# Perfect Cuboid Radical-FDRS R7

## Hyperelliptic Normal Form of the Gate-Swap Quotient and the Explicit $E_f/E_0$ Bridge

**Series:** Perfect Cuboid Global Closure  
**Branch:** Radical-FDRS / Geometry-Coupling Branch R  
**Phase:** R7 — Hyperelliptic Normalization and Shared-$E_0$ Synchronization  
**Date:** 2026-10-03  
**Status:** Exact structural theorem — not a proof of Perfect Cuboid existence or nonexistence

---

# 0. Executive summary

R6 constructed a canonical genus-$2$ quotient

$$
G_n
=
C_n/\langle\alpha\rangle
$$

of the fixed-$n$ genus-$5$ Perfect Cuboid fiber.

Its Jacobian was shown to satisfy

$$
\operatorname{Jac}(G_n)
\sim_{\mathbb Q}
E_f(N)
\times
E_0(N),
$$

where

$$
N=n^2.
$$

R7 makes this quotient completely explicit in hyperelliptic form.

Define

$$
\boxed{
B=(N-1)^4,
}
$$

and

$$
\boxed{
C=N^2+6N+1.
}
$$

Then $G_n$ is birational over $\mathbb Q$ to

$$
\boxed{
\mathcal G_N:
\qquad
H^2
=
2
\left(
t^4-B
\right)
\left(
Ct^2-B
\right).
}
$$

This is an even sextic genus-$2$ model.

It has the two standard nonhyperelliptic involutions

$$
(t,H)
\mapsto
(-t,H),
$$

and

$$
(t,H)
\mapsto
(-t,-H).
$$

The first elliptic quotient is

$$
\boxed{
H^2
=
2
(z^2-B)
(Cz-B),
\qquad
z=t^2.
}
$$

It is explicitly $\mathbb Q$-isomorphic to the full-$2$-torsion family

$$
\boxed{
E_0'(N):
\qquad
y^2
=
x
\left(
x-\frac{4N}{(1+N)^2}
\right)
(x+1).
}
$$

The second elliptic quotient is

$$
\boxed{
V^2
=
2z
(z^2-B)
(Cz-B),
\qquad
V=tH,
}
$$

and its binary-quartic invariants identify it exactly with the R5 family

$$
\boxed{
E_f(N).
}
$$

Thus the R6 Jacobian splitting

$$
E_f(N)\times E_0(N)
$$

is now visible directly from one even sextic equation.

At the historical slice

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

a rational rescaling gives the particularly simple genus-$2$ model

$$
\boxed{
Y^2
=
2
(T^4-1)
\left(
1352033T^2-950625
\right).
}
$$

Its first elliptic quotient maps directly to the historical integral curve

$$
\boxed{
E_0':
\qquad
\mathcal Y^2
=
X
(X-448^2)
(X+1073^2)
}
$$

through

$$
\boxed{
X
=
\frac{
1352033T^2-950625
}{2},
}
$$

and

$$
\boxed{
\mathcal Y
=
\frac{1352033}{4}Y.
}
$$

This is exactly the $E_0'$ model used in the earlier exact rank proof.

R7 therefore gives an explicit bridge

$$
\boxed{
g=5
\longrightarrow
g=2
\longrightarrow
E_0'
\overset{2}{\sim}
E_0
}
$$

from the new Radical-FDRS line to the historical arithmetic line.

