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

# Perfect Cuboid Radical-FDRS R0 First Experimental Pass

## Squareclass Defect Migration, Compatibility Graph, and Gaussian-Norm Constraints

**Series:** Perfect Cuboid Global Closure  
**Branch:** Radical-FDRS / Geometry-Coupling Branch R  
**Phase:** R0 Experimental Pass 1  
**Date:** 2026-10-03  
**Status:** Experimental / Methodological Result — Not a proof of Perfect Cuboid existence or nonexistence

---

# 1. Experimental Objective

This round no longer directly enumerates large integer edge lengths; it instead uses the one-dimensional squareclass state of Radical-FDRS.

The first stage restricts the squarefree alphabet:

$$
\mathcal K_6
=
\{1,2,3,5,6\}.
$$

For a normalized square-state:

$$
A+B+C=1,
$$

define six arithmetic channels:

$$
K
=
(\kappa_A,\kappa_B,\kappa_C,\kappa_D,\kappa_E,\kappa_F),
$$

where:

$$
D=A+B,
$$

$$
E=A+C,
$$

$$
F=B+C.
$$

Write each positive rational channel as:

$$
X=q^2\kappa,
$$

where:

$$
q\in\mathbb Q_{>0},
$$

and:

$$
\kappa
$$

is a positive squarefree kernel.

The terminal state of a Perfect Cuboid is:

$$
\boxed{
K_{\rm goal}
=
(1,1,1,1,1,1).
}
$$

The principal question of this round is:

> Can a squareclass defect move, split, fuse, or disappear while global conservation is preserved?

---

# 2. A First Exact Example of Defect Migration

Take:

$$
(A,B,C)
=
\left(
\frac19,\frac49,\frac49
\right).
$$

The first three basic channels are all rational squares:

$$
A=\left(\frac13\right)^2,
$$

$$
B=C=\left(\frac23\right)^2.
$$

Therefore:

$$
(\kappa_A,\kappa_B,\kappa_C)
=
(1,1,1).
$$

The three face channels:

$$
D=A+B=\frac59
=
5\left(\frac13\right)^2,
$$

$$
E=A+C=\frac59
=
5\left(\frac13\right)^2,
$$

$$
F=B+C=\frac89
=
2\left(\frac23\right)^2.
$$

Thus:

$$
\boxed{
K
=
(1,1,1,5,5,2).
}
$$

Now apply a conservation move:

$$
A'=A,
$$

$$
B'=B-\frac19,
$$

$$
C'=C+\frac19.
$$

Therefore:

$$
(A',B',C')
=
\left(
\frac19,\frac13,\frac59
\right),
$$

and:

$$
A'+B'+C'=1.
$$

The new basic squareclasses:

$$
A'=\frac19
\Rightarrow
\kappa_A'=1,
$$

$$
B'=\frac13
=
3\left(\frac13\right)^2
\Rightarrow
\kappa_B'=3,
$$

$$
C'=\frac59
=
5\left(\frac13\right)^2
\Rightarrow
\kappa_C'=5.
$$

The new face channels:

$$
D'=A'+B'=\frac49
\Rightarrow
\kappa_D'=1,
$$

$$
E'=A'+C'=\frac23
=
6\left(\frac13\right)^2
\Rightarrow
\kappa_E'=6,
$$

$$
F'=B'+C'=\frac89
\Rightarrow
\kappa_F'=2.
$$

Therefore:

$$
\boxed{
(1,1,1,5,5,2)
\longrightarrow
(1,3,5,1,6,2).
}
$$

where:

$$
\boxed{
\kappa_D:5\rightarrow1.
}
$$

That is, the defect originally in the:

$$
D
$$

channel has been repaired, but defects are simultaneously redistributed to other channels.

This yields the first experimental conclusion:

$$
\boxed{
\text{Arithmetic defect can migrate.}
}
$$

But the number of defects is not conserved:

$$
\boxed{
\text{defect count is not a conservation invariant.}
}
$$

Defects can:

- move;
- split;
- fuse;
- be redistributed to different Boolean layers.

---

# 3. Low-Height Prototypes in a Small Squareclass Alphabet

In:

$$
\mathcal K_6
=
\{1,2,3,5,6\}
$$

a bounded exact rational scan yields several low-height prototypes.

