# RCHM × Hodge Conjecture — Research Round 11
## Realizability Dominance, Information-Quotient Ladders, and Gate-Ordered Carrier Verification

**Status:** Internal research note + reproducible exact algebra experiment  
**Version:** Round 11 / v1.1  
**Date:** 2026-09-17  
**Methodological base:** RCHM; Typed Relation & Constraint Algebra; X-Layer Mathematical Completeness; Graph-Native Mathematics  
**Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–10  
**Canonical source:** UTF-8 Markdown

---

# 0. Round 11 Objective

Round 10 studied the possibility that the unknown two-dimensional non-geometric Hodge sector might create false carrier signatures.

That analysis produced an alias discriminant.

Round 11 asks a more fundamental question:

> Even if the Hodge signature creates a false carrier **pattern**, does that pattern correspond to an actual geometric carrier inside the known ambient hypersurface?

For the explicit rational-quartic experiment, the answer produces a strong simplification.

Once the level-$2$ Hodge data reconstructs the smooth quadric envelope

$$
Q,
$$

every divisor carrier inside $Q$ that is genuinely contained in the ambient quartic surface

$$
S
$$

must be an irreducible component of

$$
S\cap Q.
$$

But Round 8 already computed an exact factorization:

$$
\boxed{
S\cap Q
=
C\cup D_0.
}
$$

Therefore, inside the envelope-relative divisor universe, no cross-type signature alias can become a verified carrier unless it is one of these actual factors.

This separates two notions that must not be conflated:

$$
\boxed{
\text{signature alias}
\neq
\text{geometric witness alias}.
}
$$

---

# 1. The Explicit Envelope

The safe level-$2$ signature gives:

$$
q=xw-yz.
$$

Hence:

$$
\boxed{
Q=V(q)
\simeq
\mathbb P^1\times\mathbb P^1.
}
$$

Use the Segre coordinates:

$$
x=ac,
\qquad
y=ad,
\qquad
z=bc,
\qquad
w=bd.
$$

The quartic surface equation from Rounds 8--10 restricts to a bidegree-$(4,4)$ polynomial:

$$
f|_Q.
$$

---

# 2. Exact Ambient Restriction Factorization

Exact symbolic factorization gives:

$$
\boxed{
f|_Q
=
F_C F_D,
}
$$

where:

$$
F_C
=
a^3d-b^3c
$$

has bidegree:

$$
(3,1),
$$

and:

$$
\begin{aligned}
F_D={}&
a(c^3+cd^2-d^3)\\
&+
b(-c^3-c^2d-2cd^2+d^3)
\end{aligned}
$$

has bidegree:

$$
(1,3).
$$

Exact factorization over:

$$
\mathbb Q[a,b,c,d]
$$

shows both factors are irreducible.

Moreover:

$$
\boxed{
\gcd(F_C,F_D)=1.
}
$$

Thus the divisor:

$$
S\cap Q
$$

is reduced at the level of these two divisorial components.

---

# 3. Realizability Gate

Let:

$$
F
$$

be a candidate irreducible divisor factor on $Q$.

For:

$$
V_Q(F)
$$

to be contained in:

$$
S,
$$

one must have:

$$
\boxed{
F\mid f|_Q.
}
$$

This is not merely a heuristic.

For irreducible divisors on the smooth integral surface $Q$, containment:

$$
V_Q(F)\subseteq V_Q(f|_Q)
$$

implies that $F$ divides the section defining $S|_Q$, up to units in the Cox-coordinate description.

Therefore define:

$$
\boxed{
\operatorname{Real}_Q(F;S)
=
1
\iff
F\mid f|_Q.
}
$$

Call this the **Ambient Realizability Gate**.

---

# 4. Verified Envelope-Relative Carrier Set

Define the raw signature candidate set at type $\tau$:

$$
\mathfrak P_\tau(A)
=
\mathbb P(\ker\Theta^A_\tau).
$$

Define the realizable subset:

$$
\boxed{
\mathfrak P_\tau^{\mathrm{real}}(A;S,Q)
=
\left\{
[F]\in\mathfrak P_\tau(A):
F\mid f|_Q
\right\}.
}
$$

Because:

$$
f|_Q=F_CF_D
$$

with $F_C,F_D$ irreducible and coprime, every irreducible divisor factor of $f|_Q$ is projectively one of:

$$
\boxed{
[F_C]
\quad\text{or}\quad
[F_D].
}
$$

Hence:

$$
\boxed{
\bigcup_\tau
\mathfrak P_\tau^{\mathrm{real}}
\subseteq
\{[F_C],[F_D]\}.
}
$$

Since both genuine factors are known to remain signature candidates under enlargement of the signature space:

$$
\boxed{
\bigcup_\tau
\mathfrak P_\tau^{\mathrm{real}}
=
\{[F_C],[F_D]\}
}
$$

within the envelope-relative irreducible divisor model.

