# Perfect Cuboid G2E.5 — Bielliptic Genus-$2$ Reduction

## Exact involution, explicit elliptic quotients, and a corrected dual-quotient sieve

**Series:** Perfect Cuboid Global Closure  
**Phase:** G2E.5  
**Fixed slice:** $k=7/32$  
**Status:** exact structural theorem + implementation corrigendum  
**Date:** 2026-09-30

---

# 1. Corrigendum to the finite-field Mordell–Weil projection

G2E.4 introduced the proof-safe projection

$$
E(\mathbb F_p)
\longrightarrow
E(\mathbb F_p)[2^\infty]
$$

by writing

$$
\#E(\mathbb F_p)=2^s m,
\qquad
m\text{ odd},
$$

and multiplying by:

$$
[m].
$$

During the G2E.5 reimplementation, one checker path was found to apply:

$$
[m]
$$

to the Mordell–Weil subgroup but not to the candidate point being tested.

That is not a valid projected-membership test.

The correct condition is:

$$
\boxed{
[m]P_{\rm candidate}
\in
\langle
[m]G,\,
[m]E(\mathbb Q)_{\rm tors}
\rangle.
}
$$

Both sides must be projected by the same map.

The checker has been corrected.

---

# 2. Revalidation of G2E.4

After correcting the projected-membership test, the five clean primes from G2E.4 were recomputed from scratch.

They remain:

$$
\boxed{
11,\quad59,\quad181,\quad197,\quad1669.
}
$$

At each prime, simultaneous compatibility with the two elliptic Mordell–Weil images still leaves only:

$$
h\equiv0,\quad1,\quad-1
$$

among finite residues.

Therefore the global divisibility theorem survives the corrigendum:

$$
\boxed{
38622984917
\mid
u\,v\,(u-v)(u+v)
}
$$

for every reduced nonboundary rational point:

$$
h=\frac uv.
$$

The independent exhaustive genus-$3$ integer-square scan through denominator:

$$
20000
$$

is unaffected, because it does not use this finite-field projection machinery.

---

# 3. A smaller intermediate quotient

G2E.2 introduced the genus-$3$ fiber product:

$$
Y_1^2=f_4(t),
$$

$$
Y_2^2=f_2(t).
$$

Multiply the two square roots:

$$
Y=Y_1Y_2.
$$

This gives the genus-$2$ curve:

$$
\boxed{
H_2:
\quad
Y^2
=
-(1024t^2+49)
(49t^2-196t+1220)
(20617451839t^2-7997395778678t-779078718161).
}
$$

The sextic is separable.

Hence:

$$
\boxed{
g(H_2)=2.
}
$$

Every rational point on the genus-$3$ curve maps to a rational point on $H_2$.

Thus $H_2$ is another necessary intermediate layer for the fixed-slice Perfect Cuboid problem.

---

# 4. Exact extra involution

The branch divisor of $H_2$ has a non-hyperelliptic involution.

Define:

$$
\boxed{
\tau(t)
=
\frac{1171-98t}{975t+98}.
}
$$

Direct substitution gives:

$$
\boxed{
\tau(\tau(t))=t.
}
$$

Moreover:

$$
\boxed{
H_2(\tau(t))
(975t+98)^6
=
1073^6H_2(t).
}
$$

Therefore the Möbius involution lifts to the genus-$2$ curve.

One lift is:

$$
\boxed{
(t,Y)
\longmapsto
\left(
\tau(t),
\frac{1073^3Y}{(975t+98)^3}
\right).
}
$$

Composing with the hyperelliptic involution gives the second nontrivial bielliptic involution.

This is the geometric origin of the split Jacobian.

---

# 5. Rational fixed points

The fixed-point equation:

$$
t=\tau(t)
$$

reduces to:

$$
975t^2+196t-1171=0.
$$

Its discriminant is:

$$
4605316=2146^2.
$$

Hence the two fixed points are rational:

$$
\boxed{
t=1,
}
$$

and:

$$
\boxed{
t=-\frac{1171}{975}.
}
$$

This allows the involution to be rationally conjugated to a sign change.

