# Perfect Cuboid Radical-FDRS R13

## Kummer-Twist Equivalence of the New Quotient Gate and the Historical G2E.5 Elliptic-Chabauty Cover

**Series:** Perfect Cuboid Global Closure  
**Branch:** Radical-FDRS / Geometry-Coupling Branch R  
**Phase:** R13 — Quotient-Cover Identification and Lineage Merge  
**Date:** 2026-10-05  
**Status:** Exact structural identification — not a proof of Perfect Cuboid existence or nonexistence

---

# 0. Executive summary

R12 quotiented the two original square gates by the exact gate-swap torsion symmetry and obtained a linear square-value condition on the $2$-isogenous elliptic quotient.

After normalization, the quotient elliptic curve is exactly the historical rank-one curve

$$
\boxed{
E_0:
\qquad
y^2
=
x^3
-
2x^2
+
\lambda^2x,
}
$$

where

$$
\boxed{
\lambda
=
\frac{1352033}{950625}.
}
$$

The R12 linear gate becomes

$$
\boxed{
f_{\rm new}(x)
=
2x-\lambda^2.
}
$$

The historical G2E.5 genus-$2$ cover was controlled by

$$
\boxed{
f_{\rm old}(x)
=
2-x.
}
$$

At first sight these appear to be two different square-value problems on the same elliptic curve.

R13 proves the exact relationship.

Let

$$
\boxed{
T=(0,0)\in E_0(\mathbb Q)[2].
}
$$

Translation by $T$ is

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

Then

$$
\boxed{
\tau_T^\ast f_{\rm old}
=
\frac{
f_{\rm new}
}{
x
}.
}
$$

Equivalently, in the squareclass group of the elliptic function field,

$$
\boxed{
[f_{\rm new}]
=
[x]\,
[\tau_T^\ast f_{\rm old}].
}
$$

Thus the new and old genus-$2$ covers are not literally identical over $E_0$.

They differ by the Kummer twist

$$
\boxed{
[x].
}
$$

This distinction is essential.

However, the class $[x]$ is exactly the unramified quadratic Kummer class trivialized by the historical $2$-isogeny cover

$$
\boxed{
\phi:
E_0'\longrightarrow E_0.
}
$$

Indeed, on $E_0'$ one has

$$
\boxed{
\phi^\ast x=q^2
}
$$

for an explicit rational function $q$.

Therefore, after base change to $E_0'$, the two square-value covers become explicitly isomorphic.

On the pullback model define

$$
\boxed{
q^2=x.
}
$$

Then $E_0'$ admits the quartic representation

$$
\boxed{
w^2
=
q^4
-
2q^2
+
\lambda^2.
}
$$

The historical and new square lifts become

$$
\boxed{
r^2
=
2-q^2,
}
$$

and

$$
\boxed{
s^2
=
2q^2-\lambda^2.
}
$$

The explicit torsion-chart transformation

$$
\boxed{
q'
=
\frac{\lambda}{q},
}
$$

$$
\boxed{
w'
=
-\frac{\lambda w}{q^2},
}
$$

$$
\boxed{
r'
=
\frac{s}{q}
}
$$

maps the new pullback cover to the old pullback cover.

Its inverse is

$$
\boxed{
q
=
\frac{\lambda}{q'},
}
$$

$$
\boxed{
w
=
-\frac{\lambda w'}{q'^2},
}
$$

$$
\boxed{
s
=
\frac{\lambda r'}{q'}.
}
$$

Therefore the two previously separate research lines are now formally merged:

$$
\boxed{
\text{R12 new quotient cover}
}
$$

and

$$
\boxed{
\text{G2E.5 old elliptic-Chabauty cover}
}
$$

are two $2$-torsion/Kummer charts of the same Perfect-Cuboid lift obstruction after pullback to $E_0'$.

The correct conclusion is not that the covers are already identical over $E_0$.

The correct conclusion is:

$$
\boxed{
\text{they differ by the isogeny Kummer class }[x],
}
$$

and:

$$
\boxed{
\text{that twist becomes trivial on }E_0'.
}
$$

Thus no independent new elliptic-Chabauty problem was created by R12.

The preferred future arithmetic route is to work with one canonical cover family and use the torsion chart only when it improves local or computational behavior.

