\simeq
H^4(Y)
\oplus
\mathbb Q(-2).
}
$$

The blow-up modifies the algebraic middle cohomology but not the K3-type off-diagonal block.

This is a precise ancestry statement.

---

# 6. Bridge Reuse for $c7$

Because the hard primitive Hodge structure comes from the cubic parent:

$$
Y,
$$

the bridge compiler should not search for a new hyperkähler object attached specifically to:

$$
X_{c7}.
$$

Instead:

$$
\boxed{
X_{c7}
\to
Y
\to
F(Y),
}
$$

where:

$$
F(Y)
$$

is the Fano variety of lines of the cubic fourfold.

Thus the Round 16 cubic bridge is reusable.

Call this:

$$
\boxed{
\textbf{Inherited Bridge Reuse}.
}
$$

---

# 7. $c7$ Witness-Transfer Graph

The graph is:

$$
\boxed{
H^4_{\mathrm{hard}}(X_{c7})
\simeq
H^4_{\mathrm{hard}}(Y)
\to
H^2(F(Y)).
}
$$

The extra blow-up class:

$$
\mathbb Q(-2)
$$

is already algebraic.

Therefore the witness problem decomposes into:

$$
\boxed{
\text{parent hard block}
+
\text{easy exceptional Tate block}.
}
$$

No new bridge invention is necessary.

---

# 8. Küchle $d3$: Center-Injected Case

Küchle fourfolds of type $d3$ are isomorphic to blow-ups:

$$
\boxed{
X_{d3}
\simeq
\operatorname{Bl}_S
(\mathbb P^1)^4,
}
$$

where:

$$
S
$$

is a K3 surface.

The parent:

$$
(\mathbb P^1)^4
$$

has purely Tate cohomology.

Thus:

$$
\boxed{
h^{3,1}((\mathbb P^1)^4)=0.
}
$$

But:

$$
h^{2,0}(S)=1.
$$

Therefore the entire K3 off-diagonal block of the blow-up comes from the center.

---

# 9. Exact Hodge Contribution for $d3$

A K3 surface has:

$$
h^{2,0}=1,
\qquad
h^{1,1}=20,
\qquad
h^{0,2}=1.
$$

After the blow-up shift:

$$
(-1),
$$

the center contributes:

$$
\boxed{
h^{3,1}=1,
\qquad
h^{2,2}=20,
\qquad
h^{1,3}=1.
}
$$

Thus:

$$
\boxed{
\text{K3 block of }X_{d3}
=
H^2(S)(-1).
}
$$

This is the mirror image of $c7$.

---

# 10. Direct Witness Transfer from the Center

Let:

$$
E
=
\mathbb P(N_{S/Y})
$$

be the exceptional divisor, with:

$$
p:E\to S,
$$

and inclusion:

$$
j:E\hookrightarrow X_{d3}.
$$

The blow-up summand:

$$
H^2(S)(-1)
$$

is realized by the algebraic correspondence:

$$
\boxed{
j_*p^*.
}
$$

A rational $(1,1)$ class on $S$ is algebraic by Lefschetz $(1,1)$.

Therefore its image in:

$$
H^{2,2}(X_{d3})
$$

is algebraic.

Thus the K3 center is already a solved witness layer.

No hyperkähler bridge search is needed.

---

# 11. Two Ancestry Motifs

The pair $c7/d3$ gives two canonical ancestry motifs.

## Motif P — Parent Inheritance

$$
\boxed{
\text{K3 parent}
+
\text{Tate center}
\to
\text{same hard K3 block}.
}
$$

Example:

$$
c7.
$$

Action:

$$
\boxed{
\text{reuse parent bridge}.
}
$$

---

## Motif C — Center Injection

$$
\boxed{
\text{Tate parent}
+
\text{K3 center}
\to
\text{center-generated K3 block}.
}
$$

Example:

$$
d3.
$$

Action:

$$
\boxed{
\text{use center as direct witness bridge}.
}
$$

---

# 12. General Blow-Up Source Vector

Define the source vector:

$$
\boxed{
\mathbf s_{\mathrm{K3}}
=
(
r_{\mathrm{parent}},
r_{\mathrm{center}}
),
}
$$

where:

$$
r_{\mathrm{parent}}
=
\dim H^{3,1}(Y),
$$

and:

$$
r_{\mathrm{center}}
=
\dim H^{2,0}(Z).
$$

For:

$$
c7,
$$

we have:

$$
\boxed{
\mathbf s_{\mathrm{K3}}=(1,0).
}
$$

For:

$$
d3,
$$

we have:

$$
\boxed{
\mathbf s_{\mathrm{K3}}=(0,1).
}
$$

A mixed blow-up would have:

$$
(1,1)
$$

or higher ranks.

