# Perfect Cuboid G2E.3 — 2-Primary Mordell–Weil Compatibility Sieve

## An explicit rank-one elliptic parameterization of the genus-$3$ lift

**Series:** Perfect Cuboid Global Closure  
**Phase:** G2E.3  
**Fixed slice:** $k=7/32$  
**Status:** exact partial Mordell–Weil sieve  
**Date:** 2026-09-30

---

# 1. Goal

G2E.2 reduced the fixed-slice problem to the genus-$3$ intermediate curve

$$
C_2
\longrightarrow
C_1,
$$

where $C_1$ is a rank-one elliptic curve.

The remaining lift is a square condition on a rational function of the $C_1$ point.

This phase makes that statement completely explicit and applies a proof-safe $2$-primary Mordell–Weil sieve.

The output is not yet a complete classification of $C_2(\mathbb Q)$.

It is a substantial reduction of every hypothetical rational lift to one torsion coset and finitely many $2$-adic coefficient residue disks.

---

# 2. The first-lift quartic

The genus-$1$ first lift is

$$
Y^2
=
(1024t^2+49)
(49t^2-196t+1220).
$$

The rational point

$$
(t,Y)=(1,1073)
$$

allows an explicit quartic-to-Weierstrass transformation.

Set

$$
u=t-1.
$$

Then:

$$
Y^2
=
50176u^4
+
950625u^2
+
2092350u
+
1073^2.
$$

---

# 3. Explicit birational map to $E_0'$

Define:

$$
x
=
\frac{
2146(Y+1073+975u)
}{
u^2
},
$$

and:

$$
y
=
x
\left(
975+\frac{2146}{u}
\right).
$$

Then:

$$
\boxed{
E_0':
\qquad
y^2
=
x(x-200704)(x+1151329).
}
$$

Since:

$$
200704=448^2,
$$

and:

$$
1151329=1073^2,
$$

this is exactly the full-$2$-torsion model already used in G2D.

Conversely, away from the finite exceptional set:

$$
u
=
\frac{
2146x
}{
y-975x
},
$$

and:

$$
\boxed{
t
=
1+
\frac{
2146x
}{
y-975x
}.
}
$$

Therefore the second square lift is now an explicit rational-function condition on the rank-one Mordell–Weil group $E_0'(\mathbb Q)$.

---

# 4. The second-lift square function

The second equation of the genus-$3$ layer is:

$$
Z^2
=
-20617451839t^2
+
7997395778678t
+
779078718161.
$$

Define:

$$
\boxed{
\mathcal F(t)
=
-20617451839t^2
+
7997395778678t
+
779078718161.
}
$$

A rational point of $E_0'$ lifts to $C_2$ if and only if:

$$
\boxed{
\mathcal F
\left(
1+\frac{2146x}{y-975x}
\right)
\in
\mathbb Q^2,
}
$$

with the projective exceptional points handled separately.

Thus:

$$
\boxed{
C_2(\mathbb Q)
}
$$

is a square-value subset of a rank-one elliptic Mordell–Weil group.

---

# 5. An odd-index generator surrogate

Consider:

$$
\boxed{
G_0=(480704,-468686400)
\in E_0'(\mathbb Q).
}
$$

Its three Kummer squareclasses are:

$$
480704
=
448\cdot1073,
$$

$$
480704-200704
=
280000
=
7\cdot200^2,
$$

and:

$$
480704+1151329
=
1632033
=
1073\cdot39^2.
$$

Hence:

$$
\boxed{
\delta(G_0)
=
(7511,7,1073).
}
$$

This class is independent of the rational $2$-torsion image.

Since G2D certified:

$$
\operatorname{rank}E_0'=1
$$

and the full $2$-Selmer group has the exact size corresponding to rank one plus full $2$-torsion, $G_0$ has odd index in the free part, modulo torsion.

Consequently, at every good prime where the reduction group is a $2$-group, multiplication by the unknown odd saturation index is an automorphism.

Therefore $G_0$ may be used as a proof-safe surrogate for the primitive Mordell–Weil generator in all such reductions.

---

# 6. Why power-of-two reduction primes are useful

Write a hypothetical primitive generator as:

$$
G.
$$

Because $G_0$ has odd index, there exists odd:

$$
m
$$

and torsion:

$$
T_0
$$

such that:

$$
G_0=mG+T_0.
$$

If:

$$
E_0'(\mathbb F_p)
$$

is a finite $2$-group, multiplication by odd $m$ is invertible.

