> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# GERES P8: Arithmetic Geometry and Birational Analysis of the G2 Fiber

**G2 Fiber Arithmetic Geometry and Birational Analysis**

Version: v0.1  
Series: GERES — Global Equivalence Reconstruction and Scaling  
Phase: P8  
Nature: Genus-1 Fiber / Elliptic Model / Exact Arithmetic Analysis  
Primary seed: G2  
Canonical source of the original: UTF-8 Markdown

---

## Abstract

In bounded rational enumeration, P7 observed that:

$$
r=\frac35
$$

and:

$$
s=\frac47
$$

the two G2 coordinate fibers each had only two structural rational hits within the denominator bound:

$$
H\le500
$$

in that range.

P8 shows that this bounded sparsity is **not because the two fibers have only finitely many rational points**.

Both fibers can be put exactly into nonsingular genus-1 quartic form and transformed birationally into elliptic curves. G2 corresponds to a non-torsion rational point on each elliptic curve, so each fiber has at least infinitely many rational structural points.

The fiber with fixed:

$$
r=\frac35
$$

is:

$$
z^2
=
s^4+\frac{706}{225}s^2+1,
\qquad
z=2Cs,
$$

Its integral elliptic model:

$$
\boxed{
E_r:
Y^2
=
(X+706)(X^2-202500).
}
$$

G2 corresponds to:

$$
\boxed{
P_r=(3390,215040).
}
$$

The fiber with fixed:

$$
s=\frac47
$$

is:

$$
z^2
=
r^4+\frac{2657}{784}r^2+1,
\qquad
z=2Cr,
$$

Its integral elliptic model:

$$
\boxed{
E_s:
Y^2
=
(X+2657)(X^2-2458624).
}
$$

G2 corresponds to:

$$
\boxed{
P_s=(11032,1277640).
}
$$

Exact good-reduction checks bound the rational torsion subgroup of each curve to order $8$, realized by explicit torsion points:

$$
E_r(\mathbb Q)_{\mathrm{tors}}
\cong
\mathbb Z/2\mathbb Z
\times
\mathbb Z/4\mathbb Z,
$$

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

Neither G2 point belongs to the torsion subgroup, so:

$$
\operatorname{rank}E_r(\mathbb Q)\ge1,
$$

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

Therefore:

$$
\boxed{
\text{Both G2 structural fibers have infinitely many rational points.}
}
$$

This result revises the intuition from P7: the fibers look sparse mainly because **rational height rapidly explodes**, not because there are finitely many rational points.

But P8 also emphasizes that infinitely many structural rational points do not imply the existence of a Perfect Cuboid at all. The final open-face rational-square condition is a second independent arithmetic gate.

---

# 1. General Fixed-Parameter Fiber

Fix:

$$
\rho
=
\frac ab
\in\mathbb Q_{>0},
$$

where:

$$
\gcd(a,b)=1.
$$

Let the other world parameter be:

$$
x.
$$

Structural body condition:

$$
C^2
=
H(\rho)^2+P(x)^2.
$$

where:

$$
P(u)
=
\frac{1-u^2}{2u},
$$

$$
H(u)
=
\frac{1+u^2}{2u}.
$$

---

# 2. Quartic Form

Multiplying by:

$$
4x^2,
$$

gives:

$$
(2Cx)^2
=
x^4
+
\left(
\rho^2+\rho^{-2}
\right)x^2
+
1.
$$

Define:

$$
z=2Cx,
$$

and:

$$
A_\rho
=
\rho^2+\rho^{-2}.
$$

Then:

$$
\boxed{
\mathcal Q_\rho:
z^2=x^4+A_\rho x^2+1.
}
$$

---

# 3. Why it is Genus One

The quartic polynomial:

$$
f(x)=x^4+A_\rho x^2+1
$$

has discriminant:

$$
\operatorname{Disc}(f)
=
16(A_\rho-2)^2(A_\rho+2)^2.
$$

For:

$$
0<\rho<1,
$$

we have:

$$
A_\rho>2.
$$

Therefore:

$$
\operatorname{Disc}(f)\neq0.
$$

The hyperelliptic curve:

$$
z^2=f(x)
$$

is thus a smooth genus-one quartic.

And:

$$
(x,z)=(0,1)
$$

is a rational point, so it can be regarded as an elliptic curve.