---

# 1. R6 genus-$2$ quotient model

R6 obtained the invariant presentation

$$
\boxed{
Z^2
=
A\xi^4
+
B(\xi-1)^2,
}
$$

$$
\boxed{
W^2
=
C\xi^2
+
4Z,
}
$$

where

$$
\boxed{
A=N(N+1)^2,
}
$$

$$
\boxed{
B=(N-1)^4,
}
$$

and

$$
\boxed{
C=N^2+6N+1.
}
$$

This curve has genus

$$
2
$$

for physical nondegenerate

$$
0<N<1.
$$

---

# 2. New hyperelliptic coordinates

Define

$$
\boxed{
t
=
\frac{W}{\xi-2},
}
$$

and

$$
\boxed{
r
=
\frac{\xi}{\xi-2}.
}
$$

Then

$$
\xi
=
\frac{2r}{r-1},
$$

and

$$
W
=
\frac{2t}{r-1}.
$$

From

$$
W^2=C\xi^2+4Z
$$

we obtain

$$
\boxed{
Z
=
\frac{
t^2-Cr^2
}{
(r-1)^2
}.
}
$$

Substitute this into

$$
Z^2=A\xi^4+B(\xi-1)^2.
$$

---

# 3. The cancellation identity

After substitution, the equation becomes

$$
\left(
t^2-Cr^2
\right)^2
=
16Ar^4
+
B(r^2-1)^2.
$$

The crucial identity is

$$
\boxed{
C^2-16A=B.
}
$$

Indeed,

$$
\left(
N^2+6N+1
\right)^2
-
16N(N+1)^2
=
(N-1)^4.
$$

Therefore the coefficient of

$$
r^4
$$

cancels.

The genus-$2$ equation becomes linear in

$$
r^2.
$$

---

# 4. Solving for $r^2$

Exact simplification gives

$$
\boxed{
r^2
=
\frac{
B-t^4
}{
2(B-Ct^2)
}.
}
$$

Define

$$
\boxed{
H
=
2r(B-Ct^2).
}
$$

Then

$$
H^2
=
4r^2(B-Ct^2)^2.
$$

Substitution gives

$$
\boxed{
H^2
=
2(B-t^4)(B-Ct^2).
}
$$

Equivalently,

$$
\boxed{
H^2
=
2(t^4-B)(Ct^2-B).
}
$$

This is the R7 hyperelliptic normal form.

---

# 5. Hyperelliptic normal-form theorem

## Theorem

For every nondegenerate rational physical

$$
0<n<1,
$$

with

$$
N=n^2,
$$

the R6 genus-$2$ quotient is birational over

$$
\mathbb Q
$$

to

$$
\boxed{
\mathcal G_N:
\qquad
H^2
=
2
\left[
t^4-(N-1)^4
\right]
\left[
(N^2+6N+1)t^2-(N-1)^4
\right].
}
$$

The inverse birational reconstruction is obtained from

$$
r
=
\frac{
H
}{
2(B-Ct^2)
},
$$

followed by

$$
\xi
=
\frac{2r}{r-1},
$$

$$
W
=
t(\xi-2),
$$

and

$$
Z
=
\frac{W^2-C\xi^2}{4}.
$$

Exceptional denominator loci are treated projectively.

---

# 6. Separability

Let

$$
f_N(t)
=
2(t^4-B)(Ct^2-B).
$$

Its discriminant is

$$
\boxed{
\operatorname{Disc}_t(f_N)
=
-
2^{36}
N^4
(N-1)^{32}
(N+1)^8
(N^2+6N+1).
}
$$

For

$$
0<N<1,
$$

every factor is nonzero.

Therefore the sextic is separable throughout the physical rational region.

Hence

$$
\boxed{
g(\mathcal G_N)=2.
}
$$

---

# 7. Visible involutions

Because the sextic is even in

$$
t,
$$

define

$$
\boxed{
\iota_0:
(t,H)
\mapsto
(-t,H).
}
$$

The hyperelliptic involution is

$$
\boxed{
\iota_h:
(t,H)
\mapsto
(t,-H).
}
$$

Their product is

$$
\boxed{
\iota_f:
(t,H)
\mapsto
(-t,-H).
}
$$

Thus the genus-$2$ curve has two nonhyperelliptic involutions:

$$
\iota_0,
$$

and

$$
\iota_f.
$$

These produce the two elliptic quotient factors.

---

# 8. First elliptic quotient

Take the quotient by

$$
\iota_0.
$$

Set

$$
\boxed{
z=t^2.
}
$$

Then

$$
\boxed{
\mathcal E_0^{\rm hyp}(N):
\qquad
H^2
=
2
(z^2-B)
(Cz-B).
}
$$

This is a cubic genus-$1$ curve.

