Minimal Physical Subcarrier Candidate

Purpose

The last two screens sharpened the status of the hypercharge repair:

  • the projector term \(Y = J^{01} + \frac12 Q7 + \frac12 P_{\mathrm{aux},0}\) works on the desired slots;
  • but the same carrier still contains unwanted sectors:
    • the complementary T2 branch on the auxiliary even line, with exotic doublet charges \((\mathbf 3,\mathbf 2)_{7/6} \oplus (\mathbf 1,\mathbf 2)_{1/2};\)
    • and, if one uses the full Fock completion of the quaternionic doublet, the top-wedge sector adds another wrong-type doublet family.

So the natural next move is no longer to enlarge the operator. It is:

define the smallest non-factorized physical subcarrier that keeps the wanted sectors and removes the unwanted ones.

This note gives that candidate explicitly.

Desired content

From the previous notes, the wanted one-generation sectors are:

  1. left-handed doublets from the auxiliary even line on the T1 branch,
  2. right-handed singlets from the weak-singlet channel of the auxiliary odd sector on both T1 and T2.

So the minimal desired carrier is \(\mathcal H_{\mathrm{phys}} = \Big( T1 \otimes \mathbf 1_{\mathrm{aux}} \Big) \;\oplus\; \Big( (T1 \oplus T2)\otimes S_{\mathrm{aux}} \Big)_{j=0},\) tensored throughout with the color/lepton factor \(\mathbf 3 \oplus \mathbf 1.\)

In words:

  • keep only the T1 branch on the even auxiliary line;
  • keep only the weak-singlet channel on the odd auxiliary doublet factor.

This is the smallest carrier that retains exactly the sectors previously used in the slot-level fit.

Projector onto the physical subcarrier

The candidate is naturally encoded by a projector.

Even-line branch projector

Since J^{01} has eigenvalue -1/2 on T1 and +1/2 on T2, the projector onto T1 is \(P_{T1} = \frac12\bigl(\mathbf 1 - 2J^{01}\bigr).\)

This gives:

  • P_{T1}=1 on T1,
  • P_{T1}=0 on T2.

Odd-sector singlet projector

On the odd auxiliary sector, the weak content is \(\mathbf 2 \otimes \mathbf 2 = \mathbf 1 \oplus \mathbf 3.\)

Let C_{\mathrm{tot}} be the total weak SU(2) Casimir on \((T1 \oplus T2)\otimes S_{\mathrm{aux}}.\)

With the standard normalization:

  • C_{\mathrm{tot}} = 0 on the singlet,
  • C_{\mathrm{tot}} = 2 on the triplet.

Therefore the projector onto the singlet channel is \(P_{\mathrm{odd},0} = \mathbf 1 - \frac12 C_{\mathrm{tot}}.\)

Combined physical projector

Let P_{\mathrm{aux},0} be the projector onto the auxiliary even line and \(P_{\mathrm{aux},1} := \mathbf 1 - P_{\mathrm{aux},0}\) the projector onto the auxiliary odd doublet.

Then the minimal physical-subcarrier projector is \(P_{\mathrm{phys}} = P_{T1}\,P_{\mathrm{aux},0} \;+\; P_{\mathrm{odd},0}\,P_{\mathrm{aux},1}.\)

This is non-factorized, which is exactly the point. The successful spectrum is not living on the full tensor product. It lives on a smaller correlated subcarrier.

Hypercharge on the kept sectors

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

Even sector kept by P_{T1} P_{\mathrm{aux},0}

On the kept even line:

  • J^{01} = -1/2,
  • P_{\mathrm{aux},0}=1.

So:

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

This is exactly \((\mathbf 3,\mathbf 2)_{1/6} \oplus (\mathbf 1,\mathbf 2)_{-1/2}.\)

Odd singlet sector kept by P_{\mathrm{odd},0} P_{\mathrm{aux},1}

On the odd sector:

  • P_{\mathrm{aux},0}=0,
  • so Y = J^{01} + \frac12 Q7.

Restricting to the weak singlet channel then 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 projector-defined subcarrier reproduces exactly the intended one-generation static content.

What this removes

The projector P_{\mathrm{phys}} removes both previously identified exotic families:

  1. Complementary even branch. P_{T1} kills the T2 auxiliary-even sector \((\mathbf 3,\mathbf 2)_{7/6} \oplus (\mathbf 1,\mathbf 2)_{1/2}.\)

  2. Odd triplet channels. P_{\mathrm{odd},0} kills the triplet part of \(\mathbf 2 \otimes \mathbf 2 = \mathbf 1 \oplus \mathbf 3.\)

So this is the first candidate that actually addresses the carrier problem itself, not only the operator fit.

Fock-space reading

If the auxiliary block is taken from the low-occupancy completion of the quaternionic doublet V, \(\Lambda^0 V \oplus \Lambda^1 V,\) then:

  • P_{\mathrm{aux},0} becomes the vacuum projector P_{\mathrm{vac}},
  • P_{\mathrm{aux},1} becomes the one-particle projector.

In that language, \(P_{\mathrm{phys}} = P_{T1} P_{\mathrm{vac}} \;+\; P_{\mathrm{odd},0}\,P_{\mathrm{1p}}.\)

So the physical content can be read as:

  • left-handed doublets = T1 vacuum sector,
  • right-handed singlets = one-particle weak-singlet sector.

This is the cleanest structural interpretation reached so far.

Why this is progress

This candidate is better than the earlier ones in three ways:

  1. it explains exactly which subspace of the carrier is being used, instead of silently selecting slots;
  2. it removes the already-identified even-branch and odd-triplet exotics at the representation level;
  3. it packages the selection in canonical operators:
    • J^{01} for branch choice,
    • auxiliary even/odd projector,
    • total weak Casimir for singlet-versus-triplet choice.

So the problem has moved from “which slots were we informally using?” to a precise operator-level question.

What remains open

This note still does not finish the static sector.

The remaining burden is now:

why should the physical Hilbert space be the P_{\mathrm{phys}} subcarrier rather than the full tensor-product carrier?

More concretely, one still needs a principled reason for:

  1. selecting T1 rather than T2 on the even auxiliary line;
  2. selecting the weak singlet rather than the triplet on the odd auxiliary sector;
  3. if using the Fock-space reading, restricting to the low-occupancy block rather than the full completion.

But these are now sharply localized operator questions, not diffuse carrier ambiguity.

What is now established

The following statement is now finite:

the smallest candidate carrier that reproduces exactly the intended one-generation static content is not the full tensor product, but the correlated subcarrier cut out by \(P_{\mathrm{phys}} = P_{T1}P_{\mathrm{aux},0} + P_{\mathrm{odd},0}P_{\mathrm{aux},1}.\)

That is the first fully explicit operator-level candidate for the physical static subspace.

What remains open

The next question is:

can the subcarrier projector P_{\mathrm{phys}} be derived from the parent reduction, the eventual orientation selector, or a dynamical superselection rule, rather than imposed as the minimal fix by hand?