# Perfect Cuboid Global Closure — Phase 1 Baseline

## Experiment

**ID:** `PCGC-P1-RS60`  
**Method:** exact rational-sphere scan  
**Parameter bound:** reduced stereographic parameters with denominator at most `60`  
**Arithmetic:** exact integer / rational arithmetic only  
**Independent validation:** PASS

## Search domain

For reduced rational parameters $u,v$ with $0<u\le v<1$ and $u^2+v^2<1$, generate

$$
x=\frac{2u}{1+u^2+v^2},
$$

$$
y=\frac{2v}{1+u^2+v^2},
$$

$$
z=\frac{1-u^2-v^2}{1+u^2+v^2}.
$$

Then sort $(x,y,z)$ and reduce the common scale to a canonical primitive integer quadruple

$$
(a,b,c,g)
$$

with

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

The remaining exact Perfect-Cuboid conditions are whether

$$
a^2+b^2,\qquad a^2+c^2,\qquad b^2+c^2
$$

are all integer squares.

## Baseline result

- Reduced parameter fractions: **1,101**
- Admissible ordered parameter pairs before shape deduplication: **479,086**
- Duplicate shapes removed: **321**
- Unique canonical rational-sphere shapes: **478,765**
- Exactly 0 face-square closures: **475,302**
- Exactly 1 face-square closure: **3,462**
- Exactly 2 face-square closures: **1**
- Exactly 3 face-square closures: **0**

No Perfect Cuboid was found in this bounded parameter domain. This is **not** a nonexistence result.

## First critical square-defect sample

The unique two-face closure in this scan is

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

It satisfies

$$
448^2+840^2=952^2,
$$

$$
495^2+840^2=975^2,
$$

and

$$
448^2+495^2+840^2=1073^2.
$$

The only missing face is

$$
448^2+495^2=445729.
$$

Using the global identity,

$$
a^2+b^2=g^2-c^2=(g-c)(g+c),
$$

we obtain

$$
445729=(1073-840)(1073+840)=233\cdot1913.
$$

Thus the missing face-square condition is represented exactly by the surviving odd prime-parity defect

$$
\{233,1913\}.
$$

The other two face conditions have zero square defect.

## Structural consequence

For any exact rational/integer sphere state,

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

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

$$
b^2+c^2=g^2-a^2.
$$

Therefore the three face-square requirements can be studied as parity closure of

$$
(g-c)(g+c),\qquad
(g-b)(g+b),\qquad
(g-a)(g+a).
$$

This yields an exact square-defect representation that does not depend on floating-point distance to a square.

## Validation

The scan was implemented twice:

1. an integer-only stereographic engine;
2. an independent `Fraction`-based rational engine.

They produced exactly the same set of **478,765** canonical shapes, the same closure-count distribution, and the same two-face sample.

**Dual-engine validation:** PASS

## Next gate

Phase 1B should extend the exact state representation from `face-square PASS/FAIL` to a prime-parity square-defect signature for each of the three factors

$$
(g-a)(g+a),\quad
(g-b)(g+b),\quad
(g-c)(g+c),
$$

then compare those signatures across prime/radix channels.

The next objective is not merely to increase the denominator bound. It is to determine whether defect parity exhibits recurrent global structure that can be used as a constraint operator or a proof-safe obstruction.