Its three rational branch roots are

$$
\boxed{
z=(N-1)^2,
}
$$

$$
\boxed{
z=-(N-1)^2,
}
$$

and

$$
\boxed{
z=
\frac{
(N-1)^4
}{
N^2+6N+1
}.
}
$$

Thus the quotient has full rational $2$-torsion.

---

# 9. Explicit map to $E_0'(N)$

Define

$$
\boxed{
D=(N-1)^2,
}
$$

and

$$
\boxed{
C_N
=
\frac{
4N
}{
(1+N)^2
}.
}
$$

Set

$$
\boxed{
x
=
\frac{
Cz-D^2
}{
2D(N+1)^2
},
}
$$

and

$$
\boxed{
y
=
\frac{
CH
}{
4(N-1)^3(N+1)^3
}.
}
$$

Then exact substitution gives

$$
\boxed{
y^2
=
x(x-C_N)(x+1).
}
$$

Therefore

$$
\boxed{
\mathcal E_0^{\rm hyp}(N)
\cong_{\mathbb Q}
E_0'(N),
}
$$

where

$$
\boxed{
E_0'(N):
\qquad
y^2
=
x
\left(
x-\frac{4N}{(1+N)^2}
\right)
(x+1).
}
$$

This is the same full-$2$-torsion family found independently in R6.

---

# 10. Standard $2$-isogeny

Write

$$
E_0'(N)
:
\qquad
y^2
=
x^3+ax^2+bx,
$$

with

$$
a=1-C_N,
$$

and

$$
b=-C_N.
$$

The standard degree-$2$ isogeny with kernel

$$
(0,0)
$$

is

$$
\boxed{
\phi_N(x,y)
=
\left(
\frac{y^2}{x^2},
\frac{
y(b-x^2)
}{
x^2
}
\right).
}
$$

Its target is

$$
\boxed{
\widehat E_0(N):
\qquad
Y^2
=
X^3
-
2aX^2
+
(a^2-4b)X.
}
$$

Since

$$
a^2-4b
=
(1+C_N)^2,
$$

this target is a full explicit member of the R5

$$
E_0(N)
$$

isogeny class.

---

# 11. Normalized $E_0$ model

Because

$$
1-C_N
=
\left(
\frac{1-N}{1+N}
\right)^2,
$$

the coefficient

$$
a
$$

is itself a rational square.

After the rational scaling

$$
X=aX_0,
$$

and

$$
Y=a^{3/2}Y_0,
$$

the isogenous curve becomes

$$
\boxed{
Y_0^2
=
X_0^3
-
2X_0^2
+
\Lambda(N)^2X_0,
}
$$

where

$$
\boxed{
\Lambda(N)
=
\frac{
N^2+6N+1
}{
(N-1)^2
}.
}
$$

This is the generic normalized historical

$$
E_0
$$

model.

---

# 12. Second elliptic quotient

Take the quotient by

$$
\iota_f:
(t,H)\mapsto(-t,-H).
$$

Define

$$
\boxed{
z=t^2,
}
$$

and

$$
\boxed{
V=tH.
}
$$

Then

$$
\boxed{
\mathcal E_f^{\rm hyp}(N):
\qquad
V^2
=
2z
(z^2-B)
(Cz-B).
}
$$

This is a genus-$1$ quartic with the rational point

$$
(z,V)=(0,0).
$$

Therefore it is an elliptic curve over

$$
\mathbb Q.
$$

---

# 13. Exact invariant identification with $E_f(N)$

Let

$$
I_f(N)
=
N^4+14N^2+1,
$$

and

$$
J_f(N)
=
-2
(N^2+1)
(N^2-6N+1)
(N^2+6N+1).
$$

These are the R5 invariants of the generic family

$$
E_f(N).
$$

For the quartic

$$
\mathcal E_f^{\rm hyp}(N),
$$

exact calculation gives

$$
\boxed{
I
=
16
(N-1)^8
I_f(N),
}
$$

and

$$
\boxed{
J
=
64
(N-1)^{12}
J_f(N).
}
$$

Since

$$
16(N-1)^8
=
\left[
2(N-1)^2
\right]^4,
$$

and

$$
64(N-1)^{12}
=
\left[
2(N-1)^2
\right]^6,
$$

the canonical Jacobian is $\mathbb Q$-isomorphic to

$$
\boxed{
E_f(N).
}
$$

Thus

$$
\boxed{
\operatorname{Jac}(\mathcal G_N)
\sim_{\mathbb Q}
E_0(N)\times E_f(N).
}
$$

The R6 split is now visible directly from the hyperelliptic equation.