These include:

| $A,B,C$ | $K=(\kappa_A,\kappa_B,\kappa_C,\kappa_D,\kappa_E,\kappa_F)$ |
|---|---|
| $\left(\frac19,\frac49,\frac49\right)$ | $(1,1,1,5,5,2)$ |
| $\left(\frac14,\frac14,\frac12\right)$ | $(1,1,2,2,3,3)$ |
| $\left(\frac19,\frac13,\frac59\right)$ | $(1,3,5,1,6,2)$ |
| $\left(\frac1{49},\frac{24}{49},\frac{24}{49}\right)$ | $(1,6,6,1,1,3)$ |
| $\left(\frac1{50},\frac1{50},\frac{24}{25}\right)$ | $(2,2,6,1,2,2)$ |
| $\left(\frac13,\frac13,\frac13\right)$ | $(3,3,3,6,6,6)$ |

This shows that squarehood is not fixed to a particular layer.

For example:

$$
(1,1,1,5,5,2)
$$

represents complete closure of the basic layer, with defects in every face-layer channel.

And:

$$
(1,6,6,1,1,3)
$$

has two basic defects, but two face channels have returned to the trivial squareclass.

Therefore:

$$
\boxed{
\text{squareclass defects can move across Boolean layers.}
}
$$

---

# 4. Dual-Squareclass Compatibility

Consider a complementary pair:

$$
X+Y=1.
$$

If:

$$
X=\kappa u^2,
$$

$$
Y=\lambda v^2,
$$

where:

$$
u,v\in\mathbb Q,
$$

then:

$$
\boxed{
\kappa u^2+\lambda v^2=1.
}
$$

Therefore define:

$$
\kappa\sim\lambda
$$

if and only if:

$$
\kappa u^2+\lambda v^2=1
$$

has a nondegenerate rational solution.

In:

$$
\{1,2,3,5,6\}
$$

one obtains the following compatibility pattern:

$$
\boxed{
1\sim1,
}
$$

$$
\boxed{
1\sim2,
\quad
1\sim3,
\quad
1\sim5,
\quad
1\sim6,
}
$$

$$
\boxed{
2\sim2,
}
$$

$$
\boxed{
3\sim6,
}
$$

$$
\boxed{
5\sim5.
}
$$

The following pairs are incompatible:

$$
2\not\sim3,
$$

$$
2\not\sim5,
$$

$$
2\not\sim6,
$$

$$
3\not\sim3,
$$

$$
3\not\sim5,
$$

$$
5\not\sim6,
$$

$$
6\not\sim6.
$$

This means that the Radical-FDRS state graph is not a complete graph.

Squareclass transitions are restricted by local quadratic-form conditions.

---

# 5. A Local Exclusion Example

Suppose:

$$
2x^2+3y^2=z^2
$$

has a primitive integer solution.

Modulo:

$$
3
$$

we obtain:

$$
2x^2\equiv z^2\pmod3.
$$

However:

$$
2
$$

is a quadratic nonresidue modulo:

$$
3
$$

as displayed above.

Therefore one must have:

$$
3\mid x,
$$

and:

$$
3\mid z.
$$

Substitution back into the original equation also forces:

$$
3\mid y.
$$

This contradicts the primitive assumption.

Thus:

$$
\boxed{
2\not\sim3.
}
$$

Similar modular descents can exclude other small squareclass pairs.

The squareclass graph is therefore a constrained transition graph with arithmetic walls.

---

# 6. Sum-of-Squares Restrictions on an Isolated Face Defect

Consider a single face defect:

$$
A=x^2,
$$

$$
B=y^2,
$$

However:

$$
D=A+B
$$

is not a rational square.

If:

$$
D=\kappa z^2,
$$

then:

$$
x^2+y^2=\kappa z^2.
$$

Therefore:

$$
\boxed{
\kappa
=
\left(\frac{x}{z}\right)^2
+
\left(\frac{y}{z}\right)^2.
}
$$

That is:

$$
\kappa
$$

must be a sum of two rational squares.

For positive squarefree:

$$
\kappa,
$$

its odd prime factors must therefore satisfy the conditions of the sum-of-two-squares theorem.

