Proposition. The half-turn commutant permits mixing within a sector [fdip-003T]
AGENTDRAFTED
For the real half-turn, define its commutant by
\(\operatorname {Comm}(P)=\{R\in \operatorname {End}_{\mathbb R}(V):RP=PR\}\).
Restriction to the two sectors gives an algebra isomorphism
\[\operatorname {Comm}(P)
\cong \operatorname {End}(V_+)\times \operatorname {End}(V_-)
\cong M_2(\mathbb R)\times M_2(\mathbb R).\]
If \(R\) commutes with \(P\) and \(Pv=\varepsilon v\), then
\(PRv=RPv=\varepsilon Rv\), so it preserves each sector.
Conversely, a map preserving both sectors obeys
\(PR(v_++v_-)=Rv_+-Rv_-=RP(v_++v_-)\).
The inverse to restriction acts separately on the unique sector
components; composition and the unit are componentwise. The
explicit bases identify each endomorphism algebra
with \(M_2(\mathbb R)\).
In sector coordinates \(P=\operatorname {diag}(I_2,-I_2)\).
For \(R=\begin {pmatrix}A&B\\C&D\end {pmatrix}\), direct comparison gives
\[RP=PR\quad \Longleftrightarrow \quad 2B=2C=0
\quad \Longleftrightarrow \quad B=C=0.\]
The group-algebra image is the strictly smaller algebra
\[\{\operatorname {diag}(xI_2,yI_2):x,y\in \mathbb R\}.\]
Indeed \(aI+bP\) has \(x=a+b\), \(y=a-b\), and every pair is
obtained from \(a=(x+y)/2\), \(b=(x-y)/2\). The map swapping
\(b_{+,0},b_{+,1}\) and vanishing on \(V_-\) belongs to the
commutant but is not scalar on \(V_+\), proving strict containment.
Each sign occurs twice, so an operator commuting with \(P\) may
mix the two vectors in either sector. Symmetry therefore does not
force \(Q_\pm \) to be scalar; the
cycle precision blocks exhibit the freedom explicitly.
The group-algebra/module convention comes from
[sengupta2010representations, Sections 3.1-3.2, pp. 39-41];
the commutant calculation is direct. The
interpretation boundary remains: this finite algebra
calculation selects no continuous gauge group, physical mass spectrum
or interaction.