# 18｜Provisional Derived Family Theorem

## Statement

Let:

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

Let $\mathcal P$ be the prime set:

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

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

and:

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

is irreducible over $\mathbb F_q$.

Then $\mathcal P$ has natural density:

$$
\frac1{24}.
$$

Candidate derived conclusion:

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

where $E_q$ is the 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

Currently, it should be called:

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

rather than:

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

The reason is not that we still see obvious mathematical gaps, but that this has entered the stage of "requiring an independent referee to verify citations and conventions line-by-line."