evals/fixtures/inconclusive-continuation.md

# Recurrence program handoff

Continue this authorized research program through a proof or counterexample.
Use the existing route R and preserve a durable result and current handoff in
the temporary project. No ledger has been initialized.

## Target

Let $a_0=0$, $a_1=1$ and $a_{n+2}=2a_{n+1}+a_n$ for every integer $n\geq 0$.
Determine whether

$$
a_{n+1}a_{n-1}-a_n^2=(-1)^n
$$

for every integer $n\geq 1$.

## Route R: a uniform recurrence identity

Success means a proof for every stated index; failure means a counterexample
or a demonstrated obstruction to this mechanism. The program proposed two
continuations:

1. Derive a formula using roots of the characteristic polynomial and simplify
   the products uniformly.
2. Study the determinant of consecutive recurrence vectors and its evolution.

## Returned attempt A

The executor tried continuation 1. Its scratch work gives
$r^2-2r-1=0$ and roots $1+\sqrt{2}$ and $1-\sqrt{2}$. It did not finish the
uniform product simplification before its work package ended. Its outcome is
inconclusive, with that simplification unresolved. It supplied no counterexample
or no-go argument. Continuation 2 has no recorded attempt.

The present program has resources for another bounded attempt.