---

# 1. Historical normalized elliptic curve

The historical G2E.5 elliptic curve is

$$
\boxed{
E_0:
\qquad
y^2
=
x^3
-
2x^2
+
\lambda^2x.
}
$$

Here

$$
\boxed{
\lambda
=
\frac{
1352033
}{
950625
}.
}
$$

Equivalently,

$$
E_0:
\qquad
y^2
=
x
\left(
x^2-2x+\lambda^2
\right).
$$

The point

$$
\boxed{
T=(0,0)
}
$$

is the unique nontrivial rational $2$-torsion point on this normalized model.

---

# 2. Historical G2E.5 square-value cover

The old genus-$2$ route reduced the fixed-slice rational-point problem to:

$$
\boxed{
P\in E_0(\mathbb Q)
}
$$

such that

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

Define

$$
\boxed{
f_{\rm old}
=
2-x.
}
$$

The corresponding degree-$2$ cover is

$$
\boxed{
\mathcal H_{\rm old}:
\qquad
r^2
=
2-x.
}
$$

Its normalization is the historical genus-$2$ curve used for the elliptic-Chabauty / square-value program.

---

# 3. R12 quotient gate

R12 worked on the unnormalized $2$-isogenous quotient

$$
\widehat E:
\qquad
V^2
=
U^3
-
2MU^2
+
L^2U,
$$

with

$$
M=950625,
$$

and

$$
L=1352033.
$$

The quotient square gate was

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

Normalize by

$$
\boxed{
U=Mx.
}
$$

Because

$$
M=975^2
$$

is itself a rational square,

$$
2MU-L^2
=
M^2
\left(
2x-\frac{L^2}{M^2}
\right).
$$

Thus squarehood is equivalent to

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

Define

$$
\boxed{
f_{\rm new}
=
2x-\lambda^2.
}
$$

The R12 cover is therefore

$$
\boxed{
\mathcal H_{\rm new}:
\qquad
s^2
=
2x-\lambda^2.
}
$$

---

# 4. Translation by the rational $2$-torsion point

For a curve

$$
y^2=x^3+ax^2+bx
$$

with rational $2$-torsion point

$$
(0,0),
$$

translation by that point sends

$$
x
\longmapsto
\frac{b}{x}.
$$

For

$$
E_0,
$$

we have

$$
b=\lambda^2.
$$

Hence

$$
\boxed{
\tau_T:
(x,y)
\longmapsto
\left(
\frac{\lambda^2}{x},
-
\frac{\lambda^2y}{x^2}
\right).
}
$$

Direct substitution verifies that

$$
\tau_T^2=1.
$$

---

# 5. Transformation of the old square function

Evaluate the old function after translation:

$$
\tau_T^\ast f_{\rm old}
=
2-\frac{\lambda^2}{x}.
$$

Therefore

$$
\boxed{
\tau_T^\ast f_{\rm old}
=
\frac{
2x-\lambda^2
}{
x
}.
}
$$

Since

$$
f_{\rm new}=2x-\lambda^2,
$$

we obtain

$$
\boxed{
f_{\rm new}
=
x\,
\tau_T^\ast f_{\rm old}.
}
$$

Passing to squareclasses:

$$
\boxed{
[f_{\rm new}]
=
[x]\,
[\tau_T^\ast f_{\rm old}].
}
$$

This is the canonical R13 identity.

---

# 6. The precise relationship of the two covers

The old and new covers are therefore related by:

1. translation of the elliptic base by:
   $$
   T=(0,0);
   $$

2. multiplication of the quadratic-cover squareclass by:
   $$
   [x].
   $$

Thus they are not literally the same quadratic extension of

$$
\mathbb Q(E_0).
$$

They differ by a Kummer twist.

The twist is exactly:

$$
\boxed{
[x].
}
$$

This corrects the overly strong informal statement that the two covers are already identical on $E_0$.

---

# 7. Divisor of the Kummer twist

On

$$
E_0,
$$

the rational function

$$
x
$$

has divisor

$$
\boxed{
\operatorname{div}(x)
=
2(T)-2(O),
}
$$

where

$$
O
$$

is the elliptic identity.

Thus the divisor of

$$
x
$$

is even.

