# Phase 1C — Fixed-Hypotenuse Global Synchronization

**Experiment:** `PCGC-P1C-GSYNC-300K`  
**Bound:** $g\le 300,000$  
**Arithmetic:** exact integer arithmetic  
**Perfect Cuboid hits:** **0**

## Equivalent fixed-$g$ formulation

Define

$$
L(g)=\left\{x\in\mathbb Z_{>0}:g^2-x^2\in\mathbb Z^2\right\}.
$$

Thus $x\in L(g)$ exactly when $x$ can be a leg of an integer right triangle with hypotenuse $g$.

A Perfect Cuboid with edges $(a,b,c)$ and space diagonal $g$ satisfies

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

and

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

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

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

Therefore

$$
\text{Perfect Cuboid}
\iff
a,b,c\in L(g)
\quad\text{and}\quad
a^2+b^2+c^2=g^2.
$$

This is an exact reformulation. It converts the problem into a three-leg closure problem inside one common-hypotenuse state.

## Representation-count gate

Let

$$
g=2^\alpha
\prod_{p\equiv1\pmod4}p^{e_p}
\prod_{q\equiv3\pmod4}q^{f_q}.
$$

The number of unordered positive representations

$$
x^2+y^2=g^2
$$

is

$$
R(g)=\frac12
\left(
\prod_{p\equiv1\pmod4}(2e_p+1)-1
\right).
$$

The generated leg sets satisfy

$$
|L(g)|=2R(g).
$$

Independent enumeration versus this factorization formula:

**PASS**

A nondegenerate Perfect Cuboid needs three edge legs that cannot contain a complementary pair, so it requires at least three distinct Pythagorean representations:

$$
R(g)\ge3.
$$

Hence a necessary gate is

$$
\prod_{p\equiv1\pmod4}(2e_p+1)\ge7.
$$

This means the common space diagonal must carry enough $1\pmod4$ prime structure before a Perfect Cuboid is even eligible.

## Exact search result through the current bound

- $g$ values with at least one Pythagorean representation: **211,898**
- $g$ values with $R(g)\ge3$: **70,482**
- Fixed-$g$ candidate-pair checks: **2,776,968**
- Perfect Cuboid hits: **0**
- Exact two-channel sphere near-hits: **3,508**

No Perfect Cuboid was found for

$$
g\le 300,000.
$$

Because the fixed-$g$ formulation is exhaustive in this bound and the leg generator was cross-checked against the independent representation-count formula, this is a **certified exclusion for this $g$ range** under the stated exact implementation.

## Smallest eligible common hypotenuses

- $g=65$: $R(g)=4$, $L(g)=[16, 25, 33, 39, 52, 56, 60, 63]$
- $g=85$: $R(g)=4$, $L(g)=[13, 36, 40, 51, 68, 75, 77, 84]$
- $g=125$: $R(g)=3$, $L(g)=[35, 44, 75, 100, 117, 120]$
- $g=130$: $R(g)=4$, $L(g)=[32, 50, 66, 78, 104, 112, 120, 126]$
- $g=145$: $R(g)=4$, $L(g)=[17, 24, 87, 100, 105, 116, 143, 144]$
- $g=170$: $R(g)=4$, $L(g)=[26, 72, 80, 102, 136, 150, 154, 168]$
- $g=185$: $R(g)=4$, $L(g)=[57, 60, 104, 111, 148, 153, 175, 176]$
- $g=195$: $R(g)=4$, $L(g)=[48, 75, 99, 117, 156, 168, 180, 189]$
- $g=205$: $R(g)=4$, $L(g)=[45, 84, 123, 133, 156, 164, 187, 200]$
- $g=221$: $R(g)=4$, $L(g)=[21, 85, 104, 140, 171, 195, 204, 220]$

## Smallest exact two-channel near-hits

- $(104,153,672;697)$, $R(g)=4$
- $(117,520,756;925)$, $R(g)=7$
- $(448,495,840;1073)$, $R(g)=4$
- $(264,448,975;1105)$, $R(g)=13$
- $(264,495,952;1105)$, $R(g)=13$
- $(208,306,1344;1394)$, $R(g)=4$
- $(234,1040,1512;1850)$, $R(g)=7$
- $(312,459,2016;2091)$, $R(g)=4$
- $(896,990,1680;2146)$, $R(g)=4$
- $(333,644,2040;2165)$, $R(g)=4$

These near-hits satisfy the space-diagonal sphere equation exactly and have exactly two edge values in $L(g)$. The third edge fails the common-hypotenuse leg test, so exactly one face remains open.

## Structural consequence

The search problem has now changed from seven coupled unknown integers to:

1. choose a common $g$ that passes the representation-count gate;
2. generate the finite exact set $L(g)$;
3. search for a three-element subset whose squared sum is $g^2$;
4. reject complementary-pair degeneracy;
5. primitive-reduce and canonicalize.

This is both exact and globally synchronized around the shared space diagonal.