---

# 4. General Birational Transformation

Let:

$$
B
=
\frac{z+1}{x^2},
$$

$$
u=2B,
$$

$$
v=2x(B^2-1).
$$

Exact elimination then gives:

$$
\boxed{
v^2=(u+A_\rho)(u^2-4).
}
$$

---

# 5. Integral Model

If:

$$
\rho=\frac ab,
$$

define:

$$
d=ab,
$$

$$
A_0=a^4+b^4.
$$

Since:

$$
A_\rho
=
\frac{A_0}{d^2},
$$

further let:

$$
X=d^2u,
$$

$$
Y=d^3v,
$$

to obtain the integral elliptic model:

$$
\boxed{
E_\rho:
Y^2
=
(X+A_0)(X^2-4d^4).
}
$$

That is:

$$
Y^2
=
X^3
+
A_0X^2
-
4d^4X
-
4A_0d^4.
$$

---

# 6. Inverse Map

For a non-exceptional point:

$$
X^2\neq4d^4,
$$

first take:

$$
u=\frac{X}{d^2},
$$

$$
v=\frac{Y}{d^3}.
$$

Then:

$$
x
=
\frac{2v}{u^2-4}
=
\frac{2dY}{X^2-4d^4}.
$$

Further let:

$$
z
=
\frac u2x^2-1,
$$

Finally:

$$
C=\frac{z}{2x}.
$$

This is therefore not merely a projection, but a birational chart with an exact inverse, apart from finitely many exceptional points.

---

# 7. Built-In Rational $2$-Torsion

Integral model factorization:

$$
Y^2
=
(X+A_0)
(X-2d^2)
(X+2d^2).
$$

There are therefore naturally three nonzero rational $2$-torsion points:

$$
(-A_0,0),
$$

$$
(2d^2,0),
$$

$$
(-2d^2,0).
$$

---

# 8. Built-In Rational $4$-Torsion

Note:

$$
2d^2+A_0
=
(a^2+b^2)^2,
$$

and:

$$
2d^2-(-2d^2)
=
(2ab)^2.
$$

Thus:

$$
(2d^2,0)
$$

can be halved over $\mathbb Q$.

Let:

$$
R
=
2ab(a^2+b^2).
$$

Then the order-$4$ points include:

$$
X_\pm
=
2d^2\pm R,
$$

and:

$$
Y_+
=
\pm R(a+b)^2,
$$

$$
Y_-
=
\pm R(a-b)^2.
$$

The generic fiber therefore contains at least torsion of type:

$$
\mathbb Z/2\mathbb Z
\times
\mathbb Z/4\mathbb Z
$$

as a subgroup.

---

# 9. Fixed $r=3/5$ Fiber

Take:

$$
a=3,
\qquad
b=5.
$$

Then:

$$
d=15,
$$

$$
A_0=3^4+5^4=706.
$$

quartic:

$$
\boxed{
z^2
=
s^4+\frac{706}{225}s^2+1.
}
$$

integral elliptic curve:

$$
\boxed{
E_r:
Y^2
=
(X+706)(X^2-202500).
}
$$

Expanding:

$$
Y^2
=
X^3
+
706X^2
-
202500X
-
142965000.
$$

---

# 10. Fixed-$r$ Discriminant

Its elliptic discriminant:

$$
\Delta_r
=
1135012384604160000.
$$

Thus:

$$
E_r
$$

is nonsingular.

---

# 11. The position of G2 on $E_r$

G2:

$$
s=\frac47,
$$

$$
C=\frac{1073}{840}.
$$

Under the P8 birational map:

$$
\boxed{
P_r
=
(3390,215040).
}
$$

The P7 fixed-$r$ neighbor:

$$
s=\frac{11}{45},
$$

$$
C=\frac{221}{99}
$$

maps to:

$$
\boxed{
Q_r
=
(15750,2019600).
}
$$

Both lie exactly in:

$$
E_r(\mathbb Q).
$$

---

# 12. Torsion of $E_r$

Explicit torsion points:

$$
\mathcal O,
$$

$$
(-706,0),
$$

$$
(-450,0),
$$

$$
(450,0),
$$

$$
(-570,\pm4080),
$$

$$
(1470,\pm65280).
$$

A total of:

$$
8
$$

points.