---

# 13. Ancestry Reduction Rule

Before bridge search, run:

$$
\boxed{
\operatorname{AncestryReduce}(X).
}
$$

If:

$$
X
\simeq
\operatorname{Bl}_Z Y,
$$

compute:

$$
H^4(Y)
$$

and:

$$
H^2(Z)(-1).
$$

Then assign each difficult Hodge direction to:

- parent;
- center;
- both;
- neither.

The bridge search is run only on irreducible unresolved source blocks.

---

# 14. Why This Matters for a 64-Family Scanner

A 2026 classification lists:

$$
\boxed{
64
}
$$

families of Fano fourfolds of K3 type from a database of at least:

$$
634
$$

Fano fourfold constructions.

The authors identify the origin of most K3 structures by relating them to:

- cubic fourfolds;
- Gushel--Mukai fourfolds;
- actual K3 surfaces.

Therefore the scanner should perform ancestry/source reduction before counting a family as a new bridge-discovery target.

Otherwise the graph will contain many aliases of previously solved Hodge structures.

---

# 15. Hodge-Source Equivalence Classes

Define:

$$
X_1
\sim_{\mathrm{source}}
X_2
$$

when their hard K3-type Hodge blocks are connected through a certified chain of:

- blow-up decompositions;
- Cayley tricks;
- projective bundle formulas;
- known Hodge isomorphisms;
- algebraic correspondences.

Then the bridge compiler searches once per:

$$
\boxed{
\sim_{\mathrm{source}}
\text{ equivalence class},
}
$$

not once per geometric presentation.

This is a major search-space reduction.

---

# 16. Bridge Memoization

Store a map:

$$
\boxed{
\operatorname{BridgeCache}
:
[\text{Hodge-source class}]
\mapsto
\{\text{verified bridge paths}\}.
}
$$

For $c7$, the source key resolves to:

$$
\boxed{
\text{special cubic fourfold}.
}
$$

The cache already contains:

$$
\boxed{
\text{Fano-lines bridge}.
}
$$

So no new geometric search is launched.

For $d3$, the source key resolves to:

$$
\boxed{
\text{K3 surface}.
}
$$

The cache contains:

$$
\boxed{
H^{1,1}\to\text{divisors}.
}
$$

Again no new hyperkähler search is launched.

---

# 17. Novelty Frontier

Define the bridge novelty status:

### REUSED

The hard block is source-equivalent to a solved target.

### REDUCED

The target decomposes into known source blocks, each already handled.

### CONDITIONAL

The source is known but some transfer edge remains unresolved.

### NOVEL

No known ancestry/source reduction closes the hard block.

The bridge compiler should spend discovery budget preferentially on:

$$
\boxed{
\text{NOVEL}
}
$$

targets.

---

# 18. Küchle $c5$ as a Novelty Candidate

Among Küchle's K3-type fourfolds:

$$
c5,\quad c7,\quad d3,
$$

the latter two are known to reduce to earlier geometry:

$$
c7
\to
\text{cubic fourfold},
$$

and:

$$
d3
\to
\text{K3 center}.
$$

By contrast, the $c5$ family has long been singled out as the potentially genuinely new K3-category case.

Public 2026 research activity continues to study $c5$ geometry, including conic-bundle degenerations and even nodal surfaces.

Therefore the ancestry scanner should assign:

```text
c7: REUSED
d3: REDUCED
c5: NOVEL / ACTIVE_RESEARCH
```

The last label is a search-priority status, not a theorem that no reduction exists.

---

# 19. Novelty Is Not Truth

A target being classified:

$$
\text{NOVEL}
$$

means only:

> No source reduction has been certified in the current graph.

It does not mean:

- the family is mathematically new in every sense;
- no bridge exists;
- the K3 category is definitely new;
- a conjectured new bridge is correct.