Therefore adjoining

$$
\sqrt{x}
$$

defines an unramified quadratic extension of the elliptic function field.

This is precisely the function-field form of a degree-$2$ isogeny cover.

---

# 8. The historical $2$-isogeny cover

Return to the full-$2$-torsion model

$$
\boxed{
E_0':
\qquad
Y^2
=
X(X-A)(X+B),
}
$$

with

$$
A=448^2,
$$

$$
B=1073^2.
$$

Let

$$
M=B-A=975^2.
$$

The standard degree-$2$ isogeny is

$$
\phi:
E_0'
\longrightarrow
E_0.
$$

On the unnormalized target define

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

Then normalize

$$
\boxed{
x
=
\frac{U}{M}.
}
$$

Therefore

$$
\boxed{
x
=
\left(
\frac{Y}{975X}
\right)^2.
}
$$

Define

$$
\boxed{
q
=
\frac{Y}{975X}.
}
$$

Then

$$
\boxed{
\phi^\ast x=q^2.
}
$$

Hence the Kummer class

$$
[x]
$$

becomes trivial on

$$
E_0'.
$$

This is exactly why the old and new covers merge after pullback.

---

# 9. The isogeny cover as a Kummer trivialization

At the level of function fields,

$$
\boxed{
\mathbb Q(E_0')
=
\mathbb Q(E_0)
\left(
\sqrt{x}
\right)
}
$$

up to the standard choice of normalized isogeny model.

Therefore:

$$
\boxed{
[x]
=
1
}
$$

inside

$$
\mathbb Q(E_0')^\times/
\mathbb Q(E_0')^{\times2}.
$$

Pull back the R13 squareclass identity:

$$
[f_{\rm new}]
=
[x]
[\tau_T^\ast f_{\rm old}].
$$

On

$$
E_0',
$$

this becomes

$$
\boxed{
[\phi^\ast f_{\rm new}]
=
[
\phi^\ast\tau_T^\ast f_{\rm old}
].
}
$$

Thus the two quadratic covers become isomorphic after the isogeny base change.

---

# 10. Quartic model of the isogeny pullback

Because

$$
x=q^2,
$$

substitute into

$$
E_0:
\qquad
y^2
=
x^3-2x^2+\lambda^2x.
$$

Then

$$
y^2
=
q^2
\left(
q^4-2q^2+\lambda^2
\right).
$$

Define

$$
\boxed{
w
=
\frac{y}{q}.
}
$$

Then

$$
\boxed{
w^2
=
q^4
-
2q^2
+
\lambda^2.
}
$$

This is a genus-$1$ quartic model of the isogeny pullback.

It is convenient because the torsion translation becomes a simple reciprocal transformation in

$$
q.
$$

---

# 11. Lift of the torsion translation

Since

$$
x=q^2,
$$

the base translation

$$
x
\mapsto
\frac{\lambda^2}{x}
$$

becomes

$$
\boxed{
q
\mapsto
\frac{\lambda}{q}.
}
$$

The quartic equation is preserved by

$$
\boxed{
(q,w)
\longmapsto
\left(
\frac{\lambda}{q},
-
\frac{\lambda w}{q^2}
\right).
}
$$

Indeed:

$$
\left(
\frac{\lambda w}{q^2}
\right)^2
=
\left(
\frac{\lambda}{q}
\right)^4
-
2
\left(
\frac{\lambda}{q}
\right)^2
+
\lambda^2.
$$

Thus the torsion translation has become an explicit reciprocal involution of the pullback quartic.

---

# 12. Old square gate in $q$ coordinates

The historical gate is

$$
r^2
=
2-x.
$$

Since

$$
x=q^2,
$$

it becomes

$$
\boxed{
r^2
=
2-q^2.
}
$$

Define the old pullback cover:

$$
\boxed{
\widetilde{\mathcal H}_{\rm old}:
}
$$

$$
w^2
=
q^4-2q^2+\lambda^2,
$$

$$
r^2
=
2-q^2.
$$

---

# 13. New square gate in $q$ coordinates

The R12/R13 gate is

$$
s^2
=
2x-\lambda^2.
$$

Thus:

$$
\boxed{
s^2
=
2q^2-\lambda^2.
}
$$

Define the new pullback cover:

