A Unified Differential-Algebraic Geometric Framework for Hilbert's Third Problem

This paper establishes a complete and structurally rigorous theory for Hilbert’s third problem (scissors congruence of equal-volume polyhedra) based on first principles of differential algebra. The core innovation is the construction and systematic study of the polyhedral differential-algebraic closure KnP , a universal differentially closed field that encodes the geometry of any polyhedron (including non-convex and infinite-faced) as a differential field extension. Within this closure, classical geometric quantities—volume, Dehn invariants, tangent cone structures—are elevated to intrinsic differential-algebraic objects: cohomology classes, differential forms, and motivic measures.We prove the fundamental Differential-Algebraic Scissors Congruence Criterion (Theorem 4.1): two polyhedra P and Q are scissors congruent if and only if there exists a differential isomorphism ϕ : KnP (P) → KnP (Q) that preserves the volume cohomology class, the full tower of Dehn cohomology classes, and the motivic invariants of all tangent cones. This criterion subsumes the classical Dehn–Sydler theorem as a special case and is both necessary and sufficient in all dimensions.The constructive proof of this criterion reveals a recursive algorithm for building dissection maps. A key technical component is the explicit derivation of higher-dimensional combinatorial correction formulas γ(n)m,α,β (Theorem 5.3), which systematically describe the cross-term cancellations required when gluing local infinitesimal approximations. These coefficients are expressed in closed form via Bell polynomials, symmetric group characters, and Jacobian determinants of face normals.The theory not only provides a unified solution to Hilbert’s third problem but also naturally handles cases beyond classical methods: non-convex polyhedra (via self-intersection invariants), infinite-faced polyhedra (via ζ-regularization), and the complete invariant set in arbitrary dimensions (given by the cohomology of the closure). This work establishes differential-algebraic geometry as a foundational tool for studying discrete geometric equivalence, with implications for high-dimensional geometry, combinatorial topology, and mathematical physics.