---

# 13. Good-Reduction Torsion Bound for $E_r$

Choose good primes:

$$
p=7,
\qquad
p=11.
$$

Exact point counting gives:

$$
\#E_r(\mathbb F_7)=8,
$$

$$
\#E_r(\mathbb F_{11})=16.
$$

Good reduction is injective on prime-to-$p$ rational torsion.

Any $7$-primary torsion is excluded modulo $11$;

any $11$-primary torsion is excluded modulo $7$.

The remaining torsion order must be bounded simultaneously by:

$$
8
$$

and:

$$
16
$$

these orders.

Therefore:

$$
|E_r(\mathbb Q)_{\mathrm{tors}}|
\mid8.
$$

Since $8$ points were explicitly found in the preceding section:

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

---

# 14. G2 Point $P_r$ is non-Torsion

Since:

$$
P_r=(3390,215040)
$$

is not in the complete torsion list above,

we have:

$$
\boxed{
P_r
\text{ has infinite order.}
}
$$

Therefore:

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

---

# 15. The Fixed-$r$ Fiber has infinitely many Rational Structural Points

From:

$$
nP_r,
\qquad
n\in\mathbb Z
$$

we can generate infinitely many distinct rational elliptic points.

Apart from finitely many exceptional points, the inverse birational map gives infinitely many rational:

$$
(s,C).
$$

Therefore:

$$
\boxed{
r=\frac35
\text{ has a structural fiber with infinitely many rational points.}
}
$$

This directly corrects a possible misreading of the H500 sparsity in P7.

---

# 16. Height Explosion

Although there are infinitely many rational points,

the denominators in the G2 cyclic orbit rise rapidly.

$$
P_r
$$

itself gives:

$$
s=\frac47,
$$

$$
C=\frac{1073}{840}.
$$

Whereas:

$$
2P_r
$$

inverse-maps to:

$$
s=
\frac{32175}{60088},
$$

$$
C=
\frac{5082931681}{3866662800}.
$$

The denominator of the second cyclic point is thus already far beyond:

$$
H=500.
$$

This explains why bounded enumeration appears to contain only a few points.

---

# 17. The Height of $3P_r$

The third multiple even produces:

$$
s
=
\frac{1006111897}{40088532364}.
$$

Its body-ratio denominator already reaches:

$$
1210006480340698035240.
$$

Therefore:

$$
\boxed{
\text{rational point infinity}
\neq
\text{low-height density}.
}
$$

---

# 18. Fixed $s=4/7$ Fiber

Take:

$$
a=4,
\qquad
b=7.
$$

Then:

$$
d=28,
$$

$$
A_0=4^4+7^4=2657.
$$

quartic:

$$
\boxed{
z^2
=
r^4+\frac{2657}{784}r^2+1.
}
$$

integral elliptic curve:

$$
\boxed{
E_s:
Y^2
=
(X+2657)(X^2-2458624).
}
$$

Expanding:

$$
Y^2
=
X^3
+
2657X^2
-
2458624X
-
6532563968.
$$

---

# 19. Fixed-$s$ Discriminant

$$
\Delta_s
=
3331050959834357760000.
$$

Thus:

$$
E_s
$$

is nonsingular.

---

# 20. The position of G2 on $E_s$

G2:

$$
r=\frac35,
$$

$$
C=\frac{1073}{840}
$$

maps to:

$$
\boxed{
P_s
=
(11032,1277640).
}
$$

The P7 fixed-$s$ neighbor:

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

$$
C=\frac{1105}{448}
$$

maps to:

$$
\boxed{
Q_s
=
(68128,18120960).
}
$$

---

# 21. Torsion of $E_s$

Explicit torsion points:

$$
\mathcal O,
$$

$$
(-2657,0),
$$

$$
(-1568,0),
$$

$$
(1568,0),
$$

$$
(-2072,\pm32760),
$$

$$
(5208,\pm440440).
$$

Eight points in total.

---

# 22. Good-Reduction Torsion Bound for $E_s$

Take:

$$
p=17,
\qquad
p=19.
$$

Exact point counts:

$$
\#E_s(\mathbb F_{17})=16,
$$

$$
\#E_s(\mathbb F_{19})=24.
$$

Using good reduction in the same way,

the rational torsion order must divide:

$$
\gcd(16,24)=8
$$

and $17$ / $19$ primary exceptions are excluded.

Therefore:

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

---

# 23. G2 Point $P_s$ is non-Torsion

$$
P_s=(11032,1277640)
$$

is not in the complete torsion list.

