Generator Resolution I: Common Phase, Cross-Resolution Symmetry, and the SU(4) Closure Envelope

A repeated number is a clue only after we know what it counts. The central K bridge linking tensor factorization, exchange symmetry, triplet–singlet structure, and the relative 3+1 Abelian generator. This investigation began with an observation that almost suspiciously simple: 3 + 1 + 3 + 8 = 15. The numbers recall the dimensions of SO(3), U(1), SU(2), and SU(3), while fifteen is the dimension of SU(4). The immediate temptation is to read the equality as a decomposition of the SU(4) Lie algebra. That temptation is mathematically wrong: su(4) is simple, whereas a direct sum of so(3), u(1), su(2), and su(3) is reducible. The failure of the obvious interpretation is not the end of the inquiry. It is the point at which the inquiry becomes properly typed. The paper therefore replaces decomposition by resolution. A resolution is a specified way of distinguishing structure within a common carrier. The same four-component carrier may be examined as 3+1, as 2+2, or as 2×2. These descriptions have the same carrier dimension but preserve different relations. Their stabilizers, complements, tensor structures, and generator roles differ. The recurring numbers can therefore be meaningful without being the summands of one fixed Lie algebra decomposition. The manuscript deliberately preserves failed routes. Clifford dimension eight, a Grassmannian bridge of real dimension eight, a traceless tensor-correlation eight, and the genuine su(3) adjoint all appear. They are not the same eight. S3 permutation symmetry, A2 geometry, and positive Grassmannian structure also recur, but none of them by itself is SU(3) gauge physics. These negative results are part of the proof discipline: a repeated number is not an identification until its type, action, bracket structure, and representation content have been audited. The central constructive result is the operator K = Σ_i σ_i⊗σ_i. It is simultaneously a scalar tensor correlation and an affine form of the swap operator, K = 2P_swap − I4. Its eigenspaces are the symmetric triplet and antisymmetric singlet, so in the coupled basis K = diag(1,1,1,−3). Hence iK is exactly the relative 3+1 Abelian direction Y31. Independently, Y31 is reconstructed from the three balanced 2+2 contrast directions as Y31 = Y1 + Y2 − Y3. One operator has therefore been followed across several resolutions without pretending that those resolutions are identical. The question in ordinary languageThe paper begins with a suspiciously neat numerical pattern: three, one, three, and eight add to fifteen, while familiar symmetry structures in physics also carry generator counts three, one, three, and eight. The tempting move is to declare that the fifteen generators of SU(4) simply split into those four familiar pieces. That tempting move is wrong. The paper becomes interesting precisely because we refuse to stop there. Instead, we ask whether the same fifteen relative transformations can look different when the underlying four-component carrier is organized in different ways. A four-component object can be viewed as three plus one, as two plus two, or as two factors of two. Those are different mathematical questions asked of the same carrier. The central discovery The strongest result is not the number fifteen. It is the operator K. In the two-by-two tensor description, K is a scalar correlation built from Pauli matrices. The same K is also an affine form of the swap operator. Swap divides the carrier into a symmetric triplet and an antisymmetric singlet. In that triplet–singlet basis, iK becomes exactly the relative phase generator of the three-plus-one resolution. A second calculation reconstructs that same relative phase direction from the three possible balanced two-plus-two contrasts. Thus several apparently different descriptions meet at one explicit operator identity. This is why the paper speaks of generator resolution rather than generator coincidence.