# RCHM × Hodge Conjecture — Research Round 8
## Rational Quartic Negative Control: Envelope Recovery, Purity Breakdown, and a Two-Dimensional Non-Geometric Defect

**Status:** Internal research note + reproducible symbolic experiment  
**Version:** Round 8 / v0.8  
**Date:** 2026-09-17  
**Methodological base:** RCHM; Typed Relation & Constraint Algebra; X-Layer Mathematical Completeness; Graph-Native Mathematics  
**Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–7  
**Canonical source:** UTF-8 Markdown

---

# 0. Round 8 Objective

Round 7 established a positive closed-loop reconstruction example using a twisted cubic on a smooth quartic surface.

Round 8 performs the required negative control.

The carrier is now a smooth rational quartic curve

$$
C\subset\mathbb P^3
$$

contained in a smooth quartic surface

$$
S\subset\mathbb P^3.
$$

The literature gives two simultaneous facts:

1. the quadratic Hodge-annihilator data still agrees with the actual quadratic carrier ideal;
2. the Hodge class is **not perfect at level $3$**.

Thus this example isolates a failure mode where:

$$
\boxed{
\text{signature information increases}
}
$$

but:

$$
\boxed{
\text{geometric purity decreases}.
}
$$

The purpose of this round is to make that failure quantitative and reproducible.

---

# 1. Explicit Smooth Rational Quartic

Use homogeneous coordinates:

$$
[x:y:z:w]
$$

on:

$$
\mathbb P^3.
$$

Define:

$$
\nu_C:
\mathbb P^1
\to
\mathbb P^3
$$

by:

$$
\boxed{
[s:t]
\mapsto
[s^4:s t^3:s^3t:t^4].
}
$$

On the chart $s\neq0$, writing:

$$
u=t/s,
$$

the map becomes:

$$
u
\mapsto
[1:u^3:u:u^4].
$$

The coordinate $z/x=u$ recovers the source parameter.

On the chart $t\neq0$, writing:

$$
v=s/t,
$$

the map becomes:

$$
v
\mapsto
[v^4:v:v^3:1],
$$

and $y/w=v$ recovers the source parameter.

Therefore the map is an embedding, and its image is a smooth rational curve of degree $4$.

Call the image:

$$
C.
$$

---

# 2. Unique Quadric Envelope

Substitution into degree-$2$ forms shows that:

$$
\dim I_{C,2}=1.
$$

A generator is:

$$
\boxed{
q=xw-yz.
}
$$

Thus:

$$
Q=V(q)
$$

is the unique quadric surface containing $C$.

The quadric is smooth.

Under the Segre parametrization:

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

we have:

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

The curve $C$ is given on $Q$ by:

$$
\boxed{
a^3d-b^3c=0,
}
$$

so it has bidegree:

$$
(3,1).
$$

This agrees with the standard geometry used in the non-perfectness theorem.

---

# 3. Cubic Ideal of $C$

The degree-$3$ kernel of the parametrization has dimension:

$$
\boxed{
\dim I_{C,3}=7.
}
$$

Four dimensions are generated by:

$$
qR_1.
$$

A convenient additional set of cubic generators is:

$$
c_1=y^3-w^2z,
$$

$$
c_2=xy^2-wz^2,
$$

$$
c_3=x^2y-z^3.
$$

Thus the homogeneous ideal is generated in degrees $2$ and $3$, as expected for a smooth rational quartic.

---

# 4. Constructing an Explicit Smooth Quartic Surface

Define:

$$
Q_2
=
w^2-wx-wz-y^2+yz,
$$

$$
L_1
=
w+x-y-z,
$$

$$
L_2
=
-w-z,
$$

$$
L_3
=
x-z.
$$

Set:

$$
\boxed{
f
=
qQ_2+c_1L_1+c_2L_2+c_3L_3.
}
$$

Then:

$$
f\in I_C,
$$

so:

$$
\boxed{
C\subset S:=V(f).
}
$$

The expanded quartic is:

$$
\begin{aligned}
f={}&
w^3x-w^3z-w^2x^2-2w^2xz+2w^2z^2
-2wxy^2+2wxyz\\
&+wy^3+wyz^2+wz^3+x^3y-x^2yz
+xy^3-xy^2z-xz^3\\
&-y^4-y^2z^2+z^4.
\end{aligned}
$$

