# Phase 1I — $g=p^2q$: Quotient of the Seven-State Closure System

## Status

The first surviving common-hypotenuse factorization type is

$$
g=p^2q,
\qquad
p\ne q,
\qquad
p,q\equiv1\pmod4.
$$

Its representation multiplicity is

$$
R(g)=7.
$$

Write

$$
t=\tan\theta=\frac mn,
\qquad
s=\tan\phi=\frac uv,
$$

for the primitive Gaussian slopes of $p$ and $q$.

Define

$$
\alpha=2\theta,
\qquad
\beta=2\phi.
$$

The seven normalized even-leg states are

$$
A=\sin\alpha,
$$

$$
B=\sin2\alpha,
$$

$$
C=\sin\beta,
$$

$$
D=\sin(\alpha+\beta),
$$

$$
E=\sin(\alpha-\beta),
$$

$$
F=\sin(2\alpha+\beta),
$$

$$
G=\sin(2\alpha-\beta),
$$

with absolute value irrelevant after squaring.

A Perfect Cuboid requires an internal Pythagorean relation

$$
X_i^2+X_j^2=X_k^2
$$

among three distinct states.

There are

$$
\binom72\cdot5=105
$$

raw ordered-hypotenuse relations.

---

## 1. Conjugation quotient

The transformation

$$
\beta\mapsto-\beta
$$

preserves the underlying common-hypotenuse state and acts by

$$
D\leftrightarrow E,
\qquad
F\leftrightarrow G,
$$

while

$$
A,\ B,\ C
$$

remain fixed after squaring.

Quotienting the 105 relations by this involution leaves exactly

$$
\boxed{57}
$$

relation orbits.

This is the first exact compression of the $p^2q$ seven-state system.

---

## 2. Prime-norm obstruction I

One recurring factor is

$$
2st+t^2-1=0.
$$

It gives

$$
s=\frac{1-t^2}{2t}.
$$

Substituting

$$
t=\frac mn
$$

in lowest terms yields

$$
s=
\frac{n^2-m^2}{2mn}.
$$

For primitive opposite-parity $m,n$,

$$
\gcd(n^2-m^2,2mn)=1.
$$

Therefore the reduced Gaussian pair for $s$ would have norm

$$
(n^2-m^2)^2+(2mn)^2
=
(m^2+n^2)^2
=
p^2.
$$

But that norm must be the distinct prime $q$.

Contradiction.

Hence

$$
\boxed{
2st+t^2-1\ne0
}
$$

throughout the admissible $p^2q$ prime-slope domain.

---

## 3. Prime-norm obstruction II

Another recurring factor is

$$
st^2-2st-s-t^2-2t+1=0.
$$

Solving for $s$ gives

$$
s=
\frac{1-2t-t^2}{1+2t-t^2}.
$$

After substituting $t=m/n$, the corresponding reduced numerator-denominator pair is, up to signs and ordering,

$$
n^2-2mn-m^2,
\qquad
n^2+2mn-m^2.
$$

Their squared norm is

$$
(n^2-2mn-m^2)^2
+
(n^2+2mn-m^2)^2
=
2(m^2+n^2)^2
=
2p^2.
$$

Thus the second Gaussian norm would be

$$
q=2p^2,
$$

impossible for an odd prime.

Therefore this factor is also globally excluded.

---

## 4. Other inherited proof-safe eliminations

The quotient also inherits the earlier mixed-prime obstructions:

- sign:
  $$
  D^2-E^2=\sin(2\alpha)\sin(2\beta)>0;
  $$
- the parity contradiction generated by
  $$
  st+s+t=1;
  $$
- the $2$-adic obstruction for the biquadratic pair
  $$
  D^2+E^2=A^2
  $$
  and
  $$
  D^2+E^2=C^2;
  $$
- boundary factors such as
  $$
  t=1,\ s=1,\ st=1;
  $$
- $s=t$, which would force the two reduced primitive Gaussian slopes, and hence the norm primes, to coincide;
- irrational single-variable roots such as
  $$
  t^2+2t-1=0.
  $$

Applying only these proof-safe rules eliminates

$$
\boxed{15}
$$

of the 57 quotient orbits.

Because most non-fixed orbits have size two, this corresponds to

$$
\boxed{27}
$$

of the original 105 raw relations.

---

## 5. Current exact frontier

The surviving system therefore has

$$
\boxed{42}
$$

quotient orbits, representing

$$
\boxed{78}
$$

raw relations.

No claim is made that these 42 orbits are genuinely solvable.

They are simply the relations not yet removed by the current proof-safe elementary gates.

The full exact factorization of all 57 orbit representatives is stored in the accompanying JSON/CSV artifacts.

---

## 6. Next reduction

The next useful step is not to increase the prime scan bound.

For the 42 surviving orbits:

1. group them by polynomial symmetry and total degree;
2. test exact local solvability under primitive opposite-parity slope constraints;
3. derive $2$-adic valuation signatures of the seven actual integer even legs;
4. identify relations incompatible with the necessary internal-Pythagorean valuation pattern;
5. birationally reduce only the final residual curves.

The immediate target is to cut the 42 orbit system before any elliptic or higher-genus machinery is used.