# RCHM × Hodge Conjecture — Research Round 13
## A Non-Factor Blind Benchmark: Recovering a Single Twisted-Cubic Carrier from Syzygies

**Status:** Internal research note + reproducible blind algebra benchmark  
**Version:** Round 13 / v1.3  
**Date:** 2026-09-17  
**Methodological base:** RCHM; Typed Relation & Constraint Algebra; X-Layer Mathematical Completeness; Graph-Native Mathematics  
**Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–12  
**Canonical source:** UTF-8 Markdown

---

# 0. Round 13 Objective

Round 12 reconstructed a witness ensemble because a period-derived plane envelope factored into two conics.

That case naturally followed:

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

Round 13 studies the opposite situation:

> What if the Hodge signature does **not** produce a single factorable envelope?

The benchmark signature is a three-dimensional subspace:

$$
\boxed{
U\subset S_2
}
$$

of quadratic forms in:

$$
S=\mathbb Q[x,y,z,w].
$$

The carrier and its parametrization are hidden from the blind reconstruction script.

The script receives only:

1. a basis of $U$;
2. the ambient quartic equation.

Its job is to decide among:

$$
\boxed{
\text{single carrier},
\quad
\text{carrier ensemble},
\quad
\text{unresolved / bridge escalation}.
}
$$

It reconstructs a single twisted-cubic carrier.

---

# 1. Why This Signature Type Is Hodge-Relevant

Movasati--Sertöz study a quartic surface containing $1536$ twisted cubics.

For every twisted-cubic class $\delta$ they compute:

$$
\boxed{
\dim \widetilde I_{\delta,2}=3,
}
$$

so the period computation produces a point of:

$$
\operatorname{Gr}(3,10),
$$

because:

$$
\dim S_2=10.
$$

They reconstruct this Grassmannian point over algebraic numbers and recover:

$$
I(T).
$$

Cifani--Pirola--Schlesinger later prove a general theorem for ACM curves. A twisted cubic has:

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

For a smooth quartic surface containing it, the twisted cubic is cut out scheme-theoretically by the quadrics lying in its Hodge annihilator.

Thus the three-quadrics signature used in this round is exactly the algebraic data type produced by the period-reconstruction method for twisted cubics.

The present benchmark is synthetic and coordinate-blinded: it does not claim that its three numerical quadrics were taken from the unpublished coordinate list of the $1536$ published examples.

---

# 2. Blind Input Signature

The blind solver receives:

$$
U
=
\operatorname{span}
\{q_1,q_2,q_3\}.
$$

Explicitly:

$$
\begin{aligned}
q_1={}&
-4w^2+8wx-4wy-8wz
+10xy+6xz+y^2-7yz-6z^2,
\end{aligned}
$$

$$
\begin{aligned}
q_2={}&
2wy+6wz-8x^2+4xz+4y^2+4z^2,
\end{aligned}
$$

and:

$$
\begin{aligned}
q_3={}&
4w^2-2wy+6wz
-8x^2-2xy-2xz-6y^2-7yz+z^2.
\end{aligned}
$$

These equations are intentionally not presented in standard twisted-cubic form.

---

# 3. Ambient Quartic

The benchmark also provides a quartic:

$$
\boxed{
F=q_1Q_1+q_2Q_2+q_3Q_3,
}
$$

with:

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

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

and:

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

Therefore:

$$
F\in(q_1,q_2,q_3).
$$

The carrier cut out by the signature is automatically contained in:

$$
X=V(F).
$$

The verifier independently checks that $X$ is smooth.

---

# 4. First Decision Gate: Common-Factor / Envelope Test

Compute:

$$
\gcd(q_1,q_2,q_3).
$$

Exact computation gives:

$$
\boxed{
\gcd(q_1,q_2,q_3)=1.
}
$$

Thus there is no nontrivial common hypersurface factor analogous to the plane or quadric envelope used in Rounds 8--12.

This blocks the immediate:

$$
\text{common envelope}
\to
\text{factorization ensemble}
$$

branch.

The solver therefore escalates to the syzygy gate.

---

# 5. Linear Syzygy Gate

Search for linear forms:

$$
\ell_1,\ell_2,\ell_3\in S_1
$$

such that:

$$
\ell_1q_1+\ell_2q_2+\ell_3q_3=0.
$$

This is a linear system with:

$$
12
$$

unknown scalar coefficients.

