# RCHM × Hodge Conjecture — Research Round 7
## A Closed-Loop Carrier Reconstruction Experiment with a Twisted Cubic on a Smooth Quartic Surface

**Status:** Internal research note + reproducible symbolic experiment  
**Version:** Round 7 / v0.7  
**Date:** 2026-09-17  
**Methodological base:** RCHM; Typed Relation & Constraint Algebra; X-Layer Mathematical Completeness; Graph-Native Mathematics  
**Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–6  
**Canonical source:** UTF-8 Markdown

---

# 0. Round 7 Objective

The previous rounds developed an increasingly elaborate carrier-inference architecture:

$$
\text{relation}
\to
\text{signature}
\to
\text{inverse carrier fiber}
\to
\text{geometric realizability}
\to
\text{verified witness}.
$$

Round 7 tests whether this architecture can complete a concrete closed loop.

The experiment uses a twisted cubic:

$$
T\subset\mathbb P^3
$$

contained in a smooth quartic surface:

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

The choice is deliberate.

For a twisted cubic $T$ contained in a smooth quartic surface $X$, Cifani--Pirola--Schlesinger prove that $T$ is cut out scheme-theoretically by the quadrics in the Hodge annihilator ideal associated with its primitive class. Thus:

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

This lets us isolate the inverse-reconstruction part of the pipeline without pretending that we independently recomputed transcendental period integrals in this round.

---

# 1. Standard Twisted Cubic

Use homogeneous coordinates:

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

on:

$$
\mathbb P^3.
$$

Define:

$$
q_1
=
xz-y^2,
$$

$$
q_2
=
xw-yz,
$$

$$
q_3
=
yw-z^2.
$$

Then:

$$
\boxed{
I_T
=
(q_1,q_2,q_3).
}
$$

These are the $2\times2$ minors of:

$$
M
=
\begin{pmatrix}
x&y&z\\
y&z&w
\end{pmatrix}.
$$

The standard parametrization is:

$$
\nu_3:
\mathbb P^1
\to
\mathbb P^3,
$$

$$
[s:t]
\mapsto
[s^3:s^2t:st^2:t^3].
$$

Substitution gives:

$$
q_i\circ\nu_3=0
$$

for:

$$
i=1,2,3.
$$

Thus the carrier type is:

$$
\boxed{
\text{smooth rational normal cubic}.
}
$$

---

# 2. Constructing an Explicit Quartic Containing $T$

Choose three quadratic forms:

$$
\begin{aligned}
Q_1={}&
-2w^2-wx-2wy-2wz
+2x^2-xy+2xz+y^2-yz,
\end{aligned}
$$

$$
\begin{aligned}
Q_2={}&
w^2-2wx-wz
-2x^2-xy+2xz
+2y^2-2yz+z^2,
\end{aligned}
$$

and:

$$
\begin{aligned}
Q_3={}&
-2w^2-2wx+wy-2wz
+2x^2-xy-xz
+2y^2-2yz-2z^2.
\end{aligned}
$$

Define the quartic polynomial:

$$
\boxed{
f
=
q_1Q_1+q_2Q_2+q_3Q_3.
}
$$

Set:

$$
X=V(f)\subset\mathbb P^3.
$$

Since:

$$
f\in I_T,
$$

we immediately have:

$$
\boxed{
T\subset X.
}
$$

This part is exact and symbolic.

---

# 3. Smoothness Verification

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

$$
\frac{\partial f}{\partial x},
\qquad
\frac{\partial f}{\partial y},
\qquad
\frac{\partial f}{\partial z},
\qquad
\frac{\partial f}{\partial w}.
$$

To exclude all nonzero projective solutions, cover:

$$
\mathbb P^3
$$

by the four affine charts:

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

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

$$
J_u
=
\left(
f_x,f_y,f_z,f_w,u-1
\right).
$$

The reproducible SymPy computation included with this round returns the unit ideal in all four charts:

$$
J_x=(1),
$$

$$
J_y=(1),
$$

$$
J_z=(1),
$$

$$
J_w=(1).
$$

Hence there is no nonzero projective gradient zero.