$$
\boxed{
\widetilde{\mathcal H}_{\rm new}:
}
$$

$$
w^2
=
q^4-2q^2+\lambda^2,
$$

$$
s^2
=
2q^2-\lambda^2.
$$

---

# 14. Explicit pullback-cover isomorphism

Define

$$
\boxed{
q'
=
\frac{\lambda}{q},
}
$$

$$
\boxed{
w'
=
-\frac{\lambda w}{q^2},
}
$$

and

$$
\boxed{
r'
=
\frac{s}{q}.
}
$$

Then:

$$
w'^2
=
q'^4
-
2q'^2
+
\lambda^2.
$$

Also:

$$
r'^2
=
\frac{
2q^2-\lambda^2
}{
q^2
}
=
2-\frac{\lambda^2}{q^2}.
$$

Since:

$$
q'^2
=
\frac{\lambda^2}{q^2},
$$

we obtain:

$$
\boxed{
r'^2
=
2-q'^2.
}
$$

Therefore:

$$
\boxed{
(q,w,s)
\longmapsto
\left(
\frac{\lambda}{q},
-\frac{\lambda w}{q^2},
\frac{s}{q}
\right)
}
$$

maps:

$$
\widetilde{\mathcal H}_{\rm new}
$$

to:

$$
\widetilde{\mathcal H}_{\rm old}.
$$

---

# 15. Explicit inverse

Starting with a point:

$$
(q',w',r')
$$

on the old pullback cover, define:

$$
\boxed{
q
=
\frac{\lambda}{q'},
}
$$

$$
\boxed{
w
=
-\frac{\lambda w'}{q'^2},
}
$$

and

$$
\boxed{
s
=
\frac{\lambda r'}{q'}.
}
$$

Then:

$$
s^2
=
2q^2-\lambda^2.
$$

Composing the two transformations gives the identity.

Therefore:

$$
\boxed{
\widetilde{\mathcal H}_{\rm new}
\cong_{\mathbb Q}
\widetilde{\mathcal H}_{\rm old}.
}
$$

This is the strongest exact equivalence statement available.

---

# 16. Pointwise rational-lift equivalence on the isogeny image

Let:

$$
P\in E_0(\mathbb Q)
$$

lie in the rational image of:

$$
\phi:
E_0'(\mathbb Q)\to E_0(\mathbb Q).
$$

Then:

$$
x(P)=q^2
$$

for some:

$$
q\in\mathbb Q.
$$

Let:

$$
P_T=P+T.
$$

Then:

$$
x(P_T)
=
\frac{\lambda^2}{q^2}
=
\left(
\frac{\lambda}{q}
\right)^2.
$$

Moreover:

$$
\boxed{
f_{\rm old}(P_T)
=
\frac{
f_{\rm new}(P)
}{
q^2
}.
}
$$

Hence:

$$
\boxed{
f_{\rm new}(P)\in\mathbb Q^2
\iff
f_{\rm old}(P_T)\in\mathbb Q^2.
}
$$

Thus the new and old rational lift conditions are equivalent after torsion translation on the isogeny image.

---

# 17. Branch-point relation

The old cover branches over the two elliptic points satisfying

$$
\boxed{
x=2.
}
$$

The new cover branches over the two elliptic points satisfying

$$
\boxed{
x=\frac{\lambda^2}{2}.
}
$$

Under:

$$
\tau_T:
x\mapsto\frac{\lambda^2}{x},
$$

we have:

$$
\boxed{
2
\longmapsto
\frac{\lambda^2}{2}.
}
$$

Thus the branch divisors are exchanged by the rational $2$-torsion translation.

The remaining difference between the covers is the unramified Kummer twist:

$$
[x].
$$

---

# 18. Why the twist is unramified

Because

$$
\operatorname{div}(x)
=
2(T)-2(O),
$$

the class

$$
[x]
$$

has even divisor.

Therefore multiplying a quadratic-cover function by

$$
x
$$

does not introduce new branch points.

It changes only the unramified quadratic character.

Thus the old and new genus-$2$ covers have the same translated branch geometry but differ by the elliptic $2$-isogeny character.

This is the correct geometric description.

