Auxiliary boundary data and the failure of intrinsic coherence in a finite-nerve route for spectral causal theory

We construct boundary operators on the nerve of a finite cover using auxiliary orientation data and prove that compatible data yield a chain complex with well-defined first homology. We establish that compatible orientation data always exist, though noncanonically. We then prove the main result: for any pair of triangles sharing a boundary edge, two global witness data differing by a single vertex transposition cannot both satisfy global coherence. This demonstrates that the coherence condition is not determined by the simplicial support of the nerve. In the context of spectral causal theory, any intrinsic approach to extracting homological data must draw on structural information beyond the nerve's unordered simplex data — such as a causal partial order. All results are formalized and machine-verified in Lean 4 (63 modules, 260+ theorems, zero sorry/admit). Source code: https://github.com/davidichalfyorov-wq/sct-theory