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

# RG-SCP Note B
## Root-Bounded Division Domain
### The Root-Bounded Division Domain, Bilateral Asymmetry, and an Operator Reformulation of Goldbach Failure

**Date:** 2026-09-30  
**Document type:** Research Note / Operator Reformulation  
**Status:** Not a proof; establishing a new language for exact reformulation

---

## 0. Objective

Reformulate prime / composite in terms of:

> Whether a nontrivial division action exists within the root bound.

The core object:

$$
\boxed{
\mathcal D(n)
=
\{d:1<d\le\sqrt n,\ d\mid n\}.
}
$$

For $n>1$:

$$
\boxed{
n\in\mathbb P
\Longleftrightarrow
\mathcal D(n)=\varnothing.
}
$$

Thus primality can be described as:

$$
\boxed{
\text{root-bounded division-action-free}.
}
$$

---

## 1. The Symmetric Form of Goldbach

For:

$$
N=2m,
$$

a Goldbach pair:

$$
N=p+q
$$

can be written as:

$$
\boxed{
p=m-d,
\qquad
q=m+d.
}
$$

Thus Goldbach success is equivalent to:

$$
\boxed{
\exists d,
\quad
m-d\in\mathbb P,
\quad
m+d\in\mathbb P.
}
$$

In the language of division domains:

$$
\boxed{
\exists d,
\quad
\mathcal D(m-d)=\varnothing,
\quad
\mathcal D(m+d)=\varnothing.
}
$$

---

## 2. The Operator Form of Failure

Goldbach failure is equivalent to:

$$
\boxed{
\forall d,
\quad
\mathcal D(m-d)\neq\varnothing
\;\lor\;
\mathcal D(m+d)\neq\varnothing.
}
$$

That is:

> For every admissible symmetric offset, at least one of the two endpoints permits a nontrivial division action within its root bound.

---

## 3. The Admissible Offset Domain

For fixed $m$, define:

$$
\boxed{
\Omega_m
=
\left\{
d\in\mathbb N:
m-d\ge3,
\quad
m-d\text{ odd}
\right\}.
}
$$

Since:

$$
m-d
$$

and:

$$
m+d
$$

have the same parity, this constrains both sides simultaneously.

For the main Goldbach branch with $N>4$, only odd endpoints are needed.

Thus:

$$
\boxed{
\Omega_m
}
$$

is a finite domain.

---

## 4. Asymmetry of the Left and Right Root Bounds

For:

$$
d>0,
$$

we have:

$$
m-d<m+d.
$$

so:

$$
\boxed{
\sqrt{m-d}<\sqrt{m+d}.
}
$$

That is:

$$
\boxed{
\text{additive symmetry}
+
\text{multiplicative root-bound asymmetry}.
}
$$

This is the core tension of the new relation domain.

---

## 5. Side-Indexed Division Relation

Let:

$$
\sigma\in\{-1,+1\}.
$$

Define:

$$
\boxed{
R_m(d,\sigma,r)
}
$$

to hold when:

$$
r\in\mathbb P,
$$

$$
r\mid(m+\sigma d),
$$

and:

$$
r^2\le m+\sigma d.
$$

Meaning:

> $r$ is a proper-factor witness for $m+\sigma d$ within the complete root bound.

---

## 6. Finite Witness Universe

Within the odd-endpoint domain:

$$
m+d\le2m-3.
$$

Thus every proper-factor witness satisfies:

$$
r^2\le2m-3.
$$

So one only needs to consider:

$$
\boxed{
r\le\sqrt{2m-3}.
}
$$

Define:

$$
\boxed{
W_m
=
\{-1,+1\}
\times
\mathbb P_{\le\sqrt{2m-3}}.
}
$$

Thus $W_m$ is finite.

---

## 7. The Finite Relation-Cover Form of Failure

Goldbach failure can be rewritten as:

$$
\boxed{
\forall d\in\Omega_m,
\quad
\exists(\sigma,r)\in W_m,
\quad
R_m(d,\sigma,r).
}
$$

That is:

> Every offset is covered by at least one bounded division witness.

---

## 8. Residue-Class Form

If:

$$
r\mid(m-d),
$$

then:

$$
\boxed{
d\equiv m\pmod r.
}
$$

If:

$$
r\mid(m+d),
$$

then:

$$
\boxed{
d\equiv -m\pmod r.
}
$$

Thus each obstruction prime $r$ supplies at most two residue families in the offset domain:

$$
\boxed{
d\equiv\pm m\pmod r.
}
$$

which are then clipped by the respective left and right root bounds.

---

## 9. Root-Bounded Residue Cells

For a witness:

$$
w=(\sigma,r),
$$

define:

$$
\boxed{
O_w
=
\left\{
d\in\Omega_m:
R_m(d,\sigma,r)
\right\}.
}
$$

Each $O_w$ is:

1. a residue class;
2. a finite interval;
3. side-dependent square-root admissibility;

the intersection of these three conditions.

---

## 10. Difference from the One-Sided Complement Cover

The original RG-SCP often considers:

$$
p\mapsto N-p.
$$

Now this becomes:

$$
m-d
\quad\leftrightarrow\quad
m+d.
$$

Both endpoints share the same center:

$$
m,
$$

and differ by:

$$
2d.
$$

Thus obstruction is no longer just about which factor blocks a complement, but:

> For every symmetric offset, whether at least one of the two different left and right root-bound domains is hit.

---

## 11. The Genuine Current GAP

We still have not proved:

$$
\boxed{
\bigcup_{w\in W_m}O_w
\neq
\Omega_m.
}
$$

That is, we still have not proved:

> That bounded divisor cells cannot completely cover all offsets.

So Goldbach has not yet been proved.

But the target has explicitly become:

$$
\boxed{
\text{excluding complete root-bounded symmetric covers.}
}
$$

---

## 12. Research Value

This reformulation may have three advantages:

1. Primality becomes finite non-incidence;
2. Goldbach failure becomes a finite cover;
3. Additive symmetry and multiplicative root bounds enter the same relation object.

---

## 13. The Shortest Conclusion

Goldbach failure can be read anew as:

$$
\boxed{
\forall d\in\Omega_m,
\quad
\text{at least one side has a root-bounded proper division action}.
}
$$

Whereas success is:

$$
\boxed{
\exists d\in\Omega_m,
\quad
\text{neither side has any root-bounded proper division action}.
}
$$

This is the direct entry point for a finite closure construction.