Thus:

$$
\boxed{
P_s
\text{ has infinite order,}
}
$$

and:

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

The fixed-$s$ structural fiber therefore also has infinitely many rational points.

---

# 24. Fixed-$s$ Height Explosion

$$
2P_s
$$

inverse-maps to:

$$
r=
\frac{7616}{16095},
$$

$$
C=
\frac{348415681}{245159040}.
$$

This is already beyond P7's enumeration at:

$$
H=500
$$

that bound.

The third multiple:

$$
r=
\frac{176270965}{1984731507},
$$

And the denominator of $C$ reaches:

$$
19591630128268478280.
$$

---

# 25. P7's Sparse-Fiber conclusion must therefore be revised

P7's exact statement remains correct:

> There are only two structural hits in finite enumeration with $H\le500$.

But P8 can now add:

$$
\boxed{
\text{Both fibers globally have infinitely many rational points.}
}
$$

“Seeing only two points” is therefore fundamentally a:

$$
\boxed{
\text{height barrier},
}
$$

not rational-point finiteness.

---

# 26. This answers part of the question about “merely computing”

If rational point height grows rapidly with elliptic group multiples,

scanning solely by denominator:

$$
1,2,3,\ldots,H
$$

may fail to see the next group-generated point for a long time.

But the elliptic group law:

$$
P\mapsto2P,3P,\ldots
$$

can jump directly to those enormous-height points.

Therefore:

$$
\boxed{
\text{arithmetic structure can replace part of ambient enumeration.}
}
$$

This is the computational significance of P8 that matters more than simply “computing more.”

---

# 27. But this is still not a Perfect Cuboid

The structural fiber guarantees only:

- fixed face rational Pythagorean closure;
- moving face with $c$ rational Pythagorean closure;
- a rational body diagonal.

Perfect Cuboid further requires:

$$
D^2
=
P(\rho)^2+P(x)^2
$$

itself to be a rational square.

Therefore:

$$
\boxed{
\text{infinite structural rational points}
\not\Rightarrow
\text{one Perfect point exists}.
}
$$

---

# 28. The second Elliptic Condition

After fixing:

$$
\rho
$$

the open-face condition:

$$
D^2
=
P(\rho)^2+P(x)^2
$$

multiplied by:

$$
4x^2
$$

gives:

$$
(2Dx)^2
=
x^4
+
\left(
A_\rho-4
\right)x^2
+
1.
$$

The final Perfect gate is therefore itself another genus-one quartic:

$$
\boxed{
\mathcal Q_\rho^{\mathrm{open}}:
w^2
=
x^4+
(A_\rho-4)x^2+1.
}
$$

---

# 29. Twin-Elliptic Structure

The fixed-parameter Perfect Cuboid problem therefore does not involve just one elliptic curve.

It requires the same rational parameter:

$$
x
$$

to lie simultaneously on:

### Structural / Body Curve

$$
z^2=x^4+A_\rho x^2+1,
$$

and:

### Open-Face Curve

$$
w^2=x^4+(A_\rho-4)x^2+1.
$$

That is:

$$
\boxed{
\text{same }x
\text{ must pass two genus-one arithmetic gates.}
}
$$

---

# 30. G2 Fixed-$r$ Twin Curves

For:

$$
r=\frac35,
$$

structural curve:

$$
z^2
=
s^4+\frac{706}{225}s^2+1.
$$

open-face curve:

$$
\boxed{
w^2
=
s^4-\frac{194}{225}s^2+1.
}
$$

Its integral elliptic model can be written as:

$$
\boxed{
E_r^{\mathrm{open}}:
Y^2
=
(X-194)(X^2-202500).
}
$$

---

# 31. G2 Fixed-$s$ Twin Curves

For:

$$
s=\frac47,
$$

structural curve:

$$
z^2
=
r^4+\frac{2657}{784}r^2+1.
$$

open-face curve:

$$
\boxed{
w^2
=
r^4-\frac{479}{784}r^2+1.
}
$$

The corresponding integral elliptic model:

$$
\boxed{
E_s^{\mathrm{open}}:
Y^2
=
(X-479)(X^2-2458624).
}
$$

---

# 32. A Perfect Point is a shared-parameter point of the Twin Curves

A fixed-$r$ Perfect hit therefore requires the existence of:

$$
s\in\mathbb Q
$$

such that:

$$
s
$$

can be lifted simultaneously to:

$$
E_r(\mathbb Q)
$$

and:

$$
E_r^{\mathrm{open}}(\mathbb Q).
$$

The fixed-$s$ case is analogous.