---

# 14. R7 bielliptic dictionary

The genus-$2$ curve

$$
\mathcal G_N
$$

has the exact quotient dictionary

| Involution | Quotient | Elliptic family |
|---|---|---|
| $(t,H)\mapsto(-t,H)$ | $H^2=2(z^2-B)(Cz-B)$ | $E_0'(N)\sim E_0(N)$ |
| $(t,H)\mapsto(-t,-H)$ | $V^2=2z(z^2-B)(Cz-B)$ | $E_f(N)$ |
| $(t,H)\mapsto(t,-H)$ | genus-$0$ hyperelliptic base | — |

Therefore the genus-$2$ split is explicit at the curve-map level.

---

# 15. Historical specialization $n=7/32$

Now set

$$
\boxed{
n=\frac7{32}.
}
$$

Then

$$
\boxed{
N=\frac{49}{1024}.
}
$$

We have

$$
N-1
=
-\frac{975}{1024}.
$$

Therefore

$$
\boxed{
B
=
\left(
\frac{975}{1024}
\right)^4.
}
$$

Also

$$
\boxed{
C
=
\frac{
1352033
}{
1024^2
}.
}
$$

The historical constants emerge automatically.

---

# 16. Rationally scaled genus-$2$ model

Set

$$
\boxed{
t
=
\frac{975}{1024}T.
}
$$

Define the scaled square-root coordinate

$$
\boxed{
Y
=
\frac{
1024^4
}{
975^3
}
H.
}
$$

Then the hyperelliptic equation becomes

$$
\boxed{
Y^2
=
2
(T^4-1)
\left(
1352033T^2-950625
\right).
}
$$

This is the canonical R7 genus-$2$ model for the historical:

$$
n=\frac7{32}
$$

slice.

---

# 17. Direct map to the historical integral $E_0'$

Take the quotient by

$$
T\mapsto-T.
$$

Set

$$
Z=T^2.
$$

The elliptic quotient is

$$
Y^2
=
2
(Z^2-1)
(1352033Z-950625).
$$

Define

$$
\boxed{
X
=
\frac{
1352033Z-950625
}{2},
}
$$

and

$$
\boxed{
\mathcal Y
=
\frac{
1352033
}{4}
Y.
}
$$

Then exact substitution gives

$$
\boxed{
\mathcal Y^2
=
X
(X-448^2)
(X+1073^2).
}
$$

This is exactly the historical curve

$$
\boxed{
E_0':
\qquad
\mathcal Y^2
=
X
(X-200704)
(X+1151329).
}
$$

No invariant matching is required.

The quotient map is explicit.

---

# 18. Historical $E_0'$ recovered from one line

The identities behind the map are

$$
\boxed{
448^2=200704,
}
$$

$$
\boxed{
975^2=950625,
}
$$

$$
\boxed{
1073^2=1151329,
}
$$

and

$$
\boxed{
950625^2
+
4\cdot448^2\cdot1073^2
=
1352033^2.
}
$$

Thus the old full-$2$-torsion model is built directly into the coefficients of the new hyperelliptic quotient.

---

# 19. Explicit $2$-isogeny to the historical normalized $E_0$

Write the historical integral model as

$$
\mathcal Y^2
=
X^3
+
975^2X^2
-
448^2\cdot1073^2X.
$$

Set

$$
a=975^2,
$$

and

$$
b=-448^2\cdot1073^2.
$$

The standard $2$-isogeny is

$$
\boxed{
U
=
\frac{
\mathcal Y^2
}{
X^2
},
}
$$

$$
\boxed{
V
=
\frac{
\mathcal Y
(b-X^2)
}{
X^2
}.
}
$$

The target is

$$
V^2
=
U^3
-
2\cdot975^2U^2
+
1352033^2U.
$$

Now scale

$$
U
=
950625x,
$$

and

$$
V
=
975^3y.
$$

Then

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

This is exactly the $E_0$ model used in the historical genus-$2$ analysis.

