The Transverse Wave Theorem in Priority-Axis Z-Algebra: From Gauge Redundancy to Algebraic Necessity — Non-Closedness of the Transverse Subspace and the Dual Role of the Priority Axis

Within the rigorous axiomatic system of the priority-axis Z-algebra (a 16-dimensional real semisimple associative algebra Z ≅ M2(C)⊕M2(C)), we prove that transversality is not a gauge choice but a rigid algebraic necessity, and clarify the dual role of the priority axis. Based on the axioms Z1–Z7, rigorous proofs are given via the dual Hodge decompositions induced by the intrinsic derivations D_τ=[τ,·] and ad_k=[k,·]; all propositions beyond the proven check library of the unified framework are newly machine-verified under the V10.3 standard representation. The results show that at the linear level, longitudinal and scalar modes are algebraically forbidden, and gravitational and electromagnetic waves possess strictly two transverse polarizations; at the nonlinear level, the transverse subspace is non-closed under multiplication (V·V⊆L, i·j=k), while the longitudinal subspace L is a commutative subalgebra generated by {k,τ,J} (all 64 ordered-pair commutators vanish, machine-confirmed for the first time); the priority axis k plays the dual role of linear gatekeeper and nonlinear channel, and the Z3 bootstrap theorem shows that it is the unique self-consistent pattern among 8 coupling patterns, with all 20 newly added checks passing with exactly zero residuals. The electromagnetic transverse-wave theorem relies on the potential–field stratification picture; the fluid-dynamics applications and the related order-of-magnitude estimates are heuristic. Transversality is thus promoted from gauge redundancy to algebraic necessity, and the theory yields the falsifiable prediction of strictly two linear gravitational-wave polarizations, consistent with current LIGO/Virgo observations.