Therefore:

$$
\boxed{
X\text{ is a smooth quartic surface.}
}
$$

This is an exact rational symbolic computation, not a floating-point test.

---

# 4. Ground-Truth Carrier

The ground-truth witness is therefore:

$$
T\subset X,
$$

with:

$$
T
=
V(q_1,q_2,q_3).
$$

We now conceptually hide this carrier and retain only the Hodge-class-side signature required by the reconstruction theorem.

---

# 5. Literature-Backed Hodge Reconstruction Step

Let:

$$
\alpha(T,X)
$$

be the primitive Hodge class associated with the curve $T$ in the quartic surface $X$.

Cifani--Pirola--Schlesinger prove that a twisted cubic is an ACM curve with:

$$
s(T)=2,
\qquad
e(T)=-1,
$$

and that when such a curve lies on a smooth quartic surface, it is cut out scheme-theoretically by the quadrics in:

$$
I_{\alpha(T,X)}.
$$

Hence:

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

For our standard carrier:

$$
I_{T,2}
=
\operatorname{span}
\{q_1,q_2,q_3\}.
$$

This is the one step of the closed loop supplied by a theorem rather than newly recomputed period integrals.

---

# 6. Signature Given to the Inverse Solver

The depth-$2$ carrier signature supplied to the inverse solver is:

$$
\boxed{
U
=
\operatorname{span}
\left\{
xz-y^2,\;
xw-yz,\;
yw-z^2
\right\}
\subset
S_2.
}
$$

Its dimension is:

$$
\dim U=3.
$$

The inverse problem is now:

> Given only the three-dimensional quadratic subspace $U$, reconstruct the carrier.

---

# 7. Recovering the Determinantal Structure

The three quadrics can be recognized as:

$$
2\times2
$$

minors of:

$$
M
=
\begin{pmatrix}
x&y&z\\
y&z&w
\end{pmatrix}.
$$

Explicitly:

$$
\det
\begin{pmatrix}
x&y\\
y&z
\end{pmatrix}
=
xz-y^2
=
q_1,
$$

$$
\det
\begin{pmatrix}
x&z\\
y&w
\end{pmatrix}
=
xw-yz
=
q_2,
$$

and:

$$
\det
\begin{pmatrix}
y&z\\
z&w
\end{pmatrix}
=
yw-z^2
=
q_3.
$$

Thus the inverse signature identifies the rank-one locus of the Hankel matrix $M$.

---

# 8. Recovering the Parametrization

Rank one means the two rows are proportional.

The resulting projective locus is the third Veronese image:

$$
[s:t]
\mapsto
[s^3:s^2t:st^2:t^3].
$$

Hence the reconstructed carrier is:

$$
\boxed{
T_{\mathrm{rec}}
=
V(U)
=
\nu_3(\mathbb P^1).
}
$$

Therefore:

$$
\boxed{
T_{\mathrm{rec}}
=
T.
}
$$

The inverse fiber has collapsed to the intended carrier scheme.

---

# 9. Carrier Identifiability Depth in This Experiment

Using the Round 6 language:

### At class-only depth

The primitive class alone is not being treated here as a directly decoded geometric equation.

Thus:

$$
|\mathfrak F_0|
$$

is not assumed to be one.

### At quadratic annihilator depth

The theorem gives:

$$
I_{\alpha(T,X),2}
=
I_{T,2}.
$$

The degree-$2$ piece generates the ideal of the twisted cubic.

Therefore:

$$
\boxed{
\mathfrak F_2
=
\{[T]\}.
}
$$

For this chosen signature hierarchy:

$$
\boxed{
\iota(T)\le2.
}
$$

The exact numerical label depends on how signature levels are indexed, but the structural statement is invariant:

> the quadratic Hodge-annihilator data already suffices to uniquely reconstruct the carrier scheme.

