Full Fock Auxiliary Completion and the Top-Wedge Obstruction

Purpose

The vacuum-plus-doublet candidate improved the auxiliary \mathbf 1 \oplus \mathbf 2 story substantially:

\[\Lambda^0 V \oplus \Lambda^1 V \cong \mathbf 1 \oplus \mathbf 2\]

for the existing quaternionic doublet V \cong \mathbf 2.

But that candidate left one immediate question open:

is the low-occupancy truncation really necessary, or could one simply use the full fermionic completion \(\Lambda^\bullet V \cong \mathbf 1 \oplus \mathbf 2 \oplus \mathbf 1\) without harm?

This note answers that question. The truncation is not optional if the current hypercharge fit is kept. The full Fock completion introduces an extra top-wedge weak-doublet sector with the wrong representation type. This is an additional obstruction on top of the already-existing complementary even-branch problem recorded in kernels/even-line-exotic-branch-obstruction.md.

Full auxiliary completion

Start with the same quaternionic doublet \(V \cong \mathbf 2.\)

Its full fermionic completion is \(\Lambda^\bullet V = \Lambda^0 V \oplus \Lambda^1 V \oplus \Lambda^2 V \cong \mathbf 1_{\mathrm{vac}} \oplus \mathbf 2_{\mathrm{1p}} \oplus \mathbf 1_{\mathrm{top}}.\)

If one feeds this into the previously successful unified carrier, the natural full candidate is \(\mathcal H_{\mathrm{full}} = (T1 \oplus T2)\otimes \Lambda^\bullet V \otimes (\mathbf 3 \oplus \mathbf 1).\)

Keep the same hypercharge operator as before, now written with the vacuum projector: \(Y = J^{01} + \frac12 Q7 + \frac12 P_{\mathrm{vac}}.\)

On the full Fock space, P_{\mathrm{vac}} acts by:

  • 1 on \Lambda^0 V,
  • 0 on \Lambda^1 V,
  • 0 on \Lambda^2 V.

So the vacuum and one-particle sectors behave exactly as in the previous note. The new issue is entirely the top wedge \Lambda^2 V.

The top wedge does not create singlets; it creates extra doublets

The crucial point is that \(\Lambda^2 V \cong \mathbf 1\) is a weak singlet as an auxiliary factor.

Therefore tensoring it with T1 or T2 does not create right-handed weak singlets. It leaves the reduced Spin(2,3) branch type unchanged: \(T1 \otimes \Lambda^2 V \cong T1 \cong (\mathbf 2,-1/2),\) \(T2 \otimes \Lambda^2 V \cong T2 \cong (\mathbf 2,+1/2).\)

So the top wedge sector contributes another pair of weak doublet carriers, not another pair of singlet carriers.

That is already enough to show that the full Fock completion is structurally different from the low-occupancy candidate.

Hypercharges on the top-wedge doublets

Because P_{\mathrm{vac}} = 0 on \Lambda^2 V, the top-wedge charges are simply \(Y = J^{01} + \frac12 Q7\) on that sector.

T1 top wedge

On the T1 branch, J^{01} = -1/2. Therefore:

  • on the color-triplet slot (Q7 = 1/3), \(Y = -\frac12 + \frac16 = -\frac13;\)
  • on the singlet slot (Q7 = -1), \(Y = -\frac12 - \frac12 = -1.\)

So the T1 top wedge carries weak doublets with hypercharges \((\mathbf 3,\mathbf 2)_{-1/3} \oplus (\mathbf 1,\mathbf 2)_{-1}.\)

T2 top wedge

On the T2 branch, J^{01} = +1/2. Therefore:

  • on the color-triplet slot (Q7 = 1/3), \(Y = \frac12 + \frac16 = \frac23;\)
  • on the singlet slot (Q7 = -1), \(Y = \frac12 - \frac12 = 0.\)

So the T2 top wedge carries weak doublets with hypercharges \((\mathbf 3,\mathbf 2)_{2/3} \oplus (\mathbf 1,\mathbf 2)_0.\)

Why this is an obstruction

These top-wedge sectors are not innocuous duplicates of the ordinary left-handed seed.

They are stranger than that:

  • the T1 top wedge carries the right-handed-style hypercharge values -1/3 and -1, but on weak doublets rather than weak singlets;
  • the T2 top wedge carries the right-handed-style hypercharge values 2/3 and 0, again on weak doublets rather than weak singlets.

So the full Fock completion does not merely overcount states. It puts the right hypercharge values on the wrong SU(2) representation type.

That is a real obstruction, not just cosmetic redundancy.

What this means for the vacuum-plus-doublet candidate

The previous note suggested that \(\Lambda^{\le 1} V = \Lambda^0 V \oplus \Lambda^1 V\) is the best current candidate auxiliary block.

The present calculation sharpens that claim:

if the current hypercharge operator is kept, then some mechanism removing or neutralizing the top wedge \Lambda^2 V is necessary.

So the low-occupancy truncation is not merely a convenient reading. It is the minimal way to avoid the extra wrong-type doublet sector.

Best current escape routes

At the current kernel level there are three clear ways one could try to save the vacuum-plus-doublet idea:

  1. Literal low-occupancy truncation. Keep only \Lambda^0 V \oplus \Lambda^1 V as the physical auxiliary carrier.

  2. Even-sector identification or quotient. Use some particle-hole, Hodge-dual, or other constraint to identify or remove the top wedge singlet \Lambda^2 V relative to the vacuum singlet \Lambda^0 V.

  3. Hidden decoupling / reinterpretation. Keep the full Fock space but show that the top-wedge doublets are projected out dynamically, lifted to a very high scale, or reinterpreted as a separate unobserved sector.

None of these is established yet. But the current note tells us exactly what any successful mechanism has to accomplish.

What is now established

The following point is now closed:

the full fermionic completion \Lambda^\bullet V \cong \mathbf 1 \oplus \mathbf 2 \oplus \mathbf 1 is not by itself a clean replacement for the low-occupancy auxiliary block in the present hypercharge construction, because the top wedge \Lambda^2 V generates extra weak doublets carrying right-handed-style hypercharge values on the wrong SU(2) representation type.

That is a finite and useful obstruction.

What remains open

The next question is now sharper than before:

what mechanism, if any, removes, identifies, or dynamically neutralizes the top-wedge sector \Lambda^2 V so that the vacuum-plus-doublet auxiliary candidate can survive as a physical construction?