---

# 5. Cross-Type Signature Aliases Become Harmless After Realizability

Suppose an impurity extension creates a false signature candidate:

$$
[F_*]
\in
\mathfrak P_\tau(A_\alpha)
$$

of type, for example:

$$
(2,2),
\quad
(2,1),
\quad
(1,2),
\quad
(3,0),
\quad
(0,3),
$$

or another bidegree.

If:

$$
F_*
\not\mid
f|_Q,
$$

then:

$$
V_Q(F_*)
\not\subset S.
$$

Therefore:

$$
\boxed{
[F_*]
\text{ is a signature alias but not a witness alias}.
}
$$

The expensive cross-type alias atlas is therefore not the final witness-validation mechanism.

It is an intermediate diagnostic.

---

# 6. Verified Alias Discriminant

Round 10 defined a signature-level alias discriminant:

$$
\mathcal D_{\mathrm{alias}}^\tau.
$$

Now define the verified alias discriminant:

$$
\boxed{
\mathcal D_{\mathrm{verified}}^\tau
=
\left\{
N:
\exists [F]\neq[F_C],[F_D],
\;
F\in\mathfrak P_\tau(A_N),
\;
F\mid f|_Q
\right\}.
}
$$

For the explicit rational-quartic geometry, within irreducible divisor carriers on $Q$:

$$
\boxed{
\mathcal D_{\mathrm{verified}}^\tau
=
\varnothing
}
$$

for every carrier type other than the two actual factor types.

Thus:

$$
\boxed{
\text{cross-type verified aliasing is absent in the envelope-relative divisor universe}.
}
$$

This statement does **not** claim that every algebraic representative of the primitive Hodge class in the entire surface must lie on $Q$.

Its scope is explicitly envelope-relative.

---

# 7. Realizability Dominance

For this experiment, the logical order is:

$$
\boxed{
\text{signature candidate}
\to
\text{ambient containment}
\to
\text{verified carrier}.
}
$$

Because ambient containment is exact and cheap after the envelope is known, it dominates any cross-type signature ambiguity.

This motivates:

$$
\boxed{
\textbf{Realizability Dominance Principle}
}
$$

> If a low-cost exact geometric realizability condition is available, apply it before spending substantial computation resolving signature-level aliases.

---

# 8. Why Round 10 Still Matters

Round 10 is not invalidated.

Its alias discriminant measures:

$$
\boxed{
\text{identification robustness before geometric verification}.
}
$$

This matters when:

- the ambient containment test is expensive;
- the carrier is not a divisor on a simple envelope;
- the envelope is incomplete;
- the witness is higher codimension;
- one wishes to rank candidates before a Hilbert/Chow solve.

Thus:

$$
\boxed{
\text{alias discriminant}
=
\text{pre-verification risk},
}
$$

whereas:

$$
\boxed{
\text{realizability gate}
=
\text{existence verification}.
}
$$

Both belong in the graph, but they have different meanings.

---

# 9. Information-Quotient Ladder

The literature reconstruction setup contains several different information levels.

Let:

$$
\eta_C
$$

denote the full cohomology class of the curve.

The primitive class is obtained after quotienting by the hyperplane class:

$$
\boxed{
\eta_C
\longmapsto
\alpha(C)
=
\eta_C
\bmod
\mathbb C\eta_H.
}
$$

The annihilator is then:

$$
\boxed{
\alpha
\longmapsto
I_\alpha.
}
$$

These maps lose different information.

Call the chain:

$$
\boxed{
\eta_C
\to
\alpha(C)
\to
I_{\alpha(C)}
}
$$

the **Information-Quotient Ladder**.

---

# 10. Hyperplane Gauge Loss

Because:

$$
\alpha(C)
$$

is defined modulo the hyperplane class, replacing the full class by:

$$
\eta_C+\lambda\eta_H
$$

does not change the primitive quotient.