---

# 6. Sign coordinate

Define:

$$
\boxed{
s
=
\frac{975(t-1)}{975t+1171}.
}
$$

Then:

$$
\boxed{
s(\tau(t))=-s(t).
}
$$

The inverse transformation is:

$$
\boxed{
t
=
\frac{1171s+975}{975(1-s)}.
}
$$

After multiplying the square-root coordinate by the natural cubic denominator and dividing by:

$$
2302658
=
2\cdot1073^2,
$$

the genus-$2$ equation becomes:

$$
\boxed{
\bar Y^2
=
-
\left(
1352033^2s^2
-
2\cdot950625^2
\right)
}
$$

$$
\boxed{
\qquad\cdot
\left(
1352033^2s^4
-
2\cdot950625^2s^2
+
950625^2
\right).
}
$$

The right side is even in:

$$
s.
$$

The bielliptic structure is therefore completely explicit.

---

# 7. First elliptic quotient

Set:

$$
z=s^2.
$$

The quotient by:

$$
s\mapsto-s
$$

is:

$$
\boxed{
E_+:
\quad
y^2
=
-
\left(
1352033^2z
-
2\cdot950625^2
\right)
}
$$

$$
\boxed{
\qquad\cdot
\left(
1352033^2z^2
-
2\cdot950625^2z
+
950625^2
\right).
}
$$

Now define:

$$
\boxed{
x
=
2
-
\frac{1352033^2}{950625^2}z.
}
$$

and scale the $y$-coordinate by:

$$
\frac{1352033}{950625^3}.
$$

The equation becomes:

$$
\boxed{
y_0^2
=
x^3
-
2x^2
+
\left(
\frac{1352033}{950625}
\right)^2x.
}
$$

This is exactly the elliptic factor denoted:

$$
E_0
$$

in G2C–G2D.

An independent exact $j$-invariant calculation gives the same result.

Therefore:

$$
\boxed{
E_+\cong_{\mathbb Q}E_0.
}
$$

---

# 8. Second elliptic quotient

Compose the sign involution with the hyperelliptic involution.

The invariant coordinates are:

$$
z=s^2,
$$

and:

$$
w=s\bar Y.
$$

The second quotient is:

$$
\boxed{
E_-^{\rm quot}:
\quad
w^2
=
-z
\left(
1352033^2z
-
2\cdot950625^2
\right)
}
$$

$$
\boxed{
\qquad\cdot
\left(
1352033^2z^2
-
2\cdot950625^2z
+
950625^2
\right).
}
$$

It is a genus-$1$ quartic.

By the genus-$2$ Kani–Rosen decomposition:

$$
\boxed{
\operatorname{Jac}(H_2)
\sim_{\mathbb Q}
E_+
\times
E_-^{\rm quot}.
}
$$

G2C independently established:

$$
\operatorname{Jac}(H_2)
\sim_{\mathbb Q}
E_0\times E_-.
$$

Since:

$$
E_+\cong E_0,
$$

the complementary quotient belongs to the isogeny class of the previously certified elliptic factor:

$$
E_-.
$$

Thus the genus-$2$ split has been made explicit at the curve-map level.

---

# 9. Rank and Néron–Severi data

G2D certified:

$$
\operatorname{rank}E_0(\mathbb Q)=1,
$$

and:

$$
\operatorname{rank}E_-(\mathbb Q)=1.
$$

Therefore:

$$
\boxed{
\operatorname{rank}
\operatorname{Jac}(H_2)(\mathbb Q)
=
2.
}
$$

Because:

$$
g(H_2)=2,
$$

ordinary Chabauty again has equality:

$$
r=g.
$$

However the split product gives at least:

$$
\boxed{
\rho
\left(
\operatorname{Jac}(H_2)/\mathbb Q
\right)
\ge2.
}
$$

Hence:

$$
g+\rho-1
\ge
2+2-1
=
3.
$$

Thus:

$$
\boxed{
r=2<3.
}
$$

The quadratic-Chabauty criterion is satisfied.

This is the smallest-genus layer currently available for a depth-$2$ rational-point computation.

