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

Core results

• Determinant line (algebra proved on supplied model data). On a rank-two bundle with structure group in $\mathrm{GL}(2,\mathbb{C})$, $\wedge^2$ is the unique one-dimensional functor among the symmetric and exterior powers $\mathrm{Sym}^k$, $\wedge^k$ ($k \geq 1$; the powers $\det^b$ are also one-dimensional and lie outside this family). If the structure group lies in $\mathrm{SL}(2,\mathbb{C})$ it is a trivial line, invisible to the adjoint sector $\mathrm{Sym}^2$. Under the spin solder [H-Spin] of Q14, $\wedge^2(S_L)$ is therefore trivial and carries no abelian weight of the spin group.
• A conditional weak determinant line. Under the distinct weak factor [H-Weak] of Q14, which is supplied by no source and is not constructed here, the line $L_Y := \wedge^2(E_{\mathrm{weak}})$ with structure group $\mathrm{U}(2)$ carries the abelian weight, through the determinant character, while being trivial on $\mathrm{SU}(2)$. Within this functor route it is a conditional weak determinant line, not a line of the spinor bundle. The exponents of the twists of the fermion sectors are not fixed by [H-Weak] alone; Q14 selects them conditionally.
• Generation-level ordering. On the gauge-singlet triplet $C^3_{\mathrm{gen}} = \mathrm{span}(e_0, e_+, e_-)$, the model operator $\mathrm{diag}(1, \tfrac12+u, \tfrac12-u)$ has exit deficits $\{0, \tfrac12-u, \tfrac12+u\}$ that order three levels exactly for $0 < u < \tfrac12$, with a level ratio invariant under the spectral scale. This is algebra on the model operator and uses no hypothesis on $E_\Pi^2$.
• [H-Res]. Reading the model operator as the restriction of the squared projective residue $E_\Pi^2$ to $C^3_{\mathrm{gen}}$ is a named hypothesis, [H-Res], which is not supplied. The map from levels to generations is not established.
• No mass value. The levels are a dimensionless pattern fixed by one number $u$; whether that pattern is a mass hierarchy depends on identifications that are missing. No mass, Yukawa coupling or mixing is claimed.

Separation from the Yukawa operator

The line $L_Y$ is one ingredient of the abelian weight of the coupling, not the mass matrix. Under [H-Spin] and [H-Weak] the left fermions lie in $P_L\mathcal{S}_\Pi \otimes E_{\mathrm{weak}} \otimes L_Y^k$ and the right fermions in $P_R\mathcal{S}_\Pi \otimes L_Y^m$, the right fermion being a weak singlet that carries the hypercharge twist $L_Y^m$. The Yukawa operator is a distinct object, a projected morphism between these sectors, whose typing has one missing element and one ruled-out reading.

• Missing element. A weak linking carrier $K$, a finite-dimensional $\mathrm{U}(2)$-module with a doublet component, supplied by no source; without it an invariant coupling between the two sectors does not exist as typed. The Lorentz part is not the missing element: the sesquilinear pairing of the chiral factors contains the Lorentz scalar exactly once.
• Ruled-out reading. A linear Lorentz-equivariant map $P_L\mathcal{S}_\Pi \to P_R\mathcal{S}_\Pi$: under [H-Spin] the two chiral factors are inequivalent $\mathrm{SL}(2,\mathbb{C})$-modules, so no non-zero such map exists, by Schur's lemma.

The norm of the Yukawa operator, the orientation (sign) of $u$ and the mixing are downstream open data, and none is supplied here. Lorentz chirality, the left-admissibility of $E_\Pi$, is an input of Q14 on its orientation-compatible branch; it is distinct from the $V-A$ structure, which is the representation content of [H-Weak], and the note does not use it beyond that separation. The polar factor and the mixing are treated in the companion operator note, where the polar generator of a morphism family is shown to differ from the vanishing projection $A_\Pi^{\mathrm{odd}}$, and the physical polar class is open.

Status and open deliverable

Proved (algebra on supplied model data). The uniqueness and triviality of the determinant line within the functor family, the exit deficits, the ordering for $0 < u < \tfrac12$ and the scale invariance of the level ratio. These are checked by exact symbolic computation (23 checks).

Conditional. The triviality of $\wedge^2(S_L)$ under [H-Spin]; the abelian weight of $L_Y = \wedge^2(E_{\mathrm{weak}})$ under [H-Weak]; the reading of the levels as levels of $E_\Pi^2|_{C^3_{\mathrm{gen}}}$ under [H-Res]. Neither [H-Spin] nor [H-Weak] is supplied by a source, and $E_{\mathrm{weak}}$ is not constructed.

Open (missing elements). The factor $E_{\mathrm{weak}}$ and the spin solder; the identification [H-Res]; the weak linking carrier $K$; the exponents of the twists beyond their conditional selection by Q14; the map from levels to generations; the Yukawa operator, its norm, the sign of $u$ and the mixing, including the polar class treated in the companion note. Hence the absolute masses, the mass ratios beyond the dimensionless level structure and the generation mixing remain open.

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.

Relation to the Cosmochrony programme

This note belongs to the fermionic matter sub-programme. It uses the tensor sectors and the two hypotheses [H-Spin] and [H-Weak] of Q14, the normal form of the squared residue read through the reading map $\Phi$ of the Schur reduction, a hypothesis there of which [H-Res] is the instance where $\Phi$ is the restriction to the triplet, and the no-go of the NSR note on the deposited candidates for an independent weak multiplicity. The Yukawa operator, the mass normalisation and the mixing remain the downstream open frontier.

References

Jérôme Beau. Toward Projected Yukawa Couplings: A Conditional Weak Determinant Line and Generation-Level Ordering. Working paper, 2026. 10.5281/zenodo.20767265. Code: github.com/Cosmochrony/projected-yukawa-line.