Therefore:

$$
\langle G_0,E_0'[2]\rangle
=
\langle G,E_0'[2]\rangle
$$

inside the reduction group.

This removes the odd-saturation uncertainty from the Mordell–Weil sieve.

No assumption that $G_0$ is a primitive global generator is required.

---

# 7. Exact power-of-two reduction primes

An exact Hasse-window point-count scan through:

$$
p\le5{,}000{,}000
$$

found the following good primes for which:

$$
\#E_0'(\mathbb F_p)
$$

is a power of two.

The triples shown are:

$$
\left(
p,\,
\#E_0'(\mathbb F_p),\,
\operatorname{ord}(G_0\bmod p)
\right).
$$

$$
(3,4,2),
$$

$$
(5,8,2),
$$

$$
(11,16,8),
$$

$$
(13,8,2),
$$

$$
(23,32,16),
$$

$$
(61,64,32),
$$

$$
(109,128,32),
$$

$$
(113,128,32),
$$

$$
(233,256,16),
$$

$$
(1049,1024,128),
$$

$$
(2087,2048,1024),
$$

$$
(4027,4096,2048),
$$

$$
(4049,4096,512),
$$

$$
(4219,4096,2048),
$$

$$
(16477,16384,8192),
$$

$$
(32647,32768,16384),
$$

$$
(33013,32768,16384),
$$

$$
(65269,65536,4096),
$$

$$
(65657,65536,16384),
$$

$$
(523357,524288,65536),
$$

and:

$$
\boxed{
(4195001,4194304,262144).
}
$$

The final group order is:

$$
4194304=2^{22}.
$$

---

# 8. Mordell–Weil coefficient in the $2$-adic completion

Because the primitive generator is only known up to odd index, it is convenient to use a coefficient:

$$
c\in\mathbb Z_2.
$$

Every rational Mordell–Weil point has reductions representable in the form:

$$
cG_0+T
$$

at every power-of-two reduction prime, with one globally compatible torsion class:

$$
T\in E_0'[2].
$$

The lift condition:

$$
\mathcal F(t)\in\mathbb Q^2
$$

must reduce to a square at every such prime.

This gives exact congruence restrictions on:

$$
(c,T).
$$

---

# 9. Unique surviving torsion coset

Combining the local square conditions forces the torsion component to be:

$$
\boxed{
T_-
=
(-1151329,0).
}
$$

The other three rational $2$-torsion cosets are eliminated by the $2$-primary Mordell–Weil sieve.

Therefore every hypothetical rational point of $C_2$ must project to the single $2$-adic Mordell–Weil coset:

$$
\boxed{
cG_0+T_-.
}
$$

This is a global compatibility reduction, not a height-bounded search.

---

# 10. Coefficient residues modulo $2^{14}$

Using the power-of-two primes through:

$$
p=65657,
$$

the surviving coefficient classes are:

$$
\boxed{
c
\equiv
1,\,
5441,\,
7153,\,
9231,\,
10943,\,
16383
\pmod{16384}.
}
$$

Thus all rational lifts are already confined to six residue classes modulo:

$$
2^{14}.
$$

---

# 11. Refinement modulo $2^{16}$

Adding the good prime:

$$
p=523357,
$$

with:

$$
\#E_0'(\mathbb F_p)=2^{19},
$$

and:

$$
\operatorname{ord}(G_0\bmod p)=2^{16},
$$

gives twelve residue disks modulo:

$$
2^{16}.
$$

They are:

$$
\boxed{
c
\equiv
\pm1,\,
\pm7153,\,
\pm16383,\,
\pm16385,\,
\pm25615,\,
\pm32767
\pmod{65536}.
}
$$

The symmetry under:

$$
c\mapsto-c
$$

reflects the sign symmetry of the elliptic Mordell–Weil direction.

