# 18｜Provisional Derived Family Theorem

## Statement

令：

$$
E/\mathbb Q:
y^2=x^3+x^2+8x-16.
$$

令 $\mathcal P$ 為 prime set：

$$
q\equiv1\pmod{24},
$$

$$
\left(\frac q{29}\right)=1,
$$

且：

$$
x^3+x^2+8x-16
$$

在 $\mathbb F_q$ 不可約。

則 $\mathcal P$ 有 natural density：

$$
\frac1{24}.
$$

候選 derived conclusion：

$$
\boxed{
\forall q\in\mathcal P,\quad
\operatorname{BSD}(E_q)
}
$$

其中 $E_q$ 是 quadratic twist by $q$。

---

# Current proof status

## CLOSED

- $\mathcal P$ infinite / positive density；
- all conductor-prime splitting conditions；
- 2-division inertness；
- support-prime ordinarity；
- base BSD$(E,2)$；
- Theorem 2.14 $2$-part / nonvanishing；
- support-prime additive branch；
- good ordinary branch；
- fixed multiplicative $3/29$ branches；
- FW supersingular residual conditions；
- exhaustive prime partition。

## NEEDS INDEPENDENT REFEREE AUDIT

1. exact convention match between FW modular representation and elliptic $E[p]$ at nonsplit multiplicative witness；
2. period normalization statement for every twist $E_q$；
3. exact isogeny/optimality phrasing used in the period comparison；
4. precise citation chain for all ordinary/additive/multiplicative p-part results；
5. novelty search.

---

# Claim level

目前應稱：

$$
\boxed{
\text{Provisional Derived Theorem}
}
$$

而不是：

$$
\boxed{
\text{Established New Theorem}
}
$$

原因不是我們仍看到明顯數學缺口，而是這已經進入「需要獨立 referee逐行核對引用與 conventions」的階段。