---

# 5. Exact Smoothness Verification

Since $f$ is homogeneous over characteristic zero, a projective singular point would be a nonzero common zero of:

$$
f_x,
\qquad
f_y,
\qquad
f_z,
\qquad
f_w.
$$

The accompanying symbolic verifier covers projective space by:

$$
x=1,
\qquad
y=1,
\qquad
z=1,
\qquad
w=1.
$$

For each chart variable $u$, it computes the Gröbner basis of:

$$
(f_x,f_y,f_z,f_w,u-1).
$$

All four ideals contain:

$$
1.
$$

Therefore:

$$
\boxed{
S\text{ is a smooth quartic surface over }\mathbb Q,
}
$$

and hence over:

$$
\mathbb C.
$$

---

# 6. Exact Residual Curve on the Quadric

Restrict the quartic surface to the Segre quadric:

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

Exact symbolic factorization gives:

$$
\boxed{
f|_Q
=
(a^3d-b^3c)\,R(a,b,c,d),
}
$$

where:

$$
\begin{aligned}
R={}&
ac^3+acd^2-ad^3
-bc^3-bc^2d\\
&-2bcd^2+bd^3.
\end{aligned}
$$

Thus:

$$
Q\cap S
=
C\cup D_0,
$$

where:

$$
D_0=V(R)\subset Q.
$$

The second factor has bidegree:

$$
(1,3).
$$

Therefore:

$$
\boxed{
C+D_0\sim2H
}
$$

on the quartic surface.

This is exactly the residual geometry appearing in the literature proof of non-perfectness.

---

# 7. Explicit Parametrization of the Residual Curve

Write:

$$
R=aA(c,d)+bB(c,d),
$$

where:

$$
A(c,d)
=
c^3+cd^2-d^3,
$$

and:

$$
B(c,d)
=
-c^3-c^2d-2cd^2+d^3.
$$

Choose:

$$
[a:b]
=
[-B(c,d):A(c,d)].
$$

Under the Segre map, this gives a quartic rational parametrization:

$$
\boxed{
[c:d]
\mapsto
[-Bc:-Bd:Ac:Ad].
}
$$

The symbolic verifier confirms:

$$
\dim I_{D_0,2}=1,
$$

and:

$$
\boxed{
\dim I_{D_0,3}=7.
}
$$

The unique quadric is again:

$$
Q.
$$

---

# 8. Cubic Intersection Geometry

The symbolic computation gives:

$$
\dim I_{C,3}=7,
$$

$$
\dim I_{D_0,3}=7.
$$

Their sum has dimension:

$$
10.
$$

Hence:

$$
\dim(I_{C,3}\cap I_{D_0,3})
=
7+7-10
=
4.
$$

The common $4$-dimensional space is precisely:

$$
\boxed{
qR_1.
}
$$

Therefore:

$$
\boxed{
\dim(I_{C,3}+I_{D_0,3})=10.
}
$$

This reproduces the dimension count used in the theoretical non-perfectness argument.

---

# 9. Jacobian Cubics

For the explicit smooth quartic $S$, the four partial derivatives:

$$
f_x,
\qquad
f_y,
\qquad
f_z,
\qquad
f_w
$$

are linearly independent cubic forms.

Hence:

$$
\boxed{
\dim J_{S,3}=4.
}
$$

The exact symbolic computation gives:

$$
\boxed{
\dim
\left(
I_{C,3}
+
I_{D_0,3}
+
J_{S,3}
\right)
=
14.
}
$$

Thus the geometric sources available in the non-perfectness proof span exactly a $14$-dimensional cubic space for this explicit example.