---

# 12. Refinement modulo $2^{18}$

The exact scan also found:

$$
p=4195001
$$

with:

$$
\#E_0'(\mathbb F_p)=2^{22},
$$

and:

$$
\operatorname{ord}(G_0\bmod p)=2^{18}.
$$

Lifting the twelve modulo-$2^{16}$ disks and applying the second-square condition leaves twenty-four modulo-$2^{18}$ disks:

$$
\boxed{
\begin{aligned}
c\equiv{}&
1,\,
131073,\,
7153,\,
72689,\,
138225,\,
203761,\\
&
81919,\,
212991,\,
16385,\,
81921,\,
147457,\,
212993,\\
&
49151,\,
114687,\,
180223,\,
245759,\,
49153,\,
180225,\\
&
58383,\,
123919,\,
189455,\,
254991,\,
131071,\,
262143
\pmod{262144}.
\end{aligned}
}
$$

The sieve therefore remains nonempty, but the admissible locus is now a very thin collection of explicit $2$-adic Mordell–Weil disks.

---

# 13. Two disks are actual rational boundary lifts

The residue class:

$$
c=1
$$

corresponds to the exact rational point:

$$
G_0+T_-.
$$

Under the inverse quartic map:

$$
\boxed{
t=\frac{98}{1073}.
}
$$

The second square root is:

$$
\boxed{
Z=
\frac{1318232175}{1073}.
}
$$

The symmetric square-lift coordinates are:

$$
P=\frac{1024}{49},
$$

$$
S=\frac{1073}{49},
$$

$$
V=\frac{975}{49}.
$$

Therefore:

$$
Q=\frac{S+V}{2}
=
\frac{1024}{49}
=
\left(
\frac{32}{7}
\right)^2,
$$

and:

$$
R=\frac{S-V}{2}
=
1.
$$

Thus this point lifts all the way to the genus-$5$ curve, but one half-angle parameter equals:

$$
1.
$$

It is a degenerate boundary point.

---

# 14. The opposite boundary disk

The residue class:

$$
c=-1
$$

gives:

$$
-G_0+T_-.
$$

The inverse quartic coordinate is:

$$
\boxed{
t=
\frac{1246879}{200704}.
}
$$

The second square root is:

$$
\boxed{
Z=
\frac{1414463123775}{200704}.
}
$$

The final square coordinates are:

$$
Q=1,
$$

and:

$$
R=\frac{49}{1024}
=
\left(
\frac7{32}
\right)^2.
$$

Again the complete square-lift tower passes, but the point is degenerate because one half-angle parameter equals:

$$
1.
$$

Therefore:

$$
\boxed{
c=\pm1
}
$$

are not candidate Perfect Cuboids.

They are exactly identifiable rational boundary lifts.

---

# 15. What remains

The $2$-primary sieve has therefore achieved three things:

1. all hypothetical rational lifts lie in one torsion coset;
2. their free Mordell–Weil coefficient lies in finitely many explicit $2$-adic residue disks at every computed precision;
3. two of these branches are identified exactly with degenerate rational boundary points.

The remaining nonboundary residue disks must now be eliminated or shown to contain additional rational points.

A finite amount of denominator enumeration cannot complete this step.

The natural tools are:

- quadratic Chabauty on $C_2$;
- elliptic Chabauty for the double cover $C_2\to E_0'$;
- a Mordell–Weil sieve incorporating non-$2$-primary saturation;
- $p$-adic height compatibility among the three elliptic quotient maps.

---

# 16. Why this is stronger than the earlier denominator scan

The previous rational scan imposed a bound on the denominator of a geometric parameter.

The present result imposes no rational-height bound.

Instead it applies to the complete rank-one Mordell–Weil group after $2$-adic completion.

Therefore the statement:

$$
T=T_-
$$

and the displayed congruence restrictions on:

$$
c
$$

hold for every rational point of the intermediate curve, regardless of the size of its numerator or denominator.

This is a qualitative change from search to arithmetic compatibility.

---

# 17. Next theorem target

The immediate G2E target is now sharper:

> Prove that the only rational Mordell–Weil disks surviving the complete elliptic compatibility conditions are the known degenerate $c=\pm1$ boundary disks.

Equivalently, establish:

$$
\boxed{
C_2(\mathbb Q)
=
C_2(\mathbb Q)_{\rm boundary}
}
$$

or at minimum show that every nonboundary $C_2(\mathbb Q)$ point fails the final square lift to $C_3$.

The preferred next step is:

$$
\boxed{
\text{G2E.4 — p-adic height separation of the remaining MW disks}.
}
$$

---

**End of G2E.3**