# Perfect Cuboid G2D — Elliptic Rank Certification Checkpoint

## Status: structural reduction complete; exact rank certification not yet closed

**Series:** Perfect Cuboid Global Closure  
**Main line:** Geometry Branch G  
**Verification line:** Verification Branch A  
**Phase:** G2D  
**Date:** 2026-09-30

---

# 1. Objective

G2C reduced the fixed-$k$ genus-$5$ simultaneous closure curve to elliptic factors.

For the current slice

$$
k=\frac7{32},
$$

the initial decomposition was

$$
\operatorname{Jac}(C_k)
\sim_{\mathbb Q}
E_f^2
\times
E_0
\times
E_+
\times
E_-.
$$

The purpose of G2D is to certify the Mordell–Weil ranks of the distinct elliptic factors.

This checkpoint records what has been exact-certified and what remains open because the current runtime lacks Sage, PARI/GP, and mwrank.

---

# 2. First simplification: only three distinct Jacobian factors

The binary quartic invariants of $E_0$ and $E_+$ were computed exactly.

They satisfy

$$
I(E_0)=I(E_+),
$$

and

$$
J(E_0)=J(E_+).
$$

For the current value

$$
k=\frac7{32},
$$

the common invariants are

$$
I_0
=
-\frac{
1689198832011869237109375
}{
1208925819614629174706176
},
$$

and

$$
J_0
=
-\frac{
7531518971347664160183300933837890625
}{
664613997892457936451903530140172288
}.
$$

Hence $E_0$ and $E_+$ have the same Jacobian elliptic curve.

Therefore the G2C decomposition compresses to

$$
\boxed{
\operatorname{Jac}(C_k)
\sim_{\mathbb Q}
E_f^2
\times
E_0^2
\times
E_-.
}
$$

Thus exact rank certification requires only three distinct elliptic curves:

$$
\boxed{
E_f,\qquad E_0,\qquad E_-.
}
$$

---

# 3. The first factor $E_f$

The first gate elliptic curve is

$$
E_f:
\quad
y^2
=
(x-2)(x+2)
\left(
x+\frac{1050977}{50176}
\right).
$$

Equivalently,

$$
y^2
=
x^3
+
\frac{1050977}{50176}x^2
-
4x
-
\frac{1050977}{12544}.
$$

The rational $2$-torsion points are

$$
(2,0),
$$

$$
(-2,0),
$$

and

$$
\left(
-\frac{1050977}{50176},
0
\right).
$$

G2B exhibited the rational point

$$
T_4
=
\left(
\frac{1297}{112},
\frac{1632033}{25088}
\right)
$$

satisfying

$$
2T_4=(2,0).
$$

Therefore

$$
T_4
$$

has order $4$.

Together with one independent $2$-torsion point this gives eight rational torsion points.

Using good reductions at

$$
p=11
$$

and

$$
p=17,
$$

with

$$
\#E_f(\mathbb F_{11})=8,
$$

and

$$
\#E_f(\mathbb F_{17})=16,
$$

the rational torsion order divides

$$
\gcd(8,16)=8.
$$

Hence

$$
\boxed{
E_f(\mathbb Q)_{\mathrm{tors}}
\cong
\mathbb Z/4\mathbb Z
\times
\mathbb Z/2\mathbb Z.
}
$$

The seed point

$$
P
=
\left(
\frac{1497}{64},
\frac{2224625}{14336}
\right)
$$

is not torsion.

Therefore

$$
\boxed{
\operatorname{rank}E_f(\mathbb Q)\ge1.
}
$$

The exact rank is not yet certified in the current runtime.

---

# 4. The common factor $E_0\cong E_+$

The binary quartic Jacobian can be written in short Weierstrass form

$$
Y^2
=
X^3
-
27I_0X
-
27J_0.
$$

Exact factorization of the cubic gives

$$
X^3-27I_0X-27J_0
$$

equal, up to a nonzero rational scalar, to

$$
\left(
549755813888X
+
2711063671875
\right)
$$

times an irreducible quadratic over $\mathbb Q$.

Therefore $E_0$ has exactly one nontrivial rational $2$-torsion point:

$$
\boxed{
\left(
-\frac{2711063671875}{549755813888},
0
\right).
}
$$

An independent torsion routine over $\mathbb Q$ returns only

$$
O
$$

and this point.

Hence the currently certified torsion subgroup is

$$
\boxed{
E_0(\mathbb Q)_{\mathrm{tors}}
\cong
\mathbb Z/2\mathbb Z.
}
$$

Because $E_+$ has the same Jacobian,

$$
\boxed{
E_+(\mathbb Q)_{\mathrm{tors}}
\cong
\mathbb Z/2\mathbb Z.
}
$$

The exact free rank is not yet certified.