---

# 10. The Hodge-Annihilator Cubic Dimension

Cifani--Pirola--Schlesinger prove for a smooth rational quartic $C$ in a smooth quartic surface $S$ that:

$$
I_{\alpha(C),1}=0.
$$

Using the Gorenstein symmetry of:

$$
R/I_{\alpha(C)}
$$

with socle degree:

$$
2s-4=4,
$$

they obtain:

$$
\boxed{
\dim I_{\alpha(C),3}=16.
}
$$

They also prove:

$$
\boxed{
I_{\alpha(C),2}=I_{C,2}.
}
$$

Finally, their Theorem 5.1 proves:

$$
\boxed{
\alpha(C)\text{ is not perfect at level }3.
}
$$

---

# 11. Correct Interpretation of the Quadratic Signature

Because:

$$
I_{\alpha(C),2}=I_{C,2}
=
\langle q\rangle,
$$

the quadratic Hodge signature exactly recovers:

$$
Q=V(q).
$$

But:

$$
Q\neq C.
$$

Therefore the degree-$2$ signature reconstructs the **unique geometric envelope** of the carrier, not the carrier itself.

This motivates the distinction:

$$
\boxed{
\textbf{Carrier Envelope}
\neq
\textbf{Carrier}.
}
$$

For the rational quartic:

$$
\boxed{
E_2(C)=Q.
}
$$

This corrects any naive statement that equality of low-degree Hodge and carrier ideals automatically reconstructs the full carrier.

---

# 12. Carrier Envelope Depth

Define the depth-$m$ signature envelope:

$$
\boxed{
E_m(\alpha)
=
V
\left(
I_{\alpha,\le m}
\right),
}
$$

when the zero locus is interpreted in the declared geometric category.

The envelope may contain the carrier:

$$
C\subseteq E_m(\alpha),
$$

but equality is not automatic.

For this example:

$$
\boxed{
E_2(\alpha(C))
=
Q
\supsetneq
C.
}
$$

Hence the quadratic signature is informative but underdetermined.

---

# 13. Why More Equations Can Become Less Geometric

At degree $3$:

$$
\dim I_{\alpha(C),3}=16,
$$

while:

$$
\dim I_{C,3}=7.
$$

Therefore the Hodge-annihilator contains:

$$
\boxed{
9
}
$$

more cubic dimensions than the actual carrier ideal.

Not all of these extra cubics are spurious in the same sense: some can be explained by residual or Hodge-equivalent geometric carriers and by the Jacobian ideal.

But even after allowing all such contributions described by the theorem, a gap remains.

Thus:

$$
\boxed{
\text{more annihilator equations}
\not\Rightarrow
\text{more faithful carrier equations}.
}
$$

This is the central negative-control result.

---

# 14. Geometric Purity Space

For a class $\alpha$ on a smooth hypersurface $S$, define at level $m$:

$$
\boxed{
G_{\alpha,m}
=
J_{S,m}
+
\sum_{
D:
I_{\alpha(D)}=I_\alpha
}
I_{D,m},
}
$$

where the sum ranges over effective geometric carriers relevant at degree $m$.

The exact admissible carrier set must be specified by the application.

Then define the **Geometric Purity Defect**:

$$
\boxed{
\operatorname{PD}_m(\alpha)
=
\dim I_{\alpha,m}
-
\dim G_{\alpha,m}.
}
$$

Whenever:

$$
G_{\alpha,m}\subseteq I_{\alpha,m},
$$

we have:

$$
\operatorname{PD}_m(\alpha)\ge0.
$$

---

# 15. Exact Purity Defect of the Explicit Rational Quartic

The non-perfectness proof shows that every effective divisor capable of contributing cubic equations for the same annihilator is already represented, at cubic level, inside the span generated by:

$$
I_{C,3},
\qquad
I_{D_0,3},
$$

together with:

$$
J_{S,3}.
$$

For our explicit surface, symbolic computation gives:

$$
\dim G_{\alpha(C),3}=14.
$$

The Hodge theorem gives:

$$
\dim I_{\alpha(C),3}=16.
$$

Therefore:

$$
\boxed{
\operatorname{PD}_3(\alpha(C))
=
16-14
=
2.
}
$$

This is an exact quantitative failure certificate.

The class does not merely fail perfection abstractly.

It has a two-dimensional cubic Hodge-annihilator sector that cannot be generated by the permitted geometric carriers plus the Jacobian contribution.

