Signed Kuramoto networks inside positive ones. An Evidence Press briefing. First, the background. Picture a network of identical oscillators, such as metronomes on a shared table, each nudging its neighbours. Usually every link pulls neighbours into step. A resting state is an arrangement where all the nudges cancel, and it is stable if small disturbances stay small. In twenty twenty-three, Davide Sclosa conjectured that these resting states always form a finite union of closed manifolds: a few tidy, smooth, closed pieces. Now, the main findings. First, links that push oscillators apart can be built exactly from links that pull. Each oscillator is replaced by a ring of copies, offset by sixty degrees, and a pushing link joins copies that are half a turn apart, where a pull acts exactly like a push. Everything near every resting state, including stability, carries over unchanged. Second, using this, a network of three hundred and twenty-four oscillators with only pulling links has a whole family of stable resting states that, at one point, is shaped like a cone over a torus, not a smooth surface. So the conjecture fails as published, with closed pieces, and also if the pieces may cross themselves. But its first arXiv version said only a finite union of manifolds, without the word closed, and that version is true, by a standard theorem on algebraic sets. Why it matters: singular families of stable states are possible even when every link is attractive. The limitations: everything happens at an exact, symmetric balance, and generic small changes destroy it. This is an unrefereed, AI-assisted candidate, released on twenty-seven October twenty twenty-six. The paper, proofs and checks are linked on this page. This briefing uses a synthetic AI voice, and is not additional evidence. This briefing uses an AI-generated voice and is not additional scientific evidence.