Thus novelty is an epistemic graph status.

---

# 20. Beyond Blow-Ups

The ancestry compiler should also recognize:

### Cayley Trick

A Fano may inherit its K3 Hodge structure from a lower-dimensional zero locus.

### Projective Bundle Formula

A projective-bundle presentation may expose a hidden source.

### Hyperbolic Reduction

Conic/quadric fibrations can relate two K3-type geometries.

### Birational Model

A complicated presentation may be birational or isomorphic to a known family.

### Derived / HPD Reduction

A K3 category may be inherited even when the cohomological geometry is less direct.

These operations form an ancestry graph before the bridge graph.

---

# 21. Ancestry Graph

Define:

$$
\boxed{
\mathcal G_{\mathrm{anc}}
}
$$

with edge types:

$$
\operatorname{BlowUpOf},
$$

$$
\operatorname{CenterInjects},
$$

$$
\operatorname{CayleyEquivalent},
$$

$$
\operatorname{ProjectiveBundleOver},
$$

$$
\operatorname{HyperbolicReduction},
$$

$$
\operatorname{BirationalTo},
$$

$$
\operatorname{HPDRelated}.
$$

The graph points from complicated presentations toward source geometries.

Bridge search begins only after ancestry normalization.

---

# 22. Compiler Order Update

The bridge compiler now runs:

$$
\boxed{
\text{Target}
\to
\text{Ancestry Reduction}
\to
\text{Source Blocks}
\to
\text{Bridge Cache}
\to
\text{Novel Blocks}
\to
\text{Bridge Search}.
}
$$

This is stronger than the Round 15 order, which started directly from:

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

Now:

$$
\operatorname{BridgeSig}
$$

is computed on ancestry-normalized source blocks.

---

# 23. Search-Space Deduplication

Suppose:

$$
N
$$

geometric families reduce to:

$$
M
$$

Hodge-source equivalence classes.

Then the expensive bridge search count drops from:

$$
N
$$

to:

$$
M.
$$

Define the deduplication ratio:

$$
\boxed{
\operatorname{Dedup}
=
1-\frac{M}{N}.
}
$$

The current 64-family classification strongly suggests that:

$$
M\ll64
$$

once cubic, GM, and K3 ancestry relations are collapsed.

The exact quotient count is a future database task.

---

# 24. Hodge-Source Provenance

Every reused bridge should retain the ancestry proof:

```text
target:
  Kuchle_c7

ancestry:
  operation: blowup
  parent: cubic_C20
  center: Veronese_P2
  center_K3_rank: 0
  hard_block_source: parent

bridge_reuse:
  source: cubic_fourfold
  bridge: Fano_lines
```

Similarly:

```text
target:
  Kuchle_d3

ancestry:
  operation: blowup
  parent: (P1)^4
  center: K3_surface
  parent_K3_rank: 0
  hard_block_source: center

bridge_reuse:
  source: K3
  solved_layer: Lefschetz_1_1
```

---

# 25. Exact Prototype Checks

The accompanying program models degree-four Hodge numbers under a codimension-two blow-up.

For $c7$:

### Parent cubic hard block

$$
(h^{3,1},h^{2,2},h^{1,3})
=
(1,\ast,1).
$$

### Center $\mathbb P^2$

$$
(h^{2,0},h^{1,1},h^{0,2})
=
(0,1,0).
$$

After shift:

$$
(0,1,0).
$$

Thus the off-diagonal K3 rank is unchanged.

---

For $d3$:

### Parent $(\mathbb P^1)^4$

$$
(h^{3,1},h^{1,3})=(0,0).
$$

### Center K3

$$
(h^{2,0},h^{1,1},h^{0,2})
=
(1,20,1).
$$

After shift:

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

The target K3 block is center-generated.

---

# 26. Fourth Structural Motif

Rounds 16--18 emphasized direct bridge discovery.

Round 19 adds:

$$
\boxed{
\textbf{Inherited Bridge / Bridge Reuse}.
}
$$

This is not a new bridge geometry.

It is a new compiler behavior:

> Recognize that the hard Hodge problem has already been solved on an ancestor/source and reuse that solution.

This can be more valuable than discovering another bridge.