Formally:

$$
\boxed{
\kappa
=
2^\epsilon
\prod_i p_i,
}
$$

where:

$$
\epsilon\in\{0,1\},
$$

and:

$$
p_i\equiv1\pmod4.
$$

Thus any prime factor:

$$
p\equiv3\pmod4
$$

cannot occur to an odd power in the squarefree kernel of an isolated face defect.

---

# 7. An Immediate Correction to the Small Alphabet

Originally:

$$
\mathcal K_6
=
\{1,2,3,5,6\}.
$$

But for isolated face defects:

$$
3
$$

and:

$$
6
$$

both contain:

$$
3\equiv3\pmod4.
$$

Therefore:

$$
\boxed{
\kappa=3
}
$$

and:

$$
\boxed{
\kappa=6
}
$$

cannot be squareclasses of an isolated face defect.

The only nontrivial candidates genuinely worth pursuing in the small alphabet are therefore:

$$
\boxed{
\kappa=2,
\quad
\kappa=5.
}
$$

This is not an empirical conclusion from a bounded scan, but a local necessary condition obtained directly from the sum-of-squares structure.

---

# 8. Arithmetic Signature of the G2 Seed

The current seed:

$$
(a,b,c;g)
=
(448,495,840;1073).
$$

Its only failed face channel:

$$
D=A+B.
$$

has numerator:

$$
445729
=
233\cdot1913.
$$

Thus:

$$
\boxed{
\kappa_{\rm G2}
=
233\cdot1913.
}
$$

Note:

$$
233\equiv1\pmod4,
$$

and:

$$
1913\equiv1\pmod4.
$$

Therefore:

$$
\boxed{
233\cdot1913
}
$$

fully satisfies the sum-of-two-squares admissibility condition for an isolated face defect.

Thus the arithmetic defect of the current G2 seed is not an arbitrary squareclass.

It lies in the Gaussian sum-of-two-squares admissible sector.

---

# 9. Gaussian norm interpretation

Because:

$$
x^2+y^2
=
N_{\mathbb Q(i)/\mathbb Q}(x+iy),
$$

the isolated face defect:

$$
x^2+y^2=\kappa z^2
$$

can be rewritten as:

$$
N(x+iy)=\kappa z^2.
$$

Therefore:

$$
\boxed{
\text{face-defect squareclass}
\leftrightarrow
\text{Gaussian norm squareclass}.
}
$$

This suggests that the face-defect dynamics of Radical-FDRS may be studied further through:

$$
\mathbb Q(i)
$$

using its norms, factorization, squareclass transport, and ideal structures.

---

# 10. A New Admissible Squareclass Alphabet

If the next stage specifically studies isolated face defects, the more natural alphabet is no longer:

$$
\{1,2,3,5,6\},
$$

but:

$$
\boxed{
\mathcal K_{\rm Gauss}
=
\{
1,2,5,10,13,17,26,29,34,37,41,\ldots
\}.
}
$$

That is, positive squarefree integers whose odd prime factors all satisfy:

$$
p\equiv1\pmod4.
$$

More abstractly:

$$
\boxed{
\mathcal K_{\rm Gauss}
=
\left\{
2^\epsilon
\prod_i p_i
:
\epsilon\in\{0,1\},
\;
p_i\equiv1\pmod4
\right\}.
}
$$

The research direction should therefore advance from:

$$
\text{small radical values}
$$

to:

$$
\boxed{
\text{admissible squareclass alphabet}.
}
$$

---

# 11. Bounded one-defect test

For:

$$
\kappa=2
$$

and:

$$
\kappa=5
$$

perform a bounded integer search.

Require:

$$
a^2+c^2=e^2,
$$

$$
b^2+c^2=f^2,
$$

$$
a^2+b^2+c^2=g^2,
$$

and require:

$$
a^2+b^2=\kappa h^2.
$$

Within the test range:

$$
1\le a,b,c\le30000
$$

we have:

$$
\boxed{
\kappa=2:
\quad
0\text{ hits},
}
$$

$$
\boxed{
\kappa=5:
\quad
0\text{ hits}.
}
$$

This is only bounded negative evidence, not a proof of nonexistence.

