Fibred realizations of the prism quandle $P_7$ in the standard four-sphere

Abstract: We prove that both smooth homotopy four-spheres arising from the two framed surgeries on a section of a period-six mapping torus built from a quaternionic prism manifold with cyclic factor of order seven are standard. Their belt spheres give two smooth fibred two-knot realizations of the prism quandle of order fifty-six in the standard four-sphere. The knots have the same knot group and diffeomorphic exteriors but are inequivalent. Combined with the fibred-rigidity reduction, this yields exactly two fibred realizations in homotopy four-spheres, both standard. The proof combines a Whitehead-link model for the orbit orbifold, circle-action and surgery methods, cyclic branched-cover calculations, and a spin-bordism evaluation of a secondary invariant. This resolves the order-seven case in the classical prism-family realization problem.