Manuscript En05-C@1 · En05-C.tex · theorem · aa213ea6

Theorem: Regional algebras and their shared center

Stage D: a finite constrained Z2 model › The theta graph · lines 1025–1065 · proof included or linked

En05-C@1#thm:thetaalgebras, En05-C@1#eq:A12, En05-C@1#eq:A3

For the theta graph,

A ({1, 2}) = C^{*} (F, Q, R) = e_{+} M_{4} e_{+} \oplus e_{-} M_{4} e_{-} ≅ M_{2} \oplus M_{2},

and

A ({3}) = C^{*} (F) ≅ C^{2} .

Consequently

Z (A ({1, 2})) = A ({3}), A ({1, 2}) \cap A ({3}) = C^{*} (F) .

No spatial tensor-factor decomposition of $M$ realizes these two regional inclusions as complementary tensor factors.

Proof. An invariant Pauli monomial supported on links $1, 2$ can have $β = (0, 0, 0)$ or $β = (1, 1, 0)$ . Hence it is generated by $X_{1}, X_{2}, Z_{1} Z_{2}$ . On the physical space $X_{2} = F Q$ . Within each $f$ sector, $Q, R$ are a Pauli pair and generate $M_{2}$ . The projection $e_{f}$ selects the sector, proving (eq:A12).

On link $3$ alone, invariance permits only $1, X_{3}$ , proving (eq:A3). The center and intersection follow. Complementary tensor factors with common unit intersect only in scalars; here the intersection contains the non-scalar $F$ .

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En05-C@1#eq:A12 · line 1028
$A ({1, 2}) = C^{*} (F, Q, R) = e_{+} M_{4} e_{+} \oplus e_{-} M_{4} e_{-} ≅ M_{2} \oplus M_{2},$
En05-C@1#eq:A3 · line 1038
$A ({3}) = C^{*} (F) ≅ C^{2} .$
unlabeled · line 1043
$Z (A ({1, 2})) = A ({3}), A ({1, 2}) \cap A ({3}) = C^{*} (F) .$

Theorem: Regional algebras and their shared center

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