Quaternionic Slice Screening for the Auxiliary `1 + 2` Block

Purpose

The previous two notes isolated the remaining hypercharge burden very sharply:

the smallest unified carrier works algebraically once one allows \(Y = J^{01} + \frac12 Q7 + \frac12 P_{\mathrm{aux},0},\)
and that projector can be rewritten canonically as the Casimir-zero projector if the auxiliary block is the reducible SU(2) module \(\mathbf 1 \oplus \mathbf 2.\)

So the live question is no longer whether the projector repair is algebraically available. It is:

does the current parent-side quaternionic slice already contain the needed auxiliary \mathbf 1 \oplus \mathbf 2 block?

This note gives the answer under the present scaffold: not yet.

The natural `SU(2)` action already in the scaffold

The ambient-reduction scaffold already identifies the local quaternionic slice \(H(u,v) = \mathrm{span}_{\mathbf R}\{1,u,v,uv\}\) as the parent-side source of the visible SU(2) structure.

The key point there is:

regard H(u,v) as a complex vector space over \mathbf C_u using right multiplication by u;
let the visible SU(2) act by left multiplication with the imaginary quaternion units.

With the complex basis \(e_1 = 1, \qquad e_2 = v,\) the three left-multiplication generators act, up to the standard i convention from right multiplication by u, as the Pauli triplet: \(L_u \sim \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}, \qquad L_v \sim \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}, \qquad L_{uv} \sim \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}.\)

So under the current natural action, H(u,v) is exactly the usual complex SU(2) doublet: \(H(u,v) \cong \mathbf 2 \qquad \text{(as a complex `SU(2)` module over `\mathbf C_u`).}\)

That is already valuable. It explains the visible weak-doublet carrier internally.

But it also means something restrictive:

the current quaternionic slice gives an irreducible complex doublet, not a reducible complex \mathbf 1 \oplus \mathbf 2.

Why no `\mathbf 1 \oplus \mathbf 2` appears under this action

On the complex carrier H(u,v) \cong \mathbf 2, any operator commuting with the full visible SU(2) action must commute with L_u, L_v, and L_{uv}.

Write a general complex-linear endomorphism as \(X = \begin{pmatrix} a & b\\ c & d \end{pmatrix}.\)

Then:

commuting with L_u forces b=c=0, so X is diagonal;
commuting with L_v then forces a=d.

Therefore \(X = \lambda\,\mathbf 1.\)

So the commutant is scalar only. Equivalently, the current visible action on H(u,v) is irreducible.

This has an immediate consequence for the hypercharge problem:

there is no nontrivial SU(2)-equivariant projector inside the current complex quaternionic carrier itself.

In particular, the auxiliary projector P_{\mathrm{aux},0} from the unified-carrier fit cannot be identified with a projector already present on H(u,v) under this natural left action.

The tempting scalar-plus-triplet split is a different action

There is another familiar decomposition of quaternions: \(H(u,v) = \mathbf R\cdot 1 \oplus \mathrm{Im}\,H(u,v),\) with \(\mathrm{Im}\,H(u,v) = \mathrm{span}_{\mathbf R}\{u,v,uv\}.\)

Under quaternionic conjugation or the adjoint SU(2) action, this gives the real decomposition \(\mathbf 1 \oplus \mathbf 3.\)

But that is not the same module used elsewhere in the scaffold:

it is a real scalar-plus-triplet package;
it comes from the adjoint/conjugation action, not from the left-multiplication action that supplies the visible weak doublet;
it does not produce the required complex reducible module \mathbf 1 \oplus \mathbf 2.

So the scalar-plus-triplet split does not solve the present problem. It belongs to a different representation package.

What the screen closes

The current parent-side quaternionic story now has a clean status:

the local slice H(u,v) does supply a natural visible SU(2) doublet carrier;
under that same natural action, H(u,v) is irreducible as a complex module and therefore contains no nontrivial equivariant projector;
the familiar scalar/imaginative split of quaternions belongs to a different real adjoint action and gives \mathbf 1 \oplus \mathbf 3, not the needed complex \mathbf 1 \oplus \mathbf 2.

So the current scaffold does not yet derive the auxiliary reducible block required by the projector/Casimir hypercharge repair.

Best current interpretation

The disciplined reading is now:

the quaternionic slice still explains why a weak-doublet factor was the first natural place to look;
the successful three-term hypercharge fit remains algebraically valid;
but the present quaternionic parent geometry does not by itself justify identifying the auxiliary block with H(u,v).

So the live options are narrower than before.

Either:

the auxiliary \mathbf 1 \oplus \mathbf 2 block is a genuinely larger carrier than the bare quaternionic visible slice;
the left-handed embedding should be enlarged or repackaged so that the current projector fix is replaced by a different global operator;
or a new parent-side selection rule / auxiliary sector must be added beyond the current H(u,v) scaffold.

What is now established

The following statement is now closed at the current kernel level:

under the present ambient-reduction scaffold, the natural quaternionic slice H(u,v) furnishes the visible irreducible complex SU(2) doublet, but it does not itself realize the reducible complex auxiliary module \mathbf 1 \oplus \mathbf 2 whose Casimir-zero projector appears in the successful hypercharge repair.

That is a useful negative result. It tells us exactly what the current parent-side geometry does not yet provide.

What remains open

The next question is no longer “does the existing quaternionic slice already contain the right auxiliary module?” It does not.

The next question is:

what is the smallest principled enlargement or reinterpretation of the parent-side carrier that can produce the auxiliary \mathbf 1 \oplus \mathbf 2 block, or else remove the need for it?

Quaternionic Slice Screening for the Auxiliary 1 + 2 Block