# Perfect Cuboid G2B — Elliptic Orbit Generation and Local Mordell-Weil Sieve

## Fixed slice $k=7/32$

This phase replaces rational-denominator enumeration by exact elliptic-curve generation.

The first square gate is

$$
q^2
=
F_k(h)
=
\frac{h^2+k^2}{1+k^2h^2}.
$$

For

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

the gate is birational to

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

The current seed

$$
h=\frac47,
\qquad
q=\frac{17}{28}
$$

maps to

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

---

# 1. Exact torsion structure

The curve contains the three rational $2$-torsion points

$$
(2,0),
$$

$$
(-2,0),
$$

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

It also contains

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

with

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

and therefore

$$
4T_4=O.
$$

Together with $(-2,0)$ this gives eight distinct rational torsion points.

At the good primes $11$ and $17$,

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

and

$$
\#E(\mathbb F_{17})=16.
$$

Rational torsion injects into prime-to-$p$ good reduction. Combining the two reductions forces the rational torsion order to divide

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

Since eight rational torsion points have already been exhibited,

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

The seed point $P$ is not a torsion point.

Therefore

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

No claim is made here that the rank is exactly one.

---

# 2. The bounded-scan points are one elliptic orbit pattern

Let $Q$ denote the point corresponding to

$$
h=\frac4{33}.
$$

Exact elliptic addition gives

$$
P+Q=T_4.
$$

Thus

$$
Q=T_4-P.
$$

The four Gate-1 points found by denominator scanning up to $20{,}000$ are generated from $P$ and torsion.

Up to the square-gate symmetries

$$
h\mapsto-h
$$

and

$$
h\mapsto\frac1h,
$$

the two canonical sequences are

$$
H_n
=
\operatorname{can}(h(nP)),
$$

and

$$
J_n
=
\operatorname{can}(h(T_4-nP)).
$$

Their first values are:

$$
H_1=\frac47,
\qquad
J_1=\frac4{33},
$$

$$
H_2=\frac{33}{119},
\qquad
J_2=\frac{256}{817}.
$$

These are exactly the four points found by the previous denominator sweep.

The next values already jump beyond that search region:

$$
H_3=\frac{19796}{135293},
$$

$$
J_3=\frac{191869}{373428}.
$$

Thus elliptic generation escapes a denominator-bounded search almost immediately.

---

# 3. Exact orbit generation to index $100$

All points

$$
nP+T
$$

were generated exactly for

$$
0\le n\le100
$$

and all eight torsion points

$$
T\in E(\mathbb Q)_{\mathrm{tors}}.
$$

After quotienting the gate-preserving symmetries

$$
h\mapsto\pm h^{\pm1},
$$

this produced

$$
\boxed{
200
}
$$

distinct positive canonical Gate-1 states.

The largest denominator among these states has more than five thousand decimal digits.

Every one was checked against the second square gate using exact integer square tests.

Result:

$$
\boxed{
0
\text{ second-gate hits}.
}
$$

This is not a proof for the full elliptic curve.

It is an exact finite exclusion on the subgroup generated by the known non-torsion point $P$ and rational torsion.

---

# 4. Why group generation is fundamentally different from denominator enumeration

The previous fixed-$k$ scan tested every reduced rational

$$
h=\frac uv
$$

with denominator

$$
v\le20{,}000.
$$

Elliptic multiplication does not increase arithmetic height by walking through all smaller denominators.

Instead, the height of

$$
nP
$$

grows roughly quadratically in $n$.

Consequently, a few dozen elliptic additions already produce rational states whose numerators and denominators contain hundreds or thousands of digits.

This is precisely the desired transition:

$$
\boxed{
\text{height enumeration}
\rightarrow
\text{group-structured navigation}.
}
$$

---

# 5. Local square obstruction on the Mordell-Weil orbit

Suppose the second gate is a rational square at a point in one of the two canonical orbit families.

For every good prime $p$ at which the rational function is defined, its reduction must be a quadratic residue in

$$
\mathbb F_p.
$$

The sequence

$$
nP
$$

is periodic modulo $p$ with period

$$
\operatorname{ord}(P\bmod p).
$$

Therefore each prime gives a finite allowed set of congruence classes for $n$.

Poles are always retained and are never used as exclusions.

The resulting sieve is therefore proof-safe.

---

# 6. First canonical family

For

$$
H_n=\operatorname{can}(h(nP)),
$$

combining exact local square conditions at good primes not exceeding $5000$ gives the necessary divisibility condition

$$
\boxed{
19{,}070{,}251{,}200
\mid n.
}
$$

Thus no positive index

$$
n<
19{,}070{,}251{,}200
$$

in this known Mordell-Weil family can satisfy the second square gate.

This conclusion does not require constructing the enormous rational coordinates of those points.

---

# 7. Torsion-shift family

For

$$
J_n=\operatorname{can}(h(T_4-nP)),
$$

the same local sieve gives

$$
\boxed{
12{,}700{,}800
\mid n.
}
$$

Therefore no positive index

$$
n<
12{,}700{,}800
$$

in this torsion-shift family can satisfy the second square gate.

Again this is an exact local obstruction on the known subgroup, not a statement about all rational points on $E$.

---

# 8. Scope of the result

The free rank of

$$
E(\mathbb Q)
$$

has not yet been certified.

It is possible that:

- the rank is greater than one;
- $P$ does not generate the full free part;
- additional independent Gate-1 rational families exist.

Therefore the local sieve does **not** prove the fixed-$k$ Perfect Cuboid slice empty.

What it does prove is much narrower and still useful:

> Within the subgroup generated by the current seed point and rational torsion, the second-gate intersection is forced to extremely large elliptic index.

This makes brute denominator expansion on this orbit unnecessary.

---

# 9. Next exact target

The next target is no longer larger orbit index.

It is:

$$
\boxed{
\text{determine the full Mordell-Weil structure}
}
$$

and exploit the genus-$5$ double-square curve itself.

The accompanying G2C decomposition shows that its Jacobian splits, up to isogeny, into five elliptic factors.

That converts the next proof problem into a finite collection of elliptic-rank and covering computations.

---

**End of G2B**