This is much harder than merely finding more rational points on the structural elliptic curve.

---

# 33. Fiber Product View

The Perfect fixed fiber can be understood as the fiber product of two double covers over a shared parameter line:

$$
\boxed{
\mathcal Q_\rho
\times_{\mathbb P^1_x}
\mathcal Q_\rho^{\mathrm{open}}.
}
$$

P9 should analyze this shared cover's:

- genus;
- Jacobian;
- possible descent;
- local obstructions;
- rational points.

This may be the actual arithmetic-geometric object of the “last bit.”

---

# 34. G2 Cyclic Orbit Experiment

P8 exact inverse-maps all of:

$$
nP_r,
\qquad
n=1,\ldots,8
$$

and:

$$
nP_s,
\qquad
n=1,\ldots,8
$$

back into structural worlds.

None of the first eight points of either cyclic orbit passed the Perfect gate:

$$
N_{\mathrm{perfect}}=0.
$$

This is only a finite subgroup-orbit experiment, not a global no-hit result.

---

# 35. Height Growth is the new main computational problem

P7's denominator enumeration was:

$$
H\le500.
$$

P8 shows that the elliptic group law can directly generate rational points with denominators on the order of:

$$
10^4,
10^9,
10^{18},
\ldots
$$

these magnitudes.

Future search complexity therefore should not be measured only by:

$$
H
$$

this bound.

Canonical / logarithmic height should also be used:

$$
\hat h(P)
$$

or at least:

$$
h(X(P)).
$$

---

# 36. World-Model Interpretation

P7's world fiber now has a more precise meaning:

$$
\boxed{
\text{structural world fiber}
\cong
\text{elliptic arithmetic orbit space}
}
$$

in the birational sense.

World deformation is no longer merely:

$$
x\rightarrow x+\delta.
$$

It can directly use the elliptic group operation:

$$
P,Q
\mapsto
P+Q.
$$

---

# 37. Elliptic Move as a Global World Move

Given an elliptic point:

$$
P_E,
$$

first perform:

$$
P_E'
=
P_E+Q_E,
$$

then the inverse birational map:

$$
P_E'
\rightarrow
(x',C'),
$$

Finally, synchronously reconstruct with the world generator:

- edge;
- face diagonal;
- body diagonal;
- closure state.

This is a genuinely arithmetic-aware form of CPGD.

---

# 38. This is closer to the goal than Decimal Gradient

A decimal gradient can know only:

$$
\Delta x.
$$

The elliptic group law instead guarantees that a move remains on the exact rational structural fiber:

$$
S=0.
$$

Thus:

$$
\boxed{
\text{group law move}
\gg
\text{floating local perturbation}
}
$$

in the sense of exact-structure preservation.

This is a conclusion about semantic strength, not speed.

---

# 39. Neighbor Points need not be small multiples of G2

The low-height neighbors found in P7:

$$
Q_r,
Q_s
$$

are not among the simple cases of the small multiples checked so far:

$$
nP_r,
nP_s
$$

in those lists.

The structural fiber may therefore have:

- higher rank;
- other independent generators;
- directions beyond the G2 generator plus torsion.

P8 has not computed the full Mordell-Weil rank or basis.

---

# 40. Rank Claim Boundary

At present, only the following are safely proved:

$$
\operatorname{rank}E_r(\mathbb Q)\ge1,
$$

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

One cannot claim:

$$
\operatorname{rank}=1.
$$

Nor can one claim that $P_r$ or $P_s$ is a generator.

---

# 41. P8 Exact Artifacts

This bundle contains:

- `p8_elliptic_fiber.py`
- `p8_elliptic_summary.json`
- `p8_fixed_r_g2_orbit_first8.csv`
- `p8_fixed_s_g2_orbit_first8.csv`

The following can be replayed exactly:

- birational map;
- inverse map;
- elliptic equations;
- torsion points;
- finite-field point counts;
- G2 points;
- cyclic multiples;
- Perfect gate.

---

# 42. Core P8 results

### P8-R1

Both G2 structural fibers are smooth genus-one quartics.

