Toward Projected Yukawa Couplings: A Conditional Weak Determinant Line and Generation-Level Ordering

This note opens the mass-sector frontier of the fermionic-matter sub-programme of Cosmochrony. It derives no mass, Yukawa coupling, or mixing. It records two statements, each with the hypothesis it needs. First, algebra proved on supplied model data: on a rank-two bundle with structure group in SL(2,C), the determinant line ∧² is the unique one-dimensional functor among the symmetric and exterior powers, and it is trivial. Hence ∧²(S_L) carries no abelian weight of the spin group under the spin solder [H-Spin] of Q14. Under the distinct weak factor [H-Weak] of Q14, which is not supplied and not constructed here, the line L_Y ≔ ∧²(E_weak) with structure group U(2) carries the abelian weight: within this functor route it is a conditional weak determinant line, not a line of the spinor bundle. Second, on the generation triplet C³_gen, the model operator diag(1, 1/2 + u, 1/2 - u) has exit deficits 0, 1/2 - u, 1/2 + u that order three levels for 0 < u < 1/2, with a scale-independent ratio. Reading it as the restriction of the squared projective residue E_Π² is a named hypothesis, [H-Res], not supplied, and the map from levels to generations is not established. The Yukawa operator, which needs a weak linking carrier supplied by no source, its norm, the sign of u, and the mixing remain open, so no mass value is claimed. Interpretive outlook (a reading, not a result): where the Yukawa coupling attaches to hypercharge becomes the question of where a distinct weak factor comes from, and the levels are a dimensionless pattern whose reading as a hierarchy awaits the missing identifications.