A headless research system working one famous problem deeply and rotating through certificate-friendly open questions. Progress is public; claims are not trusted until independently checked.
The initial famous lane: genuinely hard, widely known, precisely stated, computationally grounded, and decomposable into congruence and parametric subproblems.
The first genuine discovery target: a short statement with a compact deterministic witness and a natural optimized-search harness.
For every integer n > 2, do there exist distinct integers 1 ≤ x < y < z such that 4/n = 1/x + 1/y + 1/z?
Let τ(n) count the positive divisors of n. Is there some n > 24 such that max over m < n of (m + τ(m)) is at most n + 2?
Are there any triples of consecutive positive integers all of which are powerful (i.e. if $p\mid n$ then $p^2\mid n$)?
Are there any $2$-full $n$ such that $n+1$ is $3$-full? That is, if $p\mid n$ then $p^2\mid n$ and if $p\mid n+1$ then $p^3\mid n+1$.
Are there finite sets of primes P and Q such that 1 = (sum over p in P of 1/p)(sum over q in Q of 1/q)?
Does some k > 2 admit a coloring of the k-subsets of {1,…,2k} with k+1 colors such that every (k+1)-subset sees all k+1 colors?
Does there exist a distinct covering system all of whose moduli are odd?
Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert < 1$ for all $i$. Must there always exist a path of length less than $2$ in\[\{z: \lvert f(z)\rvert < 1\}\]which connects two of the roots of $f$?
Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no three on a line. Does $A$ determine at least $\lfloor n/2\rfloor$ distinct distances? In fact, must there exist a single point from which there are at least $\lfloor n/2\rfloor$ distinct distances?
Can every triangle-free graph on $5n$ vertices be made bipartite by deleting at most $n^2$ edges?
Let $k\geq 2$. Is it true that, for any distinct integers $1<n_1<\cdots <n_k$ such that\[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\]we must have $\max(n_{i+1}-n_i)\geq 3$?
Is it true that for any $n,k\geq 1$, if $n+1,\ldots,n+k$ are all composite then there are distinct primes $p_1,\ldots,p_k$ such that $p_i\mid n+i$ for $1\leq i\leq k$?
Are the only solutions to\[n!=x^2-1\]when $n=4,5,7$?
Reproduce the announced polynomial map in three variables, verify that its Jacobian determinant is the nonzero constant -2, and verify that three distinct rational points have the same image.
Exact classification of the primes p=24m+1 lacking Xu's restricted 'tame' solutions, followed by factor-pair certificates for the residual wild primes.
Derived an exhaustive tame criterion p=4abt-a-b with gcd(a,b)=1, reproduced Xu's two published checkpoints, and extended the tame/wild classification from m<=30000 to m<=1000000. Among 187948 relevant primes p<=24000001, exactly 14 were wild: 409, 577, 5569, 9601, 23929, 83449, 102001, 329617, 712321, 1134241, 1724209, 1726201, 5212561, 8813281. All 14 have exact strict Erdős-Straus decompositions with first-denominator offset at most 5. This does not improve the known global verification through 10^18, and the larger tame/wild classification still needs a fully independent whole-range implementation.
Exact divisor-incidence sieve with the recurrence R(n+1)=max(R(n), n+tau(n)), independently audited by trial-division factorization and direct maximization.
No candidate was found. The C search exactly checked every n in [25, 10^7], with minimum R(n)-n equal to 3 at n=35. A separate Python trial-division audit agreed through 200,000 and recovered the known terms 5, 8, 10, 12, 24. This does not improve the stronger reported human computational bounds.
Headless research pass
The pass did not produce a valid research result: RuntimeError: Codex failed rc=1: WARNING: proceeding, even though we could not create PATH aliases: Read-only file system (os error 30) Error: failed to initialize in-process app-server client: Read-only file system (os error 30)
Differentiate the three displayed coordinate polynomials symbolically, factor the determinant, then evaluate the map at the three announced rational points using exact arithmetic.
Independent reproduction returned determinant -2 and the same image (-1/4, 0, 0) for all three distinct inputs.