One Number at the Founding: Θ_c Priced to a Single Unowned Input, and the Last Road to It Closed as Circular

Whether the sever angle Θ_c can be produced (a derived value for it, against the declaration that stands meanwhile at Θ_c = 2582.39 rad) has stood open in this program. The declaration is not idle bookkeeping. The produced separation law uses u(1) = 11/411; the load time τ_load = 6.4427 × 10⁻²² s, computed in an earlier paper on durations in this program, is computed from it; the scar cap of 284.88 nats and an earlier study's selected bound Q_c > 2601.10 rest on it. Promote Θ_c from declared to derived and every one of those results strengthens in a single stroke; that is why the question exists. This paper assembles the earlier work's three reductions of the problem and adds a fourth, closing exhibit. First: the denomination is owned — the phase-cell partition (one grain per 2π of phase advance) is produced in form, so deriving Θ_c is exactly deriving one number, the store's total phase advance, and nothing else. Second: an earlier search for the level law in this program located that number — it is the founding's total twist, Θ_c = ∫ ω_0, the integral of the founding profile; the drive law cannot return it, because the drive law's coordinate is the fraction φ ∈ [0,1], the total having been divided out when the drive law itself was founded; and the founding profile ω_0 is the output of a law this program does not own. Third: an earlier counting survey in this program attacked the same number in its integer form (N_c = 411) through eight rule families, fixed in advance, and returned a failure with its reasons stated — local counts at the frame's only length are O(1), global counts are astronomical, and 411 is in between, in neither regime, with the frame containing no generator at that in-between scale. Fourth — this paper's own work: the one unexamined road, through τ_load, is exhibited and closed. The load time's formula is τ_load = 2π N_c P_sev ℓ_T / c: the declared angle is its input, printed as a declaration in that earlier paper itself. The near-number that made the road look promising (the earlier study's Q_c bound, 2601.10, sitting 0.72% from Θ_c) decomposes exactly: 2601.10 = Θ_c × P_sev, the declared angle wearing a produced factor that itself rests on unproved assumptions. There is no second value of Θ_c anywhere near the earlier work. Verdict: Θ_c has not been produced. It stands as the author's declaration, and the condition that would retire it is a founding law.