---

# 16. Purity Breakdown Depth

Define:

$$
\boxed{
b_{\mathrm{pur}}(\alpha)
=
\min
\left\{
m:
\operatorname{PD}_m(\alpha)>0
\right\}.
}
$$

For the rational quartic class:

### Level 1

$$
I_{\alpha,1}=0.
$$

No defect occurs.

### Level 2

$$
I_{\alpha,2}=I_{C,2}.
$$

Since:

$$
J_{S,2}=0,
$$

the class is geometrically accounted for at this level.

### Level 3

$$
\operatorname{PD}_3(\alpha)=2.
$$

Therefore:

$$
\boxed{
b_{\mathrm{pur}}(\alpha(C))=3.
}
$$

This gives a precise location for the purity breakdown.

---

# 17. Signature Refinement vs. Carrier Fidelity

Round 6 proved an abstract monotonicity statement:

if the signature hierarchy is retained as data, then higher signature depth can only shrink an inverse fiber.

That statement remains correct.

What fails here is a different operation:

$$
\boxed{
\text{interpreting every new annihilator equation as a carrier equation}.
}
$$

These are not the same process.

Thus distinguish:

$$
\boxed{
\text{information refinement}
}
$$

from:

$$
\boxed{
\text{geometric zero-set refinement}.
}
$$

The first can be monotone.

The second can overshoot the true carrier.

---

# 18. Two Failure Modes

The rational quartic now separates two inverse-problem failures.

## 18.1 Underidentification

At degree $2$:

$$
E_2=Q\supsetneq C.
$$

The signature is too weak to isolate the carrier.

## 18.2 Overconstraint / impurity

At degree $3$, interpreting the full annihilator as carrier equations introduces non-geometric constraints.

Thus:

$$
\boxed{
\text{too little signature}
\to
\text{carrier envelope too large},
}
$$

while:

$$
\boxed{
\text{naively interpreted extra signature}
\to
\text{carrier equations too strong}.
}
$$

The correct inverse solver must navigate between these two failures.

---

# 19. Safe Reconstruction Depth

Define a level $m$ to be **carrier-safe** if all signature equations used as geometric equations are justified by geometric carriers plus declared structural ideals such as the Jacobian ideal.

Define:

$$
\boxed{
m_{\mathrm{safe}}(\alpha)
=
\max
\left\{
m:
\operatorname{PD}_j(\alpha)=0
\text{ for all }j\le m
\right\}.
}
$$

For the rational quartic:

$$
\boxed{
m_{\mathrm{safe}}(\alpha(C))=2.
}
$$

But safety does not imply identifiability.

At safe depth $2$, only the envelope $Q$ is recovered.

Therefore:

$$
\boxed{
\text{safe}
\not\Rightarrow
\text{identifying}.
}
$$

---

# 20. The Safe-but-Ambiguous Zone

This example exhibits a particularly important state:

$$
\boxed{
\text{geometrically safe but carrier-ambiguous}.
}
$$

At level $2$:

- the signature is pure;
- the unique quadric is correct;
- the carrier is not uniquely recovered.

At level $3$:

- more signature information exists;
- geometric purity breaks.

Thus there is no naive degree-by-degree path:

$$
\text{safe}
\to
\text{more informative}
\to
\text{unique carrier}
$$

using annihilator equations alone.

This is a genuine inverse-problem obstruction.

---

# 21. Reconstruction Phase Diagram

Introduce two axes:

$$
\text{ambiguity}
$$

and:

$$
\text{purity defect}.
$$

Then four regimes appear.

### Phase A — Pure and identifying

$$
A_m=0,
\qquad
\operatorname{PD}_m=0.
$$

Ideal reconstruction regime.

### Phase B — Pure but ambiguous

$$
A_m>0,
\qquad
\operatorname{PD}_m=0.
$$

The rational quartic at level $2$ belongs here.

### Phase C — Impure but apparently identifying

A dangerous regime where equations isolate a geometric-looking object but contain unjustified non-carrier information.

### Phase D — Impure and ambiguous

Signature equations are unsuitable for direct carrier recovery.