---

# 19. Relation to the R12 linear gate

R12 obtained the unnormalized quotient gate

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

Using:

$$
U=Mx,
$$

we have:

$$
2MU-L^2
=
M^2
\left(
2x-\lambda^2
\right).
$$

Since:

$$
M^2
$$

is a square,

$$
\boxed{
[
2MU-L^2
]
=
[
f_{\rm new}
].
}
$$

Thus the R12 quotient gate is exactly the new torsion chart described in R13.

No additional obstruction remains hidden in the scaling.

---

# 20. Relation to the historical G2E.5 gate

G2E.5 used:

$$
\boxed{
f_{\rm old}=2-x.
}
$$

R13 proves:

$$
\boxed{
[f_{\rm new}]
=
[x]
[
\tau_T^\ast f_{\rm old}
].
}
$$

Therefore the R12 gate and the G2E.5 gate are not separate research problems.

They are related by:

$$
\boxed{
\text{torsion translation}
+
\text{isogeny Kummer twist}.
}
$$

This formally merges the new quotient line with the historical elliptic-Chabauty line.

---

# 21. Why the distinction matters

If one incorrectly states that:

$$
f_{\rm new}
$$

and:

$$
\tau_T^\ast f_{\rm old}
$$

have the same squareclass already on:

$$
E_0,
$$

one drops the nontrivial class:

$$
[x].
$$

That would be false in general.

The correct identity is:

$$
[f_{\rm new}]
=
[x]
[
\tau_T^\ast f_{\rm old}
].
$$

Only after pullback to:

$$
E_0'
$$

does:

$$
[x]
$$

become trivial.

R13 therefore preserves the proof-safe distinction between:

- equivalence over:
  $$
  E_0;
  $$

- equivalence after the:
  $$
  2
  $$
  -isogeny base change.

---

# 22. Kummer interpretation

The class:

$$
[x]
\in
\mathbb Q(E_0)^\times/
\mathbb Q(E_0)^{\times2}
$$

is the Kummer class associated with the rational $2$-torsion point:

$$
T=(0,0).
$$

The isogeny:

$$
E_0'\to E_0
$$

is precisely the unramified quadratic cover that kills this class.

Therefore the R13 identity can be read as:

> the new square-value cover is the torsion-translated old cover twisted by the kernel Kummer character of the historical $2$-isogeny.

This is the conceptual unification.

---

# 23. Function-field diagram

The relevant fields may be organized as:

$$
\mathbb Q(E_0)
$$

with the Kummer extension:

$$
\boxed{
\mathbb Q(E_0')
=
\mathbb Q(E_0)(\sqrt{x}).
}
$$

The old cover is:

$$
\boxed{
\mathbb Q(E_0)
\left(
\sqrt{2-x}
\right).
}
$$

The new cover is:

$$
\boxed{
\mathbb Q(E_0)
\left(
\sqrt{2x-\lambda^2}
\right).
}
$$

After translating the old cover by:

$$
T,
$$

its square function becomes:

$$
\frac{
2x-\lambda^2
}{
x
}.
$$

After adjoining:

$$
\sqrt{x},
$$

the denominator squareclass disappears.

Thus the two translated extensions become isomorphic over:

$$
\mathbb Q(E_0').
$$

---

# 24. Perfect-Cuboid interpretation

The original Perfect-Cuboid fixed-fiber construction already passes through the $2$-isogeny geometry.

Therefore the relevant rational lift points are not arbitrary unrelated points of:

$$
E_0.
$$

They arise together with the isogeny/Kummer data encoded by:

$$
E_0'.
$$

On this lifted geometry:

$$
\boxed{
\text{old gate}
}
$$

and:

$$
\boxed{
\text{new gate}
}
$$

are two torsion charts of the same arithmetic obstruction.

Thus maintaining both as independent proof branches would duplicate work.

---

# 25. Canonical future cover choice

For future arithmetic work one may choose either chart.

## Historical chart

$$
\boxed{
r^2=2-x.
}
$$

Advantages:

- already connected to G2E.5;
- existing bielliptic transformations;
- existing Mordell-Weil square-value formulation;
- existing boundary anchors.

## New quotient chart

$$
\boxed{
s^2=2x-\lambda^2.
}
$$

Advantages:

- arises directly from the R12 gate-swap quotient;
- linear in the quotient coordinate;
- naturally aligned with the product-gate compression.