Thus:

$$
\boxed{
\text{primitive Hodge information does not retain the full hyperplane component}.
}
$$

Consequently numerical invariants depending on the full class, such as degree, cannot automatically be treated as recoverable from the primitive quotient alone.

This is a type issue.

---

# 11. Scalar Gauge Loss of the Annihilator

For any nonzero scalar:

$$
c,
$$

the annihilator satisfies:

$$
\boxed{
I_{c\alpha}
=
I_\alpha.
}
$$

Therefore the ideal:

$$
I_\alpha
$$

does not retain the normalization of the primitive functional.

Thus:

$$
\boxed{
\text{annihilator data is at least projective in }\alpha.
}
$$

A reconstruction algorithm that knows only:

$$
I_\alpha
$$

must not silently use invariants requiring a fixed scale of $\alpha$.

---

# 12. Three Reconstruction Modes

Define three information modes.

## 12.1 Full-Class Mode

Input includes:

$$
\eta_C.
$$

Available filters may include:

- degree;
- self-intersection;
- full intersection pairing;
- Chern-character data;
- exact lattice class.

---

## 12.2 Primitive-Class Mode

Input includes:

$$
\alpha(C)
$$

with normalization.

Hyperplane shifts are erased, but primitive pairings and normalized primitive information may remain usable.

---

## 12.3 Annihilator-Only Mode

Input includes only:

$$
I_{\alpha(C)}.
$$

At minimum, scalar normalization of $\alpha$ is erased.

Therefore fewer numerical filters are legal.

---

# 13. Typed Numerical Filtering

Let:

$$
\mathcal I
$$

denote a numerical invariant.

A filter:

$$
\mathcal I(W)=\mathcal I_{\mathrm{target}}
$$

is legal only if:

$$
\mathcal I_{\mathrm{target}}
$$

is determined by the current information mode.

Therefore:

$$
\boxed{
\operatorname{FilterAllowed}
(
\mathcal I,
\text{mode}
)
}
$$

must itself be part of the mathematical type system.

This is an explicit example of:

$$
\boxed{
\text{relation/type information controlling which mathematical inference is legal}.
}
$$

---

# 14. Example: Full-Class Filter on $Q$

Suppose the full class of the rational quartic is retained.

Then:

$$
H\cdot C=4,
$$

and since the quartic surface is a K3 surface:

$$
C^2=-2.
$$

For a divisor of bidegree:

$$
(u,v)
$$

on:

$$
Q\simeq\mathbb P^1\times\mathbb P^1,
$$

the degree is:

$$
\boxed{
d=u+v,
}
$$

and the arithmetic genus is:

$$
\boxed{
p_a=(u-1)(v-1).
}
$$

On the quartic K3 surface:

$$
D^2
=
2p_a-2.
$$

Thus:

$$
\boxed{
D^2
=
2uv-2u-2v.
}
$$

Requiring:

$$
d=4
$$

and:

$$
D^2=-2
$$

gives:

$$
u+v=4,
$$

and:

$$
(u-1)(v-1)=0.
$$

Hence:

$$
\boxed{
(u,v)\in\{(3,1),(1,3)\}.
}
$$

So in Full-Class Mode the numerical gate already eliminates:

$$
(2,2)
$$

and all other degree-$4$ bidegrees.

---

# 15. Primitive Norm Filter

For a curve $D$ of degree:

$$
d=H\cdot D
$$

on a quartic K3 surface with:

$$
H^2=4,
$$

define its orthogonal primitive projection:

$$
D_{\mathrm{prim}}
=
D-\frac d4H.
$$

Then:

$$
\boxed{
D_{\mathrm{prim}}^2
=
D^2-\frac{d^2}{4}.
}
$$

For the rational quartic:

$$
C^2=-2,
\qquad
d=4,
$$

so:

$$
\boxed{
C_{\mathrm{prim}}^2=-6.
}
$$

For a bidegree $(u,v)$ divisor on $Q$:

$$
\boxed{
D_{\mathrm{prim}}^2
=
2uv-2u-2v
-
\frac{(u+v)^2}{4}.
}
$$

Enumerating all factor bidegrees visible at total signature degree $(3,3)$, namely:

$$
0\le u,v\le3,
$$

shows that the only proper bidegrees with:

$$
D_{\mathrm{prim}}^2=-6
$$

are:

$$
\boxed{
(3,1)
\quad\text{and}\quad
(1,3).
}
$$

This is a useful filter in a normalized Primitive-Class Mode.

