Even-Sector Observable Projector Descent

Purpose

After the epsilon-channel refinement, the candidate physical-state rule has the form \(P_{\mathrm{phys}} = P_{\mathrm{obs}}P_{\mathrm{aux},0} \;+\; P_{\epsilon}P_{\mathrm{aux},1},\) where:

  • P_{\epsilon} is now essentially fixed by ordinary SU(2) representation theory on the odd auxiliary sector;
  • the real remaining ambiguity sits in the even-line selector P_{\mathrm{obs}}.

This note isolates the finite result already available in the corpus:

the even-line selector is not a new static ad hoc rule. It is exactly the reduced observable-sector projector already required by the ambient/readout scaffold.

That does not finish the derivation. But it narrows the open burden substantially.

The scaffold already contains P_{\mathrm{obs}}

The ambient-reduction scaffold introduces an observable-sector projector \(P_{\mathrm{obs}}\) selected by the forward readout orientation.

The key compatibility statement there is: \(\mathcal R_{\mathrm{op}}(P_{\Pi,-}) = P_{\mathrm{obs}}, \qquad \mathcal R_{\mathrm{op}}(P_{\Pi,+}) = 1 - P_{\mathrm{obs}},\) with the sector named T1 defined to be the reduced eigenspace selected by the physical orientation.

So once the global orientation is fixed, the static branch projector is not extra data. It is simply \(P_{\mathrm{obs}} = P_{T1}.\)

In the chosen reduced convention, \(P_{T1} = \frac12(\mathbf 1 - 2J^{01}).\)

Why this matters for the hypercharge carrier

The minimal physical-subcarrier candidate used the even-line projector \(P_{T1}P_{\mathrm{aux},0}.\)

Without the scaffold context, that can look like a local patch:

  • keep T1 on the even line because it gives the desired left-handed charges;
  • discard T2 because it gives exotic charges.

But the scaffold changes the interpretation:

the physical even-line branch is whichever J^{01} eigenspace is selected by the forward observable/readout orientation, and in the current orientation that is exactly T1.

So the even half of the static subcarrier is not a new hypercharge-specific branch rule. It is the same observable-sector rule the reduction programme already needed for zero-mass support.

Support preservation viewpoint

The scaffold’s support-compatibility conditions say that if the parent direct/readout operator is already confined to the parent negative-charge sector, \(H_{\Pi}^{(0)} = P_{\Pi,-} H_{\Pi}^{(0)} P_{\Pi,-},\) then reduction should not create spurious support on the opposite reduced sector.

Transporting this through the reduction map gives \(H_0 = P_{\mathrm{obs}} H_0 P_{\mathrm{obs}},\) with \((\mathbf 1 - P_{\mathrm{obs}}) H_0 = H_0 (\mathbf 1 - P_{\mathrm{obs}}) = 0.\)

So the even-line branch selector can be read as a support-preservation consequence:

  • parent direct/readout support is one-sided,
  • reduction preserves that one-sided support,
  • the reduced one-sided support is exactly the P_{\mathrm{obs}} sector.

This is the cleanest current reason the complementary even branch should be absent.

Rewriting the static candidate one more time

Using the scaffold notation, the physical-state projector now reads \(P_{\mathrm{phys}} = P_{\mathrm{obs}}P_{\mathrm{aux},0} \;+\; P_{\epsilon}P_{\mathrm{aux},1},\) where:

  • P_{\mathrm{obs}} comes from the ambient/readout orientation and reduced support-preservation rule,
  • P_{\epsilon} comes from the unique antisymmetric invariant scalar channel on \mathbf 2 \otimes \mathbf 2.

This is better than the earlier wording because the two halves now have distinct and coherent roles:

  • even line: observable/readout support selector,
  • odd line: invariant antisymmetric scalar-pairing selector.

What this does and does not close

This note does close the following finite point:

in the current static carrier proposal, the even-line projector is not an independent new ingredient beyond the ambient scaffold. It is the reduced observable-sector projector already required by the zero-mass/readout descent programme.

This note does not yet derive:

  1. why the ambient/readout selector picks one global orientation rather than the other;
  2. whether the same parent mechanism that fixes P_{\mathrm{obs}} also explains the low-occupancy auxiliary structure;
  3. whether the branch/readout selector and the odd \epsilon selector come from one common parent principle.

So the open burden is now mostly upstream of the static carrier, not inside it.

Best current formulation

The cleanest current statement is:

the candidate physical static subspace is selected by the observable/readout projector on the auxiliary even line and by the invariant antisymmetric SU(2) pairing projector on the auxiliary odd line.

This means the static problem has been reduced to two upstream questions:

  1. what fixes the observable/readout orientation,
  2. what justifies the auxiliary low-occupancy block if the Fock reading is used.

The carrier algebra itself is now comparatively disciplined.

What is now established

The following statement is now finite:

the even half of the current physical-subcarrier candidate, \(P_{\mathrm{obs}}P_{\mathrm{aux},0},\) is exactly the reduced observable-sector projector already built into the ambient/readout scaffold, not a separate hypercharge-specific branch rule.

That is a genuine narrowing of the remaining static ambiguity.

What remains open

The next question is:

can the same ambient/readout selector that fixes P_{\mathrm{obs}} also explain why the auxiliary odd sector should enter only through the invariant antisymmetric pairing channel, or are these two superselection inputs fundamentally separate?