### P8-R2

They are birational to:

$$
E_r:
Y^2=(X+706)(X^2-202500),
$$

$$
E_s:
Y^2=(X+2657)(X^2-2458624).
$$

### P8-R3

G2 corresponds to non-torsion points:

$$
P_r=(3390,215040),
$$

$$
P_s=(11032,1277640).
$$

### P8-R4

The torsion subgroups of both curves are:

$$
\mathbb Z/2\mathbb Z
\times
\mathbb Z/4\mathbb Z.
$$

### P8-R5

Therefore:

$$
\operatorname{rank}E_r(\mathbb Q)\ge1,
$$

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

### P8-R6

Each G2 structural fiber has infinitely many rational points.

### P8-R7

P7's bounded sparsity is a rational-height phenomenon, not rational-point finiteness.

### P8-R8

The final Perfect closure forms a second genus-one arithmetic gate.

---

# 43. The most important change in understanding in this round

In P7, we saw:

$$
H\le500
$$

with only two points.

With brute-force thinking alone, one can easily ask:

> Does that mean this route is almost running out of points?

P8's answer is instead:

$$
\boxed{
\text{No. There are infinitely many points; they just rapidly reach enormous arithmetic height.}
}
$$

This is the first very clear separation of the “order-of-magnitude problem” from the “structural problem.”

---

# 44. Direct implications for Perfect Cuboid search

If future search still only uses:

$$
q\le H,
$$

it may need to cross an enormous empty height range before seeing the next point of the elliptic orbit.

But if the elliptic model is known,

one can directly apply:

$$
P\mapsto nP.
$$

Therefore:

$$
\boxed{
\text{It is not about running the same brute force for longer,}
\text{but about changing how candidates are generated.}
}
$$

---

# 45. But the second Gate may still be very hard

Even if the structural fiber has infinitely many rational points,

the Perfect condition requires the same parameter to pass the twin elliptic gate.

The final problem may therefore shift from:

$$
\text{enormous integer enumeration}
$$

to:

$$
\boxed{
\text{rational points on a higher arithmetic fiber product.}
}
$$

This may still be difficult, but it is a low-dimensional object closer to the actual obstruction.

---

# 46. Claim Boundary

P8 does not claim:

- to have found a Perfect Cuboid;
- structural infinity implies Perfect existence;
- the full Mordell-Weil ranks are known;
- G2 points are generators;
- rational points on the twin elliptic fiber product have already been classified;
- elliptic search is necessarily faster than all traditional methods.

P8 has established exactly:

- genus-one fiber;
- birational elliptic models;
- torsion subgroup;
- G2 non-torsion;
- rank lower bound;
- infinite structural rational points;
- twin elliptic final-gate formulation.

---

# 47. Conclusion

P8 truly elevates P7's bounded world fibers into arithmetic geometry.

The most important conclusion is not:

$$
\text{“We found some more near-hits.”}
$$

Instead, it is:

$$
\boxed{
\text{The two structural fibers of G2 are themselves positive-rank elliptic objects.}
}
$$

The actual search direction should therefore no longer be merely:

$$
H=500
\rightarrow
5000
\rightarrow
50000.
$$

Instead, it should become:

$$
\boxed{
\text{elliptic group law}
\rightarrow
\text{exact structural worlds}
\rightarrow
\text{second arithmetic gate}.
}
$$

That is, what the world model requires:

$$
\boxed{
\text{First understand how the world is generated, then decide which points to compute.}
}
$$

---

## Next phase

**GERES P9 — Twin-Elliptic Perfect Gate and Fiber-Product Analysis**

Prioritize analysis of:

For fixed:

$$
r=\frac35
$$

the pair:

$$
z^2
=
s^4+\frac{706}{225}s^2+1,
$$

and:

$$
w^2
=
s^4-\frac{194}{225}s^2+1.
$$

For fixed:

$$
s=\frac47
$$

the pair:

$$
z^2
=
r^4+\frac{2657}{784}r^2+1,
$$

and:

$$
w^2
=
r^4-\frac{479}{784}r^2+1.
$$

The next step should determine:

- fiber-product genus;
- Jacobian decomposition;
- local solvability;
- rational-point obstructions;
- whether the G2 elliptic orbit can cross the second gate.

This is the low-dimensional object that should actually be studied for the last arithmetic bit:

$$
(0,1,1,1)
\rightarrow
(1,1,1,1)
$$

in this transition.