Exact computation gives:

$$
\boxed{
\dim\operatorname{Syz}_1(q_1,q_2,q_3)=2.
}
$$

One exact basis is:

$$
\begin{aligned}
s_1
={}&
\left(
-x-y-\frac12z,\;
-w+x-\frac12y-z,\;
-x-\frac12y+z
\right),
\end{aligned}
$$

and:

$$
\begin{aligned}
s_2
={}&
\left(
w+2x+z,\;
w+\frac32y+\frac32z,\;
w+y
\right).
\end{aligned}
$$

Arrange them as a:

$$
2\times3
$$

matrix:

$$
\boxed{
B=
\begin{pmatrix}
-x-y-\frac12z
&
-w+x-\frac12y-z
&
-x-\frac12y+z\\
w+2x+z
&
w+\frac32y+\frac32z
&
w+y
\end{pmatrix}.
}
$$

---

# 6. Hilbert--Burch Reconstruction

Take the three maximal minors of $B$.

Exact calculation gives:

$$
\det B_{\widehat1}
=
\frac14q_1,
$$

$$
\det B_{\widehat2}
=
-\frac14q_2,
$$

$$
\det B_{\widehat3}
=
\frac14q_3.
$$

Therefore the signature ideal is exactly the determinantal ideal:

$$
\boxed{
I
=
I_2(B).
}
$$

This is the Hilbert--Burch signature of a codimension-two ACM curve generated by three quadrics with two linear syzygies.

The carrier is not found by factoring the quadrics.

It is reconstructed through their **relation structure**.

---

# 7. Minimal Free Resolution

The recovered relation pattern gives:

$$
\boxed{
0
\to
S(-3)^2
\to
S(-2)^3
\to
S
\to
S/I
\to
0.
}
$$

This is the classical minimal free-resolution pattern of a twisted cubic.

The Hilbert polynomial is therefore:

$$
\boxed{
P_C(n)=3n+1.
}
$$

The verifier also derives this independently from a Gröbner basis.

---

# 8. Gröbner Certificate

With the chosen variable order, an exact Gröbner basis has initial monomials:

$$
\boxed{
x^2,\quad xy,\quad y^2.
}
$$

Thus:

$$
\operatorname{in}(I)
=
(x,y)^2.
$$

The number of standard monomials of degree $n$ is:

$$
(n+1)+n+n
=
3n+1.
$$

Therefore:

$$
\boxed{
H_{S/I}(n)=3n+1
}
$$

for every:

$$
n\ge0.
$$

Consequences:

$$
\deg C=3,
$$

and:

$$
p_a(C)=0.
$$

---

# 9. Nondegeneracy

The signature ideal begins in degree $2$.

Hence:

$$
\boxed{
I_1=0.
}
$$

Therefore the projective curve is not contained in a plane.

It is a nondegenerate degree-$3$ curve in:

$$
\mathbb P^3.
$$

---

# 10. Smoothness Gate

Let:

$$
J(q)
$$

be the:

$$
3\times4
$$

Jacobian matrix of:

$$
q_1,q_2,q_3.
$$

A singular point of the one-dimensional complete determinantal scheme must lie in:

$$
V(q_1,q_2,q_3)
$$

and have Jacobian rank less than $2$.

Thus add all:

$$
2\times2
$$

minors of:

$$
J(q).
$$

The verifier covers:

$$
\mathbb P^3
$$

by:

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

On every chart, the singularity ideal is the unit ideal.

Therefore:

$$
\boxed{
C\text{ is smooth}.
}
$$

---

# 11. Single-Carrier Decision

We now know:

1. $C$ is a smooth projective curve;
2. $\deg C=3$;
3. $p_a(C)=0$;
4. $C$ is nondegenerate in $\mathbb P^3$.

Therefore:

$$
\boxed{
C\text{ is a twisted cubic}.
}
$$

The inverse solver chooses:

$$
\boxed{
\textbf{Single Carrier Branch}.
}
$$

No ensemble factorization is required.

No bridge object is required.

---

# 12. Why This Decision Did Not Need Ground Truth

The hidden benchmark was generated by an unknown projective coordinate change of the standard twisted cubic.

However the blind solver does not read that coordinate change.

It uses only:

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

Its certificate is intrinsic:

- gcd/common-factor structure;
- syzygy dimension;
- Hilbert--Burch minors;
- Gröbner initial ideal;
- Hilbert polynomial;
- nondegeneracy;
- smoothness;
- ambient containment.

The coordinate transformation is stored separately only for post-hoc audit.

---

# 13. Hodge Interpretation

For a twisted cubic $T$ in a smooth quartic surface, Cifani--Pirola--Schlesinger prove that $T$ is ACM and reconstructed at level $2$ from:

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

Therefore, if a period computation outputs the three-dimensional quadratic space $U$ used above, the carrier reconstruction can proceed exactly through the syzygy gate.