---

# 10. Geometric Purity

The reconstruction succeeds because the relevant quadratic annihilator piece is geometrically pure enough to coincide with the actual quadratic carrier ideal:

$$
I_{\alpha(T,X),2}
=
I_{T,2}.
$$

This is stronger than merely having:

$$
I_{T,2}
\subseteq
I_{\alpha(T,X),2}.
$$

Thus the twisted-cubic case calibrates the favorable regime:

$$
\boxed{
\text{identifiable}
+
\text{realizable}
+
\text{geometrically pure at reconstruction degree}.
}
$$

---

# 11. Closed-Loop Verification

The complete loop is:

$$
\boxed{
T
\to
[T]
\to
\alpha(T,X)
\to
I_{\alpha(T,X),2}
\to
V(I_{\alpha(T,X),2})
\to
T_{\mathrm{rec}}.
}
$$

For this experiment:

$$
\boxed{
T_{\mathrm{rec}}=T.
}
$$

Breaking the loop into evidence types:

### Computationally verified in this round

$$
T\subset X,
$$

$$
X\text{ smooth},
$$

$$
\dim I_{T,2}=3,
$$

and the three quadrics have the twisted-cubic determinantal form.

### Supplied by established reconstruction theory

$$
I_{\alpha(T,X),2}=I_{T,2}.
$$

### Standard algebraic-geometric identification

$$
V(I_{T,2})
=
\text{the standard twisted cubic}.
$$

No step is intentionally conflated with another.

---

# 12. Why This Is a Genuine Test of the RCHM Inverse Architecture

This experiment is intentionally simple.

It does not solve a new Hodge-conjecture case.

Its role is to test whether the Round 6 carrier-inverse architecture has a nontrivial domain in which all stages can close.

It does.

The process is:

$$
\boxed{
\text{Hodge signature}
\to
\text{candidate ideal}
\to
\text{carrier type}
\to
\text{actual carrier}
\to
\text{symbolic verification}.
}
$$

Thus the architecture is at least operationally meaningful in a known reconstructible regime.

---

# 13. Negative Control: Smooth Rational Quartic

A valid reconstruction framework also needs a failure case.

Cifani--Pirola--Schlesinger prove that for a smooth rational quartic curve:

$$
C\subset S,
$$

where:

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

is a smooth quartic surface, one still has:

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

However the class is not perfect at level $3$.

Thus:

$$
\boxed{
\text{correct low-degree reconstruction}
\not\Rightarrow
\text{full signature ideal is geometrically pure}.
}
$$

This is the required negative control against a naive monotonicity assumption.

---

# 14. Positive and Negative Calibration Pair

The two carrier families therefore give:

## Twisted cubic

$$
\boxed{
\text{quadratic signature}
\to
\text{carrier reconstruction}
}
$$

with strong perfection behavior for ACM curves.

## Rational quartic

$$
\boxed{
\text{quadratic signature}
\to
\text{partial carrier reconstruction},
}
$$

but:

$$
\boxed{
\text{higher signature}
\not\to
\text{naively purer carrier ideal}.
}
$$

Together they calibrate Round 6's two independent axes:

$$
\boxed{
\text{identifiability}
\neq
\text{geometric purity}.
}
$$

---

# 15. Reconstruction Certificate

For AMRAL, a successful inverse reconstruction should emit a certificate.

For this experiment:

```text
carrier_certificate:
  ambient_space: P3
  ambient_surface_degree: 4
  surface_smooth: true
  carrier_type: twisted_cubic
  carrier_degree: 3
  carrier_genus: 0
  reconstruction_level: 2
  signature_dimension: 3
  carrier_generators:
    - x*z - y^2
    - x*w - y*z
    - y*w - z^2
  containment_verified: true
  reconstruction_theorem: CPS_Theorem_1_1 / Remark_4_2
  symbolic_verification: true
```

The certificate distinguishes theorem-backed links from newly computed links.

---

# 16. Reproducible Quartic Test Surface

The surface is represented canonically by:

$$
f=q_1Q_1+q_2Q_2+q_3Q_3.
$$

This representation is preferable to a fully expanded quartic for provenance because it makes:

$$
T\subset X
$$

immediate.

The included Python verifier also expands $f$ internally and tests all four projective charts.

