Point fibres in N* and an obstruction to coordinate decomposition

We construct in ZFC a unital embedding C(N*) → M₂(C((N*)²)) for which no inner conjugate admits a finite coordinate decomposition, even when the coordinate corners carry arbitrary matrix-valued homomorphisms. The obstruction persists in the conjugacy-invariant form of the tensor-factor formulas discussed by Ghasemi and by Vignati and Yilmaz, and on every finite power (N*)^d, d ≥ 2. Its topological ingredient is a fibre realization theorem: there are a fixed separable closed X ⊆ N* and q ∈ X such that every nonempty closed P-set of character at most ℵ₁ is a⁻¹({q}) for some continuous a:N* → X. An exact rank-two rotation criterion turns scalar nonextension on a fibre complement into the coordinate obstruction. On the positive side, for nowhere locally constant labels a,b:N* → N*, the mixed disagreement locus {(x,y): a(x) ≠ b(y)} is C*-embedded in (N*)². We also give finite-label criteria for extending compatible spectral projections through collisions, including compact parameter families, and characterize universal same-coordinate decomposition by a static conjugacy condition. GitHub repository: https://github.com/eulogica/point-fibres-coordinate-decomposition