---

# 5. The factor $E_-$

For $E_-$, the binary quartic invariants are

$$
I_-
=
\frac{
2506275714994464684481921
}{
1208925819614629174706176
},
$$

and

$$
J_-
=
\frac{
77505146247523133440975975283702719
}{
664613997892457936451903530140172288
}.
$$

The Jacobian short Weierstrass cubic factors completely over $\mathbb Q$.

Its three rational roots are

$$
-\frac{4097521685571}{549755813888},
$$

$$
\frac{8256895726659}{1099511627776},
$$

and

$$
-\frac{61852355517}{1099511627776}.
$$

Therefore $E_-$ has full rational $2$-torsion.

The three pairwise root differences become exact integer squares after the natural power-of-two scaling:

$$
16451939097801
=
4056099^2,
$$

$$
8133191015625
=
2851875^2,
$$

$$
8318748082176
=
2884224^2.
$$

This unusually strong square-difference structure is likely useful for an explicit $2$-descent.

The currently certified torsion subgroup contains

$$
\boxed{
(\mathbb Z/2\mathbb Z)^2.
}
$$

The exact free rank is not yet certified.

---

# 6. Current certified rank information

At this checkpoint:

$$
\boxed{
\operatorname{rank}E_f(\mathbb Q)\ge1.
}
$$

For

$$
E_0
$$

and

$$
E_-,
$$

no positive free-rank lower bound is claimed yet.

Thus the only proof-safe rank table is:

| Factor | Certified torsion | Rank lower bound | Rank upper bound | Exact rank |
|---|---|---:|---:|---:|
| $E_f$ | $\mathbb Z/4\times\mathbb Z/2$ | $1$ | not yet certified | open |
| $E_0\cong E_+$ | $\mathbb Z/2$ | $0$ | not yet certified | open |
| $E_-$ | contains $(\mathbb Z/2)^2$ | $0$ | not yet certified | open |

No exact rank is asserted unless lower and upper bounds are both certified and equal.

---

# 7. Why the current runtime cannot close G2D

The current execution environment was checked for:

- SageMath;
- PARI/GP;
- mwrank;
- cypari2;
- python-flint.

None is locally available.

SymPy exposes an elliptic-curve `rank` property but currently raises

$$
\texttt{NotImplementedError}.
$$

Therefore SymPy cannot be used as a rank certificate.

A remote dependency-capable backend was also considered, but that path requires additional user authorization and was not used.

Accordingly, the missing step is computational infrastructure, not an ambiguity in the mathematical target.

---

# 8. Exact backend specification for the next run

The next successful rank-certification run should use one of:

1. SageMath;
2. PARI/GP `ellrank`;
3. Cremona mwrank;
4. Magma;
5. an independently implemented full $2$-descent.

For each of

$$
E_f,E_0,E_-,
$$

the backend must report:

- minimal model;
- torsion subgroup;
- rank lower bound;
- Selmer or descent rank upper bound;
- generators when available.

The result is accepted as certified only when

$$
r_{\mathrm{lower}}
=
r_{\mathrm{upper}}.
$$

---

# 9. G2D success condition

Let

$$
r_f,
\qquad
r_0,
\qquad
r_-
$$

denote the exact ranks of the three distinct factors.

Then

$$
\operatorname{rank}\operatorname{Jac}(C_k)
=
2r_f+2r_0+r_-.
$$

Because

$$
g(C_k)=5,
$$

the first global criterion to check is:

$$
\boxed{
2r_f+2r_0+r_-<5.
}
$$

If this inequality holds, classical Chabauty-type methods become immediately plausible for the fixed slice.

If equality or a larger rank occurs, the explicit elliptic decomposition still enables:

- elliptic Chabauty;
- Mordell–Weil sieve;
- quotient-image compatibility;
- covering collections.

Thus exact rank certification is the decision point for the next proof architecture.

---

# 10. Current conclusion

The G2D structural reduction has succeeded:

$$
\boxed{
5\text{ elliptic factors}
\longrightarrow
3\text{ distinct elliptic rank problems}.
}
$$

The exact torsion structures have been substantially resolved.

The remaining unresolved task is narrow:

$$
\boxed{
\text{certify }
\operatorname{rank}E_f,
\operatorname{rank}E_0,
\operatorname{rank}E_-.
}
$$

No stronger rank claim is made in this checkpoint.

---

**End of G2D Checkpoint**