As a scanner sanity check, specifying:

$$
\kappa
=
233\cdot1913
=
445729
$$

recovers the known seed:

$$
\boxed{
(a,b,c)
=
(448,495,840),
}
$$

and:

$$
\boxed{
(e,f,g)
=
(952,975,1073).
}
$$

The scanner can therefore at least reproduce the known one-defect state.

---

# 12. Four Structural Layers Developed in This Round

After Radical-FDRS flattening, the Perfect Cuboid problem can initially be divided into four layers:

## Layer 1 — Conservation

$$
\boxed{
A+B+C=1.
}
$$

Controls the global total.

## Layer 2 — Boolean incidence

$$
D=A+B,
$$

$$
E=A+C,
$$

$$
F=B+C.
$$

Controls the subset-sum structure.

## Layer 3 — Squareclass compatibility

Controls which:

$$
\kappa_i
$$

can occur together in complementary pairs or face couplings.

## Layer 4 — Norm arithmetic

For example, face defects are constrained by the Gaussian norm structure of:

$$
\mathbb Q(i)
$$

as displayed above.

Therefore:

$$
\boxed{
\text{Perfect Cuboid}
}
$$

is no longer merely a search for large integers; it can be viewed as:

$$
\boxed{
\text{Conservation}
+
\text{Boolean incidence}
+
\text{Squareclass compatibility}
+
\text{Norm closure}.
}
$$

---

# 13. A Revised Understanding of Defect Dynamics

The experiments of this round show:

$$
\boxed{
\text{defect location is dynamic},
}
$$

However:

$$
\boxed{
\text{defect squareclass is constrained}.
}
$$

Thus one cannot study only:

$$
\text{defect count}.
$$

A more suitable state descriptor would be:

$$
\boxed{
\mathfrak S
=
(K,\Gamma,\Pi).
}
$$

where:

$$
K
=
(\kappa_1,\ldots,\kappa_6)
$$

records squareclasses;

$$
\Gamma
$$

records the Boolean / complementary compatibility graph;

$$
\Pi
$$

records prime-support data.

For example:

$$
\Pi(\kappa)
=
\{p:p\mid\kappa\}.
$$

Only in this way can one study defect:

- migration；
- splitting；
- fusion；
- annihilation；
- prime-support exchange；
- norm-class transport。

---

# 14. Recommendations for R1

The next stage does not prioritize expanding the integer scan.

It is more natural to establish:

$$
\boxed{
\text{Squareclass Compatibility / Transition Algebra}.
}
$$

The central study concerns:

$$
\kappa
\overset{\mathcal M}{\longrightarrow}
\kappa'.
$$

Study how a conservation move:

$$
(A,B,C)
\rightarrow
(A',B',C')
$$

induces:

$$
K\rightarrow K'.
$$

In particular, for the current seed:

$$
\boxed{
K_{\rm G2}
=
(1,1,1,233\cdot1913,1,1),
}
$$

study whether an admissible trajectory exists:

$$
\boxed{
(1,1,1,233\cdot1913,1,1)
\longrightarrow
\cdots
\longrightarrow
(1,1,1,1,1,1).
}
$$

The question can be subdivided further:

1. Can the defect move from $D$ to another channel?
2. Can $233$ and $1913$ be separated into different channels?
3. Does prime support have a conserved quantity?
4. Can the Gaussian norm class change under a move?
5. Does defect annihilation require two particular squareclasses to meet?
6. Is there a global obstruction invariant?

---

# 15. Conclusions of This Round

The first Radical-FDRS experimental pass supports:

$$
\boxed{
\text{Arithmetic defects can migrate across the flattened Boolean cube.}
}
$$

while also showing:

$$
\boxed{
\text{their motion is constrained by quadratic-form and norm arithmetic.}
}
$$

In particular, an isolated face defect must belong to a sum-of-two-squares admissible squareclass.

The current G2 value:

$$
\boxed{
233\cdot1913
}
$$

happens to fully meet this necessary structure.

The next research question should therefore not be merely:

$$
\text{Can another, larger near-solution be found?}
$$

but should become:

$$
\boxed{
\text{what is the transition algebra of arithmetic defects?}
}
$$

and:

$$
\boxed{
\text{can the defect }233\cdot1913\text{ be transported, factor-separated, and annihilated?}
}
$$

This will provide the formal starting point of:

$$
\boxed{
\text{Radical-FDRS R1}
}
$$

as displayed above.