---

# 20. Direct new-route $E_0$ coordinates

Combining the quotient and the $2$-isogeny gives a direct map from the R7 genus-$2$ model to historical $E_0$.

First define

$$
\boxed{
X
=
\frac{
1352033T^2-950625
}{2},
}
$$

and

$$
\boxed{
\mathcal Y
=
\frac{
1352033
}{4}
Y.
}
$$

Then the normalized $E_0$ coordinate is

$$
\boxed{
x_{\rm new}
=
\frac{
\mathcal Y^2
}{
950625X^2
}.
}
$$

The corresponding $y$-coordinate is

$$
\boxed{
y_{\rm new}
=
-
\frac{
\mathcal Y
\left(
X^2
+
448^2\cdot1073^2
\right)
}{
975^3X^2
}.
}
$$

Thus the entire new genus-$2$ route has an explicit map to the same historical rank-one elliptic curve.

---

# 21. Historical G2E.5 genus-$2$ route

The earlier genus-$2$ curve was transformed to the even model

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

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

The involution

$$
s\mapsto-s
$$

gives the quotient coordinate

$$
z=s^2.
$$

The historical map to $E_0$ is

$$
\boxed{
x_{\rm old}
=
2
-
\frac{
1352033^2
}{
950625^2
}
s^2.
}
$$

If the quotient square-root coordinate is denoted by

$$
y_{\rm q},
$$

then

$$
\boxed{
y_{\rm old}
=
\frac{
1352033
}{
950625^3
}
y_{\rm q}.
}
$$

The resulting elliptic curve is again

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

---

# 22. Shared-$E_0$ synchronization

The new R7 genus-$2$ route gives

$$
\boxed{
G_{7/32}
\longrightarrow
E_0'
\overset{2}{\longrightarrow}
E_0.
}
$$

The historical G2E.5 route gives

$$
\boxed{
H_2
\longrightarrow
E_0.
}
$$

Thus both genus-$2$ windows land on the same explicit rank-one elliptic curve.

This creates the synchronization diagram

$$
\boxed{
G_{7/32}
\longrightarrow
E_0
\longleftarrow
H_2.
}
$$

A hypothetical rational point surviving both quotient architectures must satisfy compatibility in this shared Mordell-Weil target.

---

# 23. Important compatibility warning

The two maps into

$$
E_0
$$

need not send a common genus-$5$ point to literally identical affine coordinates without accounting for the precise quotient-map normalization, signs, torsion translations, and automorphisms.

Therefore R7 does not impose the unproved condition

$$
P_{\rm new}=P_{\rm old}
$$

as a theorem.

The correct next task is to derive the induced correspondence between the two $E_0$ images from the common parent curve.

This is an exact finite-ambiguity problem.

---

# 24. Why the bridge is still useful before full synchronization

Even before the final correspondence is derived, both routes reduce rational-point questions to the same rank-one group

$$
E_0(\mathbb Q).
$$

Therefore one can compare:

- Mordell-Weil coefficient parity;
- $2$-adic residue classes;
- torsion cosets;
- local reductions modulo good primes;
- square-value lift conditions.

Any incompatibility at the $E_0$ level eliminates the higher genus candidate immediately.

Thus the shared target is already computationally valuable.

---

# 25. New genus-$2$ quotient versus old genus-$2$ quotient

The new R7 curve has Jacobian

$$
\boxed{
E_f\times E_0.
}
$$

The historical G2E.5 curve has Jacobian

$$
\boxed{
E_0\times E_-.
}
$$

Therefore they are not the same genus-$2$ reduction.

They are complementary reductions.

