Per @Hatsushira's suggesstion, I moved the init of both objects into my initializer. Compiles fine, but now I can't get any notifications. I've tried various different permutations, most of which don't compile, some of which compile but then complain at runtime about the object being referenced while not being installed on a view--another error message I can barely grasp. Welcoming any suggestions, cheers.

@main
struct ArkansasApp: App {
    @StateObject var appState: AppState
    @StateObject var arkansasScene: ArkansasScene

    init() {
        let state = AppState()
        let scene = ArkansasScene(appState: state)

        _appState = StateObject(wrappedValue: state)
        _arkansasScene = StateObject(wrappedValue: scene)
    }

    var body: some Scene {
        WindowGroup {
            ContentView()
                .environmentObject(appState)
                .environmentObject(arkansasScene)
        }
    }
}

class AppState: ObservableObject {
    @Published var numberOfStopSignsInArkansas = 400000
}

class ArkansasScene: SKScene, SKSceneDelegate, ObservableObject {
    @ObservedObject var appState: AppState

    var stopSignObserver: AnyCancellable!

    let side: Double = 1024

    init(appState: AppState) {
        self._appState = ObservedObject(initialValue: appState)

        super.init(size: CGSize(width: side, height: side))

        ping()
    }

    func ping() {
        DispatchQueue.main.asyncAfter(deadline: .now() + 1) {
            [weak self] in guard let myself = self else { return }
            myself.appState.numberOfStopSignsInArkansas += 1
            print("new value \(myself.appState.numberOfStopSignsInArkansas)")
            myself.ping()
        }
    }

    override func didMove(to view: SKView) {
        stopSignObserver = appState.$numberOfStopSignsInArkansas.sink {
            print("Got notification \($0)")
        }
    }

    required init?(coder aDecoder: NSCoder) {
        fatalError("init(coder:) has not been implemented")
    }
}

I didn't instantiate the view that contains ArkansasScene 🤦‍♂️