Overview
This companion note opens the mass-sector frontier of the fermionic matter sub-programme and closes only its first stage. It does not derive the three-generation masses, the Yukawa couplings, or the mixing. It isolates two structural results that the admissible spinor carrier already determines, and separates them sharply from the mass normalisation that does not yet follow.
The method lock is stated once and held throughout: the determinant line $L_Y$ fixes the coupling line, the squared projective residue $E_\Pi^2|_{C^3_{\mathrm{gen}}}$ fixes the generation levels, and the mass comes after.
Core results
- Projected Yukawa line. On the rank-two admissible spinor bundle $\mathcal{S}_\Pi$, the determinant line $L_Y = \wedge^2(\mathcal{S}_\Pi)$ is the unique functorial line of the carrier. For any $g \in \mathrm{SL}(2,\mathbb{C})$ its induced action on the line is $\det(g) = 1$, so $L_Y$ is invisible to the $\mathrm{SU}(2)_L$ adjoint sector $\mathrm{Sym}^2(\mathcal{S}_\Pi)$; a diagonal abelian generator acts by the trace, so $L_Y$ carries the $\mathrm{U}(1)_Y$ weight.
- Intrinsic Yukawa sector. The projected Yukawa sector is therefore not an external matter bundle, but the determinant-line sector of the admissible spinor functorial closure — the abelian half of the same closure whose symmetric half is the non-abelian gauge sector.
- Generation-level assignment. On the gauge-singlet triplet $C^3_{\mathrm{gen}} = \mathrm{span}(e_0, e_+, e_-)$ the squared residue takes the closed normal form $E_\Pi^2|_{C^3_{\mathrm{gen}}} = \mathrm{diag}(1, \tfrac12 + u, \tfrac12 - u)$, whose exit deficits $\{0, \tfrac12 - u, \tfrac12 + u\}$ order the three generation levels: $e_0$ lightest (central, non-exiting), $e_-$ heaviest for $u > 0$.
- Levels, not masses. The level ratio $(\tfrac12 + u)/(\tfrac12 - u)$ is scale-invariant, so the dimensionless level structure is fixed by the single split coefficient $u$; the absolute scale is not. The assignment map $e_0, e_+, e_- \mapsto$ three ordered levels is fixed prior to any identification with the physical $e, \mu, \tau$.
Separation from the Yukawa operator
The line $L_Y$ is the weight datum of the Yukawa coupling, not the mass matrix. The mass matrix is a distinct object, a projected morphism $Y_\Pi : \mathcal{S}_{L,\Pi} \otimes L_Y \to \mathcal{S}_{R,\Pi}$, or its spectral avatar after compression by $E_\Pi$. Closing the coupling line does not close the operator: the norm of $Y_\Pi$, the orientation (sign) of the split coefficient $u$, and the complex metaplectic mixing phase are downstream open data, and none of them is supplied here.
Status and open deliverable
Established. The functorial Yukawa line $L_Y = \wedge^2(\mathcal{S}_\Pi)$, invisible to the $\mathrm{SU}(2)_L$ adjoint and carrying the abelian weight, and the generation-level assignment from $E_\Pi^2|_{C^3_{\mathrm{gen}}}$ through the exit deficits. Both are structural and unconditional given the carrier identifications of Q14, PRS, and the Born–Infeld even-sector closure. All representation-theoretic and spectral identities are verified by exact symbolic computation.
Open. The projected Yukawa operator $Y_\Pi$, its norm, the sign of $u$, and the complex mixing phase — hence the absolute masses, the mass ratios beyond the dimensionless level structure, and the generation mixing. Front 3a is closed as a structural result: the Yukawa line and the generation ordering, not the mass values.
Relation to the Cosmochrony programme
This note belongs to the fermionic matter sub-programme. It consolidates the $\wedge^2$ half of Theorem A of Q14 as the Yukawa coupling line, and the exit-deficit ordering of the same paper, together with the squared-residue normal form of the Schur reduction and the Born–Infeld even-sector closure used in the saturation margin note. The projected Yukawa operator, the mass normalisation, and the mixing remain the downstream open frontier.
References
Jérôme Beau. Projected Yukawa Line and Generation-Level Assignment. Working paper, 2026. 10.5281/zenodo.20767265