Because the two pullbacks are isomorphic over:

$$
E_0',
$$

the choice is computational rather than conceptual.

---

# 26. Preferred canonical convention

R13 recommends keeping:

$$
\boxed{
f_{\rm old}=2-x
}
$$

as the canonical historical elliptic-Chabauty gate because the old G2E.5 infrastructure is already built around it.

The R12 gate:

$$
f_{\rm new}=2x-\lambda^2
$$

should be retained as the torsion-dual chart.

Whenever one chart is inconvenient locally, pass to the other using:

$$
x\mapsto\frac{\lambda^2}{x}.
$$

The Kummer factor:

$$
[x]
$$

must be tracked unless the calculation has already been pulled back to:

$$
E_0'.
$$

---

# 27. Boundary anchors in the two charts

Suppose:

$$
x=1
$$

is the historical boundary anchor.

Then:

$$
f_{\rm old}(1)=1.
$$

Under torsion translation:

$$
x
\mapsto
\lambda^2.
$$

At that point:

$$
f_{\rm new}(\lambda^2)
=
\lambda^2.
$$

Since:

$$
\lambda^2
$$

is a rational square, the boundary square condition is preserved.

Thus boundary anchors move naturally between the two torsion charts.

This is another consistency check of the identification.

---

# 28. Involution on the square-root coordinate

On the isogeny pullback, the reciprocal transformation is:

$$
\boxed{
q
\mapsto
\frac{\lambda}{q}.
}
$$

The old and new gate roots transform as:

$$
\boxed{
r'
=
\frac{s}{q},
}
$$

and inversely:

$$
\boxed{
s
=
\frac{\lambda r'}{q'}.
}
$$

Thus gate-root rationality is preserved exactly.

The transformation exchanges a small-$q$ region with a large-$q$ region and may therefore be useful for choosing numerically or $p$-adically favorable charts.

---

# 29. Relation to Radical-FDRS reciprocal structure

The map:

$$
q\mapsto\frac{\lambda}{q}
$$

is another manifestation of the reciprocal involutions that have appeared throughout the Radical-FDRS program.

Earlier stages produced:

- reciprocal quartics;
- reciprocal genus-$3$ curves;
- Cayley involutions;
- $m\mapsto1/m$ symmetries;
- gate-swap Möbius maps.

R13 shows that even the final elliptic-Chabauty square-value cover retains the same reciprocal architecture.

This strengthens the interpretation that reciprocal symmetry is intrinsic to the flattened Perfect-Cuboid state, not an artifact of one coordinate system.

---

# 30. No new independent genus-$2$ obstruction

The central strategic conclusion is:

$$
\boxed{
\text{R12 did not create a new independent genus-2 obstruction.}
}
$$

It rediscovered the historical G2E.5 obstruction in a torsion-dual Kummer chart.

Therefore future work should not attempt to classify the two covers separately from scratch.

Their arithmetic information must be shared.

---

# 31. Merged research lineage

The old and new paths now merge as:

$$
\boxed{
\text{G2E.5}
}
$$

$$
\longrightarrow
$$

$$
\boxed{
E_0:
\quad
r^2=2-x
}
$$

and:

$$
\boxed{
\text{R12}
}
$$

$$
\longrightarrow
$$

$$
\boxed{
E_0:
\quad
s^2=2x-\lambda^2.
}
$$

The bridge is:

$$
\boxed{
\tau_T
+
[x].
}
$$

After:

$$
E_0'\to E_0,
$$

the bridge becomes an explicit isomorphism.

Thus the two lineages are formally unified.

---

# 32. Implication for the R9-R12 finite-prime sieve

The R9-R12 sieve was performed naturally on:

$$
E_0',
$$

where the Kummer class:

$$
[x]
$$

is already trivial.

Therefore the finite-prime double-gate calculations are fully compatible with the historical G2E.5 square-value condition.

No contradictory extra condition was introduced by changing charts.

The two descriptions differ only in how the same lifted arithmetic state is presented.

---

# 33. Implication for elliptic Chabauty

The historical target was:

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

R13 shows that the new quotient target:

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

is its torsion/Kummer companion.

Thus an elliptic-Chabauty implementation can choose whichever cover:

- has simpler local splitting;
- has a better base point;
- gives smaller coefficients;
- has better reduction at the selected prime.

