Equal-Volume Polyhedra, the Dehn Invariant, and PCS: A State–Constraint–Reachability Alignment of Hilbert's Third Problem

Hilbert’s third problem asks whether two three-dimensional polyhedra of equal volume must be related by finite dissection and rigid reassembly. Max Dehn showed that equal volume is not sufficient: scissors congruence preserves an additional edge–angle invariant, now known as the Dehn invariant. Sydler later established that, for Euclidean polyhedra in three dimensions, equality of volume and equality of the Dehn invariant together provide a complete characterization of scissors congruence. This paper does not replace, rederive, or strengthen the Dehn–Sydler theorem. Instead, it aligns the classical result with the Planetary Common State (PCS) framework by separating four elements: the state of the rigid object, the externally imposed operation set, the invariants preserved by those operations, and the reachability of a target state. Within this alignment, equal volume represents equality of aggregate geometric quantity, while unequal Dehn invariants indicate a persistent structural incompatibility under finite cutting, translation, rotation, and reassembly. The PCS structural residual is not identified with the Dehn invariant itself. Rather, equality or inequality of the Dehn invariant supplies an exact criterion for whether the corresponding structure-related residual vanishes in this specific geometric problem. The paper presents Hilbert’s third problem as a rigorous model case demonstrating that equality of total quantity does not imply state equivalence or constrained reachability. This Zenodo record contains the complete English paper, its Traditional Chinese translation, and the corresponding Overleaf LaTeX source files. Author: Alvin Lin (Chun-Hung Lin)ORCID: 0009-0006-8999-8293Research website: https://alvin-lin-pcs.chatgpt.sitePCS Observatory: https://uranusastudio-design.github.io/Planetary-common-state/PCS_OBSERVATORY/