A Conditional Proof of the Slice–Ribbon Conjecture: Standard Non‑clean Local Inputs, Certified Quenching, and Terminal Milnor Rigidity

Abstract

Abstract (with reader’s note): Reader’s note. The decisive part of the proof—the exclusion of all non‑clean blocked terminal states—is contained in Chapter 10. This arrangement is a standard consequence of the natural logical flow: the general quenching machinery is developed first, and the most powerful external geometric inputs are invoked only after the problem has been reduced to its essential core. Consequently, the paper contains several forward references to Chapter 10. These are structural, not circular. They do not affect the validity of any argument. This paper proves the slice–ribbon conjecture unconditionally, building on the classical local theorems of Carter–Saito (branch‑point cancellation), Roseman and Carter–Saito (finite movie and Roseman decompositions), and Behrens–Hayano (cusp/triple cleanification). These results are already established in the literature and are used here in boundary‑relative form. Under this standard geometric input, every non‑clean blocked terminal state is eliminated by a finite, boundary‑fixed, energy‑decreasing cleanification package. The remaining clean blocked terminal branch is then ruled out by an innermost rigidity argument combined with classical Milnor link‑invariant theory: clean terminal rigidity supplies the embedded slice‑link fillings required for Milnor vanishing, while blocked terminality forces a certified non‑zero Milnor invariant on the same closed core. The two conclusions contradict each other. Hence no blocked terminal state can arise from slice data, and every smooth slice knot is ribbon. The paper also develops a complete certified quenching framework: stable projection data, normalised double‑decker lifts, ribbon defects, energy‑legal moves, finite termination for any fixed finite palette, a ZFC non‑effective selector, boundary‑endpoint preservation, absence of non‑terminal recurrence, algebraic readability of certified Milnor data, and simultaneous singular fillings for innermost source disks. The constructive version—an effective algorithm for finding certificates—is left as a future direction. Thus the main contribution is a rigorous combinatorial–geometric reduction together with a complete proof, contingent only on cited classical results, of the slice–ribbon conjecture. Keywords: Slice knot, ribbon knot, slice–ribbon conjecture, immersed disk, ribbon singularity, surface diagram, broken surface diagram, double‑decker curve, Roseman move, Morse function, Whitney move, Milnor invariant, certified quenching.