Results can be transported between charts after accounting for the isogeny Kummer class.

---

# 34. R13 theorem package

## Theorem A — Torsion translation

On:

$$
E_0,
$$

translation by:

$$
T=(0,0)
$$

is:

$$
\boxed{
(x,y)
\mapsto
\left(
\frac{\lambda^2}{x},
-
\frac{\lambda^2y}{x^2}
\right).
}
$$

## Theorem B — Kummer-twist identity

$$
\boxed{
[f_{\rm new}]
=
[x]
[
\tau_T^\ast f_{\rm old}
].
}
$$

## Theorem C — Isogeny trivialization

For:

$$
\phi:E_0'\to E_0,
$$

$$
\boxed{
\phi^\ast x=q^2.
}
$$

Hence:

$$
\boxed{
[\phi^\ast f_{\rm new}]
=
[
\phi^\ast\tau_T^\ast f_{\rm old}
].
}
$$

## Theorem D — Explicit pullback-cover isomorphism

$$
\boxed{
(q,w,s)
\mapsto
\left(
\frac{\lambda}{q},
-\frac{\lambda w}{q^2},
\frac{s}{q}
\right)
}
$$

maps the new pullback cover to the old pullback cover.

---

# 35. Proof-safety boundary

R13 does not claim that:

$$
\mathcal H_{\rm old}
$$

and:

$$
\mathcal H_{\rm new}
$$

are already $\mathbb Q$-isomorphic as covers of:

$$
E_0.
$$

The exact statement is:

$$
\boxed{
\text{they differ by the nontrivial Kummer class }[x].
}
$$

They become isomorphic after pullback to the $2$-isogeny cover:

$$
E_0'.
$$

This distinction must be preserved in any formal proof.

---

# 36. Proposed R14 program

R14 should stop duplicating the old/new genus-$2$ arithmetic and use the merged cover structure.

## R14-A — Choose the best elliptic-Chabauty chart

Compare:

$$
r^2=2-x
$$

and:

$$
s^2=2x-\lambda^2
$$

at useful primes.

Select the chart with simpler local residue disks and smaller numerical coefficient growth.

## R14-B — Transport R9-R12 residue classes

Map the surviving:

$$
E_0'
$$

Mordell-Weil residue classes into the chosen:

$$
E_0
$$

cover.

Identify exactly which genus-$2$ residue disks remain.

## R14-C — Elliptic-Chabauty / Mordell-Weil sieve

Use the certified rank-one:

$$
E_0(\mathbb Q)
$$

and classify rational lifts on the chosen genus-$2$ cover.

The R9-R12 finite-prime conditions should be imposed from the start to reduce the search to the boundary tubes.

## R14-D — Dual-chart cross-check

Any surviving candidate in one chart must map to a candidate in the torsion-dual chart after the Kummer/isogeny transformation.

Use this as an independent exact consistency test.

## R14-E — Fixed-slice theorem target

The preferred theorem remains:

> every rational lift satisfying the complete fixed-slice compatibility is boundary.

If proved, then:

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

contains no nondegenerate Perfect Cuboid point.

---

# 37. Final conclusion

R13 identifies the exact relationship between the R12 quotient square gate and the historical G2E.5 elliptic-Chabauty gate.

The old function is:

$$
\boxed{
f_{\rm old}=2-x.
}
$$

The new function is:

$$
\boxed{
f_{\rm new}=2x-\lambda^2.
}
$$

They satisfy:

$$
\boxed{
[f_{\rm new}]
=
[x]
[
\tau_T^\ast f_{\rm old}
].
}
$$

The factor:

$$
[x]
$$

is the Kummer class of the rational $2$-torsion point:

$$
T=(0,0).
$$

The historical $2$-isogeny:

$$
E_0'\to E_0
$$

trivializes precisely this class because:

$$
x=q^2
$$

on the pullback.

Therefore the two genus-$2$ square-value problems become explicitly isomorphic on the isogeny cover.

R12 and G2E.5 are not independent obstructions.

They are two torsion/Kummer charts of the same Perfect-Cuboid lift condition.

This formally merges the Radical-FDRS quotient route with the historical elliptic-Chabauty route.

That is the canonical starting point for R14.