It is not automatically legal in Annihilator-Only Mode because scale has been erased.

---

# 16. Why Both Surviving Types Are Expected

The residual curve satisfies:

$$
C+D_0\sim2H.
$$

Therefore:

$$
D_0-H
=
-(C-H)
$$

at the primitive-class level.

Thus the two bidegrees:

$$
(3,1)
$$

and:

$$
(1,3)
$$

represent opposite primitive directions.

Their annihilator ideals are insensitive to nonzero scalar rescaling, so a sign reversal is invisible at the annihilator-only level.

Therefore the survival of exactly these two types is structurally natural.

---

# 17. Type-Erasure Warning

If one begins only with:

$$
I_\alpha
$$

and then silently applies the full-class condition:

$$
\deg D=4,
$$

one has used information that may not actually be encoded in the current object.

This is a concrete mathematical example of:

$$
\boxed{
\text{type erasure causing an invalid inference}.
}
$$

X-layer completeness avoids this by retaining the ladder:

$$
\eta_C
\to
\alpha(C)
\to
I_{\alpha(C)}
$$

rather than keeping only the terminal ideal.

---

# 18. Gate-Ordered Candidate Filtration

A robust inverse solver should maintain a descending candidate family:

$$
\mathcal W_0
\supseteq
\mathcal W_{\mathrm{type}}
\supseteq
\mathcal W_{\mathrm{scope}}
\supseteq
\mathcal W_{\mathrm{envelope}}
\supseteq
\mathcal W_{\mathrm{real}}
\supseteq
\mathcal W_{\mathrm{signature}}
\supseteq
\mathcal W_{\mathrm{class}}
\supseteq
\mathcal W_{\mathrm{verified}}.
$$

The exact ordering can change when one gate depends on another.

But every removal must store:

$$
\boxed{
\text{candidate}
+
\text{gate}
+
\text{evidence}
+
\text{information mode}.
}
$$

---

# 19. Cheap Gates Before Expensive Gates

A useful default cost ordering is:

### G0 — Type compatibility

Does the candidate have the correct mathematical type?

### G1 — Scope compatibility

Is it defined in the required ambient category?

### G2 — Safe envelope

Does it lie in the safely reconstructed geometric container?

### G3 — Ambient realizability

Is it actually contained in the known ambient variety?

### G4 — Numerical invariants

When legal under the information mode, do degree/intersection invariants match?

### G5 — Signature coherence

Does it reproduce the Hodge/period signature?

### G6 — Higher-order persistence

Does it survive higher jets and scheme-level tests?

### G7 — Exact class verification

Does it represent the target class under the requested equivalence?

This order is not universal, but it implements:

$$
\boxed{
\text{cheap exact rejection before expensive ambiguous inference}.
}
$$

---

# 20. Realizability Gate as an Existence Test

The Round 11 result also clarifies RCHM's existence hierarchy.

A factor may be:

$$
\operatorname{SignatureCompatible}(F)=1,
$$

while:

$$
\operatorname{GeometricallyRealizable}(F)=0.
$$

Therefore:

$$
\boxed{
\text{signature compatibility}
\not\Rightarrow
\text{existence as a carrier}.
}
$$

This is exactly the distinction introduced in the original RCHM methodology:

$$
\operatorname{Related}
\not\Rightarrow
\operatorname{Admissible}
\not\Rightarrow
\operatorname{Witnessed}.
$$

---

# 21. The Cross-Type Alias Atlas Collapses in the Explicit Test

Round 10 suggested computing a large cross-type alias atlas.

Round 11 shows that, for verified irreducible divisor carriers on the already known quadric envelope, the atlas collapses after ambient realizability:

$$
\boxed{
\text{all valid irreducible envelope-relative carriers}
=
\{C,D_0\}.
}
$$

Thus detailed alias geometry remains relevant only for:

- pre-verification ranking;
- alternative carrier universes;
- higher-codimension carriers;
- situations without an exact ambient factorization.

This is a useful reduction of computational scope.

