Deepening 4 of 6

Parabolic Stabilizers: Sectors Are Derived, Not Assumed

This is the clean structural version of “which channels can I permute without the boundary noticing?” The answer is: the stabilizer subgroup of the observable family.

Once you define that subgroup, sectors stop being hand-labeled boxes. They become orbits of the stabilizer action. That is the right bridge from measurement interface to sector structure.

Theorem Reference

Lean anchors. parabolicStabilizer_antitone, channelSector_refines, parabolicStabilizer_subset_writable

Math statement.

\mathcal O_1 \subseteq \mathcal O_2 \;\Longrightarrow\; P_{\mathcal O_2} \le P_{\mathcal O_1}, \] \[ i \sim_{\mathcal O_2} j \;\Longrightarrow\; i \sim_{\mathcal O_1} j, \qquad \sigma \in P_{\mathcal O},\ \sigma \cdot x \neq x \;\Longrightarrow\; x \in \mathrm{BWR}(\mathcal O).

In English. Finer observables shrink the invisible permutation group. The channel sectors induced by that group can only subdivide, never coarsen. And whenever a stabilizer permutation really moves a state, that state sits in the writable region.

Physical intuition. Sectors are not primitive facts about the system. They are what remains mutually swappable after you decide what the boundary is allowed to see.

physics: symmetry-derived sectors math: subgroup-orbit decomposition engineering: permutation freedom from observables

Observables carve sectors by forbidding swaps

Refinement does not create arbitrary new sectors. It breaks old permutation freedoms by letting the boundary resolve previously hidden distinctions.

Why This Is High Value

Sector language

You get sectors from observable invariance, not by stipulation.

Relation to writable freedom

If a stabilizer permutation truly moves a state, that motion is already a writable direction.

Bridge to Coxeter language

The stabilizer is the precursor to the parabolic / orbit-coset line, where sectors become subgroup orbits.

Feynman reading: first decide what the observer cannot tell apart. Then the hidden permutation group falls out, and sectors are just its orbit structure.