An Oriented Finite-Spin Vertex with Retained Tetrahedral Geometry: construction, canonical measure, and the gluing obstruction

Abstract

We construct a finite-spin, orientation-resolved four-simplex vertex amplitude that retains the full tetrahedral intertwiner algebra. Orientation is selected by restricting the history integration to the domain on which a native Spin(4)-invariant pseudoscalar has fixed sign. Because the restriction acts on the integration variables rather than on the boundary state, the amplitude stays linear in the boundary data, no Barrett-Crane projection is applied, and the tetrahedral volume degree of freedom survives. Four properties are established: exact Spin(4) invariance of the pseudoscalar and of the orientation domain; an exact sector decomposition with antiunitary exchange of the two orientation sectors; an exact discrete representative of the canonical purely geometric measure in the native variables, matching the cubic volume power of the continuum measure rather than fitting it; and a cubic zero of the pseudoscalar on the fully degenerate rank-one locus, which reverses the local asymptotic ordering between the degenerate and geometric contributions. Global geometric dominance, stated as the central open problem in the first version of this paper, is settled at the vertex: the obstruction considered there required a per-face coherent sign that this construction does not have. What is now open is gluing. Contracting coherent intertwiners across a shared tetrahedron forces the two tetrahedral shapes to agree, while an area-Regge continuum argument requires them to differ, and the mismatch survives only at the scale of the coherent fluctuations. No claim is made to have derived four-dimensional General Relativity, and no result in the paper depends on such a claim. The derivations, code and drafting were carried out in collaboration with large language models from more than one provider, under the author's direction; the author is responsible for the content. Every load-bearing numerical claim was reproduced by an independent implementation sharing no code path, every search for the absence of something carried a positive control, and each stage was submitted to adversarial review across providers. SHA-256 checksums for the uploaded files are in SHA256SUMS.txt.