Their common factor is

$$
\boxed{
E_0.
}
$$

Together they expose all three distinct fixed-slice elliptic factors

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

---

# 26. The new hyperelliptic form explains the bielliptic split

The R7 sextic is

$$
H^2
=
2(t^4-B)(Ct^2-B).
$$

Because it is even, the two involutions

$$
(t,H)\mapsto(-t,H)
$$

and

$$
(t,H)\mapsto(-t,-H)
$$

are visible immediately.

Therefore the decomposition

$$
\boxed{
\operatorname{Jac}(\mathcal G_N)
\sim
E_0(N)\times E_f(N)
}
$$

no longer requires abstract differential or Kani-Rosen reasoning to discover.

It is encoded directly in the shape of the equation.

---

# 27. The fixed-slice constants are no longer mysterious

At

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

the coefficients

$$
975,
$$

$$
448,
$$

$$
1073,
$$

and

$$
1352033
$$

all arise canonically:

$$
975^2
=
1073^2-448^2,
$$

and

$$
1352033^2
=
975^4
+
4\cdot448^2\cdot1073^2.
$$

They are therefore structural invariants of the elliptic quotient conversion, not arbitrary large coefficients produced by symbolic elimination.

---

# 28. R7 genus ladder

The preferred fixed-slice architecture is now

$$
\boxed{
g=5
}
$$

$$
\Downarrow
$$

gate-swap quotient

$$
\boxed{
g=2
}
$$

$$
\Downarrow
$$

bielliptic quotient

$$
\boxed{
g=1
}
$$

followed by

$$
\boxed{
\text{rank-one Mordell-Weil arithmetic}.
}
$$

This route is completely explicit.

For

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

the final rank-one target is the previously certified curve

$$
E_0.
$$

---

# 29. Comparison with the historical ladder

The historical route used

$$
\boxed{
5
\longrightarrow
3
\longrightarrow
2
\longrightarrow
1.
}
$$

The new R7 route is

$$
\boxed{
5
\longrightarrow
2
\longrightarrow
1.
}
$$

Neither dominates the other in every arithmetic task.

The historical route exposes

$$
E_-.
$$

The new route exposes

$$
E_f.
$$

Both share

$$
E_0.
$$

Therefore the strongest future sieve should use both.

---

# 30. A two-window fixed-slice architecture

For the historical slice define

$$
\mathcal W_{\rm new}
=
G_{7/32},
$$

and

$$
\mathcal W_{\rm old}
=
H_2.
$$

Then

$$
\boxed{
\operatorname{Jac}(\mathcal W_{\rm new})
\sim
E_f\times E_0,
}
$$

while

$$
\boxed{
\operatorname{Jac}(\mathcal W_{\rm old})
\sim
E_0\times E_-.
}
$$

A hypothetical Perfect Cuboid point must survive both windows.

Thus the natural compatibility object is no longer only one genus-$2$ curve.

It is the pair

$$
\boxed{
(\mathcal W_{\rm new},\mathcal W_{\rm old})
}
$$

synchronized through

$$
E_0.
$$

---

# 31. Potential shared-$E_0$ fiber product

Define the schematic compatibility object

$$
\boxed{
\mathcal S
=
\mathcal W_{\rm new}
\times_{E_0}
\mathcal W_{\rm old}.
}
$$

The precise fiber product depends on the exact normalization of the two maps into

$$
E_0.
$$

R7 does not yet claim a genus or rational-point classification for

$$
\mathcal S.
$$

The point is methodological:

> the original genus-$5$ compatibility problem can now be attacked through two degree-$2$ genus-$2$ covers sharing one rank-one elliptic target.

This is a smaller and more structured synchronization problem.

---

# 32. Proof-safe status of the historical slice

The earlier certified facts remain unchanged:

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

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

and

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

Also:

$$
\operatorname{rank}
\operatorname{Jac}(C_{7/32})(\mathbb Q)
=
5.
$$

R7 adds no existence or nonexistence theorem for the Perfect Cuboid itself.

It adds a new exact quotient architecture.

