Conditional Static Spectrum Closure

Purpose

The last sequence of notes narrowed the static carrier problem substantially:

the hypercharge operator \(Y = J^{01} + \frac12 Q7 + \frac12 P_{\mathrm{aux},0}\) works on the desired sectors;
the naive full carrier contains exotic branches;
the best current physical-state candidate is the correlated subcarrier \(P_{\mathrm{phys}} = P_{\mathrm{obs}}P_{\mathrm{aux},0} \;+\; P_{\epsilon}P_{\mathrm{aux},1},\) with:
- P_{\mathrm{obs}} the observable/readout-sector projector on the even auxiliary line,
- P_{\epsilon} the odd antisymmetric SU(2)-invariant pairing projector.

At this point the right next step is to say clearly what is and is not already closed.

This note gives the clean conditional statement:

if the current operational orientation rule is accepted and the current auxiliary-sector rule is accepted, then the one-generation static spectrum and hypercharge assignment are fixed on the reduced physical subspace.

That is not a derivation from first principles. But it is stronger than a loose fit.

Inputs

Assume the following three inputs.

Input A: observable orientation rule

Use the current operational rule:

the physical forward/readout orientation is the constructive/persistent branch, equivalently the branch with \kappa_u > 0.

In the present reduced orientation convention, that fixes the observable-sector projector to be \(P_{\mathrm{obs}} = P_{T1} = \frac12(\mathbf 1 - 2J^{01}).\)

Input B: auxiliary even/odd block

Assume the reduced auxiliary sector is split into:

an even line selected by P_{\mathrm{aux},0},
an odd weak doublet selected by P_{\mathrm{aux},1} = 1 - P_{\mathrm{aux},0}.

This may be read abstractly as an auxiliary \mathbf 1 \oplus \mathbf 2 block, or concretely through the low-occupancy quaternionic-doublet completion discussed elsewhere.

Input C: odd invariant pairing rule

Assume the odd auxiliary sector enters only through the invariant antisymmetric pairing channel \(P_{\epsilon} = \frac12(\mathbf 1 - \tau) = \mathbf 1 - \frac12 C_{\mathrm{tot}}\) on \(\mathbf 2 \otimes \mathbf 2 = \mathbf 3 \oplus \mathbf 1.\)

This is the unique SU(2)-invariant scalar channel on the odd sector.

Resulting physical subspace

Under Inputs A-C, the reduced physical static subspace is \(P_{\mathrm{phys}} = P_{\mathrm{obs}}P_{\mathrm{aux},0} \;+\; P_{\epsilon}P_{\mathrm{aux},1}.\)

Equivalently:

on the even auxiliary line, keep only the observable branch;
on the odd auxiliary line, keep only the invariant antisymmetric scalar channel.

This removes:

the complementary even T2 branch \((\mathbf 3,\mathbf 2)_{7/6} \oplus (\mathbf 1,\mathbf 2)_{1/2},\)
the odd weak-triplet sector,
and, if the Fock reading is used, the full-completion exotic sectors once low occupancy is imposed.

Hypercharge on the physical subspace

Keep the same hypercharge operator \(Y = J^{01} + \frac12 Q7 + \frac12 P_{\mathrm{aux},0}.\)

Even branch

On P_{\mathrm{obs}}P_{\mathrm{aux},0} in the present convention, \(J^{01} = -\frac12, \qquad P_{\mathrm{aux},0}=1.\)

So:

color triplet: \(Y = -\frac12 + \frac16 + \frac12 = \frac16;\)
color singlet: \(Y = -\frac12 - \frac12 + \frac12 = -\frac12.\)

This gives \(Q_L : (\mathbf 3,\mathbf 2)_{1/6}, \qquad L_L : (\mathbf 1,\mathbf 2)_{-1/2}.\)

Odd branch

On P_{\epsilon}P_{\mathrm{aux},1}, one has P_{\mathrm{aux},0}=0, so \(Y = J^{01} + \frac12 Q7.\)

Restricting to the odd weak singlet channel gives:

T1 branch: \((\mathbf 3,\mathbf 1)_{-1/3} \oplus (\mathbf 1,\mathbf 1)_{-1};\)
T2 branch: \((\mathbf 3,\mathbf 1)_{2/3} \oplus (\mathbf 1,\mathbf 1)_0.\)

So the odd sector gives \(d_R,\ e_R,\ u_R,\ \nu_R\) with exactly the standard one-generation hypercharges.

Conditional closure statement

Under Inputs A-C, the reduced static content on P_{\mathrm{phys}} is exactly \((\mathbf 3,\mathbf 2)_{1/6} \oplus (\mathbf 1,\mathbf 2)_{-1/2} \oplus (\mathbf 3,\mathbf 1)_{-1/3} \oplus (\mathbf 1,\mathbf 1)_{-1} \oplus (\mathbf 3,\mathbf 1)_{2/3} \oplus (\mathbf 1,\mathbf 1)_0.\)

That is precisely the usual one-generation left-handed plus right-handed charge pattern.

So the safe conclusion is:

conditional on the repo’s current observable-branch rule and auxiliary-sector rule, the reduced static spectrum is closed at the one-generation level.

What this does not prove

This note does not prove:

that Inputs A-C follow from the parent geometry alone;
that the auxiliary low-occupancy reading is uniquely forced;
that this is the only possible static completion;
that generations, anomaly completion, or dynamics are thereby closed.

So this is a conditional closure, not an unconditional theorem.

Why this matters

This conditional statement changes the status of the static line in an important way.

Before:

there were several local fits and several local obstructions.

Now:

the remaining uncertainty has been pushed almost entirely into a short list of explicit upstream inputs.

That means the repo can now separate:

carrier algebra, which is in relatively good shape;
upstream selection principles, which are the real remaining burden.

That is real progress.

Best current formulation

The cleanest honest formulation now available is:

the Spin(2,3) static carrier is conditionally closed at the one-generation level, provided one accepts the current observable/readout-sector selector on the even line and the invariant antisymmetric pairing selector on the odd line.

That is the right current strength.

What is now established

The following point is now finite:

given \(P_{\mathrm{phys}} = P_{\mathrm{obs}}P_{\mathrm{aux},0} + P_{\epsilon}P_{\mathrm{aux},1}\) and \(Y = J^{01} + \frac12 Q7 + \frac12 P_{\mathrm{aux},0},\) the reduced static spectrum on P_{\mathrm{phys}} reproduces exactly the standard one-generation charge pattern.

What remains open

The next questions are now sharply localized:

can P_{\mathrm{obs}} be derived rather than operationally adopted?
can the auxiliary low-occupancy / even-odd split be derived rather than postulated?
do those two inputs come from one common parent principle?