The rational quartic crosses from Phase B into an impure regime at level $3$.

---

# 22. Required Algorithmic Correction

A carrier inverse solver must never perform:

$$
I_{\alpha,\le m}
\mapsto
V(I_{\alpha,\le m})
$$

and automatically call the result the witness.

Instead:

$$
\boxed{
I_{\alpha,m}
\to
\text{purity test}
\to
\text{carrier-safe subspace}
\to
\text{geometric solve}.
}
$$

If:

$$
\operatorname{PD}_m>0,
$$

the impure quotient must be quarantined:

$$
\boxed{
N_{\alpha,m}
=
I_{\alpha,m}/G_{\alpha,m}.
}
$$

Call:

$$
N_{\alpha,m}
$$

the **Non-Geometric Signature Sector** at level $m$.

For the explicit rational quartic:

$$
\boxed{
\dim N_{\alpha(C),3}=2.
}
$$

---

# 23. X-Layer Interpretation

A theorem-only record would store:

> The rational quartic Hodge class is not perfect at level $3$.

The X-layer record should instead preserve:

$$
\boxed{
\begin{aligned}
&\text{carrier type}=\text{rational quartic},\\
&\text{unique envelope}=Q,\\
&\dim I_{C,2}=1,\\
&\dim I_{C,3}=7,\\
&\dim I_{D_0,3}=7,\\
&\dim(I_{C,3}\cap I_{D_0,3})=4,\\
&\dim J_{S,3}=4,\\
&\dim G_{\alpha,3}=14,\\
&\dim I_{\alpha,3}=16,\\
&\operatorname{PD}_3=2,\\
&b_{\mathrm{pur}}=3.
\end{aligned}
}
$$

This is much more useful for AI reasoning than the final theorem statement alone.

---

# 24. Evidence-Typed Closed Loop

This round uses three evidence classes.

## Computationally verified here

- explicit rational quartic parametrization;
- unique quadratic envelope;
- cubic carrier-space dimensions;
- explicit smooth quartic surface;
- exact residual factorization;
- residual cubic-space dimension;
- cubic intersection dimension;
- Jacobian dimension;
- total geometric span dimension $14$.

## Literature theorem bridge

- $I_{\alpha(C),1}=0$;
- $I_{\alpha(C),2}=I_{C,2}$;
- $\dim I_{\alpha(C),3}=16$;
- classification of cubic-contributing same-annihilator divisors;
- non-perfectness at level $3$.

## Standard geometry

- $(3,1)$ and $(1,3)$ divisor interpretation on a smooth quadric;
- $C+D_0\sim2H$.

---

# 25. Reproducible Failure Certificate

The accompanying verifier emits:

```text
failure_certificate:
  carrier_type: smooth_rational_quartic
  ambient_surface: smooth_quartic
  unique_quadric_dimension: 1
  carrier_cubic_dimension: 7
  residual_cubic_dimension: 7
  common_cubic_dimension: 4
  geometric_cubic_span_dimension: 14
  hodge_annihilator_cubic_dimension: 16
  geometric_purity_defect: 2
  purity_breakdown_depth: 3
  safe_reconstruction_depth: 2
  level_2_result: unique_quadric_envelope
  level_3_result: non_geometric_signature_sector
```

This is the first explicit negative-control certificate in the RCHM Hodge series.

---

# 26. Comparison with Round 7

## Twisted cubic

Quadratic Hodge signature gives enough equations to cut out the full carrier:

$$
\boxed{
E_2=T.
}
$$

The class is ACM and perfect.