---

# 33. R7 canonical theorem package

For generic physical rational

$$
0<n<1,
$$

with

$$
N=n^2,
$$

R7 establishes:

### Theorem A — Hyperelliptic normal form

$$
\boxed{
G_n
\sim_{\mathbb Q}
\mathcal G_N:
\quad
H^2
=
2
\left[
t^4-(N-1)^4
\right]
\left[
(N^2+6N+1)t^2-(N-1)^4
\right].
}
$$

### Theorem B — First elliptic quotient

$$
\boxed{
\mathcal G_N/\iota_0
\cong_{\mathbb Q}
E_0'(N).
}
$$

### Theorem C — Second elliptic quotient

$$
\boxed{
\mathcal G_N/\iota_f
\cong_{\mathbb Q}
E_f(N).
}
$$

### Theorem D — Bielliptic Jacobian split

$$
\boxed{
\operatorname{Jac}(\mathcal G_N)
\sim_{\mathbb Q}
E_0(N)\times E_f(N).
}
$$

### Theorem E — Historical specialization

At

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

the first quotient maps directly to

$$
\boxed{
Y^2
=
X(X-448^2)(X+1073^2).
}
$$

---

# 34. Proposed R8 program

R8 should focus on synchronization rather than deriving another independent quotient.

## R8-A — Exact common-parent $E_0$ correspondence

Start from the original fixed genus-$5$ function field.

Compute both maps into

$$
E_0
$$

and derive the exact relation

$$
\boxed{
P_{\rm old}
=
\Theta(P_{\rm new})
}
$$

for the induced rational map or finite correspondence

$$
\Theta
$$

on

$$
E_0.
$$

Determine whether

$$
\Theta
$$

is:

- sign;
- torsion translation;
- multiplication;
- a low-degree rational correspondence.

## R8-B — Fiber-product normalization

Construct

$$
\mathcal W_{\rm new}
\times_{E_0}
\mathcal W_{\rm old}
$$

using the exact correspondence.

Determine:

- components;
- genera;
- branch structure;
- whether the original genus-$5$ curve appears as a component or cover.

## R8-C — Dual genus-$2$ Mordell-Weil sieve

Use the certified rank-one generator of

$$
E_0(\mathbb Q).
$$

For each Mordell-Weil coefficient class, require simultaneous liftability to:

$$
\mathcal W_{\rm new}
$$

and

$$
\mathcal W_{\rm old}.
$$

Intersect local conditions before checking the genus-$5$ lift.

## R8-D — Compare square-value functions

The historical genus-$2$ route requires a square-value condition on

$$
E_0.
$$

Derive the corresponding square-value function for the new genus-$2$ route.

Study simultaneous squarehood of the two rational functions on the same rank-one Mordell-Weil group.

This may reduce the entire fixed-slice problem to a double square-value problem on one elliptic curve.

## R8-E — Generalize away from $7/32$

Derive the old-style complementary genus-$2$ window generically in

$$
N.
$$

If successful, every rational fiber would possess two complementary genus-$2$ windows

$$
E_f\times E_0
$$

and

$$
E_0\times E_-.
$$

This would produce a global two-window architecture over

$$
\mathbb Q(N).
$$

---

# 35. Final conclusion

R7 converts the new R6 genus-$2$ quotient into a simple even sextic:

$$
\boxed{
H^2
=
2
\left[
t^4-(N-1)^4
\right]
\left[
(N^2+6N+1)t^2-(N-1)^4
\right].
}
$$

Its two elliptic quotients are visible by inspection.

One is the generic full-$2$-torsion family

$$
E_0'(N),
$$

and the other is

$$
E_f(N).
$$

At

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

the first quotient maps directly to the exact historical integral model

$$
\boxed{
Y^2
=
X(X-448^2)(X+1073^2).
}
$$

The standard $2$-isogeny then lands on the same normalized

$$
E_0
$$

used by the older G2E.5 genus-$2$ route.

Therefore the new and old genus-$2$ reductions share an explicit rank-one elliptic target.

The fixed-slice Perfect Cuboid problem can now be studied through two complementary genus-$2$ windows:

$$
\boxed{
E_f\times E_0
}
$$

and

$$
\boxed{
E_0\times E_-.
}
$$

The next exact target is to synchronize those two windows through their common

$$
E_0
$$

Mordell-Weil image.

This is the canonical starting point for R8.