The crucial point is:

$$
\boxed{
\text{Hodge reconstruction need not always be factor reconstruction}.
}
$$

It may instead be:

$$
\boxed{
\text{Hodge signature}
\to
\text{syzygy structure}
\to
\text{determinantal carrier}.
}
$$

---

# 14. Relation-First Reconstruction

This benchmark gives an unusually direct realization of the original RCHM principle.

The individual quadrics look opaque.

Their relation space is simple:

$$
\dim\operatorname{Syz}_1=2.
$$

Those relations form a matrix:

$$
B.
$$

The carrier is recovered from:

$$
I_2(B).
$$

Thus:

$$
\boxed{
\text{the relations among the equations identify the geometry more clearly than the equations individually}.
}
$$

This is precisely the kind of information that a theorem-only or equation-list-only representation can hide.

---

# 15. Carrier Decision Tree

Rounds 12 and 13 now give two experimentally validated branches.

## Branch A — Factorable Envelope

If a safe low-degree signature yields a hypersurface envelope and the ambient restriction factors:

$$
\boxed{
\text{Envelope}
\to
\text{Factorization}
\to
\text{Witness Ensemble}.
}
$$

Round 12 follows this branch.

---

## Branch B — Syzygetic Determinantal Carrier

If no useful common factor exists but the signature ideal has a structured low-degree syzygy module:

$$
\boxed{
\text{Signature Ideal}
\to
\text{Syzygies}
\to
\text{Hilbert--Burch Matrix}
\to
\text{Single Carrier}.
}
$$

Round 13 follows this branch.

---

## Branch C — Escalation

If neither branch closes:

$$
\boxed{
\text{higher jets}
\to
\text{Hilbert/Chow search}
\to
\text{bridge-object search}.
}
$$

This is the unresolved branch intended for future unknown Hodge classes.

---

# 16. Carrier Architecture Classifier

Define a carrier-architecture output:

$$
\boxed{
\mathsf A(U)
\in
\{
\text{Factor-Ensemble},
\text{Syzygetic-Single},
\text{Bridge/Escalate}
\}.
}
$$

The classification is not performed by visual pattern recognition.

It is certified by gates.

For Round 13:

$$
\boxed{
\mathsf A(U)=\text{Syzygetic-Single}.
}
$$

---

# 17. Syzygy Signature

Define the low-degree syzygy signature:

$$
\boxed{
\operatorname{SyzSig}(U)
=
(
\dim U,
\dim\operatorname{Syz}_1(U),
\text{generator degrees},
\text{syzygy degrees}
).
}
$$

For this benchmark:

$$
\boxed{
\operatorname{SyzSig}(U)
=
(3,2,(2,2,2),(1,1)).
}
$$

Equivalently, the minimal resolution shifts are:

$$
\boxed{
(2^3;3^2).
}
$$

This becomes a compact carrier-template key for AMRAL.

---

# 18. Relation to X-Layer Completeness

If one stored only:

$$
q_1,q_2,q_3,
$$

the most informative structure would still have to be rediscovered.

The X-layer record should also preserve:

$$
\boxed{
\operatorname{Syz}_1(U),
}
$$

the Hilbert--Burch matrix:

$$
B,
$$

the free resolution:

$$
0\to S(-3)^2\to S(-2)^3\to S,
$$

and the geometric certificate:

$$
P_C(n)=3n+1.
$$

This is another explicit reason why:

$$
\boxed{
\text{relations are first-class mathematical data}.
}
$$

---

# 19. Ambient Quartic Verification

The benchmark quartic is:

$$
F=q_1Q_1+q_2Q_2+q_3Q_3.
$$

Hence:

$$
C\subset X=V(F).
$$

The verifier independently computes the four projective gradient charts and finds no singular point.

Therefore:

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

The benchmark therefore lies in the exact geometric regime of the twisted-cubic Hodge reconstruction theorem.

---

# 20. Evidence-Typed Chain

The reconstruction graph is:

$$
U
\xrightarrow{\text{exact linear algebra}}
\operatorname{Syz}_1(U),
$$

$$
\operatorname{Syz}_1(U)
\xrightarrow{\text{Hilbert--Burch}}
I_2(B),
$$

$$
I_2(B)
\xrightarrow{\text{Gröbner}}
P_C(n)=3n+1,
$$

$$
I
\xrightarrow{\text{Jacobian test}}
\text{smooth},
$$

$$
(\deg,p_a,\text{nondegenerate},\text{smooth})
\xrightarrow{\text{classification}}
\text{twisted cubic},
$$

$$
(C,F)
\xrightarrow{\text{exact containment}}
C\subset X.
$$

The Hodge interpretation is then supplied by the external reconstruction theorem.