---

# 27. Five Current Compiler Motifs

The Hodge program now has at least five structural motifs:

### M1 — Factor Ensemble

$$
\text{signature}
\to
\text{factorization}
\to
\text{witness ensemble}.
$$

### M2 — Syzygetic Single Carrier

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

### M3 — Symmetry-Amplified Bridge

$$
\text{defect}
\to
\text{bridge seed}
\to
\text{symmetry closure}.
$$

### M4 — Canonical Incidence / Pareto Bridge

$$
\text{low-curve moduli}
\to
\text{hyperkähler/K3 bridge}.
$$

### M5 — Inherited Bridge Reuse

$$
\text{ancestry reduction}
\to
\text{known source}
\to
\text{cached bridge}.
$$

---

# 28. Scanner Priority

The corrected global order is:

$$
\boxed{
\text{Ancestry first}.
}
$$

Then:

$$
\boxed{
\text{direct reconstruction if signature is explicit}.
}
$$

Then:

$$
\boxed{
\text{canonical incidence / bridge search}.
}
$$

Finally:

$$
\boxed{
\text{abstract Kuga--Satake or high-cost fallback}.
}
$$

This ordering avoids solving the same Hodge structure repeatedly under different geometric presentations.

---

# 29. Round 19 Conclusion

Round 19 changes the K3-type family scanner from a bridge generator into a source normalizer.

The crucial formula is:

$$
\boxed{
H^4(\operatorname{Bl}_Z Y)
\simeq
H^4(Y)
\oplus
H^2(Z)(-1)
}
$$

for codimension-two blow-ups.

For Küchle $c7$:

$$
\boxed{
X_{c7}
=
\operatorname{Bl}_{\mathbb P^2}Y_{\mathrm{cubic}},
}
$$

so the center adds only $(2,2)$ Tate cohomology.

The K3 block is inherited from the cubic parent.

The compiler reuses the cubic Fano-lines bridge.

For Küchle $d3$:

$$
\boxed{
X_{d3}
=
\operatorname{Bl}_{K3}(\mathbb P^1)^4.
}
$$

The parent is Tate.

The entire K3 block is injected from:

$$
H^2(K3)(-1).
$$

The center itself gives the solved divisor witness layer.

Thus $c7$ and $d3$ should not consume new bridge-discovery budget.

The 2026 classification of $64$ K3-type Fano fourfold families makes this compiler stage essential, because most families are related to cubic, GM, or actual K3 geometries.

The resulting bridge workflow is:

$$
\boxed{
\text{geometric family}
\to
\text{ancestry graph}
\to
\text{Hodge-source blocks}
\to
\text{bridge memoization}
\to
\text{novelty frontier}.
}
$$

Under this normalization, Küchle $c5$ becomes a much more meaningful active research candidate than $c7$ or $d3$.

This is the main closure of Round 19.

---

# References

1. Marcello Bernardara, Enrico Fatighenti, Laurent Manivel, Fabio Tanturri, **Fano fourfolds of K3 type**, Annales de l'Institut Fourier, online first (2026), DOI: 10.5802/aif.3761.  
   Relevant result: list of 64 K3-type Fano fourfold families; most K3 structures are related to cubic fourfolds, Gushel--Mukai fourfolds, or actual K3 surfaces.

2. Alexander Kuznetsov, **On Küchle manifolds with Picard number greater than 1**, Izvestiya: Mathematics 79 (2015).  
   Relevant results: $c7$ is the blow-up of a cubic fourfold along a Veronese surface; $d3$ is the blow-up of $(\mathbb P^1)^4$ along a K3 surface.

3. Robert Laterveer, **Multiplicative Chow--Künneth decompositions and varieties of cohomological K3 type**, Annali di Matematica Pura ed Applicata 201 (2022).  
   Relevant material: Küchle $c7$, relation to cubic fourfolds containing a Veronese surface, and MCK decomposition.

4. Alexander Kuznetsov, **Küchle fivefolds of type c5**, Mathematische Zeitschrift 284 (2016), 1245--1278.  
   Relevant discussion: $c7$ and $d3$ reduce to known geometries, while $c5$ was identified as the potentially genuinely new K3-category case.

5. Public 2026 seminars and current work continue to investigate Küchle $c5$ geometry, including conic-bundle degenerations and even nodal surfaces. This is used only as an active-research indicator, not as a theorem status.