Why the remaining phase cannot be chosen from the declared data
Furey interface: selector obstruction
Internally exhaustedAuditor
This page exposes the assumptions, proof chain, certificate files and authority boundary behind the chapter claim.
Claim under audit
Within the declared endpoint-blind data, no CP, CPT, boundary, residual \(\mathbb Z_3\), flavour or index datum canonically selects the sterile phase reference. A new charge-minus-two reference is the minimal typed reopening datum.
Start with the picture
A perfectly round compass dial does not point north until something outside the dial supplies a north reference. Here the rephasing circle plays the role of the dial: every candidate already in the model moves with, or is blind to, the phase one is trying to choose.
Objects and assumptions
| Symmetry | A sterile Weyl field may be rephased \(N\mapsto e^{i\alpha}N\). |
|---|---|
| Charged pair | The bilinear \(NN\) transforms as \(NN\mapsto e^{2i\alpha}NN\). |
| Target variable | A proposed pair phase \(\beta\) therefore shifts as \(\beta\mapsto\beta+2\alpha\). |
| Admissible selector | It must be built from declared physical data and transform equivariantly; a coordinate convention is not a measurement. |
Certificate chain
- Any candidate made only from rephasing-invariant data is unchanged when \(\alpha\) changes.
- A selected phase \(\beta\) must change by \(2\alpha\) under the same transformation.
- An invariant input therefore cannot define an equivariant point on this phase circle. The mismatch is structural, not a failure to search a longer list of invariant combinations.
- Introducing \(R_2\) with \(R_2\mapsto e^{-2i\alpha}R_2\) makes \(R_2NN\) invariant and defines \(\beta_R=-\arg(R_2)\pmod{\pi}\). This is a valid typed remedy only when \(R_2\) is physically prepared, measured or dynamically derived and locally coupled.