---

# 22. Reproducible Exact Checks

The accompanying verifier confirms:

1. the quartic restriction to the quadric factors as $F_CF_D$;
2. $F_C$ is irreducible over $\mathbb Q$;
3. $F_D$ is irreducible over $\mathbb Q$;
4. $\gcd(F_C,F_D)=1$;
5. the bidegrees are $(3,1)$ and $(1,3)$;
6. their product has bidegree $(4,4)$;
7. all proper bidegrees $(u,v)$ with $0\le u,v\le3$ are enumerated;
8. only $(3,1)$ and $(1,3)$ have primitive square $-6$;
9. in full-class mode only $(3,1)$ and $(1,3)$ satisfy degree $4$ and self-intersection $-2$.

All calculations are exact over $\mathbb Q$.

---

# 23. New Principles

## Principle A — Signature Alias vs. Witness Alias

$$
\boxed{
\text{signature-level compatibility}
\neq
\text{geometric witness existence}.
}
$$

---

## Principle B — Realizability Dominance

If an exact ambient containment test is cheap, apply it before resolving fine signature ambiguity.

---

## Principle C — Information-Quotient Ladder

$$
\boxed{
\eta_C
\to
\alpha(C)
\to
I_{\alpha(C)}
}
$$

is a sequence of information-losing maps.

Every downstream inference must respect what information remains.

---

## Principle D — Gauge-Typed Filtering

A numerical invariant may be used as a rejection gate only when it is determined by the current information mode.

---

## Principle E — Envelope-Relative Closure

Once an exact safe envelope and ambient intersection are known, the witness search may collapse to factor/component verification rather than unrestricted carrier inference.

---

# 24. Consequence for AMRAL

AMRAL should store not merely a claim:

```text
candidate rejected
```

but:

```text
candidate_gate_event:
  candidate_ref
  gate_type
  information_mode
  required_invariant
  evidence_ref
  result
  scope
```

And every Hodge object should record:

```text
hodge_information_level:
  full_class_available
  primitive_class_available
  primitive_normalization_available
  annihilator_available
  period_vector_available
```

This prevents downstream AI reasoning from using erased information.

---

# 25. Round 11 Conclusion

Round 11 resolves the immediate cross-type alias problem for the explicit envelope-relative divisor experiment more simply than expected.

The safe level-$2$ signature identifies:

$$
Q.
$$

The known ambient quartic then gives:

$$
\boxed{
f|_Q=F_CF_D.
}
$$

Both factors are irreducible and coprime.

Therefore any verified irreducible divisor carrier contained simultaneously in:

$$
Q
$$

and:

$$
S
$$

must be one of:

$$
\boxed{
C
\quad\text{or}\quad
D_0.
}
$$

Thus unknown Hodge impurity may create additional **signature aliases**, but it cannot create additional **geometrically realizable envelope-relative irreducible divisor carriers**.

At the same time, the reconstruction problem contains a separate information hierarchy:

$$
\boxed{
\text{full class}
\to
\text{primitive class}
\to
\text{annihilator ideal}.
}
$$

Each step erases information, so admissible filtering rules must be typed by information level.

The main lesson is therefore:

$$
\boxed{
\text{Carrier inference is not one classifier.}
}
$$

It is a gate-ordered proof of legality:

$$
\boxed{
\text{type}
\to
\text{scope}
\to
\text{envelope}
\to
\text{existence}
\to
\text{signature}
\to
\text{class}
\to
\text{verified witness}.
}
$$

This is the main closure of Round 11.

---

# References

1. Maria Gioia Cifani, Gian Pietro Pirola, Enrico Schlesinger, **Reconstructing curves from their Hodge classes**, Rendiconti del Circolo Matematico di Palermo Series 2, 72 (2023), 945--958.  
   DOI: https://doi.org/10.1007/s12215-021-00704-9  
   The paper defines $\alpha(C)$ as the class modulo the hyperplane-section direction and studies its annihilator ideal.

2. Hossein Movasati, Emre Can Sertöz, **On reconstructing subvarieties from their periods**, Rendiconti del Circolo Matematico di Palermo Series 2, 70 (2021), 1441--1457.  
   DOI: https://doi.org/10.1007/s12215-020-00562-x

3. Standard K3 adjunction: on a K3 surface, a divisor $D$ satisfies $D^2=2p_a(D)-2$.

4. Standard quadric geometry: a divisor of type $(u,v)$ on $\mathbb P^1\times\mathbb P^1$ has degree $u+v$ under the Segre embedding and arithmetic genus $(u-1)(v-1)$.