## Rational quartic

Quadratic Hodge signature gives only:

$$
\boxed{
E_2=Q\supsetneq C.
}
$$

At the next degree:

$$
\boxed{
\operatorname{PD}_3=2.
}
$$

Thus the two cases separate:

$$
\boxed{
\text{reconstructible perfect carrier}
}
$$

from:

$$
\boxed{
\text{safe envelope followed by purity breakdown}.
}
$$

---

# 27. New Principles from Round 8

## Principle A — Carrier Envelope Principle

A low-degree Hodge signature may exactly recover a unique geometric container without recovering the carrier.

$$
\boxed{
\text{exact low-degree ideal equality}
\not\Rightarrow
\text{carrier identifiability}.
}
$$

---

## Principle B — Purity Breakdown Depth

The first degree with nonzero geometric purity defect is a meaningful inverse-problem invariant:

$$
\boxed{
b_{\mathrm{pur}}(\alpha)
=
\min\{m:\operatorname{PD}_m(\alpha)>0\}.
}
$$

---

## Principle C — Safe Depth vs. Identifiability

$$
\boxed{
\text{carrier-safe signature}
\not\Rightarrow
\text{carrier-identifying signature}.
}
$$

---

## Principle D — Non-Geometric Signature Sector

When:

$$
\operatorname{PD}_m>0,
$$

the quotient:

$$
\boxed{
N_{\alpha,m}
=
I_{\alpha,m}/G_{\alpha,m}
}
$$

measures signature directions that cannot be explained by the allowed geometric carriers plus structural Jacobian contributions.

---

## Principle E — More Equations Can Reduce Geometric Fidelity

Raw signature depth can increase while the interpretation of the signature as carrier equations becomes less valid.

Therefore:

$$
\boxed{
\text{information monotonicity}
\neq
\text{geometric-fidelity monotonicity}.
}
$$

---

# 28. Immediate Next Research Direction

The positive and negative controls now identify the real inverse problem.

The next useful object is not another carrier family chosen arbitrarily.

It is a **purity splitter**:

$$
\boxed{
I_{\alpha,m}
=
G_{\alpha,m}
\oplus_{\text{noncanonical}}
N_{\alpha,m}.
}
$$

The direct-sum symbol is only schematic: a canonical splitting need not exist.

The research question becomes:

> Can one detect the geometric subspace $G_{\alpha,m}$ using only Hodge/period/deformation data, before knowing the carrier?

If yes, the inverse solver could discard non-geometric annihilator directions before attempting carrier reconstruction.

This would convert the rational-quartic failure from a warning into a usable diagnostic.

---

# 29. Round 8 Conclusion

Round 8 produces the first explicit RCHM negative-control geometry.

A smooth rational quartic:

$$
C\subset S
$$

was constructed inside an explicit smooth quartic surface.

Its unique quadratic envelope is:

$$
Q=V(xw-yz).
$$

The surface intersection factors exactly as:

$$
Q\cap S=C\cup D_0,
$$

with:

$$
C\text{ of type }(3,1),
\qquad
D_0\text{ of type }(1,3).
$$

Symbolic computation gives:

$$
\dim I_{C,3}=7,
$$

$$
\dim I_{D_0,3}=7,
$$

$$
\dim(I_{C,3}\cap I_{D_0,3})=4,
$$

$$
\dim J_{S,3}=4,
$$

and:

$$
\boxed{
\dim
(
I_{C,3}+I_{D_0,3}+J_{S,3}
)
=14.
}
$$

The Hodge theorem gives:

$$
\boxed{
\dim I_{\alpha(C),3}=16.
}
$$

Therefore:

$$
\boxed{
\operatorname{PD}_3(\alpha(C))=2.
}
$$

The purity breakdown occurs exactly at:

$$
\boxed{
b_{\mathrm{pur}}=3.
}
$$

Meanwhile degree $2$ is geometrically safe but recovers only the carrier envelope:

$$
\boxed{
E_2=Q\supsetneq C.
}
$$

The central lesson is:

$$
\boxed{
\text{more Hodge-annihilator information does not automatically mean more carrier information}.
}
$$

This is the main closure of Round 8.

---

# 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.  
   arXiv: https://arxiv.org/abs/2104.05576  
   DOI: https://doi.org/10.1007/s12215-021-00704-9  
   Relevant result: Theorem 5.1 and its proof.

2. Hossein Movasati, Emre Can Sertöz, **On reconstructing subvarieties from their periods**, Rendiconti del Circolo Matematico di Palermo Series 2, 70 (2021), 1441--1457.  
   arXiv: https://arxiv.org/abs/1908.03221  
   DOI: https://doi.org/10.1007/s12215-020-00562-x