Even-Line Exotic Branch Obstruction

Purpose

The projector repair \(Y = J^{01} + \frac12 Q7 + \frac12 P_{\mathrm{aux},0}\) solved a real algebraic problem:

it fit the standard left-handed doublet charges;
it fit the standard right-handed singlet charges;
it gave the first working unified hypercharge operator on the small carrier.

But that fit was made on a selected slot assignment inside the carrier. The natural next question is:

does the same carrier automatically avoid extra unwanted states, or was the fit only partial?

This note answers that question. The fit is only partial. Even before the top-wedge issue from the full Fock completion, the even auxiliary line already contains an unused exotic T2 doublet sector.

The even auxiliary line carries both `T1` and `T2`

In the projector-repair carrier \((T1 \oplus T2)\otimes (\mathbf 1 \oplus S_{\mathrm{aux}})\otimes (\mathbf 3 \oplus \mathbf 1),\) the left-handed doublets were taken from:

the \mathbf 1 part of \mathbf 1 \oplus S_{\mathrm{aux}},
on the T1 branch.

With the fitted coefficients \(a = 1, \qquad b = \frac12, \qquad c = \frac12,\) the hypercharge operator is \(Y = J^{01} + \frac12 Q7 + \frac12 P_{\mathrm{aux},0}.\)

On the auxiliary even line one has \(P_{\mathrm{aux},0} = 1.\)

So both T1 and T2 branches on that line carry charges \(Y = J^{01} + \frac12 Q7 + \frac12.\)

The earlier fit used only the T1 part of this sector. But the T2 part is still present unless an additional rule removes it.

Charges on the even line

`T1` even line

On T1, J^{01} = -1/2, so:

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

So the T1 even line gives the desired left-handed doublets \((\mathbf 3,\mathbf 2)_{1/6} \oplus (\mathbf 1,\mathbf 2)_{-1/2}.\)

`T2` even line

On T2, J^{01} = +1/2, so:

on the color-triplet slot (Q7 = 1/3), \(Y = \frac12 + \frac16 + \frac12 = \frac76;\)
on the singlet slot (Q7 = -1), \(Y = \frac12 - \frac12 + \frac12 = \frac12.\)

So the complementary T2 even line gives \((\mathbf 3,\mathbf 2)_{7/6} \oplus (\mathbf 1,\mathbf 2)_{1/2}.\)

These are exotic weak doublets, not part of the usual one-generation matter content.

Why this matters

This is not the same problem as the later top-wedge obstruction.

The top-wedge obstruction appeared only after replacing the auxiliary \mathbf 1 \oplus \mathbf 2 block by the full fermionic completion \(\mathbf 1 \oplus \mathbf 2 \oplus \mathbf 1.\)

By contrast, the present obstruction already exists on the smaller auxiliary block itself:

one even auxiliary line,
one odd auxiliary doublet,
but two branch choices on the even line.

So the low-occupancy vacuum-plus-doublet candidate does not yet solve the full spectrum problem by itself. It solves the slot-level hypercharge fit, but not the unwanted complementary even-branch sector.

What this changes about the projector repair

The projector repair should now be read more carefully:

it is a successful selected-slot fit, not yet a full-spectrum completion of the candidate carrier.

That is still useful. It means the operator itself is plausible.

But it also means a second mechanism is required:

some rule must keep the T1 even line and remove, identify, or decouple the T2 even line.

Without that extra rule, the candidate carrier still contains exotic weak doublets.

Best current escape routes

At the current kernel level, the live ways out are:

Branch-selective even-sector rule. Only one branch of the auxiliary even line is physical in the chosen orientation.
Dynamical lifting or decoupling. The complementary T2 even-line doublets are massive, confined, or otherwise removed from the low-energy spectrum.
Non-factorized carrier refinement. The physical carrier is not the full tensor product \((T1 \oplus T2)\otimes (\mathbf 1 \oplus \mathbf 2),\) but a smaller subcarrier that keeps the wanted even sector and odd singlet channels without keeping the complementary even branch.
Orientation/readout-linked selection. The same structure that eventually chooses the physical branch orientation may also choose which even-line branch survives.

None of these is closed yet. But the obstruction makes clear what the next mechanism must do.

What is now established

The following statement is now finite:

the minimal projector/vacuum repair is not yet a full carrier solution. On the auxiliary even line it produces both the desired T1 left-handed doublets and an unwanted complementary T2 doublet sector with charges (\mathbf 3,\mathbf 2)_{7/6} \oplus (\mathbf 1,\mathbf 2)_{1/2}.

So the current fit is a selected-slot success, not a closed static spectrum.

What remains open

The next question is now:

what principled mechanism removes, identifies, or dynamically neutralizes the complementary T2 even-line sector while preserving the successful hypercharge fit on the selected slots?