---

# 17. Exact Verification Protocol

The accompanying script performs:

### V1 — Twisted cubic parametrization

Checks:

$$
q_i(s^3,s^2t,st^2,t^3)=0.
$$

### V2 — Carrier generator rank

Checks that:

$$
q_1,q_2,q_3
$$

are linearly independent in $S_2$.

### V3 — Surface containment

Checks:

$$
f\circ\nu_3=0.
$$

### V4 — Homogeneity

Checks:

$$
\deg f=4.
$$

### V5 — Smoothness

For every:

$$
u\in\{x,y,z,w\},
$$

checks:

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

### V6 — Determinantal reconstruction

Checks the three quadrics agree with the minors of:

$$
M.
$$

---

# 18. What Was Not Computed

To avoid overclaiming, this round did **not** numerically compute:

- periods of the explicit quartic;
- the transcendental integration defining the Hodge signature;
- Picard rank;
- the full Hodge lattice;
- a new proof of the twisted-cubic reconstruction theorem.

Instead the round tests the inverse carrier reconstruction pipeline using a theorem-backed signature equality.

This distinction is part of X-layer provenance completeness.

---

# 19. New RCHM Principle: Theorem-Bridged Closed Loop

A computational experiment may contain a formally identified theorem bridge.

Write:

$$
A
\xrightarrow{\mathrm{compute}}
B
\xrightarrow{\mathrm{theorem}}
C
\xrightarrow{\mathrm{compute}}
D.
$$

If the bridge theorem is explicitly sourced and its hypotheses are verified, the complete pipeline remains valid.

Thus:

$$
\boxed{
\text{closed loop}
\neq
\text{every arrow must be recomputed numerically}.
}
$$

What matters is that every arrow has a typed evidence source.

---

# 20. Evidence-Typed Edges

The graph for this round contains at least four edge types:

$$
\operatorname{SymbolicallyVerified},
$$

$$
\operatorname{TheoremGuaranteed},
$$

$$
\operatorname{StandardIdentification},
$$

$$
\operatorname{ConstructedByDesign}.
$$

This is precisely why graph-native mathematics should retain edge type.

A plain arrow:

$$
A\to B
$$

would erase the epistemic difference among these steps.

---

# 21. Next Experimental Escalation

The twisted cubic is a favorable case.

The next useful experiments should increase difficulty in stages.

### Stage A — Rational quartic negative control

Test how far the correct quadratic reconstruction survives before geometric purity fails.

### Stage B — Carrier alias

Use a fake-linear-cycle regime where first-order signatures imitate another carrier type.

### Stage C — Carrier transmutation

Start with a class represented by one carrier type and deform into a different witness carrier.

### Stage D — Unknown witness regime

Only after the reconstruction machinery survives A--C should it be trusted as a search heuristic in Hodge-conjecture cases with unknown algebraic witnesses.

---

# 22. Round 7 Conclusion

Round 7 completes the first explicit RCHM carrier-reconstruction loop.

The experiment constructs a smooth quartic:

$$
X\subset\mathbb P^3
$$

containing a standard twisted cubic:

$$
T\subset X.
$$

Known Hodge reconstruction theory gives:

$$
I_{\alpha(T,X),2}
=
I_{T,2}.
$$

The quadratic signature is:

$$
\operatorname{span}
\left\{
xz-y^2,
xw-yz,
yw-z^2
\right\}.
$$

Its common zero locus reconstructs the determinantal twisted cubic:

$$
\boxed{
T_{\mathrm{rec}}=T.
}
$$

Therefore the carrier inverse architecture is not merely descriptive.

It supports a complete theorem-bridged computational loop in a known reconstructible case.

The key methodological outcome is:

$$
\boxed{
\text{signature}
\to
\text{typed inverse fiber}
\to
\text{realizability}
\to
\text{symbolic carrier certificate}.
}
$$

The negative-control rational-quartic result simultaneously confirms that:

$$
\boxed{
\text{successful low-degree identification}
\not\Rightarrow
\text{unrestricted geometric purity}.
}
$$

This is the main closure of Round 7.

---

# 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

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