Clifford Rigidity of Transverse Combinations for Weight-4 Newforms

We uncover a family of congruence rigidity phenomena among the Fourier coefficients of weight-4 newforms of levels 8, 6, and 12. The coefficients are assembled into a single element A(n) of the even subalgebra of the real Clifford algebra Cl(2,2), and a distinguished trace pairing extracts the integer invariant T(n)=4(a(n)-b(n)+c(n)) (mod 32). This invariant exhibits a sharply rigid structure: its values lie in the discrete octad {0,4,8,12,16,20,24,28}; it is multiplicative modulo 8 and congruent to σ₃(n) for odd n coprime to 3; and it satisfies a universal 2-adic truncation. We prove a Transverse Combination Selection Theorem: for triples of newforms attached to class-number-1 definite quaternion orders, the full rigidity occurs iff the three orders occupy the canonical ramification layers of depths 0, 2, and 4, and the non-baseline orders share the same algebra discriminant. An exhaustive classification yields 120 rigid triples up to level 100, perfectly matching the theoretical count. The framework further reveals a hidden parity filtration within the 1-dimensional space S₄(Γ₀(8)) governed by the Clifford volume element, with a uniquely determined shift parameter d=23. On the geometric side, the trace invariant is reinterpreted via Arakelov intersection numbers on quaternionic Shimura curves, exposing a structural parallel with the arithmetic cancellation. These results indicate that the Clifford filtration captures arithmetic information invisible to the classical dimension formula.