---

# 21. Post-Hoc Ground-Truth Audit

For audit only, the benchmark package includes a hidden coordinate transformation:

$$
\begin{pmatrix}
X\\Y\\Z\\W
\end{pmatrix}
=
M
\begin{pmatrix}
x\\y\\z\\w
\end{pmatrix},
$$

where:

$$
M=
\begin{pmatrix}
2&1&-2&0\\
2&-1&-2&-2\\
2&2&1&0\\
-2&2&-1&-2
\end{pmatrix},
$$

and:

$$
\det M=72\neq0.
$$

Under this transform, the recovered ideal becomes the standard twisted-cubic ideal:

$$
XZ-Y^2,
$$

$$
XW-YZ,
$$

$$
YW-Z^2.
$$

The blind solver never reads this file.

---

# 22. New Principles

## Principle A — Syzygy Reconstruction Principle

When a low-degree signature does not factor usefully, the syzygy module can be a more informative carrier invariant than the generators themselves.

---

## Principle B — Relation-First Geometry

$$
\boxed{
\text{generator relations}
\to
\text{determinantal structure}
\to
\text{carrier}
}
$$

can reconstruct geometry without guessing equations in a known template.

---

## Principle C — Carrier-Architecture Branching

The inverse solver should decide whether a signature represents:

- a factorable ensemble;
- a syzygetic single carrier;
- an unresolved structure requiring escalation.

---

## Principle D — Resolution as Carrier Fingerprint

The minimal graded free resolution is a compact and coordinate-invariant carrier fingerprint.

For a twisted cubic:

$$
\boxed{
0\to S(-3)^2\to S(-2)^3\to S.
}
$$

---

## Principle E — Non-Factor Envelope Does Not Mean No Reconstruction

Failure of common-factor decomposition is not a failure of the inverse problem.

It can signal that the carrier is encoded by higher-order relations among the equations.

---

# 23. AMRAL Representation

Suggested object:

```text
carrier_architecture:
  signature_ref
  common_factor_dimension
  architecture_type
  syzygy_signature
  hilbert_burch_matrix_ref
  minimal_resolution
  hilbert_polynomial
  smoothness_status
  nondegeneracy_status
  carrier_type
  verification_status
```

For this benchmark:

```text
architecture_type: syzygetic_single
syzygy_signature:
  generators: 3
  generator_degree: 2
  linear_syzygies: 2
minimal_resolution:
  - S(-3)^2
  - S(-2)^3
  - S
hilbert_polynomial: 3*n + 1
carrier_type: twisted_cubic
```

---

# 24. Reproducible Benchmark Files

The package contains:

- `benchmark_input.json` — only the blind solver input;
- `blind_reconstruct_twisted_cubic.py` — reconstruction script;
- `ground_truth.json` — audit-only hidden coordinate transform;
- `verification_output.txt`;
- `MANIFEST.json`;
- this Markdown report.

The reconstruction script does not open or depend on `ground_truth.json`.

---

# 25. Round 13 Conclusion

Round 13 gives the first blind non-factor carrier reconstruction in the RCHM Hodge series.

Input:

$$
\boxed{
U\subset S_2,
\qquad
\dim U=3,
}
$$

plus a smooth ambient quartic.

The quadrics have no common factor:

$$
\gcd(U)=1.
$$

But their relation space satisfies:

$$
\boxed{
\dim\operatorname{Syz}_1(U)=2.
}
$$

The two syzygies form a Hilbert--Burch matrix whose maximal minors regenerate the complete signature ideal.

The resulting projective scheme has:

$$
\boxed{
P_C(n)=3n+1,
}
$$

no linear equation, and no projective singularity.

Therefore the gate system concludes:

$$
\boxed{
\text{single smooth nondegenerate rational cubic}
=
\text{twisted cubic}.
}
$$

This gives the carrier-decision chain:

$$
\boxed{
\text{Hodge quadratic signature}
\to
\text{no factor envelope}
\to
\text{linear syzygies}
\to
\text{Hilbert--Burch structure}
\to
\text{single carrier}.
}
$$

Combined with Round 12, the inverse architecture now has experimentally validated **ensemble** and **single-carrier** branches.

The next unresolved branch is:

$$
\boxed{
\text{signature not closed by factorization or syzygies}
\to
\text{bridge / higher-structure search}.
}
$$

This is the main closure of Round 13.

---

# References

1. 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  
   Relevant sections: 2.3, 3.2.

2. 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  
   Theorem 1.1: ACM reconstruction and perfectness; twisted cubic example.

3. Hilbert--Burch theorem for codimension-two perfect ideals generated by maximal minors of a presentation matrix.