---

# 10. Boundary points become a single elliptic anchor

The two known rational boundary $t$-coordinates are:

$$
t_A
=
\frac{1246879}{200704},
$$

and:

$$
t_B
=
\frac{98}{1073}.
$$

The involution exchanges them:

$$
\boxed{
\tau(t_A)=t_B,
\qquad
\tau(t_B)=t_A.
}
$$

In the sign coordinate:

$$
s_A
=
\frac{950625}{1352033},
$$

and:

$$
s_B
=
-\frac{950625}{1352033}.
$$

Hence both have the same quotient coordinate:

$$
z
=
\left(
\frac{950625}{1352033}
\right)^2.
$$

Under the explicit $E_0$ map:

$$
x
=
2
-
\frac{1352033^2}{950625^2}z,
$$

we obtain:

$$
\boxed{
x=1.
}
$$

Thus the two boundary points on the genus-$2$ curve collapse to one simple elliptic anchor on $E_0$.

This is favorable for an elliptic-Chabauty implementation.

---

# 11. Interpretation as a square-value problem on a rank-one elliptic curve

The quotient:

$$
H_2\longrightarrow E_0
$$

forgets the sign of:

$$
s.
$$

A rational elliptic point lifts back to $H_2(\mathbb Q)$ exactly when its quotient coordinate:

$$
z=s^2
$$

is itself a rational square.

Equivalently, using:

$$
x
=
2
-
\frac{1352033^2}{950625^2}z,
$$

the lift condition is:

$$
\boxed{
\frac{
950625^2
}{
1352033^2
}
(2-x)
\in
\mathbb Q^2.
}
$$

Since the prefactor is already a rational square, this reduces to:

$$
\boxed{
2-x
\in
\mathbb Q^2.
}
$$

Therefore the genus-$2$ rational-point problem becomes:

> Find all points $P\in E_0(\mathbb Q)$ for which $2-x(P)$ is a rational square.

This is an exact elliptic-Chabauty / square-value problem on a rank-one elliptic curve.

It is strictly smaller than working directly on genus $3$ or genus $5$.

---

# 12. Why this is the preferred next layer

The current hierarchy is now:

$$
C_3
\longrightarrow
C_2
\longrightarrow
H_2
\longrightarrow
E_0.
$$

The genera are:

$$
5\longrightarrow3\longrightarrow2\longrightarrow1.
$$

The final map:

$$
H_2\to E_0
$$

has degree $2$.

Thus any fixed-slice Perfect Cuboid point must first satisfy the simple rank-one elliptic square-value condition:

$$
\boxed{
2-x(P)\in\mathbb Q^2.
}
$$

After classifying those points, only the higher lifts need to be checked.

This is now the preferred computational route.

---

# 13. Corrected clean-prime verifier

A standalone C++ finite-field verifier accompanies this document.

The critical corrected rule is:

$$
\boxed{
[m]P_{\rm candidate}
\text{ must be tested against }
\langle[m]G,[m]T_{\rm tors}\rangle.
}
$$

Using this rule reproduces exactly the five G2E.4 clean primes through:

$$
p\le5000:
$$

$$
11,\quad59,\quad181,\quad197,\quad1669.
$$

The earlier clean-prime theorem is therefore retained.

The implementation correction is nevertheless recorded explicitly so later phases cannot accidentally use an asymmetric projection.

---

# 14. What remains

This phase does not yet classify:

$$
H_2(\mathbb Q).
$$

The exact new target is:

$$
\boxed{
\{P\in E_0(\mathbb Q):
2-x(P)\in\mathbb Q^2\}.
}
$$

Because:

$$
\operatorname{rank}E_0(\mathbb Q)=1,
$$

this is a one-dimensional Mordell–Weil square-value problem.

Possible exact completion methods are:

- elliptic Chabauty;
- a Mordell–Weil sieve on the square-value double cover;
- $p$-adic logarithm separation;
- a second descent on the degree-$2$ cover.

Any complete enumeration at this level immediately restricts all higher Perfect Cuboid lifts.

---

# 15. Canonical next target

The next phase is:

$$
\boxed{
\text{G2E.6 — classify }
P\in E_0(\mathbb Q)
\text{ with }
2-x(P)\in\mathbb Q^2.
}
$$

The known boundary anchor is:

$$
\boxed{
x=1,
}
$$

for which:

$$
2-x=1.
$$

The key question is whether any nonboundary rank-one elliptic point satisfies the same square-value condition and survives the subsequent $C_2$ and $C_3$ lifts.

---

**End of G2E.5**